Characterizing eigenstate thermalization via measures in the Fock space of operators

The eigenstate thermalization hypothesis (ETH) attempts to bridge the gap between quantum mechanical and statistical mechanical descriptions of isolated quantum systems. Here, we define unbiased measures for how well the ETH works in various regimes, by mapping general interacting quantum systems on regular lattices onto a single particle living on a high-dimensional graph. By numerically analyzing deviations from ETH behavior in the nonintegrable Ising model, we propose a quantity that we call the $n$-weight to democratically characterize the average deviations for all operators residing on a given number of sites, irrespective of their spatial structure. It appears to have a simple scaling form, which we conjecture to hold true for all nonintegrable systems. A closely related quantity, which we term the $n$-distinguishability, tells us how well two states can be distinguished if only $n$-site operators are measured. Along the way, we discover that complicated operators on average are worse than simple ones at distinguishing between neighboring eigenstates, contrary to the naive intuition created by the usual statements of the ETH that few-body (many-body) operators acquire the same (different) expectation values in nearby eigenstates at finite energy density. Finally, we sketch heuristic arguments that the ETH originates from the limited ability of simple operators to distinguish between quantum states of a system, especially when the states are subject to constraints such as roughly fixed energy with respect to a local Hamiltonian.

Figure 1

Schematic of the tree network, constructed by arranging operators according to their size. The identity operator is on the extreme left, followed by the single-site operators, and so on. The total number of operators is $D^{2L}$, while the fraction in the $n\mathrm{th}$ layer is $f_n=\frac{{(D^2-1)}^n}{D^{2L}}\left(\begin{array}{c}L\\ n\end{array}\right)$, which is maximum for $n=(1-1/D^2)L=\frac{3}{4}L\equiv n^{*}$ for $D=2$. Note that $f_n$ grows exponentially with $n$ for small $n$, so the apparent linear growth in the number of dots with $n$ in the figure is for ease of depiction and should not be taken literally.

Figure 2

Schematic illustration of the behavior of neighboring eigenstates ${\rho }_1$ and ${\rho }_2$ on being projected onto the space of small-$n$ (a) and large-$n$ (b) operators. ${\rho }_i^{\left(n\right)}$ is shorthand for the projections ${\vec{\psi }}_n\left({\rho }_i\right)$ defined in the text. The three directions together depict the full Hilbert space of operators, while the horizontal plane represents its projection onto the space of operators of size $n$. ${\rho }_1$ and ${\rho }_2$ are mutually orthogonal vectors in the full space. However, they appear nearly identical when projected onto simple operators as shown on the left, but look quite different for larger operators as shown on the right. The angle ${\theta }_n$ will be calculated numerically for the nonintegrable Ising model in Sec. 4.

Figure 3

$P_n\left(\mathrm{\Delta }\rho \right)$ vs $n$ [(a), (b)] and $P_n\left(\mathrm{\Delta }\rho \right)/f_n$ vs $n$ [(c), (d)] for $\sim 200$ randomly chosen pairs of neighboring eigenstates for $L=14$ sites for the ergodic [(a), (c)] and the integrable [(b), (d)] Ising model. Here, $\mathrm{\Delta }\rho ={\rho }_1-{\rho }_2$ is the difference between the density matrices of the two eigenstates. The color is proportional to the density of states, with blue (red) representing states in regions of the spectrum with low (high) density of states. The black dashed line marks $P_n\left(\mathrm{\Delta }\rho \right)=2f_n$.

Figure 4

Left: $\frac{1}{L}ln\left[g_{\infty }/g\left(E\right)\right]$ vs the energy density $E/L$ for various system sizes for the nonintegrable (a) and the integrable (d) Ising model. Except near the band edges, the curves are indistinguishable, indicating that $g\left(E\right)/g_{\infty }$ grows exponentially with $L$ with an energy density dependent exponent. Middle: $ln\left[P_n\left(\mathrm{\Delta }\rho \right)/f_n\right]$ vs $ln\left[g_{\infty }/g\left(E\right)\right]$ for various $n$ for $L=14$ for the nonintegrable (b) and the integrable (e) Ising model, and straight line fits to the ergodic data. The fits are good for $n\ge 3$. A similar fitting procedure for other system sizes yields the panels on the right, where we show that the slopes (c) and the intercepts (f) of the straight lines are simple functions of $n/L$.

Figure 5

The angle between the projections of two neighboring eigenstates onto the space of operators of size $n,{\theta }_n$ as a function of energy density for various $n$ for $L=14$ for the nonintegrable (a) and the integrable (b) Ising model. For small $n,{\theta }_n$ is small for most states in the ergodic model, but is large in the integrable one. For large $n$, it is very close to $\pi /2$ in both figures, thus proving that neighboring eigenstates are nearly orthogonal when projected onto the space of $n$-site operators with large $n$. Large ${\theta }_n$ at small $n$ and large $g\left(E\right)$ $\left[g\right(E)$ is large near the middle of the spectrum and small near the edges; see Fig. 4] is due to the vectors ${\rho }_1^{\left(n\right)}$ and ${\rho }_2^{\left(n\right)}$ becoming very small in magnitude; then the angle between them is meaningless. The sharp change in ${\theta }_n$ at the lowest and the highest energies in the ergodic case originates from the fact that these states do not satisfy the ETH at the current system size.