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

We present numerical studies of fermion and boson models with random all-to-all interactions (the Sachdev-Ye-Kitaev 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 nonzero 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.

Figure 1

Plot, adapted from Ref. [1], showing the imaginary part of the Green's function multiplied by $\sqrt{\omega }$ as a function of $\omega$ at the particle-hole symmetric point $\theta =0$. Our definition of the Green's function, Eq. (16), differs by a sign from Ref. [1].

Figure 2

Entropy computation from exact large-$N$ EOM and HTE: at high temperature, all approach the infinite-temperature limit $S/N=ln2$. HTE result fits 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 approach the infinite-temperature limit $S/N=ln2$. At low temperature, all ED results go to zero, but do approach the $N=\infty$ results with increasing $N$. Note that the limits $N\to \infty$ and $T\to 0$ do not commute, and the nonzero entropy as $T\to 0$ is obtained only when the $N\to \infty$ is taken first.

Figure 6

The difference of integrated thermal entropy between ED at different N and large-N result. The difference goes to 0 as $1/N$ approaches 0.

Figure 7

Entanglement entropy for the ground state. We divide the system into two subsystems, A and B, and we trace out part B and calculate the entropy for the reduced density matrix ${\rho }_A$. The $x$ axis is the size of subsystem A.

Figure 8

OTOC as a function of time at infinite temperature with different interaction strength $J=1$ and $J=2$. Here the total system size $N=7$.

Figure 9

OTOC as a function of time at different temperature with interaction strength $J=1$. Here the total system size $N=7$.

Figure 10

Imaginary part of Green's function for hard-core boson and fermion model. The peak near the center gets much higher in the boson model when system size gets larger. The inset figure is zoomed in near $\omega =0$.