From Ginzburg-Landau to Hilbert-Einstein via Yamabe

TL;DR

This paper reveals a deep connection between Ginzburg-Landau equations and Hilbert-Einstein gravity in higher dimensions, using mathematical results to unify phase transition theory with geometric gravity and string theory methods.

Contribution

It demonstrates that Ginzburg-Landau equations can be derived from the Hilbert-Einstein action in higher dimensions, completing Lifshitz's group-theoretical work and extending string-theoretic techniques.

Findings

Ginzburg-Landau equations derived from gravity action in dimensions ≥3

Unified phase transition theory with geometric gravity models

Extended string-theoretic path integral methods to higher dimensions

Abstract

In this work, based on some mathematical results obtained by Yamabe, Osgood, Phillips and Sarnak, we demonstrate that in dimensions three and higher the famous Ginzburg-Landau equations used in theory of phase transitions can be obtained (without any approximations) by minimization of the Riemannian-type Hilbert-Einstein action functional for pure gravity in the presence of cosmological term. We use this observation in order to bring to completion the work by Lifshitz (done in 1941) on group-theoretical refinements of the Landau theory of phase transitions. In addition, this observation allows us to develop a systematic extension to higher dimensions of known string-theoretic path integral methods developed for calculation of observables in two dimensional conformal field theories.