Quantum Strong-to-Weak Spontaneous Symmetry Breaking in Decohered One Dimensional Critical States

TL;DR

This work addresses how spontaneous symmetry breaking manifests in mixed, open quantum states by introducing quantum SWSSB in a 1D XXZ chain under symmetry-preserving local XX decoherence. Employing a Choi-state field theory together with bosonization and large-scale MPS simulations, the authors uncover a boundary-BKT transition separating a trivial Luttinger liquid phase from a SWSSB phase, accompanied by a line with a nontrivial, decoherence-dependent effective central charge $c_{\text{eff}}$. They illuminate the transition from dual viewpoints—order-parameter behavior and entropic/ recoverability measures—showing the SWSSB is intrinsically quantum and can be tuned via the Hamiltonian parameter $\Delta$ even at arbitrary decoherence; they also extend the framework to generic symmetry-preserving decoherence channels and discuss experimental routes. The findings reveal a distinct open-system mechanism for symmetry breaking in 1D critical states, with potential relevance to quantum simulators and future explorations of mixed-state phases. Overall, the paper provides a rigorous, broadly applicable description of quantum SWSSB and its rich interplay with entanglement, boundary conformal field theory, and recoverability.

Abstract

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 \emph{quantum} 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 XXZ 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. Remarkably, the SWSSB transition in our case is \emph{purely quantum} 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.

Figures (11)

• Figure 1: (a) Schematic diagram of our model describing the ground state of Hamiltonian \ref{['eq:Hxxz']} under decoherence. (b) Phase diagram of the XXZ model under XX decoherence. The system exhibits a boundary BKT phase transition at the critical point $\Delta_c = 0$. For $\Delta < 0$, the system is in a trivial Luttinger liquid phase with strong symmetry, whereas for $\Delta > 0$, it transitions into the SWSSB phase, where only weak symmetry is preserved. At the critical point $\Delta_c = 0$, the correlation length diverges as $\xi \sim (\Delta - \Delta_c)^{-\nu}$ with $1/\nu = 0$, consistent with the boundary BKT universality class. The red dashed critical line is characterized by an effective central charge $c_{\text{eff}}$ of the Choi state $|\rho\rangle\rangle$ and critical exponents of Rényi-2 correlation function, which decrease with increasing decoherence.
• Figure 2: Standard correlation function $\mathop{\operatorname{Tr}}[\hat{\rho}\hat{\sigma}^x_i\hat{\sigma}^x_j]$ at decoherence strength $p=0.1$ for various anisotropy parameters $\Delta$. The numerically extracted critical exponents show excellent agreement with theoretical predictions from Luttinger liquid theory across all parameter values. Simulations were performed for system sizes ranging from $L=32$ to $512$, we demonstrate the result of numerical results (data points) and analytical predictions (solid lines).
• Figure 3: (a)-(b) Direct calculation and data collapse of Rényi-2 correlation function for mixed states and (c)-(d) data collapse of the Binder ratio $U_4 = \frac{1}{2}\left(3 - \frac{\langle \hat{O}_{\mathrm{xFM}}^4\rangle}{\langle \hat{O}_{\mathrm{xFM}}^2\rangle^2}\right)$, where $\hat{O}_{\mathrm{xFM}}=\frac{1}{L}\sum\hat{\sigma}_i^x$, obtained through effective decoupling theory, i.e., $\ket{\Psi_0}\rightarrow\hat{K}_{X}\ket{\Psi_0}$. $U_{4}$ is a dimensionless physical quantity at the critical point artificially constructed for $\mathbb{Z}_2$ symmetry breaking. Both cases show good agreement with $\Delta_c = 0$, and the $x$-axis is chosen as $\Delta \log L$ in data collapses to be consistent with theoretical expectations. The decoherence strength is $p=0.1$ here. Simulated system size is $L=32$ to $128$ in (a)-(b) and $L=128$ to $512$ in (c)-(d). And we also show the result of data collapse when $p=0.2$ in Appendix.\ref{['appendix:F']}.
• Figure 4: (a) Results for the Rényi-2 mutual information $I^{(2)}(A:B)$ of the decohered density matrix, where $A$ and $B$ correspond to the left and right halves of the system, respectively. Theoretical predictions give $I^{(n)}(A:B) = 4\dim[\hat{\mathcal{B}}^\alpha_n]\ln{\frac{L}{a}}$ (see the main text for details). For $n=2$, when the decoherence channel is irrelevant, $4\dim[\hat{\mathcal{B}}^{X}_2] = 1/4$, where $4 \dim[\hat{\mathcal{B}}^{X}_2] = 0.26 \approx \frac{1}{4}$ at $\Delta = -0.7$; otherwise, deviations from $\frac{1}{4}$ are observed, with $4 \dim[\hat{\mathcal{B}}^{X}_2] = 0.34$ for $\Delta = 0.7$ and $4 \dim[\hat{\mathcal{B}}^{X}_2] = 0.29$ for $\Delta = 0$. . (b) Direct calculation of the effective central charge $c_{\text{eff}}$ for the Choi state at $\Delta = -0.5$. Here, $c_{\text{eff}} = 1$ remains stable across a wide range of decoherence strengths $p$, while the correlation length $\xi \sim \min[L, \frac{a\mu_0}{\mu}]$, where $\mu_0$ is a constant. Finite-size effects cause deviations from the RG fixed point prediction for large $\mu$. (c) Results for the effective central charge $c_{\text{eff}}$ of the Choi state at $\Delta = 0.5$, showing a rapid decrease of $c_{\text{eff}}$ to zero as $p$ increases. The correlation length follows $\xi \sim \min[L, \frac{a\tilde{\mu}}{\tilde{\mu}_0}]$, where $\tilde{\mu}_0$ is a constant. For small $\mu$, finite-size effects similarly lead to deviations from the RG fixed point prediction. The simulated system size is $L = 8$ to $128$ in (a) and $L = 16$ to $256$ in (b)-(c).
• Figure 5: An illustration of the geometry of the system where $A = [-R/2, R/2]$, $B = [-r - R/2, R/2 + r]$, and $C$ is the rest of the system. The quantum channel $\hat{\mathcal{N}}_A$ is the restriction of $\hat{\mathcal{N}}$ to $A$, and the recovery channel $\hat{\mathcal{R}}_{AB}$ acts on $AB$.