Time-reversal-invariant topological superconductivity induced by repulsive interactions in quantum wires

Haim, Arbel; Keselman, Anna; Berg, Erez; Oreg, Yuval

doi:10.1103/physrevb.89.220504

Cited by 115 publications

(129 citation statements)

References 46 publications

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“…It has been well demonstrated that the class DIII topological invariant can take a nontrivial value only if there exist at least two bands with opposite sign of the pairing potential [38,40,41,50,61], or, equivalently, that the anomalous component of the self-energy has both positive and negative eigenvalues. It was also shown in Ref.…”

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confidence: 99%

“…In one dimension (1D), where superconductivity is required to be induced by the proximity effect, it has been shown that a nontrivial topological phase in class DIII can be realized by proximity coupling a noninteracting multichannel Rashba nanowire to an unconventional superconductor [36][37][38][39] or to two conventional superconductors forming a Josephson junction with a phase difference of π [40]. Alternatively, an effective π -phase difference can be induced in a multichannel Rashba nanowire with repulsive electron-electron interactions [41] or in a system of two topological insulators coupled to a conventional superconductor via a magnetic insulator [42]. It has also been proposed to realize class DIII topological superconductivity in a system of two Rashba nanowires [43][44][45] or two topological insulators [46] coupled to a single conventional superconductor, but repulsive interactions are also necessary to reach the topological phase in these setups, which require a strength of induced crossed Andreev (interwire) pairing exceeding that of the direct (intrawire) pairing [47][48][49].…”

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DIII topological superconductivity with emergent time-reversal symmetry

et al. 2017

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We find a class of topological superconductors which possess an emergent time-reversal symmetry that is present only after projecting to an effective low-dimensional model. We show that a topological phase in symmetry class DIII can be realized in a noninteracting system coupled to an s-wave superconductor only if the physical time-reversal symmetry of the system is broken, and we provide three general criteria that must be satisfied in order to have such a phase. We also provide an explicit model which realizes the class DIII topological superconductor in 1D. We show that, just as in time-reversal invariant topological superconductors, the topological phase is characterized by a Kramers pair of Majorana fermions that are protected by the emergent time-reversal symmetry. DOI: 10.1103/PhysRevB.96.161407 Introduction. Topological superconductors have been intensively pursued in recent years [1][2][3] because the Majorana fermions which are localized to their boundaries have potential applications in the development of a topological quantum computer [4,5]. The most promising proposals to date for engineering topological superconductivity involve coupling a conventional superconductor either to a nanowire with Rashba spin-orbit interaction that is subjected to an external magnetic field or to a ferromagnetic atomic chain [27][28][29][30][31][32][33][34].Additionally, there have been several proposals to engineer topological superconductors in symmetry class DIII. Such systems possess both particle-hole symmetry and time-reversal symmetry [35], with the presence of time-reversal symmetry ensuring that the Majorana fermions existing at the boundaries of class DIII topological superconductors come in Kramers pairs. In one dimension (1D), where superconductivity is required to be induced by the proximity effect, it has been shown that a nontrivial topological phase in class DIII can be realized by proximity coupling a noninteracting multichannel Rashba nanowire to an unconventional superconductor [36][37][38][39] or to two conventional superconductors forming a Josephson junction with a phase difference of π [40]. Alternatively, an effective π -phase difference can be induced in a multichannel Rashba nanowire with repulsive electron-electron interactions [41] or in a system of two topological insulators coupled to a conventional superconductor via a magnetic insulator [42]. It has also been proposed to realize class DIII topological superconductivity in a system of two Rashba nanowires [43][44][45] or two topological insulators [46] coupled to a single conventional superconductor, but repulsive interactions are also necessary to reach the topological phase in these setups, which require a strength of induced crossed Andreev (interwire) pairing exceeding that of the direct (intrawire) pairing [47][48][49]. While it would be beneficial to engineer a DIII topological superconductor in a noninteracting 1D system coupled to a single conventional superconductor, as such a setup could avoid relying on unconventional superc...

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DIII topological superconductivity with emergent time-reversal symmetry

et al. 2017

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“…Hence the admissible solutions decay as exp(−|∆ 0 | r ⊥ ) in the bulk, where i z = i ∆ 0 can be identified with the EP solution with complex k ⊥ . In 1d, numerous lattice models have been studied which can support one [22] or multiple MBSs [27][28][29][30][31][32] at one edge of an open chain. For such models, one can show that the chiral MBS solutions at an edge, obtained by the transfer matrix approach formalism [22,[27][28][29]33] in the lattice space, have the same relation with the EP solutions obtained in the complex k-space.…”

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Exceptional points for chiral Majorana fermions in arbitrary dimensions

