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Degeneracy beyond the parity-symmetry protection in the Lipkin-Meshkov-Glick model

This paper demonstrates that in an anharmonic Lipkin-Meshkov-Glick model, exponential degeneracy in energy doublets can persist even when parity symmetry is broken by a first-order ground-state quantum phase transition, a phenomenon attributed to an underlying $\mathbb{Z}_2$ reflection symmetry in the classical phase space that manifests as an anti-unitary symmetry in the quantum system.

Original authors: Jamil Khalouf-Rivera, Miguel Carvajal, Francisco Pérez-Bernal

Published 2026-08-24

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CC BY 4.0

Original authors: Jamil Khalouf-Rivera, Miguel Carvajal, Francisco Pérez-Bernal

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 quantum world, the rules of symmetry often act as a silent guardian, ensuring that energy levels in a system appear in perfect pairs. Imagine a landscape where a ball can roll into one of two identical valleys; if the landscape is perfectly symmetrical, the ball has an equal chance of resting in either, creating a state of balance known as degeneracy. For decades, physicists have understood that in certain models, this pairing is protected by a specific kind of mirror symmetry called parity, which flips the system inside out. If this mirror is broken, the two valleys become different depths, the ball rolls to the deeper one, and the perfect pairing disappears. This principle has been a cornerstone of understanding how quantum systems behave, particularly in models used to describe everything from atomic nuclei to interacting spins in a chain.

However, a new study challenges the assumption that breaking this mirror symmetry must always destroy the pairing. Researchers have discovered that even when the landscape is tilted so that the two valleys are no longer identical, a different kind of hidden order can emerge to keep the energy levels paired. By tweaking a well-known theoretical model called the Lipkin-Meshkov-Glick model, the team found a way to break the traditional mirror symmetry while preserving a different, more subtle symmetry. This new symmetry acts like a reflection across a horizontal line rather than a vertical one, allowing the system to maintain its paired states in a specific high-energy region. The finding suggests that the quantum world is more flexible than previously thought, offering new ways to engineer systems where these delicate pairings survive even when the usual protections are removed.

The researchers began their investigation with a model that describes a collection of particles interacting with one another, a setup often used to simulate complex magnetic materials or nuclear behavior. In its standard form, this model exhibits a phase where the ground state, or the lowest energy configuration, splits into two identical options, a phenomenon known as a broken-symmetry phase. This splitting is usually protected by the parity symmetry mentioned earlier. The team decided to introduce a disturbance to the system, adding a term to the mathematical description that tilts the energy landscape, effectively breaking the parity symmetry. In a typical scenario, one would expect this tilt to make one energy state lower than the other, destroying the pairing.

To their surprise, the team found that while the lowest energy states did indeed separate, a different region of the system's energy spectrum behaved differently. In the high-energy excited states, the particles continued to form perfect pairs, even though the parity symmetry was gone. This occurred in a specific part of the model where the system undergoes what are called excited-state quantum phase transitions. These are moments where the nature of the system's energy levels changes abruptly as the energy increases, similar to how water changes from ice to liquid, but happening within the quantum states themselves. The researchers identified that in this high-energy region, the system possesses two symmetric peaks in its energy landscape. While the valleys at the bottom were no longer identical due to the tilt, the peaks at the top remained mirror images of each other across a horizontal axis.

This horizontal reflection is the key to the new finding. In the quantum realm, this reflection corresponds to an anti-unitary symmetry, a complex mathematical property that ensures the two states remain paired. The researchers demonstrated that this symmetry protects the degeneracy, meaning the energy difference between the paired states becomes vanishingly small as the system grows larger. They verified this by simulating the model with increasing numbers of particles, observing that the energy gap between the paired states shrank exponentially, a hallmark of true quantum degeneracy. This behavior was confirmed to be robust, persisting even when the system was pushed into a regime where the traditional parity symmetry was completely broken.

