Thermalization in a closed quantum system from randomized dynamics

TL;DR

The work shows that canonical thermal observables can emerge in a closed finite quantum system without a bath by averaging over randomized internal interactions. A target Hamiltonian $H_0$ (a 1D transverse Ising model) is perturbed by random ${\cal V}^{(j)}$ drawn from a Gaussian unitary ensemble, yielding $H=H_0+{\cal V}^{(j)}$ whose spectrum exhibits chaotic, GUE-like statistics, enabling thermalization through averaging over realizations. The main findings are that the average eigenstate occupations $w(E_\alpha)=\overline{|\langle \psi^{(j)}_\alpha|\psi_{\rm in}\rangle|^2}$ follow a Gibbs distribution with $\beta=\beta(E_{\rm in})$, that eigenstate expectation values ${\cal O}(E_\alpha)$ weight accordingly, and that finite-temperature correlation functions exhibit a coherence length $\xi$ that scales linearly with $\beta$, $\xi \approx (4c/\pi a)\,\beta$ with a finite-size offset. This approach yields thermal observables from real-time propagation in a closed system and suggests pathways for thermal-state preparation on quantum simulators without explicit baths.

Abstract

The emergence of statistical mechanics from quantum dynamics is a central problem in quantum many-body physics. Deriving observables aligned with the prediction of the canonical ensemble for a quantum system relies on the presence of a bath provided either as an external environment or as a larger part of a closed system. We demonstrate that thermal (canonical) observables for a whole closed quantum system of finite size can arise in the absence of a bath. These thermal observables stem from classical averaging over randomized unitary evolutions for a few-body system. The temperature in the canonical ensemble appears as a global constraint on the total energy of the system, determined by the choice of the initial state. From averaging randomized evolutions, we derive spin-spin correlation functions for a finite spin chain and show that they exhibit a temperature-dependent finite correlation length, in agreement with the prediction of the canonical ensemble. This establishes a method for computing thermal observables in a closed, finite-size system from real-time propagation without a bath. An implementation of this thermalization approach on a quantum computer can be utilized for thermal state preparation.

Figures (6)

• Figure 1: Thermalization from randomized dynamics. Panel (a) shows the eigenstate expectation value ${\cal O}^{(j)}_{\alpha\alpha}=\langle \psi^{(j)}_\alpha | {\cal O} |\psi^{(j)}_\alpha\rangle$ for ${\cal O}=|\varphi_\nu\rangle\langle \varphi_\nu|$ ($\nu = 500$) weighted by the eigenstate occupations in the initial state $|c^{(j)}_\alpha|^2=|\langle \psi^{(j)}_\alpha | \psi_{\rm in} \rangle|^2$, averaged over $M=200$ realizations of the system versus the many-body spectrum of $H$. For the horizontal axis, we use the mean spectrum of $H$: $E_\alpha=\overline{E^{(j)}_\alpha}$. The deviations from the mean values for each energy level are small as seen from the horizontal error bars Appendix, defined as $\delta E_\alpha=\sqrt{\overline{ |E^{(j)}_\alpha-E_\alpha|^2}/(M-1)}$, which are less than the size of the marker. The blue squares display $\overline{|c^{(j)}_\alpha|^2 {\cal O}^{(j)}_{\alpha\alpha}}$ and the red circles display $\overline{|c^{(j)}_\alpha|^2} \,\, \overline{{\cal O} ^{(j)}_{\alpha\alpha}}$: the two data sets lie right on top of each other. The inset shows $|c^{(j)}_\alpha|^2$ and ${\cal O}^{(j)}_{\alpha\alpha}$. The magenta dots denote the average values $\overline{|c^{(j)}_\alpha|^2}$, and the pink lines display every tenth realization of $|c^{(j)}_\alpha|^2$ that fluctuate from realization to realization. The green diamonds and the light green lines show the same for ${\cal O}^{(j)}_{\alpha\alpha}$. Panel (b) shows the average eigenstate occupation probabilities $w(E_\alpha)=\overline{|c^{(j)}_\alpha|^2}$. The blue squares display $w(E_\alpha)$, and the dashed red line corresponds to the Gibbs distribution with the temperature $\beta^{-1}$ determined from Eq. (\ref{['beta']}). In the inset, the green diamonds show the average eigenstate expectation value ${\cal O}(E_\alpha)=\overline{{\cal O}^{(j)}_{\alpha\alpha}}$. In panels (a) and (b), the horizontal axis in the inset and in the main figure coincide. In panel (c), the blue, red, and green histograms show the probability distributions for three eigenstates of $H$ sampled from $M$ realizations each. The correspondingly colored lines show Porter-Thomas distribution, $P(p)=e^{-p/w(E_\alpha)}/w(E_\alpha)$, for each eigenstate. All figures are plotted on a log scale except for the inset in panel (a).
• Figure 2: Occupation probabilities of eigenstates of the Ising model calculated by averaging randomized evolutions. The gray lines show the occupation probabilities for every tenth of the $200$ realizations. The blue squares are the average occupation probabilities found from the r.h.s. of Eq. (\ref{['O']}). The red circles are occupation probabilities found from Eqs. (\ref{['O_th']}-\ref{['Gibbs']}). The dashed black line shows occupations populated according to the Gibbs distribution with the temperature determined from Eq. (\ref{['beta']}).
• Figure 3: Correlation functions of the Ising model calculated by averaging randomized unitary dynamics. Panel (a) shows the average same-time spin-spin correlators as functions of the lattice sites on a log scale. The blue squares represent the correlator positioned at $i_0=0$ and the red circles and green diamonds display the correlators placed at $i_0=2$ and $i_0=-2$. The correspondingly colored lines show the exponential decay with the rate determined from fitting the cental correlator in the interval $i = [-2,2]$. In panel (b), the blue squares, red circles, green diamonds, and magenta crosses show correlator positioned at $i_0=0$ for gradually increasing initial energies. The correspondingly colored lines show the fits performed at every initial energy $E_{\rm in}$. In panel (c) we plot the coherence length $\xi$ as a function of the inverse temperature $\beta$ determined by $E_{\rm in}$ (with small uncertainties in the total energy $\langle \overline{H}\rangle$ that stem from averaging over realizations). The blue squares show $\xi$ found from the fit in the central panel for the correlators computed using the exact diagonalization (ED). The magenta circles show the same for the correlators computed with the real-time propagation (RTP). The solid blue line is the linear fit for the data and the dashed red line shows predictions for $\xi$ in the thermodynamic limit up to the finite-size offset. In the inset, the green diamonds and black dashes display $\langle \overline{H}\rangle$ and $E_{\rm in}=\langle\psi_{\rm in} |H_0|\psi_{\rm in}\rangle$ as a function of $\beta$. The horizontal axis is the same as in the main figure.
• Figure 4: Probability distribution of the ratio of consecutive level spacings. The blue histogram shows the probability distribution of $r_\alpha^{(j)}$ sampled from the $200$ realizations of the Hamiltonian $H$ with $N=11$. The dashed black line shows the probability distribution for a system with the level statistics described by the Gaussian unitary ensemble of random matrix theory. The dotted gray line is the probability distribution for a system with Poisson level statistics.
• Figure 5: Average eigenstate occupation probabilities for different initial states. Legends in panels (a)-- (c) are analogous to the legend in Fig. \ref{['fig:ETH']} b.