We investigate the nonequilibrium dynamics of strong-to-weak spontaneous symmetry breaking in many-body quantum systems undergoing decoherence from thermal pure states. For generic initial pure states with volume-law entanglement entropy, we show that the system undergoes a discontinuous dynamical phase transition at a critical time. This transition is accompanied by a singularity in the entropy of the system, which saturates to its maximum value at the same critical time. Through numerical simulations of the dephasing Ising and hard-core boson models, we establish the universality of this transition across different symmetries. Our results reveal that the dynamical emergence of a decohered mixed state from a highly entangled state is not a gradual asymptotic relaxation, but rather a sharp phase transition driven by a sudden collapse of global coherence.
We investigate the nonequilibrium dynamics of strong-to-weak spontaneous symmetry breaking in many-body quantum systems undergoing decoherence from thermal pure states. For generic initial pure states with volume-law entanglement entropy, we show that the system undergoes a discontinuous dynamical phase transition at a critical time. This transition is accompanied by a singularity in the entropy of the system, which saturates to its maximum value at the same critical time. Through numerical simulations of the dephasing Ising and hard-core boson models, we establish the universality of this transition across different symmetries. Our results reveal that the dynamical emergence of a decohered mixed state from a highly entangled state is not a gradual asymptotic relaxation, but rather a sharp phase transition driven by a sudden collapse of global coherence.
Symmetry in mixed quantum states can manifest in two distinct forms: , where each individual pure state in the quantum ensemble is symmetric with the same charge, and , which applies only to the entire ensemble. This paper explores a novel type of spontaneous symmetry breaking (SSB) where a strong symmetry is broken to a weak one. While the SSB of a weak symmetry is measured by the long-ranged two-point correlation function, the strong-to-weak SSB (SWSSB) is measured by the . We prove that SWSSB is a universal property of mixed-state quantum phases, in the sense that the phenomenon of SWSSB is robust against symmetric low-depth local quantum channels. We also show that the symmetry breaking is “spontaneous” in the sense that the effect of a local symmetry-breaking measurement cannot be recovered locally. We argue that a thermal state at a nonzero temperature in the canonical ensemble (with fixed symmetry charge) should have spontaneously broken strong symmetry. Additionally, we study nonthermal scenarios where decoherence induces SWSSB, leading to phase transitions described by classical statistical models with bond randomness. In particular, the SWSSB transition of a decohered Ising model can be viewed as the “ungauged” version of the celebrated toric-code decodability transition. We confirm that, in the decohered Ising model, the SWSSB transition defined by the fidelity correlator is the only physical transition in terms of channel recoverability. We also comment on other (inequivalent) definitions of SWSSB, through correlation functions with higher Rényi indices.
Strong-to-Weak Spontaneous Symmetry Breaking in a (2+1)D Transverse-Field Ising Model under Decoherence
Decoherence in many-body quantum systems can give rise to intrinsically mixed-state phases and phase transitions beyond the pure-state paradigm. Here we study the (2+1)D transverse-field Ising model subject to a strongly Z2-symmetric decoherence channel, with a focus on strong-to-weak spontaneous symmetry breaking (SWSSB). This problem is challenging because the relevant transitions occur in the strong-decoherence regime, beyond the reach of perturbative expansions around the pure-state limit, while conventional quantum Monte Carlo (QMC) methods are hampered by the need to access nonlinear observables and by the sign problem. We overcome these difficulties by developing a QMC algorithm that efficiently evaluates nonlinear Rényi-2 correlators in higher dimensions, complemented by an effective field-theoretic approach. We show that the decohered state realizes a rich mixed-state phase diagram governed by an effective 2D Ashkin-Teller theory. This theory enables analytical predictions for the mixed-state phases and the universality classes of the phase boundaries, all of which are confirmed by large-scale QMC simulations.
