Tunable Hilbert space fragmentation and extended critical regime

TL;DR

This work demonstrates that Hilbert-space fragmentation in the $t$-$J_z$ model can be tuned by progressively adding hopping terms, effectively erasing SLIOMs and merging fragmentation blocks. It provides analytical expressions for the number of SLIOMs and blocks and shows, via numerical simulations, that SLIOMs imprint a broad spectrum of relaxation times in local observables. A central result is the emergence of an extended critical regime, revealed by multiple peaks in the rescaled fidelity susceptibility $\chi_{\text{av}}\mu$ as system parameters cross fragmentation transitions, with the peak count increasing with system size. The findings offer a controlled pathway to ergodicity restoration and reveal a multi-step ergodicity-breaking phenomenon distinct from conventional integrability-breaking transitions.

Abstract

Systems exhibiting the Hilbert-space fragmentation are nonergodic, and their Hamiltonians decompose into exponentially many blocks in the computational basis. In many cases, these blocks can be labeled by eigenvalues of statistically localized integrals of motion (SLIOM), which play a similar role in fragmented systems as local integrals of motion in integrable systems. While a nonzero perturbation eliminates all nontrivial conserved quantities from integrable models, we demonstrate for the $t$-$J_z$ chain that an appropriately chosen perturbation may gradually eliminate SLIOMs (one by one) by progressively merging the fragmented subspaces. This gradual recovery of ergodicity manifests as an extended critical regime characterized by multiple peaks of the fidelity susceptibility. Each peak signals a change in the number of SLIOMs and blocks, as well as an ultra-slow relaxation of local observables.

Figures (9)

• Figure 1: A system with $L = 8$ sites, $N = 6$ particles and total spin $S^z = 0$. (a) Illustration of two types of particle hops. The spin pattern is indicated above the sketch of the ladder. Comparing the left and middle configurations, it is clear that the hops within the ladder legs do not modify the spin pattern. Specifically, the third and sixth particles, despite changing sites, remain the third and sixth particles. In contrast, comparing the left and right configurations reveals that the hops between the ladder legs do modify the spin pattern. The former fifth particle becomes the third, the former third becomes the fourth, and the former fourth becomes the fifth. (b) Illustration of SLIOMs. The particles at the edges of the ladder always have the same index, and so correspond to the same spin in the spin pattern, as they cannot hop to the opposite leg. They are shown with blue and red arrows. Other spins in the spin pattern are not fixed and can be flipped, as illustrated by the exemplary configurations in the middle and right sketches. They are marked with grey arrows. (c) The variation of the Hamiltonian matrix in the computational basis with the number of rungs $r$. State indices increase from left to right and from bottom to top. Grey squares mark blocks. For $r = 1$, an exemplary state from each block is shown. For $r \ge 2$, there is a single block, but clusters of matrix elements become more interconnected as $r$ increases.
• Figure 2: The time evolution of spin projections, $\langle \psi_t | S^z_i | \psi_t \rangle$, for all sites $i\le L$. We focus on a system with $L=16$ sites, $N=13$ particles, and total spin $S^z=1/2$. Results in (a) and (b) were calculated for $r=2$ ($Q=6$) and $r=4$ ($Q=2$), respectively. The initial state was the computational basis state with spin-up (spin-down) particles occupying the rightmost positions in the upper (lower) leg. The dashed lines mark the thermal prediction $(N_\uparrow-N_\downarrow)/(2L)\approx 0.031$.
• Figure 3: Rescaled fidelity susceptibilities $\chi_\text{av}\mu$ plotted against rung number $r$ for a ladder with (a) $L=12$ sites, $N=8$ particles and total spin $S^z=0$. Analogical results, but for a ladder with $L=14$ sites, $N=12$ particles, and total spin $S^z=0$, are presented in (b). We consider two energy cutoffs, $\mu = 2 \times 10^{-4}$ and $6 \times 10^{-4}$.
• Figure 4: (a) The spin-spin autocorrelation function $\text{corr}(t)$ for a ladder with $L=12$ sites, $N=8$ particles, and total spin $S^z=0$. Analogous results for a ladder with $L=14$ sites, $N=12$ particles, and total spin $S^z=0$ are shown in (b). The longest considered time corresponds to the Heisenberg time, $\tau_H=\omega_H^{-1}$, obtained from the entire energy spectrum. Different solid lines correspond to different numbers of additional rungs, $r \in \{0,1,2,3,4\}$ for $L=12$ and $r \in \{0,1,2,3,4,5\}$ for $L=14$. The long-time value of $\text{corr}(t)$ decreases with $r$. Recall that $\chi_\text{av}^\mu$ does not exhibit a peak at integer $r$. Dashed lines were calculated for fractional $r$ near the peak of $\chi_\text{av}^\mu$, i.e., $r \in \{0.1,1.2\}$ in (a) and $r \in \{0.05,1.05,2.05,3.1,4.2\}$ in (b). Here, $\text{corr}(t)$ exhibits an ultra-long relaxation time, $\tau \propto \tau_H$.
• Figure 5: (a) Spectral functions $|f(\omega)|^2$ shown for a ladder with $L=12$ near the eigenstate transition at $r=0$, i.e., $r=0$, $0.01$, $0.02$, $0.05$, $0.2$, $0.3$ and $0.6$. (b) Same results for $L=14$ and $r=1$, i.e., $r=1$, $1.004$, $1.01$, $1.015$, $1.03$, $1.1$ and $1.2$. Darker colors indicate larger $r$.