Spectral and thermodynamic properties of the Sachdev-Ye-Kitaev model

We study spectral and thermodynamic properties of the Sachdev-Ye-Kitaev model, a variant of the $k$-body embedded random ensembles studied for several decades in the context of nuclear physics and quantum chaos. We show analytically that the fourth- and sixth-order energy cumulants vanish in the limit of a large number of particles $N\to \infty$, which is consistent with a Gaussian spectral density. However, for finite $N$, the tail of the average spectral density is well approximated by a semicircle law. The specific heat coefficient, determined numerically from the low-temperature behavior of the partition function, is consistent with the value obtained by previous analytical calculations. For energy scales of the order of the mean level spacing we show that level statistics are well described by random matrix theory. Due to the underlying Clifford algebra of the model, the universality class of the spectral correlations depends on $N$. For larger energy separations we identify an energy scale that grows with $N$, reminiscent of the Thouless energy in mesoscopic physics, where deviations from random matrix theory are observed. Our results are a further confirmation that the Sachdev-Ye-Kitaev model is quantum chaotic for all time scales. According to recent claims in the literature, this is an expected feature in field theories with a gravity dual.

Figure 1

The fourth and sixth normalized energy cumulants related to the Hamiltonian (1) as a function of the system size $N$. The circles correspond to the numerical results obtained by exact diagonalization after the spectral and ensemble average. At least a total of $5\times 10^5$ eigenvalues were employed for each $N$. The solid line is the analytical prediction for the fourth [left; Eq. (8)] and sixth [right; Eq. (9)] cumulant.

Figure 2

Spectral density $\rho (E)$ as a function of the energy $E$. The solid line is the analytical prediction valid in the $N\to \infty$ limit. Circles are the numerical spectral density for the largest size $N=34$ for which we can obtain all eigenvalues of the Hamiltonian. Except for the tails, the agreement with the numerical results is very good.

Figure 3

Ensemble average of the smallest eigenvalue as a function of the system size $N$. For $N\gg 1$ we observe that it decreases linearly with $N$. This is an expected feature for a system of $N$ interacting fermions.

Figure 4

The fitted values of $\log a$ (left) and $b$ (right), defined in Eq. (10), versus $N$. The lines are the best fits to the data. In the right figure, only the points for $N\ge 28$ have been used for the fitting.

Figure 5

The specific heat as a function of the temperature for $N=28$ (left) and $N=36$ (right). The red dots represent the numerical result for the SYK model when specific heat is calculated relative to the ensemble average [see Eq. (20)], while the blue dots show the results where the free energy is calculated relative to the average energy ${\overline{E}}_p$ for each realization of the ensemble [see Eq. (24)]. The blue curve is a linear fit to the blue dots on the interval [0.0075, 0.015] for $N=36$ and a cubic polynomial fit on [0.025, 0.05] for $N=28$.

Figure 6

The exponent of the prefactor ${\beta }^{-q(N)}$ of the partition function [Eq. (18)] versus $N$ (left) and the specific heat coefficient $c(N)$ [Eq. (25)] versus $N$ (right). The exponent $q(N)$ and the specific heat coefficient $c(N)$ are fitted by a constant.

Figure 7

Level spacing distribution $P(s)$ [Eq. (29)]. Numerical results $N=28$ (circles), $N=34$ (squares), and $N=36$ (diamonds) are in excellent agreement with the predictions of random matrix theory (solid lines) corresponding to the Gaussian unitary ensemble (GUE) for $N=34$ and the Gaussian symplectic ensemble (GSE) for $N=28$, 36. We note that while for $N=28$, 34 we have taken about 70% of the available spectrum around the center of the band; for $N=36$, where we cannot diagonalize the full Hamiltonian, we consider only a total of about 15000 different eigenvalues close to the ground state from 15 disorder realizations. The universality class is controlled by the type of allowed representations of the Clifford algebra of the Majorana fermions, which is sensitive to $N$ (see Table 1 and the main text for more details). These results clearly show that the SYK model has quantum chaotic features even for large times $s\sim 1$ of the order of the Heisenberg time.

Figure 8

Level spacing distribution $P(s)$ [Eq. (29)], for $N=16$ (circles) and $N=32$ (squares). For both dimensions, the Clifford algebra admits a real representation so the expected universality class is that of the Gaussian orthogonal ensemble (GOE). Indeed, the numerical results (symbols) are in excellent agreement with the GOE prediction. Interestingly, despite the vast difference in size, we do not observe substantial deviations from the GOE prediction, even for $N=16$, where mesoscopic fluctuations are expected to be stronger.

Figure 9

Number variance ${\mathrm{\Sigma }}^2(L)$ [Eq. (31)], for $N=28$ (diamonds), $N=32$ (circles), $N=34$ (squares), which belong to the GSE, GOE and GUE universality classes, respectively (solid lines), as a function of the width $L$ of the energy interval, corresponding to a spectral window with $L$ eigenvalues on average. For sufficiently large $L$, we start observing deviations from the random matrix theory predictions.

Figure 10

Number variance ${\mathrm{\Sigma }}^2(L)$ [Eq. (31)], for $N=22$ and $N=34$, both corresponding to the GUE universality classes. We observe that deviations from the random matrix prediction occur much earlier for the smaller dimension $N=22$. This suggests that the observed deviations are due to mesoscopic fluctuations in a way reminiscent to the existence of a Thouless energy in the system. It is also an indication that the system is not ergodic and chaotic for sufficiently short times, an expected feature [20, 46] in field theories with a gravity dual.

Figure 11

Contractions contributing to the fourth- and sixth-order cumulants.