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Locality of KMS states for infinite fermionic lattice systems

This paper establishes the stability of thermal equilibrium states in infinite fermionic lattice systems under extensive perturbations by demonstrating that exponential decay of correlations and local indistinguishability in the unperturbed state lead to stretched-exponential decay in the perturbed state, utilizing Quantum Belief Propagation and Lieb-Robinson bounds to close the cycle of implications between correlation decay, local perturbation principles, and local indistinguishability.

Original authors: Lennart Becker

Published 2026-09-29

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CC BY 4.0

Original authors: Lennart Becker

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

Imagine a vast, infinite grid of tiny magnets or particles, each interacting with its neighbors. In the real world, we often study such systems in a box, but nature does not come in boxes; it extends forever. Physicists describe the state of such an infinite system at a specific temperature using a concept called a thermal equilibrium state. In this state, the system has settled down, and its properties are stable. A key feature of these stable states is how information travels. If you measure one particle, how much does that tell you about a particle far away? In a healthy, stable system, this connection should fade rapidly as the distance grows. This fading, known as the decay of correlations, is the hallmark of a system that is not stuck in a chaotic or frozen state where everything is entangled with everything else.

However, real systems are rarely perfect. They are subject to changes, or perturbations, in their underlying energy rules. A scientist might ask: if we tweak the energy of the system, perhaps by adding a new interaction that stretches across the entire grid, does the system remain stable? Does the connection between distant particles still fade away, or does the change ripple through the whole grid, destroying the local order? This question is particularly tricky when the system is large enough to have multiple possible stable arrangements, a situation known as phase coexistence, where different types of order can exist side by side. Understanding how these systems respond to change is fundamental to predicting the behavior of materials, from superconductors to magnetic storage devices.

In a new study, Lennart Becker addresses this question for a specific class of quantum systems made of fermions, the type of particles that make up electrons and protons. The research focuses on what happens when an infinite lattice of these particles, already in a stable thermal state, is subjected to a massive change in its energy landscape. Unlike previous studies that often assumed the system was unique or simple, this work allows for the possibility that the system could be in one of several different stable phases. The central achievement is a proof that if the original system has a specific type of stability—where connections between distant parts fade away quickly—then a new, slightly different system created by a large-scale energy change will also retain a similar, though slightly modified, form of stability.

The researchers did not just assume this stability would hold; they demonstrated it mathematically. They showed that even when the change to the system's energy is extensive, meaning it affects the entire infinite grid rather than just a small patch, the new state of the system still exhibits a rapid fading of connections between distant points. The rate at which these connections fade is not as fast as in the original system, but it is still strong enough to be considered a stretched-exponential decay. This means that while the influence of a local change might linger a bit longer than before, it does not spread indefinitely to disrupt the entire system. The system remains locally distinct; a change in one region does not instantly scramble the properties of a region far away.

To reach this conclusion, the team developed and applied a sophisticated mathematical tool called Quantum Belief Propagation. In simpler terms, this is a method for tracking how a change in the energy of a system propagates through its quantum state. The researchers used this tool to construct a precise map of how the system transforms from its old state to its new one. They combined this with another powerful concept known as Lieb-Robinson bounds, which act like a speed limit for how fast information can travel through a quantum system. By weaving these two ideas together, they were able to prove that the "locality" of the system—the idea that things mostly affect their immediate neighbors—is preserved even under significant stress.

The study also clarifies the relationship between different ways of measuring stability. The researchers proved that if a system is stable in one sense, meaning that adding a small piece of a perturbation has a negligible effect on distant parts, it implies that the system will also maintain its fading connections. Conversely, if the system maintains its fading connections, it implies that local changes stay local. This creates a cycle of stability: the system's ability to ignore distant disturbances is mathematically linked to its ability to keep its internal correlations short-ranged. This cycle holds true even when the system is not unique, meaning it can exist in multiple different stable forms simultaneously. This is a crucial distinction, as many real-world materials can exist in different phases, and knowing that stability principles hold across these phases is vital for a complete understanding of their behavior.

Furthermore, the work provides a way to calculate exactly how much the system's properties change when the energy is tweaked. The researchers showed that the change in the system's local behavior is directly proportional to the strength of the energy change. This linear relationship gives scientists a reliable way to predict the outcome of small adjustments without needing to solve the entire, complex system from scratch. The results are rigorous and proven, relying on the mathematical structure of the system rather than simulations or approximations. They hold true at any temperature, provided the system starts with the right kind of stability and the energy changes are not too wild.

Ultimately, this paper offers a robust framework for understanding the resilience of quantum matter. It confirms that the delicate balance of thermal equilibrium in infinite systems is not easily broken by large-scale changes. As long as the system starts with a healthy decay of correlations, it will survive the perturbation, maintaining a structure where local events remain local. This insight strengthens our theoretical foundation for studying complex materials and suggests that the principles governing stability are more universal and flexible than previously demonstrated, even in the most challenging scenarios where multiple stable states compete.

