A topological phase discrimination method based on body region measurement

Abstract

The application provides a topological phase discrimination method based on bulk region measurement. The method comprises the following steps: step 1, measuring S 21 parameters in a bulk region of a crystal with spatial symmetry, wherein the bulk region is a region except a boundary primitive cell; step 2, obtaining a field distribution with a phase from the S 21 parameters, performing Fourier transform on the field distribution, and then performing intrinsic state projection on the Fourier-transformed field distribution to obtain band structure information; and step 3, according to the band structure information, using a symmetry index theory to specifically discriminate the topological phase of the crystal. The method of the application uses the spatial symmetry of the topological structure, is based on bulk region measurement, realizes topological phase discrimination independent of boundary measurement, breaks through the limitation that traditional topological phase discrimination requires a full band gap and an interface state in a band gap, and has an immune feature to a defect state, and can adapt to a wider topological phase discrimination scene.

Description

TECHNICAL FIELD

[0001] The application belongs to the field of topological materials, and particularly relates to a topological phase discrimination method based on bulk region measurement. BACKGROUND

[0002] Since the quantum spin Hall effect was first theoretically predicted, the research on topological materials has undergone significant development. Such materials exhibit stable boundary states and quantized bulk response values, providing new possibilities for realizing coherent spin transport, quantum information storage and manipulation, and precise control of light emission. Classifying and diagnosing these topological phases is also crucial, as it not only provides guidance for designing topological devices but also drives the exploration process of new topological phases.

[0003] For topological phases with complete band gaps or interface states in band gaps, there are already a variety of effective diagnostic methods. In topological systems with boundary states in the band gap, the topological invariants used to characterize the bulk topological properties can be directly mapped to the number of interface states within the band gap, which is the famous bulk-boundary correspondence. However, in some topological systems, there are no boundary states within the band gap, making it difficult to determine the topological properties by counting boundary states, and the bulk-boundary correspondence is invalid in this case. Such band structures are widely found in a special type of topological material called topological crystalline materials, whose topological invariants are protected by spatial symmetry. However, spatial symmetry alone cannot restrict the frequency position of interface states, and certain local symmetries (such as chiral symmetry) can fix interface state modes at zero energy within the band gap, but are naturally broken in some materials (such as photonic crystals), meaning that localized boundary modes can appear in the bulk band.

[0004] There are also other existing topological diagnostic methods, such as the fractional quantum number anomaly (FCA) and the reflection spectrum method, 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 it impossible for the above methods to effectively distinguish these materials from ordinary materials. SUMMARY

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

[0006] The technical scheme adopted by the application is as follows:

[0007] A topological phase discrimination method based on bulk region measurement, comprising the following steps:

[0008] Step 1, measuring S 21Parameters, wherein the volume region is the region excluding the boundary unit cells;

[0009] Step 2, by the S 21 The parameters are used to obtain the phase field distribution, the field distribution is subjected to Fourier transform, and then the eigenstate projection is performed on the Fourier transform field distribution to obtain the band structure information.

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

[0011] Further, in step 1, the measurement apparatus includes a microwave vector network analyzer, microwave transmission lines, and microwave probes; there are two microwave transmission lines, each with one end connected to a microwave probe, and the other end connected to the output and receiving ends of the vector network analyzer, respectively. The transmitting microwave probe connected to the output end passes through the crystal and is fixed, serving as the emission source; the receiving microwave probe connected to the receiving end moves horizontally above the crystal, serving as a detection probe for measuring S. 21 parameter.

[0012] Furthermore, in step 3, the corresponding topological invariants are obtained using symmetry index theory; the values ​​of the topological invariants are calculated based on the band structure information, and then the topological phase of the crystal is specifically determined based on the values ​​of the topological invariants. When multiple topological invariants exist, topological phases with the same value for each topological invariant belong to the same category.

[0013] Furthermore, in step 2, obtaining the band structure information specifically includes: comparing the field distribution with k... s -Filter multiplication, the k s - The filter is used to extract the field distribution at a specific wave vector k; then the filtered field distributions are summed according to the following rules: atoms within a single unit cell are not added, but atoms corresponding to all units of the unit cell are added together. The summation yields the specific wave vector, k. s The field distribution at a given location is then decomposed using eigenstate basis vectors, i.e., projected onto the eigenstates, to obtain k. s The modal proportions of the field distribution at each frequency are determined. The above operation is repeated at high symmetry points for the field distribution at each frequency. Finally, the mode with the highest modal proportion is marked on the band diagram to obtain the band structure information.

