⚛️ quantum physics

Restoring thermalization in long-range quantum magnets with staggered magnetic fields

This paper demonstrates that applying a staggered magnetic field to long-range Heisenberg antiferromagnets restores thermalization by breaking permutational symmetry to create a dense energy spectrum, although quantum scar-like eigenstates can still prevent thermalization for specific initial states in the middle of the spectrum.

Original authors: Lucas Winter, Pietro Brighi, Andreas Nunnenkamp

Published 2026-09-28

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CC BY 4.0

Original authors: Lucas Winter, Pietro Brighi, Andreas Nunnenkamp

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the quiet world of isolated quantum systems, where particles interact but cannot exchange energy with the outside world, a fundamental question has long puzzled physicists: how do these systems ever settle down? In our daily lives, a hot cup of coffee cools to room temperature, and a spinning top eventually stops, because they interact with their surroundings. But in the isolated realm of quantum mechanics, where a system is perfectly sealed off, the rules are different. Scientists have found that many of these isolated systems do not cool down or settle into a calm, predictable state as expected. Instead, they can get stuck in a kind of permanent, chaotic loop, refusing to reach thermal equilibrium. This resistance to settling is often caused by the system's internal structure, specifically when particles interact over long distances or when the system possesses a high degree of symmetry, creating a rigid energy landscape that traps the system in place. Understanding why some systems thermalize while others do not is crucial for building future quantum technologies, as it determines whether we can control these systems or if they will remain stubbornly unpredictable.

A team of researchers at the University of Vienna has now demonstrated a way to break this stubbornness in a specific type of quantum magnet. They focused on a system where spins, the tiny magnetic moments of atoms, interact strongly with one another across long distances. In such systems, the energy levels are often so discrete and spaced out that the system cannot easily find a path to equilibrium. The researchers showed that by applying a specific kind of magnetic field—one that points in opposite directions for alternating rows of atoms—they could force the system to thermalize. This "staggered" magnetic field acts like a key that unlocks the rigid structure of the system. By breaking the perfect symmetry that usually keeps the system trapped, the field causes the energy levels to crowd together so densely that the system can finally explore all its possible states and settle into a stable, thermal condition.

The team used powerful computer simulations and mathematical models to watch how this system behaves over time. They started with a state where the spins were arranged in a specific, ordered pattern, similar to a checkerboard of north and south poles. Without the special magnetic field, this system would oscillate forever, never truly settling. However, with the staggered field applied, the researchers observed a clear three-stage journey toward equilibrium. First, the system oscillated in a rhythmic, pendulum-like motion. Then, these oscillations began to fade and reappear in a pattern of periodic revivals. Finally, the system settled down, with its average behavior matching what is predicted for a system in thermal equilibrium. Remarkably, the time it took for this system to settle did not depend on how many atoms were in the chain; it depended only on how much the initial state fluctuated. If the starting state had even a tiny amount of uncertainty or variation, the system would thermalize in a finite time, regardless of its size.

However, the story is not entirely uniform across all energy levels. The researchers found that while the system thermalizes for most starting conditions, there is a specific, narrow region in the middle of the energy spectrum where this rule breaks down. In this middle zone, the system fails to thermalize because of the presence of special, rare quantum states that act like islands in a sea of chaos. These states are localized around unstable points in the system's classical behavior, preventing the system from exploring the full range of possibilities. This phenomenon, known as quantum scarring, means that for these specific initial states, the system remembers its starting point forever and never reaches a true thermal state. Yet, for the vast majority of initial states, particularly those at lower or intermediate energies, the thermalization holds firm.

The implications of this work extend beyond theoretical curiosity. The researchers noted that their findings can be tested on several existing experimental platforms, including arrays of atoms trapped by lasers or atoms interacting with light inside optical cavities. These setups can already create the long-range interactions and magnetic fields required to observe this effect. By showing that a simple change in the magnetic environment can restore thermalization, the study offers a new way to control quantum systems. It suggests that even in systems where long-range interactions usually prevent equilibrium, a carefully chosen symmetry-breaking field can guide the system back to a predictable, thermal state, opening new doors for understanding and manipulating the quantum world.

Problem Statement
Quantum systems with strong long-range (LR) interactions, where interaction strength decays as $1/r^\alpha$ with $\alpha < d$ , are theoretically expected to resist thermalization. This resistance arises from two primary mechanisms: the emergence of a discrete energy spectrum in the thermodynamic limit, which protects metastable states with lifetimes scaling with system size, and the persistence of permutational symmetry, which induces structure in eigenstates and violates the Eigenstate Thermalization Hypothesis (ETH). While previous research has focused on how the interaction range $\alpha$ affects thermalization, it remains an open question whether thermalization can be restored by reducing the symmetry of the Hamiltonian itself, independent of the interaction range.

