Spontaneous Breaking of $U(1)$ Symmetry in Coupled Complex SYK Models

TL;DR

This paper investigates how coupling two complex SYK models with a quartic term can lead to spontaneous breaking of the $U(1)$ symmetry, revealing new phases and operator behaviors in the large $N$ limit.

Contribution

It introduces a coupled complex SYK model with a quartic interaction that preserves $U(1) imes U(1)$ symmetry and demonstrates spontaneous $U(1)$ symmetry breaking at low temperatures.

Findings

Operator $c_{1i}^\u2212 c_{2i}$ acquires a complex dimension outside fixed points.

Spontaneous $U(1)$ symmetry breaking occurs at low temperatures.

Large $N$ Dyson-Schwinger equations confirm symmetry breaking.

Abstract

As shown in [1], two copies of the large $N$ Majorana SYK model can produce spontaneous breaking of a $Z_2$ 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 $U(1) \times U(1)$ symmetry. We also present a tensor counterpart of this coupled model. When the coefficient $\alpha$ 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 $\alpha$. We show that the operator $c_{1i}^\dagger c_{2i}$, 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 $N$.