Breakdown of Quantum Chaos in the Staggered-Field XXZ Chain: Confinement and Meson Formation

TL;DR

This work demonstrates confinement-induced nonergodicity in a clean one-dimensional spin model: the XXZ chain with a staggered field. By combining exact diagonalization in symmetry-resolved sectors with analytic meson spectroscopy near the two-spinon threshold, the authors show a GOE-to-Poisson crossover in level statistics, correlation- and entanglement-band formation labeled by emergent domain-wall number $W$, and quantitative agreement with an analytic meson ladder near threshold. The results unify global spectral diagnostics with microscopic confinement physics, linking weak Hilbert-space fragmentation and scar-like eigenstates to meson formation in a disordered-free setting. The findings provide a practical template for mesoscale spectroscopy and dynamical probes of confinement-related nonergodicity in quantum spin chains and related materials.

Abstract

Confinement of fractionalized excitations can strongly restructure many-body spectra. We investigate this phenomenon in the gapped spin-$\frac{1}{2}$ XXZ chain subject to a staggered field, where spinons bind into domain-wall ``mesons'' deep in the antiferromagnetic phase. We present evidence that this non-integrable model exhibits both Hilbert space fractionalization and quantum scar formation as controlled by the anisotropy parameter $Δ$. Exact diagonalization across symmetry-resolved sectors reveals a crossover from Gaussian-orthogonal (chaotic) level statistics at weak anisotropy $Δ\sim 1$ to non-ergodic behavior deep in the antiferromagnetic regime $Δ\gg 1$ through scrutinizing the adjacent gap ratios, accompanied by a striking banding of eigenstates by domain-wall number in correlation and entanglement measures. The Page-like entanglement dome characteristic of chaotic spectra gives way to suppressed, band-resolved entanglement consistent with emergent quasi-conservation of domain walls. To investigate further the formation mechanism of mesonic scar states, we carry out meson spectroscopy near the two-spinon threshold and compare with the analytic ladder predicted by Rutkevich [Phys. Rev. B 106, 134405 (2022)]. We test the theory through continuum-relative bindings, an offset-removed Airy scaling collapse, and explicit two-meson thresholds that determine the number of stable meson levels. The low-lying spectrum shows close quantitative agreement, while deviations at higher energies are consistent with finite-size and subleading corrections. These results establish a unified account of confinement-induced nonergodicity and provide a template for quantitative meson spectroscopy in quantum spin chains.

Figures (11)

