Topological phase discrimination method based on body region measurement

Abstract

The invention provides a topological phase discrimination method based on body region measurement. The method comprises the following steps: step 1, measuring an S21 parameter in a body region of a crystal with spatial symmetry, the body region being a region except a boundary primitive cell; step 2, obtaining field distribution with a phase according to the S21 parameter, performing Fourier transform on the field distribution, and then performing eigenstate projection on the field distribution after Fourier transform to obtain energy band structure information; and step 3, according to the energy band structure information, utilizing a symmetry index theory to make specific discrimination on the topological phase of the crystal. According to the method, the spatial symmetry of a topological structure is utilized, based on body region measurement, topological phase discrimination independent of boundary measurement is achieved, the limitation that traditional topological phase discrimination has requirements for a full band gap and an interface state in the band gap is broken through, and the discrimination has immune characteristics for a defect state and can be suitable for wider topological phase discrimination scenes.

Description

Technical Field

[0001] The present invention belongs to the field of topological materials, and in particular relates to a topological phase discrimination method based on body region measurement. Background Art

[0002] Since the quantum spin Hall effect was first theoretically predicted, the study of topological materials has experienced significant development. Such materials exhibit stable boundary states and quantized bulk response values, which provide new possibilities for coherent spin transport, storage and manipulation of quantum information, and precise control of light emission. It is also crucial to classify and diagnose these topological phases, which can not only provide guidance for the design of topological devices, but also promote the exploration of new topological phases.

[0003] There are already several effective diagnostic methods for topological phases with complete band gaps or interface states in the band gap. In topological systems with edge states in the band gap, the topological invariants used to characterize the bulk topological properties can be directly corresponded to the number of interface states in the band gap, which is the famous bulk-edge correspondence. However, in some topological systems, there is a lack of edge states in the band gap, so it becomes difficult to judge the topological properties by counting edge states, and the bulk-edge correspondence fails in this case. This band structure is widely present in a special topological material called topological crystalline material, whose topological invariants are protected by spatial symmetry. However, spatial symmetry itself cannot limit the frequency position of interface states, and some local symmetries (such as chiral symmetry) can fix the interface state mode at zero energy in the band gap, but they are naturally broken in some materials (such as photonic crystals), which means that localized edge modes may appear in the bulk band.

[0004] There are other existing topological diagnostic schemes, such as fractional quantum number anomaly (FCA) and reflectance spectroscopy, which do not rely on the presence of interface states within the band gap but strictly require the system to have a complete band gap. However, directional band gaps are common in both natural and artificial topological crystalline materials, making the above methods ineffective in distinguishing these materials from ordinary materials. Summary of the invention

[0005] The present invention provides a topological phase discrimination method based on bulk region measurement, which aims to solve the problem of discriminating topological phases lacking full band gaps and interface states in the band gaps.

[0006] The technical solution adopted by the present invention is as follows:

[0007] A topological phase discrimination method based on volume region measurement comprises the following steps:

[0008] Step 1: Measure S in the bulk region of a crystal with spatial symmetry. 21Parameters, wherein the body region is the region excluding the boundary primitive cells;

[0009] Step 2, by the S 21 Parameters are used to obtain a field distribution with a phase, Fourier transform the field distribution, and then eigenstate projection is performed on the field distribution after the Fourier transform to obtain band structure information;

[0010] Step 3: Based on the energy band structure information, the topological phase of the crystal is specifically judged using the symmetry index theory.

[0011] Furthermore, in step 1, the measuring device includes a microwave vector network analyzer, a microwave transmission line and a microwave probe; there are two microwave transmission lines, one end of each of which is connected to a microwave probe, and the other end is respectively connected to the output end and the receiving end of the vector network analyzer, wherein the transmitting microwave probe connected to the output end passes through the crystal and is fixed as a transmitting source; the receiving microwave probe connected to the receiving end moves horizontally above the crystal as a detection probe for measuring S 21 parameter.

[0012] Furthermore, in step 3, the corresponding topological invariant is obtained by using the symmetry index theory; the value of the topological invariant is calculated according to the band structure information, and then the topological phase of the crystal is specifically judged according to the value of the topological invariant. When there are multiple topological invariants, the topological phases with the same value of each topological invariant belong to the same category.

[0013] Furthermore, in step 2, obtaining the band structure information specifically includes: s - filter multiplication, the k s -The filter is used to extract the field distribution at a certain wave vector k; then the filtered field distribution is summed, and the summation rule is: the atoms in a single primitive cell are not added, and the atoms corresponding to all primitive cells are added, and the specific wave vector, i.e. k, is obtained after the summation. s The field distribution at k is then decomposed by the eigenstate basis vector, that is, the eigenstate is projected to obtain s Repeat the above operation at high symmetry points for the field distribution at each frequency, and finally mark the mode with the highest modal ratio on the band diagram to obtain the band structure information.

