Supersymmetry in the Standard Sachdev-Ye-Kitaev Model

Supersymmetry is a powerful concept in quantum many-body physics. It helps to illuminate ground-state properties of complex quantum systems and gives relations between correlation functions. In this Letter, we show that the Sachdev–Ye–Kitaev model, in its simplest form of Majorana fermions with random four-body interactions, is supersymmetric. In contrast to existing explicitly supersymmetric extensions of the model, the supersymmetry we find requires no relations between couplings. The type of supersymmetry and the structure of the supercharges are entirely set by the number of interacting Majorana modes and are thus fundamentally linked to the model’s Altland–Zirnbauer classification. The supersymmetry we uncover has a natural interpretation in terms of a one-dimensional topological phase supporting Sachdev–Ye–Kitaev boundary physics and has consequences away from the ground state, including in $q$-body dynamical correlation functions.

Figure 1

$q$-body time-dependent correlation function at infinite temperature, averaged over an ensemble of up to $2^{16}$ Gaussian distributed $J_{qrst}$, for classes (a) D, (b) C, (c) DIII, and (d) CI. The different colors denote different $k$ and $q$, cf. inset in (a), the dashed lines represent $q=3$ and the solid lines $q=5$. The ramp shape follows the Dyson index $\beta$ and hence links to the number of supercharges. The long-time plateau ${\overline{C}}_{q,\infty }$ is studied in Fig. 2. Error bars are either smaller than the linewidth (for small $k$) or smaller than the disorder-induced fluctuations of the lines (for large $k$).

Figure 2

Normalized plateau ${\overline{C}}_{q,\infty }M/4$ of the $q$-body correlation function, averaged over an ensemble of up to $2^{14}$ Gaussian distributed $J_{qrst}$. The color encodes the number $k$ of Majoranas, cf. panels (d) and (e). In all classes, ${\overline{C}}_{\infty }M/4$ alternates with $q$ approximately as predicted in Eq. (13); the agreement is excellent when $(\genfrac{}{}{0ex}{}{k}{q})/(\genfrac{}{}{0ex}{}{k}{\lfloor k/2\rfloor })$ is close to one. In panel (d), we show that ${\overline{C}}_{\infty }M/(4c)$ [with $c$ the random matrix expectation based on Eq. (13)] increases as a function of $k$ but with a rate that decreases upon increasing $q$ [panel (e)]. Statistical error bars are smaller than the marker size.