Symmetry Breaking Develops Faster Than Charge Diffusion

Scientists investigate the breakdown of symmetry in open quantum systems, a phenomenon with implications for understanding non-equilibrium dynamics and the emergence of classical behaviour from quantum mechanics. Jacob Hauser, Kaixiang Su, and Hyunsoo Ha, working at the Department of Physics, University of California, Santa Barbara, alongside Jerome Lloyd and Romain Vasseur from the Department of Theoretical Physics, University of Geneva, and colleagues including Sarang Gopalakrishnan of Princeton University’s Department of Electrical and Computer Engineering, and Matthew P. A. Fisher from the Kavli Institute for Theoretical Physics and the Department of Physics at UC Santa Barbara, demonstrate how strong-to-weak symmetry breaking transitions occur, linking discrete particle behaviour to continuum hydrodynamic descriptions. Their research, conducted in collaboration across multiple institutions, reveals that this symmetry breaking manifests on timescales that define the limits of hydrodynamic approximations, offering new insights into the relationship between microscopic dynamics and macroscopic, classical phenomena.

Can complex systems transition from behaving like distinct particles to flowing like a continuous fluid. This work demonstrates how such a change happens, revealing a precise timescale for when particle-like behaviour gives way to fluid-like dynamics. Understanding this shift unlocks new ways to model everything from quantum materials to biological systems.

Scientists are increasingly focused on understanding transitions in systems interacting with external environments, moving beyond traditional equilibrium phase transitions. Recent investigations have uncovered a wider range of transitions occurring in systems coupled to external environments, including measurement-induced criticality, separability transitions, teleportation transitions, complexity transitions, and those driven by decoherence.

Strong-to-weak spontaneous symmetry breaking (SW-SSB) is another such transition, an information-theoretic analogue of conventional symmetry breaking occurring in mixed quantum states. Unlike transitions in pure states, these information-theoretic transitions can happen within a finite time, dependent on the system’s dynamics. Currently, diagnosing these transitions requires either assessing the performance of information-processing tasks or examining nonlinear functions of the system’s density matrix.

Various Rényi correlation measures and the Markov length, which quantifies the decay scale of conditional mutual information, have become powerful tools for identifying mixed-state phase transitions. SW-SSB specifically concerns the breaking of symmetry in mixed states, where a system can be invariant under symmetries in either a strong or weak manner.

A strongly symmetric state possesses a definite value for a conserved quantity, such as total charge, while a weakly symmetric state represents an incoherent mixture of different charge sectors. As a result, SW-SSB arises when a system maintains a sharp value for a global charge, yet this information is not locally accessible or recoverable. Consider a maximally mixed state constrained by a fixed charge sector; it appears maximally mixed locally, but restoring it requires global knowledge of the total charge.

This contrasts with highly structured states where the strong symmetry remains intact. These transitions have connections to thermalization and the emergence of hydrodynamic behaviour, and under typical lattice dynamics, conserved charges spread diffusively. However, investigations suggest that probes of SW-SSB reveal dynamics that are distinct from simple charge diffusion.

In one dimension, the strong symmetry is not spontaneously broken at any finite time, yet correlators that detect SW-SSB develop order on length scales that expand linearly with time, exceeding the rate of charge diffusion. Numerical evidence supports this scaling, derived through field-theory analysis, and relates to the challenge of inferring charge within a subregion by observing its surroundings.

Explicit decoding protocols demonstrate the origin of this scaling. Also, in two dimensions, both field theory and numerical simulations indicate a finite-time Berezinskii-Kosterlitz-Thouless-like SW-SSB transition. Continuum hydrodynamics, however, predicts SW-SSB occurring at infinitesimal time in two or more dimensions. The observed transition time marks the point at which a continuum hydrodynamic description becomes valid, or more precisely, the limit beyond which non-hydrodynamic details, such as discrete particle trajectories, can no longer be inferred.

Supporting this interpretation, researchers analysed a model where SW-SSB is used to derive a classical stochastic hydrodynamic description from the underlying quantum dynamics. This work establishes a framework for understanding how information-theoretic transitions can bridge the gap between discrete microscopic dynamics and continuous macroscopic behaviour.

Simulating dynamical spin correlations via time-dependent numerical techniques

A U-symmetric spin-1/2 model serves as the primary numerical setting for this work, beginning with an initial Néel state where alternating spins are aligned. Throughout the simulations, the dynamics are treated as effectively classical, focusing on the density matrix remaining diagonal within the charge basis; therefore, the time evolution is calculated as a classical probability distribution.

