Relevance of the eigenstate thermalization hypothesis for thermal relaxation

We study the validity of the eigenstate thermalization hypothesis (ETH) and its role for the occurrence of initial-state independent (ISI) equilibration in closed quantum many-body systems. Using the concept of dynamical typicality, we present an extensive numerical analysis of energy exchange in integrable and nonintegrable spin-$1/2$ systems of large size outside the range of exact diagonalization. In the case of nonintegrable systems, our finite-size scaling shows that the ETH becomes valid in the thermodynamic limit and can serve as the underlying mechanism for ISI equilibration. In the case of integrable systems, however, indication of ISI equilibration has been observed despite the violation of the ETH. We establish a connection between this observation and the need of choosing a proper parameter within the ETH.

Figure 1

Schematic representation of the systems for the study of energy exchange between a “left” (L) and “right” (R) part.

Figure 2

Mean energy difference $\overline{D}$ of energy eigenstates in an energy shell of width $\sigma =0.6$ around $\overline{E}$, calculated for the model in Fig. 1 with anisotropy $\Delta =0.3$ and system size $N_{\mathrm{L}}=8$. For the “average eigenstate,” the distribution of energy onto the subsystems is proportional to their sizes.

Figure 3

The parameter ${\Sigma }^{\prime }$, quantifying the degree of ETH violation, as a function of the effective dimension $d_{\mathrm{eff}}$ that increases with system sizes. The violation appears to vanish according to a power law in the limit of large system sizes, regardless of the properties of the specific models studied. For model parameters, see text.

Figure 4

The ETH parameter ${\Sigma }^{\prime }$ (circles), the scaled ETH parameter $v$ (squares), and the long-time value $r$ for the integrable Heisenberg chain in an energy window of width $\sigma =0.6$ around energy $\overline{E}=0$. All quantities are shown as a function of the inverse number of spins $1/N_{\mathrm{L}}$.

Figure 5

Real-time decay of energy-difference expectation values $d\left(t\right)$, divided by their initial values $d\left(0\right)$, for initial states in Eq. (7) and three system sizes: (a) $N_{\mathrm{L}}=4$, (b) $N_{\mathrm{L}}=6$, and (c) $N_{\mathrm{L}}=8$. While $r\left(t\right)=d\left(t\right)/d\left(0\right)$ does not vanish for long times and finite systems, comparing (a), (b), and (c) indicates a vanishing $r\left(t\right)$ for long times in the limit of large system sizes. For more details on finite-size scaling, see Fig. 4 (triangles).