⚛️ quantum physics

Revealing Many-Body Phases at finite temperatures through Quantum Coherence Distribution

This paper introduces a diagnostic framework based on the full probability distribution of quantum coherence, demonstrating that its entropy serves as a robust indicator for identifying single-particle and many-body phases at finite temperatures, even in the presence of thermal fluctuations.

Original authors: Antonio Palamara, Nicola Lo Gullo, Antonello Sindona, Francesco Plastina

Published 2026-09-04

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CC BY 4.0

Original authors: Antonio Palamara, Nicola Lo Gullo, Antonello Sindona, Francesco Plastina

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, particles do not simply sit in one place or hold one value; they can exist in a blend of possibilities at the same time. This ability to be in multiple states simultaneously is called superposition, and the measure of how strongly a system holds these blended states is known as quantum coherence. For decades, scientists have used this concept to understand everything from how energy moves in tiny systems to how information is processed in future computers. However, a major challenge remains: real-world experiments rarely happen in the perfect, frozen silence of absolute zero. As soon as heat is introduced, the environment begins to jostle the system, creating thermal noise that tends to wash out these delicate quantum effects, turning sharp, distinct boundaries between different states into blurry, indistinct transitions. This makes it incredibly difficult to tell exactly when a material shifts from one phase to another, such as when it stops conducting electricity and becomes an insulator, once the temperature rises even slightly.

A team of researchers from the University of Calabria and the Italian National Institute of Nuclear Physics has developed a new way to see through this thermal fog. Instead of looking at the average amount of quantum coherence in a system, which often gets lost in the noise, they decided to map out the entire landscape of how that coherence is distributed. Imagine taking a photograph of a crowd: looking at the average height of the people tells you something, but looking at the full spread of heights reveals the true structure of the group. The researchers applied this logic to quantum systems, specifically studying models where particles move through a lattice with a repeating, yet irregular, pattern of obstacles. They found that while the simple averages of coherence fade away as the temperature rises, a different property—the entropy, or the measure of disorder and spread, of the entire coherence distribution—remains remarkably sharp. This new metric acts as a robust beacon, clearly identifying the boundaries between different quantum phases even when the system is warm and the usual signs of quantum criticality have been blurred by heat.

To test this idea, the scientists first looked at a simpler, non-interacting model known as the Aubry-André model. In this setup, a single particle moves along a one-dimensional chain of sites, encountering a potential that varies in a quasiperiodic fashion, meaning it repeats but never quite aligns perfectly. At low temperatures, this system undergoes a clear transition: when the strength of the potential is low, the particle is free to roam across the entire chain, a state called delocalized. When the potential becomes strong enough, the particle gets trapped in a specific region, becoming localized. The researchers calculated the probability distribution of coherence for this system at various temperatures. They observed that the average amount of coherence dropped as the temperature rose, and the sharp point where the transition happened became harder to spot. However, when they calculated the entropy of the coherence distribution, the story changed. This value remained sensitive to the transition, showing a distinct change right at the critical point where the particle switches from roaming to trapped, even at temperatures where the average coherence had become too noisy to be useful.

The team then pushed their method further by introducing interactions between particles, moving from a single-particle system to a many-body system. This is a much more complex scenario where particles influence one another, leading to a richer variety of phases, including a metallic phase where particles flow freely, a charge-density-wave phase where they arrange themselves in a static pattern, and a many-body localized phase where they get stuck despite their interactions. In these simulations, the researchers again found that the standard measures of coherence, such as the average value or its variance, became increasingly unreliable as the temperature increased. The clear lines separating these different phases began to blur, making it difficult to determine the state of the system. Yet, the entropy of the coherence distribution continued to perform with striking clarity. It successfully mapped out the boundaries between the metallic, charge-density-wave, and localized phases across the entire range of temperatures they tested. The entropy acted as a stable indicator, retaining the fingerprints of the underlying quantum phases even when thermal fluctuations were strong enough to obscure the details seen in simpler measurements.

The significance of this work lies in its ability to disentangle the effects of heat from the intrinsic quantum nature of a system. By analyzing the full probability distribution rather than just a single number, the researchers showed that the entropy of coherence captures the global structure of the quantum state. This approach reveals that the quantum phases are not easily destroyed by heat; rather, their signatures are simply hidden within the broader statistical spread of the system's behavior. The findings suggest that to truly understand quantum materials in realistic, finite-temperature environments, scientists must look beyond simple averages. The entropy of the coherence distribution offers a powerful new tool that remains reliable where other methods fail, providing a clear window into the quantum world even when it is being shaken by thermal noise. This framework not only helps in identifying phases in current experiments with ultracold atoms but also opens the door to exploring how these quantum states evolve over time and how they might be manipulated in future technologies.

