The 2D effective field theory of interfaces derived from 3D field theory

TL;DR

The paper addresses how to derive a 2D effective theory for interfaces between coexisting phases from a 3D $φ^4$ field theory. It computes the one-loop energy splitting around a kink in a finite cylindrical geometry using $$-function (zeta-function) regularization to evaluate the determinant ratio, revealing a decomposition into a massless 2D conformal field theory on the interface and finite corrections. The main result is an explicit expression for the interface energy $E(L_1,L_2)=\frac{C}{[\mathrm{Im}(\tau)]^{1/2}|\eta(\tau)|^2}\,e^{-\sigma L_1 L_2}$ with $\tau=iL_1/L_2$, linking the 3D theory to a $2D$ $c=1$ free scalar CFT on the interface and validating the Gaussian capillary wave model. This provides a principled derivation of the CWM from a microscopic 3D field theory and sets the stage for exploring non-Gaussian corrections and comparisons with lattice simulations.

Abstract

The one--loop determinant computed around the kink solution in the 3D $φ^4$ theory, in cylindrical geometry, allows one to obtain the partition function of the interface separating coexisting phases. The quantum fluctuations of the interface around its equilibrium position are described by a $c=1$ two--dimensional conformal field theory, namely a 2D free massless scalar field living on the interface. In this way the capillary wave model conjecture for the interface free energy in its gaussian approximation is proved.