US20040169204A1  Multiplycomplexed onedimensional structure, multiplytwisted helix, multiplylooped ring structure and functional material  Google Patents
Multiplycomplexed onedimensional structure, multiplytwisted helix, multiplylooped ring structure and functional material Download PDFInfo
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 US20040169204A1 US20040169204A1 US10/795,870 US79587004A US2004169204A1 US 20040169204 A1 US20040169204 A1 US 20040169204A1 US 79587004 A US79587004 A US 79587004A US 2004169204 A1 US2004169204 A1 US 2004169204A1
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O=[Si]=O VYPSYNLAJGMNEJUHFFFAOYSAN 0 description 6
 229910001885 silicon dioxide Inorganic materials 0 description 6
 239000000377 silicon dioxide Substances 0 description 6
 238000004088 simulation Methods 0 description 5
 239000011343 solid materials Substances 0 description 1
 239000007787 solids Substances 0 description 6
 230000002269 spontaneous Effects 0 description 54
 229910052682 stishovite Inorganic materials 0 description 6
 239000011232 storage materials Substances 0 description 1
 239000000126 substances Substances 0 description 6
 239000000758 substrates Substances 0 description 3
 125000004434 sulfur atoms Chemical group 0 description 2
 230000002123 temporal effects Effects 0 description 1
 238000004613 tight binding model Methods 0 description 15
 229910052905 tridymite Inorganic materials 0 description 6
 238000007738 vacuum evaporation Methods 0 description 1
 238000004804 winding Methods 0 description 1
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 B—PERFORMING OPERATIONS; TRANSPORTING
 B82—NANOTECHNOLOGY
 B82Y—SPECIFIC USES OR APPLICATIONS OF NANOSTRUCTURES; MEASUREMENT OR ANALYSIS OF NANOSTRUCTURES; MANUFACTURE OR TREATMENT OF NANOSTRUCTURES
 B82Y25/00—Nanomagnetism, e.g. magnetoimpedance, anisotropic magnetoresistance, giant magnetoresistance or tunneling magnetoresistance

 B—PERFORMING OPERATIONS; TRANSPORTING
 B82—NANOTECHNOLOGY
 B82Y—SPECIFIC USES OR APPLICATIONS OF NANOSTRUCTURES; MEASUREMENT OR ANALYSIS OF NANOSTRUCTURES; MANUFACTURE OR TREATMENT OF NANOSTRUCTURES
 B82Y10/00—Nanotechnology for information processing, storage or transmission, e.g. quantum computing or single electron logic

 H—ELECTRICITY
 H01—BASIC ELECTRIC ELEMENTS
 H01F—MAGNETS; INDUCTANCES; TRANSFORMERS; SELECTION OF MATERIALS FOR THEIR MAGNETIC PROPERTIES
 H01F1/00—Magnets or magnetic bodies characterised by the magnetic materials therefor; Selection of materials for their magnetic properties
 H01F1/0036—Magnets or magnetic bodies characterised by the magnetic materials therefor; Selection of materials for their magnetic properties showing low dimensional magnetism, i.e. spin rearrangements due to a restriction of dimensions, e.g. showing giant magnetoresistivity
 H01F1/0072—Magnets or magnetic bodies characterised by the magnetic materials therefor; Selection of materials for their magnetic properties showing low dimensional magnetism, i.e. spin rearrangements due to a restriction of dimensions, e.g. showing giant magnetoresistivity one dimensional, i.e. linear or dendritic nanostructures

 H—ELECTRICITY
 H01—BASIC ELECTRIC ELEMENTS
 H01F—MAGNETS; INDUCTANCES; TRANSFORMERS; SELECTION OF MATERIALS FOR THEIR MAGNETIC PROPERTIES
 H01F1/00—Magnets or magnetic bodies characterised by the magnetic materials therefor; Selection of materials for their magnetic properties
 H01F1/0036—Magnets or magnetic bodies characterised by the magnetic materials therefor; Selection of materials for their magnetic properties showing low dimensional magnetism, i.e. spin rearrangements due to a restriction of dimensions, e.g. showing giant magnetoresistivity
 H01F1/009—Magnets or magnetic bodies characterised by the magnetic materials therefor; Selection of materials for their magnetic properties showing low dimensional magnetism, i.e. spin rearrangements due to a restriction of dimensions, e.g. showing giant magnetoresistivity bidimensional, e.g. nanoscale period nanomagnet arrays

 H—ELECTRICITY
 H01—BASIC ELECTRIC ELEMENTS
 H01L—SEMICONDUCTOR DEVICES; ELECTRIC SOLID STATE DEVICES NOT OTHERWISE PROVIDED FOR
 H01L29/00—Semiconductor devices adapted for rectifying, amplifying, oscillating or switching, or capacitors or resistors with at least one potentialjump barrier or surface barrier, e.g. PN junction depletion layer or carrier concentration layer; Details of semiconductor bodies or of electrodes thereof; Multistep manufacturing processes therefor
 H01L29/02—Semiconductor bodies ; Multistep manufacturing processes therefor
 H01L29/12—Semiconductor bodies ; Multistep manufacturing processes therefor characterised by the materials of which they are formed
 H01L29/122—Single quantum well structures
 H01L29/127—Quantum box structures

