Abstract

We present a full symmetry classification of fermion matter in and out of thermal equilibrium. Our approach starts from first principles, the ten different classes of linear and anti-linear state transformations in fermionic Fock spaces, and symmetries defined via invariance properties of the dynamical equation for the density matrix. The object of classification are then the generators of reversible dynamics, dissipation and fluctuations featuring in the generally irreversible and interacting dynamical equations. A sharp distinction between the symmetries of equilibrium and out of equilibrium dynamics, respectively, arises from the different role played by `time' in these two cases: In unitary quantum mechanics as well as in `micro-reversible' thermal equilibrium, anti-linear transformations combined with an inversion of time define time reversal symmetry. However, out of equilibrium an inversion of time becomes meaningless, while anti--linear transformations in Fock space remain physically significant, and hence must be considered in autonomy. The practical consequence of this dichotomy is a novel realization of antilinear symmetries (six out of the ten fundamental classes) in non-equilibrium quantum dynamics that is fundamentally different from the established rules of thermal equilibrium. At large times, the dynamical generators thus symmetry classified determine the steady state non-equilibrium distributions for arbitrary interacting systems. To illustrate this principle, we consider the fixation of a symmetry protected topological phase in a system of interacting lattice fermions. More generally, we consider the practically important class of mean field interacting systems, represented by Gaussian states. This class is naturally described in the language of non-Hermitian matrices, which allows us to compare to previous classification schemes in the literature.

Highlights

  • The distinction between different unitary and antiunitary symmetries [1,2,3,4] is a powerful organizing principle in the classification of quantum matter

  • Our approach starts from first principles, the ten different classes of linear and antilinear state transformations in fermionic Fock spaces, and symmetries defined via invariance properties of the dynamical equation for the density matrix

  • The practical consequence of this dichotomy is a novel realization of antilinear symmetries in nonequilibrium quantum dynamics that is fundamentally different from the established rules of thermal equilibrium

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Summary

INTRODUCTION

The distinction between different unitary and antiunitary symmetries [1,2,3,4] is a powerful organizing principle in the classification of quantum matter. The generators, X 1⁄4 XðH ; D ; P Þ can generally be expressed in terms of three subordinate operators, describing the contributions of unitary evolution, dissipation, and fluctuations to the dynamics In this representation, the symmetry criterion splits into three, individually for these operators. Given the scarcity of general principles characterizing outof-equilibrium quantum distributions, the specification of symmetry criteria universally described in terms of the generators H ; D ; Pis an important contribution of this work. At this point, we have mentioned two settings, the limit of closed system unitary dynamics and that of nonequilibrium irreversible dynamics.

Synopsis and summary of results
SYMMETRIES IN DRIVEN OPEN QUANTUM DYNAMICS
Symmetry operations in Fock space
SYMMETRIES IN MARKOVIAN DYNAMICS
Symmetries in Lindbladian dynamics
Symmetries of Gaussian states
Topology of Gaussian states
Edge state formation
CASE STUDY
Interacting model
Symmetries
Linearized model
Edge states
Adding a reversible contribution
Non-Hermitian Su-Schrieffer-Heeger model and exceptional points
BEYOND THE MARKOVIAN LIMIT
Symmetries in the Keldysh path integral
Symmetries in systems with detailed balance
Scope of the equilibrium symmetry conditions
CONCLUSIONS
Full Text
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