Full Quantum Work Statistics for Non-Homogeneous Many-Body Systems

TL;DR

The paper addresses the challenge of describing nonequilibrium thermodynamics in interacting quantum many-body systems by developing a first-principles framework (LR-thTDDFT) to compute full quantum work statistics within linear response. It expresses the dissipated-work cumulants through a relaxation function derived from density-density response functions, decomposing it into adiabatic and nonadiabatic parts and linking the nonadiabatic contribution to the imaginary part of the response function. The authors demonstrate the approach on the 1D Hubbard model with a staggered potential, revealing phase-dependent signatures across Mott, metallic, and band-insulating regimes and identifying a nonadiabatic-to-adiabatic crossover as driving becomes slower. The work provides a microscopic, transferable framework bridging thermal density functional theory and nonequilibrium work statistics, with potential extensions to frequency-dependent kernels and broad applicability to ultracold atomic and condensed-m matter systems.

Abstract

The nonequilibrium thermodynamics of interacting quantum many-body systems is investigated within the framework of thermal time-dependent density functional theory using a generalized linear-response formulation for the full quantum work statistics. A first-principles route is established to reconstruct the relaxation function that underlies linear-response theory, thereby moving beyond phenomenological descriptions and enabling a consistent evaluation of all moments of the dissipated-work distribution in interacting systems. The predictive power of the approach is demonstrated for the Hubbard model subject to a staggered external potential, where the evolution of the relaxation dynamics across the Mott-to-band-insulator crossover reveals how distinct many-body phases shape the out-of-equilibrium thermodynamic response. These results provide a microscopic and transferable framework for quantum thermodynamics in correlated systems, bridging thermal density functional theory and nonequilibrium work statistics.

Figures (3)

• Figure 1: Phase diagram of the Hubbard model with a staggered field, obtained from the dissipated work, $\beta\,\frac{\langle W_{\mathrm{diss}}\rangle}{(\delta v)^2}$, following a sudden quench of the work parameters. Results are shown for inverse temperature $\beta J = 1$ and quench amplitude $\delta\lambda = 0.01 J$ for a half-filled chain of length $\mathcal{L} = 50$. The solid line, corresponding to the 45% contour of the maximum value, serves as a visual guide to distinguish the MI, metallic (M), and BI phases.
• Figure 2: (a)-(c) First three cumulants of the dissipated-work distribution for the Hubbard Hamiltonian (\ref{['eq:H-Hubbard']}) and (d) the corresponding Fano factor, plotted as functions of the quench duration $\tau$ and the staggered potential amplitude $v_0$. Results are shown for inverse temperature $\beta J = 1$, quench amplitude $\delta\lambda = 0.01 J$, $U=1J$ and a half-filled chain of length $L = 50$. The dashed line $\tau^\ast(v_0)$ indicates the crossover between the nonadiabatic and adiabatic regimes, identified as the maximum of the absolute derivative of $k^{3}_{\normalfont\textsc{w}}$ with respect to the quench time.
• Figure 3: Benchmark of the thermal LR-TDDFT approach against exact diagonalization for the two-site Hubbard model in the canonical ensemble at half filling and zero total spin projection. Panels (a)-(c) show the first three cumulants of the dissipated-work distribution obtained from exact diagonalization, while panels (e)-(g) show the corresponding results from LR-thTDDFT. The mean relative errors are approximately $1.36\times 10^{-2}$ for the first cumulant, $2.76\times 10^{-2}$ for the second, and $1.76\times 10^{-2}$ for the third.