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

[0015]

[0016] Among them, C j,m,l (f) is the frequency f measured on the j-th unit in the primitive cell with coordinates (m,l).21 parameter; It is the Fourier transform factor, k x and k y These 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 of the unit cell.

[0017] Furthermore, step 2, specifically the eigenstate projection of the Fourier-transformed field distribution, includes:

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

[0019] Step 22, will be done by c n,j (k) represents the field distribution with frequency f and wave vector k, denoted as ψ(f,k), which is decomposed again into a 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 To obtain information on whether there is an energy band at frequency f and wave vector k, and the corresponding mode if the energy band exists;

[0024] Step 24: Based on the information obtained in Step 23, the topological invariants are calculated from the high-symmetry point band structure using the symmetry index theory to achieve topological discrimination.

[0025] This invention utilizes the spatial symmetry of topological structures and, based on volume region measurements, achieves topological phase discrimination independent of boundary measurements, overcoming the limitations of traditional topological phase discrimination methods that require full bandgap and interface states within the bandgap. Compared with existing technologies, this 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 bandgap and interface states within the bandgap; (2) This discrimination is based on a large volume region and has an immune characteristic against defect states. Attached Figure Description

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

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

[0028] Figure 3 The results are experimental discrimination results of topological and mediocre samples using the method of this invention. Detailed Implementation

[0029] The technical solution of the present invention will be further described below with reference to the accompanying drawings.

[0030] This invention provides a topological phase discrimination method based on volume region measurement, the implementation device of which is as follows: Figure 1 As shown, the system includes a microwave vector network analyzer, microwave transmission lines, and microwave probes. There are two microwave transmission lines, each with one end connected to a microwave probe and the other end connected to the output and receiving ends of the vector network analyzer, respectively. The transmitting microwave probe connected to the output end passes through the sample from the bottom and is fixed, serving as the emission source; it can be located at a 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, serving as a detection probe used to measure the S-values ​​across the entire sample area. 21 Parameters. The operating frequency range of the microwave vector network analyzer, microwave transmission line, and microwave probe covers the eigenstate energy range of the sample; in this embodiment, it is 2GHz-18GHz. The sample measured in this embodiment is a topologically or trivial photonic crystal plate composed of a ceramic pillar array, exhibiting spatial translational symmetry and fourfold rotational symmetry. The bulk region is defined as the region excluding the boundary unit cells; the unit cells must be intact and cannot be cut in half.

[0031] The S-wave velocity relative to a fixed emission source was measured above each ceramic pillar unit of the photonic crystal plate. 21 The parameters are used to obtain the phase-dependent field distribution information, which is then processed through a series of steps, including Fourier transform and eigenstate projection, to acquire the band structure information of the sample. Furthermore, symmetry index theory is used to specifically determine the topological phase of the sample. To ensure measurement accuracy, during the measurement process, the microwave probe must be stopped after it has stabilized within the sample volume region. 21 Parameter measurement.

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

[0033] (1) Fix the emitting microwave probe and use the receiving microwave probe to scan the sample area. The measured S 21 The parameters yield the field distribution, including intensity and phase. A schematic diagram of the field distribution can be found in [reference needed]. Figure 2 The first column from the left.

[0034] (2) Perform a Fourier transform on the obtained field distribution, as shown in the following formula:

[0035]

[0036] Among them, C j,m,l (f) is the frequency f of the S-value measured experimentally on the j-th pillar in the primitive cell with coordinates (m,l). 21 Parameters, i.e. Figure 2 Each small square in the first column from the left, labeled "Eigenstates Ψ within the measured volume region"; It is the Fourier transform factor.

