Cat states carrying long-range correlations in the many-body localized phase

TL;DR

The paper investigates the high-energy regime of the random-field Heisenberg chain to understand non-ergodic MBL behavior and its stability. By surveying the full spectrum with extensive disorder averaging, the authors identify rare, systemwide correlations that manifest as nearly degenerate pairs of cat states spanning the chain. They develop a toy-cat-state framework, prove its predictive power for observables such as C_{L/2}^{zz}, QMI, and KL divergences, and demonstrate that these long-range resonances dominate a broad disorder window, with an estimated upper bound h_max ≈ 20–25 for MBL stability. The findings imply that the MBL phase harbors a rich landscape of rare events that can seed ergodicity through long-range resonances, potentially reshaping the understanding of MBL transitions and intermediate phases. The work provides analytic guidance, a concrete selection algorithm, and detailed microscopic diagnostics (fluctuating spins, gap distributions, and magnetization profiles) to quantify these cat-like resonances and offers a pathway to integrate rare-event physics into MBL theory and renormalization approaches.

Abstract

Despite considerable efforts over the last decade, the high-energy phase diagram of the random-field Heisenberg chain still eludes our understanding, in particular the nature of the non-ergodic many-body localized (MBL) regime expected at strong disorder. In this work, we revisit this paradigmatic model by studying the statistics of rare atypical events of strongly correlated spin pairs traversing the entire system. They occur for unexpectedly strong disorder, i.e., in a regime where standard estimates fail to detect any instability. We then identify these very peculiar high-energy eigenstates, which exhibit system-wide ${\cal{O}}(1)$ correlations, as nearly degenerate pairs of resonant cat states of the form $|Φ_{\pm}\rangle\sim {|{α_1}\rangle}\pm {|{α_2}\rangle}$, where ${|{α_1}\rangle}$ and ${|{α_2}\rangle}$ are spin basis states. We propose a simple and generic analytical description of this new class of eigenstates that exhibit system-spanning entanglement. This analytical ansatz guides us in our search for rare hidden cat states in exponentially large many-body spectra. This also enables a systematic numerical inspection of the microscopic anatomy of these unconventional pairs, which appear in a wide range of disorder strengths. In the light of recent studies and ongoing debates on the MBL problem, our results offer new perspectives and stimulating challenges to this very active field.

Figures (24)

• Figure 1: Overview of the different regimes of the random-field Heisenberg chain model Eq. \ref{['eq:RFHCM']}, as seen from the statistics of the midchain longitudinal correlation functions $C_{L/2}^{zz}$. The colored arrow (top) shows the main physical regimes as a function of the disorder strength $h$, and below, the energy density $\epsilon$ dependence of these correlations is plotted in the various insets, where $C_{L/2}^{zz}(\epsilon)$ is shown for all the ${\cal{N}}_{\rm H}=12870$ eigenstates ($L=16$ chains), for 4 representative strengths of disorder, each illustrated by two samples. Panels (a) show the ergodic behavior at $h=2$, below $h_{\rm MBL}\sim 5$ where there are no rare events, and the 2 examples shown are typical, i.e. without anomalously large correlations. In the rare event regime, panels (b) for $h=8$ and (c) for $h=16$ show the emergence of rare eigenstates across the spectrum: two pairs of cat states with anomalously large correlations are highlighted in yellow, and enlarged in the top two sub-panels where we visualize the tiny energy gaps $\Delta\epsilon$ as small as $\sim 2\cdot 10^{-6}$ (b) and $\sim 4\cdot 10^{-8}$ (c). It is also worth noting the values of the associated correlations within each pairs: (b) $C^{zz}_{L/2}\approx (0.1043,\,-0.1057)$, and (c) $C_{L/2}^{zz}\approx (0.23678,\, 0.23686)$. Panels (d) for $h=24$ show two typical behaviors around $h_{\rm max}\sim 20-25$ where no rare events are detected, and all eigenstates host exponentially suppressed correlations (note the very small scale on the y-axis). Between the ergodic regime (expected for $h\le h_{\rm MBL}$, where $h_{\rm MBL}\sim 5$ comes from extrapolated standard estimates sierant_polynomially_2020colbois_interaction_2024) and $h_{\rm max}\sim 20-25$ (above which all eigenstates exhibit short-range correlations), there is a rather broad rare-event regime that evolves continuously from fat-tail to exponential-tail behavior for the distributions $P(\ln|4C_{L/2}^{zz}|)$colbois_statistics_2024.
• Figure 2: Average number of eigenstates (per sample) $N_\star^z$ having strong systemwide $zz$ correlations, i.e. mid-chain correlations such that $|C_{L/2}^{zz}|\ge C_{\star}$, with (a) $C_\star=0.1$ and (b) $C_\star=0.2$. Plotted against disorder strength $h$, the different curves $N_\star^z(L,h)$ for chain lengths $L\in [8,\,18]$, exhibit a clear crossing for $h\sim 20$. Inset shows how the length scale $\Lambda_z$ for $C_\star=0.1$ (rescaled by the spectral length $\Lambda_{\rm H}=\frac{1}{\ln 2}$) decays with $h$ for various fitting windows indicated on the plot. The line $\Lambda_z/\Lambda_{\rm H}=1$ is crossed for $h\sim 20$. Finite size effects become significant below $h\sim 10$.
• Figure 3: Same as Fig. \ref{['fig:Nz']} for the transverse component $N^*_x$ with (a) $C_\star=0.1$ and (b) $C_\star=0.2$. The inset shows length scale $\Lambda_x$ for $C_\star=0.1$, rescaled by the spectral length $\Lambda_{\rm H}=\frac{1}{\ln 2}$. The line $\Lambda_z/\Lambda_{\rm H}=1$ is crossed for $h\sim 10$, but with significant finite-size effects.
• Figure 4: Maximal value of the long-distance longitudinal correlation $C_{L/2}^{zz}$ per sample. The maximum is computed for each random sample over all the ${\cal{N}}_{\rm H}$ eigenstates (and the $L/2$ possible pairs), and then averaged over a large number of samples (typically $\sim 2\cdot 10^4$ for $L\le 14$, $\sim 10^4$ for $L=16$, and $3\cdot 10^3$ for $L=18$). The inset shows the typical average computed over all the eigenstates for comparison: note the very different scale on the y-axis.
• Figure 5: Same as Fig. \ref{['fig:max_Czz']} for $C_{L/2}^{xx}$ (left) and QMI$_{L/2}$ (right). The insets show zooms on the crossing regions.