New types of topological superconductors under local magnetic symmetries

Abstract

We classify gapped topological superconducting (TSC) phases of one-dimensional quantum wires with local magnetic symmetries, in which the time-reversal symmetry is broken, but its combinations with certain crystalline symmetries, such as , , and , are preserved. Our results demonstrate that an equivalent BDI class TSC can be realized in the or superconducting wire, which is characterized by a chiral Z^c invariant. More interestingly, we also find two types of totally new TSC phases in the and superinducting wires, which are beyond the known AZ class, and are characterized by a helical Z^h invariant and Z^h⊕Z^c invariants, respectively. In the Z^h TSC phase, Z pairs of Majorana zero modes (MZMs) are protected at each end. In the case, the MZMs can be either chiral or helical, and even helical-chiral coexisting. The minimal models preserving or symmetry are presented to illustrate their novel TSC properties and MZMs.

Topological superconducting phases of one-dimensional quantum wires with local magnetic symmetries are classified and new types of Majorana modes are discussed.

INTRODUCTION

Topological superconductors (TSCs) are new kinds of topological quantum states, which are fully superconducting gapped in the bulk but support gapless excitations called Majorana zero modes (MZMs) at the boundaries [1–5]. As analogues of the famous Majorana fermions [6], MZMs are their own antiparticles, and are proposed as the qubits of topological quantum computation because of their nonlocal correlation and non-Abelian statistic nature [7–10]. Hence, searching for TSC materials with MZMs is now an important topic in condensed matter physics, and a series of schemes have been proposed in the last decade, including the proximity effect on the surface of topological insulators [11–16] and the recently predicted intrinsic superconducting topological materials [17–25].

To identify whether a superconductor is topologically nontrivial, we should first ascertain to what topological classification it belongs. The topological classification can be highly enriched by symmetries, including time-reversal symmetry , particle-hole symmetry and especially the crystalline symmetries [26–35]. The topology for noninteracting Hamiltonians of the 10 Altland–Zirnbauer (AZ) classes with or without and has been well classified [26,27]. Particularly, the Bogoliubov–de Gennes (BdG) Hamiltonians of the one-dimensional (1D) superconductors, with breaking or preserving, belong to the D and DIII classes, respectively. In both cases we only have the Z₂ classification. In addition to these local symmetries, crystalline symmetries are considered for each AZ class to generalize the topological classification [28–31], and the topological crystalline superconductors protected by mirror reflection symmetry [32,33] or rotational symmetries  [34,35] have been proposed. Furthermore, the TSC phase protected by the magnetic symmetries and has been discussed in [34,36–38]. Nevertheless, the topological classification of superconductors with general magnetic symmetries is still an open question, and the corresponding theoretical analysis is necessary for understanding and searching for new magnetic TSC materials and MZMs.

In this paper, we focus on the topological phases of gapped superconducting wires with local magnetic symmetries (LMSs), in which is broken, but its combinations with certain crystalline symmetries—those leaving each site invariant, including , , and —are preserved. Our analysis shows that, with or symmetry, an effective BDI class TSC can be realized, which is characterized by a chiral Z^c topological invariant and protects an integer number of MZMs at each end. Remarkably, two totally new TSC phases are discussed in the superconducting wire with or symmetry. In the case, the BdG Hamiltonian is characterized by a helical Z^h invariant, which can protect Z pairs of MZMs at each end. The BdG Hamiltonian with symmetry possesses Z^h⊕Z^c invariants, which means that the helical and chiral MZMs can coexist in a single wire system. The minimal models with the LMSs and are presented separately, in which the TSC with helical MZMs and the TSC with helical-chiral coexisting MZMs are discussed. Our results may facilitate the ongoing search for novel TSCs.

