Vieri Mastropietro

METHODS FOR THE ANALYSIS OF THE LINDSTEDT SERIES FOR KAM TORI AND RENORMALIZABILITY IN CLASSICAL MECHANICS: A review with Some Applications

This paper consists in a unified exposition of methods and techniques of the renormalization group approach to quantum field theory applied to classical mechanics, and in a review of results: (1) a proof of the KAM theorem, by studying the perturbative expansion (Lindstedt series) for the formal solution of the equations of motion; (2) a proof of a conjecture by Gallavotti about the renormalizability of isochronous hamiltonians, i.e. the possibility to add a term depending only on the actions in a hamiltonian function not verifying the anisochrony condition so that the resulting hamiltonian is integrable. Such results were obtained first by Eliasson; however the difficulties arising in the study of the perturbative series are very similar to the problems which one has to deal with in quantum field theory, so that the use of the methods which have been envisaged and developed in the last twenty years precisely in order to solve them allows us to obtain unified proofs, both conceptually and technically. In the final part of the review, the original work of Eliasson is analyzed and exposed in detail; its connection with other proofs of the KAM theorem based on his method is elucidated.

Periodic Solutions for Completely Resonant Nonlinear Wave Equations with Dirichlet Boundary Conditions

We consider the nonlinear string equation with Dirichlet boundary conditions utt−uxx=ϕ(u), with ϕ(u)=Φu3+O(u5) odd and analytic, Φ≠0, and we construct small amplitude periodic solutions with frequency ω for a large Lebesgue measure set of ω close to 1. This extends previous results where only a zero-measure set of frequencies could be treated (the ones for which no small divisors appear). The proof is based on combining the Lyapunov-Schmidt decomposition, which leads to two separate sets of equations dealing with the resonant and non-resonant Fourier components, respectively the Q and the P equations, with resummation techniques of divergent powers series, allowing us to control the small divisors problem. The main difficulty with respect to the nonlinear wave equations utt−uxx+Mu=ϕ(u), M≠0, is that not only the P equation but also the Q equation is infinite-dimensional.

Non-Perturbative Renormalization

Renormalization Mathematical Techniques QED2, Thirring and Gross-Neveu Models Ward Identities Chiral Anomalies and the Adler-Bardeen Theorem Vanishing of Beta Function Wilson Fermion and Axiom Verification Infrared QED4 Universality in Ising Models Nonuniversality in Vertex or Askin-Teller Models The Anisotropic Ashkin-Teller Model Luttinger Liquids and Spin Chains The 1D Hubbard Model Fermi Liquid Behavior in the 2D Hubbard Model The BCS Model.

Ward Identities and Chiral Anomaly in the Luttinger Liquid

Systems of interacting non-relativistic fermions in d =1, as well as spin chains or interacting two dimensional 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.

RENORMALIZATION GROUP, HIDDEN SYMMETRIES AND APPROXIMATE WARD IDENTITIES IN THE XYZ MODEL

Using renormalization group methods, we study the Heisenberg–Ising XYZ chain in an external magnetic field directed as the z axis, in the case of small coupling J3 in the z direction. In particular, we focus our attention on the asymptotic behaviour of the spin correlation function in the direction of the magnetic field and the singularities of its Fourier transform. An expansion for the ground state energy and the effective potential is derived, which is convergent if the running coupling constants are small enough. Moreover, by using hidden symmetries of the model, we show that this condition is indeed verified, if J3 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, together with other properties, so obtaining a rather detailed description of the XYZ correlation function.

Renormalization group for one-dimensional fermions. A review on mathematical results

Section

The Two-Dimensional Hubbard Model on the Honeycomb Lattice

We consider the two-dimensional (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; analyticity 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 thermal ground state.

Functional Integral Construction of the Massive Thirring model: Verification of Axioms and Massless Limit

We present a complete construction of a Quantum Field Theory for the Massive Thirring model by following a functional integral approach. This is done by introducing an ultraviolet and an infrared cutoff and by proving that, if the “bare” parameters are suitably chosen, the Schwinger functions have a well defined limit satisfying the Osterwalder-Schrader axioms, when the cutoffs are removed. Our results, which are restricted to weak coupling, are uniform in the value of the mass. The control of the effective coupling (which is the main ingredient of the proof) is achieved by using the Ward Identities of the massless model, in the approximated form they take in the presence of the cutoffs. As a byproduct, we show that, when the cutoffs are removed, theWard Identities have anomalies which are not linear in the bare coupling. Moreover, we find for the interacting propagator of the massless theory a closed equation which is different from that usually stated in the physical literature.

Beta function and anomaly of the Fermi surface for a

We derive a perturbation theory, based on the renormalization group, for the Fermi surface of a one dimensional system of fermions in a periodic potential interacting via a short range, spin independent potential. The infrared problem is studied by writing the Schwinger functions in terms of running couplings. Their flow is described by a Beta function, whose existence and analyticity as a function of the running couplings is proved. If the fermions are spinless we prove that the Beta function is vanishing and the renormalization flow is bounded for any small interaction. If the fermions are spinning the Beta function is not vanishing but, if the conduction band is not filled or half filled and the interaction is repulsive, it is possible again to control the flow proving the partial asymptotic freedom of the theory. This is done showing that the Beta function is partially vanishing using the exact solution of the Mattis model, which is the spin analogue of the Luttinger model. In both these cases Schwinger functions are anomalous so that the system is a “Luttinger liquid”. Our results extend the work in [B.G.P.S.], where neither spin nor periodic potential were considered; an explicit proof of some technical results used but not explicitly proved there is also provided.

d=1

The Adler-Bardeen theorem has been proven only as a statement valid at all orders in perturbation theory, without any control on the convergence of the series. In this paper we prove a nonperturbative version of the Adler-Bardeen theorem in d=2 by using recently developed technical tools in the theory of Grassmann integration. The proof is based on the assumption that the boson propagator decays fast enough for large momenta. If the boson propagator does not decay, as for Thirring contact interactions, the anomaly in the WI (Ward Identities) is renormalized by higher order contributions.