 H—ELECTRICITY
 H01—BASIC ELECTRIC ELEMENTS
 H01L—SEMICONDUCTOR DEVICES; ELECTRIC SOLID STATE DEVICES NOT OTHERWISE PROVIDED FOR
 H01L51/00—Solid state devices using organic materials as the active part, or using a combination of organic materials with other materials as the active part; Processes or apparatus specially adapted for the manufacture or treatment of such devices, or of parts thereof
 H01L51/05—Solid state devices using organic materials as the active part, or using a combination of organic materials with other materials as the active part; Processes or apparatus specially adapted for the manufacture or treatment of such devices, or of parts thereof specially adapted for rectifying, amplifying, oscillating or switching, or capacitors or resistors with at least one potential jump barrier or surface barrier multistep processes for their manufacture
 H01L51/0575—Solid state devices using organic materials as the active part, or using a combination of organic materials with other materials as the active part; Processes or apparatus specially adapted for the manufacture or treatment of such devices, or of parts thereof specially adapted for rectifying, amplifying, oscillating or switching, or capacitors or resistors with at least one potential jump barrier or surface barrier multistep processes for their manufacture the devices being controllable only by variation of the electric current supplied or the electric potential applied, to one or more of the electrodes carrying the current to be rectified, amplified, oscillated or switched, e.g. twoterminal devices
 H01L51/0595—Solid state devices using organic materials as the active part, or using a combination of organic materials with other materials as the active part; Processes or apparatus specially adapted for the manufacture or treatment of such devices, or of parts thereof specially adapted for rectifying, amplifying, oscillating or switching, or capacitors or resistors with at least one potential jump barrier or surface barrier multistep processes for their manufacture the devices being controllable only by variation of the electric current supplied or the electric potential applied, to one or more of the electrodes carrying the current to be rectified, amplified, oscillated or switched, e.g. twoterminal devices molecular electronic devices
Abstract
In a multiplycomplexed onedimensional structure having a hierarchical structure in which a linear structure as an element of a onedimensional structure having a finite curvature is made of a thinner onedimensional structure having a finite curvature, at least two layers of onedimensional unit structures are bonded in at least one site. For example, in a multiplytwisted helix having a hierarchical structure in which a linear structure as an element of a spiral structure is made of a thinner spiral structure, at least two layers of the unit spiral structures are bonded in at least one site. Alternatively, in a multiplylooped ring structure having a hierarchical structure in which a linear structure as an element of a ring structure is made of a thinner ring structure, at least two layers of ring unit structures are bonded in at least one site.
Description
 This invention relates to a multiplycomplexed onedimensional structure, multiplytwisted helix, multiplylooped ring structure and functional material, especially suitable for use as highly functional materials based on a novel principle.
 For application of a solid material to electronic or optical devices, physical properties of the material may restrict its applications. For example, in case of using a semiconductor material in a light emitting device, it will be usable in a device of an emission wavelength corresponding to the band gap of the material, but some consideration will be necessary for changing the emission wavelength. Regarding physical properties related to semiconductor bands, controls by superlattices have been realized. More specifically, by changing the period of a superlattice, the bandwidth of its subband can be controlled to design an emission wavelength.
 Targeting on controlling manyelectronstate structures by material designs, the Inventor proposed manybody effect engineering by quantum dotbonded structures and has continued theoretical analyses ((1) U.S. Pat. No. 5,430,309; (2) U.S. Pat. No. 5,663,571; (3) U.S. Pat. No. 5,719,407; (4) U.S. Pat. No. 5,828,090; (5) U.S. Pat. No. 5,831,294; (6) J. Appl. Phys. 76, 2833(1994); (7) Phys. Rev. B51, 10714(1995); (8) Phys. Rev. B51, 11136(1995); (9) J. Appl. Phys. 77, 5509(1995); (10) Phys. Rev. B53, 6963(1996); (11) Phys. Rev. B53, 10141(1996); (12) Appl. Phys. Lett. 68, 2657(1996); (13) J. Appl. Phys. 80, 3893(1996); (14) J. Phys. Soc. Jpn. 65, 3952(1996); (15) Jpn. J. Appl. Phys. 36, 638(1997); (16) J. Phys. Soc. Jpn. 66, 425(1997); (17) J. Appl. Phys. 81, 2693 (1997); (18) Physica (Amsterdam) 229B, 146(1997); (19) Physica (Amsterdam) 237A, 220(1997); (20) Surf. Sci. 375, 403(1997); (21) Physica (Amsterdam) 240B, 116(1997); (22) Physica (Amsterdam) 240B, 128(1997); (23) Physica (Amsterdam) IE, 226(1997); (24) Phys. Rev. Lett. 80, 572(1998); (25) Jpn. J. Appl. Phys. 37, 863(1998); (26) Physica (Amsterdam) 245B, 311(1998); (27) Physica (Amsterdam) 235B, 96(1998); (28) Phys. Rev. B59, 4952(1999); (29) Surf. Sci. 432, 1(1999); (30) International Journal of Modern Physics B. Vol. 13, No. 21, 22, pp.26892703, 1999). For example, realization of various correlated electronic systems is expected by adjusting a tunneling phenomenon between quantum dots and interaction between electrons in quantum dots. Let the tunneling transfer between adjacent quantum dots be written as t. Then, if quantum dots are aligned in form of a tetragonal lattice, the bandwidth of one electron state is T_{eff}=4t. If quantum dots form a onedimensional chain, the bandwidth of one electron state is T_{eff}=2t. In case of a threedimensional quantum dot array, T_{eff}=6t. That is, if D is the dimension of a quantum dot array, the bandwidth of one electron state has been T_{eff}=2Dt. Here is made a review about halffilled (one electron per each quantum dot) Mott transition (also called MottHubbard transition or Mott metalinsulator transition). Let the effective interaction of electrons within a quantum dot be written as U_{eff}, then the Hubbard gap on the part of the Mott insulator is substantially described as Δ=U_{eff}−T_{eff}, and the Mott transition can be controlled by changing U_{eff }or t. As already proposed, the MottHubbard transition can be controlled by adjusting U_{eff }or t, using a field effect, and it is applicable to field effect devices (Literatures (5), (6), (11) and (14) introduced above).
 On the other hand, reviewing the equation of Δ=U_{eff}−T_{eff}−2Dt, it will be possible to control MottHubbard transition by controlling the dimensionality D of the system. For this purpose, the Applicant already proposed a fractalbonded structure that can continuously change the dimensionality, and have exhibited that MottHubbard transition is controllable by changing the fractal dimensions.
 To enable designing of wider materials, it is desired to modify and control the dimension of materials by methods different from the fractal theory. For example, for the purpose of changing the nature of phase transition, it is first conceivable to control the number of nearestneighbor elements among elements forming a material.
 On the other hand, here is changed the attention to ferromagnetic phase transition taking place in the fractalbonded structure. Of course, ferromagnetic materials are one of the most important magnetic storage materials. When using z as the number of nearestneighbor atoms, k_{B }as the Boltzmann constant, T as temperature, it is known that spontaneous magnetization M in the averaging theory describing ferromagnetic phase transition satisfies
 M=Tanh(zM/k _{B} T)
 The highest among temperatures T leading to solutions of M≠0 of this equation is the critical temperature T_{c}. As readily understood from the equation, T_{c }is proportional to z. When assuming a tetragonal lattice, since z=2D, it is expected that the critical temperature of ferromagnetic phase transition depends on the dimensionality of a material. The Inventor executed more exact Monte Carlo simulation, and showed that the critical temperature of ferromagnetic transition occurring in a fractalbonded structure could be controlled by the fractal dimensions.
 It is therefore an object of the invention to provide a multiplytwisted helix complementary with a fractalshaped material and representing a new physical property, and a functional material using the multiplytwisted helix.
 A further object of the invention is to provide a multiplylooped ring structure complementary with a fractalshaped material and representing a new physical property, and a functional material using the multiplylooped ring structure.
 A still further object of the invention is to provide a multiplycomplexed onedimensional structure complementary with a fractalshaped material and representing a new physical property, and a functional material using the multiplycomplexed onedimensional structure.
 The Inventor proposes a multiplytwisted helix as one of spatial filler structures. This is made by winding a spiral on a spiral structure as a base like a chromatin structure that a gene represents, and by repeating it to progressively fill a threedimensional space. By adjusting the spiral pitch, the spatial filling ratio can be selected, and dimensionality of a material, i.e. the number of nearestneighbor elements in this structure can be modified.
 In other words, here is proposed a multiplytwisted helix in which spirals are made up by using a spiral structure as the base and using the spiral structure as an element. In this structure including hierarchically formed multiple spirals, onedimensional vacancies penetrate the structure to form a structure as a porous material. However, by adjusting the turn pitch of the spirals, the number of nearestneighbor elements can be changed. According to researches by the Inventor, the value of critical interelectron interaction of MottHubbard metalinsulator transition in this kind of structure can be controlled by the spiral pitch.
 The multiplytwisted helical structure may be formed regularly; however, in case a multiplytwisted helical structure is actually made, bonding positions appearing among spiral layers possibly distribute randomly. The degree of the randomness can be new freedom of material designs. Taking it into consideration, for the purpose of clarifying the effect of the random distribution, exact simulation was conducted. As a result, introduction of randomness has been proved to increase the width of the MottHubbard gap and enhance the Mott insulation. Therefore, the value of critical interelectron interaction of MottHubbard metalinsulator transition can be controlled not only by controlling the degree of randomness of the spiral turn pitch but also by controlling the degree of randomness regarding interlayer bonding positions.
 Still in the multiplytwisted helix, there is also the interlayer bonding position as a control parameter, in addition to the degree of randomness regarding the spiral turn pitch and interlayer bonding positions. That is, by controlling interlayer bonding positions, desired material designs are possible. More specifically, freedom of parallel movements of interlayer bonding, that is in other words, simultaneous parallel movements of interlayer bonding positions, can be used to control the value of critical interelectron interaction of MottHubbard metalinsulator transition occurring in the structure.
 The multiplytwisted helix can be used as a magnetic material as well. That is, in the multiplytwisted helix, the critical temperature for ferromagnetic phase transition to occur can be controlled by adjusting the turn pitch.
 The inventor also proposes a multiplylooped ring structure as another spatial filler structure, which is different in structure from the multiplytwisted helix but similar in effect to same. This can be obtained by hierarchically forming rings, using a ring as a base. The number of nearestneighbor elements can be changed progressively by adjusting the number of elements of loworder rings forming highorder hierarchies. Thereby, the spatial filling ratio can be established, and the dimensionality of the material can be modified. In this multiplylooped ring structure, the above discussion is directly applicable only if the turn pitch in the multiplytwisted helix is replaced by the number of elements.
 The inventor further proposes a multiplycomplexed onedimensional structure as a more general spatial filler structure that includes both a multiplytwisted helix and a multiplylooped ring structure. This is obtained by hierarchically forming onedimensional structure systems having a finite curvature using a onedimensional structure system having a finite curvature as the base. In this case, the number of nearestneighbor elements can be changed progressively by adjusting the curvature of loworder onedimensional structure systems forming highorder hierarchies. Thereby, the spatial filling ratio can be established, and the dimensionality of the material can be modified. In this multiplycomplexed onedimensional structure, the above discussion is directly applicable only if the turn pitch in the foregoing multiplytwisted helix or the number of elements in the foregoing multiplylooped ring structure is replaced by the curvature. In this multiplycomplexed onedimensional structure, those having a finite twisting ratio correspond to multiplytwisted helixes whilst those having zero twisting ratios correspond to multiplylooped ring structures.
 The present invention has been made as a result of progressive studies based on the above review.
 That is, to overcome the aboveindicated problems, according to the first aspect of the invention, there is provided a multiplycomplexed onedimensional structure having a hierarchical structure in which a linear structure as an element of a onedimensional structure having a finite curvature is made of a thinner onedimensional structure having a finite curvature, comprising:
 at least two layers of the onedimensional structures bonded to each other in at least one site.
 According to the second aspect of the invention, there is provided a multiplycomplexed onedimensional structure having a hierarchical structure in which a linear structure as an element of a onedimensional structure having a finite curvature is made of a thinner onedimensional structure having a finite curvature, characterized in:
 exhibiting a nature regulated by setting a curvature in case the onedimensional structure is made of thinner onedimensional structures.
 According to the third aspect of the invention, there is provided a multiplycomplexed onedimensional structure having a hierarchical structure in which a linear structure as an element of a onedimensional structure having a finite curvature is made of a thinner onedimensional structure having a finite curvature, characterized in:
 having a dimensionality regulated by setting a curvature in case the onedimensional structure is made of thinner onedimensional structures.
 According to the fourth aspect of the invention, there is provided a multiplycomplexed onedimensional structure having a hierarchical structure in which a linear structure as an element of a onedimensional structure having a finite curvature is made of a thinner onedimensional structure having a finite curvature, having a random potential introduced therein, and at least two onedimensional structures bonded in at least one site, characterized in:
 a quantum chaos occurring therein being controlled by setting the intensity of the random potential, by setting the intensity of layertolayer bonding, by setting the curvature used when forming the onedimensional structure from thinner onedimensional structures, or by adding a magnetic impurity.
 Control of the quantum chaos produced typically relies on setting a bonding force between layers. Addition of magnetic impurities contributes to a decrease of the bonding force between layers and good control of the quantum chaos.
 According to the fifth aspect of the invention, there is provided a functional material including in at least a portion thereof a multiplycomplexed onedimensional structure having a hierarchical structure in which a linear structure as an element of a onedimensional structure having a finite curvature is made of thinner onedimensional structures having a finite curvature, characterized in:
 at least two layers of the onedimensional structures being bonded to each other in at least one site.
 According to the sixth aspect of the invention, there is provided a functional material including in at least a portion thereof a multiplycomplexed onedimensional structure having a hierarchical structure in which a linear structure as an element of a onedimensional structure having a finite curvature is made of thinner onedimensional structures having a finite curvature, characterized in:
 the multiplycomplexed onedimensional structure exhibiting a nature regulated by setting the curvature used when the onedimensional structure is made of thinner onedimensional structures.
 According to the seventh aspect of the invention, there is provided a functional material including in at least a portion thereof a multiplycomplexed onedimensional structure having a hierarchical structure in which a linear structure as an element of a onedimensional structure having a finite curvature is made of thinner onedimensional structures having a finite curvature, characterized in:
 the multiplycomplexed onedimensional structure having a dimensionality regulated by setting a curvature in case the onedimensional structure is made of thinner onedimensional structures.
 According to the eighth aspect of the invention, there is provided a multiplytwisted helix having a hierarchical structure in which a linear structure as an element of a spiral structure is made of thinner spiral structures, characterized in:
 at least two layers of spiral structures being bonded in at least one site.
 According to the ninth aspect of the invention, there is provided a multiplytwisted helix having a hierarchical structure in which a linear structure as an element of a spiral structure is made of thinner spiral structures, characterized in:
 exhibiting a nature regulated by setting a turn pitch in case the spiral structure is made of thinner spiral structures.
 According to the tenth aspect of the invention, there is provided a multiplytwisted helix having a hierarchical structure in which a linear structure as an element of a spiral structure is made of thinner spiral structures, characterized in:
 having a dimensionality regulated by setting a turn pitch in case the spiral structure is made of thinner spiral structures.
 According to the eleventh aspect of the invention, there is provided a multiplytwisted helix having a hierarchical structure in which a linear structure as an element of a spiral structure is made of a thinner spiral structure, having a random potential introduced therein, and at least two spiral structures bonded in at least one site, characterized in:
 a quantum chaos occurring therein being controlled by setting the intensity of the random potential, by setting the intensity of layertolayer bonding, by setting the turn pitch used when forming the spiral structure from thinner spiral structures, or by adding a magnetic impurity.
 Control of the quantum chaos produced typically relies on setting a bonding force between layers. Addition of magnetic impurities contributes to good control of the quantum chaos. Decreasing the bonding force between layers also contributes to good control of the quantum chaos.
 According to the twelfth aspect of the invention, there is provided a multiplytwisted helix having a hierarchical structure in which a linear structure as an element of a spiral structure is made of a thinner spiral structure, and having at least two layers of spiral structures bonded in at least one site, characterized in:
 the bonding performance between linear structures as elements of the spiral structure being controlled by a turn pitch in case of forming the spiral structure from thinner spiral structures, by the bonding force between the layers, or by a fluctuation in the bonding site between at least two layers of spiral structures.
 According to the thirteenth aspect of the invention, there is provided a functional material including in at least a portion thereof a multiplytwisted helix having a hierarchical structure in which a linear structure as an element of a spiral structure is made of thinner spiral structures, characterized in:
 at least two layers of spiral structures in the multiplytwisted helix being bonded in at least one site.
 According to the fourteenth aspect of the invention, there is provided a functional material including in at least a part thereof a multiplytwisted helix having a hierarchical structure in which a linear structure as an element of a spiral structure is made of thinner spiral structures, characterized in:
 the multiplytwisted helix exhibiting a nature regulated by setting a turn pitch produced when the spiral structure is made of thinner spiral structures.
 According to the fifteenth aspect of the invention, there is provided a functional material including in at least a part thereof a multiplytwisted helix having a hierarchical structure in which a linear structure as an element of a spiral structure is made of thinner spiral structures, characterized in:
 the multiplytwisted helix having a dimensionality regulated by setting a turn pitch in case the spiral structure is made of thinner spiral structures.
 According to the sixteenth aspect of the invention, there is provided a multiplylooped ring structure having a hierarchical structure in which an annular structure as an element of a ring structure is made of a thinner ring structure, characterized in:
 at least two layers of ring structures being bonded in at least one site.
 According to the seventeenth aspect of the invention, there is provided a multiplylooped ring structure having a hierarchical structure in which a linear structure as an element of a ring structure is made of a thinner ring structure, characterized in:
 exhibiting a nature regulated by setting a number of elements in case the ring structure is made of thinner ring structures.
 According to the eighteenth aspect of the invention, there is provided a multiplylooped ring structure having a hierarchical structure in which a linear structure as an element of a ring structure is made of a thinner ring structure, characterized in:
 having a dimensionality regulated by setting a number of elements in case the ring structure is made of thinner ring structures.
 According to the nineteenth aspect of the invention, there is provided a multiplylooped ring structure having a hierarchical structure in which a linear structure as an element of a ring structure is made of a thinner ring, having a random potential introduced therein, and at least two ring structures bonded in at least one site, characterized in:
 a quantum chaos occurring therein being controlled by setting the intensity of the random potential, by setting the intensity of layertolayer bonding, by setting the number elements used when forming the ring structure from thinner ring structures, or by adding a magnetic impurity.
 Control of the quantum chaos produced typically relies on setting a bonding force between layers. Addition of magnetic impurities contributes to a decrease of the bonding force between layers and good control of the quantum chaos.
 According to the twentieth aspect of the invention, there is provided a functional material including in at least a portion thereof a multiplylooped ring structure having a hierarchical structure in which a linear structure as an element of a ring structure is made of thinner ring structures, characterized in:
 at least two layers of the ring structures being bonded to each other in at least one site.
 According to the twentyfirst aspect of the invention, there is provided a functional material including in at least a portion thereof a multiplylooped ring structure having a hierarchical structure in which a linear structure as an element of a ring structure is made of thinner ring structures, characterized in:
 the multiplylooped ring structure exhibiting a nature regulated by setting the number of elements used when the ring structure is made of thinner ring structures.
 According to the twentysecond aspect of the invention, there is provided a functional material including in at least a portion thereof a multiplyloop ring structure having a hierarchical structure in which a linear structure as an element of a ring structure is made of thinner ring structures, characterized in:
 the multiplylooped ring structure having a dimensionality regulated by setting a number of elements in case the ring structure is made of thinner ring structures.
 In the present invention, in the multiplycomplexed onedimensional structure, the curvature used when a onedimensional structure in the first layer, for example, is made of a thinner onedimensional structure in the second layer lower by one layer than the first layer is set to a value different from the curvature used when a onedimensional structure in the third layer different from the first layer is made of a thinner onedimensional structure in the fourth layer lower by one layer than the third layer. This curvature may be set to a different value, depending on a difference in position in the onedimensional structure of the same layer. In the multiplytwisted helix, the turn pitch used when a spiral structure in the first layer, for example, is made of a thinner spiral structure in the second layer lower by one layer than the first layer is set to a value different from the curvature used when a spiral structure in the third layer different from the first layer is made of a thinner spiral structure in the fourth layer lower by one layer than the third layer. This turn pitch may be set to a different value, depending on a difference in position in the spiral structure of the same layer. In the multiplylooped ring structure, the number of elements used when a ring structure in the first layer is made of a thinner ring structure in the second layer lower than one layer than the first layer is set to a value different from the number of elements used when a ring structure in the third layer different by one layer from the first layer is made of a thinner ring structure in the fourth layer lower by one layer than the third layer. This number of elements may be set to a different value, depending on a difference in position in the ring structure of the same layer.
 There may be fluctuation in sites where spiral structures, ring structures or onedimensional structures bond between at least two layers. This is equivalent to introduction of randomness to sites where spiral structures, ring structures or onedimensional structures are bonded between layers. “Fluctuation” involves both spatial fluctuation (that can be reworded to disturbance or deviation) and temporal fluctuation. Any way is employable for introduction of fluctuation. For example, it may be introduced to appear in predetermined pitches. The fluctuation may be introduced by removing or adding bonds between at least two layers of spiral structures, ring structures or onedimensional structures.
 Curvature of the multiplycomplexed onedimensional structure, turn pitch of the multiplytwisted helix or number of elements of the multiplylooped ring structure is made variable under external control, for example. In a typical example of the multiplycomplexed onedimensional structure, multiplytwisted helix or multiplylooped ring structure, the onedimensional structure, spiral structure or ring structure is made of a linear formation essentially made of atoms or groups of atoms (clusters) as elements. To introduce the abovementioned fluctuation into this kind of structure, there is a method of inducing random absorption or elimination (surplus bonds or lack of bonds) of molecules in the linear formation. Since such introduction of a change or fluctuation of the curvature, turn pitch of number elements can be utilized as a kind of memory function, these multiplycomplexed onedimensional structure, multiplytwisted helix and multiplylooped ring structure can be used as memory devices.
 In the present invention, control of curvature in the multiplycomplexed onedimensional structure, turn pitch in the multiplytwisted helix or number of elements in the multiplylooped ring structure can bring about phase transition, especially metalinsulator phase transition or ferromagnetic phase transition. Critical value of the phase transition is regulated in accordance with the curvature, turn pitch or number of elements. This phase transition is controlled by control of bonding positions between onedimensional structures, helical structures or ring structures of two layers. Specifically, these positions are controlled by parallel movements of these bonds. Specifically, the multiplycomplexed onedimensional structure, multiplytwisted helix or multiplylooped ring structure includes, for example, metallic phase portions and insulating phase portions. The insulating phase portions can change their phase to the metallic phase because of their versatility. Critical temperature of ferromagnetic phase transition can be regulated by the degree of the abovementioned fluctuation. Alternatively, critical temperature for ferromagnetic phase transition can be regulated by parallel movements of bonds between at least two onedimensional structures, spiral structures or ring structures of different layers.
 In a multiplycomplexed onedimensional structure, multiplytwisted helix or multiplylooped ring structure, including onedimensional structures, helical structures or ring structures of at least two different layers, which are bonded, at least, at one position, if this at least one bond itself is made in form of a linear structure, then it is possible to control various physical phenomena that take place in the structure. For example, by setting the force of the bond made in form of a linear structure, critical temperature for ferromagnetic phase transition can be regulated. Alternatively, quantum chaos that may take place can be controlled. Further, electron state (electron correlation) can be controlled thereby to control metalinsulator phase transition.
 In a multiplycomplexed onedimensional structure, multiplytwisted helix or multiplylooped ring structure, including onedimensional structures, spiral structures or ring structures of at least two different layers, which are bonded, at least, at one position, if this at least one bond is made via an independent element, then it is possible to control various physical phenomena that take place in the structure. For example, by regulating critical temperature for ferromagnetic phase transition by making use of the criticality obtained by the structure, the structure can exhibit a stable physical property against minute structural fluctuations. Alternatively, control of the quantum chaos or control of metalinsulator phase transition is possible. Furthermore, it is possible to control the electron state thereby to control metalinsulator phase transition. These structures are not sensitive to structural fluctuations and exhibit pandemic physical properties, so it is easy to massproduce a material exhibiting a constantly uniform physical property.
 Features of the present aspect of the invention common to those of the fourth, 11th and 19th aspects of the invention are directly applicable to functional materials using any of the multiplycomplexed onedimensional structure, multiplytwisted helix or multiplylooped ring structure. Features of the present aspect of the invention common to those of the 12th aspect of the invention are also applicable to the multiplycomplexed onedimensional structure when the helical structures are replaced by onedimensional structures having a finite curvature, and to the multiplylooped ring structure when the helical structures are replaced by ring structures. They are also applicable to functional materials using the multiplycomplexed onedimensional structure, multiplytwisted helix or multiplylooped ring structure.
 According to the invention having the abovesummarized configurations, by appropriately setting curvature of the multiplycomplexed onedimensional structure, spiral turn pitch of the multiplytwisted helix or number of elements in the multiplylooped ring structure, it is possible to select a desired value of spatial filling ratio, thereby modify or control dimensionality of the material and control physical properties that the material exhibit. Especially, for example, by controlling the number of nearestneighbor elements in the multiplycomplexed onedimensional structure, multiplytwisted helix or multiplylooped ring structure, it is possible to modify phase transitional natures such as MottHubbard metalinsulator phase transition or magnetic phase transition. Further, by introducing fluctuations of bonding sites between at least two onedimensional structures, spiral structures or ring structures of different layers, that is in other words, by introducing randomness in bonding sites between onedimensional structures, spiral structures or ring structures of different layers and controlling the intensity of the randomness, it is possible to adjust the insulation performance by interelectron correlation for wider material designs. Furthermore, the quantum chaos can be controlled by appropriately setting the potential intensity and the bonding force between layers when introducing random potentials, by appropriately setting the curvature when onedimensional structures in the multiplycomplexed onedimensional structure are made of thin onedimensional structures, by appropriately setting the turn pitch when spiral structures in the multiplytwisted helix are made of thin helical structures, by appropriately setting the number of elements when ring structure in the multiplylooped ring structure are made of thin ring structures, or by addition of magnetic impurities.
 FIG. 1 is a rough diagram that shows actual bonding in a multiplytwisted helical structure in case of N=4;
 FIG. 2 is a rough diagram that schematically shows a multiplytwisted helix according to the first embodiment of the invention;
 FIG. 3 is a rough diagram that shows a density of states obtained by numerical calculation in case of N=2 in the first embodiment of the invention;
 FIG. 4 is a rough diagram that shows a density of states obtained by numerical calculation in case of N=4 in the first embodiment of the invention;
 FIG. 5 is a rough diagram that shows a density of states obtained by numerical calculation in case of N=6 in the first embodiment of the invention;
 FIG. 6 is a rough diagram that shows a density of states obtained by numerical calculation in case of N=10 in the first embodiment of the invention;
 FIG. 7 is a rough diagram that shows a density of states obtained by numerical calculation in case of N=20 in the first embodiment of the invention;
 FIG. 8 is a rough diagram that shows a density of states obtained by numerical calculation in case of U=8 in the first embodiment of the invention;
 FIG. 9 is a rough diagram that shows a density of states obtained by numerical calculation in case of U=10 in the first embodiment of the invention;
 FIG. 10 is a rough diagram that shows a density of states obtained by numerical calculation in case of U=12 in the first embodiment of the invention;
 FIG. 11 is a rough diagram that shows a onedimensional structure of a protein;
 FIG. 12 is a rough diagram that shows a multiplytwisted spiral as a twodimensional structure using disulfide bonds;
 FIG. 13 is a rough diagram that shows a density of states obtained by numerical calculation in case of U=8 and N=2 in the second embodiment of the invention;
 FIG. 14 is a rough diagram that shows a density of states obtained by numerical calculation in case of U=8 and N=4 in the second embodiment of the invention;
 FIG. 15 is a rough diagram that shows a density of states obtained by numerical calculation in case of U=8 and N=6 in the second embodiment of the invention;
 FIG. 16 is a rough diagram that shows a density of states obtained by numerical calculation in case of U=8 and N=10 in the second embodiment of the invention;
 FIG. 17 is a rough diagram that shows a density of states obtained by numerical calculation in case of U=8 and N=20 in the second embodiment of the invention;
 FIG. 18 is a rough diagram that shows a density of states obtained by numerical calculation in case of U=10 and N=2 in the second embodiment of the invention;
 FIG. 19 is a rough diagram that shows a density of states obtained by numerical calculation in case of U=10 and N=4 in the second embodiment of the invention;
 FIG. 20 is a rough diagram that shows a density of states obtained by numerical calculation in case of U=10 and N=6 in the second embodiment of the invention;
 FIG. 21 is a rough diagram that shows a density of states obtained by numerical calculation in case of U=10 and N=10 in the second embodiment of the invention;
 FIG. 22 is a rough diagram that shows a density of states obtained by numerical calculation in case of U=10 and N=20 in the second embodiment of the invention;
 FIG. 23 is a rough diagram that shows a density of states obtained by numerical calculation in case of U=8 in the second embodiment of the invention;
 FIG. 24 is a rough diagram that shows a density of states obtained by numerical calculation in case of U=10 in the second embodiment of the invention;
 FIG. 25 is a rough diagram that shows a density of states obtained by numerical calculation in case of U=12 in the second embodiment of the invention;
 FIG. 26 is a rough diagram that shows a density of states obtained by numerical calculation in case of U=8 in the third embodiment of the invention;
 FIG. 27 is a rough diagram that shows a density of states obtained by numerical calculation in case of U=10 in the third embodiment of the invention;
 FIG. 28 is a rough diagram that shows a density of states obtained by numerical calculation in case of U=12 in the third embodiment of the invention;
 FIG. 29 is a rough diagram that shows spontaneous magnetization obtained by numerical calculation in case of N=2, 3, 4, 6, 10, 20 in the fourth embodiment of the invention;
 FIG. 30 is a rough diagram that shows spontaneous magnetization obtained by numerical calculation in case of N=10, 12, 14, 16, 18, 20 in the fourth embodiment of the invention;
 FIG. 31 is a rough diagram that shows quantum level statistics upon changes of v in case of N=10 and s=1 in the fifth embodiment of the invention;
 FIG. 32 is a rough diagram that shows quantum level statistics upon changes of v in case of N=10 and s=1 in the fifth embodiment of the invention;
 FIG. 33 is a rough diagram that shows quantum level statistics upon changes of v in case of N=10 and s=1 in the fifth embodiment of the invention;
 FIG. 34 is a rough diagram that shows quantum level statistics upon changes of v in case of N=10 and s=1 in the fifth embodiment of the invention;
 FIG. 35 is a rough diagram that shows quantum level statistics upon changes of v in case of N=10 and s=1 in the fifth embodiment of the invention;
 FIG. 36 is a rough diagram that shows quantum level statistics upon changes of v in case of N=10 and s=1 in the fifth embodiment of the invention;
 FIG. 37 is a rough diagram that shows quantum level statistics upon changes of v in case of N=10 and s=1 in the fifth embodiment of the invention;
 FIG. 38 is a rough diagram that shows quantum level statistics upon changes of v in case of N=10 and s=1 in the fifth embodiment of the invention;
 FIG. 39 is a rough diagram that shows quantum level statistics upon changes of v in case of N=10 and s=1 in the fifth embodiment of the invention;
 FIG. 40 is a rough diagram that shows quantum level statistics upon changes of v in case of N=10 and s=1 in the fifth embodiment of the invention;
 FIG. 41 is a rough diagram that shows spontaneous magnetization obtained by numerical calculation in case of N=2, 3, 4, 6, 10, 20 in the sixth embodiment of the invention;
 FIG. 42 is a rough diagram that shows spontaneous magnetization obtained by numerical calculation when R is changed in case of N=4 in the sixth embodiment of the invention;
 FIG. 43 is a rough diagram that shows spontaneous magnetization obtained by numerical calculation when R is changed in case of N=6 in the sixth embodiment of the invention;
 FIG. 44 is a rough diagram that shows spontaneous magnetization obtained by numerical calculation when R is changed in case of N=10 in the sixth embodiment of the invention;
 FIG. 45 is a rough diagram that shows spontaneous magnetization obtained by numerical calculation when R is changed in case of N=20 in the sixth embodiment of the invention;
 FIG. 46 is a rough diagram that shows changes of Ω(r) when N is changed in case of R=0 and α=0 in the seventh embodiment of the invention;
 FIG. 47 is a rough diagram that shows changes of Ω(r) when R is changed in case of N=10 and α=0 in the seventh embodiment of the invention;
 FIG. 48 is a rough diagram that shows changes of Ω(r) when α is changed in case of N=10 and R=0 in the seventh embodiment of the invention;
 FIG. 49 is a rough diagram that shows a coefficient C_{1 }upon optimum approximation of Ω(r) by Equation 120 in case of R=0 and α=0 in the seventh embodiment of the invention;
 FIG. 50 is a rough diagram that shows a coefficient C_{2 }upon optimum approximation of Ω(r) by Equation 120 in case of R=0 and α=0 in the seventh embodiment of the invention;
 FIG. 51 is a rough diagram that shows a coefficient C_{1 }upon optimum approximation of Ω(r) by Equation 120 in case of N=10 and α=0 in the seventh embodiment of the invention;
 FIG. 52 is a rough diagram that shows a coefficient C_{2 }upon optimum approximation of Ω(r) by Equation 120 in case of N=10 and α=0 in the seventh embodiment of the invention;
 FIG. 53 is a rough diagram that shows a coefficient C_{1 }upon optimum approximation of Ω(r) by Equation 120 in case of N=10 and R=0 in the seventh embodiment of the invention;
 FIG. 54 is a rough diagram that shows a coefficient C_{2 }upon optimum approximation of Ω(r) by Equation 120 in case of N=10 and R=0 in the seventh embodiment of the invention;
 FIG. 55 is a rough diagram that shows spontaneous magnetization obtained by numerical calculation when α is changed in case of N=6 in the eighth embodiment of the invention;
 FIG. 56 is a rough diagram that shows spontaneous magnetization obtained by numerical calculation when α is changed in case of N=10 in the eighth embodiment of the invention;
 FIG. 57 is a rough diagram that shows spontaneous magnetization obtained by numerical calculation when α is changed in case of N=20 in the eighth embodiment of the invention;
 FIG. 58 is a rough diagram that shows quantum level statistics upon changes of v in case of N=10 and s=1 in the ninth embodiment of the invention;
 FIG. 59 is a rough diagram that shows quantum level statistics upon changes of v in case of N=10 and s=1 in the ninth embodiment of the invention;
 FIG. 60 is a rough diagram that shows quantum level statistics upon changes of v in case of N=4 and s=1 in the ninth embodiment of the invention;
 FIG. 61 is a rough diagram that shows quantum level statistics upon changes of v in case of N=4 and s=1 in the ninth embodiment of the invention;
 FIG. 62 is a rough diagram that shows quantum level statistics upon changes of N in case of v=2 and s=1 in the ninth embodiment of the invention;
 FIG. 63 is a rough diagram that shows quantum level statistics upon changes of N in case of v=2 and s=1 in the ninth embodiment of the invention;
 FIG. 64 is a rough diagram that shows quantum level statistics upon changes of N in case of v=2 and s=0.8 in the ninth embodiment of the invention;
 FIG. 65 is a rough diagram that shows quantum level statistics upon changes of N in case of v=2 and s=0.8 in the ninth embodiment of the invention;
 FIG. 66 is a rough diagram that shows quantum level statistics upon changes of N in case of v=2 and s=0.6 in the ninth embodiment of the invention;
 FIG. 67 is a rough diagram that shows quantum level statistics upon changes of N in case of v=2 and s=0.6 in the ninth embodiment of the invention;
 FIG. 68 is a rough diagram that schematically shows a multiplylooped ring structure according to the tenth embodiment of the invention;
 FIG. 69 is a rough diagram that shows a density of states obtained by numerical calculation in case of U=8 in the tenth embodiment of the invention;
 FIG. 70 is a rough diagram that shows a density of states obtained by numerical calculation in case of U=10 in the tenth embodiment of the invention;
 FIG. 71 is a rough diagram that shows a density of states obtained by numerical calculation in case of U=12 in the tenth embodiment of the invention;
 FIG. 72 is a rough diagram that shows quantum level statistics upon changes of v in case of N=6 and s=1 in the 11th embodiment of the invention;
 FIG. 73 is a rough diagram that shows quantum level statistics upon changes of v in case of N=6 and s=1 in the 11th embodiment of the invention;
 FIG. 74 is a rough diagram that shows quantum level statistics upon changes of v in case of N=10 and s=1 in the 11th embodiment of the invention;
 FIG. 75 is a rough diagram that shows quantum level statistics upon changes of v in case of N=10 and s=1 in the 11th embodiment of the invention;
 FIG. 76 is a rough diagram that shows quantum level statistics upon changes of v in case of N=20 and s=1 in the 11th embodiment of the invention;
 FIG. 77 is a rough diagram that shows quantum level statistics upon changes of v in case of N=20 and s=1 in the 11th embodiment of the invention;
 FIG. 78 is a rough diagram that shows quantum level statistics upon changes of s in case of v=2 and L=6 in the 11th embodiment of the invention;
 FIG. 79 is a rough diagram that shows quantum level statistics upon changes of s in case of v=2 and L=10 in the 11th embodiment of the invention;
 FIG. 80 is a rough diagram that shows quantum level statistics upon changes of s in case of v=2 and L=20 in the 11th embodiment of the invention;
 FIG. 81 is a rough diagram that shows a density of states obtained by numerical calculation in case of U=8 in a multiplylooped ring structure of α=0 as a comparative example in the 12th embodiment of the invention;
 FIG. 82 is a rough diagram that shows a density of states obtained by numerical calculation in case of U=10 in a multiplylooped ring structure of α=0 as a comparative example in the 12th embodiment of the invention;
 FIG. 83 is a rough diagram that shows a density of states obtained by numerical calculation in case of U=12 in a multiplylooped ring structure of α=0 as a comparative example in the 12th embodiment of the invention;
 FIG. 84 is a rough diagram that shows a density of states obtained by numerical calculation in case of U=8 in the 12th embodiment of the invention;
 FIG. 85 is a rough diagram that shows a density of states obtained by numerical calculation in case of U=10 in the 12th embodiment of the invention;
 FIG. 86 is a rough diagram that shows a density of states obtained by numerical calculation in case of U=12 in the 12th embodiment of the invention;
 FIG. 87 is a rough diagram that shows quantum level statistics upon changes of V in case of L=6 and s=1 in the 13th embodiment of the invention;
 FIG. 88 is a rough diagram that shows quantum level statistics upon changes of V in case of L=6 and s=1 in the 13th embodiment of the invention;
 FIG. 89 is a rough diagram that shows quantum level statistics upon changes of V in case of L=10 and s=1 in the 13th embodiment of the invention;
 FIG. 90 is a rough diagram that shows quantum level statistics upon changes of V in case of L=10 and s=1 in the 13th embodiment of the invention;
 FIG. 91 is a rough diagram that shows quantum level statistics upon changes of V in case of L=20 and s=1 in the 13th embodiment of the invention;
 FIG. 92 is a rough diagram that shows quantum level statistics upon changes of V in case of L=20 and s=1 in the 13th embodiment of the invention;
 FIG. 93 is a rough diagram that shows quantum level statistics upon changes of s in case of L=6 and v=2 in the 13th embodiment of the invention;
 FIG. 94 is a rough diagram that shows quantum level statistics upon changes of s in case of L=10 and v=2 in the 13th embodiment of the invention;
 FIG. 95 is a rough diagram that shows quantum level statistics upon changes of s in case of L=20 and v=2 in the 13th embodiment of the invention;
 FIG. 96 is a rough diagram that shows spontaneous magnetization obtained by numerical calculation when L is changed in case of α=0 in the 14th embodiment of the invention;
 FIG. 97 is a rough diagram that shows spontaneous magnetization obtained by numerical calculation when L is changed in case of α=0.5 in the 14th embodiment of the invention;
 FIG. 98 is a rough diagram that shows actual bonding in a helixbased hierarchy of multiple connections according to the 15th embodiment of the invention;
 FIG. 99 is a rough diagram that shows changes of the average number of nearestneighbor sites with N in a helixbased hierarchy of multiple connections and a multiplytwisted helix according to the 15th embodiment of the invention;
 FIG. 100 is a rough diagram that shows spontaneous magnetization obtained by numerical calculation in case of N=3, 4, 6, 8, 10, 20 in the 15th embodiment of the invention;
 FIG. 101 is a rough diagram that shows spontaneous magnetization obtained by numerical calculation when s is changed in case of N=3 in the 15th embodiment of the invention;
 FIG. 102 is a rough diagram that shows spontaneous magnetization obtained by numerical calculation when s is changed in case of N=4 in the 15th embodiment of the invention;
 FIG. 103 is a rough diagram that shows spontaneous magnetization obtained by numerical calculation when s is changed in case of N=6 in the 15th embodiment of the invention;
 FIG. 104 is a rough diagram that shows spontaneous magnetization obtained by numerical calculation when s is changed in case of N=8 in the 15th embodiment of the invention;
 FIG. 105 is a rough diagram that shows spontaneous magnetization obtained by numerical calculation when s is changed in case of N=10 in the 15th embodiment of the invention;
 FIG. 106 is a rough diagram that shows spontaneous magnetization obtained by numerical calculation when s is changed in case of N=20 in the 15th embodiment of the invention;
 FIG. 107 is a rough diagram that shows quantum level statistics upon changes of v in case of N=10 and s=1 in the 16th embodiment of the invention;
 FIG. 108 is a rough diagram that shows quantum level statistics upon changes of v in case of N=10 and s=1 in the 16th embodiment of the invention;
 FIG. 109 is a rough diagram that shows quantum level statistics upon changes of N in case of v=2 and s=1 in the 16th embodiment of the invention;
 FIG. 110 is a rough diagram that shows quantum level statistics upon changes of N in case of v=2 and s=1 in the 16th embodiment of the invention;
 FIG. 111 is a rough diagram that shows quantum level statistics upon changes of s in case of N=10 and v=2 in the 16th embodiment of the invention;
 FIG. 112 is a rough diagram that shows a density of states obtained by numerical calculation in case of N=10 and s=1 in the 17th embodiment of the invention;
 FIG. 113 is a rough diagram that shows a density of states obtained by numerical calculation in case of s=1 and U=12 in the 17th embodiment of the invention;
 FIG. 114 is a rough diagram that shows a density of states obtained by numerical calculation in case of N=6 and U=6 in the 17th embodiment of the invention;
 FIG. 115 is a rough diagram that shows a density of states obtained by numerical calculation in case of N=10 and U=6 in the 17th embodiment of the invention;
 FIG. 116 is a rough diagram that shows a density of states obtained by numerical calculation in case of N=20 and U=6 in the 17th embodiment of the invention;
 FIG. 117 is a rough diagram that shows actual bonding in a critical spiral structure according to the 18th embodiment of the invention;
 FIG. 118 is a rough diagram that shows spontaneous magnetization obtained by numerical calculation in case of N=3, 4, 6, 8, 10, 20 in the 18th embodiment of the invention;
 FIG. 119 is a rough diagram that shows spontaneous magnetization obtained by numerical calculation when s is changed in case of N=3 in the 18th embodiment of the invention;
 FIG. 120 is a rough diagram that shows spontaneous magnetization obtained by numerical calculation when s is changed in case of N=4 in the 18th embodiment of the invention;
 FIG. 121 is a rough diagram that shows spontaneous magnetization obtained by numerical calculation when s is changed in case of N=6 in the 18th embodiment of the invention;
 FIG. 122 is a rough diagram that shows spontaneous magnetization obtained by numerical calculation when s is changed in case of N=8 in the 18th embodiment of the invention;
 FIG. 123 is a rough diagram that shows spontaneous magnetization obtained by numerical calculation when s is changed in case of N=10 in the 18th embodiment of the invention;
 FIG. 124 is a rough diagram that shows spontaneous magnetization obtained by numerical calculation when s is changed in case of N=20 in the 18th embodiment of the invention;
 FIG. 125 is a rough diagram that shows quantum level statistics upon changes of v in case of N=10 and s=1 in the 19th embodiment of the invention;
 FIG. 126 is a rough diagram that shows quantum level statistics upon changes of v in case of N=10 and s=1 in the 19th embodiment of the invention;
 FIG. 127 is a rough diagram that shows quantum level statistics upon changes of N in case of v=2 and s=1 in the 19th embodiment of the invention;
 FIG. 128 is a rough diagram that shows quantum level statistics upon changes of N in case of v=2 and s=1 in the 19th embodiment of the invention;
 FIG. 129 is a rough diagram that shows quantum level statistics upon changes of s in case of v=2 and N=10 in the 19th embodiment of the invention;
 FIG. 130 is a rough diagram that shows a density of states obtained by numerical calculation in case of N=10 and s=1 in the 20th embodiment of the invention;
 FIG. 131 is a rough diagram that shows a density of states obtained by numerical calculation in case of s=1 and U=6 in the 20th embodiment of the invention;
 FIG. 132 is a rough diagram that shows a density of states obtained by numerical calculation in case of N=10 and U=6 in the 20th embodiment of the invention;
 FIG. 133 is a rough diagram that shows a density of states obtained by numerical calculation in case of N=10 and U=6 in the 20th embodiment of the invention;
 FIG. 134 is a rough diagram that shows a density of states obtained by numerical calculation in case of N=20 and U=6 in the 20th embodiment of the invention;
 FIG. 135 is a perspective view for explaining a manufacturing method of a multiplytwisted helix of q=2 and N=8 in the 21st embodiment of the invention;
 FIG. 136 is a perspective view for explaining a manufacturing method of a multiplytwisted helix of q=2 and N=8 in the 21st embodiment of the invention;
 FIG. 137 is a perspective view for explaining a manufacturing method of a multiplytwisted helix of q=2 and N=8 in the 21st embodiment of the invention;
 FIG. 138 is a perspective view for explaining a manufacturing method of a multiplytwisted helix of q=2 and N=8 in the 21st embodiment of the invention; and
 FIG. 139 is a perspective view for explaining a manufacturing method of a multiplytwisted helix of q=2 and N=8 in the 21st embodiment of the invention.
 Embodiments of the invention will now be explained below.
 First explained is a multiplytwisted helix according to the first embodiment of the invention.
 An electron system on a multiplytwisted spiral in the multiplytwisted helix according to the first embodiment is explained below.
 Assuming a onedimensional lattice, numbers are assigned as n= . . . , −1, 0, 1, . . . . Let the operator for generating an electron of a spin σ at the pth lattice point be Ĉ_{p,σ} ^{†}. Of course, there is the anticommutation relation
$\begin{array}{cc}\left\{{\hat{C}}_{p,\sigma},{\hat{C}}_{q,\rho}^{\u2020}\right\}={\delta}_{p,q}\ue89e{\delta}_{\sigma ,\rho}& \left(1\right)\end{array}$  Here is defined a singleband Hubbard Hamiltonian Ĥ of the electron system as follows.
$\begin{array}{cc}\hat{H}=t\ue89e\sum _{i,j,\sigma}\ue89e{\lambda}_{i,j}\ue89e{\hat{C}}_{i,\sigma}^{\u2020}\ue89e{\hat{C}}_{j,\sigma}+U\ue89e\sum _{j}\ue89e{\hat{n}}_{j,\uparrow}\ue89e{\hat{n}}_{j,\downarrow}& \left(2\right)\end{array}$  Letting electrons be movable only among neighboring sites, as λ_{p,q}
$\begin{array}{cc}{\lambda}_{p,q}={\lambda}_{q,p}=\{\begin{array}{ccccc}1& \mathrm{when}& \uf603pq\uf604=1& \text{\hspace{1em}}& \text{\hspace{1em}}\\ s& \mathrm{when}& \uf603pq\uf604=N& \text{\hspace{1em}}& \text{\hspace{1em}}\\ {s}^{2}& \mathrm{when}& \mathrm{mod}\left(p,N\right)=0& \mathrm{and}& q=p+{N}^{2}\\ {s}^{3}& \mathrm{when}& \mathrm{mod}\left(p,{N}^{2}\right)=1& \mathrm{and}& q=p+{N}^{3}\\ {s}^{4}& \mathrm{when}& \mathrm{mod}\left(p,{N}^{3}\right)=2& \mathrm{and}& q=p+{N}^{4}\\ \text{\hspace{1em}}& \vdots & \text{\hspace{1em}}& \text{\hspace{1em}}& \text{\hspace{1em}}\\ 0& \mathrm{otherwise}& \text{\hspace{1em}}& \text{\hspace{1em}}& \text{\hspace{1em}}\end{array}& \left(3\right)\end{array}$  is employed. Assume hereunder that s=1. However, mod(a, b) is the remainder as a result of division of a by b. Here is made a review about the average number of nearestneighbor sites. It is
$\begin{array}{cc}z=2+2+\frac{2}{N}+\frac{2}{{N}^{2}}+\dots & \left(4\right)\\ \text{\hspace{1em}}\ue89e=2+\frac{2\ue89eN}{N1}& \left(5\right)\end{array}$  Apparently, it can be any value from the value of a threedimensional tetragonal lattice, namely, z=6 when N=2, to the value of a twodimensional tetragonal lattice, namely, z=4 when N→∞. A multiplytwisted helical structure is defined by the definition of the nearestneighbor sites. FIG. 1 schematically show how the actual bonding is when N=4. FIG. 1A is for N pitch, FIG. 1B is for N^{2 }pitch, and FIG. 1C is for N^{3 }pitch. When the structure is folded such that the nearestneighbor sites become spatially closer, the multiplytwisted spiral is obtained as shown in FIG. 2. FIG. 2, however, embellishes it to provide an easier view. In this case, the onedimensional chain is bonded at two right and left sites, and the chain forms spirals of the N pitch. Since the spirals form spirals of the N^{2 }pitch, the term of S^{2 }enters as a result of transfer of electrons between adjacent spirals (Equation 3). Then, the spirals form spirals of a larger, i.e. N^{2}, pitch.
 Here is defined a pin σ electron density operator of the jth site {circumflex over (n)}_{j,σ}=ĉ_{j,ρ} ^{554 }Ĉ_{j,σ} and its sum {circumflex over (n)}_{j}=Σ_{σ}{circumflex over (n)}_{j,σ}.
 For the purpose of defining a temperature Green's function, here is introduced a grand canonical Hamiltonian {circumflex over (K)}=Ĥ−μ{circumflex over (N)} where {circumflex over (N)}=Σ_{j}{circumflex over (n)}_{j}. In the half filled taken here, chemical potential is μ=U/2. The halffilled grand canonical Hamiltonian can be expressed as
$\begin{array}{cc}\hat{K}=t\ue89e\sum _{i,j,\sigma}\ue89e{\lambda}_{j,i}\ue89e{\hat{t}}_{j,i,\sigma}+U/2\ue89e\sum _{i}\ue89e\left({\hat{u}}_{i}1\right)& \left(6\right)\end{array}$  Operators {circumflex over (t)}_{j,i,σ}, ĵ_{j,i,σ}, û_{i }and {circumflex over (d)}_{i,σ} are defined beforehand as
 {circumflex over (t)} _{j,i,σ} =Ĉ _{j,σ} ^{†} Ĉ _{i,σ} +Ĉ _{i,σ} ^{†} Ĉ _{j,σ} (7)
 ĵ _{j,i,σ} =ĉ _{j,q} ^{†} ĉ _{i,σ} −ĉ _{i,σ} ^{†} ĉ _{j,σ} (8)
 û _{i} =ĉ _{i,↑} ^{†} ĉ _{i,↑} ^{†} ĉ _{i,↓} ĉ _{i,↓} ĉ _{i,↑} ĉ _{i,↑} ^{†} ĉ _{i,↓} ĉ _{i,↓} ^{†} (9)
 {circumflex over (d)} _{i,σ} =ĉ _{i,σ} ^{†} ĉ _{i,σ} −ĉ _{i,σ} ĉ _{i,σ} ^{†} (10)
 If the temperature Green function is defined for operators Â and {circumflex over (B)} given, taking τ as imaginary time, it is as follows.
$\begin{array}{cc}\u3008\hat{A};\hat{B}\u3009={\int}_{0}^{\beta}\ue89e\text{\hspace{1em}}\ue89e\uf74c\tau \ue89e\u3008{T}_{\tau}\ue89e\hat{A}\ue8a0\left(\tau \right)\ue89e\hat{B}\u3009\ue89e{\uf74d}^{\uf74e\ue89e\text{\hspace{1em}}\ue89e{\omega}_{n}\ue89e\tau}& \left(11\right)\end{array}$  The onsite Green function
 is especially important because, when analytic continuation is conducted as iω_{n}→ω+iδ for a small δ,
$\begin{array}{cc}{D}_{j}\ue8a0\left(\omega \right)=\sum _{\sigma =\uparrow ,\downarrow}\ue89e\mathrm{Im}\ue89e\text{\hspace{1em}}\ue89e{G}_{j,\sigma}\ue8a0\left(\omega +\uf74e\ue89e\text{\hspace{1em}}\ue89e\delta \right)& \left(13\right)\end{array}$ 
 becomes the density of states of the system. For later numerical calculation of densities of states, δ=0.0001 will be used. Further let the total number of sites be n=10001.

