We study the properties of one-dimensional topological superconductors under the influence of generic boundary conditions mimicking the coupling with external environments. We identify a general four-parameter classification of the boundary effects and show that particle-hole and reflection symmetries can be broken or preserved by appropriately fixing the boundary parameters. When the particle-hole symmetry is broken, the topological protection of the edge modes is lost due to the hybridization with the external degrees of freedom. We assess the robustness of the edge modes in the various regimes by considering different quantifiers of topological properties. In particular, we investigate the resilience of the long-distance, edge-to-edge quantum mutual information and squashed entanglement, measuring the nonlocal correlations of the Majorana excitations. Besides their relevance for the open dynamics of topological systems, these results may provide a useful guide to the appropriate embedding of low-dimensional topological systems on nanodevices in realistic conditions.
We study the properties of one-dimensional topological superconductors under the influence of generic boundary conditions mimicking the coupling with external environments. We identify a general four-parameter classification of the boundary effects and show that particle-hole and reflection symmetries can be broken or preserved by appropriately fixing the boundary parameters. When the particle-hole symmetry is broken, the topological protection of the edge modes is lost due to the hybridization with the external degrees of freedom. We assess the robustness of the edge modes in the various regimes by considering different quantifiers of topological properties. In particular, we investigate the resilience of the long-distance, edge-to-edge quantum mutual information and squashed entanglement, measuring the nonlocal correlations of the Majorana excitations. Besides their relevance for the open dynamics of topological systems, these results may provide a useful guide to the appropriate embedding of low-dimensional topological systems on nanodevices in realistic conditions.
Identifying entanglement-based order parameters characterizing topological systems, in particular topological superconductors and topological insulators, has remained a major challenge for the physics of quantum matter in the last two decades. Here we show that the end-to-end, long-distance, bipartite squashed entanglement between the edges of a many-body system, defined in terms of the edge-to-edge quantum conditional mutual information, is the natural nonlocal order parameter for topological superconductors in one dimension as well as in quasi-one-dimensional geometries. For the Kitaev chain in the entire topological phase, the edge squashed entanglement is quantized to ln(2)/2, half the maximal Bell-state entanglement, and vanishes in the trivial phase. Such a topological squashed entanglement exhibits the correct scaling at the quantum phase transition, is stable in the presence of interactions, and is robust against disorder and local perturbations. Edge quantum conditional mutual information and edge squashed entanglement defined with respect to different multipartitions discriminate topological superconductors from symmetry breaking magnets, as shown by comparing the fermionic Kitaev chain and the spin-1/2 Ising model in transverse field. For systems featuring multiple topological phases with different numbers of edge modes, like the quasi-1D Kitaev ladder, topological squashed entanglement counts the number of Majorana excitations and distinguishes the different topological phases of the system. In fact, we show that the edge quantum conditional mutual information and the edge squashed entanglement remain valid detectors of topological superconductivity even for systems, like the Kitaev tie with long-range hopping, featuring geometrical frustration and a suppressed bulk-edge correspondence.
Topological superconductors are believed to host exotic quasiparticle excitations known as Majorana zero modes (MZMs), with much of the evidence based on BCS mean-field theory. The direct application of mean-field arguments is tenuous in finite, isolated systems relevant in some experiments. Here, we develop a new correlation-based method for identifying MZMs in interacting, number-conserving systems. Using the density matrix renormalization group, we study fermion number-conserving models with long-range interactions, which under periodic boundary conditions exhibit robust topological and nontopological superconductivity, tuned by the strength of interaction [Ortiz et al., Phys. Rev. Lett. 113, 267002 (2014)PRLTAO0031-900710.1103/PhysRevLett.113.267002]. We find evidence that, on the topological side, Majorana edge modes appear in open chains, manifesting as the vanishing of the energy splitting between odd- and even-parity ground states with increasing system size. Additionally, off-diagonal two-point correlation functions show nonlocal, parity-dependent edge effects. These correlations reveal the spatial structure of Majorana modes in the many-body wave function. We show that the correlation diagnostic applies broadly, including to short-range interacting models, where topological superconductivity is more fragile due to the absence of a bulk excitation gap.
