Abstract
We construct a Hamiltonian lattice regularisation of the N-flavour Gross-Neveu model that manifestly respects the full O(2N) symmetry, preventing the appearance of any unwanted marginal perturbations to the quantum field theory. In the context of this lattice model, the dynamical mass generation is intimately related to the Coleman-Mermin-Wagner and Lieb-Schultz-Mattis theorems. In particular, the model can be interpreted as lying at the first order phase transition line between a trivial and symmetry-protected topological (SPT) phase, which explains the degeneracy of the elementary kink excitations. We show that our Hamiltonian model can be solved analytically in the large N limit, producing the correct expression for the mass gap. Furthermore, we perform extensive numerical matrix product state simulations for N = 2, thereby recovering the emergent Lorentz symmetry and the proper non-perturbative mass gap scaling in the continuum limit. Finally, our simulations also reveal how the continuum limit manifests itself in the entanglement spectrum. As expected from conformal field theory we find two conformal towers, one tower spanned by the linear representations of O(4), corresponding to the trivial phase, and the other by the projective (i.e. spinor) representations, corresponding to the SPT phase.
Highlights
This prescription differs from the typical regularisation using Wilson fermions, and takes a different prescription for the mass term as proposed in the original staggered formulation by Susskind. This prescription is well known in the context of the SSH model, but a Gross-Neveu type interaction resulting from it had, to the best of our knowledge, not been considered. By studying this lattice model in the limit of large N — where mean field theory becomes exact — as well as at N = 2 using matrix product state (MPS) simulations, we have established that its low energy behaviour replicates all the features expected from the field theory
We argued how the resulting lattice model lies at the first order phase transition between a trivial and topological insulator, and much of the degeneracies in both the excitation and entanglement spectrum can be reinterpreted from that perspective
We observed that the non-perturbative behaviour of this marginally relevant interaction makes it especially challenging to accurately probe the continuum limit
Summary
We first provide a short introduction to the GN model and its symmetries before porting it to the lattice. The GN model has an additional Z2 chiral symmetry that acts as ψ → γ5ψ and prohibits perturbative contributions to the condensate σ = c∈N ψcψc , or a perturbative mass term. Unlike in conventional (i.e. Isingtype) Z2 symmetry breaking, where the kink from one vacuum to the other is unique, in the case of GN the kinks are of the Callen-Coleman-Gross-Zee type [77] and bind massless fermions They transform according to the fundamental spinor representations [47]. This is similar to Jackiw-Rebbi kinks [78] and we will interpret this from a condensed matter perspective as the protected gapless edge modes on the interface between a trivial and SPT phase, when constructing the lattice model. A more standard yet involved Feynman diagram calculation of the scattering matrix in appendix A proves that this relation is valid for all values of N
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