Depinning in integer-restricted Gaussian Fields and BKT phases of two-component spin models

Abstract

It is shown that the Villain model of two-component spins over two dimensional lattices exhibits slow, non-summable, decay of correlations at any temperature at which the dual integer-valued Gaussian field exhibits depinning. For the latter, we extend the recent proof by P. Lammers of the existence of a depinning transition in the integervalued Gaussian field in two-dimensional graphs of degree three to all doubly-periodic graphs, in particular to Z 2 . Taken together these two statements yield a new perspective on the Berezinskii-Kosterlitz-Thouless phase transition in the Villain model, and complete a new proof of depinning in two-dimensional integer-valued height functions.

The Discrete Gaussian model, I. Renormalisation group flow at high temperature

“…The rigorous approach of [42] (see also [43]) uses a multiscale resummation based on conditional expectations and Jensen's inequality. For a recent exposition as well as recent extensions and applications of this approach, see [46,47,55,67], and for recent alternative approaches to the proof of the existence of the Kosterlitz-Thouless transition, see also [2,58,59,66]. These approaches have many appealing features which include that they apply quite robustly to various models, but they are not precise enough to derive scaling limits or sharp asymptotics of correlation functions, or to study the (expected) critical curve -the Kosterlitz-Thouless transition line, see Fig.…”

“…The scaling limit of the infinite-volume limit of the Discrete Gaussian model on Z 2 , or the consideration of mesoscopic test functions on the torus, requires additional analysis and the extension is considered in the companion paper [15]. To state the torus scaling limit, let T 2 = (R/Z) 2 and for f ∈ C ∞ (T 2 ) with…”

“…Remark 1.2. For the standard range-ρ distribution J = J ρ , there exists C > 0 such that for any δ > 0, ρ 2 C| log δ| and L = L(J ρ , δ), the conclusion of Theorem 1.1 holds with β 0 (J ρ ) = (1 + δ)β free (J ρ ), where for any step distribution J, we define β free (J) := 8πv 2 J .…”

“…We will bound the terms in log G j+s (ϕ+ξ, X) one by one, see (5.16). First, ∇ϕ 2 L 2 (X) will be isolated from ∇(ϕ + ξ) 2 L 2 (X) . Let B ∈ B j+s (X) and without loss of generality, let B, l i (i = 1, 2, 3, 4) be as above but B = [1, L j+s ] 2 .…”

The Discrete Gaussian model, II. Infinite-volume scaling limit at high temperature

“…To incorporate the effect of the scaledependent external fields, we need an extension of the norms and regulators that take the external field into account. In the context of Section 2, we note that by translation invariance of the Discrete Gaussian model on the torus Λ N , we may assume that f is centred with respect to the block decomposition; that is, supp(f ) and supp(∆f ) are contained in the box m+[0, 1 4 L j f ) 2 , where m is (one of) the lattice points closest to the center of some block B ∈ B j for all j f j N . In particular, then, by Lemma 2.2, for all scales j N , there is a block B ∈ B j such that whenever L C, N 5 (supp(u j )) ⊂ B where N k (X) denotes the set of points with ℓ 1 -distance at most k from the set X.…”