 As the equation of motion of the onsite Green function,
$\begin{array}{cc}\uf74e\ue89e\text{\hspace{1em}}\ue89e{\omega}_{n}\ue89e\u3008{\hat{c}}_{j,\sigma};{\hat{c}}_{j,\sigma}^{\u2020}\u3009=1+t\ue89e\sum _{p,j}\ue89e{\lambda}_{p,j}\ue89e\u3008{\hat{c}}_{p,\sigma};{\hat{c}}_{j,\sigma}^{\u2020}\u3009+\frac{U}{2}\ue89e\u3008{\hat{d}}_{j,\sigma}\ue89e{\hat{c}}_{j,\sigma};{\hat{c}}_{j,\sigma}^{\u2020}\u3009& \left(16\right)\end{array}$  is obtained. Then, the approximation shown below is introduced, following Gros ((31) C. Gros, Phys. Rev. B50, 7295(1994)). If the site p is the nearestneighbor site of the site j, the resolution
$\begin{array}{cc}\u3008{\hat{c}}_{p,\sigma};{\hat{c}}_{j,\sigma}^{\u2020}\u3009\to t\ue89e\u3008{\hat{c}}_{p,\sigma};{\hat{c}}_{p,\sigma}^{\u2020}\u3009\ue89e\u3008{\hat{c}}_{j,\sigma};{\hat{c}}_{j,\sigma}^{\u2020}\u3009& \left(17\right)\end{array}$  is introduced as the approximation. This is said to be exact in case of infinitedimensional Bethe lattices, but in this case, it is only within approximation. Under the approximation, the following equation is obtained.
$\begin{array}{cc}\left({\mathrm{\uf74e\omega}}_{n}{t}^{2}\ue89e{\Gamma}_{j,\sigma}\right)\ue89e{G}_{j,\sigma}=1+\frac{U}{2}\ue89e\u3008{\hat{d}}_{j,\sigma}\ue89e{\hat{c}}_{j,\sigma};{\hat{c}}_{j,\sigma}^{\u2020}\u3009\ue89e\text{}\ue89e\mathrm{where}& \left(18\right)\\ {\Gamma}_{j,\sigma}=\sum _{p}\ue89e{\lambda}_{p,j}\ue89e{G}_{p,\sigma}& \left(19\right)\end{array}$  was introduced. To solve the equation, obtained, {circumflex over (d)}_{j,−σ}ĉ_{j,σ};ĉ_{j,σ} ^{†} has to be analyzed. In case of halff models, this equation of motion is
$\begin{array}{cc}{\mathrm{\uf74e\omega}}_{n}\ue89e\u3008{\hat{d}}_{j,\sigma}\ue89e{\hat{c}}_{j,\sigma};{\hat{c}}_{j,\sigma}^{\u2020}\u3009=\frac{U}{2}\ue89e{G}_{j,\sigma}2\ue89et\ue89e\sum _{p}\ue89e{\lambda}_{p,j}\ue89e\u3008{\hat{j}}_{p,j,\sigma}\ue89e{\hat{c}}_{j,\sigma};{\hat{c}}_{j,\sigma}^{\u2020}\u3009+t\ue89e\sum _{p}\ue89e{\lambda}_{p,j}\ue89e\u3008{\hat{d}}_{j,\sigma}\ue89e{\hat{c}}_{p,\sigma};{\hat{c}}_{j,\sigma}^{\u2020}\u3009& \left(20\right)\end{array}$  Here again, with reference to the Gros logic, approximation is introduced. It is the following translation.
 By executing this translation, the following closed equation is obtained.
$\begin{array}{cc}\left({\mathrm{\uf74e\omega}}_{n}{t}^{2}\ue89e{\Gamma}_{j,\sigma}\right)\ue89e{G}_{j,\sigma}=1+\frac{{\left(U/2\right)}^{2}}{{\mathrm{\uf74e\omega}}_{n}{t}^{2}\ue89e{\Gamma}_{j,\sigma}2\ue89e{t}^{2}\ue89e{\Gamma}_{j,\sigma}}\ue89e{G}_{j,\sigma}& \left(23\right)\end{array}$  Here is assumed that there is no dependency on spin. That is, assuming
 G _{j} =G _{j,↑} =G _{j,↓} (24)
 the following calculation is executed.

