Supersymmetric SYK models

TL;DR

This work introduces and analyzes supersymmetric generalizations of the SYK model with N Majorana fermions, deriving the large-N effective action and solving for IR scaling dimensions under SUSY constraints. The N=1 case yields Δ_ψ = 1/6, Δ_b = 2/3 with unbroken SUSY at large N but nonperturbative SUSY breaking at finite N, while the N=2 case preserves SUSY and exhibits a large ground-state degeneracy captured by a generalized Witten index. Low-energy dynamics are governed by a super-Schwarzian action, and the four-point function reveals a ladder spectrum of bosonic and fermionic operators consistent with supersymmetric multiplets, including special modes from residual IR symmetries. Overall, the paper extends the SYK paradigm to SUSY, highlighting rich IR structures, entropy calculations, and potential implications for holographic duals of SUSY black holes.

Abstract

We discuss a supersymmetric generalization of the Sachdev-Ye-Kitaev model. These are quantum mechanical models involving $N$ Majorana fermions. The supercharge is given by a polynomial expression in terms of the Majorana fermions with random coefficients. The Hamiltonian is the square of the supercharge. The ${\cal N}=1$ model with a single supercharge has unbroken supersymmetry at large $N$, but non-perturbatively spontaneously broken supersymmetry in the exact theory. We analyze the model by looking at the large $N$ equation, and also by performing numerical computations for small values of $N$. We also compute the large $N$ spectrum of "singlet" operators, where we find a structure qualitatively similar to the ordinary SYK model. We also discuss an ${\cal N}=2$ version. In this case, the model preserves supersymmetry in the exact theory and we can compute a suitably weighted Witten index to count the number of ground states, which agrees with the large $N$ computation of the entropy. In both cases, we discuss the supersymmetric generalizations of the Schwarzian action which give the dominant effects at low energies.

Figures (6)

• Figure 1: Thermal entropy obtained by numerically solving the large $N$ equations of motion (\ref{['EOM12']})(\ref{['EOM34']}). At high temperatures we have just the log of the dimension of the Hilbert space, ${S \over N } = {1 \over 2 } \log 2$. The zero temperature entropy is approximately ${S \over N } \sim 0.2745+0.0005$, where the error is estimated by the convergence of the FFT (Fast Fourier Transform) algorithm. The analytical result ${S \over N } =\frac{1}{2}\log{\left[2\cos{\frac{\pi}{6}}\right]}$ (\ref{['GSEntropy']}) also lies in this range .
• Figure 2: Imaginary time Green's function at $T=0$ for $N=24$ Majorana fermions averaged over 100 samples. The blue solid line is $G_{bb}(\tau)$, and the pink dashed line is $-\partial_{\tau}G_{\psi}(\tau)$.
• Figure 3: Ground state energy as a function of $N$ in a log-linear plot, where we have averaged over 100 samples. The plot is compatible with an exponential decrease of $E_0$ with $N$. Notice also the structure in $E_0$ dependent on $N$ (mod 8).
• Figure 4: Imaginary time Green's function at $T=0$ for $N=24$ Majorana fermions averaged over 100 samples. Left panel: blue solid line is $G_{\psi \psi}(\tau)$, red dotted-dashed line is the conformal solution $G_{\psi \psi}^c(\tau)$ in Eq. (\ref{['Gbc']}); right panel: blue solid line is $G_{bb}(\tau)$, red dotted-dashed line is the conformal solution $G_{bb}^c(\tau)$ in Eq. (\ref{['Gbc']}).
• Figure 5: Imaginary time Green's function at finite temperature for $N=20$ Majorana fermions averaged over 100 samples. Left panel is $G_{\psi\psi}(\tau)$ while right panel is $G_{bb}(\tau)$. Solid lines are the exact diagonalization result; dashed lines are conformal results as in Eq. (\ref{['GpsiGb']}); dotted line are large $N$ result by numerically solving Eq. (\ref{['EOM12']}) and Eq. (\ref{['EOM34']}). Different colors correspond to different interaction strength: blue one is $\beta J=5$; red one is $\beta J=20$ and black one is $\beta J=200$.