Subsystem eigenstate thermalization hypothesis

Motivated by the qualitative picture of canonical typicality, we propose a refined formulation of the eigenstate thermalization hypothesis (ETH) for chaotic quantum systems. This formulation, which we refer to as subsystem ETH, is in terms of the reduced density matrix of subsystems. This strong form of ETH outlines the set of observables defined within the subsystem for which it guarantees eigenstate thermalization. We discuss the limits when the size of the subsystem is small or comparable to its complement. In the latter case we outline the way to calculate the leading volume-proportional contribution to the von Neumann and Renyi entanglment entropies. Finally, we provide numerical evidence for the proposal in the case of a one-dimensional Ising spin chain.

Figure 1

(a) Values of $ln\left({\sigma }_{m,n}\right)$ with superimposed linear fit functions $-{\alpha }_{m}n+{\beta }_m$ for $m=1\cdots 8$, $n=12\cdots 17$, and $\mathrm{\Delta }E=0.1n$, $g=1.05,h=0.1$. The slope of linear functions ${\alpha }_m$ for all $m$ is within $5\%$ close to the theoretical value $ln\left(2\right)/2$. (b) The maximum value of $\mathrm{Tr}\left(\mathrm{\Delta }{\rho }_a^2\right)$ over all eigenstates inside the central band $|E_a|\le \mathrm{\Delta }E=0.1n$.

Figure 2

(a) Plot of $L_A\left(E\right)$ vs $\epsilon =E/n$ for $n=15$ and $n=17$. (b) $M_{m=1,2,3}$ all decrease exponentially with $n$. Here $\mathrm{\Delta }E$ is chosen to be equal to $0.1n$ and $h=0.1$.

Figure 3

(a) Probability distribution $P\left(\mathrm{\Delta }R\right)$ of the deviation $\mathrm{\Delta }R=\mathrm{\Delta }{R}_a^{11}$ corresponding to the matrix element $\langle {\mathcal{E}}_1\left|{\rho }_a^{m=1}\right|{\mathcal{E}}_1\rangle$ for $\mathrm{\Delta }E=0.1n$ and $h=0.1$. It is superimposed with a Gaussian distribution fit. The vertical axis is the number of energy eigenstates within the energy shell $|E_a|<\mathrm{\Delta }E$ with a particular value of $\mathrm{\Delta }R$. All matrix elements of ${\rho }_E^{m=1,2,3}$ show almost identical behavior. (b) Linear behavior of $ln\left({\sigma }_n\right)$ as a function of system size $n$ for two matrix elements $\mathrm{\Delta }{R}^{11}$ and $\mathrm{\Delta }{R}^{12}$ for $m=1$ and $h=0.1$. Because of the approximate equality ${\rho }_C\approx {\rho }_G$ the typical magnitude of the diagonal terms of ${\rho }_a$ is much larger than the off-diagonal ones. There is no qualitative difference between different matrix elements. Results for $m=2,3$ are similar.

Figure 4

(a) Dependence of $ln{\sigma }_{m,17}$ on the subsystem size $m$ with the superimposed linear fit $-4.455+0.219m$. (b) Comparison of matrix elements of ${\rho }_a^A,{\rho }_C^A,{\rho }_G^A$ and the quasiclassical result (16) which we refer to as ${\rho }_Q^A$. Blue dots are matrix elements $\langle {\mathcal{E}}_1\left|{\rho }_a^{m=8}\right|{\mathcal{E}}_1\rangle$ as a function of energy per site $\epsilon =E_a/n$ for $h=0.1$ and $n=17$. We see that $\langle {\mathcal{E}}_1\left|{\rho }_a^{m=8}\right|{\mathcal{E}}_1\rangle$ follows the semiclassical result $\langle {\mathcal{E}}_1\left|{\rho }_Q^{m=8}\right|{\mathcal{E}}_1\rangle$ as given by (16) well, while it differs significantly from $\langle {\mathcal{E}}_1|{\rho }_C^{m=8}|{\mathcal{E}}_1\rangle \approx \langle {\mathcal{E}}_1|{\rho }_G^{m=8}|{\mathcal{E}}_1\rangle$, which lie on top of each other. Quasiclassical result (16) is calculated using density of states $\mathrm{\Omega }$ specified in Appendix Sec. pp1-s1. Other matrix elements show similar behavior.

Figure 5

The density of states of the spin chain system for $g=1.05,h=0.1,n=17$. The horizontal axis is energy per site $\epsilon =E/n$. The yellow bars which fill the plot are the histogram for the density of states calculated using direct diagonalization. The blue solid line is a theoretical fit by the binomial distribution function (A2) with $\kappa \approx 0.3489$; see Appendix Sec. pp1-s1.