Finite-size scaling analysis of the eigenstate thermalization hypothesis in a one-dimensional interacting Bose gas

By calculating correlation functions for the Lieb-Liniger model based on the algebraic Bethe ansatz method, we conduct a finite-size scaling analysis of the eigenstate thermalization hypothesis (ETH), which is considered to be a possible mechanism of thermalization in isolated quantum systems. We find that the ETH in a weak sense holds in the thermodynamic limit even for an integrable system, although it does not hold in the strong sense. Based on the result of the finite-size scaling analysis, we compare the contribution of the weak ETH to thermalization with that of yet another thermalization mechanism, the typicality, and show that the former gives only a logarithmic correction to the latter.

Figure 1

(a) Real and (b) imaginary parts of the expectation values of ${\Psi }^{†}\left(x\right)\Psi \left(0\right)$ plotted against the eigenenergy for $N=5$ (squares), 10 (triangles), 15 (asterisks), $N=20$ (crosses), $N=25$ (pluses), and $N=30$ (dots) at $x=L/2N$. For each $N$, all the energy eigenstates in the energy window $[E_g,E_g+10]$ are shown, where $E_g$ is the ground-state energy. The fluctuations of the data around their mean values, which we call the ETH noise, become smaller with increasing size of the system. However, a pair of side branches enclosed by circles emerge for $N=20$, which causes the jump in Fig. 2 (see the text for details).

Figure 2

Support of the ETH noise for real (pluses) and imaginary (asterisks) parts of the correlation function as functions of the size of the system between $N=5$ and 32. The jumps at $N=16$ originate from rare states, which prevent the supports from decreasing monotonically and invalidates the strong ETH. See the text for details.

Figure 3

Variance of the ETH noise for real (pluses) and imaginary (asterisks) parts of the correlation function as functions of the size of the system between $N=5$ and 32. The ETH noise decays as $\sigma \propto N^{-\alpha }$, where the least-squares fit gives $\alpha =1.43\pm 0.04$ and $0.786\pm 0.008$ for the real and imaginary parts, respectively.