Symmetry breaking has been a central theme in classifying quantum phases and phase transitions. Recently, this concept has been extended to the mixed states of open systems, attracting considerable attention due to the emergence of novel physics beyond closed systems. In this work, we reveal a new type of phase transition in mixed states, termed strong-to-weak spontaneous symmetry breaking (SWSSB). Using a combination of field theory calculations and large-scale matrix product state simulations, we map out the global phase diagram of the X X Z critical spin chain under local strong symmetry preserving decoherence, which features an SWSSB phase and a trivial Luttinger liquid phase, separated by a straight critical line that belongs to the boundary Berezinskii-Kosterlitz-Thouless universality class with a varying effective central charge. Importantly, we analyze this transition from two complementary perspectives: on one hand, through the behavior of order parameters that characterize the symmetry breaking; on the other hand, from a quantum information viewpoint by studying entropic quantities and the concept of quantum recoverability. This dual approach allows us to provide a more comprehensive understanding of the phase structure and the nature of the transition. Remarkably, the SWSSB transition in our case is in the sense that it can only be driven by tuning the Hamiltonian parameter even under arbitrary decoherence strength, fundamentally distinguishing it from the decoherence-driven SWSSB transitions extensively discussed in previous literature. Importantly, our unified theoretical framework is applicable to a broad class of one-dimensional quantum systems, including spin chains and fermionic systems, whose low-energy physics can be described by Luttinger liquid theory, under arbitrary symmetry-preserving decoherence channels. Finally, we also discuss the experimental relevance of our theory on quantum simulator platforms.
Symmetry-based classification of quantum phases of matter is one of the most foundational organizing principles in physics; however, an analogous framework for mixed, decohered quantum states has only begun to emerge. A central new concept is strong-to-weak spontaneous symmetry breaking (SW-SSB), a sharp transition in mixed quantum states that is invisible to any observable linear in the density matrix and that has since been predicted across a broad class of open and monitored quantum systems. It also provides a unifying language for phenomena as disparate as the decodability of topological quantum memories and the emergence of classical hydrodynamics from decohered quantum dynamics. Here we report the first experimental observation of SW-SSB, in dephased single-component fermionic matter imaged by a quantum gas microscope. A quantum-classical estimator built on a machine-learned Gaussian reference state gives direct access to the nonlinear Rényi-1 and Rényi-2 correlators that diagnose SW-SSB, and reveals long-range Rényi order in the dephased Fermi liquid. Adding a commensurate superlattice drives the underlying fermions through a metal-to-insulator transition that, after full dephasing, manifests as a sharp SW-SSB phase transition. Our results uncover the symmetry principle behind information-theoretic transitions in open quantum systems, and extend Landau's symmetry paradigm into the regime of real, decohering quantum devices.
Quantum phases and their transitions play a pivotal role to characterize macroscopic matter. Recent studies have revealed that critical transitions reminiscent of equilibrium phase transitions can occur even out of equilibrium, as a singularity of dynamical counterpart of free energy. In this paper, we report a novel mechanism of dynamical phase transitions caused by spontaneous hidden antiunitary-symmetry breaking in periodically driven unitary dynamics. We characterize dynamical phases in such systems with nonunitary many-body operator obtained by switching space and time and find that its symmetry constrains the nature of phases and their transitions. In particular, at some parameters of stroboscopic Ising models, dynamical free energy density exhibits a critical transition with novel singularity absent in equilibrium. The criticality and its universal exponent originate from a many-body exceptional point of the nonunitary operator associated with spontaneous antiunitary-symmetry breaking. We also show that the generalized correlation function has the divergent correlation length at transition and exhibits oscillatory long-range order after symmetry breaking. Our results open up a quest for hitherto unknown phases with their hidden connection to nonunitary physics in macroscopic nonequilibrium systems, which can be tested in state-of-the-art quantum simulators.
In a quantum many-body system coupled to the environment, its steady state can exhibit spontaneous symmetry breaking when a control parameter exceeds a critical value. In this study, we consider spontaneous symmetry breaking in nonsteady modes of an open quantum many-body system. Assuming that the time evolution of the density matrix of the system is described by a Markovian master equation, the dynamics of the system is fully characterized by the eigenmodes and spectrum of the corresponding time evolution superoperator. Among the nonsteady eigenmodes with finite lifetimes, we focus on the eigenmodes with the highest frequency, which we call the most coherent mode. For a dissipative spin model, it is shown that the most coherent mode exhibits a transition from a disordered phase to a symmetry-broken ordered phase, even if the steady state does not show singular behavior. We further argue that the phase transition of the most coherent mode induces a qualitative change in the decoherence dynamics of highly entangled states, i.e., the Schr\"odinger's cat states.