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Fermionic functional integrals and the renormalization group
 CRM Monograph Series, 16, Providence RI, AMS
, 2002
"... Abstract 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 nonrigorous application of this technique one assumes, based on one’s physica ..."
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Abstract 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 nonrigorous application of this technique one assumes, based on one’s physical intuition, that only a certain finite dimensional subspace (usually of dimension three or less) is important. These notes concern a technique for justifying this approximation in a broad class of Fermionic models used in condensed matter and high energy physics. These notes expand upon the Aisenstadt Lectures given by J. F. at the Centre de
Determinant Bounds and the Matsubara UV Problem of Many–Fermion Systems
 Comm. Math. Phys
, 2008
"... Dedicated to Jürg Fröhlich in celebration of his 61 st birthday It is known that perturbation theory converges in fermionic field theory at weak coupling if the interaction and the covariance are summable and if certain determinants arising in the expansion can be bounded efficiently, e.g. if the co ..."
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Dedicated to Jürg Fröhlich in celebration of his 61 st birthday It is known that perturbation theory converges in fermionic field theory at weak coupling if the interaction and the covariance are summable and if certain determinants arising in the expansion can be bounded efficiently, e.g. if the covariance admits a Gram representation with a finite Gram constant. The covariances of the standard many–fermion systems do not fall into this class due to the slow decay of the covariance at large Matsubara frequency, giving rise to a UV problem in the integration over degrees of freedom with Matsubara frequencies larger than some Ω (usually the first step in a multiscale analysis). We show that these covariances do not have Gram representations on any separable Hilbert space. We then prove a general bound for determinants associated to chronological products which is stronger than the usual Gram bound and which applies to the many–fermion case. This allows us to prove convergence of the first integration step in a rather easy way, for a short–range interaction which can be arbitrarily strong, provided Ω is chosen large enough. Moreover, we give – for the first time – nonperturbative bounds on all scales for the case of scale decompositions of the propagator which do not impose cutoffs on the Matsubara frequency. present address: Institut für Mathematik, Universität Mainz 1 1 Gram representations and determinant bounds Let X be a set and M: X 2 → C, (x, y) ↦ → M(x, y). We call M an (X×X)matrix and use the notation M = (Mxy)x,y∈X (if X = {1,..., n}, we call it as usual an (n × n)–matrix). Definition 1.1 Let M be an (X × X)matrix. A triple (H, v, w), where H is a Hilbert space and v and w are maps from X to H, is called a Gram representation of M if ∀ x, x ′ ∈ X: Mxx ′ = 〈vx, wx ′ 〉 (1) and if there is a finite constant γM> 0 such that sup x∈X max{‖vx‖, ‖wx‖} ≤ γM. (2) γM is called the Gram constant of M associated to the Gram representation (H, v, w).
The ground state construction of bilayer graphene
"... We consider a model of halffilled bilayer graphene, in which the three dominant SlonczewskiWeissMcClure hopping parameters are retained, in the presence of short range interactions. Under a smallness assumption on the interaction strength U as well as on the interlayer hopping , we construct th ..."
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We consider a model of halffilled bilayer graphene, in which the three dominant SlonczewskiWeissMcClure hopping parameters are retained, in the presence of short range interactions. Under a smallness assumption on the interaction strength U as well as on the interlayer hopping , we construct the ground state in the thermodynamic limit, and prove its analyticity in U, uniformly in . The interacting Fermi surface is degenerate, and consists of eight Fermi points, two of which are protected by symmetries, while the locations of the other six are renormalized by the interaction, and the effective dispersion relation at the Fermi points is conical. The construction reveals the presence of different energy regimes, where the effective behavior of correlation functions changes qualitatively. The analysis of the crossover between regimes plays an important role in the proof of analyticity and in the uniform control of the radius of convergence. The proof is based on a rigorous implementation of fermionic renormalization group methods, including determinant estimates for the renormalized expansion. Table of contents
HEIGHT FLUCTUATIONS IN INTERACTING DIMERS
"... Abstract. We consider a nonintegrable model for interacting dimers on the twodimensional square lattice. Configurations are perfect matchings of Z2, i.e. subsets of edges such that each vertex is covered exactly once (“closepacking ” condition). Dimer configurations are in bijection with discret ..."
