Spontaneous spin polarization at the edge of the chiral superconducting states in a spin Hall metal

We discuss the chiral superconducting state in a square lattice system with an intrinsic spin-orbit coupling causing the spin Hall effect. We estimate the spontaneous spin polarization (local moment) 〈mi 〉 at the i-th site and its sum M = ∑ i 〈mi 〉 in ribbons. In the zigzag ribbon, we see 〈mi 〉 ≠ 0 near the edge. The energy spectra of the chiral edge states show the spin splitting and then we have non-zero M. Moreover, we find that M is enhanced significantly at a topological phase transition point, where the energy gap is closing. We also have non-zero 〈mi 〉 near the edge of the straight ribbon, but in contrast, there is no spin splitting in the spectra and M vanishes.


Introduction
The chiral superconductor is the topological superconducting state with broken time-reversal symmetry supporting the chiral edge state at the sample boundary and has been keeping a lot of attention. The role of the spin-orbit coupling in a superconductor has also been studied intensively, and its relation to the chiral edge state is the central issue of this paper. In this point of view, Imai, Wakabayashi and Sigrist argued the spontaneous spin polarization due to the chiral edge states in the model of the chiral superconductor Sr 2 RuO 4 [1]. The same mechanism was also examined in the ordered honeycomb network superconductor SrPtAs, which is a potential candidate for the chiral d-wave superconductor [2,3,4,5]. Both systems possess the spin-orbit coupling which gives in the normal state the spin-dependent Aharonov-Bohm phase to an electron and causes the spin Hall effect.
To gain the insight into this intriguing phenomena, it would be important to examine a minimal model intensively. We then utilize the simple model on the square lattice proposed by Bernevig, Hughes and Zhang (BHZ) to discuss the quantum spin Hall insurator in the HgTe quantum wells [6,7].

The chiral p-wave pairing in the BHZ metal
In the BHZ model we have four electron fields on a cite r; ψ s↑,↓ (r), ψ p + ↑ (r), and ψ p − ↓ (r), where p ± = p x ± ip y . We consider the metallic state by shifting the Fermi level and introduce the chiral p-wave order parameter. The mean field Hamiltonian is where the up (down) spin is denoted by + (−),σ = −σ, and θ δ = θ −δ +π = 0, π/2 for δ = e x , e y . The term t sp is the spin-orbit coupling, which gives the spin-dependent Aharonov-Bohm phase to an electron hopping along a certain closed path and causes the spin Hall effect. We may diagonalize H BHZ by the Fourier transformation in the bulk, and 3D plots of the energy bands and Fermi surfaces are given in Figures 1(a) and 1(b). The topology of the chiral p-wave state is characterized by the Chern integer N Ch [8,9], which counts the phase winding of the k-space gap functions around the Fermi surface and gives the number of the topologically protected chiral edge states in a finite-size system.

Spontaneous spin polarization at the edge
Besides the spontaneous charge current, we would have the spin current and therefore spin at the edge due to the spin orbit coupling [1,5]. In this paper, we focus on the spin polarization in our model and discuss its enhancement.
We examine straight and zigzag ribbons (see Figure 2), and obtain the energy spectra and spin polarization (local moment) at the i-th site in the unit cell (i = 1, ..., N ) where ⟨...⟩ denotes the expectation value in the superconducting ground state and k wave number. We verify that ⟨m i ⟩ would be zero in the absence of the spin-orbit coupling t sp . We then use the parameters t ss = t pp = 1.0, t sp = 0.1, µ = −1.5, δµ = 1.0, which give the bulk Fermi surfaces shown in Figure 1(b) , and ∆ = 1.0 and N = 100. The results are summarized in Figure 3. We have N Ch = 4 and four chiral states at each edge. In the straight ribbon, all edge spectra are degenerate with respect to spin degrees of freedom. The spin polarization ⟨m i ⟩ arises near the edge, whereas its sum vanishes completely. On the other hand, in the zigzag case the degeneracy is lifted for the edge spectra with respect to spin. We thus have a difference between the total occupation numbers for up and down spin and M ̸ = 0 besides the spin polarization near the edge.
We then examine the µ dependence of M . The result is summarized in Figure 4. We see a sharp enhancement slightly below µ = −1, where the topological phase transition occurs and N Ch shows a jump from +4 to 0 with increasing µ. This enhancement comes from the fact that the split edge spectra become almost flat when the gaps at k = ±π become tiny, and we have a large difference between the occupation numbers of up and down spin.   by Imai, Wakabayashi and Sigrist in the model for the chiral superconductor Sr 2 RuO also applied to SrPtAs, which is a potential candidate for the chiral d-wave supercon In this paper, we estimate the spin polarization in our model and discuss its enhancem We examine straight and zigzag ribbons and obtain the energy spectrum and spin po (local moment) at the i-th site We then examine the µ dependence of M. The result is summarized in Figure 5. sharp enhancement at the vicinity of µ = −1, where the topological phase transition o N Ch shows a jump from +4 to 0 with increasing µ.

Summary Acknowledgment
This work was supported by JSPS KAKENHI Grant Number JP20K03826.  by Imai, Wakabayashi and Sigrist in the model for the chiral superconductor Sr 2 RuO also applied to SrPtAs, which is a potential candidate for the chiral d-wave supercon In this paper, we estimate the spin polarization in our model and discuss its enhancem We examine straight and zigzag ribbons and obtain the energy spectrum and spin po (local moment) at the i-th site ⟨m i ⟩ = σ σ ⟨ψ † isσ ψ isσ ⟩ + ⟨ψ † ipσσ ψ ipσσ ⟩ , where ⟨...⟩ de expectation value in the superconducting ground state. We use the parameters t ss = t sp = 0.1, µ = −1.5, δµ = 1.0 and ∆ = 1.0. The results are summarized in Figures As we see, N Ch = 4 in this case and we have four chiral edge spectrum with spect In the straight ribbon, the spectrum are completely degenerate with respect to spin. polarization ⟨m i ⟩ arises near the edge but its sum M = N i=1 ⟨m i ⟩ = 0. On the other the zigzag case the degeneracy is lifted for the edge states and M ̸ = 0.
We then examine the µ dependence of M. The result is summarized in Figure 5. sharp enhancement at the vicinity of µ = −1, where the topological phase transition o N Ch shows a jump from +4 to 0 with increasing µ.

Summary Acknowledgment
This work was supported by JSPS KAKENHI Grant Number JP20K03826.

Summary and discussion
We discuss the chiral p-wave pairing in the BHZ spin Hall metal and estimate the spin polarization ⟨m i ⟩ in Eq. (2) and its sum M in Eq. (3) in the ribbon geometries. The result depends on the shape of the ribbon significantly. We have ⟨m i ⟩ around the edge but no M in the straight ribbon. In the zigzag ribbon, on the other hand, we have both since the energy Chemical potential ..⟩ denotes the cting ground state. We use the parameters t ss = t pp = 1.0, ∆ = 1.0. The results are summarized in Figures ?? and and we have four chiral edge spectrum with spectral flows. are completely degenerate with respect to spin. The spin ge but its sum M = N i=1 ⟨m i ⟩ = 0. On the other hand, in ted in the energy spectrum of the edge state and M ̸ = 0. nce of M. The result is summarized in Figure . We see a AKENHI Grant Number JP20K03826.  We would discuss elsewhere the relation between the ribbon geometry and the spin splitting in the edge spectra [10].