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

AI summary

Overview

Research area: Quantum many-body physics and quantum information — specifically, using quantum coherence (a thermodynamic/information-theoretic quantity) as a diagnostic of quantum phases and phase transitions at nonzero temperature.

Technical level: Advanced. The paper assumes familiarity with density matrices, relative entropy, projective measurement, free-fermion lattice models, and many-body phase diagrams.

Scope (one sentence): The authors define a full probability distribution of stochastic quantum coherence, show that its entropy survives thermal fluctuations far better than any of its moments, and demonstrate this on the non-interacting and interacting Aubry–André localization models.

What This Paper Is About

Quantum phase transitions are defined at zero temperature, but real experiments — cold atoms in optical lattices, for instance — always run at finite temperature, where thermal fluctuations smear sharp critical lines into broad crossovers. Most coherence-based diagnostics used so far rely on average quantities, which lose their sharp signatures quickly as temperature rises. The authors ask whether the entire probability distribution of quantum coherence, not just its mean, carries phase information that survives heating.

Key Contributions

1. A distribution-level diagnostic for coherence. The authors build a two-point-measurement (TPM) probability distribution of a stochastic coherence variable, $c_{n\alpha} = \ln p_n - \ln p_\alpha$, and show its first moment exactly reproduces the relative entropy of coherence.
2. An entropy-based order indicator. They introduce the Shannon entropy $\textsc{H}_c$ of the coherence distribution and prove it decomposes into the von Neumann entropy of the state plus the average coherence of its eigenstates minus a non-negative degeneracy correction $\Delta \geq 0$ — hence it is bounded above by $H_c^u = S(\hat\rho) + \sum_n p_n C[|r_n\rangle\langle r_n|]$.
3. A single-particle demonstration. In the non-interacting Aubry–André model on $N = 10^3$ sites, they show the variance peaks at the $v_0 = 2$ localization transition, while $\textsc{H}_c$ keeps a sharp visible change across the transition even at temperatures where the moments have lost it.
4. A many-body extension. In the interacting Aubry–André model with nearest-neighbor interactions at half filling ($N = 12$, $N_p = N/2$), they map out the metallic phase (MP), Anderson band insulator (ABI) and charge-density-wave (CDW) phases and show $\textsc{H}_c$ still resolves all three at inverse temperature $\beta J = 1$.

Main Findings

• First moment tracks delocalization. Average coherence is larger in the extended phase and decreases with temperature; it vanishes in the fully localized limit $v_0 \to \infty$, where the Hamiltonian commutes with every local projector.
• Variance peaks at the transition. The second central moment $\sigma_C^2$ develops a pronounced peak at $v_0 = 2$ in the non-interacting model, fed by both thermal fluctuations (eigenstate populations) and quantum fluctuations (non-commutativity of Hamiltonian and local projectors). The thermal part takes over as $T$ approaches $J$, broadening and eventually erasing the peak.
• Moments wash out; entropy does not. In the non-interacting model the entropy is larger in the delocalized phase (broader distribution) and drops sharply on entering the localized phase; the sharp change "remains clearly visible over the entire temperature range considered."
• The entropy encodes eigenstate fractality. At high temperature the excess $H_c^u - S(\hat\rho_\beta)$ scales as $\langle D_n\rangle \ln N$, where $\langle D_n\rangle$ is the thermally averaged fractal dimension of the eigenstates. Figure 2 reports excellent agreement between this quantity and $\langle D_n\rangle$ extracted from finite-size scaling.
• Three many-body phases resolved at zero temperature. Ground-state coherence approaches its maximum in the MP (sharp peak, small variance), is strongly suppressed in the ABI where $v_0/U \gg 1$, and takes an intermediate, nearly constant value in the CDW where $v_0/U \ll 1$; the variance is largest in the crossovers. Phase boundaries are drawn from the maxima of $\partial C[\hat\rho]/\partial v_0$ and $\partial C[\hat\rho]/\partial U$.
• Finite temperature blurs the coherence peak. Comparing the ground state with a thermal state at $\beta J = 1$ shows the metallic-phase coherence peak broadening substantially, making boundaries hard to read from the first moment alone.
• Entropy resolves the full phase diagram at $\beta J = 1$. $\textsc{H}_c$ continues to identify MP, ABI and CDW across the $(U, v_0)$ plane, indicating its dependence on temperature — equivalently on the von Neumann entropy of the thermal state — is much weaker than that of the individual moments.

