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arXiv:2411.05364 · Physical review. B./Physical review. B

Strong-to-weak symmetry breaking and entanglement transitions

Langxuan Chen, Ning G. Sun, Pengfei Zhang

Published Feb 25, 2025• 10 citations
arXivHTMLPDF

Abstract

When interacting with an environment, the entanglement within quantum many-body systems is rapidly transferred to the entanglement between the system and the bath. For systems with a large local Hilbert space dimension, this leads to a first-order entanglement transition for the reduced density matrix of the system. On the other hand, recent studies have introduced a new paradigm for classifying density matrices, with particular focus on scenarios where a strongly symmetric density matrix undergoes spontaneous symmetry breaking to a weak symmetry phase. This is typically characterized by a finite R\'enyi-2 correlator or a finite Wightman correlator. In this Letter, we study the entanglement transition from the perspective of strong-to-weak symmetry breaking, using solvable complex Brownian Sachdev-Ye-Kitaev models. We perform analytical calculations for both the early-time and late-time saddles. The results show that while the R\'enyi-2 correlator indicates a transition from a symmetric to symmetry-broken phase, the Wightman correlator becomes finite even in the early-time saddle due to the single-replica limit, demonstrating that the first-order transition occurs between a near-symmetric phase and a deeply symmetry-broken phase in the sense of a Wightman correlator. Our results provide another viewpoint on the entanglement transition under symmetry constraints and can be readily generalized to systems with repeated measurements.

DOI
10.1103/physrevb.111.l060304

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When interacting with an environment, the entanglement within quantum many-body systems is rapidly transferred to the entanglement between the system and the bath. For systems with a large local Hilbert space dimension, this leads to a first-order entanglement transition for the reduced density matrix of the system. On the other hand, recent studies have introduced a new paradigm for classifying density matrices, with particular focus on scenarios where a strongly symmetric density matrix undergoes spontaneous symmetry breaking to a weak symmetry phase. This is typically characterized by a finite R'enyi-2 correlator or a finite Wightman correlator. In this Letter, we study the entanglement transition from the perspective of strong-to-weak symmetry breaking, using solvable complex Brownian Sachdev-Ye-Kitaev models. We perform analytical calculations for both the early-time and late-time saddles. The results show that while the R'enyi-2 correlator indicates a transition from a symmetric to symmetry-broken phase, the Wightman correlator becomes finite even in the early-time saddle due to the single-replica limit, demonstrating that the first-order transition occurs between a near-symmetric phase and a deeply symmetry-broken phase in the sense of a Wightman correlator. Our results provide another viewpoint on the entanglement transition under symmetry constraints and can be readily generalized to systems with repeated measurements.
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.
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.
Entanglement is one of the most important concepts in quantum physics. We review recent progress in understanding the quantum entanglement in many-body systems using large- N solvable models: the Sachdev—Ye—Kitaev (SYK) model and its generalizations. We present the study of entanglement entropy in the original SYK model using three different approaches: the exact diagonalization, the eigenstate thermalization hypothesis, and the path-integral representation. For coupled SYK models, the entanglement entropy shows linear growth and saturation at the thermal value. The saturation is related to replica wormholes in gravity. Finally, we consider the steady-state entanglement entropy of quantum many-body systems under repeated measurements. The traditional symmetry breaking in the enlarged replica space leads to the measurement-induced entanglement phase transition.
The preparation of long-range entangled (LRE) states via quantum measurements is a promising strategy, yet its stability against realistic, non-commuting measurement noise remains a critical open question. Here, we systematically investigate the rich phase structure emerging from a minimal model of competing, non-commuting weak measurements: nearest-neighbor Ising (ZiZjZ_iZ_jZi​Zj​) and single-qubit transverse (XiX_iXi​) operators. We analyze three experimentally relevant scenarios based on which measurement outcomes are read out: complete readout, no readout, and partial readout. Using a replica mean-field theory for higher dimensions, complemented by numerical simulations in one dimension, we derive the complete finite-time and stationary phase diagrams. Our analysis reveals a striking dependence on the readout protocol. Complete readout yields a direct transition between a short-range entangled (SRE) phase and a pure LRE phase. No readout (pure decoherence) precludes entanglement but exhibits a strong-to-weak spontaneous symmetry breaking (SWSSB) transition into a classically ordered mixed state. Most intriguingly, partial readout interpolates between these limits, featuring a mixed-state phase transition where the system can become trapped in the SWSSB phase or, for weaker non-commutativity, undergo successive symmetry breaking to reach a mixed LRE phase. A novel technical contribution is the use of a channel-fidelity-based partition function that allows us to simultaneously characterize both entanglement and SWSSB order, revealing a deep interplay between them in the replica limit. These results provide a cohesive picture for understanding measurement phase transitions, SWSSB, and mixed-state phase transitions, offering crucial insights for designing robust state preparation protocols on noisy quantum devices.