 that were obtained by numerical calculation are shown below. FIG. 3 shows DOS in case of N=2. As shown in FIG. 3, when N=2, the density of state D(ω=0) under the Fermi energy ω=0 exists in case of U<8, and the system behaves as a metal. On the other hand, when U=10, a region where DOS has disappeared exists near the Fermi energy, and the system behaves as a Mott insulator. FIG. 4 shows DOS in case of N=4. As shown in FIG. 4, when U=8, D(ω=0) becomes substantially zero, the system is closer to a Mott insulator. When the models of N=6 (FIG. 5), N=10 (FIG. 6) and N=20 (FIG. 7) are compared, this tendency is enhanced as N increases. Regarding the case of changing N while fixing U=8, DOS is shown in FIG. 8. From FIG. 8, the system is apparently a metal when N=2, and it is easily observed that a change to a Mott insulator occurs as N increases. Examples of U=10 and U=12 are shown in FIGS. 9 and 10, respectively. Since a value twice the value of ω rendering DOS be zero is the gap of a Mott insulator (Hubbard gap), it is appreciated that the width of the Hubbard gap increases with N. Therefore, it has been confirmed that MottHubbard transition can be controlled by adjusting N.
 In this manner, when the turn pitch is controlled and designed, the system can behave as a metal or can behave an insulator. Therefore, a material having a plurality of regions having different turn pitches can be a superstructure having various regions including metallic regions and insulating regions, and this enables richer controls and designs of physical properties. For example, if a material is designed to have regions of metallic/insulating/metallic phases, a device behaving as a diode can be realized. Further, if the insulator region is made to changeable in phase to a metal under external control, the material can behave as a transistor.
 A material containing the multiplytwisted helix as a part thereof could be made. For example, a material in which a structure other than multiplytwisted helical structures is added as an electrode to a multiplytwisted helical structure will be useful.
 In the foregoing explanation, widths of N, N and N^{3 }were assumed as turn pitches of multiplytwisted spirals; however, this assumption is only for simplicity, and there are various other possibilities. For example, in a pthorder spiral, it will be possible to select N^{P}+Δ_{p }(where Δ_{p }is a random number of −N^{P−1}<Δ_{P}<N^{p−1}). This is realized when a randomness regarding positions of bonding points is introduced in a highorder spiral. This randomness gives a qualitative difference to phase transition of a system. It will be explained later in greater detail with reference to the second embodiment.
 A method of fabricating such a multiplytwisted spiral will now be explained. If a genetic engineering method is used, DNA having a predetermined sequence can be made. α helix is known as one of the most popular proteins. This is a spiral structure in which four amino acids form one turn. DNA can be made such that cysteine, one of one of amino acids and having sulfur atoms, is introduced into the a helix. When this DNA is copied by mRNA to activate the genes to synthesize a protein, its primary structure is as shown in FIG. 11. On the other hand, it is known that sulfur atoms of cysteine bond one another to form a disulfide linkage, and contribute to form highorder proteins. It is expected that the multiplytwisted spirals will be formed as a secondary structure using the disulfide linkage (FIG. 12).
 Next explained is a multiplytwisted helix according to the second embodiment of the invention.
 Explained below is an electron system on a multiplytwisted spiral in a multiplytwisted helix having introduced randomness regarding bonding positions between spiral structures of different layers.
 Assuming a onedimensional lattice, numbers are assigned as n= . . . , −1, 0, 1, . . . . Let the operator for generating an electron of a spin σ at the pth lattice point be Ĉ_{p,σ} ^{†}. Of course, there is the anticommutation relation
 {Ĉ _{p,σ} ,Ĉ _{q,ρ} ^{†}}=δ_{p,q}δ_{σ,ρ} (25)
 Here is defined a singleband Hubbard Hamiltonian Ĥ of the electron system as follows.
$\begin{array}{cc}\hat{H}=t\ue89e\sum _{i,j,\sigma}\ue89e{\lambda}_{i,j}\ue89e{\hat{C}}_{i,\sigma}^{\u2020}\ue89e{\hat{C}}_{j,\sigma}+U\ue89e\sum _{j}\ue89e{\hat{n}}_{j,\uparrow}\ue89e{\hat{n}}_{j,\downarrow}& \left(26\right)\end{array}$  Letting electrons be movable only among neighboring sites, as λ_{p,q}
$\begin{array}{cc}{\lambda}_{p,q}={\lambda}_{q,p}=\{\begin{array}{cc}1& \mathrm{when}\ue89e\text{\hspace{1em}}\ue89e\uf603pq\uf604=1\\ s& \mathrm{when}\ue89e\text{\hspace{1em}}\ue89e\uf603pq\uf604=N\\ {s}^{2}& \mathrm{when}\ue89e\text{\hspace{1em}}\ue89e\mathrm{mod}\ue8a0\left(p,N\right)=0\ue89e\text{\hspace{1em}}\ue89e\mathrm{and}\ue89e\text{\hspace{1em}}\ue89eq=p+{N}^{2}+{N}_{r}\\ {s}^{3}& \mathrm{when}\ue89e\text{\hspace{1em}}\ue89e\mathrm{mod}\ue8a0\left(p,{N}^{2}\right)=1\ue89e\text{\hspace{1em}}\ue89e\mathrm{and}\ue89e\text{\hspace{1em}}\ue89eq=p+{N}^{3}+{N}^{2}\ue89er\\ {s}^{4}& \ue89e\mathrm{when}\ue89e\text{\hspace{1em}}\ue89e\mathrm{mod}\ue8a0\left(p,{N}^{3}\right)=2\ue89e\text{\hspace{1em}}\ue89e\mathrm{and}\ue89e\text{\hspace{1em}}\ue89eq=p+{N}^{4}+{N}^{3}\ue89er\\ \text{\hspace{1em}}& \vdots \\ 0& \mathrm{otherwise}\end{array}& \left(27\right)\end{array}$  is employed. Note that mod(a, b) is the remainder as a result of division of a by b. r is a random variable that satisfies
 −1<r<1 (28)


 When R=1, distribution becomes uniform, and dispersion decreases as R decreases. At the limit of R→0, randomness disappears. In the following simulation, calculation is made for the cases of R=1, R=½, R={fraction (1/4)}, R=⅛, R={fraction (1/16)} and R=0. s=1 is used below.


 In the half filled taken here, chemical potential is μ=U/2. The halffilled grand canonical Hamiltonian can be expressed as
$\begin{array}{cc}\hat{K}=t\ue89e\text{\hspace{1em}}\ue89e\sum _{i,j,\sigma}\ue89e{\lambda}_{j,i}\ue89e{\hat{t}}_{j,i,\sigma}+U/2\ue89e\sum _{i}\ue89e\left({\hat{u}}_{i}1\right)& \left(32\right)\end{array}$  Operators {circumflex over (t)}_{j,i,σ}, ĵ_{j,i,σ}, û_{i }and {circumflex over (d)}_{i,σ} are defined beforehand as
 {circumflex over (t)} _{j,i,σ} =Ĉ _{j,σ} ^{†} Ĉ _{i,σ} +Ĉ _{i,σ} ^{†} Ĉ _{j,σ} (33)
 ĵ _{j,i,σ} =ĉ _{j,q} ^{†} ĉ _{i,σ} −ĉ _{i,σ} ^{†} ĉ _{j,σ} (34)
 û _{i} =ĉ _{i,↓} ^{†} ĉ _{i,↑} ĉ _{i,↓} ^{†} ĉ _{i,↓} +ĉ _{i,↑ĉ} _{i,↑ĉ} _{i,↓} ĉ _{i,↓} ^{†} (35)
 {circumflex over (d)} _{i,σ} =ĉ _{i,σ} ^{†} ĉ _{i,σ} −ĉ _{i,σ} ĉ _{i,σ} (36)
 If the temperature Green function is defined for operators Â and {circumflex over (B)} given, taking τ as imaginary time, it is as follows.
 The onsite Green function
 is especially important because, when analytic continuation is conducted as iω_{n}ω+iδ for a small δ,
$\begin{array}{cc}{D}_{j}\ue8a0\left(\omega \right)=\sum _{\sigma =\uparrow ,\downarrow}\ue89e\mathrm{Im}\ue89e\text{\hspace{1em}}\ue89e{G}_{j,\sigma}\ue8a0\left(\omega +\mathrm{\uf74e\delta}\right)& \left(39\right)\end{array}$ 
 becomes the density of states of the system. For later numerical calculation of densities of states, δ=0.0001 will be used. Further let the total number of sites be n=10001.

 As the equation of motion of the onsite Green function,
$\begin{array}{cc}\uf74e\ue89e\text{\hspace{1em}}\ue89e{\omega}_{n}\ue89e\u3008{\hat{c}}_{j,\sigma};{\hat{c}}_{j,\sigma}^{\u2020}\u3009=1+t\ue89e\sum _{p,j}\ue89e{\lambda}_{p,j}\ue89e\u3008{\hat{c}}_{p,\sigma};{\hat{c}}_{j,\sigma}^{\u2020}\u3009+\frac{U}{2}\ue89e\u3008{\hat{d}}_{j,\sigma}\ue89e{\hat{c}}_{j,\sigma};{\hat{c}}_{j,\sigma}^{\u2020}\u3009& \left(42\right)\end{array}$  is obtained. Then, the approximation shown below is introduced, following Gros ((31) C. Gros, Phys. Rev. B50, 7295(1994)). If the site p is the nearestneighbor site of the site j, the resolution
 is introduced as the approximation. This is said to be exact in case of infinitedimensional Bethe lattices, but in this case, it is only within approximation. Under the approximation, the following equation is obtained.
$\begin{array}{cc}\left({\mathrm{\uf74e\omega}}_{n}{t}^{2}\ue89e{\Gamma}_{j,\sigma}\right)\ue89e{G}_{j,\sigma}=1+\frac{U}{2}\ue89e\u3008{\hat{d}}_{j,\sigma}\ue89e{\hat{c}}_{j,\sigma};{\hat{c}}_{j,\sigma}^{\u2020}\u3009& \left(44\right)\end{array}$ 
 was introduced. To solve the equation obtained, {circumflex over (d)}_{j,−σ}ĉ_{j,σ};ĉ_{j,σ} ^{†} has to be analyzed. In case of halffilled models, this equation of motion is
$\begin{array}{cc}\uf74e\ue89e\text{\hspace{1em}}\ue89e{\omega}_{n}\ue89e\u3008{\hat{d}}_{j,\sigma}\ue89e{\hat{c}}_{j,\sigma};{\hat{c}}_{j,\sigma}^{\u2020}\u3009=\frac{U}{2}\ue89e{G}_{j,\sigma}2\ue89et\ue89e\text{\hspace{1em}}\ue89e\sum _{p}\ue89e{\lambda}_{p,j}\ue89e\u3008{\hat{j}}_{p,j,\sigma}\ue89e{\hat{c}}_{j,\sigma};{\hat{c}}_{j,\sigma}^{\u2020}\u3009+t\ue89e\sum _{p}\ue89e{\lambda}_{p,j}\ue89e\u3008{\hat{d}}_{j,\sigma}\ue89e{\hat{c}}_{p,\sigma};{\hat{c}}_{j,\sigma}^{\u2020}\u3009& \left(46\right)\end{array}$  Here again, with reference to the Gros logic, approximation is introduced. It is the following translation.
 By executing this translation, the following closed equation is obtained.
$\begin{array}{cc}\left({\mathrm{\uf74e\omega}}_{n}{t}^{2}\ue89e{\Gamma}_{j,\sigma}\right)\ue89e{G}_{j,\sigma}=1+\frac{{\left(U/2\right)}^{2}}{{\mathrm{\uf74e\omega}}_{n}{t}^{2}\ue89e{\Gamma}_{j,\sigma}2\ue89e{t}^{2}\ue89e{\Gamma}_{j,\sigma}}\ue89e{G}_{j,\sigma}& \left(49\right)\end{array}$  Here is assumed that there is no dependency on spin.
 That is, assuming
 G_{j}=G_{j,↑}=G_{j,↓} (50)
 the following calculation is executed. In the models currently under consideration, there exist three parameters of R that determine interelectron interaction U, turn pitch N and random distribution. Although t=1, generality is not lost.
 Let effects of randomness be discussed. Aspects of DOS under U=8 are shown in FIG. 13 (N=2), FIG. 14 (N=4), FIG. 15 (N=6), FIG. 16 (N=10), and FIG. 17 (N=20). As R increases, randomness also increases. Randomness gives almost no influences to metallic states (FIG. 13). As N increases and the nature of multiplytwisted spirals is introduced, a tail appears on DOS under R=0. This tail narrows the Hubbard gap, but as the randomness increases, the tail disappears and the insulation performance increases, as understood from the figure. Aspects of DOS under U=10 are shown in FIG. 18 (N=2), FIG. 19 (N=4), FIG. 20 (N=6), FIG. 21 (N=10), and FIG. 22 (N=20). In FIG. 18, it should be remarked that influences of randomness are not so large. In this case, the system is already a Mott insulator, the value of N is small. Therefore, there exists no tail structure derived from the multiplytwisted helical structure. On the other hand, as N increases, a tail is produced, and thereafter disappears due to randomness as understood from the figure.
 It has been confirmed in this manner that, when the turn pitch is controlled and designed, the system can behave as a metal or can behave an insulator, and its insulation performance is enhanced by randomness.
 For the purpose of comparison with a system in which randomness has been introduced to bonds between different layers of the multiplytwisted helical structure of different layers, an absolute randombonding system was analyzed. This was directed to Mott transition on lattices in which bonds are formed absolutely randomly under the constraint that each of10001 sites has at least four nearestneighbor sites. As the average, the number of nearestneighbor sites was determined as
 Z=4+L/5 (51)
 Therefore, when L=0, it becomes a number of nearestneighbor sites equivalent to that of a twodimensional tetragonal lattice. When L=10, it becomes a number of nearestneighbor sites equivalent to that of a threedimensional tetragonal lattice. Aspects of DOS of this randombonding system are shown in FIG. 23 (U=8), FIG. 24 (U=10) and FIG. 25 (U=12). Changes of DOS are smooth, apparently unlike the foregoing multiplytwisted helical structure.
 Next explained is an electron system on a multiplytwisted spiral in a multiplytwisted helix according to the third embodiment.
 Assuming a onedimensional lattice, numbers are assigned as n= . . . , −1, 0, 1, . . . . Let the operator for generating an electron of a spin (at the pth lattice point be ĉ_{p,σ} ^{†}. Of course, there is the anticommutation relation
 {ĉ _{p,σ},ĉ_{q,ρ} ^{†}}=δ_{p,q}δ_{σ,ρ} (52)
 Here is defined a singleband Hubbard Hamiltonian Ĥ of the electron system as follows.
$\begin{array}{cc}\hat{H}=t\ue89e\sum _{i,j,\sigma}\ue89e{\lambda}_{i,j}\ue89e{\hat{c}}_{i,\sigma}^{\u2020}\ue89e{\hat{c}}_{j,\sigma}+U\ue89e\sum _{j}\ue89e{\hat{n}}_{j,\uparrow}\ue89e{\hat{n}}_{j,\downarrow}& \left(53\right)\end{array}$  Letting electrons be movable only among neighboring sites, as λ_{p,q}
$\begin{array}{cc}{\lambda}_{p,q}={\lambda}_{q,p}=\{\begin{array}{ccccc}1& \mathrm{when}& \uf603pq\uf604=1& \text{\hspace{1em}}& \text{\hspace{1em}}\\ s& \mathrm{when}& \uf603pq\uf604=N& \text{\hspace{1em}}& \text{\hspace{1em}}\\ {s}^{2}& \mathrm{when}& \mathrm{mod}\left(p,N\right)=0& \mathrm{and}& q=p+{N}^{2}+f\ue8a0\left(N\right)\\ {s}^{3}& \mathrm{when}& \mathrm{mod}\left(p,{N}^{2}\right)=1& \mathrm{and}& q=p+{N}^{3}+f\ue8a0\left({N}^{2}\right)\\ {s}^{4}& \mathrm{when}& \mathrm{mod}\left(p,{N}^{3}\right)=2& \mathrm{and}& q=p+{N}^{4}+f\ue8a0\left({N}^{3}\right)\\ \text{\hspace{1em}}& \vdots & \text{\hspace{1em}}& \text{\hspace{1em}}& \text{\hspace{1em}}\\ 0& \mathrm{otherwise}& \text{\hspace{1em}}& \text{\hspace{1em}}& \text{\hspace{1em}}\end{array}& \left(54\right)\end{array}$  is employed. Note that mod(a, b) is the remainder as a result of division of a by b. In this definition, an arbitrary function f(x) is introduced to provide a generalized model. For example, a model given by
 f(x)=0 (55)
 is the model analyzed in the first embodiment. Whereas, a model given by
 f(x)=rx (56)
 where r is a random variable that satisfies
 −1<r<1 (57)

