Fock space fragmentation in quenches of disordered interacting fermions

Abstract

Hilbert space fragmentation, as it is currently investigated, primarily originates from specific kinematic constraints or emergent conservation laws in many-body systems with translation invariance. It leads to non-ergodic dynamics and possible breakdown of the eigenstate thermalization hypothesis. Here, we demonstrate that also in disordered systems, such as the XXZ model with random on-site fields, fragmentation appears as a natural concept offering fresh perspectives, for example, on many-body delocalization (MBdL). Specifically, we split the Fock-space into subspaces, potential-energy shells, which contain the accessible phase space for the relaxation of a quenched initial state. In this construction, dynamical observables reflect properties of the shell geometry, e.g., the drastic sample-to-sample fluctuations observed in the weak disorder regime, $W<W_c$, represent fluctuations of the mass of the shell. Upon crossing over from weak to strong disorder, $W>W_c$, the potential-energy shell decays into fragments; we argue that, unlike percolation, fragmentation is a strong-coupling scenario with turn-around flow: $W_c(L)$ diverges with increasing system size. We conjecture that the slowing down of the relaxation dynamics reported in traditional MBdL studies is (essentially) a manifestation of Fock-space fragmentation introduced here.

Figures (12)

• Figure 1: Fock space structure for a system of size $L {=} 10$, shown for three different disorder realizations at $W {=} 1.5$. The leftmost site corresponds to the Néel state, and the states to the right are ordered by increasing Hamming distance from it. The vertical arrangement of the sites is arbitrary. The red dots indicate the potential-energy shell; the connectivity of the basis states introduced by the hopping term in Eq. \ref{['e1']} is indicated by lines. Upper inset: The distribution $\mathcal{P}(\varepsilon_b)$ of the Fock-space energy density $\varepsilon_b{=}E_b/L$, where $E_b$ is defined in Eq. \ref{['e3a']}; the red vertical line marks the energy of the Néel state, and the shaded region indicates the energy variance, $\Delta_\text{N\'eel}$. Lower inset: Imbalance relaxation $I(t)$ for the corresponding sample.
• Figure 2: Shows the distribution of the normalized mass of the potential-energy shell, $N_\text{sh}\,$/$N_\text{FS}\,$, for different $W, L$.
• Figure 3: Similar data as in Fig. \ref{['f1']} for disorder strength $W{=}5.5$ for $L=10$. With $W$ growing beyond $W\approx 3.5$, the energy shell is seen to become increasingly fragmented. Correspondingly, $I(t)$ does not decay. The resulting oscillations in $I(t)$, which is a hallmark of fragmentation, are analyzed in detail in Fig. \ref{['fig:resonance']}.
• Figure 4: The size of the cluster that contains the initial state, $N_\text{csh}\,$, normalized to the size of the corresponding potential-energy shell, $N_\text{sh}\,$, as a function of the disorder strength $W$ for system sizes $L{=}12,18,24$. Inset: The corresponding fluctuations of the ratio $N_\text{csh}\,$/$N_\text{sh}\,$.
• Figure 5: Direct comparison of variance and mean values, re-plotting the data Fig. \ref{['f4']}. An approximate collapse is achieved after normalizing the ordinate with the heuristic factor of $\sqrt{L}$. Inset: Zoomed left part of the main plot, showing the turn-around flow.