Topology of One-Dimensional Quantum Systems Out of Equilibrium

Abstract

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.

Tenfold Way for Quadratic Lindbladians

“…Introduction.-Topological band theory was developed to predict and explain robust features in the electronic structure of insulators and superconductors close to their ground states [1,2]. While these ideas have already found fundamental applications in quantum metrology [3] and quantum computation [4], there has been a recent effort to understand the role of topology in the dynamics of many-body systems in highly nonequilibrium environments [5][6][7][8][9][10][11][12].…”

Interacting symmetry-protected topological phases out of equilibrium

“…We can also make connection with our previous results on non-equilibrium classifications of free-fermion systems [29,30] using the Jordan-Wigner transform approach [44]. In that context, the fermion systems belong to one of 10 Altland-Zirnbauer symmetry classes [56], which can then be re-interpreted as symmetry groups of the auxiliary spin system (although in some cases one needs to specify whether the free Hamiltonian represents a superconductor or an insulator [57]).…”

“…The key difference between the equilibrium and nonequilibrium deformation procedures is the possibility that unitary (non-adiabatic) evolution between symmetryrespecting SRE wavefunctions |Ψ 1 and |Ψ 2 may proceed via states which at intermediate times do not respect all the symmetries of the Hamiltonian [29]. In particular, those symmetries that are realised antiunitarily (e.g.…”

“…In addition to these works, which describe features of the system's trajectory over time, the instantaneous topological properties of wavefunctions undergoing time evolution have also been (a) -Equilibrium [27,28]. In this approach, the time dependence of non-interacting bulk indices can be understood, allowing one to systematically characterise non-interacting free fermion systems far from equilibrium [29,30]. However, the techniques applied in these previous studies are specific to non-interacting systems, and do not generalize to strongly interacting SPT phases.…”

“…It may be possible to construct a unitary evolution between two symmetric wavefunctions where the wavefunction at intermediate times does not respect the symmetry of the governing Hamiltonian. This phenomenon -which we have called dynamically induced symmetry breaking [29] -leads to a reduced non-equilibrium classification compared to that in equilibrium.…”