Method and system for determining aging number of quantum anomalous Hall insulator based on topological defect, computer equipment, storage medium and program product

Abstract

The invention provides a quantum anomalous Hall insulator aging number determination method and system based on topological defects, computer equipment, a storage medium and a program product. The method comprises the steps that the topological charge of each topological defect of two belt models is determined; determining the contribution of the corresponding topological charge to the aging number so as to determine the aging number of the valence band or the conduction band; fourier transform is carried out on the Hamiltonian to obtain a tight constraint Hamiltonian in a real space; performing one-dimensional Fourier transform on the tightly bound Hamiltonian in the real space to obtain a one-dimensional effective Hamiltonian; judging whether the number of the energy-gap-free edge states is equal to the absolute value of the conduction band aging number or not; if so, taking the number of the edge states without the energy gap as the aging number of the quantum abnormal Hall insulator; and if not, reconstructing the Hamiltonian of the two-band model and the expression of the component x and the component y of the Hamiltonian with respect to the two-dimensional momentum, and calculating the momentum again. According to the method, a large number of high aging models can be constructed, and the research efficiency is improved.

Description

Technical Field

[0001] The present invention belongs to the field of quantum computing technology, and in particular relates to a method, system, computer equipment, storage medium and program product for determining the Chern number of a quantum anomalous Hall insulator based on topological defects. Background Art

[0002] Topological materials are an important research direction in the current field of condensed matter physics. A topological material is characterized by non-zero topological invariants in the internal energy bands; correspondingly, there is a special band structure on the surface. These structures have extremely high application potential, especially for the development of low-power, high-performance devices, quantum computers, and spin electronics. In this field, quantum anomalous Hall insulators are typical representatives. Because they can achieve quantization of Hall resistance and gapless edge states without an external magnetic field, they have higher application potential and have received widespread attention from theorists and experimentalists.

[0003] It is observed that the Hall conductance has a quantized platform: Ce 2 / h, and the longitudinal resistance disappears, which means that no dissipation occurs when electrons move along the edge channels. Where e is the electron charge, h is Planck's constant, and C is the topological invariant - Chern number, which determines the number of edge channels. However, the contact resistance between the metal electrode and the edge channel is an unavoidable drawback of the material. In quantum anomalous Hall insulators, the minimum contact resistance is h / (Ce 2 ). It can be seen that increasing the Chern number will greatly reduce the contact resistance and significantly improve the performance of the device. Therefore, the research on high Chern number structures is an important direction in this field.

[0004] At present, most of the methods for studying high Chern number models are trial and error. The research steps are to select a lattice structure and then try to write the Hamiltonian, calculate the Chern number of the model, and analyze the band structure. This method requires assuming the Hamiltonian first, and then calculating the Chern number. If it does not meet the requirements, it is necessary to re-assume a new Hamiltonian and then recalculate the Chern number. However, the Chern number needs to be calculated once for each Hamiltonian, which takes a lot of time to screen out a suitable model, and it cannot be replicated in large quantities. Summary of the invention

[0005] In view of the deficiencies in the prior art, the present invention provides a method, system, computer equipment, storage medium and program product for determining the Chern number of a quantum anomalous Hall insulator based on topological defects.

[0006] In a first aspect, the present invention provides a method for determining the Chern number of a quantum anomalous Hall insulator based on topological defects, comprising:

[0007] S1, determine the topological charge of each topological defect in the two-band model;

[0008] S2, determining the contribution of the corresponding topological charge to the Chern number according to the value of the z component of the Hamiltonian at each topological defect, so as to determine the Chern number of the valence band or conduction band;

[0009] S3, according to the type of two-dimensional lattice structure, perform Fourier transform on the Hamiltonian to obtain the tight-binding Hamiltonian in real space;

[0010] S4, using the mixed boundary conditions to perform a one-dimensional Fourier transform on the tight-binding Hamiltonian in real space to obtain a one-dimensional effective Hamiltonian;

[0011] S5, determining whether the number of gapless edge states is equal to the absolute value of the conduction band Chern number; wherein the gapless edge state is the gapless edge state of the image corresponding to the eigenvalue of the one-dimensional effective Hamiltonian;

[0012] S6, if equal to, the number of gapless edge states is taken as the Chern number of the quantum anomalous Hall insulator;

[0013] S7, if not equal, reconstructs the Hamiltonian of the two-band model and the expressions of its x-component and y-component with respect to the two-dimensional momentum, and returns to execute the operation of S1.