Technical Summary: Revealing Many-Body Phases at Finite Temperatures through Quantum Coherence Distribution

Problem Statement
Quantum phase transitions are fundamentally defined at zero temperature, yet realistic experimental platforms, such as ultracold atomic gases in optical lattices, operate at finite temperatures. At non-zero temperatures, thermal fluctuations progressively obscure the sharp signatures of quantum criticality, replacing critical lines with finite-width crossover regions. While quantities like quantum discord and teleportation-based schemes have shown some resilience to thermal noise, there remains a need for robust diagnostic tools capable of identifying quantum phases and critical points in finite-temperature many-body systems. Existing coherence-based approaches typically rely on average quantities (first moments), which may lose sensitivity to criticality as thermal fluctuations increase.

Methodology
The authors introduce a diagnostic framework based on the full probability distribution of quantum coherence, rather than just its average value. This approach is rooted in nonequilibrium thermodynamics and utilizes a two-point measurement (TPM) protocol:

First Measurement: Performed in the eigenbasis of the system's state $\hat{\rho}$ (energy measurement for thermal states).
Second Measurement: Performed in a chosen basis $\{|\alpha\rangle\}$ (e.g., local real-space basis).

From these measurements, a stochastic coherence variable $c_{n\alpha} = \ln p_n - \ln p_\alpha$ is defined, where $p_n$ are eigenvalues of $\hat{\rho}$ and $p_\alpha$ are diagonal elements in the chosen basis. This yields a probability distribution $P(c)$ .

The framework analyzes:

Moments of the distribution: The first moment $\langle c \rangle$ recovers the standard relative entropy of coherence $C[\hat{\rho}]$ . Higher-order moments (e.g., variance) characterize specific statistical properties.
Entropy of the distribution ( $H_c$ ): Defined as $H_c = -\sum_c P(c) \ln P(c)$ , this quantity provides a global measure of the information encoded in the stochastic coherence. The authors derive that $H_c$ is bounded by the sum of the von Neumann entropy of the state and the average coherence of its eigenstates, corrected for the degeneracy of coherence values.

The framework is applied to two paradigmatic models:

Non-interacting Aubry-André (AA) model: A single-particle system with a quasiperiodic potential exhibiting a localization-delocalization transition.
Interacting Aubry-André model: A many-body system of spinless fermions with nearest-neighbor interactions, exhibiting a richer phase diagram including Metallic (MP), Anderson Band Insulator (ABI), and Charge-Density-Wave (CDW) phases.

Key Results

Non-interacting AA Model:
- Moments: The average coherence (first moment) decreases with temperature and vanishes in the fully localized limit. The variance (second moment) exhibits a peak at the critical point ( $V_0 = 2$ ) at low temperatures. However, as temperature increases, thermal fluctuations dominate, broadening the crossover region and washing out these critical signatures.
- Entropy: In contrast to the moments, the entropy of the coherence distribution ( $H_c$ ) retains a sharp signature of the localization transition across the entire temperature range studied. $H_c$ is larger in the delocalized phase (broad distribution) and decreases sharply in the localized phase (narrow distribution).
- Fractal Connection: The authors show that the upper bound of the coherence entropy relates to the thermal average of the fractal dimension of the eigenstates, $\langle D_n \rangle$ . In the high-temperature limit, the entropy contribution from eigenstate coherences scales with the system size $N$ as $\langle D_n \rangle \ln N$ , directly capturing the multifractal nature of the spectrum near the transition.
Interacting AA Model:
- Ground State: At zero temperature, the first two moments of the coherence distribution (evaluated in the many-body real-space configuration basis) successfully distinguish the MP, ABI, and CDW phases. The variance peaks near phase boundaries.
- Finite Temperature: Similar to the non-interacting case, thermal fluctuations progressively obscure the phase boundaries when relying solely on the average coherence or variance. The coherence peak associated with the metallic phase broadens significantly.
- Robustness of Entropy: The entropy of the coherence distribution remains a robust indicator, clearly distinguishing the three phases across the $(U, V_0)$ phase diagram even at finite temperatures ( $T=J$ ). Its dependence on temperature is significantly weaker than that of the individual moments.

Significance and Claims
The paper claims that the entropy of the coherence distribution is a significantly more powerful and robust indicator of underlying quantum phases than any individual statistical moment (such as the mean or variance) when thermal fluctuations are present.

Disentangling Fluctuations: The framework provides a natural way to disentangle quantum and thermal fluctuations. While moments capture specific statistical properties that are easily washed out by thermal noise, the entropy probes the coherence distribution as a whole, retaining sensitivity to the structural changes induced by phase transitions.
General Applicability: The method is demonstrated to work for both single-particle localization transitions and complex many-body phases, suggesting its utility as a general diagnostic tool for finite-temperature quantum systems.
Experimental Relevance: The authors note that while the current work is theoretical, the framework motivates the development of experimentally accessible protocols (e.g., quench-based schemes) to reconstruct the coherence distribution, potentially enabling direct experimental tests of these robust signatures.

The work does not claim to solve all problems of finite-temperature detection but specifically highlights the superior robustness of coherence distribution entropy over traditional coherence measures in the presence of thermal noise.

Technical Summary: Revealing Many-Body Phases at Finite Temperatures through Quantum Coherence Distribution

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