  The concept of geometrical frustration in condensed matter physics refers to
the fact that a system has a locally preferred structure with an energy density
lower than the infinite ground state. This notion is however often used in a
qualitative sense only. In this article, we discuss a quantitative definition
of geometrical frustration in the context of lattice models of binary spins. To
this aim, we introduce the framework of local energy landscapes, within which
frustration can be quantified as the discrepancy between the energy of locally
preferred structures and the ground state. Our definition is scale-dependent
and involves an optimization over a gauge class of equivalent local energy
landscapes, related to one another by local energy displacements. This ensures
that frustration depends only on the physical Hamiltonian and its range, and
not on unphysical choices in how it is written. Our framework shows that a
number of popular frustrated models, including the antiferromagnetic Ising
model on a triangular lattice, only have finite-range frustration: geometrical
incompatibilities are local and can be eliminated by an exact coarse-graining
of the local energies.