[0014] Optionally, determining the topological charge of each topological defect of the two-band model includes:

[0015] Construct the x-component h of the Hamiltonian for the two-band model x and the y component h y About the two-dimensional momentum (k x , k y )

[0016] will satisfy h x = 0 and h y =0(k x ,k y ) as the coordinates of the topological defects in the two-band model;

[0017] Get the two-dimensional vector (h x ,h y ); when all arrows in the vector distribution map are away from the coordinate point, the topological charge corresponding to the coordinate point is +1; when all arrows in the vector distribution map converge to the coordinate point, the topological charge corresponding to the coordinate point is +1; when the arrows in the vector distribution map are away from the coordinate point in one direction and converge to the coordinate point in another direction, the topological charge corresponding to the coordinate point is -1; when the arrows in the vector distribution map surround the coordinate point, the topological charge corresponding to the coordinate point is -1.

[0018] Optionally, determining the contribution of the corresponding topological charge to the Chern number according to the value of the Hamiltonian z component at each topological defect to determine the Chern number of the valence band or the conduction band includes:

[0019] Construct the z component h of the Hamiltonian of the two-band model z Expression of

[0020] In h z When it is greater than 0, the Chern number C of the conduction band is calculated according to the following formula N :

[0021]

[0022] Among them, L N is the total number of defects on the conduction band; W l is the topological charge of the lth defect;

[0023] In h z When it is less than 0, the Chern number C of the conduction band is calculated according to the following formula S :

[0024]

[0025] Among them, L S is the total number of defects on the valence band.

[0026] In a second aspect, the present invention provides a Chern number determination system for a quantum anomalous Hall insulator based on topological defects, comprising:

[0027] A first determination module is used to determine the topological charge of each topological defect of the two-band model;

[0028] A second determination module is used to determine the contribution of the corresponding topological charge to the Chern number according to the value of the Hamiltonian z component at each topological defect, so as to determine the Chern number of the valence band or the conduction band;

[0029] A first Fourier transform module is used to perform Fourier transform on the Hamiltonian according to the type of the two-dimensional lattice structure to obtain a tight-binding Hamiltonian in real space;

[0030] The second Fourier transform module is used to perform a one-dimensional Fourier transform on the tight-binding Hamiltonian in the real space using the mixed boundary conditions to obtain a one-dimensional effective Hamiltonian;

[0031] A judgment module is used to judge whether the number of gapless edge states is equal to the absolute value of the conduction band Chern number; wherein the gapless edge state is the gapless edge state of the image corresponding to the eigenvalue of the one-dimensional effective Hamiltonian;

[0032] A third determination module is used to determine the number of gapless edge states as the Chern number of the quantum anomalous Hall insulator when the judgment module determines that the number of gapless edge states is equal to the absolute value of the conduction band Chern number;

[0033] The fourth determination module is used to determine the Hamiltonian of the reconstructed two-band model and the expression of its x-component and y-component about the two-dimensional momentum when the judgment module determines that the number of gapless edge states is not equal to the absolute value of the conduction band Chern number, and execute the operation of the first determination module.

[0034] Optionally, the first determining module includes:

[0035] The first building block is used to construct the x component h of the Hamiltonian of the two-band model x and the y component h y About the two-dimensional momentum (k x ,k y )

[0036] The first determining unit is used to satisfy h x = 0 and h y =0(k x ,k y ) as the coordinates of the topological defects in the two-band model;

[0037] Acquisition unit, used to obtain a two-dimensional vector (h x ,h y ); when all arrows in the vector distribution map are away from the coordinate point, the topological charge corresponding to the coordinate point is +1; when all arrows in the vector distribution map converge to the coordinate point, the topological charge corresponding to the coordinate point is +1; when the arrows in the vector distribution map are away from the coordinate point in one direction and converge to the coordinate point in another direction, the topological charge corresponding to the coordinate point is -1; when the arrows in the vector distribution map surround the coordinate point, the topological charge corresponding to the coordinate point is -1.

[0038] Optionally, the second determining module includes:

[0039] The second building block is used to construct the z component h of the Hamiltonian of the two-band model z Expression of

[0040] The first calculation unit is used to calculate the z When it is greater than 0, the Chern number C of the conduction band is calculated according to the following formula N :

[0041]

[0042] Among them, L N is the total number of defects on the conduction band; W l is the topological charge of the lth defect;

[0043] The second computing unit is used to calculate the z When it is less than 0, the Chern number C of the conduction band is calculated according to the following formula S :

[0044]

[0045] Among them, L S is the total number of defects on the valence band.

