Spontaneous Generation of a Crystalline Ground State in a Higher Derivative Theory

TL;DR

This paper demonstrates that a higher derivative Lifshitz scalar model can spontaneously form a crystalline ground state with lattice structure, breaking translation symmetry and exhibiting stable phonon-like excitations.

Contribution

It introduces the concept of spontaneous lattice formation in a non-relativistic higher derivative scalar field theory, linking microscopic interactions to macroscopic crystalline order.

Findings

Ground state has a lattice structure with discrete translation symmetry.

Crystalline ground state is stable below a critical velocity.

Fluctuations exhibit phonon-like dispersion under certain conditions.

Abstract

The possibility of Spontaneous Symmetry Breaking in momentum space in a generic Lifshitz scalar model - a non-relativistic scalar field theory with higher spatial derivative terms - has been studied. We show that the minimum energy state, the ground state, has a lattice structure, where the translation invariance of the continuum theory is reduced to a discrete translation symmetry. The scale of translation symmetry breaking (or induced lattice spacing) is proportional to the inverse of the momentum of the condensate particle. The crystalline ground state is stable under excitations below a certain critical velocity. The small fluctuations above the ground state can have a phonon like dispersion under suitable choice of parameters. At the beginning we have discussed the effects of next to nearest neighbour interaction terms in a model of linear triatomic molecule depicted by a linear system of three particles of same mass connected by identical springs. This model is relevant since in the continuum limit the next to nearest neighbour interaction terms generate higher (spatial) derivative wave equation, the main topic of this paper.