[0037] k x and k y Let these represent the two components of the wave vector k, respectively. Figure 2 The middle is represented by the second column from the left, k. s - Filter, which is related to C j,m,l The product of (f) in Figure 2 This is represented by the third column "k" s -Filtered field distribution Ψ s ", where k s Also a Fourier transform factor, k s - The filter is used to extract the field distribution at a specific 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 j-th pillar in the unit cell. The summation over the energy band numbers n yields... Figure 2 In the fifth column, the summation rule is: atoms within a single unit cell are not added, but atoms corresponding to all unit cells are added. This Fourier transform is performed only at high symmetry points (HSPs), which in this example are points X, M, and Γ, i.e. Figure 2 Chinese K s - The filter subscript s takes values ​​of X, M, and Γ.

[0038] (3) Perform eigenstate projection on the field distribution after Fourier transform. In this embodiment, orthogonal and complete eigenstate basis vectors are selected: {|s>, |p1>, |p2>, |d>}; the field distribution after Fourier transform is ∑ n c n,j (k)δ(ff n (k) can be simply denoted as ψ(f,k), and this field distribution can be decomposed again into a linear combination of basis vectors: Among them, C s , C d Let be the coefficients of the eigenstate basis vectors. For the eigenstate basis vectors, find the field distribution ψ(f,k) with frequency f and wave vector k. The strength of the state is like Figure 2 The fifth column from the left, "Modal Ratio," shows that, according to the modal ratio, if the intensity D of a certain mode... i The highest value indicates that this mode dominates at frequency f and wave vector k. Fourier transforms and eigenstate projections are performed on the field distribution at each measured frequency at high-symmetry points X, M, and Γ. Finally, the mode with the highest modal proportion is labeled on the band structure, resulting in the sixth band structure.

[0039] Based on the above measurement and processing methods, the experimental results of this embodiment are as follows: Figure 3 As shown. Figure 3 (b)-(d) and (f)-(h) respectively show the intensity distributions of the topological and trivial sample s, p, and d states in the HSP. s , D d Variation with frequency. Each peak in the figure represents the band distribution at that frequency of the high symmetry point, and the mode is the mode represented by the corresponding color, i.e., s, p, or d state.

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

[0041] (1) Represent the high symmetric point (HSP) as Π (q) It satisfies the relation: R q Π (q) =Π (q) (modek0), where modek0 represents the modulus of the reciprocal lattice vector, R q For the q-fold rotation operator acting on the crystal momentum, this means that in Π (q) The point, q-fold rotation operator commutes with the Bloch-Hamiltonian. Therefore, the Bloch-Hamiltonian's energy eigenstates in the HSP are also rotation operators. The eigenstates.

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

[0043]

[0044] Given a band subspace, we can compare the variations of these rotational eigenvalues ​​at different HSPs. If the eigenvalues ​​differ at different HSPs, it indicates that these bands possess nontrivial topological properties.

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

[0046]

[0047] in Indicates having eigenvalues The number of bands below the band gap, Π can take values ​​of X, M, Indicates having eigenvalues The number of energy bands below the band gap.

[0048] This method can be applied to crystals with various symmetries. The photonic crystal in this embodiment has time-reversal symmetry and C4 rotational symmetry. Referring to symmetry index theory, the topological invariant polarization P... (4) and angular charge It can be represented as:

[0049]

[0050] Where a x and a y Let X and M be the basis vectors of the unit cell, and let X and M be the height-symmetric points with coordinates (π,0) and (π,π) in the reciprocal space, respectively. The subscripts and superscripts have the same meanings as described above. q denotes q-fold rotational symmetry, p denotes the p-th eigenvalue of this rotational symmetry, and mod 1 indicates that the calculation result is modulo 1. According to The s, p, and d states can all be relabeled as eigenvalues ​​of the rotation operator. Figure 3 The eigenvalues ​​of the 4-fold or 2-fold rotation-symmetric operators corresponding to each state are plotted. Substituting these values ​​into the formula yields: For topological samples, This indicates that the sample possesses a first-order topological boundary state. This indicates that the sample possesses high-order topological angular states, while the trivial sample has both values ​​of 0, exhibiting only bulk states and lacking boundary and angular states. Therefore, different topological phases can be classified based on the calculated values ​​of the topological invariants. It is important to note that the topological classification here is not necessarily limited to two types (topological or trivial). This embodiment is merely illustrative; more detailed categories can be derived. For example, for some structures, the topological invariant polarization P can be calculated. (4) =1 / 2, angular charge They are then classified into another category. Structures that are identical for every topological invariant belong to the same category.

