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 $\mathbb{Z}_2$-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.
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 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.
Discovering and categorizing quantum orders in mixed many-body systems are currently one of the most important problems. Specific types of decoherence applied to typical quantum many-body states can induce a novel kind of mixed state accompanying characteristic symmetry orders, which has no counterparts in pure many-body states. We study phenomena generated by the interplay between two types of decoherence applied to the one-dimensional transverse field Ising model (TFIM). We show that in the doubled Hilbert space formalism, the decoherence can be described by a filtering operation applied to matrix product states (MPS) defined in the doubled Hilbert system. The filtering operation induces specific deformation of the MPS, which approximates the ground state of a certain parent Hamiltonian in the doubled Hilbert space. In the present case, such a parent Hamiltonian is the quantum Ashkin-Teller model, having a rich phase diagram with critical lines and quantum phase transitions. By investigating the deformed MPS, we find various types of mixed states emergent from the ground states of the TFIM and clarify phase transitions between them. In that study, strong and weak Z2 symmetries play an important role, for which we introduce efficient order parameters, such as R\'enyi-2 correlators, entanglement entropy, etc., in the doubled Hilbert space.
Strong-to-weak spontaneous symmetry breaking (SWSSB) is a novel phase transition in mixed states associated with the loss of global charge information. In this work we study fermions with U(1) symmetry under infinite density dephasing and derive the SWSSB properties of the resulting mixed states. Our central observation is that at infinite dephasing every Renyi correlator is exactly the spin-spin correlator of a compact XY model, whose Boltzmann weight is the squared full-counting-statistics generating function of a classical ensemble built from the parent state. For the Renyi-2 correlator it is the full counting statistics of the parent state itself. We complement the infinite dephasing mapping with a novel diagrammatic expansion for the Renyi-1 correlator, applicable to any diagonal mixed state, which computes the couplings of the Renyi-1 XY model. We also study a replica field theory for even Renyi index quantities, applicable at finite dephasing. Our analysis indicates that fully dephased metals display long-range SWSSB in d≥2, as confirmed numerically and experimentally in d=2, and quasi-long-range SWSSB in d=1. For insulating parent states both an SWSSB and a trivial phase are possible at infinite dephasing depending on UV details. Long-range SWSSB in d≥3, or quasi-long-range SWSSB in d=2, survives above a nonuniversal, Renyi-index-dependent threshold and is lost when the parent state is deep in the insulating phase, with the exception of quantum Hall insulators which generically have quasi-long-range SWSSB. For even Renyi indices we also determine the stability of the SWSSB phase away from infinite dephasing and derive the universality class of the SWSSB transition.
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.
Discovering mixed state quantum orders is an ongoing issue. Recently, it has been recognized that there are (at least) two kinds of symmetries in the mixed state: strong and weak symmetries. Under symmetry-respective decoherence, spontaneous strong-to-weak symmetry breaking (SSSB) can occur. This paper provides a scheme to describe SSSB and other decoherence phenomena in the mixed state by employing the stabilizer formalism and the efficient numerical algorithm of Clifford circuits. We present two systematic numerical studies. In a two-dimensional circuit with a stochastic Ising type decoherence, an SSSB phase transition is clearly observed and its criticality is elucidated by the numerical methods. In particular, we calculate R\'enyi-2 correlations and estimate critical exponents of the SSSB transition. For the second system, we introduce an idea of subgroup SSSB. As an example, we study a system with symmetry-protected-topological order provided by both one-form and zero-form symmetries, and observe how the system evolves under decoherence. After displaying numerical results, we show that viewpoint of percolation is quite useful to understand the SSSB transition, which is applicable for a wide range of decohered states. Finally, we comment on SSSB of one-form-symmetry exemplifying toric code.