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.
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.
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.
Symmetry classification is crucial in understanding universal properties of quantum matter. Recently, the scope of symmetry classification has been extended to open quantum systems governed by the Lindblad master equation. However, the classifications of Lindbladians and steady states remain largely separate, because the former requires the non-Hermitian classification framework, while the latter relies on the classification scheme for Hermitian matrices. In this paper we build connections between symmetry classes of quadratic Lindbladians and their steady states, despite their different classification frameworks. We classify the full matrix representation of generic quadratic Lindbladians with particle conservation, showing they fall into 27 non-Hermitian symmetry classes. Among these, 22 classes lead to an infinite-temperature steady state. The remaining five classes have one-to-one correspondence with five steady-state Hermitian symmetry classes. Numerical simulations of random Lindbladian dynamics confirm the convergence to the correct steady-state symmetry classes at long time.
We develop a general framework for classifying fermionic dynamical systems with measurements using symmetry and topology. We introduce two complementary classification schemes based on the Altland-Zirnbauer tenfold way: (1) the many-body evolution operator (mEO) symmetry class, which classifies fermionic dynamics at the many-body level and naturally extends to interacting dynamics, and (2) the single-particle transfer matrix symmetry class, which classifies free-fermion dynamics at the single-particle level and connects to Anderson localization physics. In the free-fermion limit, we show that these two frameworks are equivalent via a dynamical bulk-boundary correspondence: The topology of the dynamical system's spacetime bulk determines the topology of the area-law-entangled steady-state ensemble living on its temporal boundary. Next, we prove that symmetry-invariant, postselection-free Gaussian measurements are realizable in only four of the ten mEO classes (A, AI, BDI, D); the remaining six require either postselection or interacting (non-Gaussian) measurements. Building on these results, we construct general postselection-free topological adaptive circuits that realize topological dynamical phases in any spatial dimension for the four admissible mEO classes. These circuits simultaneously provide a protocol for preparing and stabilizing free-fermion topological states in all ten symmetry classes. As a concrete demonstration, we construct and simulate 2 + 1 d adaptive circuits that realize mEO-class-A topological dynamics, steering toward a steady-state ensemble of Chern insulators in O ( 1 ) circuit depth. Finally, we numerically characterize topological phase transitions, dynamical domain-wall modes, and robustness to coherent noise, identifying finite error thresholds at which trajectory-resolved and trajectory-averaged quantities undergo distinct phase transitions.
Symmetries play a pivotal role in our understanding of the properties of quantum many-body systems. While there are theorems and a well-established toolbox for systems in thermal equilibrium, much less is known about the role of symmetries and their connection to dynamics out of equilibrium. This arises due to the direct link between a system’s thermal state and its Hamiltonian, which is generally not the case for nonequilibrium dynamics. Here we present a pathway to identify the effective symmetries and to extract them from data in nonequilibrium quantum many-body systems. Our approach is based on exact relations between correlation functions involving different numbers of spatial points, which can be viewed as nonequilibrium versions of (equal-time) Ward identities encoding the symmetries of the system. We derive symmetry witnesses, which are particularly suitable for the analysis of measured or simulated data at different snapshots in time. To demonstrate the potential of the approach, we apply our method to numerical and experimental data for a spinor Bose gas. We investigate the important question of a dynamical restoration of an explicitly broken symmetry of the Hamiltonian by the initial state. Remarkably, it is found that effective symmetry restoration can occur long before the system equilibrates. We also use the approach to define and identify spontaneous symmetry breaking far from equilibrium, which is of great relevance for applications to nonequilibrium phase transitions. Our work opens new avenues for the classification and analysis of quantum as well as classical many-body dynamics in a large variety of systems, ranging from ultracold quantum gases to cosmology.
We perform a systematic symmetry classification of many-body Lindblad superoperators describing general (interacting) open quantum systems coupled to a Markovian environment. Our classification is based on the behavior of the many-body Lindbladian under antiunitary symmetries and unitary involutions. We find that Hermiticity preservation reduces the number of symmetry classes, while trace preservation and complete positivity do not, and that the set of admissible classes depends on the presence of additional unitary symmetries: in their absence or in symmetry sectors containing steady states, many-body Lindbladians belong to one of ten non-Hermitian symmetry classes; if however, there are additional symmetries and we consider non-steady-state sectors, they belong to a different set of 19 classes. In both cases, it does not include classes with Kramer's degeneracy. Remarkably, our classification admits a straightforward generalization to the case of non-Markovian, and even non-trace-preserving, open quantum dynamics. While the abstract classification is completely general, we then apply it to general (long-range, interacting, spatially inhomogeneous) spin-1/2 chains. We explicitly build examples in all ten classes of Lindbladians in steady-state sectors, describing standard physical processes such as dephasing, spin injection and absorption, and incoherent hopping, thus illustrating the relevance of our classification for practical physics applications. Finally, we show that the examples in each class display unique random-matrix correlations. To fully resolve all symmetries, we employ the combined analysis of bulk complex spacing ratios and the overlap of eigenvector pairs related by symmetry operations. We further find that statistics of levels constrained onto the real and imaginary axes or close to the origin are not universal due to spontaneous breaking of Lindbladian PT symmetry.
Symmetry plays a fundamental role in understanding complex quantum matter, particularly in classifying topological quantum phases, which have attracted great interests in the recent decade. An outstanding example is the time-reversal invariant topological insulator, a symmetry-protected topological (SPT) phase in the symplectic class of the Altland-Zirnbauer classification. We report the observation for ultracold atoms of a noninteracting SPT band in a one-dimensional optical lattice and study quench dynamics between topologically distinct regimes. The observed SPT band can be protected by a magnetic group and a nonlocal chiral symmetry, with the band topology being measured via Bloch states at symmetric momenta. The topology also resides in far-from-equilibrium spin dynamics, which are predicted and observed in experiment to exhibit qualitatively distinct behaviors in quenching to trivial and nontrivial regimes, revealing two fundamental types of spin-relaxation dynamics related to bulk topology. This work opens the way to expanding the scope of SPT physics with ultracold atoms and studying nonequilibrium quantum dynamics in these exotic systems.