Field theory approach to eigenstate thermalization in random quantum circuits

Eigenstate thermalization hypothesis (ETH) explains the emergence of the statistical mechanics behavior in isolated quantum systems, and relates thermalization to the statistics of energy eigenstates. In this work, we investigate the statistics of quasienergy eigenstates in Floquet random quantum circuits where each qudit is coupled with all its neighboring qudits by independent Haar random unitaries at arbitrary and distinct substeps. Within the supersymmetric sigma-model framework, we prove that the correlation function of the quasienergy eigenstates agrees with that of the circular unitary ensemble to the leading order in the Hilbert space dimension N. This result shows that the matrix elements of local traceless operators in the quasienergy eigenbasis have small variance of the order of 1/N, consistent with ETH. Moreover, the eigenstate correlation function allows for the investigation of the temporal relaxation of physical observables to their thermal expectation values.