Abstract
We investigate two sparse Sachdev-Ye-Kitaev (SYK) systems coupled by a bilinear term as a holographic quantum mechanical description of an eternal traversable wormhole in the low temperature limit. Each SYK system consists of N Majorana fermions coupled by random q-body interactions. The degree of sparseness is captured by a regular hypergraph in such a way that the Hamiltonian contains exactly k N independent terms. We improve on the theoretical understanding of the sparseness property by using known measures of hypergraph expansion. We show that the sparse version of the two coupled SYK model is gapped with a ground state close to a thermofield double state. Using Krylov subspace and parallelization techniques, we simulate the system for q = 4 and q = 8. The sparsity of the model allows us to explore larger values of N than the ones existing in the literature for the all-to-all SYK. We analyze in detail the two-point functions and the transmission amplitude of signals between the two systems. We identify a range of parameters where revivals obey the scaling predicted by holography and signals can be interpreted as traversing the wormhole.
Highlights
Some of the main features of the two coupled SYK system are that it has a ground state close to a thermofield double (TFD) with a temperature that depends on the strength of the coupling [8, 13, 14] and that there is an energy gap above the ground state that plays an important role in the characterization of the dynamics of the model
While it has been understood that the ground state of the two coupled SYK system is approximately the TFD, it is not obvious that the same happens when we introduce sparsity in the SYK model
In this work we investigated a variant of the SYK model defined on a sparse regular hypergraph
Summary
The Sachdev-Ye-Kitaev (SYK) model [1, 2] has been a successful toy model of lower dimensional quantum black holes. The model was proposed by Kitaev inspired by a spin quantum Heisenberg magnet with Gaussian distributed interactions previously studied by Sachdev and Ye [27]. The gravitational description is given by a nearly AdS2 solution of two-dimensional JT gravity [11]. A variant of the SYK model, dubbed sparse SYK, was recently proposed as an effective theory for the all-to-all SYK with the advantage of allowing for more efficient computer simulation [6] (see [7]). We will briefly review the sparse SYK model and some basic definitions pertaining hypergraphs which will be helpful to describe the structure of the interactions in the model
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