Symmetry Breaking in Coupled SYK or Tensor Models

Abstract

We study a large-N tensor model with O(N)3 symmetry containing two flavors of Majorana fermions, ψ1abc and ψ2abc. We also study its random counterpart consisting of two coupled Sachdev-Ye-Kitaev (SYK) models, each containing NSYK Majorana fermions. In these models, we assume tetrahedral quartic Hamiltonians which depend on a real coupling parameter α. We find a duality relation between two Hamiltonians with different values of α, which allows us to restrict the model to the range of -1≤α≤1/3. The scaling dimension of the fermion number operator Q=iψ1abcψ2abc is complex and of the form 1/2+if(α) in the range -1≤α<0, indicating an instability of the conformal phase. Using Schwinger-Dyson equations to solve for the Green functions, we show that in the true low-temperature phase this operator acquires an expectation value, which demonstrates the breaking of an antiunitary particle-hole symmetry and other discrete symmetries. We also calculate spectra of the coupled SYK models for values of NSYK where exact diagonalizations are possible. For negative α, we find a gap separating the two lowest energy states from the rest of the spectrum, leading to an exponential decay of the zero-temperature correlation functions. For NSYK divisible by 4, the two lowest states have a small splitting. They become degenerate in the large-NSYK limit, as expected from the spontaneous breaking of a Z2 symmetry.