Field Theory Approach to Eigenstate Thermalization in Random Quantum Circuits

Yunxiang Liao; Victor Galitski

Abstract

We use a field-theoretic method to explore the statistics of eigenfunctions of the Floquet operator for a large family of Floquet random quantum circuits. The correlation function of the quasienergy eigenstates is calculated and shown to exhibit random matrix circular unitary ensemble statistics, which is consistent with the analogue of Berry’s conjecture for quantum circuits. This quantity determines all key metrics of quantum chaos, such as the spectral form factor and thermalizing time-dependence of the expectation value of an arbitrary observable. It also allows us to explicitly show that the off-diagonal matrix elements of local operators possess a variance exponentially small in system size, an essential aspect of the eigenstate thermalization hypothesis (ETH). These results provide analytical evidence for ETH in the family of Floquet random quantum circuits. An outstanding open question for this and most of other sigma-model calculations is a mathematically rigorous proof of the validity of the saddle-point approximation in the large-N limit.

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