Emergent Dynamical Translational Symmetry Breaking as an Order Principle for Localization and Topological Transitions

TL;DR

The work defines dynamical translational symmetry (DTS) via long-time dynamics and introduces the time-averaged local translational contrast (TLTC), denoted by $\mathcal{C}_a^{(O)}$. It proves that $\lim_{T\to\infty} \mathcal{C}_a^{(O)}(T)=0$ in extended or ergodic phases and that finite TLTC signals localization, memory retention, or boundary confinement. By analyzing the Aubry-Andr\é model, the interacting AA model, and the SSH model, the authors show TLTC captures Anderson localization, many-body localization, and topological transitions within a unified dynamical-symmetry framework. The approach extends the Landau paradigm to nonequilibrium quantum matter and offers experimentally accessible diagnostics via time-averaged local observables across platforms.

Abstract

Localization transitions represent a fundamental class of continuous phase transitions, yet they occur without any accompanying symmetry breaking. We resolve this by introducing the concept of dynamical translational symmetry (DTS), which is defined not by the Hamiltonian but by the long-time dynamics of local observables. Its order parameter, the time-averaged local translational contrast (TLTC), quantitatively diagnoses whether evolution restores or breaks translational equivalence. We demonstrate that the TLTC universally captures the Anderson localization transition, the many-body localization transition, and topological phase transitions, revealing that these disparate phenomena are unified by the emergent breaking of DTS. This work establishes a unified dynamical-symmetry framework for phases transitions beyond the equilibrium paradigm.

Figures (3)

• Figure 1: (a) Time evolution of the TLTC indicator $\mathcal{C}_a^{(P)}(T)$ for representative $\lambda/J=1,2,3$. The inset shows the corresponding long-time averaged site probabilities $\overline{P_i}$. (b) Long-time value $\mathcal{C}_a^{(P)}(T=1000)$ versus $\lambda/J$. Here we fix $J=1$, $L=610$, $i_0=L/2$, $a=1$, use a time integration step of $dt=0.2$, and employ open boundary conditions (OBC).
• Figure 2: (a) Time evolution of the TLTC $\mathcal{C}_a^{(P)}(T)$ (solid) and its site-averaged counterpart $\overline{\mathcal{C}}_a^{(P)}(T)$ (dashed) in the interacting AA model at $\lambda/J=1$ (red) and $\lambda/J=4$ (blue). (b) Comparison between the instantaneous and time-averaged TLTC, $\mathcal{C}_a^{(P)}(t)$ and $\mathcal{C}_a^{(P)}(T)$, and the density imbalance, $\mathcal{I}(t)$ and $\mathcal{I}(T)$, as functions of the quasiperiodic potential strength $\lambda/J$. All four quantities change from vanishing to finite values at approximately the same critical $\lambda/J$. Here we fix $J=1$, $V=1$, $L=14$, $N=7$, $i_0=7$, $a=1$, $dt=0.5$, and use OBC.
• Figure 3: (a) Time evolution of the TLTC $\mathcal{C}_a^{(P)}(T)$ in the SSH model for $J_2/J_1=0.5$ (red) and $J_2/J_1=1.5$ (blue), with the particle initially localized at the boundary site. (b) The long-time value $\mathcal{C}_a^{(P)}(T{=}1000)$ as a function of $J_2/J_1$. Here we fix $J_1=1$, $L=600$, $i_0=1$, $a=1$, $dt=0.5$, and use OBC.