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Bulk and Boundary Critical Behavior at Lifshitz Points
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
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 -axial Lifshitz points. The possible boundary critical
behavior depends on whether the surface plane is perpendicular to one of the
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.Comment: Invited contribution to STATPHYS 22, to be published in the
Proceedings of the 22nd International Conference on Statistical Physics
(STATPHYS 22) of the International Union of Pure and Applied Physics (IUPAP),
4--9 July 2004, Bangalore, Indi