Magnetic domains stabilized by symmetry-protected zero modes

arXiv:2604.15510 · quant-ph, cond-mat.quant-gas, cond-mat.stat-mech, cond-mat.str-el · Submitted 2026-04-16 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: Today's paper: "Magnetic domains stabilized by symmetry-protected zero modes".

Mira: The gist: Symmetry-protected zero modes in the XX model give rise to an infinite lifetime of ferromagnetic domains and this effect is sustained by chiral symmetry.

Kai: First, who's behind it and why it matters.

Title and authors: Mira: Before we go into the details, let's look at who wrote this paper and what that title actually says about what they found >

Kai: The paper is titled "Magnetic domains stabilized by symmetry-protected zero modes" and it was written by Pavel Kos, Dominik Wild, Kristian Knakkergaard Nielsen >

Lev: So we have these authors from different groups working on quantum systems, which usually means they're looking at different sides of the same coin >

Mira: That title really emphasizes that the stability isn't just some random thing; it’s tied to a specific symmetry feature in the system's structure >

Kai: It suggests that this non-ergodicity is fundamentally linked to how those zero modes are protected by chiral symmetry in the XX model on coupled chains >

Mira: That sounds like they are using topological features, not just simple energy minimization, to stabilize these domains >

Lev: And that's a key point for me because if it’s topological protection, it might be more robust against noise than just being a local energy minimum >

The paper's summary: Kai: So what does the paper actually show about the physics? It demonstrates 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 >

Mira: And they attribute this persistence to those exponentially many zero modes that are protected by chiral symmetry >

Lev: Okay, so the numbers here are important. They mention that when both the number of x and y chains, Nx and Ny, are even, the trace of that chiral symmetry operator Cˆ is equal to negative one over forty-two times N over two > <ref:2604.15510#pg2>

Kai: That’s a concrete calculation they did to show how many zero modes there are in that sector >

Mira: And they also found something about the entanglement entropy dynamics; for non-ergodic initial states, the entanglement entropy settles at values significantly below the Page value >

Lev: That means instead of scrambling toward a universal value, it gets stuck at a lower plateau that depends on the initial state itself >

Kai: So what does this mean for someone who just listens to the show? It means you can have magnetic domains stay separated for a very long time, even when you expect them to mix up quickly >

Mira: Right. It suggests these systems are much more stable than we might think in terms of their long-term dynamics >

The paper's improvements: Kai: Now, the paper discusses how they can improve this understanding or what the next steps could be. They identify a localization transition at a critical coupling between the chains >

Mira: They use the Lanczos algorithm to study this, and they find that for J⊥ less than or equal to Jc⊥ which is about zero point five times J∥, the sum of overlap with the initial state doesn't converge as N goes to infinity >

Lev: So if we want to keep this non-ergodic behavior, we need the coupling between chains, that's J⊥, to be stronger than that critical value >

Kai: That’s right. And when J⊥ is greater than Jc⊥, the sum converges and the zero mode keeps a nonzero overlap with the initial state as Nx goes to infinity >

Mira: This implies you have some control over these properties by tuning that coupling parameter between your chains >

Conclusion: Kai: So to wrap up this paper on "Magnetic domains stabilized by symmetry-protected zero modes," the main implication is that we can engineer systems where magnetic domains stay separated for a long time due to those zero modes >

Mira: It also shows that these specific scars, like the rung-ferromagnetic states, can have subvolume law entanglement entropy scaling as ln Nx >

Lev: I think for us in terms of building things, it means we need to be careful about the coupling J⊥ because if it's too weak, the dynamics quickly relax into thermal equilibrium >

Kai: So the paper gives us a way to diagnose whether our experimental setup is actually exhibiting this symmetry protection or if it's just standard thermalization >

Mira: And that methodology might help us design new quantum simulators where we can reliably find these non-thermalizing states because of what they show about the even and odd effects on the zero mode degeneracy >

Lev: Yeah, so to recap, this paper shows how symmetry protection stabilizes domains by controlling the coupling J⊥ and identifying specific scar states like those rung-ferromagnetic ones >

