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, ψabc1 and ψabc2 . We also study its random counterpart consisting of two coupled Sachdev-Ye-Kitaev models, each one 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 = iψabc1 ψabc2 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. This demonstrates the breaking of an anti-unitary 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; this leads to 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.