Supersymmetric Sachdev-Ye-Kitaev models

We discuss a supersymmetric generalization of the Sachdev-Ye-Kitaev (SYK) 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 $\mathcal{N}=1$ model with a single supercharge has unbroken supersymmetry at large $N$, but nonperturbatively 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 $\mathcal{N}=2$ 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.

Figure 1

Thermal entropy obtained by numerically solving the large $N$ equations of motion (2.11) and (2.12). At high temperatures we have just the log of the dimension of the Hilbert space, $\frac{S}{N}=\frac{1}{2}\log 2$. The zero temperature entropy is approximately $\frac{S}{N}\sim 0.2745+0.0005$, where the error is estimated by the convergence of the FFT (fast Fourier transform) algorithm. The analytical result $\frac{S}{N}=\frac{1}{2}\log [2\cos \frac{\pi }{6}]$ (2.33) 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: blue solid line is $G_{\psi \psi }(\tau )$, red dotted-dashed line is the conformal solution $G_{\psi \psi }^c(\tau )$ in Eq. (3.2). Right: blue solid line is $G_{bb}(\tau )$, red dotted-dashed line is the conformal solution $G_{bb}^c(\tau )$ in Eq. (3.2).

Figure 5

Imaginary time Green’s function at finite temperature for $N=20$ Majorana fermions averaged over 100 samples. Left: $G_{\psi \psi }(\tau )$; right: $G_{bb}(\tau )$. Solid lines are the exact diagonalization result; dashed lines are conformal results as in Eq. (3.3); dotted line are large $N$ result by numerically solving Eqs. (2.11) and (2.12). Different colors correspond to different interaction strength: blue is $\beta J=5$; red is $\beta J=20$, and black is $\beta J=200$.

Figure 6

(a) Diagram contributing to a correction to the fermion propagator. Full lines are fermions and dotted lines are bosons. (b) Correction to the boson propagator. (c) A simple ladder diagram contributing to the four-point function in the fermionic channel, where the intermediate state obtained when we cut the ladder is a fermion. (d)–(f) Diagrams contributing in the bosonic channel, with either a pair of bosons or a pair of fermions. The full ladders are obtained by iterating these diagrams. These are the diagrams for $\widehat{q}=3$ and they look similar in the general case.