Effective dimension, level statistics, and integrability of Sachdev-Ye-Kitaev-like models*

Recently, quantum many-body systems exhibiting fast scrambling (delocalization of quantum information) have attracted much attention. A well-known example is the Sachdev-Ye-Kitaev (SYK) model, describing random all-to-all four-body interactions among fermions. Here, we introduce a variant of the SYK model, which we refer to as the Wishart SYK model [1]. The model includes the original SYK model without quenched disorder as a special case.
We investigate the Wishart SYK model for complex fermions and that for hard-core bosons. In both cases, the ground state of the model is massively degenerate. We numerically study the out-of-time-ordered correlators (OTOC) and level statistics. Then we find that the OTOC in the fermionic case exhibits large temporal fluctuations at late times, which is attributed to a small effective dimension of the initial state. We also show that the level statistics of the fermionic (bosonic) Wishart SYK model obeys the Poisson (GOE or GUE) distribution. This is consistent with the integrability of the fermionic model, which can be proved by mapping it to a known integrable model.
[1] E. Iyoda, H. Katsura, and T. Sagawa, Phys. Rev. D 98, 086020 (2018).