[0046] In a third aspect, the present invention provides a computer device comprising a processor and a memory; wherein, when the processor executes a computer program stored in the memory, the steps of the method for determining the Chern number of a quantum anomalous Hall insulator based on topological defects described in the first aspect are implemented.

[0047] In a fourth aspect, the present invention provides a computer-readable storage medium for storing a computer program; when the computer program is executed by a processor, the steps of the method for determining the Chern number of a quantum anomalous Hall insulator based on topological defects described in the first aspect are performed.

[0048] In a fifth aspect, the present invention provides a computer program product, comprising computer executable instructions or a computer program, which, when executed by a processor, implements the steps of the method for determining the Chern number of a quantum anomalous Hall insulator based on topological defects described in the first aspect.

[0049] The present invention provides a method, system, computer device, storage medium and program product for determining the Chern number of a quantum anomalous Hall insulator based on topological defects. In the method, starting from an existing model, the topological defects are derived from the existing model. A high Chern number result can be first given by changing the position, quantity and other information of the topological defects, and then the Hamiltonian and lattice structure can be inverted. Moreover, multiple new models can be constructed from an existing model, and then the Hamiltonian can be inverted to realize multiple high Chern number models. The present invention can construct a large number of high Chern number models, provide reference for the screening of experimental materials, and improve research efficiency. BRIEF DESCRIPTION OF THE DRAWINGS

[0050] In order to more clearly illustrate the technical solution of the present invention, the drawings required for use in the embodiments are briefly introduced below. Obviously, the drawings described below are only embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative work.

[0051] Figure 1 A schematic flow chart of a method for determining the Chern number of a quantum anomalous Hall insulator based on topological defects provided in an embodiment of the present invention;

[0052] Figure 2 Vector distribution diagrams of four defects provided in the embodiments of the present invention;

[0053] Figure 3 An image of the energy eigenvalue of a one-dimensional effective Hamiltonian provided by an embodiment of the present invention;

[0054] Figure 4 The defect coordinates, topological charge and h provided in the embodiment of the present invention z The corresponding relationship diagram of

[0055] Figure 5 The two-dimensional vector field (h x ,h y )=(λ2sin2k x ,λ2sin2k y ) distribution map;

[0056] Figure 6 An image of the energy eigenvalue of the reconstructed one-dimensional effective Hamiltonian provided by an embodiment of the present invention;

[0057] Figure 7 A schematic diagram of the structure of a Chern number determination system for a quantum anomalous Hall insulator based on topological defects provided in an embodiment of the present invention. DETAILED DESCRIPTION

[0058] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0059] Example 1

[0060] like Figure 1 As shown, this embodiment provides a method for determining the Chern number of a quantum anomalous Hall insulator based on topological defects, comprising:

[0061] S1, determine the topological charge of each topological defect in the two-band model.

[0062] At present, all systems that realize quantum anomalous Hall state can use the mass-carrying two-dimensional Dirac equation to describe the energy band at the Fermi level under the adiabatic approximation. The system corresponding to this equation has only two energy bands (conduction band and valence band). Therefore, quantum anomalous Hall insulators are generally studied in the two-band model. The general expression of the Hamiltonian of the two-band model is:

[0063] H1=εI+h·σ.

[0064] Where H1 is the Hamiltonian of the two-band model; ε represents the curvature of the energy band; I represents the two-dimensional unit matrix; h represents the two-dimensional momentum (k x ,k y ) determines the three-dimensional vector, h=(h x ,h y ,hz ), the x-component, y-component and z-component of the Hamiltonian of the two-band model are h x 、h y and h z ;σ is the Pauli matrix, σ=(σ x ,σ y ,σ z );

[0065]

[0066] Here, i is an imaginary unit.

[0067] Substituting the Pauli matrix into H1, we find that there are two energy bands: E + The corresponding energy band is the conduction band; the corresponding energy band of E- is the valence band.

[0068] In this step, we select the existing two-band model with a Chern number of ±1 and construct the x component h of the Hamiltonian of the two-band model. x About the first momentum k x The expression of , and the y component h y About the second momentum k y expression.