• Figure 1: The spin-$\frac{1}{2}$ XXZ spin chain and its zero field phase diagram. (a) A one-dimensional chain of $S=\frac{1}{2}$ spins that interact via couplings $J$ and $\Delta$ in the presence of a staggered magnetic field. (b) At zero staggered magnetic field $(h=0)$ the Hamiltonian $\mathcal{H}(J,\Delta,0)$ realizes three distinct phases depending on the anisotropy parameter $\Delta$: the ferromagnetic phase at $\Delta > 1$, the critical phase at $-1 < \Delta < 1$, and the gapped antiferromagnetic (AF) phase at $\Delta < -1$. The candidate region for the occurrence of scar-like meson states is the gapped AF phase.
• Figure 2: Average level-spacing ratio $\langle r \rangle$ versus anisotropy $\Delta$ for the spin-$\frac{1}{2}$ XXZ chain $\mathcal{H}(1, \Delta, h)$ in a staggered magnetic field $h$, resolved in the symmetry sector with $(S^{z}_{\rm total}, P, \mathcal{I}, \mathcal{C}_{\rm flip}) = (0,0,+1,+1)$. (a) Average adjacent gap ratio $\langle r \rangle$ as a function of anisotropy $\Delta$ for chain lengths $N=18,20,22$ at $h=0.3$. The horizontal lines indicate the Gaussian orthogonal ensemble (GOE) value $\langle r \rangle_{\rm GOE} \approx 0.528$ and the Poisson value $\langle r \rangle_{\rm Poisson} \approx 0.386$. For $N=22$ we observe that for $\Delta \gtrsim -2.3$ (orange region), $\langle r \rangle$ approaches the GOE prediction, consistent with quantum chaotic behavior, while for $\Delta \lesssim -2.3$ (blue region), $\langle r \rangle$ decreases, signaling a crossover towards nonergodic behavior. (b) Distribution $P(r)$ of the energy level spacing ratios $r$ for $N=22$, $\Delta=-1.1$, and $h=0.3$. The histogram is compared to the Poisson (magenta) and Wigner-Dyson (orange) distributions, with $\langle r \rangle \approx 0.532$ indicating spectral statistics close to GOE. (c) Same as (b) but for $\Delta=-4.5$, deep in the antiferromagnetic phase. Here $P(r)$ deviates from Wigner-Dyson statistics and shifts towards Poisson-like behavior, with $\langle r \rangle \approx 0.47$, reflecting the onset of spectral clustering and emergent integrability associated with quasi-conserved quantities in the confined phase.
• Figure 3: Finite-size scaling of the transition anisotropy $\Delta_{\mathrm{tra}}(N)$ for the spin-$\frac{1}{2}$ XXZ chain $\mathcal{H}(1, \Delta, h)$ in a staggered magnetic field $h=0.3$, evaluated in the symmetry sector $(S^{z}_{\rm total}, P, \mathcal{I}, \mathcal{C}_{\rm flip}) = (0,0,+1,+1)$. The transition points $\Delta_{\mathrm{tra}}(N)$ are extracted from the crossover in the average level-spacing ratio $\langle r \rangle$ for system sizes $N=18,20,22$ [see Fig. \ref{['figure2']}(a)] and plotted as a function of $1/N$. The linear fit (solid line) extrapolates to $\Delta_{\rm tra}(\infty) = -4.19 \pm 0.115$ in the thermodynamic limit, indicating the onset of nonergodic behavior and confinement deep in the antiferromagnetic phase.
• Figure 4: Correlation measure, staggered magnetization, and entanglement entropy across the spectrum of the staggered-field XXZ chain. (a) Correlation measure $C^{zz}_j(\lvert\psi_j\rangle,\lvert\psi_0\rangle)$ vs. energy per site $E/N{+}E_{\rm shift}$ for $N=20$, $h=0.3$, and anisotropies $\Delta=-1.1,-2.6,-3.1,-4.5$ with respective energy shift $E_{\rm shift} = 0.0, 2.0, 4.0, 6.0$. For small $|\Delta|$, $C^{zz}_j$ forms a single broad cloud while for larger $|\Delta|$ the spectrum splits into distinct branches associated with different domain-wall numbers $W$ (banding). (b) Staggered magnetization $\langle\sigma^z_i\rangle_{\rm stag}$ for the same system, which remains small and disordered at weak anisotropy and becomes structured as $|\Delta|$ grows, consistent with strengthening antiferromagnetic correlations. (c) Von Neumann entanglement $S^{\rm vN}$ for the same parameters. At small $|\Delta|$, $S^{\rm vN}$ exhibits a volume-law "dome" consistent with quantum chaos while for larger $|\Delta|$, clear bands of reduced entropy emerge, signaling the onset confinement and nonergodicity [also see Fig. \ref{['figure5']}(c)]. A dashed horizontal line marks the Page benchmark for a half chain ($N_A=N_B=10$) entanglement entropy of $S_{\rm Page}=10\ln (2)-\frac{1}{2}\approx 6.43$.
• Figure 5: Banded structure at strong anisotropy ($\Delta=-4.5$) and its entanglement signature. (a) Correlation measure $C^{zz}_{j}(|\psi_{j}\rangle, |\psi_{0}\rangle)$ versus energy per site $E/N$ for eigenstates of the $N=20$ chain at $h=0.3$ deep in the antiferromagnetic phase at anisotropy $\Delta =−4.5$. The eigenstates segregate cleanly into flat bands labeled by the number of domain walls $W$, highlighting the quasi-conserved nature of $W$ in the deep antiferromagnetic phase. Notably, the one-meson ($W{=}2$) band sits at $C^{zz}_{j}(|\psi\rangle, |\psi_{0}\rangle) \simeq +4$, reflecting exactly two aligned bonds (two kinks). (b) $\langle\sigma^z_i\rangle_{\rm stag}$ for the same parameters follows the same band pattern, further validating the domain-wall identification. (c) The von Neumann entanglement entropy $S^{\rm vN}$ exhibits a corresponding banded structure: low-lying $W=2$ mesons have markedly reduced entanglement compared to the chaotic background.