Dynamics of entanglement asymmetry for space-inversion symmetry of free fermions on honeycomb lattices

TL;DR

This work investigates how space-inversion symmetry breaking, encoded in the sublattice on-site imbalance of free fermions on a honeycomb lattice, relaxes after a quench to the inversion-symmetric point. By combining dimensional reduction with Gaussian-state methods, it derives analytical expressions for the time evolution of the entanglement asymmetry, revealing that subsystem geometry (stripe width parity) and Dirac-point physics control symmetry restoration. Remarkably, for even transverse width, a flat band at fixed $k_y$ leads to a macroscopic occupation of zero-velocity modes, preventing symmetry restoration and yielding a finite late-time entanglement asymmetry that can approach $\\ln 2$ in the thermodynamic limit. The results highlight the critical role of band structure in nonequilibrium symmetry dynamics and suggest feasible cold-atom experiments to observe these effects via higher-order Rényi entanglement measures.

Abstract

We study the entanglement asymmetry for the space-inversion symmetry of free fermions on a two-dimensional honeycomb lattice with an on-site energy imbalance between the two sublattices. We show that the entanglement asymmetry of a local subsystem exhibits nonanalytic dependence on the energy imbalance, due to the presence of Dirac points in the Brillouin zone. We also study the quench dynamics from the ground state into the inversion-symmetric point at which the energy imbalance vanishes. Under certain conditions on the subsystem geometry, the entanglement asymmetry relaxes to a finite value after the quench, revealing that the inversion-symmetry breaking in the initial ground state can persist even under the symmetric dynamics. We attribute the absence of symmetry restoration to the presence of a flat energy dispersion (flat band) in a specific direction.

Figures (5)

• Figure 1: Schematic illustration of the transformation from the honeycomb to the brickwork lattices.
• Figure 2: The energy spectra of the Hamiltonian in Eq. \ref{['eq:H_0']} for several fixed values of $L_y$. We set $J=1$ in all the plots.
• Figure 3: Schematic illustration of the bipartition of the whole system into the subsystems. The blue region represents subsystem $A$ and the cross symbol at the center indicates the coordinate origin.
• Figure 4: The $n=2$ Rényi entanglement asymmetry of the ground state \ref{['eq:Psi_0']}. The solid curves and symbols are the analytical result in Eq. \ref{['eq:REA_t=0']} with $X_0$ obtained using Eq. \ref{['eq:J_0']} and the exact results obtained numerically by evaluating Eq. \ref{['eq:Z_nk']}, respectively. The black dashed curves denote the asymptotic approximations of Eq. \ref{['eq:REA_t=0']} with Eqs. \ref{['eq:J_0_1']} and \ref{['eq:J_0_2']}. We take $\ell=100$ for all the plots.
• Figure 5: (a)-(c) Time evolution of the $n$-th order Rényi entanglement asymmetry after the quench into the Hamiltonian \ref{['eq:H']} starting from the ground state of the Hamiltonian \ref{['eq:H_0']} for several finite $M/J$ values. The solid curves denote the analytical result in Eq. \ref{['eq:REA_t']} with Eq. \ref{['eq:J_z']}. The symbols are the exact value of the Rényi entanglement asymmetry obtained by numerically evaluating the charged moments using Eq. \ref{['eq:Z_nk']}. The dashed lines correspond to Eqs. \ref{['eq:REA_large t']} and \ref{['eq:REA_inf']} that predict the asymptotic form of the entanglement asymmetry at large times. We take $\ell=100$ in all the plots. (d) $\Upsilon (v)$ defined in Eq. \ref{['eq:Ups']} that characterizes how much the quasiparticle pairs with the longitudinal group velocity $v$ contribute to the entanglement asymmetry. To numerically evaluate $\Upsilon(v)$, we approximate the Dirac delta function in Eq. \ref{['eq:Ups']} by the Lorentz function $\delta_\eta(x)=\eta/[\pi(x^2+\eta^2)]$ with $\eta=10^{-3}$.