Identification of topological superconductivity in magnetic impurity systems using bulk spin-polarization

Magnetic impurities on the surface of spin-orbit coupled but otherwise conventional superconductors provide a promising way to engineer topological superconductors with Majorana bound states as for boundary modes. In this work, we show that the spin-polarization in the interior of both one-dimensional impurity chains and two-dimensional islands can be used to determine the topological phase, as it changes sign exactly at the topological phase transition. This offers an independent probe of the topological phase, beyond the zero-energy Majorana bound states appearing at the boundaries of the topological region.

Magnetic impurities on the surface of spin-orbit coupled but otherwise conventional superconductors provide a promising way to engineer topological superconductors with Majorana bound states as for boundary modes. In this work, we show that the spin-polarization in the interior of both one-dimensional impurity chains and twodimensional islands can be used to determine the topological phase, as it changes sign exactly at the topological phase transition. This offers an independent probe of the topological phase, beyond the zero-energy Majorana bound states appearing at the boundaries of the topological region.
Topological states of matter have been at the center of attention in condensed matter physics for the past decade [1][2][3][4]. The notion of topology as a classifier uses non-local properties, such as the Berry phase, and is thus fundamentally different from the traditional Landau-Ginzburg paradigm for phase transitions building on local order parameters [5,6]. In terms of realizing topological superconductivity, different platforms have already been proposed, such as spin-orbit coupled nanowires in proximity to superconductors or nanostructures created by depositing magnetic atoms on the surface of conventional superconductors [7][8][9][10][11][12][13][14][15][16].
Based on the bulk-boundary correspondence, zero-energy Majorana bound states (MBSs) appear at the end-points of many one-dimensional (1D) topological superconductors. The appeal of MBSs is particularly strong considering that they might be suitable for topological quantum computation [17,18]. Several experiments have already reported zero-energy peaks in both impurity chains and nanowires [8,10,11,14], suggestive of non-trivial topology and MBSs. For impurity chains, finite in-plane spin-polarization of the MBSs has further been used to differentiate MBS from trivial states [19,20].
In this work we consider both 1D impurity chains and 2D islands and show that the spin-polarization of the low-energy states, measured in the interior, or bulk, and along the direction of the magnetic impurity moments, can be used to uniquely determine the topological state of the system. These low-energy states are formed from hybridizing Yu-Shiba-Russinov (YSR) states [21][22][23], which arise within the energy gap for magnetic impurities in conventional superconductors. An individual YSR state has a spin-polarization whose direction is set by its energy being positive or negative. We show that for both ferromagnetic chains and islands deposited on superconductors with large spin-orbit coupling (SOC), this spin-polarization remains and encodes the topological phase transition (TPT) as an interchange of the spin-polarization between negative and positive low-energy states. We also show that the same interchange takes place for impurity chains with a helical spin structure if using the locally defined magnetic moment direction as the basis for the spin-polarization. Thus our results establish that the spin-polarized local density of states (SP-LDOS), measurable using spin-polarized scanning tunneling spectroscopy (STS) [12,24,25], provides a powerful tool to verify non-trivial topology, entirely independent of MBSs.
Model.-To avoid unnecessary complexities, we keep our model simple yet capturing all important features. For the substrate we consider a square lattice with lattice constant a = 1 with Rashba SOC and conventional s-wave superconducting pairing. The full mean-field Hamiltonian reads H = H sub + H imp where, Here c iα (c † iα ) is the creation (annihilation) operator at site i = (i x , i y ) with spin α ∈ {↑, ↓} and σ = (σ x , σ y , σ z ) with σ i the Pauli spin matrices. The chemical potential is t ii = 2µ [26] and we restrict the range of hopping to nearest neighbors: t i j = t with t = 1. Rashba SOC is present due to inversion symmetry breaking at the surface and determined by λ R , while the superconductor order is given by ∆. Furthermore, we assume the magnetic impurities behave as classical moments [27,28], such that J S = V Zn acts as a local Zeeman field in direction n specified on each impurity site [22]. Inspired by different experimental setups, we study both ferromagnetic chains with all local moments perpendicular to the substrate (n =ẑ) and spin-helical chains where the moments lie in x-y plane (n ⊥ẑ), see Fig 1. We also consider ferromagnetic 2D impurity islands. Without loss of generality, we assume that the impurity chains are oriented along the x-axis while the impurity island forms a circle, with both systems embedded in the middle of a large square lattice. We are here primarily concerned with the SP-LDOS: and measurable using STS with spin-polarized tips. We calculate the SP-LDOS within a Bogoliubov-de Gennes (BdG) formulation of Eq. (1) by using a Chebyshev expansion of the Green's function [29][30][31][32].
