Spin-charge separation and quantum spin Hall effect of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta$$\end{document}β-bismuthene

Multiple works suggest the possibility of classification of quantum spin Hall effect with magnetic flux tubes, which cause separation of spin and charge degrees of freedom and pumping of spin or Kramers-pair. However, the proof of principle demonstration of spin-charge separation is yet to be accomplished for realistic, ab initio band structures of spin-orbit-coupled materials, lacking spin-conservation law. In this work, we perform thought experiments with magnetic flux tubes on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta$$\end{document}β-bismuthene, and demonstrate spin-charge separation, and quantized pumping of spin for three insulating states, that can be accessed by tuning filling fractions. With a combined analysis of momentum-space topology and real-space response, we identify important role of bands supporting even integer invariants, which cannot be addressed with symmetry-based indicators. Our work sets a new standard for the computational diagnosis of two-dimensional, quantum spin-Hall materials by going beyond the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Z}_{2}$$\end{document}Z2 paradigm and providing an avenue for precise determination of the bulk invariant through computation of quantized, real-space response.

In contrast to C GS , C R,GS describes quantized, non-Abelian Berry flux 2πC R,GS through 2D BZ.The definition of non-Abelian Berry flux for spin-orbit-coupled materials has many subtleties due to the absence of continuous spin rotation symmetry or spin conservation law.Kane and Mele formulated Z 2 -classification of QSH effect of T -symmetric systems, i.e., odd vs. even integer distinction of C R,GS 4 .For generic T -symmetric systems, the Z 2 invariant (−1) C R,GS can be calculated from the gauge-invariant spectrum of Wilson loops (Wannier charge centers) of non-Abelian Berry connection [11][12][13][14] .Furthermore, for inversion-symmetric materials, it can be easily identified from a symmetry-based indicator, which is the product of parity eigenvalues at timereversal-invariant momentum points 8 .
To go beyond Z 2 -classification of C R,GS without relying on spin and momentum conservation laws, various authors have considered the role of generalized TBCs [15][16][17] .However, the insertion of spin-gauge flux and the implementation of spin-dependent TBC requires detailed understanding of underlying basis states and the mechanism of violation of spin-conservation law.Hence, its application has practical limitation.
To overcome this challenge, Qi and Zhang 18 , and Ran et.al. 19 proposed diagnosis of Z 2 QSH states with magnetic flux tubes (i.e., gauging of conserved quantity).Employing 4-band models of QSH states with |C R,GS | = 1, they have shown that a flux tube, carrying half of flux quantum φ = φ0 2 (T -invariant, π flux) binds two degenerate, zero-energy, mid-gap states.At halffilling, one of these modes is occupied, and the ground state exhibits 2-fold-degeneracy (SU (2)-doublet).Consequently, the flux tube remains charge-neutral, and carries spin quantum number ± 1 2 .When both modes are occupied (empty), the flux tube carries electric charge −e (+e), and spin quantum number 0 (SU (2)-singlets).Such states can be accessed by doping insulators with one electron (hole).This solitonic mechanism of spincharge separation (SCS) is similar to what is known for polyacetelene 20,21 and topologically ordered, correlated systems 22 .When T is broken by generic values of φ, the bound modes and the half-filled ground state become non-degenerate.But the flux tube continues to bind spin and no electric charge.By adiabatically tuning φ from 0 to φ 0 , quantized pumping of spin (one Kramers-pair) can be observed [see Appendix A].
While Refs.18 and 19 and subsequent works [23][24][25] advanced conceptual understanding of QSH effect, they relied on idealized models of decoupled Chern insulators, carrying opposite Chern numbers.Due to the underlying U (1) spin-rotation symmetry, these models admit Zclassification of C R,GS = C GS,↑ = −C GS,↓ , and the spin-Hall conductivity σ z xy = 2C R,GS .The SCS for such systems would be governed by SU (2|C R,GS |)-multiplets [see Appendix A].Since C R,GS for decoupled models is easily calculated, the analysis of SCS only serves academic interest.