 is the model analyzed in the second embodiment.
 A model analyzed in the third embodiment is one given by
 f(x)−Ωx (59)
 in which Ω=−½. That is, here is taken a model based on the multiplytwisted helix according to the second embodiment but having the random variable r fixed to −½. In other words, it can be considered that interlayer bonds have entirely moved in parallel from the system of f(x)=0.
 Here is defined a pin a electron density operator of the jth site {circumflex over (n)}_{j,σ} =ĉ _{j,σ} ^{†} ĉ _{j,σ} and its sum {circumflex over (n)}_{j}=Σ_{o} {circumflex over (n)} _{j,σ}.
 For the purpose of defining a temperature Green's function, here is introduced a grand canonical Hamiltonian {circumflex over (K)}=Ĥ−μ{circumflex over (N)} where {circumflex over (N)}=Σ_{j}{circumflex over (n)}_{j}. In the half filled taken here, chemical potential is μ=U/2. The halffilled grand canonical Hamiltonian can be expressed as
$\begin{array}{cc}\hat{K}=t\ue89e\sum _{i,j,\sigma}\ue89e{\lambda}_{j,i}\ue89e{\hat{t}}_{j,i,\sigma}+U/2\ue89e\sum _{i}\ue89e\left({\hat{u}}_{i}1\right)& \left(60\right)\end{array}$  Operators {circumflex over (t)}_{j,i,σ}, ĵ_{j,i,σ}, û_{i }and {circumflex over (d)}_{i,σ} are defined beforehand as
 {circumflex over (t)} _{j,i,σ} =ĉ _{j,σ} ^{†} ĉ _{i,σ} +ĉ _{i,σ} ^{†} ĉ _{j,σ} (61)
 ĵ _{j,i,σ} =ĉ _{j,σ} ^{†} ĉ _{i,σ} −ĉ _{i,σ} ^{†} ĉ _{j,σ} (62)
 û _{i} =ĉ _{i,↑} ^{†} ĉ _{i,↑} ĉ _{i,↓} ^{†} ĉ _{i,↓} +ĉ _{i,↑} ĉ _{i,↑} ^{†} ĉ _{i,↓} ĉ _{i,↓} ^{†} (63)
 {circumflex over (d)} _{i,σ} =ĉ _{i,σ} ^{†} ĉ _{i,σ} −ĉ _{i,σ} ĉ _{i,σ} (64)
 If the temperature Green function is defined for operators Â and {circumflex over (B)} given, taking τ as imaginary time, it is as follows.
 The onsite Green function
 is especially important because, when analytic continuation is conducted as iω_{n}→ω+iδ for a small δ,
$\begin{array}{cc}{D}_{j}\ue8a0\left(\omega \right)=\sum _{\sigma =\uparrow ,\downarrow}\ue89e\mathrm{Im}\ue89e\text{\hspace{1em}}\ue89e{G}_{j,\sigma}\ue8a0\left(\omega +\uf74e\ue89e\text{\hspace{1em}}\ue89e\delta \right)& \left(67\right)\end{array}$ 
 becomes the density of states of the system. For later numerical calculation of densities of states, δ=0.0001 will be used. Further let the total number of sites be n=10001.

 As the equation of motion of the onsite Green function,
$\begin{array}{cc}\uf74e\ue89e\text{\hspace{1em}}\ue89e{\omega}_{n}\ue89e\u3008{\hat{c}}_{j,\sigma};{\hat{c}}_{j,\sigma}^{\u2020}\u3009=1+t\ue89e\sum _{p,j}\ue89e{\lambda}_{p,j}\ue89e\u3008{\hat{c}}_{p,\sigma};{\hat{c}}_{j,\sigma}^{\u2020}\u3009+\frac{U}{2}\ue89e\u3008{\hat{d}}_{j,\sigma}\ue89e{\hat{c}}_{j,\sigma};{\hat{c}}_{j,\sigma}^{\u2020}\u3009& \left(70\right)\end{array}$  is obtained. Then, the approximation shown below is introduced, following Gros ((31) C. Gros, Phys. Rev. B50, 7295(1994)). If the site p is the nearestneighbor site of the site j, the resolution
 is introduced as the approximation. This is said to be exact in case of infinitedimensional Bethe lattices, but in this case, it is only within approximation. Under the approximation, the following equation is obtained.
$\begin{array}{cc}\left({\mathrm{\uf74e\omega}}_{n}{t}^{2}\ue89e{\Gamma}_{j,\sigma}\right)\ue89e{G}_{j,\sigma}=1+\frac{U}{2}\ue89e\u3008{\hat{d}}_{j,\sigma}\ue89e{\hat{c}}_{j,\sigma};{\hat{c}}_{j,\sigma}^{\u2020}\u3009\ue89e\text{}\ue89e\mathrm{where}& \left(72\right)\\ {\Gamma}_{j,\sigma}=\sum _{p}\ue89e{\lambda}_{p,j}\ue89e{G}_{p,\sigma}& \left(73\right)\end{array}$  was introduced. To solve the equation obtained, {circumflex over (d)}_{j,−σ}ĉ_{j,σ};ĉ_{j,σ} ^{†} has to be analyzed. In case of halffilled models, this equation of motion is
$\begin{array}{cc}\uf74e\ue89e\text{\hspace{1em}}\ue89e{\omega}_{n}\ue89e\u3008{\hat{d}}_{j,\sigma}\ue89e{\hat{c}}_{j,\sigma};{\hat{c}}_{j,\sigma}^{\u2020}\u3009=\frac{U}{2}\ue89e{G}_{j,\sigma}2\ue89et\ue89e\text{\hspace{1em}}\ue89e\sum _{p}\ue89e{\lambda}_{p,j}\ue89e\u3008{\hat{j}}_{p,j,\sigma}\ue89e{\hat{c}}_{j,\sigma};{\hat{c}}_{j,\sigma}^{\u2020}\u3009+t\ue89e\sum _{p}\ue89e{\lambda}_{p,j}\ue89e\u3008{\hat{d}}_{j,\sigma}\ue89e{\hat{c}}_{p,\sigma};{\hat{c}}_{j,\sigma}^{\u2020}\u3009& \left(74\right)\end{array}$  Here again, with reference to the Gros logic, approximation is introduced. It is the following translation.
 By executing this translation, the following closed equation is obtained.
$\begin{array}{cc}\left(i\ue89e\text{\hspace{1em}}\ue89e{\omega}_{n}{t}^{2}\ue89e{\Gamma}_{j,\sigma}\right)\ue89e{G}_{j,\sigma}=1+\frac{{\left(U/2\right)}^{2}}{i\ue89e\text{\hspace{1em}}\ue89e{\omega}_{n}{t}^{2}\ue89e{\Gamma}_{j,\sigma}2\ue89e{t}^{2}\ue89e{\Gamma}_{j,\sigma}}\ue89e{G}_{j,\sigma}& \left(77\right)\end{array}$  Here is assumed that there is no dependency on spin. That is, assuming
 G_{j}=G_{j,↑}=G_{j,↓} (78)
 the following calculation is executed. In the models currently under consideration, there exist parameters of the interelectron interaction U and the turn pitch N. For the following calculation, s=t=1 is used.
 Let discussion be progressed to a case using f(x)=−x/2. Aspects of DOS under N=2, 3, 4, 6, 10, 20 are shown in FIG. 26 (U=8), FIG. 27 (U=10) and FIG. 28 (U=12). It will be appreciated that the Mott transition is qualitatively controlled by N similarly to FIG. 8. However, they are different quantitatively. Especially, the Hubbard band tail structure remarkable when N is large is small in FIG. 27. Therefore, it is possible to modify the electron state in the multiplytwisted helix by changing Ω in f(x)=Ωx.
 Next explained is a multiplytwisted helix according to the fourth embodiment of the invention. In the fourth embodiment, ferromagnetic phase transition is analyzed by using a Monte Carlo simulation to demonstrate that critical temperature for ferromagnetic phase transition can be controlled by adjusting the turn pitch of the multiplytwisted helix.
 Explanation is first made about a spin system on a multiplytwisted spiral in the multiplytwisted helix.

 Sp=1 or −1 is used as the spin variable, and an Ising model is handled. As J_{p,q}
$\begin{array}{cc}{J}_{p,q}={J}_{q,p}=\{\begin{array}{cc}1\ue89e\text{\hspace{1em}}& \mathrm{when}\ue89e\text{\hspace{1em}}\ue89e\uf603pq\uf604=1\\ s& \mathrm{when}\ue89e\text{\hspace{1em}}\ue89e\uf603\text{\hspace{1em}}\ue89epq\uf604=N\\ {s}^{2}& \mathrm{when}\ue89e\text{\hspace{1em}}\ue89e\mathrm{mod}\ue89e\text{\hspace{1em}}\ue89e\left(p,N\right)=0\ue89e\text{\hspace{1em}}\ue89e\mathrm{and}\ue89e\text{\hspace{1em}}\ue89eq=p+{N}^{2}\\ {s}^{3}& \mathrm{when}\ue89e\text{\hspace{1em}}\ue89e\mathrm{mod}\ue89e\text{\hspace{1em}}\ue89e\text{\hspace{1em}}\left(p,{N}^{2}\right)=1\ue89e\text{\hspace{1em}}\ue89e\mathrm{and}\ue89e\text{\hspace{1em}}\ue89eq=p+{N}^{3}\\ {s}^{4}& \mathrm{when}\ue89e\text{\hspace{1em}}\ue89e\mathrm{mod}\ue89e\text{\hspace{1em}}\ue89e\text{\hspace{1em}}\left(p,{N}^{3}\right)=2\ue89e\text{\hspace{1em}}\ue89e\mathrm{and}\ue89e\text{\hspace{1em}}\ue89e\text{\hspace{1em}}\ue89eq=p+{N}^{4}\\ \text{\hspace{1em}}& \vdots \\ 0& \mathrm{otherwise}\end{array}& \left(80\right)\end{array}$  is employed. Note that mod(a, b) is the remainder as a result of division of a by b. Positional relation between spins is determined by J_{p,q}, and the multiplytwisted helix is defined thereby. Partition function
$\begin{array}{cc}Z=\sum _{\left\{{s}_{p}\right\}}^{\text{\hspace{1em}}}\ue89e\text{\hspace{1em}}\ue89e{\uf74d}^{\beta \ue89e\text{\hspace{1em}}\ue89eH}& \left(81\right)\end{array}$  is introduced, and physical property values are calculated at finite temperatures by using statistical mechanics. T is the temperature, and β=1/T and k_{B}=1 are used. Expected value concerning an arbitrary function f(S_{j}) of the spin variable is calculated by
$\begin{array}{cc}\u3008f({S}_{p)}\u3009=\frac{1}{Z}\ue89e\sum _{\left\{{S}_{p}\right\}}^{\text{\hspace{1em}}}\ue89e\text{\hspace{1em}}\ue89ef\ue8a0\left({S}_{j}\right)\ue89e{\uf74d}^{\beta \ue89e\text{\hspace{1em}}\ue89eH}& \left(82\right)\end{array}$ 
 Hereunder, n=10001 is used.
 FIG. 29 is a result of calculation of spontaneous magnetization by using the Metropolis method that is one of the Monte Carlo methods. 100000 is used as the Monte Carlo step. Here are shown values obtained by setting N=2, 3, 4, 5, 10, 20 as the turn pitch. The multiplytwisted helix of N=2 exhibits a critical temperature substantially equivalent to that of a threedimensional system, but as N increases, the critical temperature for spontaneous magnetization to disappear gradually decreases.
 The multiplytwisted helixes in which N<10 exhibit behaviors similar to those of a tetragonal lattice in terms of dependency of spontaneous magnetization upon temperature; however, when the regions of N>10 are remarked, there are remarkable differences in dependency of spontaneous magnetization upon temperature. FIG. 30 shows a result of calculation of spontaneous magnetization in multiplytwisted helixes of N=10, 12, 14, 16, 18, 20 in those regions by the same method as used in FIG. 29. As the temperature rises, spontaneous magnetization first decreases as a differentiable continuous function, and thereafter, spontaneous magnetization disappears with a differentiable drop. It suggests that this magnetic transition is primary phase transition.
 Let a physical interpretation be given. A multiplytwisted helix may be considered to be made of spirals as onedimensional elements. Its diameter is written as R. Of course, R is a quantity of the same degree as the turn pitch N. If the correlation length of a spin system is smaller than R, the system is considered to be behaving onedimensionally. Since no order parameter is established under one dimension, the system is always a disorderly system at finite temperatures. However, when the spin correlation length exceeds R, spin correlation occurs between spiral structures as elements, and the system comes to behave as a threedimensional substance, exhibiting a threedimensional order. That is, it can be appreciated that primary phase transition from the onedimensional disorderly phase to the threedimensional orderly phase.
 Referring to FIG. 30, there are fluctuations in temperature for phase transition to occur, and the values do not appear to flatly decrease with N; however, this is also a feature of primary phase transition.
 Next explained is a multiplytwisted helix according to the fifth embodiment of the invention. In the fifth embodiment, quantum level statistics will be analyzed concerning quantum states in multiplytwisted helixes to which random potentials are introduced, thereby to demonstrate that quantum chaos can be controlled.
 Explained is an electron system on a multiplytwisted spiral in a multiplytwisted helix according to the fifth embodiment.
 Assuming a onedimensional lattice, numbers are assigned as n− . . . , −1, 0, 1, . . . . Let the operator for generating a quantum at the pth lattice point be Ĉ_{p} ^{†}. Of course, let this operator satisfy the anticommutation relation
 {Ĉ _{p} ,Ĉ _{q} ^{†}}=δ_{p,q} (84)
 This quantum is a fermion having no free spin freedom. This corresponds to analysis of an electron in a solid in which spin orbit interaction can be disregarded.
 Here is defined the Hamiltonian Ĥ of the electron system as follows.
$\begin{array}{cc}\hat{H}=\sum _{i,j}^{\text{\hspace{1em}}}\ue89e{\lambda}_{i,j}\ue89e{\hat{c}}_{i}^{\u2020}\ue89e{\hat{c}}_{j}+\sum _{j}^{\text{\hspace{1em}}}\ue89e{v}_{j}\ue89e{\hat{c}}_{j}^{\u2020}\ue89e{\hat{c}}_{j}& \left(85\right)\end{array}$  Letting electrons be movable only among neighboring sites, as λ_{p,q},
$\begin{array}{cc}{\lambda}_{p,q}={\lambda}_{q,p}=\{\begin{array}{cc}1& \mathrm{when}\ue89e\text{\hspace{1em}}\ue89e\uf603p,q\uf604=1\\ s& \mathrm{when}\ue89e\text{\hspace{1em}}\ue89e\uf603pq\uf604=N\\ {s}^{2}& \mathrm{when}\ue89e\text{\hspace{1em}}\ue89e\mathrm{mod}\ue89e\text{\hspace{1em}}\ue89e\text{\hspace{1em}}\left(p,N\right)=0\ue89e\text{\hspace{1em}}\ue89e\mathrm{and}\ue89e\text{\hspace{1em}}\ue89e\text{\hspace{1em}}\ue89eq=p+{N}^{2}+f\ue8a0\left(N\right)\\ {s}^{3}& \mathrm{when}\ue89e\text{\hspace{1em}}\ue89e\mathrm{mod}\ue89e\text{\hspace{1em}}\ue89e\text{\hspace{1em}}\left(p,{N}^{2}\right)=1\ue89e\text{\hspace{1em}}\ue89e\mathrm{and}\ue89e\text{\hspace{1em}}\ue89eq=p+{N}^{3}+f\ue8a0\left({N}^{2}\right)\\ {s}^{4}& \mathrm{when}\ue89e\text{\hspace{1em}}\ue89e\mathrm{mod}\ue89e\text{\hspace{1em}}\ue89e\text{\hspace{1em}}\left(p,{N}^{3}\right)=2\ue89e\text{\hspace{1em}}\ue89e\mathrm{and}\ue89e\text{\hspace{1em}}\ue89eq=p+{N}^{4}+f\ue8a0\left({N}^{3}\right)\\ \text{\hspace{1em}}& \vdots \\ 0& \mathrm{otherwise}\end{array}& \left(86\right)\end{array}$ 
 where int(x) is the maximum integer that does not exceed x. The total number of sites is M, and a periodical boundary condition is introduced beforehand to p=1, 2, . . . , M. The second term of the Hamiltonian is the term of the random potential. For each site, the random variable
$\begin{array}{cc}\frac{v}{2}<{v}_{j}<\frac{v}{2}& \left(88\right)\end{array}$  is generated to form the Hamiltonian. The variable breadth v of the random potential is useful as a parameter determining the degree of the randomness. Here, N, which determines the turn pitch of multiplytwisted spirals, s, which determines the interlayer bonding force, and v, which determines the intensity of the random potential, are varied as parameters.

 where m=1, 2, . . . , M.


