= Solution
<Spontaneous symmetry breaking> occurs when the <statistical Hamiltonian> is invariant under a <symmetry (physics)> but a chosen equilibrium pure state is not. A small selecting field or a <boundary condition> chooses one of several states related by the <symmetry (physics)>. For a finite unconstrained system the symmetric ensemble generally has zero <order parameter>; spontaneous order is defined by taking the <thermodynamic limit> before removing the selecting field. A ferromagnet with a selected magnetization direction is the standard example.
If the broken <symmetry (physics)> is continuous, nearby symmetry-related states cost no energy for a uniform displacement along the manifold of states. Locality and a nonzero stiffness then make a slowly varying displacement cost <gradient> energy rather than a mass term. These gapless long-wavelength fluctuations are <Goldstone modes>. In the conventional relativistic internal-symmetry setting their count is the number of broken generators. Nonrelativistic dynamics can pair broken generators, as in a ferromagnet; gauge <symmetries> and broken space-time <symmetries> require additional qualifications. A discrete broken <symmetry (physics)> alone does not produce a <Goldstone mode>.
For an interface with a selected mean height, the continuous displacement is a rigid translation normal to the mean plane. The tension gives a stiffness for spatially varying translations. The following calculation concerns this static Goldstone sector. It determines the excitation energy and equilibrium fluctuations, but a frequency dispersion requires a separate inertial or dissipative dynamics. In low dimensions thermal roughness also limits the existence of a strictly localized flat interface in an infinite system.
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