Kai: We're going to keep an eye on this methodology for assessing localization in the thermodynamic limit because it seems like a new diagnostic tool for complex quantum many-body systems >

Mira: That’s what we'll be focusing on as we look at how these principles apply beyond just the spin ladder geometry >

Pavel Kos, Dominik S. Wild, Kristian Knakkergaard Nielsen

Max Planck Institute of Quantum Optics · Department of Physics, Faculty of Mathematics and Physics, University of Ljubljana · Google Quantum AI · Niels Bohr Institute, University of Copenhagen

quant-ph, cond-mat.quant-gas, cond-mat.stat-mech, cond-mat.str-el

Submitted: 2026-04-16

Updated: 2026-10-14

Comments: 8 pages, 5 figures + Supplemental Material

Journal ref: Phys. Rev. Lett. 137, 160402 (2026)

DOI: 10.1103/29qr-n6mw

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 75/100

The gist: The gist: Symmetry-protected zero modes in the XX model give rise to an infinite lifetime of ferromagnetic domains and this effect is sustained by chiral symmetry.

Key concepts

Non-Ergodicity: This describes a state where the system's evolution never explores all possible states, meaning it gets stuck in specific patterns indefinitely. In this paper, it means magnetic domains persist for arbitrarily long times instead of quickly relaxing to a uniform state.
Chiral Symmetry: This is a specific type of symmetry in the model that protects certain zero modes. These protected modes are like special, stable directions in the system's energy landscape that prevent the system from easily changing its configuration.
Quantum Many-Body Scars: These are specific, localized states within a complex quantum system that exhibit unusual behavior, such as perfect correlations. The paper shows these scars are linked to the persistence of zero modes and can cause slow thermalization when symmetry is broken.
Localization Transition: This is a critical point in the coupling between chains where the system's behavior changes drastically. At this point, the system transitions from being able to move freely (delocalized) to becoming trapped in specific patterns (localized), which is studied using the Lanczos algorithm.

Summary

The gist: Symmetry-protected zero modes in the XX model give rise to an infinite lifetime of ferromagnetic domains and this effect is sustained by chiral symmetry.

Understanding Non-Ergodicity

The paper demonstrates 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. This effect arises due to exponentially many zero modes protected by chiral symmetry. The authors establish that degenerate, symmetry-protected subspaces can give rise to thermodynamically stable non-ergodic dynamics in experimentally accessible quantum systems.

Model and Initial States

The study focuses on spin-1/2 particles in a square lattice described by the XX model with Hamiltonian Hˆ = Xλ=∥,⊥⟨i,j⟩λJλhSˆxi Sˆxⱼ + Sˆyi Sˆyⱼ i. The system is analyzed on coupled chains where coupling between the chains leads to persistent domain walls for arbitrarily long times. For a single chain with J⊥ = 0, the dynamics map to free fermions, showing undamped ballistic spin motion. When nonzero J⊥ is introduced, the magnetization settles into a non-homogeneous profile, which contradicts expectations from ETH.

Localization Transition Analysis

The localization transition in the thermodynamic limit is identified at a critical coupling between the chains. This transition is analyzed using the Lanczos algorithm to compute the effective one-dimensional tight-binding model derived from an initial domain wall state K1⟩. The analysis shows that for J⊥ ≤ Jc⊥ ≃ 0.5J∥, the sum n1 = sum n1=1 cj/c12 ∼ n1−γ does not converge for n → ∞. Consequently, the overlap with the initial state K1⟩ vanishes as Nx → ∞. Conversely, when J⊥ > Jc⊥, the sum converges and the zero mode retains a nonzero overlap with the initial state as Nx → ∞.

Symmetry Protection and Zero Modes

The non-ergodicity is sustained by a chiral symmetry defined by Cˆ = XˆIˆSˆ, where Sˆ = Qi∈E (2Ŝzi). This symmetry implies a lower bound on the number of zero modes in the zero magnetization sector d(E = 0) ≥ tr(Cˆ) = 2N/2 for even Nx, Ny. The calculation shows that when both Nx and Ny are even, tr(Cˆ) = (−1)N/42 N/2. When one of the dimensions is odd, tr(Cˆ) = 0.

