Numerical study of fermion and boson models with infinite-range random interactions

Abstract

We present numerical studies of fermion and boson models with random all-to-all interactions (the SYK models). The high temperature expansion and exact diagonalization of the N -site fermion model are used to compute the entropy density: our results are consistent with the numerical solution of N = ∞ saddle point equations, and the presence of a non-zero entropy density in the limit of vanishing temperature.The exact diagonalization results for the fermion Green's function also appear to converge well to the N = ∞ solution. For the hard-core boson model, the exact diagonalization study indicates spin glass order. Some results on the entanglement entropy and the out-of-time-order correlators are also presented.

Higher dimensional generalizations of the SYK model

“…This is simply the sum over the replicated action obtained for each site separately, for a direct comparison see for example the discussion in [6,15]. Note that the saddle point equations set…”

“…The quadratic term for the fermions is interpreted as a hopping term in this language). However, in our case using the ordinary field theory 6 The actual integral we get has the contour from 0 to ∞ at an angle θ = − πi 2(2γ+1) in the complex plane. Since there are no poles and the contour at ∞ vanishes for γ > 0, we can rotate it back to real axis.…”

“…In particular, studying the 4-point function would allow us to probe the chaotic behaviour of the system and the spatial spread of chaos, as manifested by the butterfly velocity. It might also be interesting to compute entanglement entropy (perhaps numerically as in [6]). More ambitiously, one can hope that subtle issues like the information paradox [30,31] might be clearer if one has solvable models capturing the relevant physics of higher dimensional versions of the AdS/CFT correspondence.…”

“…The Sachdev-Ye-Kitaev (SYK) model [1,2] has received much attention recently [3][4][5][6][7][8]. It is a simple model with solvable aspects which exhibits interesting connections to quantum chaos [9][10][11][12][13], black hole physics and quantum gravity in 1+1 dimensions [14][15][16][17][18][19][20].…”

Approximating the Sachdev-Ye-Kitaev model with Majorana wires

“…As one notable example, Majorana-fermion zero modes [1,2] capture the essence of non-Abelian statistics and topological quantum computation [3,4], and correspondingly now form the centerpiece of a vibrant experimental effort [5][6][7][8][9][10][11][12][13][14][15][16]. More recently, randomly interacting Majorana fermions governed by the "Sachdev-YeKitaev (SYK) model" [17][18][19] were shown to exhibit sharp connections to chaos, quantum-information scrambling, and black holes-naturally igniting broad interdisciplinary activity (see, e.g., [20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39]). The goal of this Rapid Communication is to exploit hardware components of a Majorana-based topological quantum computer for a tabletop implementation of the SYK model, thus uniting these very different topics.…”