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Physical review. B./Physical review. B

Extension of the SYK model to 1+1 dimensions in the strong coupling limit

Andy A. V. Yunge, Alessandro De Martino, A. Tagliacozzo

Published Aug 23, 2023• 1 citation
Publisher

Abstract

We study a $1+1$ (space-time)-dimensional extension of the $0+1$-dimensional Sachdev-Ye-Kitaev model for $N$ Majorana fermions, with random all-to-all quartic interactions, averaged over disorder. At large interaction couplings and large $N$, the conformal symmetry of the effective action emerges, which is not broken spontaneously as in the original $0+1$ SYK model. Two-point correlators are obtained from a coupling expansion of the Schwinger-Dyson equations. For $N=4$, the model can be mapped onto complex fermions and solved exactly via the bosonization technique, featuring two branches of excitations: a gapped ``pseudocharge'' mode and a gapless ``pseudospin'' mode. We give an approximate analytic form of the two-point correlators at large distances and zero temperature for $N=4$, which is adopted heuristically to evaluate an approximation to the large-$N$ free energy in the zero temperature limit, numerically. The fact that this energy displays an absolute minimum at a finite value of the gap, though in a restricted range of parameters, suggests that a gapped branch of excitations is also present in our extension of the model at least in that range of parameters.

DOI
10.1103/physrevb.108.075428

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Abstract As shown in [1], two copies of the large N Majorana SYK model can produce spontaneous breaking of a Z2 symmetry when they are coupled by appropriate quartic terms. In this paper we similarly study two copies of the complex SYK model coupled by a quartic term preserving the U(1) × U(1) symmetry. We also present a tensor counterpart of this coupled model. When the coefficient α of the quartic term lies in a certain range, the coupled large N theory is nearly conformal. We calculate the scaling dimensions of fermion bilinear operators as functions of α. We show that the operator c1i†c2i {c}_{1i}^{\dagger }{c}_{2i} c1i†​c2i​ c 1 i † c 2 i , which is charged under the axial U(1) symmetry, acquires a complex dimension outside of the line of fixed points. We derive the large N Dyson-Schwinger equations and show that, outside the fixed line, this U(1) symmetry is spontaneously broken at low temperatures because this operator acquires an expectation value. We support these findings by exact diagonalizations extrapolated to large N.
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