Topological phases which host Majorana fermions can not be identified via local order parameters. We give simple nonlocal order parameters to distinguish quasi-one-dimensional (1D) topological superconductors of spinless fermions, for any interacting model in the absence of time reversal symmetry. These string or ``brane'' order parameters are natural for measurements in cold atom systems using quantum gas microscopy. We propose them as a way to identify symmetry-protected topological phases of Majorana fermions in cold atom experiments via bulk rather than edge degrees of freedom. Subsequently, we study two-dimensional (2D) topological superconductors via the quasi-1D limit of coupling N identical chains on the cylinder. We classify the symmetric, interacting topological phases protected by the additional ZN translation symmetry. The phases include quasi-1D analogs of (i) the p+ip chiral topological superconductor, which can be distinguished up to the 2D Chern number mod 2, and (ii) the 2D weak topological superconductor. We devise general rules for constructing nonlocal order parameters which distinguish the phases. These rules encode the signature of the fermionic topological phase in the symmetry properties of the terminating operators of the nonlocal string or brane. The nonlocal order parameters for some of these phases simply involve a product of the string order parameters for the individual chains. Finally, we give a physical picture of one of the topological phases as a condensate of certain defects, which motivates the form of the nonlocal order parameter and is reminiscent of higher dimensional constructions of topological phases.
The time evolution of topological systems is an active area of interest due to their expected applications in fault-tolerant quantum computing. Here, we analyze the dynamics of a noninteracting spinless fermion chain in its topological phase, quenched out of equilibrium by a Hamiltonian belonging to the same symmetry class. Due to particle-hole symmetry, the bulk properties of the system remain intact throughout its evolution. However, the boundary properties may be drastically altered, with the initially localized Majorana edge modes extending across the chain. Up to a timescale t∗, identified by area-law behavior of the entanglement entropy, these extended edge modes are an example of exotic effects in topological systems out of equilibrium. Further, while local disorder can be utilized to preserve localization and increase t∗, we still identify nontrivial dynamics in the Majorana polarization and Loschmidt echo.
Parafermions are the simplest generalizations of Majorana fermions that realize topological order. We propose a less restrictive notion of topological order in one-dimensional open chains, which generalizes the seminal work by Fendley [J. Stat. Mech. (2012) P11020]. The first essential property is that the ground states are mutually indistinguishable by local, symmetric probes, and the second is a generalized notion of zero edge modes which cyclically permute the ground states. These two properties are shown to be topologically robust, and applicable to a wider family of topologically ordered Hamiltonians than has been previously considered. As an application of these edge modes, we formulate a notion of twisted boundary conditions on a closed chain, which guarantees that the closed-chain ground state is topological, i.e., it originates from the topological manifold of the open chain. Finally, we generalize these ideas to describe symmetry-breaking phases with a parafermionic order parameter. These exotic phases are condensates of parafermion multiplets, which generalize Cooper pairing in superconductors. The stability of these condensates is investigated on both open and closed chains.
Topological superconductors represent one of the key hosts of Majorana-based topological quantum computing.Typical scenarios for one-dimensional (1D) topological superconductivity assume a broken gauge symmetry associated to a superconducting state.However, no interacting 1D many-body system is known to spontaneously break gauge symmetries.Here, we show that zero modes emerge in a many-body system without gauge symmetry breaking and in the absence of superconducting order.In particular, we demonstrate that Majorana zero modes of the symmetry-broken superconducting state are continuously connected to these zero-mode excitations, demonstrating that zero-bias anomalies may emerge in the absence of gauge symmetry breaking.We demonstrate that these many-body zero modes share the robustness features of the Majorana zero modes of symmetry-broken topological superconductors.We further show that the interface between the interacting model and a 1D topological superconductor does not support Majorana modes.We introduce a bosonization formalism to analyze these excitations and show that a ground state analogous to a topological superconducting state can be analytically found in a certain limit.Our results demonstrate that robust Majorana-like zero modes may appear in a many-body system without gauge symmetry breaking, thus introducing a family of protected excitations with no single-particle analogs.