Methodology in Plain English

The authors imagine measuring a quantum state twice. The first measurement reads out which energy eigenstate the system occupies (for a thermal state, this is effectively an energy measurement). The second measurement asks a different question — "which lattice site, or which many-body configuration, is occupied?" The mismatch between the two answers, written as a log-ratio of probabilities, defines a single number: the stochastic coherence for that pair of outcomes. Repeating over all outcome pairs builds a full probability distribution, subject to the fluctuation relation $\langle e^{-c}\rangle = 1$.

Crucially, the average of this distribution is the standard relative-entropy measure of coherence, and the whole distribution is accessible through a characteristic function expressed as $\mathrm{Tr}{\hat\rho e^{i\nu\ln\hat\rho} e^{-i\nu\ln\Delta_\alpha[\hat\rho]}}$. Beyond the mean and variance, the authors compute the Shannon entropy of the distribution as a single global number.

They then diagonalize the Aubry–André Hamiltonian — a one-dimensional lattice with an incommensurate cosine potential of strength $v_0$ and incommensurability $\gamma = (\sqrt5 - 1)/2$ — to build thermal states at various temperatures, and repeat for the interacting version with a nearest-neighbor density-density term $U\sum_i \hat n_i \hat n_{i+1}$ at half filling. Coherence is always evaluated in the basis natural to the phase: site basis for the single particle, many-body occupation-pattern basis (dimension $\binom{N}{N_p}$) for the interacting case.

Why This Matters

The work reframes a thermodynamic quantity — the entropy of a coherence distribution — as a finite-temperature order parameter, offering an alternative to the handful of known quantities (quantum discord, teleportation-based schemes) that survive thermal smearing. It also gives a clean information-theoretic bridge between thermal mixedness, quantum fluctuations, and the multifractal structure of critical eigenstates.

Potential applications:

• Ultracold-atom quantum simulators. Optical-lattice experiments with tunable Feshbach interactions are the natural testbed for the Aubry–André physics studied here.
• Benchmarking quantum hardware. A distribution-level coherence measure could certify whether a device retains genuinely quantum correlations as its effective temperature rises.
• Quantum thermodynamics of devices. Coherence already underpins work in energy conversion, fluctuation relations, and irreversibility; a full distribution sharpens those analyses.
• Detecting many-body localization in noisy intermediate-scale devices. Distinguishing MBL-like from thermalizing behavior is a persistent experimental challenge where robustness to temperature matters.

Industry relevance: quantum computing and simulation platforms (superconducting, neutral-atom, trapped-ion) need practical, temperature-tolerant verification metrics; the entropy of the coherence distribution is a candidate that does not require full state tomography of individual moments.

Future Directions

1. Nonequilibrium dynamics. The authors propose tracking the time evolution of the coherence distribution to look for dynamical phase transitions, which the equilibrium analysis does not cover.
2. Experimentally reconstructing the distribution. They suggest quench-based protocols for measuring the distribution, which would enable direct experimental tests of these predictions.
3. Scaling beyond $N = 12$. Finite-temperature many-body results rest on exact diagonalization at $N = 12$ at half filling; whether $\textsc{H}_c$ remains as clean at larger sizes is not established here.
4. Beyond Aubry–André. Whether the entropy of the coherence distribution is equally robust in models without a natural preferred basis — or where the right basis is not known in advance — remains open, since the authors stress that the choice of basis is what makes the critical signatures appear.

Target Audience

Quantum many-body theorists and quantum-information researchers working on finite-temperature phases, many-body localization, and quantum thermodynamics; ultracold-atom experimentalists looking for measurable order parameters; and graduate students with a background in open quantum systems and statistical mechanics who want a worked example of an information-theoretic diagnostic applied to a canonical localization model.

Authors’ abstract

We introduce a diagnostic framework based on the full probability distribution of quantum coherence to probe single-particle and many-body phases at finite temperature. We first apply this approach to the localization transition in the non-interacting Aubry-André model and then extend the analysis to its interacting counterpart, whose phase diagram is substantially richer. We show that the moments of the coherence distribution provide clear signatures of quantum phase transitions and capture the emergence of many-body phases. More importantly, we find that the entropy of the coherence distribution emerges as an even more powerful indicator, retaining clear signatures of the underlying quantum phases while remaining remarkably robust against increasing thermal fluctuations.