Eigenstate Thermalization Hypothesis and Free Probability

Quantum thermalization is well understood via the eigenstate thermalization hypothesis (ETH). The general form of ETH, describing all the relevant correlations of matrix elements, may be derived on the basis of a “typicality” argument of invariance with respect to local rotations involving nearby energy levels. In this Letter, we uncover the close relation between this perspective on ETH and free probability theory, as applied to a thermal ensemble or an energy shell. This mathematical framework allows one to reduce in a straightforward way higher-order correlation functions to a decomposition given by minimal blocks, identified as free cumulants, for which we give an explicit formula. This perspective naturally incorporates the consistency property that local functions of ETH operators also satisfy ETH. The present results uncover a direct connection between the eigenstate thermalization hypothesis and the structure of free probability, widening considerably the latter’s scope and highlighting its relevance to quantum thermalization.

Figure 1

Impact of the local rotational invariance of $A_{ij}$ on the correlations between three matrix elements. The operator $A$ in the energy eigenbasis is depicted as a random matrix with a band structure. To each matrix element $A_{ij}$ is associated with a “small” $U$ (box on the diagonal) which acts as a pseudorandom unitary matrix. Matrix elements with different indices (a) are characterized by different $U$ and their average vanishes. When the indices are repeated on a loop (b) the $U$ appear in pairs and yield a finite result.

Figure 2

ETH diagrams (a) and noncrossing partitions (b) for $q=4$. (a1)–(a6) Loop and cactus diagrams that contribute to ETH correlators. The arrow indicates the presence of a time dependence. With $\times [n]$ we indicate that there are $n$ cyclic permutations. (a7) Noncactus diagram. (b1)–(b6) Noncrossing partitions for $q=4$. Each of the blocks contributes with a free cumulant $k_n$, where $n$ is the number of points in the block. For completeness, we also represent the crossing partition after the dashed line.