Abstract
As shown in [1], two copies of the large N Majorana SYK model can produce spontaneous breaking of a Z2 symmetry when they are coupled by appropriate quartic terms. In this paper we similarly study two copies of the complex SYK model coupled by a quartic term preserving the U(1) × U(1) symmetry. We also present a tensor counterpart of this coupled model. When the coefficient α of the quartic term lies in a certain range, the coupled large N theory is nearly conformal. We calculate the scaling dimensions of fermion bilinear operators as functions of α. We show that the operator {c}_{1i}^{dagger }{c}_{2i} , which is charged under the axial U(1) symmetry, acquires a complex dimension outside of the line of fixed points. We derive the large N Dyson-Schwinger equations and show that, outside the fixed line, this U(1) symmetry is spontaneously broken at low temperatures because this operator acquires an expectation value. We support these findings by exact diagonalizations extrapolated to large N.
Highlights
Of fixed points when α is positive, while a gapped Z2 symmetry breaking phase appears when α is negative [1]
As shown in [1], two copies of the large N Majorana SYK model can produce spontaneous breaking of a Z2 symmetry when they are coupled by appropriate quartic terms
In this paper we study two copies of the complex SYK model coupled by a quartic term preserving the U(1) × U(1) symmetry
Summary
The Hamiltonian coupling them is (1.1), where Jij,kl is the random Gaussian complex tensor with zero mean Jij,kl = 0; it satisfies Jij,kl = Jk∗l,ij in order for the Hamiltonian to be Hermitian. We assume anti-symmetry in the first and second pairs of indices: Jij,kl = −Jji,kl = −Jij,lk. The definition of the random tensor Jij,kl is incomplete. Which is analogous to the Z4 symmetry which played an important role in [1] Another important symmetry is the particle-hole symmetry c1i ↔ c†1i , c2i ↔ c†2i , Jij,kl → Ji∗j,kl. Since the random coupling Jij,kl is complex, the U(1)+ and U(1)− are on a different footing: the charge conjugation acting on the second flavor c2i, C2†c2iC2 = c†2i (2.6). We can clearly see that V (Gσσ ) is invariant under the global U(2) transformations G(τ1, τ2) → U †G(τ1, τ2)U
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