1D ferromagnetic system.-To understand the behavior of the spin-polarization, we start by studying a pure 1D ferromagnetic system, which we obtain by shrinking the superconducting substrate along the y-axis to a unit cell width. We also apply periodic boundary conditions in the x-direction and Fourier transform to arrive at the 1D BdG Hamiltonian and Nambu spinor Ψ T k = (c k↑ , c k↓ , c † −k↓ , −c † −k↑ ). Here σ i (τ i ) are Pauli matrices in spin (particle-hole) space, the SOC is L R = 2λ R sin k x , normal band dispersion ξ k = −2t cos k x − µ, and V Z the impurity-induced uniform Zeeman field in the +ẑ-direction. Diagonalizing this Hamiltonian, we find four bands: Since ∆ is k-independent, it is straightforward to show that L R has to vanish at the closing points of the energy gap [33,34]. Thus, gap closings occur at the high symmetry points k x ∈ {0, ±π} and for V c ± Z = (±2 − µ) 2 + ∆ 2 . These gap closings are TPTs with a topological phase appearing between V c − Z and V c + Z [34]. In Fig. 2(a-c), we plot the two lowest energy bands, also referred to as YSR bands due to their impurity origin, as a function of V Z across the lowest TPT. Here, we initially choose a chemical potential at the bottom of the normal state band, µ = −2. Then, the superconducting gap opens in the vicinity of k x = 0, where also the first TPT occurs at V c − Z = ∆. In the trivial phase, V Z < ∆, we find a spin-polarization close to k x = 0, such that the negative and positive energy YSR bands are spinpolarized along theẑ (same as impurity moment) and −ẑ directions, respectively. At the TPT, the energy gap closes and the spin-polarization vanishes at the lowest energies, whereas in the topological phase, V Z > ∆, the z-axis spin-polarization of the positive and negative bands is interchanged. As a consequence, the SP-DOS along the z-axis, plotted in Fig. 2(df), has a positive peak (spin-up, red) for negative energies in the trivial phase but a negative peak (spin-down, blue) in the topological phase, clearly showing how the SP-DOS is interchanged at the TPT and thus offering an easily measurable signature of the TPT.
A physical explanation of the spin interchange between the lowest order energy bands at the TPT is conceived by recalling, that each magnetic impurity in a conventional superconductor induces a pair of fully spin-polarized YSR states [35]. For small V Z , the spin-polarization of a YSR state with negative (positive) energy is aligned (anti-aligned) with the local moment of the impurity, here in +ẑ direction. Increasing V Z , a level crossing takes place and the spin-polarization of the negative and positive energy states is interchanged. When arranging magnetic impurities into a chain, the YSR states overlap and instead form two fully spin-polarized YSR bands. In the absence of SOC, the same spin-interchange effect as for the single impurity YSR states appears also for the YSR bands. However, finite SOC leads to an admixture of spin-up and spin-down states. Still, spin is good quantum number at the high symmetry points k x ∈ {0, ±π}, since the SOC contribution vanishes at these points. Thus, close to k x = 0, and thus at the TPT, we expect a spin-interchange for a chain, as also seen in Fig. 2.