TOPOLOGICAL CLASSIFICATION OF GAPPED SUPERCONDUCTING WIRE

We first introduce the LMSs for a magnetic superconducting wire along the direction. Among the 1D space groups (the so-called rod group) [39], the local symmetry operators include the mirror reflection M_x and the n-fold rotation C_nz with n = 2, 3, 4, 6. Combined with , we obtain four types of LMSs, , , and , as tabulated in Table 1. We consider a 1D BdG Hamiltonian preserving . Note that the operation of does not change the positions of electrons. Hence, it acts on the BdG Hamiltonian like a time-reversal operator

(1)

Here, LMS takes the form with being the complex conjugate operator and U being a unitary matrix determined by the spatial operation and spin flipping. We employ the convention that and set , where the Pauli matrix τ_x acts on the particle-hole degree of freedom. Combining and leads to a chiral symmetry . Both and act on the BdG Hamiltonian as

(2)

(3)

Table 1.

The topological classification of the 1D gapped superconducting systems with the LMSs , , and , respectively. 2×AIII form a helical Z^h classification.

	(n = 2)	(n = 2)	(n = 4)	(n = 6)
	1	1	−1	1
	1	1	1	1
	1	1	−1	1
Invariant	Z c	Z c	Z h	Z h ⊕Zc
	(BDI)	(BDI)	(2×AIII)	(2×AIII ⊕ BDI)

The chiral symmetry has a series of eigenvalue pairs ±s₁, ±s₂, … and it can take a block-diagonal form as , where the subscript ±s₁ denotes the direct sum of eigenvector spaces |s₁〉 and | − s₁〉. The anticommute relation (3) means that H_BdG(k) can be block diagonalized according to the eigenvalues of . In other words, H_BdG(k) can adopt the form . Hence, the topological classification of the whole Hamiltonian is decomposed into examining the topology of each block and their compatibility. For each block Hamiltonian , its topology is equivalent to either the BDI or the AIII class, depending on the chiral symmetry eigenvalue s. To be specific, when s is a real number, is invariant under or , which means that it belongs to the BDI class and possesses a Z invariant expressed as = N_s − N_−s, where the N_±s are the numbers of MZMs with chiral symmetry eigenvalue ±s, respectively. Additionally, when s is a complex number, is transformed into under or . Hence, the () belongs to the AIII class that is characterized by a Z invariant = N_s − N_−s (), which is equal to the number of MZM pairs on each wire end.

We next consider the compatibility between the different MZMs possessing different eigenvalues. To do this, we introduce a coupling term m|s₁〉〈s₂|, which satisfies the chiral symmetry, i.e. . Here m is a perturbation parameter, and |s₁〉 and |s₂〉 are the eigenstates of . Then we see that m can be nonzero only when , which means that MZMs within one block having chiral eigenvalues s and −s can couple to each other and be eliminated. However, MZMs from different blocks are noninterfering due to the protection of . Therefore, the topological classification of the whole BdG Hamiltonian is determined by the summation of the topology for each block. We summarize the topological classification of 1D gapped superconductors in Table 1 and analyse each case in the following.

1. and cases. These two cases are equivalent to the BDI class with and . The chiral topological invariant = N₁ − N₋₁ ∈ Z is given by the winding number [5,26]

   (4)

   Here W(k) is a unitary matrix that diagonalizes the BdG Hamiltonian and θ(k) is the phase angle of Det[W(k)]. The identity Tr[ln(W)] = ln(Det[W]) is used to derive the above equation. These results agree well with the conclusions reached in previous studies [34,36–38,40,41].

2. case. The chiral symmetry satisfies and has eigenvalues ±e^±iπ/4 (see Fig. 1). We can conclude that the topological invariants are given by (or . The TSC phase is hence characterized by the helical topological invariant ∈ Z, which means that the MZMs always appear in Kramers pairs. This is obviously different from the chiral Z invariant in the BDI class, in which the MZMs can arise one by one as Z increases. To distinguish the chiral Z and helical Z invariants, we use Z^c and Z^h in the following. The Z^h TSC phase of the -preserving wire can be understood from the following perspective. The BdG Hamiltonian can be block diagonalized into two sectors according to the eigenvalues ±i of as H_i(k)⊕H_−i(k). Both and can map these two sectors to each other. However, their combination, i.e. the chiral symmetry , keeps each sector invariant. As a consequence, each sector belongs to the AIII class, whose Z^c topological invariant can be calculated by exploiting (4). Yielding to the symmetry, the Z^c invariants of two sectors must be equal, which finally gives a Z^h invariant for the whole BdG Hamiltonian. That is, the topological invariant is given by the winding number of each C₂ eigenvalue sector as defined in (4).