Technical Summary: Locality of KMS states for infinite fermionic lattice systems

Problem Statement
This work investigates the stability of locality properties—specifically the decay of correlations, the principle that "local perturbations perturb locally" (LPPL), and local indistinguishability (LI)—for thermal equilibrium states (KMS states) of infinitely extended fermionic lattice systems under Hamiltonian perturbations. A central challenge addressed is the behavior of these states under extensive perturbations, where the perturbation $V$ cannot generally be represented as a bounded element of the quasi-local algebra. Such perturbations can alter bulk behavior and induce phase transitions. The paper aims to establish quantitative bounds on how these perturbations affect the state, particularly in regimes where multiple equilibrium phases coexist (i.e., where the KMS state is not unique), without requiring the initial or perturbed states to be unique.

Methodology
The approach relies on three main mathematical pillars:

Quantum Belief Propagation (QBP): The author develops an algebra-valued formulation of QBP directly on the underlying $C^*$ -algebra. While previous work (e.g., by Ejima and Ogata) established QBP for $W^*$ -dynamical systems using modular theory, this paper constructs the implementing operator $\eta_P$ as a norm-convergent Dyson series within the original $C^*$ -algebra. This operator implements the state change under a bounded perturbation $P$ via $\omega_P(A) = \omega(\eta_P A \eta_P^*)$ .
Lieb–Robinson Bounds: These bounds are combined with the QBP representation to derive quantitative locality estimates. The paper utilizes fermionic Lieb–Robinson bounds (specifically those from [TW25]) to account for the non-commutativity of observables on disjoint fermionic regions, employing even perturbations and parity-preserving conditional expectations.
Araki Perturbation Theory: The framework uses Araki's construction to define KMS states for perturbed dynamics, approximating extensive perturbations by bounded operators on finite patches and taking limits.

The analysis distinguishes between short-range (exponentially decaying) and long-range (polynomially decaying) interactions and tracks the dependence on inverse temperature $\beta$ .

Key Contributions and Results

QBP on $C^*$ -Dynamical Systems: Theorem 3.1 establishes the existence of a bounded operator $\eta_P$ in the $C^*$ -algebra that transports a KMS state $\omega$ to the perturbed state $\omega_P$ . This provides a constructive proof of the state change without relying on modular theory, facilitating direct norm estimates.
Implications between Locality Notions: The paper establishes a cycle of implications for perturbations of the tracial state (and more general states) under specific decay and uniformity assumptions:
- Decay of Correlations $\implies$ LPPL: Theorem 4.10 shows that if a path of KMS states satisfies uniform decay of correlations, then the change in local expectations under a bounded perturbation decays with distance. Theorem 4.11 extends this to cases where only the initial state satisfies clustering, at the cost of exponential dependence on the perturbation norm.
- LPPL $\implies$ Local Indistinguishability (LI): Theorem 4.12 demonstrates that uniform LPPL over finite patches implies LI, which controls the effect of adding perturbation terms far from an observable.
- LI $\implies$ Decay of Correlations: Theorem 4.13 shows that for perturbations of the tracial state, LI implies a decay of correlations for the resulting KMS state, uniformly over finite patches and the limiting state.
Stability under Extensive Perturbations: Theorem 4.14 is a primary stability result. It shows that if an initial KMS state $\omega$ (which may already be interacting) exhibits exponential decay of correlations and satisfies exponential LI with respect to an extensive perturbation $V$ , then the perturbed state $\omega_V$ exhibits stretched-exponential decay of correlations. Crucially, this result does not require the uniqueness of the KMS state, allowing for the study of stability in the presence of phase coexistence.
Continuity and Differentiability: Theorem 4.16 establishes that under uniform clustering conditions over finite patches and coupling strengths, the path of local equilibrium expectations $s \mapsto \omega_{sV}$ is continuously differentiable. The derivative is given by an absolutely convergent QBP formula, and the change in expectations is bounded linearly in the interaction norm of the perturbation.

Significance and Claims
The paper claims to close the cycle of implications between decay of correlations, LPPL, and LI for perturbations of the tracial state under stated decay, support, and uniformity assumptions, extending these results from finite-volume systems to infinitely extended fermionic systems.

The author emphasizes that their purpose is to establish conditional implications rather than to verify these hypotheses in specific physical models, noting that proving locality for physically interesting models is a formidable task. The results are significant because they:

Allow for the study of stability in the presence of phase coexistence, where the KMS state is not unique.
Provide quantitative bounds for extensive perturbations, which are necessary to describe changes in bulk Hamiltonian parameters.
Extend previous results (e.g., [Cap+25]) from finite-volume Gibbs states and spin systems to infinite-volume fermionic systems and quasi-local perturbations.
Offer a differentiable path of local expectations, providing a rigorous foundation for linear response theory in this context.

The work holds for arbitrary inverse temperatures $\beta$ provided the stated hypotheses on interaction decay and clustering are met. The results apply to both short- and long-range interactions, with the long-range estimates relying on specific fermionic Lieb–Robinson bounds.

Technical Summary: Locality of KMS states for infinite fermionic lattice systems

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