Many-Body Characterization of Particle-Conserving Topological Superfluids

Abstract

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...

Random-matrix theory of Majorana fermions and topological superconductors

“…It is reassuring that for special choices of the pairing interaction the Kitaev model of topological superconductivity can be solved exactly, without recourse to the mean-field approximation (Ortiz et al, 2014). Considering a chain of L sites with nearest-neighbor hopping energy t 0 and pairing interaction g 0 η(n − m) between sites n and m, the Hamiltonian takes the form…”

Multiparticle content of Majorana zero modes in the interactingp-wave wire

“…There is therefore ample reason to explore how much of this quasi-particle picture remains valid beyond the confines of mean-field superconductivity. For example, considerable progress has been made towards developing number preserving theories of the Majorana modes, [15][16][17][18][19][20] as well as a growing body of work which examines how free-topological superconducting phases are affected by the addition of interacting electron-electron terms. [21][22][23][24][25][26][27][28][29][30][31][32][33] One aspect of this latter story is concerned with the stability and structure of the Majorana zero-modes themselves and how they are affected by the presence of density-density interaction terms that break the exactly solvable nature of the underlying model.…”

Localized Majorana-Like Modes in a Number-Conserving Setting: An Exactly Solvable Model

“…Therefore, approximations such as bosonization [10][11][12], or numerical approaches [13] were invoked. An exactly solvable model of a topological superconductor in a number conserving setting, which would directly complement Kitaev's scenario, is missing (see however [14]). …”

“…Kitaev's model is an effective mean-field model and its Hamiltonian does not commute with the particle number operator. Considerable activity has been devoted to understanding models supporting Majorana edge modes in a number-conserving setting [10][11][12][13][14], as in various experimental platforms (e.g. solid state [10,11] or ultracold atoms [12,13]) this property is naturally present.…”