3. case. We have and . As illustrated in Fig. 1, the chiral symmetry has eigenvalues ±e^±iπ/3, ±1. The topology is characterized by Z^h⊕Z^c invariants that are given by and N₁ − N₋₁, respectively. Similar to the case, the BdG Hamiltonian can be block diagonalized as according to the eigenvalues e^±i2π/3, 1 of . The and sectors both belong to the AIII class, forming a Z^h classification together, whereas the H₁ sector itself forms a Z^c classification (i.e. BDI class) with and an effective . Therefore, the topology of the whole BdG Hamiltonian is classified by Z^h⊕Z^c, whose topological invariants (^h, ^c) are given by the winding numbers of the and H₁ sectors, respectively. As a consequence, in a 1D superconducting wire with the LMS , the helical and chiral MZMs can coexist. Such novel TSC phase stimulate further interests in the manipulation of such helical-chiral coexisting MZMs [42–44].

Figure 1.

The eigenvalues of and their transformations in the and cases. Complex conjugating partners s and s^* are related by the LMSs and always coexist. A perturbation term can be introduced to couple the chiral states with opposite eigenvalues, as illustrated by the red, blue and black double-head arrows.

MODEL REALIZATION

To illustrate the TSC phase with the LMS , we construct a 1D antiferromagnetic chain along the direction, as shown in Fig. 2(a), where each unit cell contains four subsites and each subsite is occupied by one spin polarized s orbital. We consider that the intra-cell coupling between the same spin states is much larger than the spin-orbit coupling, and thus the four orbitals are well split into two double-degenerate manifolds, as illustrated in Fig. 2(a). More details of the full model have been given in the online supplementary material. Here, to capture the topological phase of the model, we take the |p_x, ↑〉 and |p_y, ↓〉 subspaces to build an effective tight-binding model. Up to the nearest-neighbor hopping, it can be written as

(5)

where t = |t|e^iα is the complex hopping, μ is the chemical potential and σ acts on the orbital degree of freedom of the |p_x, ↑〉 and |p_y, ↓〉 states. The is given by . Note that the hopping terms between opposite spins are prohibited by the C₂ symmetry. The s-wave pairing Hamiltonian takes the form

(6)

with Δ = |Δ|e^iφ. The pairing terms between the same spin are also prohibited by the C₂ symmetry.

Figure 2.

(a) A -preserving superconducting wire aligned along the direction, in which the red and blue dots denote the spin up (+) and spin down (−) polarized s orbitals, respectively. The intra-cell coupling between the same spin orbitals is much larger than the spin-orbit coupling, which split the four states into one symmetric (SY) manifold and one antisymmetric (AS) manifold. Both manifolds are double degenerate. For simplicity, only the AS manifold is considered in our tight-binding model (5). (b) The topological phase diagram of (8) as the function of μ and δ, in which 0, ±1 are the winding numbers, μ is the chemical potential and δ = π/2 + φ − α is the phase difference between the coefficients of τ_y and τ.

In the Nambu basis , and are given by and , respectively, which give . The BdG Hamiltonian anticommutes with and takes a block-diagonal form as

(7)

with

(8)

Then the spectrum is given by

(9)

Note that the two blocks in (7) are Kramers pairs related by the symmetry and have the same winding number. A straightforward way to determine the topology is to calculate the winding number using (4) for the upper or lower blocks. Here we provide a much simpler way to obtain by analogizing the coefficients of the block Hamiltonians with elliptically polarized lights, whose electric field is described by E_x = A_xcos (kz − ωt), E_y = A_ycos (kz − ωt + δ). In the following we analyze the winding number of H_i(k), where the coefficients of the Pauli matrices are h − μ/2 = |t|cos (k + α) and h_y = |Δ|sin (k + φ). When |t||sin δ| > μ/2 (<μ/2), the parameter curve of h_y(k) and h(k) will (not) wind around the zero point h = h_y = 0 (we assume that μ > 0 for simplicity), and the superconducting wire is in a topological nontrival (trivial) phase. Furthermore, when δ ∈ (0, π) [δ ∈ (−π, 0)], we have a left-handed (right-handed) parameter curve, and the topological phase is characterized by winding number +1 ( − 1). The phase diagram in the δ − μ parameter space is plotted in Fig. 2(b). We point out that, when next-nearest-neighbor hopping and pairing are considered, the competition with nearest-neighbor hopping and pairing gives rise to the opportunity for TSC phase with higher winding numbers.

