A large number of Majorana fermions with interactions coupling four of them at a time can exhibit interesting quantum dynamics. Models of this kind include the Sachdev-Ye-Kitaev (SYK) model, where the coefficients of quartic interactions are randomly distributed, and the Tensor models, where they respect continuous symmetries. These models exhibit approximate invariance under scaling of the time and have power law fall-off of the correlation functions.

In this talk we will discuss a pair of SYK or Tensor models coupled by the quartic interactions, and show that they produce a richer set of phenomena. These include a line of fixed points, where critical exponents vary along the line and formally acquire imaginary parts outside it. For one sign of the coupling constant, the approximate scale invariance continues to hold. For the other, a gap opens in the energy spectrum, resulting in exponential fall-off of correlation functions. This is indicative of breaking of a discrete symmetry. Thus, our quantum mechanical model exhibits dynamical phenomena characteristic of higher dimensional quantum field theories. Furthermore, the gapped phase of our model may be dual to a certain traversable wormhole in two-dimensional space-time.

The talk will end with a similar discussion of a pair of complex SYK models coupled by a quartic interaction which preserves the U(1) x U(1) symmetry. For a range of parameters, this model gives rise to breaking of one of the U(1) symmetries. This is demonstrated via an analysis of the large N Dyson-Schwinger equations, as well as by Exact Diagonalizations of the finite N Hamiltonians.