For realistic band structures of spin-orbit-coupled materials, various crystalline-symmetry allowed hybridization terms destroy U (1) spin-conservation law.Since there is no simple theoretical framework for computing C R,GS of such systems, the demonstration of SCS would allow unambiguous diagnosis of |C R,GS | or N-classification of QSH insulators.Moreover, Zclassification can be accomplished by measuring spin expectation values, during the process of spin-pumping.Recently, we have addressed the stability of SCS for topologically non-trivial planes of 3D Dirac semimetals (4-band model) 26 , and 3-fold symmetric planes (8-band model) 27 of octupolar topological insulators 28 .These models support SCS respectively controlled by SU (2) and SU (4) multiplets.Moreover, the quantized pumping of spin occurs even in the absence spin-rotation symmetry and gapless, helical edge-states.
Encouraged by these results, in this work, we perform proof of principle demonstration of SCS for realistic, ab initio band structures.For concreteness, we focus on a single (111)-bilayer of elemental bismuth (Bi), also known as β-bismuthene, as a suitable material platform.The analysis of SCS will be guided by the calculation of gauge-invariant magnitudes of relative Chern numbers of constituent bands (|C R,n | for n-th band) 29 .Thus, the importance of Z 2 -trivial bands, possessing even integer winding number (C R,j = 2s j = 0) will be critically addressed.We also present a brief contrasting study of β-antimonene, which does not exhibit SCS.
The crystal structure of β-bismuthene is described by buckled honeycomb layers with space group P 6/mcc.The material supports T and space-inversion (P) sym- FIG. 2. Spin Hall conductivity σ j xy of β-bismuthene from first principles calculations, as a function of energy.Three insulating states can be identified from direct gaps in density of states.In addition to showing plateau-like features, σ z xy shows sharp change of sign, when bands 4 and 5 become occupied.While these results cannot capture precise topological properties, they indicate topological non-triviality of these Z2-trivial bands.metries, leading to the two-fold Kramers degeneracy of all energy bands throughout the hexagonal BZ of Fig. 1(a).Using the crystal structure and lattice constants from Ref. 40, the ab initio band structure has been calculated with Quantum Espresso [41][42][43] .The band structure along high-symmetry path Γ − M − K − Γ is shown in Fig. 1(b).As these bands are well separated from other bands, an accurate 12-band, Wannier tight-binding (TB) model has been constructed from p x,y,z orbitals from each layer.We have included spin-orbit coupling for all calculations and utilized a 40 x 40 x 1 Monkhorst-Pack grid of k-points and a plane wave cutoff of 100 Ry.The model construction and topological analysis are performed with Wannier90 and Z2pack 12,14,44 .
When the Fermi level is tuned inside direct gaps between bands (i) 3 and 4, (ii) 4 and 5, and (iii) 5 and 6, we find three insulators at filling fractions 1/2, 2/3, and 5/6.From the parity eigenvalues shown in Fig. 1(b), we find that only bands 2 and 6 possess non-trivial Z 2 -index ν 0,n = 1, and all three insulators admit non-trivial Z 2index ν 0,GS = 1.The results of edge-states calculations, using iterative Greens function method 45 and Wannier Tools 46 are displayed in Figs.1(d) and 1(c).All three insulators support gapless edge modes, which can cross the Fermi level 2 or 6 times 34 .Whether |C R,GS | = 1 or 3 cannot be determined from edge-spectrum.
It is instructive to compute spin Hall conductivity, following the current state-of-the-art of computational materials science, [47][48][49] and the results are shown in Fig. 2. Due to the non-conservation of spin, this method cannot capture quantization of spin Hall effect.But it provides rough guidance for understanding qualitative properties of three insulators.Notice that σ z xy displays plateau- , respectively for 1/2, 2/3, and 5/6 filled insulators.While it is possible to compute signed C R,n by adding a small time-reversal symmetry breaking field, we will resolve uncertainties with SCS.