 is calculated. The staircase function obtained is converted by using a procedure called unfolding such that the density of states becomes constant in average. Using the quantum level obtained in this manner, the nearestneighbor level spacing distribution P(s) and the Δ_{3 }statistics of Dyson and Mehta are calculated as quantum level statistics. As taught in a literature ((32) L. E. Reichl, The transition to chaos: in conservative classical systems: quantum manifestations (Springer, N.Y., 1992; (33) F. Haake, Quantum Signatures of chaos, (SpringerVerlag, 1991)), by using these statistics, it can be detected whether quantum chaos has been generated or not. It is also known that a quantum chaotic system is sensitive to perturbation from outside similarly to the classical chaotic system, and analysis of quantum chaos is important as a polestar of designs of nonlinear materials.
 In case of an integrable system, nearestneighbor level spacing distribution P(s) and Δ_{3 }statistics are those of Poisson Distribution
 In case the system currently reviewed exhibits quantum chaos, it becomes GOE (Gaussian orthogonal ensemble) distribution
$\begin{array}{cc}{P}_{\mathrm{GOE}}\ue8a0\left(s\right)=\frac{\pi \ue89e\text{\hspace{1em}}\ue89es}{2}\ue89e{\uf74d}^{\pi \ue89e\text{\hspace{1em}}\ue89e{s}^{2}/4}& \left(94\right)\\ {\Delta}_{3}\ue8a0\left(n\right)=\frac{1}{{\pi}^{2}}\ue8a0\left[\mathrm{log}\ue8a0\left(2\ue89e\pi \ue89e\text{\hspace{1em}}\ue89en\right)+\gamma \frac{{\pi}^{2}}{8}\frac{5}{4}\right]+O\ue8a0\left({n}^{1}\right)& \left(95\right)\end{array}$  where γ is the Euler's constant.
 Since the multiplytwisted helix analyzed here is made up of M=10001 sites, there are M=10001 eigen states in this quantum system. Among them, the following quantum level statistics values were calculated on the basis of energy eigen values concerning 2001 states from the 4001st to the 6001st states from the ground state.
 FIG. 31 and FIG. 32 show quantum level statistics of multiplytwisted helixes of N=10 while fixing the interlayer bonding to s=1 and changing the ram potential intensity v. FIG. 31 shows P(s) whilst FIG. 32 show Δ_{3 }statistics values. When v is small, the system is in a metallic state, and the quantum level statistics is that of a quantum chaotic system of GOE. As v increases, the quantum level statistics changes toward the Poisson distribution. This is typical Anderson transition ((34) Phys. Rev. 109, 1492 (1958); (35) Phys. Rev. Lett. 42, 783(1979); (36) Rev. Mod. Phys. 57. 287(1985)).
 FIG. 33 and FIG. 34 show quantum level statistics of multiplytwisted helixes of N=4 while fixing the interlayer bonding to s=1. It is recognized that, along with changes from v=2 to v=22, the system changes from an approximately quantum chaotic system to an approximately integrable system. What should be noted is the difference the quantum level statistics values of v=2 in FIGS. 33 and 34 (N=4) and the quantum level statistics values of v=2 in FIGS. 31 and 32 (N=10).
 FIGS. 35 and 36 shows quantum level statistics values obtained when fixing v=2 and s=1 and changing N. For example, reviewing Δ_{3}(n) in detail, it is recognized that the quantum level statistics value of v=2 in the multiplytwisted helix of N=4 is larger than that in case of N=10. Apparently, as N decreases, quantum exhibits a tendency of localization.
 In case of s=1 in FIGS. 35 and 36, the change of the quantum statistics value by the change of N was small. This means that the system lacks modulability from the viewpoint of the purpose of controlling the quantum chaos. Let it tried to change s to improve that point. FIG. 37 and FIG. 38 show quantum level statistics values obtained when fixing v=2 and s=0.8 and changing N. FIG. 39 and FIG. 40 show quantum level statistics values obtained when fixing v=2 and s=0.6 and changing N. Similarly to the cases of s=1, quantum exhibits the tendency of localization as N decreases; however, the breadth of the change is larger than that in case of s=1. It has been confirmed that various quantum systems can be obtained from substantial Poisson distribution to substantial GOE distribution.
 Next explained is a multiplytwisted helix according to the sixth embodiment of the invention. In the sixth embodiment, it is demonstrated that ferromagnetic transition in a multiplytwisted helix can be controlled not only by the number of turns but also by randomness of interlayer bonding.
 Explanation is made about a spin system on a multiplytwisted spiral in the multiplytwisted helix according to the sixth embodiment.

 Here is employed a threedimensional vector having values at vertices of a regular octahedron. That is, S_{p}=(1, 0, 0), (−1, 0, 0), (0, 1, 0), (0, −1, 0), (0, 0, 1), (0, 0, −1) are allowable as the spin value.
 As J_{p,q}
$\begin{array}{cc}{J}_{p,q}={J}_{q,p}=\{\begin{array}{cc}1& \mathrm{when}\ue89e\text{\hspace{1em}}\ue89e\uf603pq\uf604=1\\ s& \mathrm{when}\ue89e\text{\hspace{1em}}\ue89e\uf603pq\uf604=N\\ {s}^{2}& \mathrm{when}\ue89e\text{\hspace{1em}}\ue89e\mathrm{mod}\ue8a0\left(p,N\right)=0\ue89e\text{\hspace{1em}}\ue89e\mathrm{and}\ue89e\text{\hspace{1em}}\ue89eq=p+{N}^{2}+f\ue8a0\left(N\right)\\ {s}^{3}& \mathrm{when}\ue89e\text{\hspace{1em}}\ue89e\mathrm{mod}\ue8a0\left(p,{N}^{2}\right)=1\ue89e\text{\hspace{1em}}\ue89e\mathrm{and}\ue89e\text{\hspace{1em}}\ue89eq=p+{N}^{3}+f\ue8a0\left({N}^{2}\right)\\ {s}^{4}& \ue89e\mathrm{when}\ue89e\text{\hspace{1em}}\ue89e\mathrm{mod}\ue8a0\left(p,{N}^{3}\right)=2\ue89e\text{\hspace{1em}}\ue89e\mathrm{and}\ue89e\text{\hspace{1em}}\ue89eq=p+{N}^{4}+f\ue8a0\left({N}^{3}\right)\\ \text{\hspace{1em}}& \vdots \\ 0& \mathrm{otherwise}\end{array}& \left(97\right)\end{array}$  is employed. Note that mod(a, b) is the remainder as a result of division of a by b. Positional relation between spins is determined by J_{p,q}, and the multiplytwisted helix is defined thereby. What is analyzed here is a model in which randomness is defined in f(x) as defined as follows.
 f(x)=sign (rx)int(rx) (98)

 The term int(x) is the largest integer that does not exceed x. r is the random variable that satisfies
 −1<r<1 (100)

 R is the parameter that characterizes the probability distribution. Although the average value of the distribution is always <r>=0, self multiplication of dispersion is given by
$\begin{array}{cc}\u3008{r}^{2}\u3009={\int}_{1}^{1}\ue89e\text{\hspace{1em}}\ue89e\uf74cr\ue89e\text{\hspace{1em}}\ue89e{r}^{2}\ue89e{P}_{R}\ue8a0\left(r\right)=\frac{1}{12/R}& \left(102\right)\end{array}$  and dispersion increases as R increases. Under R=0, randomness disappears. On the other hand, in case of R=1, the distribution becomes most even, and the randomness is largest in the model. The random variable is generated for each interlayer bond, and randomness is introduced to bonding positions, thereby to produce the multiplytwisted helix to be analyzed.



 Hereunder, n=10001 is used.
 Spontaneous magnetization is calculated by using the Metropolis method that is one of the Monte Carlo methods. 100000 was used as the Monte Carlo steps. First, FIG. 41 shows spontaneous magnetization under no randomness of R=0. Here are shown the cases setting the turn pitch as N=2, 3, 4, 6, 10, 20 as the turn pitch. The multiplytwisted helix of N=2 exhibits a critical temperature substantially equivalent to that of a threedimensional system, but as N increases, the critical temperature for spontaneous magnetization to disappear gradually decreases. This result qualitatively coincides with spontaneous magnetization in the Ising model in the fourth embodiment in which only 1 or −1 is available as the spin.
 Here is made a review about the effect of randomness. Spontaneous magnetization in cases using R=⅛, {fraction (1/4)}, {fraction (1/2)} and 1 as the value of R is shown below. FIG. 42 shows one under N=4, FIG. 43 shows one under N=6, FIG. 44 shows one under N=10, and FIG. 45 shows one under N=20. With each value of N, critical temperature changes as R increases. One of its reasons might be attenuation of the characterized nature of the magnetization curve due to introduction of randomness. When N was small, superperiod originally unique to the multiplytwisted helix did not affect so much, for example, in FIG. 42 or FIG. 43, so the magnetization curve itself is not different so much from the case of a tetragonal lattice. In this case, although the critical temperature decreases because of introduction of R>0, the shape of the magnetization curve itself does not change substantially. On the other hand, when N is large, magnetization curves under R=0 in FIGS. 44 and 45, for example, were considerably different from that of a tetragonal lattice. In case of R>0, since the system progressively changes while losing its unique structure, the breadth of the decrease of the critical temperature by introduction of randomness is large. Through the analysis shown above, it has confirmed that critical temperature for ferromagnetic transition can be controlled by introduction of randomness by R>0.
 Next explained is a multiplytwisted structure according to the seventh embodiment of the invention. In the seventh embodiment, a simple cellular automaton is assumed in the multiplytwisted structure, and simulations of its dynamics are performed to introduce interelement spacing. Based on the spacing, interelement bonding property is investigated to show that the bonding property can be modified.
 Assuming a onedimensional lattice made up of elements, numbers are assigned as p= . . . , −1, 0, 1, . . . . A multiplytwisted structure is defined by the bonding among elements. For the purpose of defining nearestneighbor elements,
$\begin{array}{cc}{J}_{p,q}={J}_{q,p}=\{\begin{array}{cc}1& \mathrm{when}\ue89e\text{\hspace{1em}}\ue89e\uf603pq\uf604=1\\ s& \mathrm{when}\ue89e\text{\hspace{1em}}\ue89e\uf603pq\uf604=N\\ {s}^{2}& \mathrm{when}\ue89e\text{\hspace{1em}}\ue89e\mathrm{mod}\ue8a0\left(p,N\right)=0\ue89e\text{\hspace{1em}}\ue89e\mathrm{and}\\ \text{\hspace{1em}}& \text{\hspace{1em}}\ue89eq=p+{N}^{2}+f\ue8a0\left(N\right)\\ {s}^{3}& \mathrm{when}\ue89e\text{\hspace{1em}}\ue89e\mathrm{mod}\ue8a0\left(p,{N}^{2}\right)=1\ue89e\text{\hspace{1em}}\ue89e\mathrm{and}\\ \text{\hspace{1em}}& q=p+{N}^{3}+f\ue8a0\left({N}^{2}\right)\\ {s}^{4}& \mathrm{when}\ue89e\text{\hspace{1em}}\ue89e\mathrm{mod}\ue8a0\left(p,{N}^{3}\right)=2\ue89e\text{\hspace{1em}}\ue89e\mathrm{and}\ue89e\text{\hspace{1em}}\\ \text{\hspace{1em}}& q=p+{N}^{4}+f\ue8a0\left({N}^{3}\right)\\ \text{\hspace{1em}}& \vdots \\ 0& \mathrm{otherwise}\end{array}& \left(106\right)\end{array}$  is introduced, and s=1 is used hereunder. Assume that the pth element and the qth element are the nearestneighbor elements with each other when J_{p,q}=1. The following function is employed as f(x).
 f(x)=sign(rx)int(rx)−int(αx) (107)

 The term int(x) is the largest integer that does not exceed x. r is the random variable that satisfies
 1−<r<1 (109)

 R is the parameter that characterizes the probability distribution. Although the average value of the distribution is always +r,=0, self multiplication of dispersion is given by
$\begin{array}{cc}\u3008{r}^{2}\u3009={\int}_{1}^{1}\ue89e\text{\hspace{1em}}\ue89e\uf74c{\mathrm{rr}}^{2}\ue89e{P}_{R}\ue8a0\left(r\right)=\frac{1}{1+2/R}& \left(111\right)\end{array}$  and dispersion increases as R increases. Under R=0, randomness disappears. On the other hand, in case of R=1, the distribution becomes most even, and the randomness is largest in the model. The random variable is generated for each interlayer bond, and randomness is introduced to bonding positions, thereby to produce the multiplytwisted structure to be analyzed. α>0 causes parallel movement of the distribution. In the multiplytwisted structure, there are parameters of the turn pitch N, interlayer bonds distribution width R, interlayer bonds distribution position α.
 Analysis made below is to introduce a certain sense of spacing between sites that are not nearestneighbor sites and to measure the intersite bonding property on the basis of the number sites existing at the distance from a certain site. For this purpose, a simple cellular automaton (CA) is introduced. A variable of σ_{n}=1 or 0 is introduced to the nth site. Let this variable vary with time t=0, 1, . . . , and let it written as σ_{n}(t).

 is used. As the CA dynamics
$\begin{array}{cc}{\sigma}_{n}\ue8a0\left(t+1\right)=\Theta \left(\sum _{m}\ue89e{J}_{n,m}\ue89e{\sigma}_{m}\ue8a0\left(t\right)\right)\ue89e\text{}\ue89e\mathrm{is}\ue89e\text{\hspace{1em}}\ue89e\mathrm{used},\mathrm{where}& \left(113\right)\\ \Theta \ue8a0\left(x\right)=\{\begin{array}{cc}1& \mathrm{when}\ue89e\text{\hspace{1em}}\ue89ex>0\\ 0& \mathrm{when}\ue89e\text{\hspace{1em}}\ue89ex\le 0\end{array}& \left(114\right)\end{array}$  Through this time development, the time t when the qth site first becomes σ_{q}(t)=1 is used to define the distance Δ_{p,q }between the pth site and the qth site.
 Let a physical interpretation be given to the distance Δ_{p,q}. There are a plurality of paths connecting the pth site and the qth site via the nearestneighbor sites, and the shortest one is the abovementioned distance. In a continuous threedimensional space (x, y, z), it corresponds to a case where x+y+z is employed as the distance from the origin. In CA mentioned above, propagation to the nearestneighbor site occurs during the time width 1, and the time required for σ_{p}(0)=1 having localized in the pth site to reach the qth site was measured to be used as the distance. Of course, the distance introduced here satisfies the axioms
 Δ_{p,q}≧0 (115)
 Δ_{p,q}=Δ_{q,p} (116)
 Δ_{p,r}≦Δ_{p,q}+Δ_{q,r }
 By execution of this CA simulation, arbitrary (p, q) distance can be determined.
 The quantity remarked in this seventh embodiment is the total number ω(r; p_{j}) of the site q at the distance Δ_{pj,q}=r from an arbitrary p_{j}th site.

 to be obtained by using a multiplytwisted structure made up of M=10001 sites and averaging M_{s}=1000 samples selected as p_{j }sites from them. It will be readily understood that this quantity becomes
 Ω(r)=Const.+4r ^{2} (119)
 in case of a threedimensional tetragonal lattice.
 Let Ω(r) in the multiplytwisted structure be calculated. In FIG. 46, Ω(r) in case of R=0 and α=0 was plotted for various N. This system is a multiplytwisted structure similar to that taken for analysis of Mott transition in the first embodiment. In FIG. 47, Ω(r) in case of N=10 and α=0 was plotted for various R. This system is a multiplytwisted structure similar to that taken for analysis of Mott transition in the second embodiment. In FIG. 48, Ω(r) in case of N=19 and R=0 was plotted for various α. This system is a multiplytwisted structure similar to that taken for analysis of Mott transition in the third embodiment. In these figures, Ω(r) in a region of the level of r<10 is described well by quadratic functions of r. Hereunder, this is generally developed as
 Ω(r)=C _{0} +C _{1} r ^{2} +C _{2} r ^{4} (120)
 and discussed while using coefficients C_{1 and C} _{2}.
 FIG. 49 and FIG. 50 respectively show the coefficients C_{1 }and C_{2 }in the case where Ω(r) under 0<r<10 is optimally approximated by Equation 120 when R=0 and α=0 in FIG. 46. FIG. 51 and FIG. 52 respectively show the coefficients C_{1 }and C_{2 }in the case where Ω(r) under 0<r<10 is optimally approximated by Equation 120 when N=10 and α=0 in FIG. 47. FIG. 53 and FIG. 54 respectively show the coefficients C_{1 }and C_{2 }in the case where Ω(r) under 0<r<10 is optimally approximated by Equation 120 when N=10 and R=0 in FIG. 48.
 In the cases of FIG. 46 and FIG. 47, the Coefficient C_{1 }of O(r^{2}) is modified well by N and R as shown in FIGS. 49 and 51. As appreciated from FIGS. 50 and 52, since C_{2 }is very small, Ω(r) is described well by quadratic functions. This demonstrates that these multiplytwisted structures have threedimensional properties, but they are controlled in number of sites in the region separated by the distance r from a certain site by modulation of its coefficients. Therefore, physical phenomena appearing on the structures are controllable. What can be said from those results is that, even in the order of r˜10, evaluation of the average number of the nearestneighbor sites is directly valid and modulability in the multiplytwisted helix extends even globally. In case of FIG. 48, as apparent from FIG. 53 and FIG. 54, there is almost no change in C_{1 }even with changes of a unlike the foregoing two examples. On the other hand, C_{2 }varies to a certain degree, and its sign is inverted. Therefore, modulability under changes of a of FIG. 48 results in changes of the coefficient of O(r^{4}).
 Next explained is a multiplytwisted structure according to the eighth embodiment of the invention. In the eighth embodiment, it is demonstrated that ferromagnetic transition in a multiplytwisted structure can be controlled by parallel movements of interlayer bonds.

