Does the SYK model have a spin glass phase?

TL;DR

The paper argues that the SYK model does not exhibit a spin glass phase at low temperatures. It combines analytic replica-off-diagonal-mode calculations in both nearly-conformal and Schwarzian limits with numerical tests of spectral statistics and ground-state fluctuations, finding no spin-glass instability and RMT-consistent level statistics down to low energies. The numerical results also show a Gaussian distribution of the ground-state energy, in line with random-matrix expectations. Overall, the findings support the paramagnetic, chaotic phase of SYK across accessible temperatures and large-$N$ regimes, consistent with its holographic interpretation.

Abstract

We argue that the Sachdev-Ye-Kitaev model has no spin glass phase, based on calculations involving both the nearly-conformal limit and the strongly-coupled Schwarzian limit of the model. This conclusion is supported by numerical computations of eigenvalue statistics with up to 46 Majorana fermions. In addition, we find numerically that the distribution of the ground state energy is Gaussian.

Figures (8)

• Figure 1: The density of states for SYK near the edge of the spectrum, with $N=42$ Majorana fermions and 800 realizations, each with about 1,200 eigenvalues at each edge of the spectrum. (a) The density of states. (b) The density of states, with the energies of each realization shifted by its respective ground state energy. The fit is to a power law for the range $E-E_0<0.01$, and gives $\rho \sim (E-E_0)^{0.49}$. (The best-fit exponent varies between $0.4-0.6$ depending on the choice of range.)
• Figure 2: The Spectral Form Factor (denoted $g$ in Cotler:2016fpe) with $N=42$ Majorana fermions and $\beta=50$, using the same data as in Figure \ref{['fig:spec']}.
• Figure 3: Level spacing statistics for the edge of the SYK spectrum, with $N=46$ Majorana fermions and 355 realizations. (a) The spacing distribution for the two lowest levels, compared with the RMT prediction for the corresponding GUE ensemble. (b) The distribution of $\log(r_n)$ (described in the text) computed over the lowest 20 energy levels, compared with the RMT prediction (blue) and with the prediction for uncorrelated energies (gray).
• Figure 4: The ground state distribution for SYK with $N=32$ Majorana fermions, with statistics collected over $10^4$ realizations. Solid lines show the Gaussian and Tracy-Widom distributions with mean and variance chosen to fit the data.
• Figure 5: Ground state mean and variance as a function of $N$, computed over $10^4$ realizations for each value of $N$. (a) The mean ground state energy, with a linear fit to $\langle E_0 \rangle = -0.043 N - 0.12$. (b) The variance of the ground state energy, with a power-law fit to $\mathrm{Var}(E_0) \sim N^{-3.43}$.