[0014] Furthermore, in step 2, Fourier transform of the field distribution follows the following formula:

[0015]

[0016] Among them, C j,m,l (f) is the S with frequency f measured on the jth unit in the primitive cell with coordinates (m, l)21 parameter; is the Fourier transform factor, k x and k y Respectively represent the two components of the wave vector k; c n,j (k)δ(ff n (k)) represents the field of the state with frequency f and wave vector k in the nth energy band on the jth unit in the primitive cell.

[0017] Furthermore, in step 2, performing eigenstate projection on the field distribution after Fourier transformation specifically includes:

[0018] Step 21, select a set of orthogonal complete eigenstate basis vectors:

[0019] Step 22, c n,j The field distribution with frequency f and wave vector k represented by (k) is denoted as ψ(f,k) and decomposed again into the linear combination of the basis vectors in step 21:

[0020]

[0021] The coefficients are obtained by the following formula:

[0022]

[0023] Step 23, calculate the field distribution ψ(f,k) with frequency f and wave vector k Strength of state Obtain information on whether there is an energy band at a frequency of f and a wave vector of k, and the corresponding mode of the energy band;

[0024] Step 24, based on the information obtained in step 23, using the symmetry index theory, the topological invariant is calculated from the band structure of the high symmetry point to achieve topological discrimination.

[0025] The present invention utilizes the spatial symmetry of the topological structure and realizes topological phase discrimination based on bulk region measurement without relying on boundary measurement, breaking through the limitation of the traditional topological phase discrimination method that requires full band gap and mid-band gap interface states. Compared with the prior art, the present invention has the following significant advantages: (1) It can adapt to a wider range of topological phase discrimination scenarios, including the discrimination of materials lacking full band gap and mid-band gap interface states; (2) The discrimination is based on a large area of ​​bulk region and has the characteristics of immunity to defect states. BRIEF DESCRIPTION OF THE DRAWINGS

[0026] Figure 1 It is a schematic diagram of a device for implementing the topological phase discrimination method based on body region measurement of the present invention;

[0027] Figure 2It is a schematic diagram of the principle of the topological phase discrimination method based on body region measurement of the present invention;

[0028] Figure 3 It is the experimental discrimination result of topological and mediocre samples using the method of the present invention. DETAILED DESCRIPTION

[0029] The technical solution of the present invention is further described below in conjunction with the accompanying drawings.

[0030] The present invention provides a topological phase discrimination method based on volume region measurement, and its implementation device is as follows: Figure 1 As shown, it includes a microwave vector network analyzer, a microwave transmission line and a microwave probe; there are two microwave transmission lines, one end of each of which is connected to a microwave probe, and the other end is connected to the output end and the receiving end of the vector network analyzer. The transmitting microwave probe connected to the output end passes through the sample from the bottom and is fixed. As a transmitting source, it can be located in the corner of the sample; the receiving microwave probe connected to the receiving end moves horizontally above the sample and maintains a constant distance from the sample surface. It is used as a detection probe to measure S in the entire body area of ​​the sample. 21 Parameters. Among them, the operating frequency range of the microwave vector network analyzer, microwave transmission line and microwave probe covers the eigenstate energy range of the sample, which is 2GHz-18GHz in this embodiment. The sample measured in this embodiment is a topological or mediocre photonic crystal slab, which is composed of a ceramic column array and has spatial translational symmetry and four-fold rotational symmetry. The body region is defined as the region excluding the boundary primitive cell, and the primitive cell must be complete and cannot be cut in half.

[0031] By measuring S relative to a fixed emission source above each ceramic column unit of the photonic crystal slab, 21 The parameters are used to obtain the field distribution information with phase, and then a series of processing is performed on the obtained field distribution, including Fourier transform and eigenstate projection, to obtain the band structure information of the sample. Furthermore, the symmetry index theory is used to make a specific judgment on the topological phase of the sample. To ensure the accuracy of the measurement, during the measurement process, when the microwave probe moves in the sample body area, it is necessary to perform S 21 Measurement of parameters.

[0032] Based on the above topological phase discrimination method, this embodiment specifically includes the following steps:

[0033] (1) Fix the transmitting microwave probe and use the receiving microwave probe to scan the sample area. 21 The parameters obtained include the field distribution of intensity and phase. The schematic diagram of the field distribution is shown in Figure 2 The first column from the left.