[0051] Therefore, this invention utilizes the spatial symmetry of topological structures and, based on volume region measurements, achieves topological phase discrimination that does not rely on boundary measurements, overcoming the limitation of traditional topological phase discrimination that requires the full bandgap and interface states within the bandgap.

Claims

1. A method for determining topological phase based on measurement of a body region, characterized by, The method comprises the following steps: Step 1, measuring S in a bulk region of a crystal having spatial symmetry 21 parameters, wherein the bulk region is a region other than a boundary unit cell; Step 2, obtaining a field distribution with a phase from the S 21 The parameter obtains a field distribution with a phase, Fourier transforms the field distribution, and then projects the Fourier-transformed field distribution on an eigenstate to obtain energy band structure information; wherein obtaining the energy band structure information specifically comprises: multiplying the field distribution by k s 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 the field distribution at a specific wave vector, i.e., k s s The field distribution at k is decomposed by an eigenstate basis vector, i.e., the eigenstate is projected to obtain the modal proportion of the field distribution at k; the field distribution at each frequency is repeatedly operated at a high-symmetry point as above, and finally the modal with the highest modal proportion is marked on the energy band diagram to obtain the energy band structure information;​​ In step 3, according to the band structure information, the symmetry indicator theory is used to obtain corresponding topological invariants; according to the band structure information, the numerical value of the topological invariants is calculated, and then the topological phase of the crystal is specifically distinguished according to the numerical value of the topological invariants.

2. The method of claim 1, wherein the topological phase is determined based on a measurement of a bulk region. The device used for measurement in step 1 includes a microwave vector network analyzer, two microwave transmission lines and two microwave probes. One end of each of the two microwave transmission lines is connected to one of the two microwave probes, and the other end of each of the two microwave transmission lines is connected to the output end and the receiving end of the vector network analyzer respectively. The transmitting microwave probe connected to the output end penetrates 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 detecting probe for measuring S 21 parameters.

3. The method of claim 2, wherein the topological phase is determined based on the bulk region measurement. In step 1, the working frequency range of the microwave vector network analyzer, the microwave transmission line and the microwave probe covers the intrinsic state energy range of the crystal.

4. The method of claim 2, wherein the topological phase is determined based on the bulk region measurement. The detection probe maintains a constant distance from the crystal surface when moving in the bulk region.

5. The method of claim 1, wherein the method is based on a topological phase discrimination method of measuring a body region. In step 3, when there are multiple topological invariants, the topological phases with the same numerical value of each topological invariant belong to the same category.

6. The method of claim 1, wherein the method is based on a topological phase discrimination method of measuring a body region. In step 2, the summation rule used for summing the filtered field distribution is that the atoms in a single unit cell are not added, and the atoms corresponding to all unit cells are added.

7. The method of claim 1, wherein the topological phase is determined based on a measurement of a body region. In step 2, the Fourier transform of the field distribution follows the formula: where C j,m,l (f) is the S 21 parameter; is the Fourier transform factor, k x and k y represent two components of the wave vector k; c n,j (k) δ(f - f n (k)) represents the field of the state with frequency f and wave vector k in the n-th band on the j-th cell in the primitive cell.

8. The method of claim 1, wherein the method is based on a topological phase discrimination method of measuring a body region. In step 2, the eigenstate projection of the Fourier-transformed field distribution specifically comprises: Step 21, select a set of orthonormal complete eigenstate basis vectors: Step 22, the field distribution characterized by a frequency f and a wave vector k is denoted as ψ(f, k), and is re-expressed as a linear combination of the basis vectors in Step 21: n,j (k) The field distribution characterized by a frequency f and a wave vector k is denoted as ψ(f, k), and is re-expressed as a linear combination of the basis vectors in Step 21: The coefficient is obtained by the following formula: Step 23, calculate the field distribution ψ(f, k) at frequency f and wave vector k the intensity of the state obtain information on whether there is a band at frequency f and wave vector k and the corresponding mode of the band if it exists; In step 24, according to the information obtained in step 23, the topological invariants are calculated from the high-symmetry-point band structure by using the symmetry indicator theory, and the topological discrimination is realized.

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