Numerical study of fermion and boson models with infinite-range random interactions

TL;DR

The paper investigates fermionic and bosonic SYK models with infinite-range random interactions using a combination of large-N saddle-point analysis, high-temperature expansions, and exact diagonalization at finite N. It demonstrates that the fermionic model exhibits a non-Fermi liquid with finite ground-state entropy, and that ED results converge toward the large-N predictions for Green's functions, entropy, and entanglement, validating the analytic approach. In contrast, the hardcore boson SYK model shows spin-glass order, signaling replica-symmetry breaking in the large-N limit. The study also explores OTOCs to characterize chaos, highlighting the role of system size and temperature, and provides insights into the distinct many-body physics of fermionic versus bosonic SYK models. Together, these results bridge finite-size numerics with analytic large-N theory and clarify the presence of spin-glass tendencies in the bosonic variant.

Abstract

We present numerical studies of fermion and boson models with random all-to-all interactions (the SYK models). The high temperature expansion and exact diagonalization of the $N$-site fermion model are used to compute the entropy density: our results are consistent with the numerical solution of $N=\infty$ saddle point equations, and the presence of a non-zero entropy density in the limit of vanishing temperature. The exact diagonalization results for the fermion Green's function also appear to converge well to the $N=\infty$ solution. For the hard-core boson model, the exact diagonalization study indicates spin glass order. Some results on the entanglement entropy and the out-of-time-order correlators are also presented.

Figures (10)

• Figure 1: Figure, adapted from Ref sachdev1992gapless, showing the imaginary part of Green's function multiplied by $\sqrt{\omega}$ as a function of $\omega$ at particle-hole symmetric point $\theta=0$. Our definition of the Green's function, Eq. (\ref{['Green function']}), differs by a sign from Ref. sachdev1992gapless.
• Figure 2: Entropy computation from exact large $N$ EOM and HTE: at hight temperature, all approaches the infinite temperature limit $S/N=\ln 2$. HTE result fit the exact result quite well for $T/J>0.6$.
• Figure 3: Imaginary part of the Green's function in real frequency space from large $N$ and exact diagonalization. The inset figure is zoomed in near $\omega=0$.
• Figure 4: The difference of integrated spectral function between ED at different N and large N result. The difference appears to be tending to 0 as $N$ approaches infinity.
• Figure 5: Thermal entropy computation from ED, and large $N$. At high temperature, all results the infinite temperature limit $S/N=\ln 2$. At low temperature, all ED results go to zero, but do approach the $N = \infty$ results with increasing $N$. Note that the limits $N \rightarrow \infty$ and $T \rightarrow 0$ do not commute, and the non-zero entropy as $T \rightarrow 0$ is obtained only when the $N\rightarrow \infty$ is taken first