In general, the chemical potential is, however, not at the bottom of the band and we depict a more general situation in Fig. 3 for finite doping. Here, inner and outer band gaps are typically found, labeled by ∆ 1 and ∆ 2 in Fig. 3(a), which are attributed to two helical bands. Starting from small Zeeman fields, the inner gap ∆ 1 shrinks and eventually closes at a TPT at k x = 0 for increasing V Z , see Fig. 3(b), while ∆ 2 remains essentially unaffected [36]. In the trivial phase, we find dominant spin-up polarization for the lowest-energy states, which at the TPT even generates a single sharp peak. The latter is due to the negative energy YSR band having a completely hole-like spin-down component around k x = 0 at the TPT, thus giving no contribution to the DOS. To quantify this behavior and compare it to Fig. 2, we study in detail the Hamiltonian Eq. (2) at k x = 0 for a general µ. We then find that the spin-polarization of negative and positive energy YSR bands at k x = 0 relies only on the ratio of η = |2 + µ|/∆. Whenever η < 1, as in Fig. 2, we find an electron-like behavior for both bands at and around the TPT. However, for η > 1, as in Fig. 3, the spin-down (spin-up) states become completely hole-like (electron-like), see Supplementary material (SM) for more details [37].
Increasing V Z further into the topological phase, ∆ 1 opens again and rapidly becomes larger than ∆ 2 . Therefore, in the topological phase, the sharp spin-up peak moves to higher positive energies, while only spin-down polarization remains at lower energies. Most notably, at negative energies the SP-LDOS is always spin-down polarized beyond the TPT, since the states associated with ∆ 1 are always hole-like for negative energies. Thus, the TPT is inherently connected with an interchange of the bulk spin-polarization for the lowest negative energy bands, as schematically indicated with colored arrows in Fig. 3(d-f) and in full agreement with the earlier results in Fig. 2. Further increasing V Z , the ∆ 2 gap eventually closes at k x = ±π in a second TPT, also with a spin-interchange of the YSR bands, see SM [37].
Ferromagnetic impurity chain.-Having understood the purely 1D limit, we perform numerical calculations for a finite ferromagnetic impurity chain with magnetic moments in thê z-direction (FMC) embedded in a 2D superconducting substrate, see Fig.1(a). The superconducting substrate consists of L = 201 lattice sites in the direction of the l = 101 long chain, and L ⊥ = 21 sites perpendicular to the chain. We here set µ = −2, which puts the 2D system well within a finite doping regime. For visualization purposes, we set ∆ = 0.4, however, the same conclusions hold for smaller values. In the topological phase we find MBSs at the end-points of the chain with a −0.5 significant x-axis spin-polarization, in agreement with earlier results [19,38,39].
Beyond the MBSs spin-polarization, we also find strong zaxis spin-polarization of the in-gap YSR states throughout the chains, and particularly in the central regions of the chains. Focusing on the lowest energy YSR states, we see in Fig. 4(b) that in the trivial phase, the negative energy subgap states are dominantly spin-polarized along theẑ-direction, i.e. parallel to the impurity spin (red), while positive energy states are mostly aligned along −ẑ-direction (blue). Remarkably, in the topological phase this spin-polarization is reversed as seen in Fig. 4(d). Here we point out that in the topological phase the spin-polarization of the positive energy YSR states depends on the Zeeman field and these states can again switch spinpolarization with increasing V Z . However, the negative energy YSR states are always anti-parallel to the impurity moment in the topological phase, see SM [37]. Thus, the spinpolarization in the interior of the chain of the lowest energy YSR states at negative energies acts as a probe of the bulk topology.
We also perform a T-matrix analysis based on an equivalent continuum model for FMCs with varying inter-impurity distances. We find that for dense impurity chains, with a realistic inter-impurity distance of d ∼ 5 Å, the spin-polarization of the low-energy YSR states shows the same spin-interchange in the lowest-energy YSR states as in Fig. 4(b,d). For dilute FMCs, the spin-polarization continues to signal the TPT, although the weak hybridization causes the signal to slightly fade away with increasing inter-impurity distance, see SM [37]. This illustrates how the spin-polarization pinpoints the TPT independent of the inter-impurity distance.
