Translation symmetry restoration in integrable systems: the noninteracting case

Abstract

The study of symmetry restoration has recently emerged as a fruitful means to extract high-level information on the relaxation of quantum many-body systems. However, while the restoration of internal symmetries has been investigated intensively, that of spatial symmetries has hitherto only been considered in the context of random unitary circuits. Here we present a complementary study of translation symmetry restoration in integrable systems. In particular, we consider a one-dimensional chain of spinless, non-interacting fermions quenched from a $ν>1$ shift invariant state, and follow the local restoration of one-site shift invariance using the Frobenius distance $ΔF_A$ between the state on a subsystem and its symmetrised counterpart. Distinct from the case of random unitary circuits, where symmetry restoration occurs abruptly for times proportional to the subsystem size, we find that symmetry here is restored smoothly and over timescales of the order of the subsystem size squared. We also find that the so-called `quantum Mpemba effect' is readily observed. Most importantly, we show that - in contrast to the case of continuous internal symmetries - this discrete symmetry restoration is not qualitatively described by a quasiparticle picture for $ΔF_A$, and therefore goes beyond the hydrodynamic description. Our results can be directly extended to higher dimensions.

Figures (6)

• Figure 1: The Frobenius distance $\Delta F_A(t)$, obtained via exact diagonalisation using Gaussian methods for subsystem size $l=100$ and different initial states. The states are of the form $\ket{\psi_{\nu}(\lambda)}$ where $\nu$ is the translational invariance and $\lambda$ is the superposition parameter, and their exact definitions are given in Eq. \ref{['eq:states']}. Crossings between the Frobenius distance of different states, which correspond to instances of QME, are indicated by the shaded orange boxes. Inset: Main plot extended to late times, showing that no further crossings occur.
• Figure 2: Different bipartitions of a correlated multiplet for a $\nu=4$ initial state with respect to subsystem $A$. The multiplet is produced in a unit cell of size $\Delta$ and each mode $k_n(p) = p +2\pi n/\nu$ propagates with fixed velocity $v_n(p) = 2\sin(p +2\pi n/\nu)$. (a) A bipartition in which two modes are inside the subsystem. The purple dimension line indicates a relative velocity between adjacent modes, which will be important for later discussion. (b) The bipartition in which all modes are inside the subsystem. This contributes to the Frobenius distance but not to the entanglement entropy (see main text). (c) A bipartition in which a single mode is inside the subsystem. This contributes to the entanglement entropy but not to the Frobenius distance.
• Figure 3: (a) Comparison of exact diagonalisation (data points) and quasiparticle method (solid lines) for $-\log[(\nu-1) - \nu \Delta F_A(t)^2]/l$ as subsystem size is increased, for $\psi_2(0.15)$. For (b) (the $\psi_4(1)$ state), refer to the Supplemental Material.
• Figure 4: (a) The difference between exact diagonalisation and quasiparticle solutions to the Frobenius distance, $\Delta F_A(t) -\Delta F_{A,\text{QP}}(t)$, for different initial states and subsystem size $l=100$. (b) The leading non-extensive correction $\delta^{(0,1)}_A(t)$, defined in Eq. \ref{['eq:corrections']}, for each of these inital states and subsystem size $l=200$. This panel is displayed on log-log scale at late times only, and the slopes of the dashed lines approximate the decay of the correction at large times as $t^{-\alpha}$. Inset (a): Example, shown for $t=20\,000$, of the linear scaling of the difference $\Delta F_A(t) -\Delta F_{A,\text{QP}}(t)$ with system size.
• Figure 5: The Frobenius distance in the scaling limit $t/l^2$ for different initial states and subsystem sizes. (a) compares the method of exact diagonalisation (data points) at increasing subsystem size to the functional form $\phi_{\rm QP}(\xi)$ given by Eq. \ref{['eq:scalingfunction']} obtained in the $l\to\infty$ limit of the quasiparticle method (solid lines), while (b) shows $\phi_{\rm QP}(\xi)$ at earlier times. Inset (a): Example, shown for $t/l^2=2$, of the scaling with system size compared to the prediction of $\phi_{\rm QP}(\xi)$, as defined via the normalised function $\mathcal{F}_A(t,l) = [\Delta F_A(t,l) - \phi_{\text{QP}}(\xi)] / [\Delta F_A(t,l_{min}) - \phi_{\text{QP}}(\xi)]$. The data are well described by a power-law fit of the form $al^{-b} + c$, where going from top to bottom we find $a=1.13, 2.02, 3.98$, $b=1.02, 0.97,1.03$, and $c=0.95,0.91, 0.78$. Inset (b): The main plot of (b) extended to late times, showing that no further crossings occur.