Bulk and Boundary Critical Behavior at Lifshitz Points

TL;DR

This paper reviews recent advances in understanding bulk and boundary critical phenomena at Lifshitz points, emphasizing field-theoretic approaches, boundary conditions, and the impact of surface orientation on universality classes.

Contribution

It provides a systematic survey of field-theoretic renormalization group results for Lifshitz points and explores boundary critical behavior depending on surface orientation.

Findings

Dimensionality expansions align with Monte Carlo results.

Boundary universality classes depend on surface orientation.

Distinct boundary conditions arise for perpendicular and parallel surface planes.

Abstract

Lifshitz points are multicritical points at which a disordered phase, a homogeneous ordered phase, and a modulated ordered phase meet. Their bulk universality classes are described by natural generalizations of the standard $\phi^4$ model. Analyzing these models systematically via modern field-theoretic renormalization group methods has been a long-standing challenge ever since their introduction in the middle of the 1970s. We survey the recent progress made in this direction, discussing results obtained via dimensionality expansions, how they compare with Monte Carlo results, and open problems. These advances opened the way towards systematic studies of boundary critical behavior at $m$-axial Lifshitz points. The possible boundary critical behavior depends on whether the surface plane is perpendicular to one of the $m$ modulation axes or parallel to all of them. We show that the semi-infinite field theories representing the corresponding surface universality classes in these two cases of perpendicular and parallel surface orientation differ crucially in their Hamiltonian's boundary terms and the implied boundary conditions, and explain recent results along with our current understanding of this matter.