Magnetic domains stabilized by symmetry-protected zero modes

1. Problem Statement

The central problem addressed is the breakdown of thermalization in closed quantum many-body systems. According to the Eigenstate Thermalization Hypothesis (ETH), isolated quantum systems should thermalize, meaning local observables eventually relax to values predicted by a thermal ensemble determined solely by the initial energy.

While several mechanisms are known to violate ETH—such as integrability, many-body localization (MBL), Hilbert-space fragmentation (HSF), and quantum many-body scars (QMBS)—these typically rely on disorder, specific constraints, or rare initial states. The authors investigate whether symmetry-protected zero modes can induce robust, thermodynamically stable non-ergodic behavior in a clean, non-integrable system (the XX model) without disorder. Specifically, they ask: Can magnetic domain walls persist indefinitely in a coupled spin ladder system?

2. Methodology

The authors employ a combination of analytical and numerical techniques:

• Model: They study the spin-1/2 XX model on a rectangular lattice ($N_x \times N_y$) with nearest-neighbor couplings $J_\parallel$ (along chains) and $J_\perp$ (between chains). The Hamiltonian is:
  $$\hat{H} = \sum_{\lambda=\parallel,\perp} \sum_{\langle i,j \rangle_\lambda} J_\lambda (\hat{S}^x_i \hat{S}^x_j + \hat{S}^y_i \hat{S}^y_j)$$
  The total magnetization $S^z$ is conserved.
• Initial States: They focus on "rung-ferromagnetic" domain wall states where all spins are up on the left half and down on the right half, with perfect ferromagnetic correlations on each rung ($|\Uparrow \dots \Uparrow \Downarrow \dots \Downarrow\rangle$).
• Lanczos Algorithm: A core analytical tool used to map the dynamics of the initial state onto a Krylov basis $\{|K_n\rangle\}$. This transforms the many-body problem into an effective 1D tight-binding model. The authors analyze the Lanczos coefficients ($\beta_j$) to determine the localization properties of the state in the Krylov space.
• Symmetry Analysis: They identify a chiral symmetry operator $\hat{C} = \hat{X}\hat{I}\hat{S}$ (involving spin flip, spatial inversion, and sublattice parity) that anticommutes with the Hamiltonian ($\{\hat{C}, \hat{H}\} = 0$). This guarantees the existence of zero-energy modes.
• Numerical Simulations: Exact diagonalization (ED) is used for systems up to $N=8 \times 2$ to compute entanglement entropy and spectral properties. Time-evolution simulations are performed for larger systems ($N=14 \times 2$) to observe long-time dynamics.

3. Key Contributions

1. Discovery of a New Non-Ergodic Mechanism: The paper identifies a mechanism for non-ergodicity distinct from MBL, HSF, or scars. It arises from an exponentially large subspace of zero-energy modes protected by chiral symmetry in the XX model on coupled chains.
2. Localization Transition: Using the Lanczos algorithm, the authors identify a critical coupling $J_c^\perp \approx 0.5 J_\parallel$. Below this threshold, the system thermalizes; above it, the system enters a localized regime where domain walls persist indefinitely.
3. Role of Chiral Symmetry: They prove that the non-ergodicity is strictly tied to chiral symmetry. Perturbations that break this symmetry (e.g., antiferromagnetic defects or specific ZZ interactions) restore thermalization, while symmetry-conserving perturbations leave the effect robust.
4. Parity Dependence: The phenomenon is highly sensitive to the lattice geometry. It persists for even $N_y$ (number of legs) but vanishes for odd $N_y$, a direct consequence of the Witten index and the chiral symmetry structure.

4. Key Results

• Stable Magnetic Domains: For coupled chains with $J_\perp \gtrsim 0.5 J_\parallel$ and even $N_x, N_y$, a domain wall initial state does not thermalize. Instead, the magnetization profile $m_z(x, t)$ settles into a non-homogeneous, time-averaged profile that retains the memory of the initial domain wall.
• Lanczos Coefficient Analysis:
  • In the localized regime ($J_\perp > J_c^\perp$), the Lanczos coefficients $\beta_j$ exhibit pronounced oscillations between even and odd indices.
  • This leads to a power-law decay of the zero-mode wavefunction coefficients $|c_j|^2 \sim j^{-\gamma}$ with $\gamma > 1$. This ensures the overlap between the initial state and the zero-energy subspace remains finite in the thermodynamic limit ($N \to \infty$).
  • In the delocalized regime ($J_\perp < J_c^\perp$), $\gamma \le 1$, causing the overlap to vanish as $N \to \infty$, leading to thermalization.
• Entanglement Entropy: Non-ergodic states exhibit low bipartite entanglement entropy ($S_{vN}$) that depends on the initial state, significantly below the Page value (thermal expectation). Thermalizing states approach the Page value.
• Spectral Properties:
  • For the pure XX model with even $N_x, N_y$, there are exponentially many zero-energy states ($d(E=0) \ge 2^{N/2}$).
  • The long-time magnetization is determined by the projection of the initial state onto this zero-mode subspace.
• Perturbation Response:
  • Symmetry-Breaking: Introducing antiferromagnetic defects in the initial state or adding ZZ interactions ($\Delta \neq 0$) that break chiral symmetry destroys the zero-mode degeneracy. This leads to thermalization (complete melting of domains) or, in the case of rung ZZ couplings, persistent oscillations associated with "cage-type" scars.
  • Symmetry-Conserving: Perturbations that respect the chiral symmetry (e.g., next-nearest neighbor XX couplings on opposite sublattices) preserve the non-ergodic behavior.

5. Significance

• Theoretical Impact: This work establishes that degenerate, symmetry-protected subspaces can act as a robust mechanism for preventing thermalization in clean, non-integrable systems. It challenges the universality of ETH in systems with specific symmetries and high degeneracy.
• Experimental Relevance: The XX model is readily implementable in current quantum simulators, including ultracold atoms in optical lattices, superconducting qubits, and Rydberg atom arrays. The predicted non-ergodic behavior is accessible with current technology.
• Diagnostic Tool: The authors propose the Lanczos-based analysis of structured mobility (oscillating coefficients) as a powerful diagnostic tool to detect localization transitions in the thermodynamic limit for a wide range of quantum systems.
• Distinction from Scars: Unlike Quantum Many-Body Scars, which typically involve a sparse set of non-thermal states embedded in a thermal spectrum, this mechanism relies on an extensive (exponentially large) subspace of zero-energy states, making the non-ergodicity thermodynamically stable rather than a rare exception.

In conclusion, the paper demonstrates that symmetry-protected zero modes can stabilize magnetic domains against thermalization, offering a new paradigm for understanding non-ergodic dynamics in quantum matter.