[0069] For example, the established quantum anomalous Hall model in the square lattice is selected to construct the x component h of the Hamiltonian of the two-band model. x and the y component h y About the two-dimensional momentum (k x , k y ) expression:

[0070]

[0071] will satisfy h x = 0 and h y =0(k x ,k y ) as the coordinates of the topological defects in the two-band model, that is, the coordinates of the four topological defects are obtained: (0,0), (π,π), (0,π) and (π,0).

[0072] Get the two-dimensional vector (h x ,h y ); when all arrows in the vector distribution map are away from the coordinate point, the topological charge corresponding to the coordinate point is +1; when all arrows in the vector distribution map converge to the coordinate point, the topological charge corresponding to the coordinate point is +1; when the arrows in the vector distribution map are away from the coordinate point in one direction and converge to the coordinate point in another direction, the topological charge corresponding to the coordinate point is -1; when the arrows in the vector distribution map surround the coordinate point, the topological charge corresponding to the coordinate point is -1.

[0073] like Figure 2 As shown, use Mathematica and other software to draw a two-dimensional vector (h x ,h y ), analyze the type of each defect from the vector distribution map, and obtain the corresponding topological charge. Taking the coordinate point (0,0) as an example, there are four types of defects (each vector map corresponds to one type): Figure 2 In (a), all arrows are away from (k x ,k y )=(0,0) point, which means this is a source point with a topological charge of +1; Figure 2 In (b), all arrows point to (k x ,k y )=(0,0) point converges, which means this is a sink point with a topological charge of +1; Figure 2 In (c), the arrow moves away from (k x ,k y )=(0,0) point, converging in the other direction, indicating that this is a saddle point with a topological charge of -1; Figure 2 In (d), the arrows surround (k x ,k y )=(0,0), which means this is a vortex with a topological charge of +1.

[0074] S2, determining the contribution of the corresponding topological charge to the Chern number according to the value of the z component of the Hamiltonian at each topological defect, so as to determine the Chern number of the valence band or conduction band.

[0075] The topological defect coordinates of the Hamiltonian in S1 are given by h x =h y =0,h z ≠0 is certain, and h z The positive or negative value of is used to determine the position of the defect. The Chern number of the valence band and the conduction band is equal to the topological sum of the topological defects on the corresponding energy band. By changing the number and position of defects to change or increase the contribution of topological charge to the Chern number, a high Chern number result is obtained, and then the corresponding Hamiltonian is inferred, a high Chern number model is realized, and finally its band structure is analyzed.

[0076] Exemplarily, this step includes constructing the z component h of the Hamiltonian of the two-band model z The expression is:

[0077] h z =m+cosk x +cosk y .

[0078] Where m is the exchange field, and h at each defect is obtained z: The coordinate at (0,0) is m+2; the coordinate at (π,π) is m-2; the coordinate at (0,π) is m; the coordinate at (π,0) is m.

[0079] In h z When it is greater than 0, the Chern number C of the conduction band is calculated according to the following formula N :

[0080]

[0081] Among them, L N is the total number of defects on the conduction band; W l is the topological charge of the lth defect.

[0082] In h z When it is less than 0, the Chern number C of the conduction band is calculated according to the following formula S :

[0083]

[0084] Among them, L S is the total number of defects on the valence band.

[0085] It can be seen that the value of m determines h z The sign of , thus determining the location of each defect:

[0086] When m>2, every point satisfies h z > 0, indicating that all defects are located on the conduction band, so C N =+1+1-1-1=0, C S =0.

[0087] When 0<m<2, h at coordinate (π,π) z <0, so the defect at coordinate (π,π) is in the valence band, and the other defects are in the conduction band, so C N =+1-1-1=-1, C S =+1.

[0088] When -2<m<0, h at coordinate (0,0) z > 0, so the defect at coordinate (0,0) is in the conduction band, and the other defects are in the valence band, so C N = +1, C S =+1-1-1=-1.

[0089] When m<-2, all defects are located in the valence band, so C N =0, C S =+1+1-1-1=0.

[0090] S3, according to the type of two-dimensional lattice structure, perform Fourier transform on the Hamiltonian to obtain the tight-binding Hamiltonian in real space.

[0091] In this step, the Hamiltonian H1 is Fourier transformed in the square lattice to obtain the tight-binding Hamiltonian H in real space. r for:

[0092]

[0093] Among them, c + (x,y) and c(x,y) are the creation and annihilation operators at the lattice point with coordinates (x,y), respectively; hc represents the Hermitian conjugation of the first two terms.

