Eigenstate Thermalization, Random Matrix Theory, and Behemoths

The eigenstate thermalization hypothesis (ETH) is one of the cornerstones of contemporary quantum statistical mechanics. The extent to which ETH holds for nonlocal operators is an open question that we partially address in this Letter. We report on the construction of highly nonlocal operators, behemoths, that are building blocks for various kinds of local and nonlocal operators. The behemoths have a singular distribution and width $w\sim {\mathcal{D}}^{-1}$ ($\mathcal{D}$ being the Hilbert space dimension). From there, one may construct local operators with the ordinary Gaussian distribution and $w\sim {\mathcal{D}}^{-1/2}$ in agreement with ETH. Extrapolation to even larger widths predicts sub-ETH behavior of typical nonlocal operators with $w\sim {\mathcal{D}}^{-\delta }$, $0<\delta <1/2$. This operator construction is based on a deep analogy with random matrix theory and shows striking agreement with numerical simulations of nonintegrable many-body systems.

Figure 1

Schematic showing the space of operators $\overline{\mathbb{O}}$ having elements zero or unity in the configuration basis. The operators within $\overline{\mathbb{O}}$ are organized into classes distinguished by the scaling of the width $\sigma \sim {\mathcal{D}}^{-\delta }$ of matrix element distributions in the eigenstate basis. The number of each class of operators is shown using the big-$O$ notation.

Figure 2

Probability distributions of matrix elements of Behemoth operators, for two different many-body systems, compared with GOE and GUE predictions. (a),(b) Spin-$1/2$ chain with anisotropic Heisenberg ($XXZ$) couplings. Nearest-neighbor (NN) and next-nearest-neighbor (NNN) coupling strengths ($J_{1,2}$) and anisotropies (${\mathrm{\Delta }}_{1,2}$) are indicated. (c) Bose-Hubbard ladder (geometry in sketch) subject to magnetic field. Solid lines in (a),(b),(c) are predictions from Eqs. (4), (5), and (7), respectively.

Figure 3

(a),(b),(c) Distributions for Hermitian operators ($2n$-point correlators) of varying locality, from behemoth (a) to 2-point correlator (c). Number of nonzero terms $M=2M^{\prime }$ in the operator matrix are shown. Solid lines are RMT predictions, Eq. (6). (d) Width of distributions for behemoths and local operators. The $\sim {\mathcal{D}}^{-1}$ line is the RMT prediction for behemoths, Eq. (5). RMT prediction for local operators (solid line) falls below the data, consistent with panel (c). (e) Width of distributions for two dense operators with $M={\mathcal{D}}^{1+\beta }$, showing the predicted sub-ETH scaling. A dashed line for ETH scaling is also shown.

Figure 4

Distributions for hermitian counterparts of behemoths. (a),(b) A spinless-fermion chain with NN and NNN interactions $V$ and $V^{\prime }$, subject to Gaussian disorder of strength $W$. (a) In the ergodic phase, distribution is exponential as predicted by RMT, Eq. (5). (b) In the many-body localized phase, the distribution is a power law. (c) Integrable $XXZ$ spin chain, showing deviation from RMT prediction.