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Abstract. We consider a nonintegrable model for interacting dimers on the twodimensional square lattice. Configurations are perfect matchings of Z2, i.e. subsets of edges such that each vertex is covered exactly once (“closepacking ” condition). Dimer configurations are in bijection with discrete height functions, defined on faces ξ of Z2. The noninteracting 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 Gaussian Free Field, asymptotically as ξ−η  → ∞. We prove that the same holds for small nonzero interactions, as was conjectured in the theoretical physics literature. Remarkably, dimerdimer 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 lattice language, the height difference hξ − hη takes the form of a nonlocal fermionic operator, consisting of a sum of fermionic monomials along an arbitrary path connecting ξ and η. As in the noninteracting case, this pathindependence plays a crucial role in the proof. 1. Introduction and
Expectations of Fermionic Fields With N Components
, 1998
"... In this paper we show with a concrete exemple that the behaviour in the number N of independent components (flavours) of the expectation of fermionic fields can be different from the one suggested by its expansion in terms of Feynman graphs. To prove this result we study the asymptotic behaviour of ..."
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In this paper we show with a concrete exemple that the behaviour in the number N of independent components (flavours) of the expectation of fermionic fields can be different from the one suggested by its expansion in terms of Feynman graphs. To prove this result we study the asymptotic behaviour of a non autonomous discrete time map.
Constructive Renormalization Theory
, 1999
"... These notes are the second part of a common course on Renormalization Theory given with Professor P. da Veiga 1. I emphasize here the rigorous nonperturbative or constructive aspects of the theory. The usual formalism for the renormalization group in field theory or statistical mechanics is reviewe ..."
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These notes are the second part of a common course on Renormalization Theory given with Professor P. da Veiga 1. I emphasize here the rigorous nonperturbative or constructive aspects of the theory. The usual formalism for the renormalization group in field theory or statistical mechanics is reviewed, together with its limits. The constructive formalism is introduced step by step. Taylor forest formulas allow to perform easily the cluster and Mayer expansions which are needed for a single step of the renormalization group in the case of Bosonic theories. The iteration of this single step leads to further difficulties whose solution is briefly sketched. The second part of the course is devoted to Fermionic models. These models are easier to treat on the constructive level, so they are very well suited to beginners in constructive theory. It is shown how the Taylor forest formulas allow to reorganize perturbation theory nicely in order to construct the GrossNeveu2 model without any need for cluster or Mayer expansions. Finally applications of this technique to condensed matter and renormalization group around Fermi surface are briefly reviewed. 1 The Renormalization Group: an overview 1.1 Functional Integration and its problems In this section we restrict ourselves to the bosonic φ4 field theory in d Euclidean space time dimensions. The model, introduced in P. Da Veiga’s lectures, is defined by the (formal) measure dµC(φ)e −S(φ) ∫, S(φ) = λ φ 4 (x)d d x (1.1) where dµC(φ) represents the Gaussian measure for the free field. Gaussian measures are characterized by their covariance, or propagator, which for a massive theory is, in Fourier space: C(p) = (p 2 + m 2) −1, (1.2) and S is the (bare) action. In dimension d = 2,3 the model is superrenormalizable, and its rigorous construction was the first major achievement of constructive theory [GJ]. In
Positivity and convergence in fermionic quantum field theory
, 1999
"... We derive norm bounds that imply the convergence of perturbation theory in fermionic quantum field theory if the propagator is summable and has a finite Gram constant. These bounds are sufficient for an application in renormalization group studies. Our proof is conceptually simple and technically el ..."
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We derive norm bounds that imply the convergence of perturbation theory in fermionic quantum field theory if the propagator is summable and has a finite Gram constant. These bounds are sufficient for an application in renormalization group studies. Our proof is conceptually simple and technically elementary; it clarifies how the applicability of Gram bounds with uniform constants is related to positivity properties of matrices associated to the procedure of taking connected parts of Gaussian convolutions. This positivity is preserved in the decouplings In fermionic field theories with an infrared and an ultraviolet cutoff, perturbation theory converges. Perturbation theory in bosonic theories always diverges. When representing the perturbation series in terms of Feynman graphs, this is often stated in the way that, although there are as many
Bosonic Monocluster Expansion
, 2000
"... We compute connected Green's functions of a Bosonic field theory with cutoffs by means of a "minimal" expansion which in a single move, interpolating a generalized propagator, performs the usual tasks of the cluster and Mayer expansion. In this way it allows a direct construction of t ..."
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We compute connected Green's functions of a Bosonic field theory with cutoffs by means of a "minimal" expansion which in a single move, interpolating a generalized propagator, performs the usual tasks of the cluster and Mayer expansion. In this way it allows a direct construction of the infinite volume or thermodynamic limit and it brings constructive Bosonic expansions closer to constructive Fermionic expansions and to perturbation theory.