[0094] S4, using the mixed boundary conditions, a one-dimensional Fourier transform is performed on the tight-binding Hamiltonian in real space to obtain a one-dimensional effective Hamiltonian.

[0095] In this step, the mixed boundary condition is to take one direction of the lattice as periodic (infinite length) and the other direction as finite length. r Perform a one-dimensional Fourier transform to obtain a one-dimensional effective Hamiltonian H e , H e is a large matrix.

[0096] In this embodiment, the y direction is assumed to be periodic, the x direction is assumed to be finite length, and the H r Perform a one-dimensional Fourier transform to obtain a one-dimensional effective Hamiltonian H e :

[0097]

[0098] in, and The coordinates are x (momentum is k y ) is the lattice point creation and annihilation operator. The Hamiltonian is expanded into a large matrix whose dimension is the number of lattice points in the x direction.

[0099] S5, determining whether the number of gapless edge states is equal to the absolute value of the conduction band Chern number; wherein the gapless edge state is the gapless edge state of the image corresponding to the eigenvalue of the one-dimensional effective Hamiltonian.

[0100] Using Mathematica software to obtain H e The eigenvalue E of and draw its image, such as Figure 3 As shown, where m is -1.5, it can be found that a gapless boundary state appears, which is consistent with the Chern number C S = corresponds to +1.

[0101] S6, if equal to, then the number of gapless edge states is taken as the Chern number of the quantum anomalous Hall insulator.

[0102] S7, if not equal, reconstructs the Hamiltonian of the two-band model and the expressions of its x-component and y-component with respect to the two-dimensional momentum, and returns to execute the operation of S1.

[0103] In this step, h is reconstructed x and h y The expression of h is added to the equation x = 0 and h y =0, that is, increasing the number of defects, so that there are more defects on the energy band, so that more topological charge contributes to the Chern number, achieving a high Chern number.

[0104] In this embodiment, the reconstructed h x and h y The expression for two-dimensional momentum is:

[0105]

[0106] Wherein, λ2 is the spin-orbit coupling strength between the next-nearest neighboring lattice points, and in this embodiment, λ2 is set to 0.5.

[0107] Make h x = 0 and h y = 0 to redefine the coordinates of the topological defect.

[0108] The recalculated Chern numbers of the two energy bands are:

[0109]

[0110] The defect position, topological charge and h obtained in the calculation z exist Figure 4 is summarized in , where the topological charge is derived from the vector field (h x ,h y )=(λ2sin2k x ,sin2k y ) is obtained from the distribution of Figure 5 shown.

[0111] According to h z , ε (here 0), reconstructed h x And the reconstructed h y Write down the Hamiltonian of the new model

[0112] H2=λ2sin2k x σ x +λ2sin2k y σ y +(m+coskx +cosk y )σ z

[0113] like Figure 6 As shown in Figure 2, when m = 0.5, three gapless edge states appear, which are related to C S =-0.5, which indicates that the new model can achieve a high Chern number.

[0114] The real-space tight-binding Hamiltonian of Hamiltonian H2 is: The one-dimensional effective Hamiltonians are:

[0115]

[0116] For H r Perform a one-dimensional Fourier transform and obtain the one-dimensional effective Hamiltonian:

[0117]

[0118] Where λ3 is the transition strength between the next nearest neighboring lattice points. This term is introduced to make the Hamiltonian itself more physically meaningful. After Fourier transformation, it gives the ε term, which, as mentioned above, has no effect on the Chern number.

[0119] In summary, the method for determining the Chern number of a quantum anomalous Hall insulator based on topological defects provided in this embodiment starts from an existing model. The topological defects originate from the existing model. By changing the position, number and other information of the topological defects, a high Chern number result can be first given, and then the Hamiltonian and lattice structure can be inferred. Moreover, multiple new models can be constructed from an existing model, and then the Hamiltonian can be inferred to achieve multiple high Chern number models. A large number of high Chern number models can be constructed, which provides a reference for the screening of experimental materials and improves research efficiency.

