Universal Dynamical Scaling of Strong-to-Weak Spontaneous Symmetry Breaking in Open Quantum Systems

Abstract

Strong-to-weak spontaneous symmetry breaking (SWSSB) defines a mixed-state phase of matter--without a pure-state counterpart--in which nonlinear observables such as the Rényi-2 correlator develop long-range order while conventional linear correlations remain short-ranged. Here we study the emergence of SWSSB in one-dimensional open quantum systems governed by Lindbladian evolution, where the transition time diverges with system size and SWSSB appears only asymptotically in the steady state. By tracking the late-time growth of the Rényi-2 correlation length, we uncover a universal dynamical regime controlled purely by the symmetry class of the Lindbladian. Contrary to the conventional expectation that late-time dynamics are governed by the low-lying Liouvillian spectrum, we find that the time dependence of the SWSSB transition--exponential versus algebraic--is dictated solely by symmetry, independent of details of the Lindbladian, including whether the Liouvillian spectrum is gapped or gapless. For $\mathbb{Z}_2$-symmetric dynamics, the Rényi-2 correlation length grows exponentially in time--even when the spectrum is gapless--yielding an effective transition time $t_c \propto \operatorname{ln} L$ and enabling rapid preparation of the $\mathbb{Z}_2$ SWSSB steady state. In contrast, U(1)-symmetric dynamics exhibit algebraic scaling, $t_c \propto L^α$, with a filling-dependent dynamical exponent: ballistic growth ($α\approx 1$) at finite filling crosses over to diffusive scaling ($α= 2$) in the zero-filling limit. These results establish symmetry--rather than spectral gap structure--as the controlling principle for SWSSB late-time dynamical scaling, and open a new route to nonequilibrium symmetry breaking in open quantum systems.

Figures (4)

• Figure 1: Gapless $\mathbb{Z}_2$ symmetric Lindbladian model [$\gamma_1=\gamma_2=1.0$]. (a) $L^{(1)}$ moves the $z$-domain wall by one unit length, while $L^{(2)}$ annihilates a pair of adjacent domain walls. The dashed lines represent domain walls. (b) Finite-size scaling of the first 50 eigenvalues (except for zeros) of the Liouvillian restricted to the diagonal sector $\Xi$ for system size up to $L=20$. The gap size scales as $\delta\propto L^{-2}$ [blue dashed line]. (c) shows the structure of the density matrix in the $z$ basis. The slow gapless mode is contributed by the sector with $\mathbf{z}=\pm\mathbf{z}'$. (d) TEBD simulation of a $L=150$ chain shows that the late-time dynamical scaling is $\xi\propto\exp(\gamma t)$.
• Figure 2: (a) The full Liouvillian spectrum of strongly symmetric U(1) model in Eq. \ref{['eq_U1Model']} with $\gamma=1.0$ and size $N=8$. In the pure dephasing limit [$J=0$], the Liouvillian eigenvalues [blue dots] are highly degenerate dots at $-\gamma n/2$ for $n=1,\cdots,N$. With weak coherent dynamics [$J=0.1$], the degeneracies are lifted [red dots]. (b) Liouvillian eigenvalues [red dots] of the slow subspace near the steady state [blue dot]. (c) When only a single particle is filled, $R_2(r,t)$ follows Eq. \ref{['eq:single_magnonR2']} [blue line], matching well with TEBD results [red dots]; (d) At finite filling [$\nu=1/3$], $R_2(r,t)$ decays exponentially in $r$ at any $t$. We extract $\xi$ from the linear fitting $\ln R_2=-r/\xi$. (e) Log-log plot of $\xi(t)$ at $\xi=1/3$. At late time, the TEBD numerical results can be fitted by a linear line of slope 1 [blue dashed line], suggesting ballistic propagation. (f) The propagation speed $v$ of the ballistic front for different filling factors $\nu$. $v$ increases roughly following a parabolic curve peaked at $\nu=1/2$. The $\nu>1/2$ part is symmetric due to particle-hole symmetry.
• Figure 3: (a) Implementation of the Choi-Jamiołkowski isomorphism using the combiner tensors [marked in blue]. (b) Sketch of a Trotter step. The evolution operator during a time step $\delta t$ is decomposed into even and odd layers, where each layer is composed of local gates which can be sequentially applied to the MPS.
• Figure 4: Universality of the ballistic dynamical scaling. (a) $\ln\xi$-$\ln t$ plot when the interaction is turned on [$L=201,J=0.1,\gamma=1.0,V=0.1$]. As the unity-slope fit suggests, the dynamical scaling is ballistic. (b) $\ln\xi$-$\ln t$ plot when $\gamma$ and $J$ are comparable with each other [$L=101,J=0.5,\gamma=0.1,V=0$].