Exponential decay of transverse correlations for O(N) spin systems and related models

Abstract

We prove exponential decay of transverse correlations in the Spin O(N) model for arbitrary non-zero values of the external magnetic field and arbitrary spin dimension $$N > 1$$ N > 1 . Our result is new when $$N > 3$$ N > 3 , in which case no Lee–Yang theorem is available, it is an alternative to Lee–Yang when $$N = 2, 3$$ N = 2 , 3 , and also holds for a wide class of multi-component spin systems with continuous symmetry. The key ingredients are a representation of the model as a system of coloured random paths, a ‘colour-switch’ lemma, and a sampling procedure which allows us to bound from above the ‘typical’ length of the open paths.

Griffiths inequalities for the $O(N)$-spin model

“…We can also prove Theorem 1.1 for more general boundary conditions, however this is very similar to the case of an inhomogeneous external magnetic field, which we present in Section 4. The random path model used for the proof has a corresponding extension to this case, as explained in [13]. Because we already need to make slight modifications to the proof for the cases η = free or η = + and the cases A ∩ B = ∅ or A ∩ B = ∅, we will describe the O(N )-spin model with external field, the corresponding changes to the random path model, and the differences in the proof in Section 4.…”

“…The purpose of introducing this representation is to allow us to take advantage of the proven connections between this model and the O(N )-spin model that we require. The reader is encouraged to consult [12] or [13] for an alternative introduction of the model.…”

“…x η G,N,J , the case of homogeneous coupling constants J i e = β ≥ 0 for all e ∈ E, i ∈ [N ] and free or periodic boundary conditions, the proof can be found in [12] for the case |A| = 2 and in [13] for the general case, even with an external field. The extension to inhomogeneous coupling and + boundary condition is straightforward.…”

“…The difference with previous representations is that realisations can be defined purely in terms of objects local to individual edges and vertices. This distinction has allowed the use important tools such as reflection positivity [12,15,17], a proof of exponential decay of transverse correlations in the presence of an external magnetic field [13] that goes beyond results proven using the well-known Lee-Yang method, and a new proof of the BKT phase transition for the O(2)-spin model in two dimensions [14].…”

Macroscopic loops in the Bose gas, Spin O(N) and related models

“…our model corresponds to the Brydges, Fröhlich and Spencer representation of the Spin O(N) model with inverse temperature λ ≥ 0 [12] (note a correction of the original definition in [13, eq. (6.18)]), see also [4,13,27,29] for further connections between the Spin O(N) model and random loops and for a definition of the spin O(N) model.…”