Many-Body Localization Dynamics from Gauge Invariance

We show how lattice gauge theories can display many-body localization dynamics in the absence of disorder. Our starting point is the observation that, for some generic translationally invariant states, the Gauss law effectively induces a dynamics which can be described as a disorder average over gauge superselection sectors. We carry out extensive exact simulations on the real-time dynamics of a lattice Schwinger model, describing the coupling between U(1) gauge fields and staggered fermions. Our results show how memory effects and slow, double-logarithmic entanglement growth are present in a broad regime of parameters—in particular, for sufficiently large interactions. These findings are immediately relevant to cold atoms and trapped ion experiments realizing dynamical gauge fields and suggest a new and universal link between confinement and entanglement dynamics in the many-body localized phase of lattice models.

Figure 1

(a) Schematic of a ($1+1$)-d lattice gauge theory, with matter fields ${\psi }_n$ defined on the vertices and pairs of conjugate variables $\{U_n,L_n\}$ defined on bonds. (b) Typical initial states used in the simulations: Fermions are arranged in an alternate pattern of empty and occupied sites (corresponding the bare vacuum), while gauge fields are in an equal weight superposition of electric field eigenstates with eigenvalues $(0,\pm 1)$. (c) Schematics of the real-time dynamics, which, starting from $|\mathrm{\Psi }{⟩}_0$, can be decomposed in an $\mathcal{N}$ superselection sector (see the text).

Figure 2

Dynamics of the overall staggered fermion occupation (a) and central staggered fermion occupation (b) from a bare vacuum initial state for different values of coupling strength parameter $J$, and $N=26$. (c),(d) Final time value of $\nu (t)$ and $\mu (t)$, respectively, averaged from $t=50$ to $t=100$ for different system sizes up to $N=30$ and two limiting values of $J$. The absolute error due to averaging realizations is shown as vertical bars for each point.

Figure 3

Time evolution of the bipartite von Neumann entanglement entropy $S_A(t)$ from a bare vacuum initial state. (a) $S_A(t)$ for different values of coupling strength parameter $J$ and $N=18$. The dashed line refers to the value $N\log (2)/2-1/2$ of the entanglement entropy as expected for a random state. (b) $S_A(t)$ on intermediate time scales for $J=1$ and different system sizes $N$. (c) Long-time growth of $S_A(t)$ for $J=10$ including a fit (solid line) $f(t)=a\log [\log (wt)]+b$ to the $N=14$ data. (d) The same data are plotted on a double logarithmic scale.

Figure 4

(a) Asymptotic long-time dynamics of the entanglement entropy density $S_A(t)$ for different system sizes $N$ at $J=1$ suggesting saturation to an extensive value by plotting the entanglement entropy density $S_A(t)/N$ (b).