 Here are employed threedimensional vectors having values at vertices of a regular octahedron. That is, S_{p}=(1, 0, 0), (−1, 0, 0), (0, 1, 0), (0, −1, 0), (0, 0, 1), (0, 0, −1) are allowable as the spin value.
 As J_{p,q}
$\begin{array}{cc}{J}_{p,q}={J}_{q,p}=\{\begin{array}{ccccc}1& \mathrm{when}& \uf603pq\uf604=1& \text{\hspace{1em}}& \text{\hspace{1em}}\\ s& \mathrm{when}& \uf603pq\uf604=N& \text{\hspace{1em}}& \text{\hspace{1em}}\\ {s}^{2}& \mathrm{when}& \mathrm{mod}\ue8a0\left(p,N\right)=0& \mathrm{and}& q=p+{N}^{2}+f\ue8a0\left(N\right)\\ {s}^{3}& \mathrm{when}& \mathrm{mod}\ue8a0\left(p,{N}^{2}\right)=1& \mathrm{and}& q=p+{N}^{3}+f\ue8a0\left({N}^{2}\right)\\ {s}^{4}& \mathrm{when}& \mathrm{mod}\ue8a0\left(p,{N}^{3}\right)=2& \mathrm{and}& q=p+{N}^{4}+f\ue8a0\left({N}^{3}\right)\\ \text{\hspace{1em}}& \vdots & \text{\hspace{1em}}& \text{\hspace{1em}}& \text{\hspace{1em}}\\ 0& \mathrm{otherwise}& \text{\hspace{1em}}& \text{\hspace{1em}}& \text{\hspace{1em}}\end{array}& \left(122\right)\end{array}$  is employed. Note that mod(a, b) is the remainder as a result of division of a by b. Positional relation between spins is determined by J_{p,q}, and the multiplytwisted structure is defined thereby. What is analyzed here is a model undergoing parallel movements as defined by
 f(x)=−int(αx) (123)
 The term int(x) is the largest integer that does not exceed x. In case of α=0, the system exhibits the same structure as that of the fourth embodiment.

 is introduced, where T is the temperature, and β=1/T and k_{B}=1 are used. Expected value concerning an arbitrary function f(S_{j}) of the spin variable is calculated by
$\begin{array}{cc}\u3008f\ue8a0\left({S}_{j}\right)\u3009=\frac{1}{Z}\ue89e\sum _{\left\{{S}_{p}\right\}}^{\text{\hspace{1em}}}\ue89ef\ue8a0\left({S}_{j}\right)\ue89e{\uf74d}^{\beta \ue89e\text{\hspace{1em}}\ue89eH}& \left(125\right)\end{array}$ 
 Hereunder, n=10001 is used.
 Spontaneous magnetization is calculated by using the Metropolis method that is one of the Monte Carlo methods. Using 100000 as the Monte Carlo steps, three different turn pitches, N=6, 10, 20, were analyzed. FIGS. 55 through 57 show results of Monte Carlo simulations conducted by using α=0, 0.1, 0.2, 0.3, 0.4, 0.5. FIG. 55 is the result of N=6, FIG. 56 is the result of N=10, and FIG. 57 is the result of N=20. Under each of N, critical temperature, i.e. the temperature for spontaneous magnetization to disappear, decreases as a increases. This phenomenon would be a result of a change of the positional relation between nearestneighbor sites along with an increase of a, which makes the spin order fragile.
 It has been confirmed through the analysis that critical temperature for ferromagnetic transition can be controlled by parallel movement of interlayer bonds by α>0.
 Next explained is a multiplytwisted helix according to the ninth embodiment of the invention. In the ninth embodiment, it is demonstrated that quantum chaos in a multiplytwisted helix having introduced a random potential can be controlled by a random magnetic field that can be realized by magnetic impurities.
 A quantum system on a multiplytwisted spiral in the multiplytwisted helix according to the ninth embodiment will be explained below.
 Assuming a onedimensional lattice, numbers are assigned as p= . . . , −1, 0, 1, . . . . Let the operator for generating a quantum at the pth lattice point be Ĉ_{p} ^{†}. Of course, there is the anticommutation relation
 {ĉ _{p,} ĉ _{p} ^{†}}=δ_{p,q} (127)
 This quantum is a fermion having no free spin freedom. This corresponds to analysis of an electron in a solid in which spin orbit interaction can be disregarded.
 Here is defined the Hamiltonian Ĥ of the electron system as follows.
$\begin{array}{cc}\hat{H}=\sum _{p,q}^{\text{\hspace{1em}}}\ue89e{t}_{p,q}\ue89e{\lambda}_{p,q}\ue89e{\hat{c}}_{p}^{\u2020}\ue89e{\hat{c}}_{p}\ue89e\sum _{p}\ue89e{v}_{p}\ue89e{\hat{c}}_{p}^{\u2020}\ue89e{\hat{c}}_{p}& \left(128\right)\end{array}$  Letting electrons be movable only among neighboring sites, as λ_{p,q},
$\begin{array}{cc}{\lambda}_{p,q}={\lambda}_{q,p}=\{\begin{array}{ccccc}1& \mathrm{when}& \uf603pq=1\uf604& \text{\hspace{1em}}& \text{\hspace{1em}}\\ s& \mathrm{when}& \uf603pq=N\uf604& \text{\hspace{1em}}& \text{\hspace{1em}}\\ {s}^{2}& \mathrm{when}& \mathrm{mod}\ue8a0\left(p,N\right)=0& \mathrm{and}& q=p+{N}^{2}+f\ue8a0\left(N\right)\\ {s}^{3}& \mathrm{when}& \mathrm{mod}\ue8a0\left(p,{N}^{2}\right)=1& \mathrm{and}& q=p+{N}^{3}+f\ue8a0\left({N}^{2}\right)\\ {s}^{4}& \mathrm{when}& \mathrm{mod}\ue8a0\left(p,{N}^{3}\right)=2& \mathrm{and}& q=p+{N}^{4}+f\ue8a0\left({N}^{3}\right)\\ \text{\hspace{1em}}& \vdots & \text{\hspace{1em}}& \text{\hspace{1em}}& \text{\hspace{1em}}\\ 0& \mathrm{otherwise}& \text{\hspace{1em}}& \text{\hspace{1em}}& \text{\hspace{1em}}\end{array}& \left(129\right)\end{array}$ 
 where int(x) is the maximum integer that does not exceed x. t_{p,q }is introduced for the random magnetic field. Here is determined as
 t _{p,q}=exp(iθ _{p,q}) (131)
 where θ_{p,q}=−θ_{q,p }is a random real number that satisfies
 0<θ_{p,q}<2π (132)
 Due to hopping to the nearestneighbor sites, the random phase factor θ_{p,q }is resuffixed, depending the sites. If the phase factor is integrated in the loop making one turn around a lattice point, a magnetic flux passing through the loop is obtained. This means a magnetic field is locally introduced to the random distribution of 0<θ_{p,q}<2π this magnetic field is absolutely random in both intensity and direction, and if it is averaged spatially, it becomes a zero magnetic field.
 Using M as the total number of sites, a periodical boundary condition is introduced before hand to p=1, 2, . . . , M. The second term of the Hamiltonian is the term of the random potential. For each site, the random variable
$\begin{array}{cc}\frac{v}{2}<{v}_{p}<\frac{v}{2}& \left(133\right)\end{array}$  is generated to form the Hamiltonian. The variable breadth v of the random potential is useful as a parameter determining the degree of the randomness. Here, N, which determines the turn pitch of multiplytwisted spirals, s, which determines the interlayer bonding force, and v, which determines the intensity of the random potential, are varied as parameters.

 where m=1, 2, . . . , M.

 and its staircase function
 λ(ε)=∫_{−∞} ^{E} dηρ(η) (136)
 is calculated. The staircase function obtained is converted by using a procedure called unfolding such that the density of states becomes constant in average. Using the quantum level obtained in this manner, the nearestneighbor level spacing distribution P(s) and the Δ_{3 }statistics of Dyson and Mehta are calculated as quantum level statistics. As taught by Literatures (32) and (33), by using these statistics, it can be detected whether quantum chaos has been generated or not. It is also known that a quantum chaotic system is sensitive to perturbation from outside similarly to the classical chaotic system, and analysis of quantum chaos is important as a polestar of designs of nonlinear materials.
 In case of an integrable system, nearestneighbor level spacing distribution P(s) and Δ_{3 }statistics are those of Poisson Distribution
 In case of quantum chaos under a magnetic field, it becomes GUE (Gaussian unitary ensemble) distribution
$\begin{array}{cc}{P}_{\mathrm{GUE}}\ue8a0\left(s\right)=\frac{32\ue89e{s}^{2}}{{\pi}^{2}}\ue89e{e}^{\frac{4\ue89e\mathrm{s2}}{\pi}}& \left(139\right)\\ {\Delta}_{3}\ue8a0\left(n\right)=\frac{1}{2\ue89e{\pi}^{2}}\ue8a0\left[\mathrm{log}\ue8a0\left(2\ue89e\pi \ue89e\text{\hspace{1em}}\ue89en\right)+\gamma \frac{5}{4}\right]+O\ue8a0\left({n}^{1}\right)& \left(140\right)\end{array}$  where γ is the Euler's constant.
 Since the multiplytwisted structure analyzed here is made up of M=10001 sites, there are M=10001 eigen states in this quantum system. Among them, the following quantum level statistics values were calculated on the basis of energy eigen values concerning 2001 states from the 4001st to the 6001st states from the ground state.
 FIG. 58 and FIG. 59 show quantum level statistics of multiplytwisted structures of N=10 while fixing the interlayer bonding to s=1 and changing the random potential intensity v. FIG. 58 shows P(s) whilst FIG. 59 shows Δ_{3 }statistics values. When v is small, the system is in a metallic state, and the quantum level statistics is that of a quantum chaotic system of GUE. As v increases, the quantum level statistics changes toward the Poisson distribution. This is typical Anderson transition under a magnetic field ((34) Phys. Rev. 109, 1492 (1958); (35) Phys. Rev. Lett. 42, 783(1979); (36) Rev. Mod. Phys. 57. 287(1985)).
 FIG. 60 and FIG. 61 show quantum level statistics of multiplytwisted structures of N=4 while fixing the interlayer bonding to s=1. It is recognized that, along with changes from v=2 to v=22, the system changes from an approximately quantum chaotic system to an approximately integrable system. What should be noted is the difference the quantum level statistics values of v=2 in FIGS. 60 and 61 (N=4) and the quantum level statistics values of v=2 in FIGS. 58 and 59 (N=10).
 FIGS. 62 and 63 shows quantum level statistics values obtained when fixing v=2 and s=1 and changing N. For example, reviewing Δ_{3}(n) in detail, it is recognized that the quantum level statistics value of v=2 in the multiplytwisted structure of N=4 is larger than that in case of N=10. Apparently, as N decreases, quantum exhibits a tendency of localization.
 In case of s=1 in FIGS. 62 and 63, the change of the quantum statistics value by the change of N was small. This means that the system lacks modulability from the viewpoint of the purpose of controlling the quantum chaos. Let it tried to change s to improve that point. FIG. 64 and FIG. 65 show quantum level statistics values obtained when fixing v=2 and s=0.8 and changing N. FIG. 66 and FIG. 67 show quantum level statistics values obtained when fixing v=2 and s=0.6 and changing N. Similarly to the cases of s=1, quantum exhibits the tendency of localization as N decreases; however, the breadth of the change is larger than that in case of s=1. As seen from N=2 to N=20, it has been confirmed that various quantum systems can be obtained from substantial Poisson distribution to substantial GUE distribution.
 Next explained is a multiplylooped ring structure according to the tenth embodiment of the invention.
 Explained below is an electron system on a multiplylooped ring in the multiplylooped ring structure according to the tenth embodiment.
 Using a ring having L sites in one cycle, multiplylooped rings are defined hierarchically. Assuming q hierarchies or layers, the total number of sites is L^{q}. The sites are described by a qdimensional cube.
 a=(a_{1},a_{2},a_{3 }. . . a_{q}) (141)
 where a_{k }is an integer that satisfies 1≦a_{k}≦L.
 Let the operator for generating an electron of spin σ at the ath lattice point be Ĉ_{a,σ} ^{†}. Of course, there is the anticommutation relation
 {ĉ _{a,σ} , ĉ _{b,ρ} ^{†}}=δ(a−b)δ(σ−ρ) (142)


 Here is defined a singleband Hubbard Hamiltonian Ĥ of the electron system as follows.
$\begin{array}{cc}\hat{H}=t\ue89e\sum _{a,b,\sigma}\ue89e{\lambda}_{a,b}\ue89e{\hat{c}}_{a,\sigma}^{\uparrow}\ue89e{\hat{c}}_{b,a}+U\ue89e\sum _{a}\ue89e{\hat{n}}_{a,\uparrow}\ue89e{\hat{n}}_{a,\downarrow}& \left(145\right)\end{array}$  Letting electrons be movable only among neighboring sites, λ_{a,b }defines their bonding relation. First introduced is the following function.
$\begin{array}{cc}\xi \ue8a0\left(a,b;k,L\right)=\{\begin{array}{ccccc}1& \mathrm{when}& \uf603{a}_{k}{b}_{k}\uf604=1& \mathrm{or}& \uf603{a}_{k}{b}_{k}\uf604=L1\\ 0& \mathrm{otherwise}& \text{\hspace{1em}}& \text{\hspace{1em}}& \text{\hspace{1em}}\end{array}& \left(146\right)\end{array}$  This function has been introduced to provide a periodical boundary condition when loops of the period L are formed in the kth layer.
 ζ(a,b;k,q)=δ(a _{1} −b _{1})δ(a _{2} −b _{2}) . . . δ(a _{k−1} −b _{k−1})×Δ(a _{k+1} −b _{k+1})δ(a _{k+2} −b _{k+2}) . . . (a _{q} −b _{q}) (147)

 has been introduced to prevent bonds from concentrating at a particular site. This definition, however, is for q>k, and η(a, b; q, q)=1 is affixed beforehand. Using these functions
$\begin{array}{cc}{\lambda}_{a,b}={\lambda}_{b,a}=\sum _{k=1}^{q}\ue89e\xi \ue8a0\left(a,b;k,L\right)\ue89e\zeta \ue8a0\left(a,b;k,q\right)\ue89e\eta \ue8a0\left(a,b;k,q\right)\ue89e{s}^{qk}& \left(149\right)\end{array}$  is defined.
 Assume hereunder that s=1. However, Let a review be made about the average number of the nearestneighbor sites. When the total number of bonds
 {dot over (N)}_{rmbond} =L ^{q} +L ^{q} +L ^{q−1} + . . . +L ^{2} (150)


 and it is known that it can take any value from the value of a threedimensional tetragonal lattice of z=6 when L=2 to the value of a twodimensional tetragonal lattice under z=4 when L→∞. A multiplylooped ring structure is defined by using the definition of the nearestneighbor sites. FIG. 68A schematically shows the multiplylooped ring structure. FIG. 68B shows a ring of L=12 sites, FIG. 68C shows a torus of L^{2 }sites, FIG. 68D shows a triple multiplylooped ring structure of L^{3 }sites.

 In this manner, sites of the multiplylooped ring structure can be equated with a onedimensional lattice of 1≦j≦L_{q}. Here is defined a pin σ electron density operator of the jth site {circumflex over (n)}_{j,σ}=ĉ_{j,σ} ^{†}Ĉ_{j,σ} and its sum {circumflex over (n)}_{j}=Σ_{σ}{circumflex over (n)}_{j,σ}.
 For the purpose of defining a temperature Green's function, here is introduced a grand canonical Hamiltonian {circumflex over (K)}=Ĥ−μ{circumflex over (N)} where {circumflex over (N)}=Σ_{j}{circumflex over (n)}_{j}. In the half filled taken here, chemical potential is μ=U/2. The halffilled grand canonical Hamiltonian can be expressed as
$\begin{array}{cc}\hat{K}=t\ue89e\sum _{i,j,\sigma}\ue89e\text{\hspace{1em}}\ue89e{\lambda}_{j,i}\ue89e{\hat{t}}_{i,j,\sigma}+U/2\ue89e\sum _{i}\ue89e\text{\hspace{1em}}\ue89e\left({\hat{u}}_{i}1\right)& \left(154\right)\end{array}$  Operators {circumflex over (t)}_{j,i,σ}, ĵ_{j,i,σ, û} _{i }and {circumflex over (d)}_{i,σ} are defined beforehand as
 {circumflex over (t)} _{j,i,σ} =ĉ _{j,σ} ^{†} ĉ _{i,σ} +ĉ _{i,σ} ^{†} ĉ _{j,σ} (155)
 ĵ _{j,i,σ} =ĉ _{j,σ} ^{†} ĉ _{i,σ} =ĉ _{i,σ} ^{†} ĉ _{j,σ} (156)
 û _{i} =ĉ _{i,↑} ^{†} ĉ _{i,↑} ĉ _{i,↓} ĉ _{i,↓} +ĉ _{i,↑} ĉ _{i,↑} ^{†} ĉ _{i,↓} ĉ _{i,↓} ^{†} (157)
 {circumflex over (d)} _{i,σ} =ĉ _{i,σ} ^{†} ĉ _{i,σ} −ĉ _{i,σ} ĉ _{i,σ} ^{†} (158)
 If the temperature Green function is defined for operators Â and {circumflex over (B)} given, taking τ as imaginary time, it is as follows.
 The onsite Green function


 becomes the density of states of the system. For later numerical calculation of densities of states, δ=0.0001 will be used.