III. SPIN CHARGE SEPARATION
To study real-space topological response we insert a flux tube at the center of 2D system.The hopping matrix element H ij connecting lattice sites r i and r j is modified to H ij e iφij .Working with Coulomb gauge, we define the Peierls phase factor We first perform exact diagonalization of gauged Hamiltonian for 24 × 24 unit cells, under periodic boundary conditions (PBC), yielding N = 6912 eigenstates.When the total number of electrons N e = N 2 , 2N 3 , 5N 6 , and φ = φ 0 /2, we find two-fold-degenerate mid-gap states bound to the flux tube, leading to two-fold degeneracy of ground states [see Figs.3(a)-3(b)].When φ = φ 0 /2, flux tube breaks time-reversal-symmetry and the degeneracy of bound states (ground state) is lifted.By holding N e fixed at commensurate values, and varying φ from 0 to φ 0 , we observe pumping of one Kramers pair. 18ext we compute the induced electric charge on fluxtube for φ = φ 0 /2.This calculation is done in two steps. 50For a given number of electrons, by summing over all occupied states, we evaluate the area charge densities σ 1 (r i , N e ), and σ 0 (r i , N e ), respectively in the presence and absence of flux tube.The induced charge density is defined as δσ(r i , N e ) = σ 1 (r i , N e ) − σ 0 (r i , N e ), and the total induced charge within a circle of radius r, centered at the flux tube is determined from δQ(r, N e ) = |ri|<r δσ(r i , N e ).In order to achieve sufficient numerical accuracy, induced charge calculations are performed for a system size of 60 × 60 unit cells, yielding N = 43, 200 eigenstates.The results are displayed in Figs.3(d , one of the degenerate mid-gap modes is occupied (halfoccupation of mid-gap states), and we find δQ(r, N e ) = 0, which remains unchanged for generic values of φ.For N e = N 2 ± 1, 2N 3 ± 1, 5N 6 ± 1, the bound modes become completely occupied (+) and empty (−), and the max-imum values of δQ(r, N ) saturate to quantized results ∓e, respectively.
Therefore, we can conclude that each non-trivial insulator supports |C R,GS | = 1, which can only be consistent with the following assignments of signed relative Chern numbers defined with respect to a global spin quantization axis for all bands.When bands 4 and 5 are occupied, C R,GS will change sign.This assertion can be further substantiated by evaluating expectation values of spin operators 1 6×6 ⊗ σ ≡ σ for bound states.We have computed expectation values for φ = ( 1 2 − )φ 0 and → 0 + , such that the bound states are infinitesimally split in energy.The expectation values for σ x,y are negligibly small.As occupied and unoccupied modes support opposite signs for ψ n | σ z |ψ n , we are only showing the results for unoccupied modes in Figs.3(g)-3(i).All signs become reversed for φ = ( 12 + )φ 0 , as a consequence of spin-pumping.Therefore, flux tube for 1/2-, 2/3-, 5/6filled insulators respectively support +, −, + signs for ψ n | σ z |ψ n , when φ < φ 0 /2.

IV. CONCLUSIONS
In summary, we have shown that bilayer bismuth is a suitable platform for studying spin charge separation as a universal topological response of quantum spin Hall insulators.The combined analysis of non-Abelian Berry phase in momentum space and real space topological response clearly identify non-trivial topology of bands that carry even integer winding numbers.Topology of such bands are not easily detected by symmetry-based indicators.In Appendix C we also analyze bulk topology of β-antimonene, which supports Z 2 -trivial ground state.With first principles based calculations of spin Hall conductivity and insertion of magnetic flux tube we show that β-antimonene is not a quantum spin Hall insulator ( i.e., C R,GS = 0).
In this work, we have only gauged a conserved quantity (electric charge) to identify the presence or absence of spin-pumping.This can be reliably used for many candidate materials for quantum spin Hall effect.Guided by the results of this work, one can further pursue insertion of spin-gauge flux (gauging of non-conserved quantity) to demonstrate pumping of electric charge (δQ = 2eC R,GS ), which directly tracks signed relative/spin Chern number.Due to technical subtleties and numerical cost of such calculations, such thought experiments on real materials would be reported in a future work.II.Along blue, magenta, and green lines, the bulk band gap can close at Γ, M , and X points, respectively.Red dots denote multi-critical points.