Methodology
The authors investigate a one-dimensional long-range Heisenberg antiferromagnet subjected to a staggered magnetic field. The system consists of two sublattices, $A$ and $B$ , coupled via power-law interactions. The Hamiltonian includes an antiferromagnetic exchange term and a staggered field term that breaks the inherent symmetry between sublattices and the rotation symmetry of the Heisenberg interactions.

The study employs a dual approach:

Self-consistent Mean-Field Theory: The authors utilize the Schwinger boson mapping and Jordan-Wigner transformation to map the spin system to fermions. By applying mean-field decoupling to string operators (justified by the suppression of local fluctuations in LR systems), they derive an effective quadratic Hamiltonian. This allows for the analytical diagonalization of the spectrum and the calculation of collective observables.
Exact Diagonalization (ED): Numerical simulations are performed on finite systems to validate the analytical predictions. The Hamiltonian (3) is diagonalized for systems up to hundreds of spins (e.g., $N=500$ for $\alpha=0$ in Fig. 1), while finite-size scaling for finite $\alpha$ dynamics utilizes smaller system sizes (e.g., $N=18$ in Supplemental Material Fig. 2). The ED is conducted within specific magnetization sectors ( $S_z$ ) and utilizes symmetries such as translational invariance to reduce the Hilbert space dimension.

The dynamics are probed using mixed Néel states, defined as statistical ensembles of pure Néel states with Gaussian-distributed orientations. This allows for the independent tuning of fluctuations ( $\sigma$ ) in the initial state, distinct from system size.

Key Contributions and Results

Restoration of Thermalization via Symmetry Breaking: The central finding is that applying a staggered magnetic field breaks the full permutational symmetry of the system, reducing it to bipartite symmetry. This symmetry breaking lifts the degeneracy of the energy spectrum. While the spectrum remains composed of discrete subspaces, the number of these subspaces scales as $O(N^2)$ (compared to $O(N)$ in permutation-invariant LR systems). Consequently, the collective spectrum becomes dense in the thermodynamic limit ( $N \to \infty$ ), with average energy spacing scaling as $O(1/N)$ . This dense spectrum enables thermalization for initial states with finite fluctuations.
Three-Stage Equilibration Dynamics: The time evolution of the Néel parameter $\langle n_z \rangle$ exhibits three distinct dynamical stages:
1. Short-time: Oscillatory mean-field dynamics resembling a classical pendulum with frequency $\omega_p = \sqrt{Jh}$ .
2. Intermediate-time: Periodic revivals and decay of oscillations. The decay time $T_{dec}$ is inversely proportional to the initial state fluctuations of the Néel parameter ( $T_{dec} \propto 1/\sigma_{n_z}$ ) but is independent of system size $N$ .
3. Long-time: Equilibration on average to the microcanonical ensemble. The equilibration time $T_{eq}$ scales as $1/\sigma_{sz}^2$ , depending on the fluctuations of the uniform magnetization. Crucially, for states with finite fluctuations, equilibration occurs in finite time regardless of system size, contrasting with the metastability observed in permutation-invariant LR systems where equilibration times diverge with $N$ .
Ergodicity Breaking at the Separatrix: While low-to-intermediate energy states thermalize (long-time averages match the microcanonical ensemble), states near the separatrix energy (the boundary between bound and free classical trajectories) exhibit ergodicity breaking. In this regime, the long-time average depends on the initial state. This behavior is attributed to "quantum scar-like" eigenstates that are localized at the unstable fixed point of the classical phase space (the upright pendulum). These states exhibit low entanglement entropy and anomalously large overlaps with specific initial states, preventing thermalization.
Finite $\alpha$ Effects: For finite interaction ranges ( $0 < \alpha < 1$ ), the collective spin lengths on sublattices $A$ and $B$ ( $S_A^2, S_B^2$ ) are no longer conserved and decay over time. However, the lifetime of these collective spins scales as $N^{3/2}$ , diverging in the thermodynamic limit. Thus, in the thermodynamic limit, the dynamics for $0 < \alpha < 1$ effectively match those of the fully connected limit ( $\alpha = 0$ ), where the Néel parameter equilibrates while collective spin lengths remain frozen.

Significance
The paper demonstrates that thermalization in strong long-range quantum magnets can be restored for a broad class of initial states by introducing a staggered magnetic field. This mechanism works by breaking permutational symmetry to generate a dense energy spectrum, thereby circumventing the discrete spectrum and metastability typical of LR systems. The results highlight that equilibration times depend on the fluctuations of the initial state rather than the system size, provided the spectrum is dense. Furthermore, the work identifies a specific mechanism for ergodicity breaking in the middle of the spectrum via scar-like states localized at unstable classical points. The authors note that their proposal requires only local staggered fields and long-range interactions, making it experimentally feasible on platforms such as trapped ions, Rydberg atom arrays, and cavity-coupled atoms.

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