Joel Feldman

A renormalizable field theory: the massive Gross-Neveu model in two dimensions

The Euclidean massive Gross-Neveu model in two dimensions is just renormalizable and asymptotically free. Thanks to the Pauli principle, bare perturbation theory with an ultra-violet cut-off (and the correct ansatz for the bare mass) is convergent in a disk, whose radius corresponds by asymptotic freedom to a small finite renormalized coupling constant. Therefore, the theory can be fully constructed in a perturbative way. It satisfies the O.S. axioms and is the Borel sum of the renormalized perturbation expansion of the model

Perturbation theory around nonnested Fermi surfaces. I. Keeping the Fermi surface fixed

The perturbation expansion for a general class of many-fermion systems with a nonnested, nonspherical Fermi surface is renormalized to all orders. In the limit as the infrared cutoff is removed, the counterterms converge to a finite limit which is differentiable in the band structure. The map from the renormalized to the bare band structure is shown to be locally injective. A new classification of graphs as overlapping or nonoverlapping is given, and improved power counting bounds are derived from it. They imply that the only subgraphs that can generater factorials in ther th order of the renormalized perturbation series are indeed the ladder graphs and thus give a precise sense to the statement that “ladders are the most divergent diagrams.” Our results apply directly to the Hubbard model at any filling except for half-filling. The half-filled Hubbard model is treated in another place.

The perturbatively stable spectrum of a periodic Schrödinger operator

b~F #. For a generic lattice F, every nonzero eigenvalue has multiplicity exactly two. Let q(x) be a function on L2(Ra/F), the Hilbert space of real-valued, square integrable functions on the torus Ra/F. Then, the spectrum of the self-adjoint operator A +q(x) acting on LZ(IRa/F) is discrete. Fix e>0. We call an eigenv a l u e b 2 o f A , having multiplicity m, stable under the perturbation q if A + q

Fermionic functional integrals and the renormalization group

The Renormalization Group is the name given to a technique for analyzing the qualitative behaviour of a class of physical systems by iterating a map on the vector space of interactions for the class. In a typical non-rigorous application of this technique one assumes, based on one’s physical intuition, that only a certain nite dimensional subspace (usually of dimension three or less) is important. These notes concern a technique for

Regularity of interacting nonspherical Fermi surfaces: The full self-energy

Regularity of the deformation of the Fermi surface under short-range interactions is established to all orders in perturbation theory. The proofs are based on a new classification of all graphs that are not doubly overlapping. They turn out to be generalized RPA graphs. This provides a simple extension to all orders of the regularity theorem of the Fermi surface movement proven in [FST2]. Models in which S is not symmetric under the reflection p → −p are included. 1 feldman@math.ubc.ca, http://www.math.ubc.ca/∼feldman/ 2 manfred@math.ethz.ch, http://www.math.ethz.ch/∼manfred/manfred.html 3 trub@math.ethz.ch 2 c2-dol.tex, Version October 15, 1997

Regularity of the moving Fermi surface: RPA contributions

Regularity of the deformation of the Fermi surface under short-range interactions is established for all contributions to the RPA self-energy (it is proven in an accompanying paper that the RPA graphs are the least regular contributions to the self-energy). Roughly speaking, the graphs contributing to the RPA self-energy are those constructed by contracting two external legs of a four-legged graph that consists of a string of bubbles. This regularity is a necessary ingredient in the proof that renormalization does not change the model. It turns out that the self-energy is more regular when derivatives are taken tangentially to the Fermi surface than when they are taken normal to the Fermi surface. The proofs require a very detailed analysis of the singularities that occur at those momenta p where the Fermi surface S is tangent to S + p. Models in which S is not symmetric under the reflection p → − p are included.

SINGLE SCALE ANALYSIS OF MANY FERMION SYSTEMS PART 1: INSULATORS

We construct, using fermionic functional integrals, thermodynamic Greens functions for a weakly coupled fermion gas whose Fermi energy lies in a gap. Estimates on the Greens functions are obtained that are characteristic of the size of the gap. This prepares the way for the analysis of single scale renormalization group maps for a system of fermions at temperature zero without a gap.

Complex Bosonic Many-Body Models: Overview of the Small Field Parabolic Flow

This paper is a contribution to a program to see symmetry breaking in a weakly interacting many boson system on a three-dimensional lattice at low temperature. It provides an overview of the analysis, given in Balaban et al. (The small field parabolic flow for bosonic many-body models: part 1—main results and algebra, arXiv:1609.01745, 2016, The small field parabolic flow for bosonic many-body models: part 2—fluctuation integral and renormalization, arXiv:1609.01746, 2016), of the ‘small field’ approximation to the ‘parabolic flow’ which exhibits the formation of a ‘Mexican hat’ potential well.

An inversion theorem in Fermi surface theory

We prove a perturbative inversion theorem for the map between the interacting and the noninteracting Fermi surface for a class of many fermion systems with strictly convex Fermi surfaces and short-range interactions between the fermions. This theorem gives a physical meaning to the counterterm function K that we use in the renormalization of these models: K can be identified as that part of the self--energy that causes the deformation of the Fermi surface when the interaction is turned on.

Renormalization in classical mechanics and many body quantum field theory

We attempt to give apedagogical introduction to perturbative renormalization. Our approach is to first describe, following Linstedt and Poincaré, the renormalization of formal perturbation expansions for quasi-periodic orbits in Hamiltonian mechanics. We then discuss, following [FT1, FT2], the renormalization of the formal ground state energy density of a many Fermion system. The construction of formal quasi-periodic orbits is carried out in detail to provide a relatively simple model for the considerably more involved, and perhaps less familiar, perturbative analysis of a field theory.As we shall see, quasi-periodic orbits and many Fermion systems have a number of important features in common. In particular, as Poincaré observed in the classical case and [FT1, FT2] pointed out in the latter, the formal expansions considered here both contain divergent subseries.