Spin-helical impurity chain.-Next we discuss a spinhelical impurity chain (SHC), also likely experimentally realized [11,16]. As illustrated in Fig. 1(b), for an impurity located at x i the local moment is in-plane and given by S (i) = S x cos (k h x i ) , S y sin (k h x i ) , 0 , where k h = 2π/ with being the pitch of the spin-helix [40]. In Fig. 4(e,g) we plot the x-axis SP-LDOS which demonstrate how the MBSs appear in the topological phase at the end-points of the chain, but notably their spin-polarization is no longer constant. Also, the spin-texture of the low-lying YSR states in the chain interior are alternating between up and down directions for x, y-axes SP-LDOS, following the pitch of the spin-helix of the implanted magnetic impurities. However, motivated by the fact that a SHC is equivalent to a FMC plus an additional SOC [41], we find a way to map the SP-LDOS and still identify the TPT: We evaluate the SP-LDOS along the SHC where at each lattice point i the spin-polarization is projected on S (i): . As illustrated in Fig. 4(f,h) this spin-projection is successful in providing a clear spin-polarization signature of the TPT. Concentrating on the low-energy YSR states at negative energies, this spin-projected SP-LDOS changes from being dominantly spin-up (red) in the trivial phase to spin-down (blue) in the topological phase. Thus spin-projected SP-LDOS for the SHC can be used in exactly the same way as the out-of-plane SP-LDOS for the FMC in predicting the topological phase only based on bulk signatures and fully independent from the existence of the MBSs. Notably, for both the FMC and SHC the relevant spin-polarization direction for predicting the TPT is defined by orientation of the impurity moments in the normal phase and can thus be a priori experimentally determined.
Ferromagnetic impurity island.-Inspired by recent work on 2D impurity islands [13,15,42], we also investigate a ferromagnetic impurity island with all magnetic moments in thê z-direction and on the surface of an s-wave superconductor with SOC. In Fig. 5 we show the SP-LDOS along a line cutting through the impurity island for a spin-polarization along the x-(a,c) and z-axis (b,d) on both sides of the TPT. In the topological phase, chiral edge states appear at the island's boundary, with distinctive x-axis spin-polarization, in agreement with earlier results [42]. By contrast, an interchange of the z-axis spin-polarization of the lowest energy YSR states at negative energies is present across the TPT in the middle of the island: in the trivial (topological) phase these YSR states are (anti-)aligned with the local moment of the magnetic impurities, exactly in the same fashion as for the 1D chains. This result is not limited to the particular parameter choices of Fig. 5. In fact, assuming less doping in the normal state results in an even more pronounced and clear-cut spininterchange signature for the TPT. Thus, measurements of the SP-LDOS along the magnetic impurity direction, allows for determination of the topological phase for both 1D impurity chains and 2D islands.
Conclusion.-In this work we perform analytical and nu- In this Supplementary Material (SM) we provide additional results and analysis supporting the conclusions in the main text. In particular, in Section I we analyze in more detail the purely 1D ferromagnetic system, in Section II we provide additional data for the ferromagnetic chain (FMC) embedded in a surrounding superconductor, and finally in Section III we provide the method and results for the complementary T-matrix calculations.

I. 1D FERROMAGNETIC SYSTEM
In this section we provide additional analytical results supporting the main text conclusions for the spin-polarization of the 1D ferromagnetic system, and in particular in relation to the topological phase transitions (TPTs). A TPT is accompanied by gap closings at high symmetry points. Here we consider the Bogoliubov-de Gennes (BdG) Hamiltonian of a 1D ferromagnetic system H = k Ψ † k H 1D (k)Ψ k with (Eq. (2) in the main text): and Nambu spinor Ψ T k = (c k↑ , c k↓ , c † −k↓ , −c † −k↑ ). It is straightforward to show that by increasing V Z , the Yu-Shiba-Rusinov (YSR) band gap closes subsequently at Γ (k x = 0) and M (k x = ±π) points in the first Brillouin zone [1,2]. We plot the phase diagram for this Hamiltonian in Fig. S1(a). By increasing V Z from zero and for µ < 0, first gap closing occurs at k x = 0 (yellow line) and the second gap closing occurs at k x = ±π (green line), with the grey region in between being the topological phase. For µ > 0, only the order of the TPTs is inverted and therefore, without restricting our results, we assume µ < 0 in the following.
