We uncover a topological classification applicable to open fermionic systems governed by a general class of Lindblad master equations. These 'quadratic Lindbladians' can be captured by a non-Hermitian single-particle matrix which describes internal dynamics as well as system-environment coupling. We show that this matrix must belong to one of ten non-Hermitian Bernard-LeClair symmetry classes which reduce to the Altland-Zirnbauer classes in the closed limit. The Lindblad spectrum admits a topological classification, which we show results in gapless edge excitations with finite lifetimes. Unlike previous studies of purely Hamiltonian or purely dissipative evolution, these topological edge modes are unconnected to the form of the steady state. We provide one-dimensional examples where the addition of dissipators can either preserve or destroy the closed classification of a model, highlighting the sensitivity of topological properties to details of the system-environment coupling.

We study the topological properties of one-dimensional systems undergoing unitary time evolution. We show that symmetries possessed both by the initial wave function and by the Hamiltonian at all times may not be present in the time-dependent wave function-a phenomenon which we dub "dynamically induced symmetry breaking." This leads to the possibility of a time-varying bulk index after quenching within noninteracting gapped topological phases. The consequences are observable experimentally through particle transport measurements. With reference to the entanglement spectrum, we explain how the topology of the wave function can change out of equilibrium, both for noninteracting fermions and for symmetry-protected topological phases protected by antiunitary symmetries.

We establish the existence of a topological classification of many-particle quantum systems undergoing unitary time evolution. The classification naturally inherits phenomenology familiar from equilibrium -it is robust against disorder and interactions, and exhibits a non-equilibrium bulkboundary correspondence, which connects bulk topological properties to the entanglement spectrum. We explicitly construct a non-equilibrium classification of non-interacting fermionic systems with non-spatial symmetries in all spatial dimensions (the 'ten-fold way'), which differs from its equilibrium counterpart. Direct physical consequences of our classification are discussed, including important ramifications for the use of topological zero-energy bound states in quantum information technologies.

The second law of thermodynamics points to the existence of an 'arrow of time', along which entropy only increases. This arises despite the time-reversal symmetry (TRS) of the microscopic laws of nature. Within quantum theory, TRS underpins many interesting phenomena, most notably topological insulators [1][2][3][4] and the Haldane phase of quantum magnets [5,6]. Here, we demonstrate that such TRS-protected effects are fundamentally unstable against coupling to an environment. Irrespective of the microscopic symmetries, interactions between a quantum system and its surroundings facilitate processes which would be forbidden by TRS in an isolated system. This leads not only to entanglement entropy production and the emergence of macroscopic irreversibility [7,8], but also to the demise of TRS-protected phenomena, including those associated with certain symmetry-protected topological phases. Our results highlight the enigmatic nature of TRS in quantum mechanics, and elucidate potential challenges in utilising topological systems for quantum technologies.Many isolated systems possess features that rely on symmetries of their Hamiltonian. Most strikingly, in many-body systems the presence of symmetries leads to new phases of matter, including symmetry-protected topological phases (SPTs) [9,10]. SPTs exhibit many remarkable features, such as the emergence of topological bound states (e.g. Majorana zero modes [11]), which have potential applications in quantum information processing [12,13].An important practical question, which we address in this Letter, is whether symmetry-protected phenomena such as these can persist in realistic scenarios where the system is weakly coupled to an environment. Previous studies of topology in open systems begin with an approximate equation of motion for the system (e.g. non-Hermitian Hamiltonian [14] or Lindblad master equation [15][16][17]). Instead, our starting point is the full systemenvironment Hamiltonian

We investigate the robustness of Majorana edge modes under disorder and interactions. We exploit a recently found mapping of the interacting Kitaev chain in the symmetric region (μ = 0, t = ) to free fermions. Extending the exact solution to the disordered case allows us to calculate analytically the topological phase boundary for all interaction and disorder strengths, which has been thought to be only accessible numerically. We discover a regime in which moderate disorder in the interaction matrix elements enhances topological order well into the strongly interacting regime U > t. We also derive the explicit form of the many-body Majorana edge wave function, revealing how it is dressed by many-particle fluctuations from interactions. The qualitative features of our analytical results are valid beyond the fine-tuned integrable point, as expected from the robustness of topological order and as corroborated here by an exact diagonalization study of small systems. DOI: 10.1103/PhysRevB.96.241113 Majorana edge modes in condensed matter physics have recently received a great deal of attention [1] primarily due to their applications in topological quantum computation [2]. In a seminal paper [3], Kitaev introduced the minimal model of a one-dimensional (1D) p-wave superconducting wire, now known as the Kitaev chain. One of its remarkable properties is the presence of zero-energy states localized at the two ends of the chain. Paired together, these Majorana edge modes can form a qubit which is largely protected from decoherence due to its nonlocal nature.Compelling experimental evidence of their existence has been reported in semiconducting nanowires in proximity to s-wave superconductors [4][5][6][7] and in ferromagnetic atomic chains [8][9][10]. However, there remains a possibility that the zero-bias conductance peak measured in these experiments is due to disorder rather than due to Majorana modes [11][12][13], and so it is important to include disorder in theoretical investigations. Additionally, the nature of these experimental platforms inevitably leads to the presence of interactions between the low-energy degrees of freedom [14][15][16].The majority of analytical studies of the Kitaev chain have focused on the clean, noninteracting case [1]. Beyond this, for clean, interacting chains, only few exact results are known [17][18][19][20], and numerical/perturbative studies have shown that Majorana edge modes can be stable up to moderate interaction strengths [14,15,[21][22][23]. Similarly, a number of works on noninteracting, disordered/quasiperiodic chains find a relatively broad parameter region of stability [24][25][26]. The combined effect of interactions and disorder in Kitaev chains has recently been studied numerically [27,28], as well as through a weak-disorder renormalization group (RG) approach [29]. However, an analytic treatment of both strong interactions and strong disorder has been thought to be impossible.In this Rapid Communication, we investigate analytically the combined effects of disorder an...