The study also explored how the system behaves as it moves between different energy regions. By tracking the average position and momentum of the particles, the team mapped out how the system transitions from a state where it sits in a single valley to a state where it explores two peaks. They found that the system's behavior is governed by two distinct critical energy points. Below the first point, the particles are confined to the valleys. Between the two points, the system behaves in a complex, non-degenerate way. But once the energy rises above the second critical point, the particles enter the region where the two peaks exist, and the pairing reappears, protected by the new horizontal symmetry. This was confirmed through various measurements, including the system's sensitivity to small changes in its parameters, which showed sharp peaks exactly where the phase transitions were predicted to occur.

The implications of this work extend beyond just this specific model. The researchers noted that similar phenomena might exist in other physical systems, such as superconducting circuits or parametric oscillators, where controlling symmetry is crucial for quantum computing. In quantum annealing, a process used to solve complex optimization problems, breaking symmetries is sometimes necessary to avoid getting stuck in unwanted states. This study suggests that one could break the usual parity symmetry to guide the system, without losing the beneficial paired states in the high-energy regions. This opens a door to designing quantum systems that are more flexible and resilient, capable of maintaining specific quantum properties even when the environment is perturbed.

The team's results were obtained through rigorous numerical simulations, calculating the energy levels of the system with extreme precision. They checked their findings by varying the number of particles and the strength of the interactions, ensuring that the observed pairing was not a fluke but a fundamental feature of the model under these conditions. The study confirms that while the traditional mirror symmetry was broken, a different, more subtle symmetry took its place to protect the quantum states. This discovery adds a new layer to our understanding of how quantum systems organize themselves, showing that degeneracy can survive in unexpected ways, guided by symmetries that are not immediately obvious.

In the end, the work provides a clear example of how quantum mechanics can surprise us. It shows that the rules governing the pairing of energy states are more nuanced than a simple yes-or-no on symmetry. By tilting the landscape just right, the researchers found a way to break one rule while keeping another, preserving the delicate balance of the quantum world. This insight could prove valuable for future technologies that rely on the precise control of quantum states, offering a new tool for engineers and physicists to manipulate the behavior of matter at its most fundamental level. The study stands as a testament to the power of exploring the edges of known models, revealing that even in a tilted landscape, the quantum world can find a way to keep its pairs together.

Problem Statement
In quantum mechanics, degeneracy patterns typically arise from system symmetries. In the well-known Lipkin-Meshkov-Glick (LMG) model, the broken-symmetry phase is characterized by doubly-degenerate states (parity doublets) protected by parity symmetry ( $\mathbb{Z}_2$ ). However, a fundamental question remains: can such exponential degeneracy (quasidegeneracy) persist if the parity symmetry is explicitly broken? While one-dimensional systems generally do not exhibit true degeneracy (except in specific frameworks), they display an exponential approach between adjacent levels known as quasidegeneracy. This paper investigates whether a parity-breaking Hamiltonian within the LMG framework can still support a degenerate excited-state phase and, if so, what symmetry protects it.

Methodology
The authors employ a combination of mean-field analysis and exact diagonalization to study a modified LMG Hamiltonian.

Model Construction: The study begins with the standard LMG Hamiltonian and introduces an anharmonic term ( $\hat{n}^2$ ) to induce a second excited-state quantum phase transition (ESQPT). Crucially, a parity-breaking interaction term, $\hat{V}_{int}$ , is added to the Hamiltonian. This results in a deformed Hamiltonian with three control parameters ( $\xi, \alpha, \delta$ ) and four interaction terms.
Mean-Field Analysis: Using the coherent-state (intrinsic-state) formalism, the authors map the quantum system to its classical limit in phase space. They analyze the classical energy surfaces to identify stationary points (minima, maxima, saddle points) and separatrices that define different dynamical phases. This allows them to visualize how the system's topology changes as control parameters vary, specifically looking for symmetries in the phase space that persist despite the breaking of parity.
Quantum Characterization: To characterize the quantum phase transitions and the nature of the degeneracy, the authors calculate several observables for systems with varying particle numbers ( $N$ $N$ ):
- Energy Gaps: They compute the energy difference between adjacent levels to verify the exponential scaling of the gap ( $e^{-aN}$ ) as $N$ increases, confirming quasidegeneracy.
- Expectation Values: They analyze the expectation values of the number operator ( $\hat{n}$ ) and angular momentum projections ( $\hat{J}_x, \hat{J}_y$ ) to map the system's behavior across different energy regimes.
- Quantum Fidelity Susceptibility (QFS): They calculate the QFS to detect the non-analyticities associated with ESQPTs. To handle the degeneracy in the perturbation theory, they diagonalize the interaction part of the Hamiltonian within the degenerate subspace.

