Lorentz-invariant scalar-field theories in $d+1$ dimensions with second-order derivative terms are unable to support static soliton solutions that are both finite in energy and stable for $d>2$, a result known as Derrick’s theorem. Lifshitz theories, which introduce higher-order spatial derivatives, need not obey Derrick’s theorem. We construct stable, finite-energy, static soliton solutions in Lifshitz scalar-field theories in $3+1$ dimensions with a dynamical critical exponent $z=2$. We exhibit three generic types: nontopological point defects, topological point defects, and topological strings. We focus mainly on Lifshitz theories that are defined through a superpotential and admit Bogomolnyi-Prasad-Sommerfield solutions. These kinds of theories are the bosonic sectors of supersymmetric theories derived from the stochastic dynamics of a scalar field theory in one higher dimension. If nature obeys a Lifshitz field theory in the ultraviolet, then the novel topological defects discussed here may exist as relics from the early universe. Their discovery would prove that standard field theory breaks down at short distance scales.

• Received 9 November 2010

DOI:https://doi.org/10.1103/PhysRevD.83.025007