Stability and Scars

The authors identify two sets of exact quantum many-body scars at E = 0 with perfect rung-ferromagnetic and rung-antiferromagnetic correlations. These states exhibit subvolume law entanglement entropy, scaling as ∼ ln Nx for the rung-ferromagnetic states. The presence of these scars is linked to the persistence of zero modes, which are related to Fock space cages and interference-caged quantum many-body scars. Explicit symmetry breaking restores relaxation in two ways: either complete melting of the domains as for the full XXZ model or persistent oscillations at arbitrarily long times associated with the continued presence of cage-type quantum many-body scars.

Generalizability

Numerical evidence on lattices like N = 6 × 3 and 4 × 4 geometries shows that scar states remain in these cases, again with a striking even/odd effect. This demonstrates that the findings are not specific to the spin ladder. The paper concludes by suggesting that this methodology may provide a novel diagnostic tool to assess the stability of localization phenomena in the thermodynamic limit and that similar behavior may be realized for many other quantum many-body systems.

Free Fermions Comparison

For free fermions with nearest neighbor hoppings J∥/2, J⊥/2, the rung-ferromagnetic initial states behave exactly like a collection of Ny uncoupled one-dimensional chains. This contrasts sharply with the strong dependency observed in the XX model. For free fermions initialized in one of the column-filled states, dynamics are completely independent of the number of rows, acting as a one-dimensional chain of length Nx.

Finite Size Corrections

The finite size corrections to the implemented Lanczos algorithm are shown to be small. The zero mode coefficients show minimal dependency on system size. The asymptotic scaling of the coefficients is given by (2βj/J∥)2 = Ny(j − 1) for J⊥ = 0.

Exact Quantum Many-Body Scars

The rung-ferromagnetic states RF(n)⟩ are eigenstates of the pure XX model with energy E = 0. These states feature perfect ferromagnetic correlations along all rungs of the ladder, which in turn inactivates the rung XX coupling. The presence of ZZ rung couplings leads to an equal energy shift of all these Nx + 1 states to E = Nx∆⊥J⊥/4 due to the perfect ferromagnetic correlations on each rung.

Rung-Antiferromagnetic States

The paper also finds scars featuring perfect rung-antiferromagnetic states RAe⟩ and RAo⟩. These states have exactly opposite energy E = −Nx∆⊥J⊥/4 to the ferromagnetic states RF(n)⟩ in the S z = 0 magnetization sector when ∆∥ = 0. These constructions show that at the same energy, E = 0, there are local correlations that strongly deviate from ETH.

Towards Two Dimensions

Numerical evidence on N = 6 × 3 and 4 × 4 geometries shows a striking even/odd effect in entanglement entropies. The zero mode degeneracy vanishes completely for Ny = 3 and N↑ = N/2, whereas it jumps back up to d = 306 states for the 4 × 4 square. This also manifests in magnetization dynamics, where the initial domain wall state retains a stable magnetization profile only for even cases Ny = 2, 4. The authors suggest that magnetic domains will slowly fade as the two-dimensional thermodynamic limit is approached.

Free Fermions Dynamics

For free fermions, the time-evolved state remains a product state and the only dependence of the coupling along the columns of the lattice, J⊥, is in front of a phase factor. This means that for free fermion models initialized in one of the column-filled states, dynamics are completely independent of the number of rows.

Hamiltonian Decomposition

The Hamiltonian can be written as two commuting terms Hˆ = Hˆ∥ + Hˆ⊥ with Hˆ⊥ = Pky ε⊥(ky)Nˆky. The time-evolved state is then Ψ(t)⟩ = e −iPky ε⊥(ky)t e −iHˆ∥tΨ(0)>. This structure ensures the dynamics are governed by the 1D chain length Nx.

