Symmetry Breaking in Coupled SYK or Tensor Models

摘要

We study a large- N tensor model with O ( N ) 3 symmetry containing two flavors of Majorana fermions, ψ 1 a b c and ψ 2 a b c . We also study its random counterpart consisting of two coupled Sachdev-Ye-Kitaev (SYK) models, each containing N SYK 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 = i ψ 1 a b c ψ 2 a b c is complex and of the form 1 / 2 + i f ( 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 N SYK 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 N SYK divisible by 4, the two lowest states have a small splitting. They become degenerate in the large- N SYK limit, as expected from the spontaneous breaking of a Z 2 symmetry.