Finite-Size and Finite-Temperature Effects in the Conformally Invariant O(N) Vector Model for 2<d<4

TL;DR

This paper investigates how finite-size and finite-temperature effects influence the operator product expansion in the conformally invariant O(N) vector model within a specific dimensional range, revealing conditions for consistency with bulk calculations.

Contribution

It provides a leading-order analysis of the OPE in a finite geometry, connecting finite-size scaling and phase transition conditions in the model.

Findings

Finite-size effects are consistent with bulk OPE under specific conditions.

Finite-temperature phase transition conditions emerge from the analysis.

The study clarifies the role of geometry in conformal field theories.

Abstract

We study the operator product expansion (OPE) of the auxiliary scalar field \lambda(x) with itself, in the conformally invariant O(N) Vector Model for 2<d<4, to leading order in 1/N in a strip-like geometry with one finite dimension of length L. We show that consistency of the finite-geometry OPE with bulk OPE calculations requires the physical conditions of, either finite-size scaling at criticality, or finite-temperature phase transition.