n 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.
Alexander Altland, Michael Fleischhauer, and Sebastian Diehl
Symmetry classes of open fermionic quantum matter
Phys. Rev. X 11,021037 (2021); https://journals.aps.org/prx/abstract/10.1103/PhysRevX.11.021037