Long-range resonances in quasiperiodic many-body localization

TL;DR

The paper addresses how long-range resonances affect many-body localization in a deterministic quasiperiodic system by analyzing mid-spectrum eigenstates of a quasiperiodic Heisenberg chain with $h_i = h \cos(2\pi \beta i + \phi)$. Beyond standard diagnostics, it reveals fat-tailed distributions of long-distance spin correlations in a strong quasiperiodic regime, attributable to rare resonant cat-like eigenstates that generate substantial long-range entanglement. These anomalous states persist in a regime where conventional metrics already suggest MBL, signaling an instability of the localized phase driven by long-range resonances rather than Griffiths regions. The work bridges quasiperiodic and random-model instabilities, and proposes density-correlation measurements in ultracold atoms as a practical probe of these long-range resonances with potential experimental relevance.

Abstract

We investigate long-range resonances in quasiperiodic many-body localized (MBL) systems. Focusing on the Heisenberg chain in a deterministic Aubry-André potential, we complement standard diagnostics by analyzing the structure of long-distance pairwise correlations at high energy. Contrary to the expectation that the ergodic-MBL transition in quasiperiodic systems should be sharper due to the absence of Griffiths regions, we uncover a broad unconventional regime at strong quasiperiodic potential, characterized by fat-tailed distributions of longitudinal correlations at long distance. This reveals the presence of atypical eigenstates with strong long-range correlations in a regime where standard diagnostics indicate stable MBL. We further identify these anomalous eigenstates as quasi-degenerate pairs of resonant cat states, which exhibit entanglement at long distance. These findings advance the understanding of quasiperiodic MBL and identify density-correlation measurements in ultracold atomic systems as a probe of long-range resonances.

Figures (9)

• Figure 1: Overview of the results for the QP Heisenberg spin chain model (Eq. \ref{['eq:H']} with $\Delta=1$). (a) QP field profile $h_i$ for $h=3.6$ and $\phi \approx 0.0554$. The red dots mark lattice sites $i$ for $L=22$. (b) Probability distribution of the gap ratio $P(r)$ for system sizes $L=16,18,20,22$ at field strengths $h=1.8$, $3.6$, and $6$. For $h=1.8$, the data approach GOE statistics with increasing $L$, while for $h=3.6$ and $h=6$, they follow Poisson statistics, signaling MBL. (c) Corresponding distributions of the half-chain entanglement entropy $P(\mathrm{EE})$ for the same parameters. $P(\mathrm{EE})$ shows volume-law behavior for $h=1.8$ and shifts toward localized behavior for $h=3.6$ and $h=6$. (d) Distribution of the rescaled longitudinal correlations $P(\ell^z)$, where $\ell^z=-\ln|4C^{zz}_{L/2}|$. For $h=1.8$, $P(\ell^z)$ is characteristic of the ergodic phase. At $h=3.6$, it develops pronounced fat tails that decay slowly with $L$, revealing eigenstates with strong long-range correlations. For $h=6$, $P(\ell^z)$ exhibits an exponential tail, consistent with a fully localized regime. (e) Schematic phase diagram vs.$h$, showing the ergodic phase, a fat-tail regime where the typical longitudinal correlation length $\xi_{\mathrm{typ}}^z$ increases with system size, and an exponential-tail regime where $\xi_{\mathrm{typ}}^z$ remains constant. The finite-size critical region $h_{\rm c}^*(L_{\max})$, obtained from the crossings between the largest available system sizes ($L=20$ and $22$), is indicated by the colored symbols, while the extrapolated critical point $h_{\rm c}^{\mathrm{standard}}=3.13(32)$, obtained from all the standard observables, is marked by the white star (see End Matter).
• Figure 2: Scaled (a) participation and (b) half-chain entanglement entropies, and (c) typical longitudinal ($\xi_{\mathrm{typ}}^z$) and transverse ($\xi_{\mathrm{typ}}^x$) correlation lengths, all plotted against the QP field strength $h$. The dashed lines in (a,b) indicate the crossings between consecutive system sizes, and the shaded gray regions mark the critical regime of the ergodic–MBL transition obtained from the extrapolated finite-size crossing points SM. In (c), the dashed line denotes the Aubry–André localization length $\xi^{\mathrm{AA}} =(\ln h)^{-1}$. Inset: $\overline{\ln |C^{zz}_{L/2}|}$ is shown vs.$L$ for $h=3.6$, with dotted lines representing 4-point fits.
• Figure 3: (a) Maximum longitudinal correlation $\max\limits_i |C_{i,i+L/2}^{zz}|$vs. energy $E$ near the middle of the spectrum for $L=22$, $h=3.6$, and $\phi \approx 0.0554$ (corresponding to Fig. \ref{['fig:fig1']}(a)). Two eigenstates with exceptionally large central correlations are highlighted by orange circles. The inset shows the real-space correlations $|C_{i,i+L/2}^{zz}|$ for these two marked states. (b) Local magnetization profiles $\langle S_i^z \rangle$ for the same nearly degenerate eigenstates. The blue squares correspond to the state with $|C_{L/2}^{zz}| \approx 0.2286$, and the red diamonds to the state with $|C_{L/2}^{zz}| \approx 0.2295$.
• Figure EM1: Finite-size scaling analysis of the MBL transition for the Heisenberg quasiperiodic chain using the half-chain entanglement entropy, EE$/S_{\rm RMT}$, for system sizes $L=10, 12, \dots, 22$. The panels show the crossing points $h_{\rm c}(L, L+2)$ for consecutive system sizes. The critical point for each pair of system sizes is determined by fitting curves in the gray region and finding their intersection, which is marked by the green diamond. We observe a non-monotonic drift of these points with increasing system size (see Fig. \ref{['fig:s4']}).
• Figure EM2: Extrapolation of crossing points for standard observables (PE, EE, EM, GR) for the quasiperiodic Heisenberg chain. The plot shows the critical field $h_{\rm c}$vs. the inverse average system size $1/L_{\rm avg}$. The dashed lines represent the corresponding linear fits. The key also holds for the finite-size crossings shown in Fig. \ref{['fig:fig1']}(e).