Numerical Verification of the Fluctuation-Dissipation Theorem for Isolated Quantum Systems

The fluctuation-dissipation theorem (FDT) is a hallmark of thermal equilibrium systems in the Gibbs state. We address the question whether the FDT is obeyed by isolated quantum systems in an energy eigenstate. In the framework of the eigenstate thermalization hypothesis, we derive the formal expression for two-time correlation functions in the energy eigenstates or in the diagonal ensemble. They satisfy the Kubo-Martin-Schwinger condition, which is the sufficient and necessary condition for the FDT, in the infinite system size limit. We also obtain the finite size correction to the FDT for finite-sized systems. With extensive numerical works for the $XXZ$ spin chain model, we confirm our theory for the FDT and the finite size correction. Our results can serve as a guide line for an experimental study of the FDT on a finite-sized system.

Figure 1

Mean values (lines) and standard deviations (error bars) of $g_{ij}(\omega )$’s for ${\beta }_T=0.3$ at $L=22$ (dashed) and 24 (solid). The indicator function from the averaged correlation functions is plotted with symbols.

Figure 2

FDT indicator functions $g_{11}$ for $\widehat{A}=\widehat{B}={\widehat{O}}_1$ (left column) and $g_{22}$ for ${\widehat{O}}_2$ (right column). The target inverse temperatures are ${\beta }_T=0.1$ (top) and 0.5 (bottom), which are marked with horizontal lines. The lattice sizes are $L=18$ (dotted), 20 (dashed dotted), 22 (dashed), and 24 (solid). Predictions from the leading order finite size effect for $L=24$ are drawn with symbols.

Figure 3

${\beta }_{\mathrm{FDT}}$ vs ${\beta }_T$ for $\widehat{A}=\widehat{B}={\widehat{O}}_1$ in (a) and ${\widehat{O}}_2$ in (b). Inset illustrates the $1/L$ dependence of ${\beta }_{\mathrm{FDT}}$ at ${\beta }_{\mathrm{T}}=0.1$, 0.3, 0.5 with lines as guides to the eyes.

Figure 4

FDT indicator functions $g_{12}(\omega )$ in (a) and $g_{13}(\omega )$ in (b) for $L=24$ and ${\beta }_T=0.1$ (solid), 0.3 (dashed), 0.5 (dashed dotted). Also shown are the finite size correction form with symbols. The overlap functions ${\mathcal{R}}_{12}(\omega )$ and ${\mathcal{R}}_{13}(\omega )$ are shown in (c) and (d) with the same parameter values.