Using renormalization group methods, we study the Heisenberg-Ising XY Z chain in an external magnetic field directed as the z axis, in the case of small coupling J 3 in the z direction. We study the asymptotic behaviour of the spin space-time correlation function in the direction of the magnetic field and the singularities of its Fourier transform.The work is organized in two parts. In the present paper an expansion for the ground state energy and the effective potential is derived, which is convergent if the running coupling constants are small enough. In the subsequent paper, by using hidden symmetries of the model, we show that this condition is indeed verified, if J 3 is small enough, and we derive an expansion for the spin correlation function. We also prove, by means of an approximate Ward identity, that a critical index, related with the asymptotic behaviour of the correlation function, is exactly vanishing.1.4 In this paper we develop a rigorous renormalization group analysis for the XY Z Hamiltonian in its fermionic form (some "not optimal" bounds for the correlation function Ω 3 (x) were already found in [M2]). As we said before, Ω 3 (x) can be obtained from the exact solution only in the case J 3 = 0, when the fermionic theory is a non interacting one. In particular, if x = (x, 0) and |ux| << 1, (1.8) and a more detailed analysis of the "small" terms in the r.h.s. (in order to prove that their derivatives of order n decay as |x| −n ), show that Ω 3 (x, 0) is a sum of "oscillating" functions with frequencies (np F )/π mod 1, n = 0, ±1, where p F = arccos(−h); this means that its Fourier transform has to be a smooth function, even for u = 0, in the neighborhood of any momentum k = 0, ±2p F . These frequencies are proportional to p F , so they depend only on the external magnetic field h.If J 3 = 0, a similar property is satisfied for the leading terms in the asymptotic behaviour, as we shall prove, but the value of p F depends in general also on u and J 3 . For example, if u = 0, the Hamiltonian (1.5) is equal, up to a constant, to the Hamiltonian of a free fermion gas with Fermi momentum p F = arccos(J 3 − h) plus an interaction term proportional to J 3 .As it is well known, the interaction modifies the Fermi momentum of the system by terms of order J 3 and it is convenient (see [BG], for example), in order to study the interacting model, to fix the Fermi momentum to an interaction independent value, by adding a counterterm to the hamiltonian. We proceed here in a similar way, that is we fix p F and h 0 so thatand we look for a value of ν, depending on u, J 3 , h 0 , such that, as in the J 3 = 0 case, the leading terms in the asymptotic behaviour of Ω 3 L,β (x) can be represented as a sum of oscillating functions with frequencies (np F )/π mod 1, n = 0, ±1.

Systems of interacting non relativistic fermions in d = 1, as well as spin chains or interacting bidimensional Ising models, verify an hidden approximate Gauge invariance which can be used to derive suitable Ward identities. Despite the presence of corrections and anomalies, such Ward identities can be implemented in a Renormalization Group approach and used to exploit nontrivial cancellations which allow to control the flow of the running coupling constants; in particular this is achieved combining Ward identities, Dyson equations and suitable correction identities for the extra terms appearing in the Ward identities, due to the presence of cutoffs breaking the local gauge symmetry. The correlations can be computed and show a Luttinger liquid behavior characterized by non universal critical indices, so that the general Luttinger liquid construction for one dimensional systems is completed without any use of exact solutions. The ultraviolet cutoff can be removed and a Quantum Field Theory corresponding to the Thirring model is also constructed. 11/ottobre/2018; 1:16 = 1 Lβ k∈D e iσkxψ[h,0]σ k,ω

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.

We consider a non-integrable model for interacting dimers on the two-dimensional square lattice. Configurations are perfect matchings of Z2, i.e. subsets of edges such that each vertex is covered exactly once ("close-packing" condition). Dimer configurations are in bijection with discrete height functions, defined on faces Ξ of Z2. The non-interacting model is "integrable" and solvable via Kasteleyn theory; it is known that all the moments of the height difference hΞ - hη converge to those of the massless Gaussian Free Field (GFF), asymptotically as |Ξ -η|→∞.We prove that the same holds for small non-zero interactions, as was conjectured in the theoretical physics literature. Remarkably, dimer-dimer correlation functions are instead not universal and decay with a critical exponent that depends on the interaction strength. Our proof is based on an exact representation of the model in terms of lattice interacting fermions, which are studied by constructive field theory methods. In the fermionic language, the height difference hΞ - hη takes the form of a non-local operator, consisting of a sum of monomials along an arbitrary path connecting Ξ and η. As in the non-interacting case, this path-independence plays a crucial role in the proof

The behaviour of correlation functions of d = 1 interacting fermionic systems is determined by a small number of critical indices. We prove that one of them is exactly zero. As a consequence, the behavior of the Fourier transform of the density-density correlation at zero momentum is qualitatively unaffected by the interaction, contrary to what happens at ±2p F , if pF is the Fermi momentum. The result is obtained by implementing Ward identities in a Renormalization Group approach.