[0120] Example 2

[0121] Based on the same inventive concept as Example 1, this embodiment provides a Chern number determination system for a quantum anomalous Hall insulator based on topological defects. Since the principle of solving the problem by this system is similar to the aforementioned Chern number determination method for a quantum anomalous Hall insulator based on topological defects, the implementation of this system can refer to the implementation of the Chern number determination method for a quantum anomalous Hall insulator based on topological defects.

[0122] like Figure 7 As shown, the Chern number determination system of quantum anomalous Hall insulator based on topological defects includes:

[0123] The first determination module 10 is used to determine the topological charge of each topological defect in the two-band model.

[0124] The second determination module 20 is used to determine the contribution of the corresponding topological charge to the Chern number according to the value of the Hamiltonian z component at each topological defect, so as to determine the Chern number of the valence band or the conduction band.

[0125] The first Fourier transform module 30 is used to perform Fourier transform on the Hamiltonian according to the type of the two-dimensional lattice structure to obtain a tight-binding Hamiltonian in real space.

[0126] The second Fourier transform module 40 is used to perform a one-dimensional Fourier transform on the tight-binding Hamiltonian in the real space using the mixed boundary condition to obtain a one-dimensional effective Hamiltonian.

[0127] The judgment module 50 is used to judge whether the number of gapless edge states is equal to the absolute value of the conduction band Chern number; wherein the gapless edge state is the gapless edge state of the image corresponding to the eigenvalue of the one-dimensional effective Hamiltonian.

[0128] The third determination module 60 is used to determine the number of gapless edge states as the Chern number of the quantum anomalous Hall insulator when the judgment module determines that the number of gapless edge states is equal to the absolute value of the conduction band Chern number.

[0129] The fourth determination module 70 is used to determine the Hamiltonian of the reconstructed two-band model and the expression of its x-component and y-component about the two-dimensional momentum when the judgment module determines that the number of gapless edge states is not equal to the absolute value of the conduction band Chern number, and execute the operation of the first determination module.

[0130] Exemplarily, the first determining module includes:

[0131] The first building block is used to construct the x component h of the Hamiltonian of the two-band model x and the y component h y About the two-dimensional momentum (k x ,k y ) expression.

[0132] The first determining unit is used to satisfy h x = 0 and h y =0(k x ,k y ) as the coordinates of the topological defects in the two-band model.

[0133] Acquisition unit, used to obtain a two-dimensional vector (h x ,h y); when all arrows in the vector distribution map are away from the coordinate point, the topological charge corresponding to the coordinate point is +1; when all arrows in the vector distribution map converge to the coordinate point, the topological charge corresponding to the coordinate point is +1; when the arrows in the vector distribution map are away from the coordinate point in one direction and converge to the coordinate point in another direction, the topological charge corresponding to the coordinate point is -1; when the arrows in the vector distribution map surround the coordinate point, the topological charge corresponding to the coordinate point is -1.

[0134] Exemplarily, the second determining module includes:

[0135] The second building block is used to construct the z component h of the Hamiltonian of the two-band model z expression.

[0136] The first calculation unit is used to calculate the z When it is greater than 0, the Chern number C of the conduction band is calculated according to the following formula N :

[0137]

[0138] Among them, L N is the total number of defects on the conduction band; W l is the topological charge of the lth defect.

[0139] The second computing unit is used to calculate the z When it is less than 0, the Chern number C of the conduction band is calculated according to the following formula S :

[0140]

[0141] Among them, L S is the total number of defects on the valence band.

[0142] For more specific working processes of the above modules, please refer to the corresponding contents disclosed in Example 1, which will not be repeated here.

[0143] Example 3

[0144] This embodiment provides a computer device, including a processor and a memory; wherein, when the processor executes a computer program stored in the memory, the steps of the method for determining the Chern number of a quantum anomalous Hall insulator based on topological defects described in Example 1 are implemented.

[0145] For more specific details of the above method, please refer to the corresponding contents disclosed in Example 1, which will not be repeated here.

[0146] Example 4

[0147] This embodiment provides a computer-readable storage medium for storing a computer program; when the computer program is executed by a processor, the steps of the method for determining the Chern number of a quantum anomalous Hall insulator based on topological defects described in Example 1 are implemented.

[0148] For more specific details of the above method, please refer to the corresponding contents disclosed in Example 1, which will not be repeated here.

[0149] Example 5

[0150] This embodiment provides a computer program product, including computer executable instructions or a computer program. When the computer executable instructions or the computer program are executed by a processor, the steps of the method for determining the Chern number of a quantum anomalous Hall insulator based on topological defects described in Example 1 are implemented.