 As the equation of motion of the onsite Green function,
$\begin{array}{cc}{\mathrm{i\omega}}_{n}<{\hat{c}}_{j,\sigma};{\hat{c}}_{j,\sigma}^{\u2020}>=1+t\ue89e\sum _{p,j}\ue89e\text{\hspace{1em}}\ue89e{\lambda}_{p,j}<{\hat{c}}_{p,\sigma};{\hat{c}}_{j,\sigma}^{\u2020}>+\frac{U}{2}<{\hat{d}}_{j,\sigma}\ue89e{\hat{c}}_{j,\sigma};{\hat{c}}_{j,\sigma}>& \left(164\right)\end{array}$  is obtained. Then, the approximation shown below is introduced, following Gros ((31) C. Gros, Phys. Rev. B50, 7295(1994)). If the site p is the nearestneighbor site of the site j, the resolution
 is introduced as the approximation. This is said to be exact in case of infinitedimensional Bethe lattices, but in this case, it is only within approximation. Under the approximation, the following equation is obtained.
$\begin{array}{cc}\left({\mathrm{i\omega}}_{n}{t}^{2}\ue89e{\Gamma}_{j,\sigma}\right)\ue89e{G}_{j,\sigma}=1+\frac{U}{2}<{\hat{d}}_{j,\sigma}\ue89e{\hat{c}}_{j,\sigma};{\hat{c}}_{j,\sigma}^{\u2020}>\text{\hspace{1em}}\ue89e\text{where}& \left(166\right)\\ {\Gamma}_{j,\sigma}=\sum _{p}\ue89e\text{\hspace{1em}}\ue89e{\lambda}_{p,j}\ue89e{G}_{p,\sigma}& \left(167\right)\end{array}$  was introduced. To solve the equation obtained, {circumflex over (d)}_{j,−σ}ĉ_{j,σ};ĉ_{j,σ} ^{†} has to be analyzed. In case of halffilled models, this equation of motion is
$\begin{array}{cc}{\mathrm{i\omega}}_{n}<{\hat{d}}_{j,\sigma}\ue89e{\hat{c}}_{j,\sigma};{\hat{c}}_{j,\sigma}^{\u2020}>=\frac{U}{2}\ue89e{G}_{j,\sigma}2\ue89et\ue89e\sum _{p}\ue89e\text{\hspace{1em}}\ue89e{\lambda}_{p,j}<{\hat{j}}_{p,j,\sigma}\ue89e{\hat{c}}_{j,\sigma};{\hat{c}}_{j,\sigma}^{\u2020}>+t\ue89e\sum _{p}\ue89e\text{\hspace{1em}}\ue89e{\lambda}_{p.j}<{\hat{d}}_{j,\sigma}\ue89e{\hat{c}}_{p.,\sigma};{\hat{c}}_{j,\sigma}^{\u2020}>& \left(168\right)\end{array}$  Here again, with reference to the Gros logic, approximation is introduced. It is the following translation.
 By executing this translation, the following closed equation is obtained.
$\begin{array}{cc}\left({\mathrm{i\omega}}_{n}{t}^{2}\ue89e{\tau}_{j,\sigma}\right)\ue89e{G}_{j,\sigma}=1+\frac{{\left(U/2\right)}^{2}}{{\mathrm{i\omega}}_{n}{t}^{2}\ue89e{\Gamma}_{j,\sigma}2\ue89e{t}^{2}\ue89e{\Gamma}_{j,\sigma}}\ue89e{G}_{j,\sigma}& \left(171\right)\end{array}$  Here is assumed that there is no dependency on spin. That is, assuming
 G_{j}=G_{j, }=G_{j, } (172)
 the following calculation is executed.
 Values of DOS obtained by numerical calculation are shown below. The numerical calculation was conducted as to L=3, 4, 5, 6, 10, 20. In case of L=3, q=8 is used, and the total number of sites is n=6561. Values of the following table were used, respectively.
L q n 3 8 6561 4 7 16384 5 6 15625 6 5 7776 10 4 10000 20 3 8000  Setting U=8 in FIG. 69, U=10 in FIG. 70, U=12 in FIG. 71, and setting L=3, 4, 5, 6, 10, 20 for each value of U, numerical calculation was executed.
 In FIG. 69, in case of L=3, the density of state D(ω=0) under the Fermi energy ω=0 exists, and the system behaves as a metal. On the other hand, in FIGS. 70 and 71, a region where DOS has disappeared exists near the Fermi energy, and the system behaves as a Mott insulator. When densities of states corresponding to different values of L are remarked in FIG. 69, in case of L>4, the system has changed to an insulator. Therefore, in case of U=8, Mott metalinsulator transition has been confirmed to occur when L changes.
 In case of U≧10, the system behaves as a Mott insulator under all analyzed values of L, but since the value corresponding to a double of the value of o) rendering DOS be zero is the gap as the Mott insulator (Hubbard gap), it is appreciated that the width of the Hubbard gap increases with L. Therefore, it has been confirmed that MottHubbard transition can be controlled by adjusting L.
 In this manner, when the number of elements L of the multiplylooped ring is controlled and designed, the system can behave as a metal or can behave an insulator. Therefore, a material having a plurality of regions having different numbers of elements can be a superstructure having various regions including metallic regions and insulating regions, and this enables richer controls and designs of physical properties. For example, if a material is designed to have regions of metallic/insulating/metallic phases, a device behaving as a diode can be realized. Further, if the insulator region is made to changeable in phase to a metal under external control, the material can behave as a transistor. A material containing the multiplylooped ring structure as a part thereof could be made. For example, a material in which a structure other than the multiplylooped ring structure is added as an electrode to a multiplylooped ring structure will be useful.
 Next explained is a multiplylooped ring structure according to the 11th embodiment of the invention. In the 11th embodiment, quantum level statistics will be analyzed concerning quantum states in the multiplylooped ring structure to which random potentials are introduced, thereby to demonstrate that quantum chaos can be controlled.
 Explained below is an electron system on a multiplylooped ring in the multiplylooped ring structure according to the 11th embodiment.
 Using a ring having L sites in one cycle, multiplylooped rings are defined hierarchically. Assuming q hierarchies or layers, the total number of sites is L_{q}. The sites are described by a qdimensional cube.
 a=(a _{1} ,a _{2} ,a _{3 } . . . a _{q}) (173)
 where a_{k }is an integer that satisfies 1≦a_{k}≦L.
 Let the operator for generating a quantum at the ath lattice point be Ĉ_{a} ^{†}. Of course, let the operator satisfy the anticommutation relation
 {ĉ _{a} ,ĉ _{b} ^{†}}=δ(a−b) (174)
 This quantum is a fermion having no free spin freedom. This corresponds to analysis of an electron in a solid in which spin orbit interaction can be disregarded. Here is used
$\begin{array}{cc}\delta \ue8a0\left(ab\right)=\prod _{j=1}^{q}\ue89e\text{\hspace{1em}}\ue89e\delta \ue8a0\left({a}_{j}{b}_{j}\right)& \left(175\right)\end{array}$ 
 Letting electrons be movable only among neighboring sites, λ_{a,b }determines their bonding relations. First introduced is the following function.
$\begin{array}{cc}{\Xi}^{(\pm )}\ue8a0\left(a,b;k,L\right)=\{\begin{array}{ccc}1& \mathrm{when}& {b}_{k}{a}_{k}=\pm 1\\ 1& \mathrm{when}& {b}_{k}{a}_{k}=\pm \left(1L\right)\\ 0& \mathrm{otherwise}& \text{\hspace{1em}}\end{array}& \left(177\right)\end{array}$  This function has been introduced to provide a periodical boundary condition and bonds between adjacent loops in the kth layer. Next introduced is
$\begin{array}{cc}\Psi \ue8a0\left(a,b;k\right)=\prod _{j=1}^{k1}\ue89e\text{\hspace{1em}}\ue89e\delta \ue8a0\left({a}_{j}{b}_{j}\right)& \left(178\right)\end{array}$  This is a function that takes 1 only when having the same suffix concerning layers beyond the kth layer. Furthermore, by using
$\begin{array}{cc}Q\ue8a0\left(a,b;k,q\right)=\{\begin{array}{ccc}\prod _{j=2}^{qk}\ue89e\text{\hspace{1em}}\ue89e\delta \ue8a0\left({a}_{k+j}j\right)\ue89e\delta \ue8a0\left({b}_{k+j}j\right)& \mathrm{when}& q>k+1\\ 1& \mathrm{when}& q\le k+1\end{array}& \left(179\right)\\ {R}^{(\pm )}\ue8a0\left(x;\alpha ,L\right)=\delta \ue8a0\left(\mathrm{mod}\ue8a0\left(x1\pm \frac{\sigma \ue89e\text{\hspace{1em}}\ue89eL}{2},L\right)\right)& \left(180\right)\end{array}$  also introduced is
 Φ^{(±)}(a,b;k,q)=R ^{(∓)}(a _{k+1} ;α,L)R ^{(±)}(b _{k+1} ;α,L)Q(a,b;k,q) (181)
 Using these functions,
$\begin{array}{cc}{\lambda}_{a,b}={\lambda}_{b,a}=\sum _{\sigma =+,}^{q}\ue89e\text{\hspace{1em}}\ue89e\sum _{k=1}^{q}\ue89e\text{\hspace{1em}}\ue89e{\Xi}^{\left(\sigma \right)}\ue8a0\left(a,b;k,L\right)\ue89e{\phi}^{\left(\sigma \right)}\ue8a0\left(a;k,q\right)\ue89e\Psi \ue8a0\left(a,b;k\right)\ue89e{s}^{qk}& \left(182\right)\end{array}$  is defined. When placing α=0, the above definition results in the same as that already discussed in the tenth embodiment. When α>0, the same effects as parallel movements of interlayer bonds in the multiplytwisted helix is expected. Regarding its effects to Mott metalinsulator transition, detailed explanation will be made later in the second embodiment. Hereunder, using α=0.5, calculation will be conducted.

 In this manner, sites of the multiplylooped ring structure can be equated with a onedimensional lattice of 1≦j≦L_{q}. The Hamiltonian Ĥ of this quantum system is defined as
$\begin{array}{cc}\hat{H}=\sum _{a,b}\ue89e{\lambda}_{a,b}\ue89e{\hat{c}}_{a}^{\u2020}\ue89e{\hat{c}}_{b}+\sum _{a}\ue89e{v}_{a}\ue89e{\hat{c}}_{a}^{\u2020}\ue89e{\hat{c}}_{a}& \left(184\right)\end{array}$ 
 is generated to form the Hamiltonian. The variable breadth v of the random potential is useful as a parameter determining the degree of the randomness.

 where m=1, 2, . . . , n.

 and its staircase function
 λ(ε)=∫_{−∞} ^{ε} dηρ(η) (188)
 is calculated. The staircase function obtained is converted by using a procedure called unfolding such that the density of states becomes constant in average. Using the quantum level obtained in this manner, the nearestneighbor level spacing distribution P(s) and the Δ_{3 }statistics of Dyson and Mehta are calculated as quantum level statistics. As taught by Literatures (32) and (33), by using these statistics, it can be detected whether quantum chaos has been generated or not. It is also known that a quantum chaotic system is sensitive to perturbation from outside similarly to the classical chaotic system, and analysis of quantum chaos is important as a polestar of designs of nonlinear materials.
 In case of an integrable system, nearestneighbor level spacing distribution P(s) and Δ_{3 }statistics are those of Poisson Distribution
 In case the system discussed here exhibits quantum chaos, it becomes GOE distribution
$\begin{array}{cc}{P}_{\mathrm{GOE}}\ue8a0\left(s\right)=\frac{\pi \ue89e\text{\hspace{1em}}\ue89es}{2}\ue89e{\uf74d}^{\pi \ue89e\text{\hspace{1em}}\ue89es\ue89e\text{\hspace{1em}}\ue89e2/4}& \left(191\right)\\ {\Delta}_{3}\ue8a0\left(n\right)=\frac{1}{{\pi}^{2}}\ue8a0\left[\mathrm{log}\ue89e\left(2\ue89e\text{\hspace{1em}}\ue89e\pi \ue89e\text{\hspace{1em}}\ue89en\right)+\gamma \frac{{\pi}^{2}}{8}\frac{5}{4}\right]+O\ue8a0\left({n}^{1}\right)& \left(192\right)\end{array}$  where γ is the Euler's constant.
 In this section, L, which determines the turn pitch of the multiplylooped ring structure, s, which determines the interlayer bonding force, and v, which determines the intensity of the random potential, are varied as parameters. When L=6, 10, 20 is used as the turn pitch, the values
L q n 6 5 7776 10 4 10000 20 3 8000  were used as the total number of sites. Statistics values are calculated by using 2001 quantum levels, in total, taking 1000 states on each side from the band center among n states used for each L.
 FIG. 72 and FIG. 73 show quantum level statistics of multiplylooped ring structures of L=6 while fixing the interlayer bonding to s=1 and changing the random potential intensity v. FIG. 74 and FIG. 75 show quantum level statistics of multiplylooped ring structures of L=10 while fixing the interlayer bonding to s=1 and changing the random potential intensity v. FIG. 76 and FIG. 77 show quantum level statistics of multiplylooped ring structures of L=20 while fixing the interlayer bonding to s=1 and changing the random potential intensity v. FIGS. 72, 74 and76 show P(s) whilst FIGS. 73, 75 and 77 show Δ_{3 }statistics values. In each of FIGS. 72 through 77, when v is small, the system is in a metallic state, and the quantum level statistics is that of a quantum chaotic system of GOE. As v increases, the quantum level statistics changes toward the Poisson distribution. This is typical Anderson transition ((34) Phys. Rev. 109, 1492 (1958); (35) Phys. Rev. Lett. 42, 783(1979); (36) Rev. Mod. Phys. 57. 287(1985)).
 Through a careful review of the case of v=2, it is confirmed that the chaotic tendency is enhanced as L increases. On the other hand, in case of v=22, as L increases, the system asymptotically changes toward the Poisson distribution. That is, as L increases, the level statistics values of this quantum system become more sensitive to changes of v.
 Similarly to the multiplytwisted helix, here is explained is a result of analysis of a system in which the interlayer bonding intensity s is decreased. While fixing v=2, FIG. 78 shows Δ_{3 }statistics values of multiplylooped ring structures of L=6, FIG. 79 shows Δ_{3 }statistics values of multiplylooped ring structures of L=10, and FIG. 80 shows Δ_{3 }statistics values of multiplylooped ring structures of L=20. When s decreases from 1, the interlayer bonding becomes weaker, and the spatial dimension reduces. Therefore, it results in enhancing localization. It is appreciated from FIGS. 78 through 80 that the system changes from a quantum chaotic sate to a considerably localized state along with changes from s=1 to s=0.4. The degree of the changes becomes larger as L becomes smaller. This is because, when L is small, because of a small diameter of the basic double ring, this is substantially a onedimensional tube, but in contrast, when L is large, because of a large diameter of the basic double ring, this itself forms a twodimensional sheet.
 Next explained is a multiplylooped ring structure according to the 12th embodiment of the invention. In the 12th embodiment, parallel movement of interlayer bonds in multiplylooped ring structures are introduced. Thereby, it is demonstrated that, by the use of this freedom, the nature of Mott metalinsulator transition occurring in multiplylooped ring structures can be controlled and interelectron correlation intensity can be designed.
 Explained below is an electron system on a multiplylooped ring in the multiplylooped ring structure according to the 12th embodiment.
 Using a ring having L sites in one cycle, multiplylooped rings are defined hierarchically. Assuming q hierarchies or layers, the total number of sites is L_{q}. The sites are described by a qdimensional cube.
 a=(a _{1} ,a _{2} ,a _{3 } . . . a _{q}) (193)
 where a_{k }is an integer that satisfies 1≦a_{k}≦L.
 Let the operator for generating an electron of spin σ at the ath lattice point be Ĉ_{a,σ} ^{†}. Of course, there is the anticommutation relation
 {ĉ _{z,σ} ,ĉ _{b,ρ} ^{†}}=δ(a−b)δ(σ−ρ) (194)


 Here is defined a singleband Hubbard Hamiltonian Ĥ of the electron system as follows.
$\begin{array}{cc}\hat{H}=t\ue89e\sum _{a,b,\sigma}\ue89e{\lambda}_{a,b}\ue89e{\hat{c}}_{a,\sigma}^{\uparrow}\ue89e{\hat{c}}_{b,\sigma}+U\ue89e\sum _{a}\ue89e{\hat{n}}_{a,\uparrow}\ue89e{\hat{n}}_{a,\downarrow}& \left(197\right)\end{array}$  Letting electrons be movable only among neighboring sites, λ_{a,b }defines their bonding relation. First introduced is the following function.
$\begin{array}{cc}{\Xi}^{\left(x\right)}\ue8a0\left(a,b;k,L\right)=\{\begin{array}{ccc}1& \mathrm{when}& {b}_{k}{a}_{k}=\pm 1\\ 1& \mathrm{when}& {b}_{k}{a}_{k}=\pm \left(1L\right)\\ 0& \mathrm{otherwise}& \text{\hspace{1em}}\end{array}& \left(198\right)\end{array}$ 
 is a function that takes 1 only when having the same suffix concerning layers beyond the kth layer. Furthermore, by using
$\begin{array}{cc}Q\ue8a0\left(a,b;k,q\right)=\{\begin{array}{ccc}\prod _{j=2}^{qk}\ue89e\delta \ue8a0\left({a}_{k+j}j\right)\ue89e\delta \ue8a0\left({b}_{k+j}j\right)& \mathrm{when}& q>k+1\\ 1& \mathrm{when}& q\le k+1\end{array}& \left(200\right)\\ {R}^{(\pm )}\ue8a0\left(x;\alpha ,L\right)=\delta \ue8a0\left(\mathrm{mod}\ue8a0\left(x1\pm \frac{\alpha \ue89e\text{\hspace{1em}}\ue89eL}{2},L\right)\right)& \left(201\right)\end{array}$  also introduced is
 Φ^{(±)}(a,b;k,q)=R ^{(∓)}(a _{k+1} ;α,L)R ^{(±)}(b _{k+1} ;α,L)Q(a,b;k,q) (202)
 Using these functions,
$\begin{array}{cc}\begin{array}{c}{\lambda}_{a,b}=\ue89e{\lambda}_{b,a}\\ =\ue89e\sum _{\sigma =+,}^{q}\ue89e\sum _{k=1}^{q}\ue89e{\Xi}^{\sigma}\ue8a0\left(a,b;k,L\right)\ue89e{\Phi}^{\left(\sigma \right)}\ue8a0\left(a;k,q\right)\ue89e\Psi \ue8a0\left(a,b;k\right)\ue89e{s}^{qk}\end{array}& \left(203\right)\end{array}$  is defined. Hereunder, s=1 is assumed.

 In this manner, sites of the multiplylooped ring structure can be equated with a onedimensional lattice of 1≦j≦L_{q}. Now defined are the operator of density of electrons with spin σ in the jth site
 {circumflex over (n)} _{j,σ} =ĉ _{j,σ} ^{†} ĉ _{j,σ} (205)
 and the sum thereof.
 {circumflex over (n)} _{j}=Σ_{σ} {circumflex over (n)} _{j,σ} (206)
 For the purpose of defining a temperature Green's function, here is introduced a grand canonical Hamiltonian
 {circumflex over (K)}=Ĥ−μ{circumflex over (N)} (207)
 where
 {circumflex over (N)}=Σ _{j} {circumflex over (n)} _{j} (208)
 In the half filled taken here, chemical potential is μ=U/2. The halffilled grand canonical Hamiltonian can be expressed as
$\begin{array}{cc}\hat{K}=t\ue89e\sum _{i,j\ue89e\text{\hspace{1em}}\ue89e\sigma}\ue89e{\lambda}_{j,i}\ue89e{\hat{t}}_{j,i,\sigma}+U/2\ue89e\sum _{i}\ue89e\left({\hat{u}}_{i}1\right)& \left(209\right)\end{array}$  Operators {circumflex over (t)}_{j,i,σ}, ĵ_{j,i,σ}, û_{i }and {circumflex over (d)}_{i,σ} are defined beforehand as
 {circumflex over (t)} _{j,i,σ} =ĉ _{j,σ} ^{†} ĉ _{i,σ} +ĉ _{i,σ} ^{†} ĉ _{j,σ} (210)
 ĵ _{j,iσ} =ĉ _{j,σ} ^{†} ĉ _{i,σ} −ĉ _{i,σ} ^{†} ĉ _{j,σ} (211)
 û _{i} =ĉ _{i,↑} ^{†} ĉ _{i,↑} ĉ _{i,↓} ^{†} ĉ _{i,↓} +ĉ _{i,↑} ĉ _{i,↑} ^{†} ĉ _{i,↓} ĉ _{i,↓} ^{†} (212)
 {circumflex over (d)} _{i,σ} =ĉ _{i,σ} ^{†} ĉ _{i,σ} =ĉ _{i,σ} ĉ _{i,σ} ^{†} (213)
 If the temperature Green function is defined for operators Â and {circumflex over (B)} given, taking τ as imaginary time, it is as follows.
 The onsite Green function
 is important for calculation of densities of states.

 As the equation of motion of the onsite Green function,
$\begin{array}{cc}i\ue89e\text{\hspace{1em}}\ue89e{\omega}_{n}\ue89e\u3008{\hat{c}}_{j,\sigma};{\hat{c}}_{j,\sigma}^{\u2020}\u3009=1+t\ue89e\sum _{p,j}\ue89e{\lambda}_{p,j}\ue89e\u3008{\hat{c}}_{p,\sigma};{\hat{c}}_{j,\sigma}^{\u2020}\u3009+\frac{U}{2}\ue89e\u3008{\hat{d}}_{j,\sigma}\ue89e{\hat{c}}_{j,\sigma};{\hat{c}}_{j,\sigma}^{\u2020}\u3009& \left(217\right)\end{array}$  is obtained. Then, the approximation shown below is introduced, following Gros ((31) C. Gros, Phys. Rev. B50, 7295(1994)). If the site p is the nearestneighbor site of the site j, the resolution
 is introduced as the approximation. This is said to be exact in case of infinitedimensional Bethe lattices, but in this case, it is only within approximation. Under the approximation, the following equation is obtained.
$\begin{array}{cc}\left(i\ue89e\text{\hspace{1em}}\ue89e{\omega}_{n}{t}^{2}\ue89e\text{\hspace{1em}}\ue89e{\Gamma}_{j,\sigma}\right)\ue89e{G}_{j,\sigma}=1+\frac{U}{2}\ue89e\u3008{\hat{d}}_{j,\sigma}\ue89e{\hat{c}}_{j,\sigma \ue89e\text{\hspace{1em}}};{\hat{c}}_{j,\sigma}^{\u2020}\u3009\ue89e\text{\hspace{1em}}\ue89e\mathrm{where}& \left(219\right)\\ {\Gamma}_{j,\sigma}=\sum _{p}\ue89e{\lambda}_{p,j}\ue89e{G}_{p,\sigma}& \left(220\right)\end{array}$ 