Approximating the Sachdev-Ye-Kitaev model with Majorana wires

Abstract

The Sachdev-Ye-Kitaev (SYK) model describes a collection of randomly interacting Majorana fermions that exhibits profound connections to quantum chaos and black holes. We propose a solid-state implementation based on a quantum dot coupled to an array of topological superconducting wires hosting Majorana zero modes. Interactions and disorder intrinsic to the dot mediate the desired random Majorana couplings, while an approximate symmetry suppresses additional unwanted terms. We use random matrix theory and numerics to show that our setup emulates the SYK model (up to corrections that we quantify) and discuss experimental signatures.

Figures (2)

• Figure 1: (a) Device that approximates the SYK model using topological wires interfaced with a 2D quantum dot. The dot mediates disorder and four-fermion interactions among Majorana modes $\gamma_{1,\ldots,N}$ inherited from the wires, while Majorana bilinears are suppressed by an approximate time-reversal symmetry. (b) Energy levels pre-hybridization. The dot-Majorana hybridization energy $\lambda$ is large compared to $N \delta \epsilon_{\rm typ}$, where $N$ is the number of Majorana modes and $\delta \epsilon_{\rm typ}$ is the typical dot level spacing; this maximizes leakage into the dot. (c) Energy levels post-hybridization. The $N$ absorbed Majorana modes enhance the energy $\epsilon$ to the next excited dot state via level repulsion; four-Majorana interactions occur on a scale $J < \epsilon$.
• Figure 2: (a) Average absorption of Majorana wavefunctions into the dot versus the hybridization strength $\lambda$ with $N = 16$ zero modes. Inset: probability density for a Majorana wavefunction swallowed and randomized by the dot of size $51 \times 51$. (b) Enhanced level repulsion of the first excited dot state $\epsilon$ by $N$ absorbed Majorana modes; cf. Figs. \ref{['Setup']}(b) and (c). (c) Histogram of $J_{ijkl}$ couplings obtained from local current-current interactions on a dot of size $21 \times 21$, together with a Gaussian fit (solid line). (d) Scaling of the variance $\propto \bar{J}^2$ of these couplings versus $N_{\rm dot}$.