Symmetry Protected Zero Modes

The non-ergodic magnetic domains are sustained by a chiral symmetry Cˆ = XˆIˆSˆ. The analysis shows that this chiral symmetry implies a lower bound on the number of zero modes in the zero magnetization sector d(E = 0) ≥ tr(Cˆ) = 2N/2 for even Nx, Ny. This suggests that similar behavior may be realized for many other quantum many-body systems.

Explicit Symmetry Breaking

Explicit symmetry breaking restores the relaxation of the domains in two ways: (1) complete melting of the domains as for the full XXZ model, or (2) persistent oscillations at arbitrarily long times associated with cage-type quantum many-body scars. The analysis based on Lanczos algorithm provides a powerful and intuitive framework revealing an intriguing localization effect in terms of structured mobility.

Outlook

Looking ahead, the methodology may provide a novel diagnostic tool to assess the stability of localization phenomena in the thermodynamic limit and similar behavior may be realized for many other quantum many-body systems. The authors thank Nicolas Loizeau, Fan Yang, Klaus Mølmer, Immanuel Bloch, Simon Karch, Marko Ljubotina and Cheryne Jonay for stimulating input.

Data Availability

The data files to produce all figures is openly available. Source code will be shared upon reasonable request to K.K.N.

References Cited

[1] J. M. Deutsch, Phys. Rev.

Improvements for AI systems

textbfIncreased Robustness Against Thermalization in Closed Quantum Systems: The AI can now model non-ergodic magnetic domains in XX models on coupled chains for arbitrarily long times by exploiting symmetry-protected zero modes, as demonstrated by the finding that domain-wall initial states retain an inhomogeneous magnetization profile for arbitrarily long times. This allows the system to predict stable, non-thermalizing dynamics in experimentally accessible quantum simulators where ETH is expected to hold.]
textbfEnhanced Characterization of Localization Transitions: The AI can identify and predict a localization transition at a critical coupling between chains by analyzing the Lanczos algorithm results, specifically identifying when the sum converges for the zero mode overlap, which occurs when "J⊥ > Jc⊥." This enables precise control over system parameters to maintain or break non-ergodic behavior.]
textbfRefined Initial State Selection and Perturbation Analysis: The AI can distinguish between perturbations that restore thermalization and those that do not, by recognizing the distinction between antiferromagnetic defects in the initial state (which restore slow thermalization) and symmetry-conserving perturbations (which remain robust for non-ergodicity). This capability allows for targeted experimental design to probe the underlying symmetry structure of quantum many-body systems.]
textbfDiscovery and Exploitation of Symmetry Protected Subspaces: The AI can identify the chiral symmetry operator, Cˆ = XˆIˆSˆ, which is shown to stabilize the non-ergodic magnetic domains by implying that there are at least 2N/2 zero modes for even Nx, Ny. This allows the system to leverage topological features for long-lived metastable states in quantum simulations.]
textbfModeling of Quantum Many-Body Scars: The AI can identify and predict specific exact quantum many-body scars, such as rung-ferromagnetic states that exhibit subvolume law entanglement entropy, scaling as ∼ ln Nx, allowing for the simulation of these rare, non-thermalizing states in larger geometries.]
textbfAdaptability Across Lattice Geometries: The AI can generalize its findings beyond the spin ladder to two dimensions by showing that the non-ergodicity is not specific to the N = Nx × 2 spin ladder, noting the striking even/odd effect observed in entanglement entropies and magnetization dynamics for geometries like N = 6 × 3 and N = 4 × 4.]
textbfPredictive Power for Free Fermion Models: For free fermion systems, the AI can predict that there is no dependency on J⊥ for these initial states, contrasting this with the strong dependency seen in the XX model, providing a simplified predictive framework for systems described by Eq. (S24). This allows faster simulation of simpler quantum models.]

Abstract

Understanding mechanisms for the breakdown of thermalization in closed quantum systems is a central problem in quantum many-body physics. We demonstrate strong nonergodic 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 zero modes protected by chiral symmetry. Using an analysis based on the Lanczos algorithm, we identify evidence for 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 nonergodic dynamics in experimentally accessible quantum systems.