Key Results

Existence of Degeneracy without Parity: The study demonstrates that the inclusion of the parity-breaking term $\hat{V}_{int}$ does not eliminate the exponential degeneracy in the high-energy excited states. Even though the Hamiltonian no longer conserves parity, the system retains a phase with doubly-degenerate states (quasidegenerate doublets).
Identification of the Protecting Symmetry: The mean-field analysis reveals that while parity symmetry ( $\Pi: (q, p) \to (-q, -p)$ ) is broken, the system retains a reflection symmetry $S: (q, p) \to (q, -p)$ in the classical phase space. This corresponds to an anti-unitary symmetry in the quantum system, specifically a reflection of the $y$ -component of the angular momentum ( $\hat{J}_y \to -\hat{J}_y$ ). This anti-unitary $\mathbb{Z}_2$ symmetry is identified as the mechanism protecting the degeneracy.
ESQPT Structure: The system exhibits a rich spectrum with a second-order ground-state quantum phase transition (GSQPT) and two distinct ESQPTs.
- The first ESQPT is associated with a saddle point at the origin. When $\alpha=0$ and $\delta=0$ , this saddle point separates two symmetric wells; for non-zero $\delta$ , the wells have different depths, and a first-order GSQPT occurs at $\delta=0$ where the two wells coexist.
- The second ESQPT is associated with the system boundary (separatrix) where two symmetric maxima appear.
- The degeneracy is preserved for energies above the critical energy of the second ESQPT, where the system orbits around the two symmetric maxima protected by the $S$ symmetry.
Scaling Behavior: Numerical results confirm that the energy gap between the degenerate doublets decays exponentially with the system size ( $N$ ), consistent with the behavior observed in parity-conserving models. The Quantum Fidelity Susceptibility shows distinct peaks corresponding to the two ESQPTs, with the second ESQPT being particularly sensitive to the symmetry-protected degeneracy.

Significance and Claims
The authors claim that this work extends the understanding of degeneracy in quantum phase transitions beyond the standard requirement of parity conservation.

Theoretical Insight: The paper establishes that a degenerate quantum phase can exist in a parity-breaking Hamiltonian if an alternative anti-unitary symmetry (specifically a reflection symmetry in phase space) is present. This provides a new mechanism for protecting quasidegeneracy in one-dimensional systems.
Relevance to Quantum Technologies: The authors suggest that these findings are relevant for quantum computing paradigms, such as quantum annealing. In such contexts, extra interactions are sometimes added to break parity to avoid degeneracies that complicate adiabatic approximations or perturbation theory. This work indicates that simply breaking parity may not be sufficient to remove degeneracy if other symmetries (like the anti-unitary $S$ symmetry) remain intact.
Generalizability: The authors note that similar degenerate phases protected by anti-unitary symmetries have been observed in other systems, such as superconducting circuits and the parity-asymmetric Kerr parametric oscillator, suggesting this is a broader phenomenon in quantum many-body physics.

In summary, the paper demonstrates that the Lipkin-Meshkov-Glick model, when deformed to break parity, still supports a degenerate excited-state phase. This degeneracy is not accidental but is rigorously protected by an anti-unitary reflection symmetry derived from the system's classical limit, offering a refined perspective on the relationship between symmetries and spectral degeneracy in quantum phase transitions.

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