At the Γ point the spin-orbit term, namely L R = 2λ R sin k x , vanishes and the 1D Hamiltonian takes a particularly simple form where ξ 0 = −2−µ is the kinetic energy at k x = 0. Diagonalizing the Hamiltonian Eq. (S2), we find two spin-up and two spin-down FIG. S1. Phase diagram of 1D ferromagnetic system (a) and energy of the YSR bands at high symmetry points of k x = 0 (solid) and k x = ±π (dashed) as a function of V Z (b). Gray regions mark the topological phase. Here ∆ = 0.1 in (a). arXiv:1906.02639v1 [cond-mat.supr-con] 6 Jun 2019 where we define ε 0 ≡ ξ 2 0 + ∆ 2 and the eigenvalues are given by E ± ↑ = V Z ± ε 0 and E ± ↓ = −V Z ± ε 0 . We depict the eigenvalues in Fig. S1(b). Focusing on the lowest energy states, E + ↓ and E − ↑ cross each other at zero energy at V Z = ε 0 . The absolute values of other two eigenvalues increases with increasing V Z , and thus never enter the subgap region. In the same fashion, we find the eigenstates of the Hamiltonian Eq. S1 at k x = ±π where again two branches enter the subgap region, shown in dashed lines in Fig. S1(b).
We next concentrate on the electronic part of the wave function and evaluate the expectation value of the spin operator along theẑ-direction given by ρ z = σ z (τ 0 + τ z )/2 for the two states we are interested in: Although the energy of these two eigenstates varies with V Z , their spin-polarization does not depend on V Z . We plot the spinpolarization given by Eq. S4 in Fig. S2 as a function of the chemical potential µ and for several different values of ∆. The figure clearly shows that for a chemical potential at the bottom of the normal band, i.e. µ = −2, both subgap states acquire a finite electronic spin-polarization. However, moving away from the bottom of the band, the spin-up state (red) becomes fully electron-like and consequently fully spin-polarized, while the spin-down state (blue) becomes fully hole-like and thus rapidly loses its spin-polarization. In fact, for smaller, and thus more realistic, ∆ this change in spin-polarization is even sharper. As a consequence of the spin-polarization being only dependent on the chemical potential µ and superconducting order parameter ∆, we define a new variable η = |2 + µ|/∆ for which we identify two limits: 1. η 1: Both positive-and negative-energy YSR bands have finite spin-polarization, with opposite spin-orientations. Fig. 2 in the main text belongs to this case, since there the chemical potential is µ = −2, and thus η = 0.
2. η 1: Only the spin-up state is dominantly electron-like and acquires a large spin-polarization, while the other state is dominantly hole-like, thus achieving only very minor spin-polarization. Fig. 3 in the main text belongs to this case, since µ = −1.85 and η = 15.
Since the denominator of η is ∆, which is generally the by far smallest energy scale in the problem, these are the only two relevant limits and values in-between would generally require extreme fine-tuning of the chemical potential.

