The Two-Dimensional Hubbard Model on the Honeycomb Lattice

Abstract

We consider the 2D Hubbard model on the honeycomb lattice, as a model for a single layer graphene sheet in the presence of screened Coulomb interactions. At half filling and weak enough coupling, we compute the free energy, the ground state energy and we construct the correlation functions up to zero temperature in terms of convergent series; analiticity is proved by making use of constructive fermionic renormalization group methods. We show that the interaction produces a modification of the Fermi velocity and of the wave function renormalization without changing the asymptotic infrared properties of the model with respect to the unperturbed non-interacting case; this rules out the possibility of superconducting or magnetic instabilities in the ground state. We also prove that the correlations verify a Ward Identity similar to the one for massless Dirac fermions, up to asymptotically negligible corrections and a renormalization of the charge velocity.

Universality of the Hall Conductivity in Interacting Electron Systems

“…Roughly, the strategy will consist in: (i) reformulating the correlation functions in terms of a Grassmann integral, in the limit where a suitable cutoff function is removed; (ii) proving the analyticity of the Grassmann integral, uniformly in the cutoff parameter; (iii) using Vitali's theorem on the convergence of holomorphic functions (also known as Vitali-Porter theorem, or Weierstrass' theorem), to conclude that the correlations themselves are analytic. The analysis of this section is a straightforward adaptation of previous works, see, e.g., [34, Appendices B,C,D] or [33,Section 6] for two recent reviews in the context of graphene with short-range interactions, and is included here just for the sake of self-containedness.…”

“…A convenient representation of the truncated expectation, due to Battle, Brydges and Federbush [7,21,22], is the following (for a proof, see, e.g., [32,33]). For a given (ordered) set of indices P = (f 1 , .…”

Anomalous Behavior in an Effective Model of Graphene with Coulomb Interactions

“…This was expected on the basis of a power counting analysis [22][23][24]; recently, it has been rigorously proven in [18,19], where the convergence of the perturbative series was established, using the methods of constructive Quantum Field Theory (QFT) and by taking into full account the lattice effects (i.e., by considering the Hubbard model on the honeycomb lattice).…”

“…Compared to other RG approaches, such as those in [32,35], the advantage of the constructive methods we adopt is that they allow us to get a rigorous and complete treatment of the effects of the cut-offs and a full control on the perturbative expansion via explicit bounds at all orders; quite remarkably, in certain cases, such as the ones treated in [6,11,12,7,18,19], these methods even provide a way to prove the convergence of the resummed perturbation theory. Using these methods, we construct a renormalized expansion, allowing us to express the Schwinger functions, from which the physical observables can be computed, as series in the effective couplings (the effective charges and the effective photon masses, also called in the following the running coupling constants), with finite coefficients at all orders, admitting explicit N !…”

Intermediate and spin-liquid phase of the half-filled honeycomb Hubbard model

“…A large number of numerical methods have been applied to study the honeycomb Hubbard model [1][2][3][4]11,[14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32]. The work by Meng et al used zero-temperature auxiliary-field (determinant) quantum Monte Carlo (AFQMC) [4].…”