[0151] For more specific details of the above method, please refer to the corresponding contents disclosed in Example 1, which will not be repeated here.

[0152] In this specification, each embodiment is described in a progressive manner, and each embodiment focuses on the differences from other embodiments. The same or similar parts between the embodiments can be referred to each other. For the systems, devices, storage media, and computer program products disclosed in the embodiments, since they correspond to the methods disclosed in the embodiments, the description is relatively simple, and the relevant parts can be referred to the method part description.

[0153] Those skilled in the art can clearly understand that the technology in the embodiments of the present invention can be implemented by means of software plus a necessary general hardware platform. Based on this understanding, the technical solution in the embodiments of the present invention is essentially or the part that contributes to the prior art can be embodied in the form of a software product, which can be stored in a storage medium such as ROM / RAM, a disk, an optical disk, etc., and includes a number of instructions for a computer device (which can be a personal computer, a server, or a network device, etc.) to execute the methods described in each embodiment of the present invention or some parts of the embodiments.

[0154] In some embodiments, computer executable instructions may be in the form of a program, software, software module, script or code, written in any form of programming language (including compiled or interpreted languages, or declarative or procedural languages), and may be deployed in any form, including as a stand-alone program or as a module, component, subroutine or other unit suitable for use in a computing environment.

[0155] As an example, computer-executable instructions may, but need not, correspond to a file in a file system, may be stored as part of a file that stores other programs or data, such as in one or more scripts in a HyperText Markup Language (HTML) document, in a single file dedicated to the program in question, or in multiple coordinated files (e.g., files storing one or more modules, subroutines, or code portions).

[0156] As an example, computer executable instructions may be deployed to be executed on one electronic device, or on multiple electronic devices located at one site, or on multiple electronic devices distributed at multiple sites and interconnected by a communication network.

[0157] The present invention has been described in detail above in conjunction with specific implementations and exemplary examples, but these descriptions cannot be understood as limiting the present invention. Those skilled in the art understand that, without departing from the spirit and scope of the present invention, a variety of equivalent substitutions, modifications or improvements may be made to the technical solution of the present invention and its implementation methods, all of which fall within the scope of the present invention. The scope of protection of the present invention shall be subject to the attached claims.

Claims

1. A method for determining the Chern number of a quantum anomalous Hall insulator based on topological defects, characterized in that: include: S1, determine the topological charge of each topological defect in the two-band model; S2, determining the contribution of the corresponding topological charge to the Chern number according to the value of the z component of the Hamiltonian at each topological defect, so as to determine the Chern number of the valence band or conduction band; S3, according to the type of two-dimensional lattice structure, perform Fourier transform on the Hamiltonian to obtain the tight-binding Hamiltonian in real space; S4, using the mixed boundary conditions to perform a one-dimensional Fourier transform on the tight-binding Hamiltonian in real space to obtain a one-dimensional effective Hamiltonian; S5, determining whether the number of gapless edge states is equal to the absolute value of the conduction band Chern number; wherein the gapless edge state is the gapless edge state of the image corresponding to the eigenvalue of the one-dimensional effective Hamiltonian; S6, if equal to, the number of gapless edge states is taken as the Chern number of the quantum anomalous Hall insulator; S7, if not equal, reconstructs the Hamiltonian of the two-band model and the expressions of its x-component and y-component with respect to the two-dimensional momentum, and returns to execute the operation of S1.

2. The method for determining the Chern number of a quantum anomalous Hall insulator based on topological defects according to claim 1, characterized in that: The determining of the topological charge of each topological defect of the two-band model comprises: Construct the x-component h of the Hamiltonian for the two-band model x and the y component h y About the two-dimensional momentum (k x , k y ) will satisfy h x = 0 and h y =0(k x ,k y ) as the coordinates of the topological defects in the two-band model; Get the two-dimensional vector (h x ,h y ); when all arrows in the vector distribution map are away from the coordinate point, the topological charge corresponding to the coordinate point is +1; when all arrows in the vector distribution map converge to the coordinate point, the topological charge corresponding to the coordinate point is +1; when the arrows in the vector distribution map are away from the coordinate point in one direction and converge to the coordinate point in another direction, the topological charge corresponding to the coordinate point is -1; when the arrows in the vector distribution map surround the coordinate point, the topological charge corresponding to the coordinate point is -1.

