Eigenstate thermalization hypothesis and out of time order correlators

The eigenstate thermalization hypothesis (ETH) implies a form for the matrix elements of local operators between eigenstates of the Hamiltonian, expected to be valid for chaotic systems. Another signal of chaos is a positive Lyapunov exponent, defined on the basis of Loschmidt echo or out of time order correlators. For this exponent to be positive, correlations between matrix elements unrelated by symmetry, usually neglected, have to exist. The same is true for the peak of the dynamic heterogeneity length ${\chi }_4$, relevant for systems with slow dynamics. These correlations, as well as those between elements of different operators, are encompassed in a generalized form of ETH.

Figure 1

Products corresponding to three-point functions (an alternative representation to Fig. 5): a loop and two cacti.

Figure 2

Diagrams corresponding to four-point functions. All the diagrams are cacti apart from (f).

Figure 3

A rotation group with three independent $\mathrm{\Delta }*\mathrm{\Delta }$ unitary transformation matrices.

Figure 4

A dotted line in the diagram on the left column means that the indices are in the same energy interval, and in a diagram on the right column that they are in the same interval but are not equal. In the diagrams of the right columns a solid line stands for a Kronecker delta, imposing the equality of indices. A double line imposes the same equality twice (e.g., ${\delta }_{ij}{\delta }_{ji}$), while a blue dot a delta with equal indices (e.g., ${\delta }_{ii}$): They are indicated only for counting purposes, to show that there are three Kronecker symbols in each diagram.

Figure 5

The final situation: Products as in diagram (a), diagrams (c)–(e), and diagram (b) have different scaling functions.

Figure 6

Products corresponding to four-point functions.

Figure 7

The function ${\widehat{G}}_{\beta }\left(\omega \right)$ for a system of size $N=12$ and where we took $\mathrm{\Delta }=0.3$ and $\beta$ corresponding to ${\epsilon }_{+}$ in the middle of the spectrum.

Figure 8

The function $G_{\beta }\left(t\right)$ for a system of size $N=10$ and where we took $\mathrm{\Delta }=0.3$ and $\beta =0$.

Figure 9

The contribution to the OTOC of all terms except the one with all the indices different, the Fourier transform of $O_{\beta }(t,0,t,0)-D_{\beta }(t,0,t,0)$. The curve is very similar to the two-point curve, and is hence unimportant for the Lyapunov times. The system of size $N=12$ and we took $\mathrm{\Delta }=0.3$ and ${\epsilon }_{+}$ in the middle of the spectrum.