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 antilinear state transformations in fermionic Fock spaces, and symmetries defined via invariance properties of the dynamical equation for the density matrix. The object of classification is 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 “microreversible” thermal equilibrium, antilinear transformations combined with an inversion of time define time-reversal symmetry. However, out of equilibrium an inversion of time becomes meaningless, while antilinear 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 nonequilibrium 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 nonequilibrium 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.
- Received 2 August 2020
- Revised 1 February 2021
- Accepted 23 March 2021
DOI:https://doi.org/10.1103/PhysRevX.11.021037
Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
Published by the American Physical Society
Physics Subject Headings (PhySH)
Popular Summary
In the study of quantum physics, one of the most powerful organizing principles is that of symmetry classifications. Here, particles and systems are grouped together by some feature that is unchanged after a transformation. For example, a movie of a quantum mechanical system in equilibrium looks the same when played forward or backward—such a system is called time-reversal symmetric. But recent realizations of novel phases of quantum matter in experiments require a generalization of these symmetry classifications to driven systems coupled to their environment, where the external drive does not allow these systems to thermally equilibrate. Here, we provide such a generalization for fermionic quantum matter.
Our approach to the classification problem is bottom up, starting from ten fundamental symmetry transformations in fermionic state space. We pay particular attention to certain symmetries that, in equilibrium, are associated with time reversal. In systems that are out of equilibrium, time is not reversible, so such symmetries require a complete rethinking.
Building on this framework, we uncover a fundamental distinction between the incarnations of the ten state-space symmetry classes, depending on whether the dynamics proceeds in or out of equilibrium. This gives rise to 20 dynamical symmetry classes: ten each for equilibrium and nonequilibrium dynamics. The transformation laws obtained out of equilibrium are sharply distinct from the known ones at equilibrium. Using the example of an interacting quantum wire, we show how this new understanding may help to engineer and manipulate a topological phase.
Our analysis provides a comprehensive description of symmetry classes, in and out of equilibrium, which expressly includes systems with interactions. From a more applied perspective, the work offers a concrete transformation rule book for the building blocks of dynamical evolution subject to symmetry constraints.