Mandal¹

2015

EPL

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Certain real parameters of a Hamiltonian, when continued to complex values, can give rise to singular points called exceptional points (EP 's), where two or more eigenvalues coincide and the complexified Hamiltonian becomes non-diagonalizable. We show that for a generic d-dimensional topological superconductor/superfluid with a chiral symmetry, one can find EP 's associated with the chiral zero energy Majorana fermions bound to a topological defect/edge. Exploiting the chiral symmetry, we propose a formula for counting the number (n) of such chiral zero modes. We also establish the connection of these solutions to the Majorana fermion wavefunctions in the position space. The imaginary parts of these momenta are related to the exponential decay of the wavefunctions localized at the defect/edge, and hence their change of sign at a topological phase transition point signals the appearance or disappearance of a chiral Majorana zero mode. Our analysis thus explains why topological invariants like the winding number, defined for the corresponding Hamiltonian in the momentum space for a defectless system with periodic boundary conditions, captures the number of admissible Majorana fermion solutions for the position space Hamiltonian with defect(s). Finally, we conclude that EP 's cannot be associated with the Majorana fermion wavefunctions for systems with no chiral symmetry, though one can use our formula for counting n, using complex k solutions where the determinant of the corresponding BdG Hamiltonian vanishes. Introduction -The Hamiltonian operator can contain certain real parameters, which on being continued to complex values, give rise to singular points where the operator becomes non-diagonalizable. These are called exceptional points (EP 's), at which two or more of the eigenvalues coalesce and the norm of at least one eigenvector of the complexified Hamiltonian vanishes [1][2][3][4][5][6][7]. The concept of EP 's is similar to that of a degeneracy point, but with the important difference that all the energy eigenvectors cannot be made mutually orthogonal. In previous works, EP 's have been used [8][9][10][11][12] to describe topological phases of matter associated with zero energy Majorana bound states (MBSs) in one-dimensional (1d) topological superconductors/superfluids.The first-quantized Hamitonians describing fully gapped noninteracting topological insulators and superconductors in d-dimensions can be classified into ten symmetry classes [13,14] in terms of nonspatial symmetries, i.e., symmetries that act locally in the position space, namely time-reversal symmetry (TRS), particlehole symmetry (PHS), and chiral symmetry. Recently, it has been realized [15][16][17][18][19][20][21] that the complete classification should include topological states protected by crystalline symmetries (such as mirror reflections and rotations), which are spatial symmetries acting nonlocally in the position space.A superconductor is described by a Bogoliubov de Gennes (BdG) Hamiltonian (H BdG ), which has an exact PHS, in add...

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“…The criterion exploits the behavior of the ground state energy of a system of N, and N AE 1 particles, for both periodic and antiperiodic boundary conditions. The emergence of topological order in a superconducting wire, closed in a ring and described in mean field, is associated [6,[39][40][41][42][43][44]. Any spin-active superconductor, topologically trivial or not, may experience a crossing of the ground state energies for even and odd number of electrons [45][46][47][48].…”

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Many-Body Characterization of Particle-Conserving Topological Superfluids

et al. 2014

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What distinguishes trivial superfluids from topological superfluids in interacting many-body systems where the number of particles is conserved? Building on a class of integrable pairing Hamiltonians, we present a number-conserving, interacting variation of the Kitaev model, the Richardson-Gaudin-Kitaev chain, that remains exactly solvable for periodic and antiperiodic boundary conditions. Our model allows us to identify fermion parity switches that distinctively characterize topological superconductivity (fermion superfluidity) in generic interacting many-body systems. Although the Majorana zero modes in this model have only a power-law confinement, we may still define many-body Majorana operators by tuning the flux to a fermion parity switch. We derive a closed-form expression for an interacting topological invariant and show that the transition away from the topological phase is of third order. DOI: 10.1103/PhysRevLett.113.267002 PACS numbers: 74.20.-z, 03.65.Vf, 74.45.+c, 74.90.+n In recent years, the physics of Majorana zero-energy modes has become a key subfield of condensed matter physics [1][2][3][4]. On the theory side, a central result is the bulkboundary correspondence [5] that associates Majorana zero modes to the boundary of (or defects in) a topologically nontrivial superconductor, with the Kitaev chain as a prototypical example [6]. The mathematical formalism underlying this correspondence relies on the symmetries and topological invariants of the Bogoliubov-de Gennes equation [7], a mean-field description of the superconducting state in which the conservation of the number of fermions (a continuous symmetry) is broken down to a discrete symmetry, the conservation of fermion-number parity. Majorana zero modes are directly linked to the spontaneous breaking of this residual discrete symmetry [8].As the experimental side of Majorana physics continues to develop [9][10][11][12][13][14], it becomes crucial to unveil how much of the mean-field picture survives beyond its natural limits. This has motivated recent studies [15][16][17][18][19][20][21], with the focus on the anomalous 2Φ 0 ¼ h=e flux periodicity of the Josephson effect-the hallmark of a topological superconductor [6].A main thrust of this Letter is the characterization of interacting many-body, number-conserving, topological superconductors, or superfluids, leading to a subsequent analysis on the meaning of Majorana zero modes beyond mean field. The theoretical study of any interacting quantum system is hampered by the exponential growth of the Hilbert space with the number of particles. An additional complication of superconducting systems is the lack of simple principles to guide the design of particlenumber conserving models, in which the phase of the order parameter is not a good quantum number. To overcome both obstacles, we have constructed an exactly solvable, number-conserving variation of the Kitaev chain. Because our model belongs to a class of integrable pairing models [22-24] based on the s-wave reduced BCS Hamiltonian firs...

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Time-reversal-invariant topological superconductivity induced by repulsive interactions in quantum wires

Cited by 115 publications

References 46 publications

DIII topological superconductivity with emergent time-reversal symmetry

DIII topological superconductivity with emergent time-reversal symmetry

Exceptional points for chiral Majorana fermions in arbitrary dimensions

Many-Body Characterization of Particle-Conserving Topological Superfluids

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