Symmetry Breaking in Coupled SYK or Tensor Models

• Main Authors: Jaewon Kim, Igor R. Klebanov, Grigory Tarnopolsky, Wenli Zhao
• Format: Article
• Language: English
• Published: American Physical Society 2019-05-01
• Series: Physical Review X
• Online Access: http://doi.org/10.1103/PhysRevX.9.021043

We study a large-N tensor model with O(N)^{3} symmetry containing two flavors of Majorana fermions, ψ_{1}^{abc} and ψ_{2}^{abc}. 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=iψ_{1}^{abc}ψ_{2}^{abc} 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 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.