Abstract
We define and study various tensorial generalizations of the Gross-Neveu model in two dimensions, that is, models with four-fermion interactions and G3 symmetry, where we take either G = U(N) or G = O(N). Such models can also be viewed as two-dimensional generalizations of the Sachdev-Ye-Kitaev model, or more precisely of its tensorial counterpart introduced by Klebanov and Tarnopolsky, which is in part our motivation for studying them. Using the Schwinger-Dyson equations at large-N, we discuss the phenomenon of dynamical mass generation and possible combinations of couplings to avoid it. For the case G = U(N),we introduce an intermediate field representation and perform a stability analysis of the vacua. It turns out that the only apparently viable combination of couplings that avoids mass generation corresponds to an unstable vacuum. The stable vacuum breaks U(N)3 invariance, in contradiction with the Coleman-Mermin-Wagner theorem, but this is an artifact of the large-N expansion, similar to the breaking of continuous chiral symmetry in the chiral Gross-Neveu model.
Highlights
That going one more step up in the rank, i.e. considering tensor fields, things simplify again, not to the level of vector fields, making tensor fields the good candidates for models with a new and manageable, yet non-trivial, large-N limit
We recall how the latter should not be understood as Goldstone bosons since any apparently broken continuous symmetry must be restored in d = 2 [37, 38], and we describe the mechanism of symmetry restoration at play in this model
In many respects the Dirac models with U(N )3 symmetry turn out to be still very similar to the usual Gross-Neveu model: they are asymptotically free and they have a non-perturbative mass gap. For such models we have introduced an intermediate matrix field representation, by means of which we have discovered a first order phase transition to a phase with apparent breaking of the U(N )3 symmetry, and with an effective action for the would-be Goldstone modes given by a complex Grassmannian non-linear sigma model
Summary
In order to reduce the number of invariants, we can demand that the action be invariant under simultaneous and identical permutations of all the tensor indices, known as color symmetry. Such symmetry requires repackaging the I1X, interactions as. The most general renormalizable interacting action compatible with Euclidean, chiral, U(N ) and color symmetries contains six independent couplings, and reads. In constructing the interacting part of the action one encounters a new invariant, besides those we discussed for the Dirac case above This is because here graphs representing the interaction vertex no longer need to be bipartite — recall that ψ = ψtγ, and .
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