3. The method for determining the Chern number of a quantum anomalous Hall insulator based on topological defects according to claim 1, characterized in that: The step of determining the contribution of the corresponding topological charge to the Chern number according to the value of the Hamiltonian z component at each topological defect to determine the Chern number of the valence band or the conduction band includes: Construct the z component h of the Hamiltonian of the two-band model z Expression of In h z When it is greater than 0, the Chern number C of the conduction band is calculated according to the following formula N : Among them, L N is the total number of defects on the conduction band; W l is the topological charge of the lth defect; In h z When it is less than 0, the Chern number C of the conduction band is calculated according to the following formula S : Among them, L S is the total number of defects on the valence band.

4. A Chern number determination system for quantum anomalous Hall insulators based on topological defects, characterized in that: include: A first determination module is used to determine the topological charge of each topological defect of the two-band model; A second determination module is used to determine the contribution of the corresponding topological charge to the Chern number according to the value of the Hamiltonian z component at each topological defect, so as to determine the Chern number of the valence band or the conduction band; A first Fourier transform module is used to perform Fourier transform on the Hamiltonian according to the type of the two-dimensional lattice structure to obtain a tight-binding Hamiltonian in real space; The second Fourier transform module is used to perform a one-dimensional Fourier transform on the tight-binding Hamiltonian in the real space using the mixed boundary conditions to obtain a one-dimensional effective Hamiltonian; A judgment module is used to judge whether the number of gapless edge states is equal to the absolute value of the conduction band Chern number; wherein the gapless edge state is the gapless edge state of the image corresponding to the eigenvalue of the one-dimensional effective Hamiltonian; A third determination module is used to determine the number of gapless edge states as the Chern number of the quantum anomalous Hall insulator when the judgment module determines that the number of gapless edge states is equal to the absolute value of the conduction band Chern number; The fourth determination module is used to determine the Hamiltonian of the reconstructed two-band model and the expression of its x-component and y-component about the two-dimensional momentum when the judgment module determines that the number of gapless edge states is not equal to the absolute value of the conduction band Chern number, and execute the operation of the first determination module.

5. The Chern number determination system of a quantum anomalous Hall insulator based on topological defects according to claim 4, characterized in that: The first determining module comprises: The first construction unit is used to construct the x-component and y-component of the Hamiltonian of the two-band model with respect to the two-dimensional momentum (k x ,k y ) The first determining unit is used to satisfy h x = 0 and h y =0(k x ,k y ) as the coordinates of the topological defects in the two-band model; Acquisition unit, used to obtain a two-dimensional vector (h x ,h y ); when all arrows in the vector distribution map are away from the coordinate point, the topological charge corresponding to the coordinate point is +1; when all arrows in the vector distribution map converge to the coordinate point, the topological charge corresponding to the coordinate point is +1; when the arrows in the vector distribution map are away from the coordinate point in one direction and converge to the coordinate point in another direction, the topological charge corresponding to the coordinate point is -1; when the arrows in the vector distribution map surround the coordinate point, the topological charge corresponding to the coordinate point is -1.

6. The Chern number determination system of a quantum anomalous Hall insulator based on topological defects according to claim 4, characterized in that: The second determining module comprises: The second building block is used to construct the z component h of the Hamiltonian of the two-band model z Expression of The first calculation unit is used to calculate the z When it is greater than 0, the Chern number C of the conduction band is calculated according to the following formula N : Among them, L N is the total number of defects on the conduction band; W l is the topological charge of the lth defect; The second computing unit is used to calculate the z When it is less than 0, the Chern number C of the conduction band is calculated according to the following formula S : Among them, L S is the total number of defects on the valence band.

7. A computer device, characterized in that: It comprises a processor and a memory; wherein, when the processor executes the computer program stored in the memory, the steps of the method for determining the Chern number of a quantum anomalous Hall insulator based on topological defects described in any one of claims 1 to 3 are implemented.

8. A computer-readable storage medium, characterized in that: Used to store computer programs; when the computer programs are executed by a processor, the steps of the method for determining the Chern number of a quantum anomalous Hall insulator based on topological defects described in any one of claims 1 to 3 are implemented.

9. A computer program product, characterized in that The method comprises computer executable instructions or a computer program, which, when executed by a processor, implements the steps of the method for determining the Chern number of a quantum anomalous Hall insulator based on topological defects as described in any one of claims 1 to 3.

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