Quasi-adiabatic thermal ensemble preparation in the thermodynamic limit

TL;DR

The paper analyzes a quasi-adiabatic protocol to prepare finite-temperature ensembles in the thermodynamic limit by evolving a noninteracting thermal state toward an interacting target. It shows that in nonintegrable spin chains, local observables can be captured with a single entropy-controlling parameter at high temperature, but the required operation time scales exponentially with precision; time averaging does not mitigate this in the thermodynamic limit. In contrast, integrable models like the transverse-field Ising chain require extensive tuning of initial-state parameters to approach Gibbs values, and the operation time grows linearly with system size, with additional finite-time corrections governed by Kibble-Zurek scaling. Overall, the results delineate when quasi-adiabatic ensemble preparation can be efficient and when integrability imposes fundamental limitations, guiding future explorations in higher dimensions and Bethe-ansatz–solvable systems.

Abstract

We investigate a quasi-adiabatic thermal process for preparing finite-temperature ensembles in the thermodynamic limit. The process gradually transforms a thermal ensemble of a noninteracting system into that of an interacting system of interest over a finite operation time, with the temperature controlled by parameters associated with the entropy of the initial state. We analyze this process in both nonintegrable and integrable spin chains with translational invariance. For the nonintegrable case, numerical simulations show that the thermal properties of local observables are accurately reproduced with a single parameter in the high temperature regime, although the operation time increases exponentially with precision. In contrast, for the integrable transverse-field Ising model, we analytically show that an extensive number of parameters tied to local conserved quantities is generally necessary, and that the operation time increases linearly with system size, diverging in the thermodynamic limit. These results clarify the potential and limitations of the quasi-adiabatic thermal process for an ensemble preparation and highlight the role of integrability in determining its efficiency.

Figures (4)

• Figure 1: System size dependence of $s_N(\rho_\mathrm{f}(\infty)\|\rho_\mathrm{g}(\beta))$. Squares (red), circles (green), and triangles (blue) denote the entropy density $0.2$, $0.4$, and $0.6$, respectively.
• Figure 2: Dependence of the inverse of $s_N(\rho_\mathrm{f}(\tau)\|\rho_\mathrm{g}(\beta))$ on operation time $\tau$ for various system sizes. The dotted line guides the scaling $s_N(\rho_\mathrm{f}(\tau)\|\rho_\mathrm{g}(\beta)) \sim (\ln\tau)^{-1}$.
• Figure 3: $\tau_\mathrm{a}$-Dependence of $s_N(\bar{\rho}(\tau,\tau_\mathrm{a}\| \bar{\rho}(\tau,\infty))$ at $\tau=10$. The dashed lines provide bounds: $r_N(\bar{\rho}(\tau,\tau_\mathrm{a}\| \bar{\rho}(\tau,\infty))$.
• Figure 4: Dependence of $s_N(\rho_\mathrm{f}(\tau)\|\rho_\mathrm{g}(\beta))$ on $\epsilon=\tau/N$ for various values of $\tau$ in (a) paramagnetic phase $(J, \Gamma)=(1, 2)$ and (b) ferromagnetic phase $(J, \Gamma)=(1, 0.5)$. $\beta = 1$ and $\beta_\mathrm{i}$ is chosen so that $\sum_{i=1}^N h_i=N$. The arrow indicates the value of $s_\mathrm{asy}=\lim_{\epsilon \to 0}(\lim_{\tau \to \infty} s_{\tau/\epsilon} (\rho_\mathrm{f}(\tau)\|\rho_\mathrm{g}(\beta)))$. (inset) $\tau$ dependence of the deviation, $s_{\tau/\epsilon}(\rho_\mathrm{f}(\tau)\|\rho_\mathrm{g}(\beta))-s_\mathrm{asy}$, at $\epsilon=0.2$