The Small Field Parabolic Flow for Bosonic Many-Body Models: Part 2—Fluctuation Integral and Renormalization

Abstract

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 is part of an analysis of the "small field" approximation to the "parabolic flow" which exhibits the formation of a "Mexican hat" potential well. Here we complete the analysis of a renormalization group step, started in [7], by "evaluating" the fluctuation integral and renormalizing the chemical potential.

The Small Field Parabolic Flow for Bosonic Many-body Models: Part 1—Main Results and Algebra

“…For constant, not too big, fields ψ(x) = ψ and ψ * (x) = ψ * , the background field φ n is again constant, again obeys φ * n = φ * n and is approximately equal to ψ . See [15,Remark 1.1]. So ψ * − Q n φ * , Q n ψ − Q n φ 0 ≈ 0 , and, in this case, the dominant part of the effective action A n (ψ * , ψ, φ * n , φ n , µ n , V n )…”

“…In this paper we argue that, for n ≤ n p (with n p specified in Definition 1.11.b), the function (ST) n e A 0 (ψ * , ψ) has -up to errors which can reasonably be expected to be nonperturbatively small (see [14] or [8, §2.2.2])a logarithm whose dominant term, A n , described in Definition 1.5.b below, exhibits a much deeper potential well. See (1.8).…”

“…with a number µ n , which we call the renormalized chemical potential at scale n, a quartic monomial V n close to the monomial V (u) n of Definition 1.5.b, which we call the renormalized interaction at scale n, a "perturbative correction" p n (ψ * , ψ) , and a "small field cut off function" χ n which is discussed in [14]. The renormalized chemical potential µ n will grow with n like L 2n (µ 0 − µ * ); see Theorem 1.17.…”

“…where φ * n , φ n are the background fields of Definition 1.5, R n (φ * , φ) is a low degree polynomial in fields on the fine lattice X n , and E n (ψ * , ψ) is an analytic function on a neighborhood of the origin in H (n) 0 × H (n) 0 . Some motivation for the need to distinguish between "high degree" and "low degree" monomials and for our choice of the particular form of the "low degree monomials" is provided in [13,Remark 5.4].…”

“…The subscript u stands for "undifferentiated" and the subscript sp stands for "spatial". A polynomial of type p is a sum of monomials of type p. R n will be a sum, over p ∈ D, of polynomials,R The motivation for this choice of D is provided in [13,Remark 5.4].…”