In the nontrivial TSC phase, the open quantum wire traps integer pairs of MZMs at its ends. By using t = 1, Δ = 1.3e^iπ/3, μ = 0.2, we observe two pairs of MZMs in total on the open wire spectrum, as shown in Fig. 3(b), which is in contrast with the gapped bulk spectrum in Fig. 3(a). These MZMs can also be solved from the continuous low-energy model [45]. Here, we need only consider H_i(k) since the other block in (7), as well as its zero energy solution, can be obtained by a transformation. By assuming that the wire is placed on the > 0 side, the low-energy massive Dirac Hamiltonian close to k = π/2 is given by

(10)

Its zero energy solution Ψ₁ and the -related partner are given by

(11)

(12)

These two states are the eigenstates of the chiral symmetry with eigenvalues e^±iπ/4, respectively. Therefore, they are immune to perturbations preserving (and ). Their combinations give the protected MZMs as γ₁ = Ψ₁ + Ψ₂ and γ₂ = i(Ψ₁ − Ψ₂).

Figure 3.

The bulk spectrum and MZMs in the -preserving TSC model. (a) The gapped bulk spectrum of the -preserving TSC phase with t = 1, Δ = 1.3e^iπ/3, μ = 0.2. (b) The corresponding spectrum of (a) with an open boundary on both sides, in which four MZMs appear at zero energy.

The -preserving BdG Hamiltonian can be easily generalized to a invariant quantum wire. For this purpose, we assume that the chiral symmetry is expressed as . The BdG Hamiltonian can then be written in three blocks with

(13)

The first two blocks are Kramers pairs and take the same form as in (8), while the last block is transformed to itself under or . For this BdG Hamiltonian, the topology is characterized by Z^h⊕Z^c numbers, which correspond to the number of helical and chiral MZMs, respectively. The topological phase diagram of the helical part Hamiltonian is the same as in Fig. 2(b). The chiral part is determined by the winding number of H₁(k), which gives a nontrivial TSC phase when |t′| > μ/2.

CONCLUSION

We have classified the TSC phases of quantum wires with LMSs. In the case of or , an equivalent BDI class TSC can be realized [37,38,40]. More importantly, we find two new types of TSC phases in the superconducting wire with or , which are beyond the already known AZ classes and can be characterized by Z^h or Z^h⊕Z^c topological invariants, respectively. These results not only enrich the variety of the 1D TSC, but also provide luxuriant building blocks for the construction of new type 2D and 3D TSCs, by following the general method proposed in [46]. For example, one can couple the 1D TSCs in the y direction to construct a 2D TSC. The high symmetry lines k_y = 0 and k_y = π in momentum space preserve the 1D LMS. With proper parameters, the k_y = 0 and k_y = π lines can belong to distinct topological phases, and result in the gapless propagating Majorana edge states connecting the conducting bands and valence bands. The superconductivity and antiferromagnetism coexisting SmOFeAs [47,48] with a proper magnetic configuration satisfying C₄T symmetry is a possible material to study the 1D TSC phase on its high-symmetry lines.

FUNDING

This work was supported by the National Key Research and Development Program of China (2018YFA0307000) and the National Natural Science Foundation of China (11874022).

AUTHOR CONTRIBUTIONS

Z.-D.S. and G.X. proposed the project. J.-Y.Z. carried out the topological classification and conceived the model. J.-Y.Z., Q.X., Z.-D.S. and G.X. analyzed the results. All authors contributed to the manuscript writing.

Conflict of interest statement. None declared.