Nonequilibrium symmetry-protected topological order: emergence of semilocal Gibbs ensembles

Abstract

We consider nonequilibrium time evolution in quantum spin chains after a global quench. Usually a nonequilibium quantum many-body system locally relaxes to a (generalised) Gibbs ensemble built from conserved operators with quasilocal densities. Here we exhibit explicit examples of local Hamiltonians that possess conservation laws with densities that are not quasilocal but act as such in the symmetry-restricted space where time evolution occurs. Because of them, the stationary state emerging at infinite time can exhibit exceptional features. We focus on a specific example with a spin-flip symmetry, which is the commonest global symmetry encountered in spin-$1/2$ chains. Among the exceptional properties, we find that, at late times, the excess of entropy of a spin block triggered by a local perturbation in the initial state grows logarithmically with the subsystem's length. We establish a connection with symmetry-protected topological order in equilibrium at zero temperature and study the melting of the order induced either by a (symmetry-breaking) rotation of the initial state or by an increase of the temperature.

Figures (16)

• Figure 1: Panel (a): odd operator $\boldsymbol{\bf O}_{\rm o}(x)$ ($\boldsymbol{\bf O}_{\rm o}(y)$) with support to the left (right) of the site $j$ commutes (anticommutes) with the string $\boldsymbol{\bf \Pi}^z(j)$. For simplicity we assume $x\ll j\ll y$. Panel (b): even operator with support that does not include the site $j$ commutes with the string $\boldsymbol{\bf \Pi}^z(j)$. Restricting to the space of even operators, the strings behave as local objects.
• Figure 2: Quasilocal theory (red line) and even semilocal theory (blue dashed line). Each of them contains operators that are able to represent local observables, provided that some of the operators of the other theory are excluded (even semilocal and odd quasilocal operators can not simultaneously represent local observables --- see Fig. \ref{['fig:half_infinite_string']}).
• Figure 3: Relaxation of $\langle\boldsymbol{\bf \sigma}_{\ell}^z\rangle$ towards $\langle\boldsymbol{\bf \sigma}_\ell^z\rangle_{\rm GGE}\approx 0.466\, (0.717)$ (dashed lines) in the XY model, computed from Eq. \ref{['eq:GGE_correlation']}. The dots represent the results of the iTEBD simulation, while the solid curve is the exact time evolution using Eq. \ref{['eq:symbol_correlation']}. Parameters in Eqs. \ref{['eq:XY_model']} and \ref{['eq:init_state']} are $J_x=1$, $J_y=2$, and $\theta =0.9\,(0.3)$. The iTEBD evolution uses a second order Trotter scheme with two-site quantum gates, time step $\delta t=0.01$, Schmidt values cutoff $10^{-6}$ and maximal allowed bond dimension $M_{\rm max}=800$.
• Figure 4: Relaxation of $\langle\boldsymbol{\bf \sigma}_{\ell}^x\rangle$ towards $\langle\boldsymbol{\bf \sigma}_\ell^x\rangle_{\rm GGE}\approx 0$ (dashed lines) in the XY model. Parameters are the same as in Fig. \ref{['fig:tilted_xy_Sz']}.
• Figure 5: Numerical (iTEBD) time evolution of $\braket{\boldsymbol{\bf \sigma}_\ell^z}$ and GGE predictions ($\braket{\boldsymbol{\bf \sigma}_\ell^z}_{\rm GGE}\approx0.038$, $0.0012$, and $0.000078$ for $\theta=0.9$, $0.3$, and $0.15$, respectively) in the dual XY model. Relaxation towards the GGE prediction is slower for smaller $\theta$. Parameters of the Hamiltonian \ref{['eq:dual_xy']} are $J_x=1$, $J_y=2$. The iTEBD uses second order Trotter scheme with four-site quantum gates, $\delta t=0.01$, Schmidt values cutoff $10^{-6}$, and maximum bond dimension $M_{\rm max}=1000$. The data converge up to times $t\sim 4.5$ (see Appendix \ref{['app:extrapolation_of_tebd_times']}).