Leading and Beyond Leading-Order Spectral Form Factor in Chaotic Quantum Many-Body Systems Across All Dyson Symmetry Classes

Abstract

We show the emergence of random matrix theory (RMT) spectral correlations in the chaotic phase of generic, periodically kicked, interacting, quantum many-body systems by analytically calculating the spectral form factor (SFF), 𝐾⁡(𝑡), up to two leading orders in time, 𝑡. We explicitly consider the presence or absence of time-reversal (𝒯) symmetry to investigate all three Dyson symmetry classes, namely, the circular orthogonal ensemble (COE) for 𝒯-invariant systems with 𝒯2 =1, the circular unitary ensemble (CUE) in the absence of 𝒯 symmetry, and the circular symplectic ensemble (CSE) for 𝒯-invariant systems with 𝒯2 =−1. Our derivation applies the random phase approximation to perform ensemble averages, and we give its justification for generic many-body interactions. We express the SFF as a sum of contributions of certain diagrams. While the number of diagrams that contribute to the SFF in the first and second orders in 𝑡 for the COE and CUE classes is finite, the number of such diagrams is exponentially large in 𝑡 for the CSE class. Thus, we have developed a new diagrammatic technique that uses reduced diagrams to incorporate contributions from many diagrams. For the COE class, we show that beyond Thouless time 𝑡*, the SFF takes the form 𝐾⁡(𝑡) ≃2⁢𝑡 −2⁢𝑡2/𝒩 up to second order in time, where 𝒩 is the Hilbert space dimension. For the CUE class, we show that 𝐾⁡(𝑡) ≃𝑡 beyond 𝑡*, and there is no universal term in the second order, unlike the COE case. For the CSE class, we show that 𝐾⁡(𝑡) ≃2⁢𝑡 +2⁢𝑡2/𝒩 up to two orders in time beyond 𝑡*. The above findings of the SFF up to second order in time are in agreement with the RMT predictions for these classes. In all three cases, the scaling of 𝑡* with system size 𝐿 is determined by the eigenvalues of a doubly stochastic matrix ℳ. For strongly interacting fermionic chains, ℳ is 𝑆⁢𝑈⁡(2) invariant in all three cases, leading to 𝑡* ∝𝐿2 in the presence of 𝑈⁡(1) symmetry. In the absence of 𝑈⁡(1) symmetry, we find 𝑡* ∝𝐿0, due to the gapped, nondegenerate, second-largest eigenvalue of ℳ, or 𝑡* ∝ln⁡(𝐿), due to the gapped second-largest eigenvalue with degeneracy proportional to 𝐿𝜁, 𝜁 >0. Our calculation of the SFF is plausible in higher space dimensions as well, where similar system-size scalings of 𝑡* can be obtained.