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arXiv:1908.06875 · Physical Review Research

Interacting symmetry-protected topological phases out of equilibrium

Max McGinley, Nigel R. Cooper

Published Dec 26, 2019• 19 citations
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Abstract

The authors make use of concepts and methods from the theory of topological phases to understand the dynamics of generic quantum many-body systems undergoing unitary time-evolution. They develop a topological classification scheme for wavefunctions far from equilibrium, and show that this classification can be used to predict a number of universal phenomena in certain non-equilibrium regimes. This classification is explicitly derived for strongly interacting bosonic systems in all spatial dimensions.

DOI
10.1103/physrevresearch.1.033204

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The authors make use of concepts and methods from the theory of topological phases to understand the dynamics of generic quantum many-body systems undergoing unitary time-evolution. They develop a topological classification scheme for wavefunctions far from equilibrium, and show that this classification can be used to predict a number of universal phenomena in certain non-equilibrium regimes. This classification is explicitly derived for strongly interacting bosonic systems in all spatial dimensions.
The topological features of quantum many-body wave functions are known to have profound consequences for the physics of ground-states and their low-energy excitations. We describe how topology influences the dynamics of many-body systems when driven far from equilibrium. Our results are succinctly captured by a nonequilibrium topological classification that can be used to predict universal aspects of generic isolated quantum systems as they evolve unitarily in time. By analogy to the classifications used to describe systems in equilibrium, we consider two short-ranged entangled wave functions to be topologically equivalent if they can be interconverted via finite-time unitary evolution governed by a symmetry-respecting Hamiltonian. We demonstrate that this definition captures the salient features of these systems in a broad range of nonequilibrium scenarios. As well as providing conceptual insights into the constraints imposed by topology on many-body dynamics, we discuss the practical implications of our findings. In particular, we show that the characteristic zero-frequency spectroscopic peaks associated with topologically protected edge modes will be broadened by external noise only when the system is trivial in the nonequilibrium classification.
Topological phases of matter are understood to be characterized by particular configurations of entanglement encoded within the ground states of many-body quantum systems. In this thesis, we discuss how novel phenomena associated with topological phases can arise in systems that are driven out of equilibrium or coupled to their surroundings, which are not described by ground state physics. Firstly, we consider systems undergoing unitary time evolution due to some external driving. We show how the entanglement features of the time-dependent wavefunction can be topologically characterized, and demonstrate that these topological properties are reflected in the dynamics of various quantities usually associated with topological phases in equilibrium. We introduce a new non-equilibrium topological classification scheme which can be used to predict whether or not the topological features of a given system will be preserved under time evolution. In brief, this classification captures the fact that certain symmetries (namely antiunitary symmetries) are inevitably broken once the system is driven out of equilibrium; thus any topological phenomena protected by such symmetries cannot be expected to persist in non-equilibrium scenarios. Secondly, we investigate how topologically non-trivial systems are affected by dissipative effects, i.e.~coupling to an external environment. We demonstrate that system-environment interactions facilitate processes in the system that effectively break any antiunitary symmetries, regardless of the symmetries of the microscopic Hamiltonian. Accordingly, those phases that were shown to be unstable against time-dependent driving are also fragile against coupling to an environment. To illustrate the consequences of this fragility, we consider the effects of dissipation on the coherence properties of topological bound states, as well as the conductance properties of chiral and helical topological edge modes. We find that the decoherence rate of the former and the deviation from quantized conductivity of the latter are (not) exponentially suppressed in the inverse temperature when the phase in question is protected by unitary (antiunitary) symmetries. These results regarding open systems can be connected to the same non-equilibrium classification developed in the context of isolated systems undergoing unitary dynamics. Our findings highlight a distinction between topological phases that are either robust or fragile against non-equilibrium effects and/or system-environment coupling. The ramifications for the use of such systems in quantum technologies are discussed.
The authors have discovered a topological classification of quantum many-body systems undergoing dynamical evolution, which can be applied to insulators and superconductors driven out of equilibrium. This classification is analogous to the equilibrium classification of topological insulators, which explains the remarkably quantized experimental signatures that these phases are known for. The nonequilibrium classification successfully predicts whether topological edge modes remain stable under external time-dependent fluctuations. This, in turn, has ramifications for the use of such systems in quantum technologies.
The authors deal with the fundamental problem of a rigorous classification of quantum topological phases. Here, they extend to the realm of interacting fermions in two dimensions techniques, developed for bosons, whereby they introduce extrinsic defects carrying symmetry fluxes into the topological phase. The result is a complete classification of symmetry-protected topological phases of interacting fermions in two dimensions. In addition, the classification of interacting fermion systems with time-reversal symmetry is also developed. The advances, reported here, are immediately relevant for the selection of reliable candidates for topologically protected storage and communication.
We give a brief overview on interacting symmetry-protected topological (SPT) phases. In particular, we will focus on various cases where the classification of non-interacting fermionic systems breaks down in the presence of interactions. The examples include topological (and trivial) superconductors in one, two and three spatial dimensions, protected by various symmetries such as time-reversal, spin parity, and reflection symmetries. Common to these examples is the ‘collapse’ of the non-interacting classification, where an integer classification (‘ ’) at non-interacting level reduces to or in the presence of interactions. We make use of these examples to illustrate various approaches to interacting SPT phases.
We investigate topological signatures in the short-time non-equilibrium dynamics of symmetry protected topological (SPT) systems starting from initial states which break the protecting symmetry. Naïvely, one might expect that topology loses meaning when a protecting symmetry is broken. Defying this intuition, we illustrate, in an interacting Su-Schrieffer-Heeger (SSH) model, how this combination of symmetry breaking and quench dynamics can give rise to both single-particle and many-body signatures of topology. From the dynamics of the symmetry broken state, we find that we are able to dynamically probe the equilibrium topological phase diagram of a symmetry respecting projection of the post-quench Hamiltonian. In the ensemble dynamics, we demonstrate how spontaneous symmetry breaking (SSB) of the protecting symmetry can result in a quantized many-body topological `invariant' which is not pinned under unitary time evolution. We dub this `dynamical many-body topology' (DMBT). We show numerically that both the pure state and ensemble signatures are remarkably robust, and argue that these non-equilibrium signatures should be quite generic in SPT systems, regardless of protecting symmetries or spatial dimension.