Asymmetry and dynamical criticality

TL;DR

This work addresses how to quantify dynamical symmetry breaking and restoration during dynamical quantum phase transitions in the Lipkin-Meshkov-Glick model. It introduces asymmetry monotones based on the $\ell_1$-norm of the commutator with symmetry generators, and analyzes the time evolution under quenches of the transverse field $h$ across an anisotropy parameter $\gamma$. The authors show that the time-averaged asymmetry $\overline{F_L(\rho)}$, evaluated for the collective generators $J_x,J_y,J_z$, tracks the dynamical critical point, correlates with the dynamical order parameter and entropy production, and reveals a critical region whose location shifts with $\gamma$ and vanishes near the isotropic limit $\gamma\to1$. This establishes asymmetry as a unifying, physically transparent diagnostic connecting symmetry, information-theoretic coherence, and nonequilibrium thermodynamics in DQPTs, with implications for experimental observation.

Abstract

Symmetries play a central role in both equilibrium and nonequilibrium phase transitions, yet their quantitative characterization in dynamical quantum phase transitions (DQPTs) remains an open challenge. In this work, we establish a direct connection between symmetry properties of a many-body model and measures of quantum asymmetry, showing that asymmetry monotones provide a robust and physically transparent indicator of dynamical quantum criticality. Focusing on the quenched Lipkin-Meshkov-Glick model, we demonstrate that asymmetry measures associated with collective spin generators faithfully capture the onset of DQPTs, reflecting the dynamical restoration or breaking of underlying symmetries. Remarkably, the time-averaged asymmetry exhibits clear signatures of the dynamical critical point, in close correspondence with both the dynamical order parameter and the behavior of entropy production. We further uncover a quantitative link between asymmetry generation and thermodynamic irreversibility, showing that peaks in asymmetry coincide with maximal entropy production across the transition. Our results position asymmetry as a unifying concept bridging symmetry, information-theoretic quantifiers, and nonequilibrium thermodynamics in dynamical quantum phase transitions, providing a powerful framework for understanding critical dynamics beyond traditional order parameters.

Figures (7)

• Figure 1: Dynamical order parameter $\overline{\langle J_z \rangle}$ for the model described by the Hamiltonian \ref{['H_LMG']} as a function of the transverse magnetic field $h$, for quench from $h_0=0$ to $h$, and for different total angular momentum. The convergence to the critical point of the model indicates the critical phenomenon in the thermodynamic limit.
• Figure 2: Time evolution of the asymmetry measure $F_L(\rho)$, Eq. (\ref{['FL_function']}), for $L = J_x$, $J_y$, and $J_z$ shown in the first, second, and third columns, respectively, and for two different anisotropy parameters: $\gamma = 0.2$ (top row) and $\gamma = 0.8$ (bottom row). In all plots, two quenches are considered: $h = 0.2$ (red solid line) and $h = 0.8$ (blue dotted line).
• Figure 3: Time average of the asymmetry measure $\overline{F_L(\rho)}$ in terms of quench parameter for the three generators, $(a) \,\, L=J_x$, $(b) \,\, L=J_y$ and $(c)\,\,L=J_z$, of the rotation group and two anisotropy parameters: $\gamma = 0.2$ (blue dotted line) and $\gamma = 0.8$ (red solid line).
• Figure 4: Time average of Entropy Production $\overline{\langle\Sigma\rangle}$, according reference Nascimento2024, for $j=100$ and two different anisotropy parameters: $\gamma=0.2$ (blue dotted line) and $\gamma=0.8$ (red solid line).
• Figure 5: Time average of Entropy Production $\overline{\langle\Sigma\rangle}$ in function of $\gamma$ and $h$, for $j=100$. We vary $\gamma \in (0,1)$ and $h \in (0,1)$.