Magnetic domains stabilized by symmetry-protected zero modes

Pavel Kos, Dominik S. Wild, Kristian Knakkergaard Nielsen

Apr 16, 2026arXiv:2604.15510v1

quant-phcond-mat.quant-gascond-mat.stat-mechcond-mat.str-el

Scorecard· 5/16

8.0/10 impact

Abstract

Understanding mechanisms for the breakdown of thermalization in closed quantum systems is a central problem in quantum many-body physics. We demonstrate strong non-ergodic behavior in the XX model on coupled chains, where domain-wall initial states retain an inhomogeneous magnetization profile for arbitrarily long times. We find that this effect arises due to exponentially many zero modes protected by chiral symmetry. Using an analysis based on the Lanczos algorithm, we identify a localization transition in the thermodynamic limit at a critical coupling between the chains. We further show that antiferromagnetic defects in the initial state and symmetry-breaking perturbations restore slow thermalization, whereas it remains robust for symmetry-conserving perturbations. These results establish that degenerate, symmetry-protected subspaces can give rise to thermodynamically stable non-ergodic dynamics in experimentally accessible quantum systems.

AI Impact Assessments

(3 models)

Scientific Impact Assessment

Core Contribution

This paper identifies a new mechanism for the breakdown of thermalization in closed quantum systems: symmetry-protected zero modes arising from chiral symmetry in the XX model on rectangular lattices. The central finding is that domain-wall initial states on coupled chains (e.g., spin ladders) retain inhomogeneous magnetization profiles indefinitely, despite the system being non-integrable and otherwise exhibiting diffusive, ergodic behavior at high temperature. The mechanism is distinct from the four established paradigms of non-ergodicity (integrability, MBL, Hilbert space fragmentation, quantum many-body scars), as it relies on an exponentially large degenerate zero-energy subspace protected by a chiral symmetry $\hat{C} = \hat{X}\hat{I}\hat{S}$ , with a rigorous lower bound of $2^{N/2}$ zero modes for even $N_{x}, N_{y}$ via the Witten index.

The key physical insight is elegant: domain-wall states with rung-ferromagnetic correlations have substantial (~54% for $J_\perp = J_\parallel$ ) overlap with this degenerate zero-mode subspace, and it is precisely this overlap that dictates the long-time magnetization profile. Antiferromagnetic defects or odd lattice dimensions destroy this effect, providing sharp predictions.

Methodological Rigor

The paper employs a multi-pronged approach:

1. Exact diagonalization (up to $N = 8 \times 2$ ) provides full spectral decomposition, entanglement entropy analysis, and direct comparison with ETH predictions.

2. Lanczos/Krylov analysis (up to $N = 14 \times 2$ ) constitutes perhaps the most innovative methodological element. By mapping the dynamics to an effective 1D tight-binding model in Krylov space, the authors identify a localization transition at $J_\perp^c \approx 0.5 J_\parallel$ . The "structured mobility" picture—alternating even/odd Lanczos coefficients producing power-law decay $|c_j|^2 \sim j^{-\gamma}$ with a critical exponent $\gamma_c = 1$ —is both intuitive and analytically tractable through the double-linear approximation [Eq. (4)].

3. Symmetry analysis rigorously establishes the lower bound on zero modes using the Witten index. The derivation in the supplemental material is thorough, including the even/odd parity effect and separate bounds for different magnetization sectors.

4. Perturbation analysis systematically tests robustness by adding various $n$ th-nearest-neighbor couplings and ZZ interactions, demonstrating that chiral-symmetry-preserving perturbations maintain the effect while symmetry-breaking ones restore thermalization.

One limitation is the accessible system sizes. While $14 \times 2$ is reasonable for exact methods, extrapolation to the thermodynamic limit relies on the Lanczos coefficient scaling analysis, which, though convincing, could benefit from further validation. The finite-size analysis in the supplemental material shows minimal corrections for the three sizes tested, but the range ( $N_{x} = 10, 12, 14$ ) is narrow.

Potential Impact

Experimental relevance: The XX model is directly realizable in optical lattice experiments (Bose-Hubbard at strong interactions), superconducting qubit arrays, and Rydberg/dipolar platforms. The predicted signatures—persistent magnetization domains from simple product state initializations—are measurable with current technology. The sharp dependence on even/odd $N_{y}$ provides a distinctive experimental fingerprint.

Theoretical implications: The Lanczos-based localization diagnostic represents a potentially powerful new tool. The authors explicitly suggest it could assess stability of localization phenomena across a broader class of systems. The connection between chiral symmetry and non-ergodic dynamics could inspire searches for similar phenomena in other models with sublattice or chiral symmetries (e.g., bipartite lattice models, SSH-type models in many-body settings).

Relation to existing work: The paper carefully positions itself relative to MBL, HSF, scars, and integrability. The mechanism is genuinely distinct—it doesn't require disorder, doesn't fragment the full Hilbert space, and affects a macroscopic fraction of states (the zero-mode subspace grows exponentially). The observation that the free-fermion version ( $J_\perp = 0$ ) shows no such effect, while the interacting (hard-core boson) version does, highlights that this is an intrinsically interacting phenomenon.

Timeliness & Relevance

The paper addresses a highly active area—mechanisms beyond ETH—at a time when experiments on quantum simulators are probing these questions directly. Recent works on Hilbert space fragmentation and quantum scars have generated enormous interest, and this paper adds a qualitatively new entry to the catalog. The connection to the Witten index and supersymmetric quantum mechanics adds interdisciplinary depth.

Strengths

Clean, sharp physical mechanism with rigorous mathematical underpinning (Witten index bound)

Novel Krylov-space diagnostic for thermodynamic stability, with clear analytical understanding

Strong experimental relevance across multiple platforms

Systematic perturbation analysis distinguishing symmetry-preserving vs. breaking effects

Interesting even/odd parity effect in

N_{y}

providing non-trivial predictions

Supplemental material is exceptionally thorough, including analytical proofs of Lanczos coefficient scaling

Limitations

System sizes are modest; the thermodynamic limit argument, while compelling, relies on extrapolation of Lanczos coefficient scaling

The 2D thermodynamic limit (

N_x, N_y \to \infty

) appears to weaken the effect (requiring stronger

J_\perp

for

N_{y} = 4

), suggesting the phenomenon may be primarily quasi-1D

The connection between the double-linear model for

\beta_j

and the actual Lanczos coefficients is approximate; deviations at larger

J_\perp/J_\parallel

are not systematically characterized

The paper does not discuss potential connections to lattice gauge theories or constrained models where similar chiral symmetries arise, which could broaden impact

The interplay with finite temperature or mixed-state dynamics is not explored

Overall Assessment

This is a high-quality paper presenting a genuinely novel mechanism for non-ergodic behavior, supported by rigorous analysis and with clear experimental implications. The combination of symmetry-based analytical arguments with the Krylov-space localization diagnostic is methodologically innovative. The main question regarding ultimate impact concerns whether the effect survives in truly two-dimensional systems or remains a quasi-1D phenomenon.

Rating:7.8/ 10

Significance 8Rigor 7.5Novelty 8.5Clarity 8.5

Generated Apr 20, 2026

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