arXiv subjects
Publications and source records attributed to Eugene Trubowitz.
We rigorously construct and control a generic class of spatially homogeneous (Bianchi VIII and Bianchi IX) vacuum spacetimes that exhibit the oscillatory BKL phenomenology. We investigate the causal structure of these spacetimes and show that there is a "particle horizon".
The Einstein vacuum equations in the formulation developed by Newman, Penrose [NP] and Friedrich [Fr] are expressed in terms of a Lie superbracket. Differential identities are derived from the super Jacobi identity. This perspective clarifies the covariance properties of the equations. The equations are intended as a tool for the analytic study of vacuum spacetimes.
The generic renormalization group map associated to a weakly coupled system of fermions at temperature zero is treated by supplementing the methods of Part 1. The interplay between position and momentum space is captured by `sectors'. It is shown that the difference between the complete four legged vertex and its `ladder' part is irrelevant for the sequence of renormalization group maps.
A self contained analysis demonstrates that the sum of all particle-hole ladder contributions for a two dimensional, weakly coupled fermion gas with a strictly convex Fermi curve at temperature zero is bounded. This is used in our construction of two dimensional Fermi liquids.
An estimate on the operator norm of an abstract fermionic renormalization group map is derived. This abstract estimate is applied in another paper to construct the thermodynamic Green's functions of a two dimensional, weakly coupled fermion gas with an asymmetric Fermi curve. The estimate derived here is strong enough to control everything but the sum of all quartic contributions to the Green's functions.
We improve on the abstract estimate obtained in Part 1 by assuming that there are constraints imposed by `overlapping momentum loops'. These constraints are active in a two dimensional, weakly coupled fermion gas with a strictly convex Fermi curve. The improved estimate is used in another paper to control everything but the sum of all ladder contributions to the thermodynamic Green's functions.
In a series of ten papers, of which this is the first, we prove that the temperature zero renormalized perturbation expansions of a class of interacting many-fermion models in two space dimensions have nonzero radius of convergence. The models have "asymmetric" Fermi surfaces and short range interactions. One consequence of the convergence of the perturbation expansions is the existence of a discontinuity in the particle number density at the Fermi surface. Here, we present a self contained formulation of our main results and give an overview of the methods used to prove them.
Using results established in other papers in our series, we prove the existence of the infinite volume, temperature zero, thermodynamic Green's functions of a two dimensional, weakly coupled fermion gas with an asymmetric Fermi curve and short range interactions. This is done by showing that our sequence of renormalization group maps converges.
We show that the particle number density derived from the thermodynamic Green's function at temperature zero constructed in the second part of this series has a jump across the Fermi curve, a basic property of a Fermi liquid. We further show that the two particle thermodynamic Green's function at temperature zero has the regularity behavior expected in a Fermi liquid.
For a two dimensional, weakly coupled system of fermions at temperature zero, one principal ingredient used to control the composition of the associated renormalization group maps is the careful counting of the number of quartets of sectors that are consistent with conservation of momentum. A similar counting argument is made to show that particle-particle ladders are irrelevant in the case of an asymmetric Fermi curve.
The first renormalization group map arising from the momentum space decomposition of a weakly coupled system of fermions at temperature zero differs from all subsequent maps. Namely, the component of momentum dual to temperature may be arbitrarily large - there is no ultraviolet cutoff. The methods of Part 1 are supplemented to control this special case.
We construct, using fermionic functional integrals, thermodynamic Green's functions for a weakly coupled fermion gas whose Fermi energy lies in a gap. Estimates on the Green's 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.
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 the Fermi surface is not symmetric under the reflection p to -p are included.
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.
The perturbation expansion for a general class of many-fermion systems with a non-nested, non-spherical 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 non-overlapping is given, and improved power counting bounds are derived from it. They imply that the only subgraphs that can generate $r$ factorials in the $r^{\rm th}$ 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.
In these notes we give a very rough sketch of nonperturbative methods of many body quantum field theory that are powerful enough to rigorously control weak coupling instabilities in condensed matter phyics, for example, the Cooper instability in an electron - phonon system.