⚛️ quantum physics

Tunable quantum Mpemba effect in long-range interacting systems

This paper demonstrates that the tunable quantum Mpemba effect in one-dimensional long-range interacting spin systems arises from the interplay between external magnetic fields, interaction range, and dynamical symmetry restoration, with the effect being observable only when interactions are sufficiently short-ranged to circumvent the Hohenberg-Mermin-Wagner theorem.

Original author: Andrew Hallam, Matthew Yusuf, Aashish A. Clerk, Ivar Martin, Zlatko Papić

Published 2026-11-11

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CC BY 4.0

Original author: Andrew Hallam, Matthew Yusuf, Aashish A. Clerk, Ivar Martin, Zlatko Papić

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 author. For technical accuracy, refer to the original paper. Read full disclaimer

In the vast landscape of physics, there is a fundamental rule that governs how matter settles down: systems left to their own devices tend to lose their special arrangements and drift toward a uniform, balanced state. Imagine a room full of people shouting in different directions; eventually, the noise smooths out into a steady hum. In the quantum world, where particles behave like waves and spins like tiny magnets, this smoothing process is called thermalization. Usually, the more chaotic or "out of balance" a system starts, the longer it takes to settle. However, nature sometimes plays tricks. There is a curious phenomenon, named after a famous paradox about water freezing, where a system that starts in a more extreme state can actually reach equilibrium faster than one that starts closer to balance. This is the quantum Mpemba effect. It suggests that in the microscopic realm, being further from the goal can sometimes be the fastest way to get there. The key to this speed often lies in symmetry, a concept describing how a system looks the same after being rotated or flipped. When a system breaks this symmetry at the start, it must eventually restore it to reach equilibrium. The speed of this restoration determines whether the Mpemba effect occurs.

A team of researchers has now mapped out exactly when this effect happens and, more importantly, when it stops. They focused on a one-dimensional chain of tiny magnetic spins, a model system that can be realized in real-world experiments with trapped ions or polar molecules. In their setup, these spins interact with each other over varying distances. Sometimes they only talk to their immediate neighbors, but in other cases, they can feel the pull of spins far down the line, with the strength of that pull fading gradually according to a specific mathematical rule. The researchers placed this chain in a strong magnetic field and watched what happened when they tilted the spins to start them off in a chaotic state. They discovered that the ability of the system to exhibit the Mpemba effect depends on a delicate tug-of-war between the strength of the magnetic field and how far the spins can reach to interact with one another.

The study reveals that for the Mpemba effect to work, the system needs a temporary, hidden order. When the magnetic field is very strong, it forces the spins to behave as if they have a specific type of rotational symmetry, even if the underlying rules of the system do not strictly possess it. This creates a "pre-thermal" state where the system is stable enough to allow the symmetry to be restored dynamically. In this regime, if the interactions between spins are short-ranged, the system behaves like a fluid that can flow back to balance, and the Mpemba effect appears: the more chaotic starting state races ahead to equilibrium. However, the researchers found that if the interactions are too long-ranged, this flow is blocked. The long-range connections lock the spins into a rigid, ordered pattern that refuses to let go of its initial broken symmetry. In this case, the system gets stuck, and the Mpemba effect vanishes. The more chaotic state no longer has the advantage; it simply takes longer to settle, just as one would expect in normal physics.

The team did not just observe this switch; they charted the precise boundaries where it happens. By simulating the system with different starting energies and interaction ranges, they created a detailed map of the conditions required for the effect to emerge. They found that the "speed limit" for the Mpemba effect is not fixed; it shifts depending on how much energy the system starts with. If the initial state is very energetic, the system can tolerate longer-range interactions before the effect breaks down. If the energy is lower, the interactions must be kept very short for the effect to survive. This means the phenomenon is tunable. By adjusting the strength of the magnetic field or the distance over which particles interact, scientists could theoretically turn the Mpemba effect on or off at will. The researchers also noted that this behavior is tied to a famous principle in physics which states that certain types of order cannot exist in low-dimensional systems unless the interactions are long-range. Their work shows that this same principle acts as a gatekeeper for the Mpemba effect, preventing it from occurring when the interactions are strong enough to create a permanent, broken symmetry.

This research is significant because it moves the Mpemba effect from a theoretical curiosity to a controllable feature of quantum systems. The findings suggest that in future experiments with quantum simulators, scientists could design systems to either exploit this effect for faster cooling and state preparation or suppress it to maintain specific quantum states for longer periods. The study confirms that the effect is not a universal rule but a fragile phenomenon that exists only within a specific window of conditions. It highlights the deep connection between how a system settles down over time and the fundamental symmetries that govern its structure. By understanding these limits, researchers can better predict how quantum materials will behave when pushed out of balance, offering new insights into the management of quantum information and the design of next-generation quantum technologies. The work serves as a clear demonstration that in the quantum world, the path to equilibrium is not just about how far you have to go, but about the rules of the road you are traveling on.