Phase
Parity eigenvalues Let us consider the following Bernevig-Hughes-Zhang (BHZ) model 5 of sp hybridization on a square lattice where A representative phase diagram is shown in Fig. 4, and the pattern of parity eigenvalues and the bulk invariant are listed in Table II.As phases 3, 4, 6, and 7 (2 and 5) , magnetic π-flux tube would bind 2 (4) zero-energy bound states.Therefore, SCS would be controlled by SU (2) and SU (4) multiplets, respectively (see Fig. 5).When flux φ is tuned from 0 to φ 0 , one and two units of spin (Kramers-pair) would be pumped.
After Fourier transformation, we obtain tight-binding model H 0,ij in real-space.In the presence of magnetic flux tube, placed at origin, the matrix elements H 0,ij between different lattice sites can be replaced by H 0,ij e iφij , with φ ij = φ φ0 rj ri ẑ×r r 2 • dl.The SCS and spin-pumping for Phases 3, 4, 6, and 7 are controlled by SU (2) multiplets.In Fig. 6, we show the results for Phase 3 and Phase 4. The results of SCS governed by SU (4) multiplets are displayed in Fig. 7, clearly showing that 2 Kramers-pair are being pumped.Due to the enhanced degeneracy of bound states, the maximum induced electric charge can now oscillate between 0, ±e, and ± 2e.
There are many ways to break U − (1) spin rotation symmetry.For example, we can modify H j as such that the 2-fold Kramers-degeneracy is preserved.The momentum dependent function d 4 (k) = t d,1 (cos 2k x − cos 2k y ) and d 5 (k) = t d,2 sin k x sin k y maintain 4-fold rotation symmetry and also vanish at the time-reversal-invariant momentum points.Following Ref. 26, it can be shown that C R,GS and SCS for all phases remain unchanged.But d-wave perturbations destroy gapless edge-states.If B 1g term is changed to (cos k x − cos k y ), X points cannot participate in band inversion.Thus, states with even integer winding number become trivialized and no longer display SCS and spin-pumping.
Therefore, we can conclude that flux tube can perform N and Z classification of quantum spin Hall states, irrespective of spin-rotation symmetry.We will further substantiate this conclusion by studying momentum space topology and real space response of (111)-bilayer of antimony (antimonene) in Sec. C.

Appendix B: Magnitude of relative Chern numbers
Recently, the in-plane Wilson loop has been utilized to quantify magnitude of SU (2) Berry flux of constituent Kramers-degenerate bands of two-dimensional first and higher-order topological insulators. 26,29The in-plane Wilson loop of n-th band measures SU (2) Berry phase accrued upon parallel transport along a nonintersecting closed contour C. It is defined by 8. (a) Schematic of path (abcdef ) for calculating in-plane Wilson loop.The size k0 is increased from 0 to k b .The results for bands 1-6 are shown in (b)-(g), respectively.Trivial bands 1 and 3 do not show winding of θ.For bands 2 and 6, possessing non-trivial Z2 index, θ winds once.For Z2-trivial bands 4 and 5, θ winds twice.Therefore, bands 1 through 6 support relative Chern numbers |CR,n| = 0, 1, 0, 2, 2, 1, respectively, as listed in Table I.

Appendix C: Analysis of β-antimonene
In contrast to β-bismuthene, the occupied subspace of single layer of (111) antimony (β-antimonene) supports trivial Z 2 -classification with ν 0,GS = 0. Whether the ground state supports quantum spin Hall effect can be directly addressed by combined analysis of momentum space topology and real-space response.The bulk band structure and N-classification of constituent bands are shown in Fig. 9(a) and Fig. 9(b), respectively.We have used the lattice parameters given by Mounet et.al. 40  The first principles calculations show that all three components of spin Hall conductivity vanish for the halffilled insulating state (see Fig. 9(c)).To unambiguously probe topological response, we have performed thought experiments with flux tube for a system size of 24 × 24 unit cells, under periodic boundary conditions.The spectrum does not show any mid-gap bound states for N e = N/2 and no spin-pumping is observed (see Fig. 9(d) ), implying C R,GS = 0. Therefore, (C R,2 , C R,3 ) = ±(1, −1) are the possible assignments of signed relative Chern numbers.Due to the lack of any further direct band gaps, we do not pursue the analysis for other filling fractions.

FIG. 1 .
FIG. 1.(a) Hexagonal Brillouin zone of β-bismuthene.(b) Band structure along high-symmetry path Γ − M − K − Γ and energies are measured with respect to a reference value E0 = 0. Bands are numbered according to their energies at Γ point, such that En(Γ) < En+1(Γ).Parity eigenvalues ±1 at time-reversal-invariant-momentum points are denoted by red and blue dots, respectively.The dashed lines correspond to three representative values of Fermi energy, tuned in direct band gaps, leading to 1/2-, 2/3-, and 5/6-filled insulators.They support non-trivial Z2-invariant ν0,GS = 1.(c,d) Spectral density on the surface under open boundary conditions along (c) y-axis and (d) x-axis.The mid-gap edge-modes, connecting bulk valence and conduction bands imply first-order topology of insulating states.