[0034] (2) Perform Fourier transform on the obtained field distribution, the formula is as follows:

[0035]

[0036] Among them, C j,m,l (f) is the S with frequency f measured experimentally on the jth column in the primitive cell with coordinates (m, l) 21 Parameters, i.e. Figure 2 Each small square in the first column from the middle left, "Eigenstate Ψ in the measured volume region"; is the Fourier transform factor,

[0037] k x and k y Represent the two components of the wave vector k, respectively. Figure 2 It is reflected as the second column k from the left s -Filter, which is connected to C j,m,l The product of (f) is Figure 2 It is reflected in the third column "k s - Filtered field distribution Ψ s ”, where k s Also the Fourier transform factor, k s - The filter is used to extract the field distribution at a certain wave vector k; c n,j (k)δ)ff n (k)) represents the field of the state with frequency f and wave vector k in the nth energy band on the jth column in the primitive cell. After summing the energy band number n, we get Figure 2 In the fifth column, the summation rule is: atoms in a single primitive cell are not added, and atoms corresponding to all primitive cells are added. The Fourier transform is only performed on high symmetry points (HSP), which are X, M, and Γ points in the embodiment, that is, Figure 2 Medium k s -The filter subscript s takes values ​​of X, M, Γ.

[0038] (3) Performing eigenstate projection operation on the field distribution after Fourier transformation. In the embodiment, an orthogonal and complete eigenstate basis vector is selected: {|s>, |p 1 >,|p 2 >,|d>};The field distribution after Fourier transformation ∑ n c n,j (k)δ(ff n (k)) is simply denoted as ψ(f,k), and the field distribution can be decomposed into a linear combination of basis vectors: Among them, C s , C d is the coefficient of the eigenstate basis vector, For the eigenstate basis vector; find the field distribution ψ(f,k) with frequency f and wave vector k The intensity of the state is like Figure 2 As shown in the fifth column from the left, "Modal Ratio", according to the modal ratio, if the intensity D of a certain mode i The value is the highest, indicating that this mode is dominant at the frequency f and the wave vector k. The field distribution at each frequency measured is Fourier transformed and eigenstate projected at high symmetry points X, M, and Γ. Finally, the mode with the highest modal ratio is marked on the band diagram to obtain the sixth column of band structures.

[0039] Based on the above measurement processing method, the experimental results of this embodiment are as follows Figure 3 shown. Figure 3 (b)-(d) and (f)-(h) show the intensity distributions of s, p, and d states of topological samples and ordinary samples at HSP, respectively. s , D d Each peak in the figure indicates that there is an energy band distribution at the frequency of the high symmetry point, and the mode is the mode represented by the corresponding color, that is, s, p or d state.

[0040] In this embodiment, the symmetry-indicator invariants theory (see Physical Review B 108, 085116 (2023) pages 6-7) is used, which is briefly summarized as follows:

[0041] (1) Denote the high symmetry point (HSP) as Π (q) , which satisfies the relationship: R q Π (q) =Π (q) (modek 0 ), where modek 0 Represents the modulus of the reciprocal lattice vector, R q is the q-rerotation operator acting on the crystal momentum, which means that in Π (q) The q-fold rotation operator commutes with the Bloch Hamiltonian. Therefore, the energy eigenstates of the Bloch Hamiltonian at HSP are also rotation operators. eigenstate of .

[0042] (2) is expressed by the following formula In HSPΠ (q) Rotational eigenvalues ​​at :

[0043]

[0044] Given a band subspace, we can compare how these rotational eigenvalues ​​change at different HSPs. If the eigenvalues ​​differ at different HSPs, this indicates that the bands have nontrivial topological properties.

[0045] (3) Based on this, the integer topological invariant is defined by comparing the rotation eigenvalues ​​of the Π point and the reference point Γ = (0,0):

[0046]

[0047] in It means that it has eigenvalue The number of bands below the energy gap, Π can be taken as X, M, It means that it has eigenvalue The number of energy bands below the energy gap.

[0048] This method can be applied to crystals of various symmetries. The photonic crystal in this embodiment has time reversal symmetry and C 4 Rotational symmetry, see Symmetry index theory, topological invariant polarization P (4) and angular charge It can be expressed as:

[0049]

[0050] where a x and a y is the basis vector of the primitive cell, X and M are the high symmetric points with coordinates (π, 0) and (π, π) in the reciprocal space respectively, and the meanings of the upper and lower scripts are the same as above. q represents q-fold rotational symmetry, p represents the pth eigenvalue of the rotational symmetry, and mod 1 represents the calculation result modulo 1. The s, p, and d states can all be recalibrated as rotation operator eigenvalues. Figure 3 The eigenvalues ​​of the 4-fold or 2-fold rotational symmetry operators corresponding to each state are marked in the formula. Substituting into the formula, we can get: For topological samples, This indicates that the sample has a first-order topological boundary state. This indicates that the sample has a high-order topological corner state, while the two values ​​of the mediocre sample are both 0, with only bulk states, no boundary states and corner states. Therefore, different topological phases can be classified according to the calculated values ​​of the topological invariants. It should be noted that the topological classification here is not necessarily limited to two types (topological or mediocre). This embodiment is only an example, and it can also be divided into more detailed categories. For example, for some structures, the topological invariant polarization P (4) =1 / 2, angular charge Structures that are identical for each topological invariant belong to the same category.