II. FERROMAGNETIC IMPURITY CHAIN
In this section we provide additional data on the spin-polarization for a FMC. In particular, we show in Fig. S3 and Fig. S4 the spin-polarized local density of states (SP-LDOS) for a gradual increase of the magnetic impurity moment V Z . In Fig. S3 we trace through the transition from the topologically trivial into the non-trivial phase at V Z = 2.1 and we plot both the SP-LDOS along the x-axis (S3(a-f)) and z-axis (S3(g-l)). In the trivial phase, the negative low-energy subgap states possess a spin-up polarization along the z-axis and by increasing V Z these states approach the Fermi level, see Figs. S3(g-i). At the TPT, these states finally cross the Fermi level and go to positive energy, see Figs. S3(j-l). The low-energy spin-down states have a complete reversed behavior, where they start from positive energy in the trivial phase and move on to negative energy in the topological phase. Therefore, the spin-polarization of both negative and positive low-energy states shows a spin-interchange across the TPT. This effect coincides with the appearance of Majorana bound states (MBSs) in the topologically non-trivial phase at the end-points of the impurity chain (see Figs. S3(d-f)).
If we continue increasing the Zeeman field V Z further, as plotted in Fig. S4, the spin-up polarized states move up to higher energies while some spin-down states with positive energy move down towards the Fermi level, see Figs. S4(g-i). Therefore, in the topological phase, the positive-energy states close to Fermi level are not necessarily spin-up polarized. However, the negative-energy states remain spin-down polarized, and thus still clearly signal the topological phase. Finally, at V Z ∼ 3.4, the second TPT from topological into trivial phase occurs, see Fig. S4(j). After the second TPT, there are no MBSs at the end-points of the impurity chain and also almost all the YSR states at positive (negative) energies are spin-up (spin-down) polarized. Thus for the second TPT the spin-polarization cannot be used to determine the topological phase. This is however not a limitation, since this regime requires such large magnetic moments as to completely suppress superconductivity.

III. T-MATRIX FORMALISM
In this section we show a T -matrix solution for magnetic impurity chains absorbed on a superconducting substrate. This formalism is efficient when the number of impurities is small, or at least discrete, and embedded in a continuum. Generally, we can write the Green's function of the system H = H 0 + V as Here H 0 and G 0 are the Hamiltonian and Green's function of the system without impurities, respectively, T = (V −1 −G 0 ) −1 is the T -matrix, which include all the effects of the impurities encoded in V. In the above equations, all the elements are matrices for a multi-impurity system. In particular, we consider a superconducting substrate with spin orbit coupling (SOC) λ. The substrate Hamiltonian in the Nambu basis can be written as where the SOC is modelled by H SOC = λ(p y σ x − p x σ y )τ 0 , while ξ k is the normal-state dispersion relation for free electrons.
Here σ and τ are the Pauli matrices in the spin and Nambu basis, respectively, with the latter explicitly written in matrix form for the first part of Eq. (S6). Similar to the lattice calculations in the main text, we replace the effect of the impurities by an effective Zeeman field V Z along the z-axis, thus ignoring dynamical processes, such as the Kondo effect. The Hamiltonian for the impurities can therefore be written as The dressed Green's function G(r i , r j , ω) can now be provided in terms of the bare Green's function G 0 (r i − r j , ω) through the T-matrix, where the bare propagator are expressed in terms of Hankel functions [3,4]. As in the main text, we consider a chain of magnetic impurities absorbed on the superconducting substrate, with the effective Zeeman field of all impurities pointing in theẑ-direction and with equal magnitude. We show the resulting LDOS and z-axis spin-polarization in Fig. S5. For small V Z the system remains in the trivial phase and the YSR states are gapped without any states emerging deep within the gap. By increasing V Z , the YSR states approach the Fermi level and eventually cross each other and the system is driven through the TPT. In the topological phase MBSs emerge at the chain end-points, while the electronic structure remains gapped around the center of the chain. Following the z-axis spin-polarization we find in the trivial phase that the negative energy states (see red arrows) have a spin-up polarization, whereas positive energy states have spin-down polarization. By tuning the parameters such that the system goes through the TPT, the spin-polarization of the low-energy YSR states is interchanged. This change in spin-polarization becomes more visible in the dense impurity chain where inter-impurity scattering is stronger, as in Fig. S5. For dilute chains the spin-polarization also signals the TPT, despite weak hybridization of YSR states, as shown in Fig. S6. In this case, the spin-polarization signal slightly fades away, nevertheless, it still shows a clear signature of TPT.