Technical Summary: Tunable Quantum Mpemba Effect in Long-Range Interacting Systems

Problem Statement
The Quantum Mpemba Effect (QME) describes a counterintuitive phenomenon where a quantum system initially farther from thermal equilibrium relaxes to equilibrium faster than one starting closer to equilibrium. While the QME has been observed in various systems, including those with continuous symmetries like U(1), its interplay with Spontaneous Symmetry Breaking (SSB) in long-range interacting systems remains unclear. Specifically, it is unknown whether sufficiently long-range interactions, which can stabilize SSB phases even in low dimensions (evading the Hohenberg-Mermin-Wagner theorem), can suppress or halt the QME. Furthermore, the role of initial state energy density and the robustness of the effect in systems with anisotropic couplings (lacking explicit U(1) symmetry) require investigation.

Methodology
The authors investigate the QME in a one-dimensional spin-1/2 XYZ model with power-law decaying interactions ( $J/|i-j|^\alpha$ ) subject to a longitudinal magnetic field ( $h_z$ ).

Model Regime: The study focuses on the prethermal regime generated by large field strengths ( $h_z \gg J$ ), where an emergent U(1) symmetry arises despite the anisotropic couplings ( $J_x \neq J_y$ ). In this regime, the system is governed by an effective prethermal Hamiltonian ( $\hat{H}_{eff}$ ) that supports a continuous time crystal (CTC) phase.
Diagnostics: The primary metric for the QME is the entanglement asymmetry ( $\Delta S_A$ ), defined as the difference between the von Neumann entropy of the reduced density matrix and its symmetry-projected counterpart. The QME is identified when the ratio of asymmetries for two initial states (one more out-of-equilibrium than the other) crosses unity, indicating the initially "hotter" state thermalizes faster.
Initial States: The study utilizes two classes of initial states:
1. Uniformly tilted product states (Eq. 9).
2. Tilted Néel states (Eq. 11).
  These states are chosen to break the U(1) symmetry initially.
Numerical Techniques: Dynamics are simulated in the thermodynamic limit using the Time-Dependent Variational Principle (TDVP) for translation-invariant infinite Matrix Product States (iMPS). Long-range interactions are approximated as a sum of exponential terms. Static properties and Symmetry Breaking (SSB) boundaries are analyzed using finite-size Density Matrix Renormalization Group (DMRG) calculations to determine the effective central charge ( $c_{eff}$ ) and entanglement spectra.

Key Contributions and Results

Phase Diagram of QME and SSB: The authors construct a dynamical phase diagram as a function of the magnetic field strength ( $h_z$ $h_{z}$ ) and the interaction range exponent ( $\alpha$ $α$ ).
- QME Existence: The QME is observed only when the field is sufficiently large to establish the prethermal regime and the interactions are sufficiently short-ranged (large $\alpha$ ). In this regime, the system effectively restores the U(1) symmetry dynamically.
- Suppression by SSB: When interactions become sufficiently long-ranged (small $\alpha$ ), the system enters a symmetry-broken phase. In this regime, dynamical symmetry restoration is halted, and consequently, the QME ceases to exist.
Dependence on Initial State Energy: The boundary for the QME ( $\alpha_M$ $α_{M}$ ) depends on the energy density of the initial state.
- High-energy density states (e.g., tilted Néel states) require shorter interaction ranges (smaller $\alpha$ ) to stabilize SSB and halt the QME compared to low-energy density states (tilted product states).
- This results in an intermediate regime ( $\alpha_M < \alpha < \alpha_c$ ) where the ground state is ordered (SSB present), but excited states accessed dynamically still permit a finite Mpemba time because the buildup of order takes longer than the relaxation time.
Role of Anisotropy and Symmetry: The study confirms that the QME can emerge in models without explicit U(1) symmetry (the generic XYZ model) provided a strong field induces an emergent symmetry.
Comparison with Existing Work: Unlike previous studies focusing on finite-size systems or very long-range interactions ( $\alpha < 1$ ) driven by quantum fluctuations, this work operates in the thermodynamic limit over a broader range of $\alpha$ , explicitly linking the halting of the QME to the thermodynamic stability of the SSB phase.

Significance and Claims
The paper claims to establish a direct link between the Hohenberg-Mermin-Wagner theorem, spontaneous symmetry breaking, and the dynamical Quantum Mpemba effect. The authors demonstrate that the QME is not a universal feature of all long-range systems but is "tunable" and can be suppressed by stabilizing symmetry-broken phases through long-range interactions.

The authors assert that their results provide a "dynamical phase diagram" for symmetry restoration, clarifying that the QME is contingent on the system being in a symmetry-restoring phase. They note that while the effect has been observed in small-scale trapped-ion experiments (Ref. [48]), those studies did not explore the suppression of the effect via SSB or generic anisotropic couplings. The work suggests that the QME can be controlled in experimental platforms (trapped ions, polar molecules, NV centers) by tuning interaction ranges and field strengths.

Finally, the authors modestly extend these findings to higher dimensions, suggesting that similar halting of the QME should occur in $d=2$ long-range systems but may be absent in $d=3$ short-range systems where SSB is possible even without long-range forces. They propose that the relatively short timescales of the Mpemba effect make it a suitable candidate for investigation on current noisy quantum hardware.

Technical Summary: Tunable Quantum Mpemba Effect in Long-Range Interacting Systems

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