Strong-to-Weak Symmetry Breaking in Open Quantum Systems: From Discrete Particles to Continuum Hydrodynamics

TL;DR

This work investigates SW-SSB in U(1)-symmetric open quantum systems using three complementary models, revealing dimension-dependent dynamics: in 1D SW-SSB is not finite-time but exhibits ballistic growth of nonlinear SW-SSB scales; in 2D there is a finite-time BKT-like SW-SSB transition with a universal Rényi-2 stiffness jump, while a continuum hydrodynamic description emerges at late times; in higher dimensions a Model-F type classical hydrodynamics governs long-wavelength behavior after SW-SSB. The authors develop a worldline/density-matrix framework, analyze both dissipative and decohered rotor/Model-F dynamics, and provide both numerical and analytical RG treatments. They show that discreteness of charge is essential for finite-time SW-SSB and demonstrate connections between SW-SSB, information-theoretic measures (Rényi correlators, CMI), and the emergence of hydrodynamics, with experimental protocols proposed for probing these phenomena. The results offer a unified view of how information-theoretic order, decoding tasks, and hydrodynamic descriptions arise in open quantum systems across dimensions, and they illuminate the conditions under which continuum hydrodynamics becomes the appropriate long-wavelength theory.

Abstract

We explore the onset of spontaneous strong-to-weak symmetry breaking (SW-SSB) under U(1)-symmetric (i.e., charge-conserving) open-system dynamics. We define this phenomenon for quantum states and classical probability distributions, and explore it in three complementary models, one of which exhibits nontrivial quantum coherence at short times. Our main conclusions are as follows. In one dimension, the strong symmetry is not spontaneously broken at any finite time; however, correlators probing strong-to-weak symmetry breaking develop order on length scales that grow linearly in time, parametrically faster than charge diffusion. We provide numerical evidence for this scaling in multiple distinct probes of SW-SSB, and derive it from a field-theory analysis. Moreover, we relate this scaling to the problem of inferring the charge inside a subregion by measuring its surroundings, and construct explicit decoding protocols that illustrate its origin. In two dimensions, field theory and numerical simulations support a finite-time Berezinskii-Kosterlitz-Thouless-like SW-SSB transition. Within continuum hydrodynamics, by contrast, SW-SSB happens at infinitesimal time in two or more dimensions. The SW-SSB transition time can thus be interpreted as marking the emergence of a continuum hydrodynamic description, or (more precisely) the timescale beyond which non-hydrodynamic information such as discrete particle worldlines can no longer be inferred. We support this picture by analyzing a model in which we exploit SW-SSB to derive a classical stochastic hydrodynamic description from the underlying quantum dynamics.

Figures (15)

• Figure 1: Illustration of the "two-step" dynamical process underlying SW-SSB in an open quantum system. In the density matrix representation of a quantum system, each particle has two worldlines, corresponding to the ket and bra space of the density matrix, or the forward and backward paths in the Keldysh formalism. At a crossover time scale $t^\ast$, the two worldlines merge together, meaning the density matrix becomes approximately diagonal. In two and three dimensions, at a later time, $t_c \gtrapprox t^\ast$, there is a finite-time phase transition whereupon the merged worldlines intertwine, exchange and condense, losing track of their original positions, marking the onset of SW-SSB and emergent classical hydrodynamics.
• Figure 2: CMI geometries in 1d. (a) In the first case, regions $A$, $B$, and $C$ cover the entire system. This is the standard geometry for CMI in the context of 1d mixed state phases and recoverability. (b) In the second case, the three regions do not cover the entire system, with regions $A$ and $C$ typically of size $O(1)$. In both cases, we denote the separation between regions $A$ and $C$ as $R_B = \abs{B}$.
• Figure 3: Worldline representation of the local $U(1)$ charge dynamics (with time running upward), illustrated for the spin-$1/2$ model. The gray arrows indicate the Néel initial state $\hat{\rho}_0$. Panel (a) shows the local charge expectation value, where the final-time boundary is free and is probed by the insertion of a charge operator $\hat{q}_r$. Panels (b) and (c) show the corresponding representation for the Rényi-$2$ correlator $C^{(2)}(i,j)$, where the temporal boundaries are both $\hat{\rho}_0$ at the bottom and the top. They correspond to the denominator and numerator of $C^{(2)}(i,j)$, respectively; in (c) the operator insertions $\hat{S}_i^+$ and $\hat{S}_j^-$ create and annihilate a defect worldline.
• Figure 4: Spatial charge profiles over time from a domain wall initial state, with $L = 100$ and $\gamma = 0.1$ from $t = 10$ to $t = 260$. In the inset, the horizontal axis is not scaled and it is apparent that the initial domain wall melts over time, with charge spreading from the left to the right. When the horizontal axis is rescaled by $\sqrt{t}$, the charge profiles at different times all collapse to one curve. This indicates a scaling form $\expval{S^z_i(t)} = f(x/\sqrt{t})$ over this time range, indicative of diffusion. In both cases, $x=0$ is taken to be halfway between site $50$ and site $51$.
• Figure 5: (a) Evolution of Rényi-$1$ and Rényi-$2$ correlations over time, with $L=128$ and with $\gamma = 0.1$. In the inset, $[C^{(2)}(x;t)]^{1/2}$ is given on a semi-log plot vs. $x$ for a range of times, demonstrating that $[C^{(2)}(x;t)]^{1/2} \propto e^{-x/(2\,\xi^{(2)}(t))}$. Upon rescaling the horizontal axis, we see that both $C^{(1)}(x;t)$ and $[C^{(2)}(x;t)]^{1/2}$ are functions of $x/t$, and specifically that $C^{(1)}(x;t) \propto [C^{(2)}(x;t)]^{1/2} \propto e^{-x/(2\,\xi^{(2)}(t))}$. This implies that $\xi^{(1)}(t) = 2\,\xi^{(2)}(t) \propto t$. (b) The infinite MPS Rényi-$2$ transfer matrix gap is given on a log-log scale. The data indicates saturation to $t^{-1}$ scaling. Specifically, applying a linear fit to the data from $t=2$ to $t=20$ yields a slope of $0.97 \approx 1$.