Absence of Ergodicity without Quenched Disorder: From Quantum Disentangled Liquids to Many-Body Localization

We study the time evolution after a quantum quench in a family of models whose degrees of freedom are fermions coupled to spins, where quenched disorder appears neither in the Hamiltonian parameters nor in the initial state. Focusing on the behavior of entanglement, both spatial and between subsystems, we show that the model supports a state exhibiting combined area and volume-law entanglement, being characteristic of the quantum disentangled liquid. This behavior appears for one set of variables, which is related via a duality mapping to another set, where this structure is absent. Upon adding density interactions between the fermions, we identify an exact mapping to an $XXZ$ spin chain in a random binary magnetic field, thereby establishing the existence of many-body localization with its logarithmic entanglement growth in a fully disorder-free system.

Figure 1

Time evolution of entanglement entropy after a quench from a charge density wave state. The results are obtained using ED for $h/J=20$, $\mathrm{\Delta }=0$. (a) The von Neumann bipartite entanglement entropy $S(t)$ of a half-system for $N=8$, 10, 12. (inset) The time $T_{\mathrm{area}}$ for which the area-law plateau persists (dashed line of main plot) as a function of $h/J$ compared with $(h/J{)}^2$. (b) Comparison between PBEEs $S_{\mathrm{PBEE}}^f(t)$ and $S_{\mathrm{PBEE}}^{\sigma }(t)$ and the entanglement of the full system $S(t)$ for $N=12$. (inset) The long time-limit $S(t\to \infty )$ (computed at $Jt\sim {10}^{12}$) as a function of system size. PBEE results are scaled by factors of 4.3 and 1.9, respectively.

Figure 2

The time evolution of the PBEE starting from a charge density wave state for $h/J=20$, $\mathrm{\Delta }=0$, and $N=8$, 10, 12, obtained using ED. (a) $S_{\mathrm{PBEE}}^c(t)$ for $c$ fermions. (b) $S_{\mathrm{PBEE}}^q(t)$ for the conserved charges $q_i$, see text.

Figure 3

Quantum quench from an initial charge density wave state, $h/J=20$. (a) Von Neumann entanglement entropy computed by ED for $N=12$ (thin, light line) and an MPS algorithm for $N=20$ (thick, dark line) with $\mathrm{\Delta }/J=0$, 0.1. The spatial bipartition is taken along the central bond. (inset) The same data on a semi-log plot. (b) Density imbalance $\mathrm{\Delta }\rho (t)\propto \sum _j|⟨0|{\widehat{n}}_j(t)-{\widehat{n}}_{j+1}(t)|0⟩|$ and the time-averaged value $(1/t){\int }_0^t\mathrm{d}\tau \mathrm{\Delta }\rho (\tau )$ (dashed lines) after the same quench.