Quantum-Coherent Thermodynamics: Leaf Typicality via Minimum-Variance Foliation

TL;DR

The paper develops a framework to extend thermodynamics beyond equilibrium by foliating quantum state space into minimum-variance leaves, each labeled by an optimal pure-state family and connected to an effective Hamiltonian $H_ ho$. Within a leaf, the least-biased, norm- and energy-constrained state is the leaf canonical ensemble, with Gibbs weights defined via the leaf's $H_{ ho_0}$ and a reference leaf barycenter; the commuting leaf yields the standard Gibbs form. A quantitative measure of energy coherence is introduced through leaf entropy $rak I( ext{L})$, and simple examples (qubits and qutrits) illustrate the geometric structure of leaves. The leaf typicality hypothesis posits that local observables depend only on the leaf and energy and can be captured by evolving a representative pure state from the optimal leaf ensemble, supported by numerical tests in nonintegrable spin chains and discussions of potential open-system dynamics and emergent foliations on subsystems. The work opens a route to non-equilibrium thermodynamics where coherence and leaf structure regulate relaxation and equilibration, with several avenues for future exploration including dynamics toward commuting leaves and integrability constraints.

Abstract

Equilibrium statistical ensembles commute with the Hamiltonian and thus carry no coherence in the energy eigenbasis. We develop a thermodynamic framework in which energy fluctuations can retain genuinely quantum-coherent contributions. We foliate state space into "minimum-variance leaves," defined by minimizing the average energy variance over all pure-state decompositions, with the minimum set by the quantum Fisher information. On each leaf we construct the least-biased state compatible with normalization and mean energy, defining a leaf-canonical ensemble. The Gibbs ensemble is recovered on the distinguished commuting leaf, while generic states are organized by their leaf label. This structure provides a natural setting to extend eigenstate thermalization beyond equilibrium via a "leaf typicality" hypothesis. According to that hypothesis, under unitary time evolution local observables depend only on the leaf and energy and, at all times, are reproduced by evolving a representative (pure) state drawn from the optimal ensemble.

Figures (6)

• Figure 1: Representation of the foliation of the state space of a single spin-$1$, viewed from two angles. The subspace is constrained by the condition $\tr[\rho\lambda_j]=0$ for $j\in\{2,4,5,6,7\}$, with Hamiltonian given by $H=\frac{1}{2}\lambda_3+\frac{3}{2}\sqrt{3}\lambda_8$. The colors encode the degree of incoherence: the lightest yellow triangle marks the commuting leaf, while progressively darker (browner) shades correspond to increasing quantum coherence. Black curves represent the corresponding leaf-canonical ensembles, interpolating between the pure state at $\beta\rightarrow-\infty$ and the pure state at $\beta\rightarrow \infty$.
• Figure 2: Left: Typicality diagnostics (see text) for the local observables $O\in\{\sigma_\ell^{z},\,\sigma_\ell^{z}\sigma_{\ell+1}^{z}\}$ in the min-variance ensemble associated with the nonintegrable Hamiltonian \ref{['eq:H']} with parameters $(g,h,D)=(\frac{\sqrt5+5}{8},\frac{\sqrt5}{2},\frac{\pi}{20})$. The diagnostics are shown for three thermal states $\rho \propto e^{-\beta H_0}$, with $\beta\in\{0.25,0.75,1.75\}$, generated by $H_0$ of the same form \ref{['eq:H']} with parameters $(0,\frac{3}{2},0)$ and chain's lengths $L=\log_2 d\in\{6,8,10,12\}$. Increasing line thickness corresponds to larger $L$. Dashed curves represent the commuting-leaf benchmark ($\beta=0$, i.e., $\rho\propto \mathrm{I}$). For comparison, dotted curves show the commuting-leaf diagnostics for the integrable Hamiltonian $H_0$, for which ETH fails Steinigeweg2013Eigenstate. Right: Comparison of the exact time evolution of various operators under Hamiltonian $H$ (solid lines), starting from $\rho(0)\propto e^{-\beta H_0}$ with $\beta=0.5$ ($\mathfrak I[\mathcal{L}_H(\rho_{\beta})]\approx 0.71 \log d$), against predictions (markers) from the time evolution of the closest-energy representative state for $L=12$. Error bars denote the $68\%$ confidence interval based on the fraction of outliers in the shell.
• Figure S1: Typicality diagnostics (see main text) for all local observables with support on $1$ or $2$ neighboring sites in the min-variance ensemble associated with the nonintegrable Hamiltonian \ref{['eq:H']} with parameters $(g,h,D)=(\frac{\sqrt5+5}{8},\frac{\sqrt5}{2},\frac{\pi}{20})$. The state is thermal, $\rho \propto e^{-\beta H_0}$, with inverse temperature $\beta=0.25$ and $H_0$ has the same form \ref{['eq:H']} with parameters $(0,\frac{1}{2},0)$. The considered chain's lengths are $L=\log_2 d\in\{6,8,10,12\}$. Increasing line thickness corresponds to larger $L$. Dashed curves represent the commuting-leaf benchmark ($\beta=0$, i.e., $\rho\propto \mathrm{I}$). The energy incoherence is $\mathfrak{I}[\mathcal{L}_H(\rho_\beta)]\approx 0.97 \log d$.
• Figure S2: The same as in Fig. \ref{['f:ETH0dot25']} with $\beta=0.75$ (the energy incoherence is $\mathfrak{I}[\mathcal{L}_H(\rho_\beta)]\approx 0.76 \log d$).
• Figure S3: The same as in Fig. \ref{['f:ETH0dot25']} with $\beta=1.75$ (the energy incoherence is $\mathfrak{I}[\mathcal{L}_H(\rho_\beta)]\approx 0.24 \log d$).