[0051] Therefore, the present invention utilizes the spatial symmetry of the topological structure and realizes topological phase discrimination independent of boundary measurement based on bulk region measurement, breaking through the limitation that traditional topological phase discrimination requires full band gap and interface states in the band gap.

Claims

1. A topological phase discrimination method based on volume region measurement, characterized in that: The method comprises the following steps: Step 1: Measure S in the bulk region of a crystal with spatial symmetry. 21 Parameters, wherein the body region is the region excluding the boundary primitive cells; Step 2, by the S 21 Parameters are used to obtain a field distribution with a phase, Fourier transform the field distribution, and then eigenstate projection is performed on the field distribution after the Fourier transform to obtain band structure information; Step 3: Based on the energy band structure information, the topological phase of the crystal is specifically judged using the symmetry index theory.

2. A topological phase discrimination method based on volume region measurement according to claim 1, characterized in that: In step 1, the measuring device includes a microwave vector network analyzer, a microwave transmission line and a microwave probe; there are two microwave transmission lines, one end of each of which is connected to a microwave probe, and the other end is respectively connected to the output end and the receiving end of the vector network analyzer, wherein the transmitting microwave probe connected to the output end passes through the crystal and is fixed as a transmitting source; the receiving microwave probe connected to the receiving end moves horizontally above the crystal as a detection probe for measuring S 21 parameter.

3. A topological phase discrimination method based on volume region measurement according to claim 2, characterized in that: In the step 1, the operating frequency range of the microwave vector network analyzer, microwave transmission line and microwave probe covers the energy range of the crystal eigenstate.

4. The method for topological phase discrimination based on volume region measurement according to claim 2, characterized in that: The detection probe maintains a constant distance from the crystal surface while moving within the bulk region.

5. The method for topological phase discrimination based on volume region measurement according to claim 1, characterized in that: In the step 3, the corresponding topological invariant is obtained by using the symmetry index theory; the value of the topological invariant is calculated according to the band structure information, and then the topological phase of the crystal is specifically judged according to the value of the topological invariant.

6. A topological phase discrimination method based on volume region measurement according to claim 5, characterized in that: In step 3, when there are multiple topological invariants, topological phases with the same value of each topological invariant belong to the same category.

7. The method for topological phase discrimination based on volume region measurement according to claim 1, characterized in that: In step 2, obtaining the band structure information specifically includes: s - filter multiplication, the k s -The filter is used to extract the field distribution at a certain wave vector k; then the filtered field distribution is summed to obtain a specific wave vector, namely k s The field distribution at k is then decomposed by the eigenstate basis vector, that is, the eigenstate is projected to obtain s The modal ratio of the field distribution at the frequency; repeat the above operation at high symmetry points for the field distribution at each frequency, and finally mark the mode with the highest modal ratio on the band diagram to obtain the band structure information.

8. A topological phase discrimination method based on volume region measurement according to claim 7, characterized in that: In step 2, the summation rule adopted for summing the filtered field distribution is: atoms in a single primitive cell are not added together, but atoms corresponding to all primitive cells are added together.

9. The method for topological phase discrimination based on volume region measurement according to claim 7, characterized in that: In step 2, Fourier transform of the field distribution follows the following formula: Among them, C j,m,l (f) is the S with frequency f measured on the jth unit in the primitive cell with coordinates (m, l) 21 parameter; is the Fourier transform factor, k x and k y Respectively represent the two components of the wave vector k; c n,j (k)δ(ff n (k)) represents the field of the state with frequency f and wave vector k in the nth energy band on the jth unit in the primitive cell.

10. A topological phase discrimination method based on volume region measurement according to claim 9, characterized in that: In step 2, performing eigenstate projection on the field distribution after Fourier transformation specifically includes: Step 21, select a set of orthogonal complete eigenstate basis vectors: Step 22, c n,j The field distribution with frequency f and wave vector k represented by (k) is denoted as ψ(f,k) and decomposed again into the linear combination of the basis vectors in step 21: The coefficients are obtained by the following formula: Step 23, calculate the field distribution ψ(f,k) with frequency f and wave vector k Strength of state Obtain information on whether there is an energy band at a frequency of f and a wave vector of k, and the corresponding mode of the energy band; Step 24, based on the information obtained in step 23, using the symmetry index theory, the topological invariant is calculated from the band structure of the high symmetry point to achieve topological discrimination.

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