We study strong-to-weak spontaneous symmetry breaking (SWSSB) at finite temperature and its implications in holography. SWSSB is a novel critical phenomenon that arises only in mixed states. We demonstrate SWSSB with the BTZ black hole and show that, in the bulk, it is realized through wormhole geometries connecting the two copies in the thermofield-double construction. We point out that SWSSB can be recast as an emergence of global symmetries under ensemble averaging. We further examine the interplay between SWSSB and the swampland program, and propose a new swampland conjecture for quantum-gravity theories involving ensemble averages.
We study strong-to-weak spontaneous symmetry breaking (SWSSB) at finite temperature and its implications in holography. SWSSB is a novel critical phenomenon that arises only in mixed states. We demonstrate SWSSB with the BTZ black hole and show that, in the bulk, it is realized through wormhole geometries connecting the two copies in the thermofield-double construction. We point out that SWSSB can be recast as an emergence of global symmetries under ensemble averaging. We further examine the interplay between SWSSB and the swampland program, and propose a new swampland conjecture for quantum-gravity theories involving ensemble averages.
It is widely believed that there are no exact global symmetries in quantum gravity. We review recent works showing that global symmetries can nevertheless emerge after ensemble averages of theories, as encountered in holography. As a precision case study, we discuss the ensemble average of two-dimensional Narain-type conformal field theories associated with an even quadratic form of general signature, which is computed by the Siegel-Weil formula and interpreted as a sum over geometries in a three-dimensional Abelian Chern-Simons theory. Global symmetries of the bulk anyons can emerge after the average, as T-dualities relating different theories in the ensemble are "folded" into symmetries of a single theory (duality origami). We discuss the relation to the Swampland program, including the emergence of global symmetries at infinite distance. Finally, we recast strong-to-weak spontaneous symmetry breaking (SWSSB) of mixed states as an emergence of symmetries in ensemble averages; in holography, wormholes connecting the two sides of a thermofield-double construction provide its bulk realization. This motivates a new Swampland conjecture.
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.
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.
Inspired by the hunt for new phases of matter in quantum mixed states, it has recently been proposed that the equivalence of microcanonical and canonical ensembles in statistical mechanics is a manifestation of strong-to-weak spontaneous symmetry breaking (SWSSB) in an underlying many-body quantum description. Here, we build an effective field theory for SWSSB of a global U(1) symmetry; the answer exactly reproduces the Schwinger-Keldysh effective field theory of diffusion for the conserved charge. We conclude that hydrodynamics can be understood as a theory of superfluidity for the broken strong symmetry: a nonvanishing susceptibility is a measurable order parameter for SWSSB, the diffusion mode is the Goldstone boson of the spontaneously broken continuous symmetry, and a generalization of Goldstone's theorem implies that the diffusion mode is always long lived. This perspective provides a transparent physical explanation for the unusual reparameterization symmetries, which are a necessary ingredient of Schwinger-Keldysh effective field theories for normal fluids.
Strong-to-weak spontaneous symmetry breaking (SW-SSB) has recently emerged as a useful framework for studying phases of matter in open systems, quantum or classical. Beginning with the simple idea of extending symmetry breaking to general mixed states, and the familiar equivalence between canonical and grand-canonical ensembles in statistical mechanics, the concept has grown into a unifying perspective connecting many different ideas in physics, including topological orders, emergent hydrodynamics, and information-theoretic characterization of phases of matter. This review provides a bird's-eye view of some of these recent developments.
Symmetry plays a fundamental role in quantum many-body physics, and a central concept is spontaneous symmetry breaking, which imposes crucial constraints on the possible quantum phases and their transitions. Recent developments have extended the discussion of symmetry and its breaking to mixed states, enhancing our understanding of novel quantum phases that have no counterpart in pure states. Specific attention has been paid to scenarios where a strongly symmetric density matrix exhibits spontaneous symmetry breaking to weak symmetry, characterized by the fidelity correlator. In this work, we propose the Wightman correlator as an alternative diagnostic tool. This construction relies on the introduction of the thermofield double state for a generic density matrix, which maps the strong symmetry of the density matrix to the doubled symmetry of the pure state, allowing the Wightman correlator to emerge naturally as a standard probe of symmetry breaking. We prove the equivalence between the Wightman function and the fidelity correlator in defining strong-to-weak symmetry breaking, and examine explicit examples involving spin glasses, thermal density matrices, and the decohered Ising model. Additionally, we discuss a susceptibility interpretation of the Wightman correlator.