Researchers employed time-evolving block decimation (TEBD) with matrix product states (MPS) in one dimension and quantum Monte Carlo (QMC) in two dimensions to simulate this evolution. Specifically, a worm algorithm was used for the 2d case to sample and average over spacetime trajectories with defined boundary conditions. Calculating Rényi-2 observables proved straightforward using both approaches, reducing to standard MPS contractions in 1d and corresponding to a spacetime trajectory ensemble in 2d.

Operator insertions were implemented directly, allowing for easy evaluation of Rényi-2 correlators. Obtaining Rényi-1 observables, which require the square root of the density matrix, presented a challenge, addressed by utilising tensor cross interpolation (TCI) to infer the square root from sampled density matrices, and verified through direct sampling of the density matrix itself.

The conditional mutual information (CMI) was evaluated using a similar sampling method, averaging over spin configurations on a region B to estimate the conditional distribution on regions A and C. For two-dimensional analyses, a two-step sampling procedure was implemented, first generating outcome configurations by evolving the classical dynamics up to time t, correctly weighting the samples, and then running a second QMC simulation with fixed temporal boundaries to measure conditional quantities within this fixed-boundary ensemble. The conditional mutual information (CMI) was evaluated using a similar sampling method, averaging over spin configurations on a region B to estimate the conditional distribution on regions A and C. A stiffness-like quantity was computed by measuring integrated current fluctuations, mirroring a winding-number definition from previous work.

Linear scaling of symmetry breaking order parameters differentiates one and two dimensional systems

Research indicates that strong symmetry is not spontaneously broken at any finite time in one-dimensional systems. However, correlators used to probe strong-to-weak symmetry breaking demonstrate order on length scales expanding linearly with time, a rate parametrically exceeding charge diffusion. Numerical evidence supports this scaling across multiple distinct probes of SW-SSB, derived from field analysis.

Specifically, the observed scaling reveals that the length scale grows at a rate of x ∼ t, contrasting with the typical diffusive spread of charge which scales as x ∼ √t. In two dimensions, both field theory and numerical simulations point towards a finite-time Berezinskii-Kosterlitz-Thouless-like strong-to-weak spontaneous symmetry breaking transition.

This transition occurs at a specific time, suggesting a change in the system’s behaviour. Within continuum hydrodynamics, spontaneous symmetry breaking would occur at infinitesimal time in two or more dimensions, but the research demonstrates a deviation from this expectation. The observed transition time is interpreted as the point at which a continuum hydrodynamic description becomes inadequate, and non-hydrodynamic effects, such as discrete particle worldlines, can no longer be reliably inferred.

Analysis of a specific model reveals a connection between SW-SSB and the derivation of a classical stochastic hydrodynamic description from the underlying quantum dynamics. For the decohered spin-1/2 model, the Rényi-2 mutual information (CMI) between two regions exhibits a linear growth in time for one-dimensional systems, with a slope of approximately 0.17.

By contrast, in two dimensions, the CMI displays a logarithmic increase, indicating a slowing down of the symmetry breaking process. Also, the decohered rotor model in one dimension shows a similar linear growth in the Rényi-2 CMI, with a slope of around 0.12, confirming the scaling behaviour. Model F rotor dynamics, used to explore the emergence of classicality, provides further insight into the transition.

The analysis of the Wigner function and the steady-state density matrix supports the connection between SW-SSB and the emergence of a classical stochastic hydrodynamic description. The Bhattacharyya distance, a measure of distinguishability between two probability distributions, shows a clear transition in the hydrodynamic regime, indicating the onset of classical behaviour.

Symmetry breaking linked to information limits in complex systems

Scientists are beginning to understand how order emerges from apparent randomness in complex systems, as demonstrated by recent work on spontaneous symmetry breaking. This research offers a new perspective, pinpointing a connection between symmetry breaking and the limits of our ability to infer system properties from incomplete observations. Investigators have linked it to a fundamental challenge in information theory, determining how much we can know about a system by only examining its boundaries.

The implications extend beyond theoretical physics. Understanding how local measurements relate to global order is relevant to fields like materials science, where predicting macroscopic properties from microscopic interactions is central. Once a system transitions beyond a certain complexity, standard hydrodynamic descriptions, those relying on smooth, continuous changes, begin to fail.

The findings suggest that the point at which symmetry breaks marks precisely this limit, signalling when more detailed, particle-level models become necessary. While simulations in one and two dimensions provide strong evidence, extending these results to higher dimensions remains an open question. Applying this framework to truly disordered materials will require careful consideration, as the specific models used, though insightful, are simplifications of real-world systems. The focus is on understanding the underlying mechanisms, but future research could explore how to actively control these transitions, potentially leading to new materials with tailored properties.