2 FIG. 4 .
FIG. 4. Phase diagram of four-band model for M = 1.Parity eigenvalues and bulk winding numbers are listed in TableII.Along blue, magenta, and green lines, the bulk band gap can close at Γ, M , and X points, respectively.Red dots denote multi-critical points.

and Γ 3 =FIG. 5 .(FIG. 6 .
FIG. 5. Schematic of spin-charge separation and induced quantum numbers for magnetic π flux tube.The occupation number of mid-gap states is denoted by 0 and 1.For half-filled systems, the number of added electrons Ne = 0, and the induced electric charge δQ = 0.By adding one electron or hole one can access δQ = ∓e on flux tube for CR,GS = ±1.Additional charge quantum numbers are found for CR,GS = ±2.

FIG. 7 .
FIG. 7. Spin-charge separation for Phase 2 and Phase 5, possessing |CR,GS| = 2.All calculations are perfomed for a system size of 24 × 24 lattice sites, under periodic boundary conditions.(a) Local density of states on flux tube as a function of φ/φ0.Both branches of spectra, which traverse the bulk gap are two-fold degenerate, implying pumping of two Kramers pairs.(b) At φ = φ0/2, they lead to four zero-energy bound states, and 6-fold degeneracy of the half-filled ground state.(c) Induced electric charge (in units of −e) on π-flux tube within a radius r, and Ne denotes the number of doped electrons.The maximum induced charge saturates to quantized values ±2e for the non-degenerate states (SU (4)-singlets), and ±e for the four-fold degenerate states (SU (4) quartets), and 0 for the half-filled state (SU (4) sextet), respectively.
B1) where A ss j,n (k) = −i ψ n,s (k)|∂ j ψ n,s (k) describes components of SU (2) Berry connection, ∂ j = ∂ ∂kj , ψ n,s=±1 (k) are degenerate eigenfunctions of n-th band, and P indicates path-ordering.While the angle θ n measures gauge-invariant magnitude of non-Abelian flux enclosed by C, the three-component unit vector Ωn depends on gauge choice.Following the convention of defining gauge-invariant eigenvalues of Wilson lines or Wannier center charges, we analyze eigenvalues of Im(Ln(W n )) ≡ ±|θ n | mod π.In-plane loops are calculated with Wannier90 44 and Z2Pack software packages, by following C 3 -symmetry preserving contour, shown in Fig. 8(a).The area enclosed by the contour is systematically increased from zero to the area of first Brillouin zone.The number of winding of θ n corresponds to the absolute value of relative Chern number |C R,n |.The results for bands 1-6 are shown in Fig. 8(b)-8(g).

3 1 TABLEFIG. 9 .
FIG. 9. (a) Band structure of β-antimonene, along high-symmetry path of hexagonal Brillouin zone.The bands are numbered according to their energies at Γ point, and parity eigenvalue +1 (−1) at time-reversal-invariant momentum points are denoted by red (blue) dots.(b) Summary of momentum-space topology of constituent bands, where ν0,n, C3,n, and CR,n respectively denote the Z2 index, 3-fold rotation eigenvalue, and the relative Chern number of n-th Kramers-degenerate bands.(c) First principle calculations of spin Hall conductivity.When the Fermi level is tuned inside direct band gap, all three components of spin Hall conductivity vanish for the insulating state.(c) Local density of states on the magnetic flux tube in the vicinity bulk band gap does not show any spin-pumping, which shows that the net relative Chern number CR,GS = 0.

TABLE I .
29tural number classification of relative Chern numbers of constituent Kramers-degenerate bands of βbismuthene.The 3-fold rotation eigenvalues and Z2-indices are listed for convenience.The symmetry data of rotation and parity eigenvalues are insufficient to distinguish between bands possessing |CR,n| = 0 and 2. This question can be conclusively answered by identifying the magnitude of relative Chern number of Kramers-degenerate bands (|C R,n |) by computing inplane Wilson loops of SU (2) Berry connection29.The results are displayed in TableIand the details of calculations are presented in Appendix B. We see that bands 4 and 5 carry even integer invariants.While they do not change odd integer classification of C R,GS , they can change the magnitude and the sign of C R,GS .TableIsuggests the following possibilities: like features, when E is tuned in direct band gaps, and changes sign when Z 2 -trivial bands 4 and 5 become occupied.Are bands 4 and 5 topologically trivial?