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arXiv:math-ph/0001031 · Communications on Pure and Applied Mathematics

An inversion theorem in Fermi surface theory

Joel Feldman, Manfred Salmhofer B, Eugene Trubowitz B

Published Jan 1, 2000• 20 citations
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Abstract

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. © 2000 Wiley & Sons, Inc.

DOI
10.1002/1097-0312(200011)53:11<1350::aid-cpa2>3.0.co;2-d

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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. © 2000 Wiley & Sons, Inc.
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 fact provides a simple extension to all orders of the regularity theorem of the Fermi surface movement proven in [2]. Models in which S is not symmetric under the reflection p → −;p are included. © 1999 John Wiley & Sons, Inc.
The naive perturbation expansion for many-fermion systems is infrared divergent. One can remove these divergences by introducing counterterms. To do this without changing the model, one has to solve an inversion equation. We call this procedure Fermi surface renormalization (FSR). Whether or not FSR is possible depends on the regularity properties of the fermion self--energy. When the Fermi surface is nonspherical, this regularity problem is rather nontrivial. Using improved power counting at all orders in perturbation theory, we have shown sufficient differentiability to solve the FSR equation for a class of models with a non-nested, non-spherical Fermi surface. I will first motivate the problem and give a definition of FSR, and then describe the combination of geometric and graphical facts that lead to the improved power counting bounds. These bounds also apply to the four--point function. They imply that only ladder diagrams can give singular contributions to the four--point function.
The naive perturbation expansion for many-fermion systems is infrared divergent. One can remove these divergences by introducing counterterms. To do this without changing the model, one has to solve an inversion equation. We call this procedure Fermi surface renormalization (FSR). Whether or not FSR is possible depends on the regularity properties of the fermion self-energy. When the Fermi surface is nonspherical, this regularity problem is rather nontrivial. Using improved power counting at all orders in perturbation theory, we have shown sufficient differentiability to solve the FSR equation for a class of models with a non-nested, non-spherical Fermi surface. I will first motivate the problem and give a definition of FSR, and then describe the combination of geometric and graphical facts that lead to the improved power counting bounds. These bounds also apply to the four-point function. They imply that only ladder diagrams can give singular contributions to the four-point function.
Divergencies appearing in perturbation expansions of interacting many-body systems can often be removed by expanding around a suitably chosen renormalized (instead of the noninteracting) Hamiltonian. We describe such a renormalized perturbation expansion for interacting Fermi systems, which treats Fermi surface shifts and superconductivity with an arbitrary gap function via additive counterterms. The expansion is formulated explicitly for the Hubbard model to second order in the interaction. Numerical solutions of the self-consistency condition determining the Fermi surface and the gap function are calculated for the two-dimensional case. For the repulsive Hubbard model close to half-filling we find a superconducting state with d-wave symmetry, as expected. For Fermi levels close to the van Hove singularity a Pomeranchuk instability leads to Fermi surfaces with a broken square lattice symmetry, whose topology can be closed or open. For the attractive Hubbard model the second-order calculation yields an s-wave superconductivity with a weakly momentum dependent gap, whose size is reduced compared to the mean-field result.
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 to -p are included.