Universality and Thouless energy in the supersymmetric Sachdev-Ye-Kitaev Model

TL;DR

The paper analyzes the supersymmetric SYK model (q=3) to understand how discrete chiral and superconducting symmetries affect spectral properties and chaos. Using a Q-Hermite density, resolvent methods, and exact diagonalization, it shows that near E ≈ 0 the average density grows exponentially with N while the chiral condensate vanishes, and the microscopic spectral density is universal and described by appropriate chiral RMT ensembles within a Thouless energy that scales linearly with N. Level statistics and the spectral form factor reveal ergodic behavior consistent with random matrix theory at short energies/times, with deviations beyond the Thouless energy characterized by a quadratic growth of the number variance and a 1/t^2 decay of g(t) before the Thouless time. These results bolster the view that quantum black holes are ergodic and can be classified by random matrix theory, while clarifying the scales at which universal behavior emerges in supersymmetric holographic models.

Abstract

We investigate the supersymmetric Sachdev-Ye-Kitaev (SYK) model, $N$ Majorana fermions with infinite range interactions in $0+1$ dimensions. We have found that, close to the ground state $E \approx 0$, discrete symmetries alter qualitatively the spectral properties with respect to the non-supersymmetric SYK model. The average spectral density at finite $N$, which we compute analytically and numerically, grows exponentially with $N$ for $E \approx 0$. However the chiral condensate, which is normalized with respect the total number of eigenvalues, vanishes in the thermodynamic limit. Slightly above $E \approx 0$, the spectral density grows exponential with the energy. Deep in the quantum regime, corresponding to the first $O(N)$ eigenvalues, the average spectral density is universal and well described by random matrix ensembles with chiral and superconducting discrete symmetries. The dynamics for $E \approx 0$ is investigated by level fluctuations. Also in this case we find excellent agreement with the prediction of chiral and superconducting random matrix ensembles for eigenvalues separations smaller than the Thouless energy, which seems to scale linearly with $N$. Deviations beyond the Thouless energy, which describes how ergodicity is approached, are universality characterized by a quadratic growth of the number variance. In the time domain, we have found analytically that the spectral form factor $g(t)$, obtained from the connected two-level correlation function of the unfolded spectrum, decays as $1/t^2$ for times shorter but comparable to the Thouless time with $g(0)$ related to the coefficient of the quadratic growth of the number variance. Our results provide further support that quantum black holes are ergodic and therefore can be classified by random matrix theory.

Figures (6)

• Figure 1: Average spectral density of the supercharge $Q$ (with Hamiltonian $H = Q^2$) of the SYK model Eq. (\ref{['hami']}) for $q=3$ and $N=34$. We find excellent agreement between analytical Q-Hermite result Eq. (\ref{['dqher']}) and the approximation Eq. (\ref{['rhoasym']}). However, agreement with the exact diagonalization results (solid dots), employing $5\times10^6$ eigenvalues, is only qualitative.
• Figure 2: Resolvent $iG(is)$ Eq. (\ref{['resd']}) for the SYK model for $q=3$ as a function of the parameter $s$ for different value of $N$ compared with the analytical result Eq. (\ref{['resa']}). After a finite size scaling analysis, we find that the $N$-dependence of $\log G(0)$ is well fitted by $\log G(0) \approx -0.0695 N$ in agreement with the value of the q-dependent part of the zero temperature entropy (\ref{['entropy']}). We find excellent agreement with the analytical expression Eq. (\ref{['resa']}) based on the Q-Hermite density except for very small $s$.
• Figure 3: Microscopic spectral density $\rho_M$ near $E=0$ in units of the mean level spacing $\Delta$. We compare $\rho_M$ for the supercharge of the SYK model with $q=3$, see Eq. (\ref{['hami']}), with the predictions of random matrix theory (solid curves). As was shown in li2017, SYK models with different values of $N$ have different discrete global symmetries and therefore must be compared to random matrix ensembles belonging to the corresponding universality classes. For instance, $N = 18$ and $N=26$ belongs to the $DIII$ ensemble, $N = 20$ and $N=28$ to the chiral symplectic ensemble (chGSE), $N = 24$ to the chiral orthogonal ensemble (chGOE), and $N = 22$ to the $CI$ ensemble. Interestingly, we observe an excellent agreement with the random matrix prediction for the first few low lying eigenvalues. The oscillations for the microscopic spectral density of the SYK model in the case of $DIII$ and chGSE case (bottom row), are eventually washed out as the effect of discrete symmetries is strong only very close to $E =0$. Analytical expressions for all random matrix ensembles can be found in Refs. verbaarschot1993nagao1995ivanov2002.
• Figure 4: Number variance $\Sigma^2(L)$ Eq. (\ref{['eq:nv']}) for small $L$ and different $N$'s. Excellent agreement with random matrix theory is observed for sufficiently small $L$. This is an indication that SYK for odd $q$ is also quantum chaotic for sufficiently long times (small energies). The point in which deviations are observed, especially its scaling with $N$, provides useful information on the quantum dynamics prior to complete relaxation. Our numerical results are consistent with a point of departure that roughly scales with $N$.
• Figure 5: Number variance $\Sigma^2(L)$ Eq. (\ref{['eq:nv']}) for $N = 30$ and $N=34$. Deviations from the logarithmic growth predicted by random matrix theory are clearly observed. We have fitted the growth (solid curves) to a function $\sim L^\alpha$. For $N$ sufficiently large $\alpha \approx 2$ ($\alpha \approx 1.99$ for $N=34$ and $\alpha \approx 1.98$ for $N=30$). For sufficiently small $N$, we find a bit slower growth but we believe that this is a finite size effect of no physical relevance.