Bruno Nachtergaele
The spectral gap for some spin chains with discrete symmetry breaking
(150K, Plain TeX)
ABSTRACT. We prove that for any finite set of generalized valence bond solid (GVBS)
states of a quantum spin chain there exists a translation invariant
finite-range Hamiltonian for which this set is the set of ground states.
This result implies that there are GVBS models with arbitrary broken
discrete symmetries that are described as combinations of lattice
translations, lattice reflections, and local unitary or anti-unitary
transformations. We also show that all GVBS models that satisfy some
natural conditions have a spectral gap. The existence of a spectral gap is
obtained by applying a simple and quite general strategy for proving lower
bounds on the spectral gap of the generator of a classical or quantum spin
dynamics. This general scheme is interesting in its own right and
therefore, although the basic idea is not new, we present it in a
system-independent setting. The results are illustrated with an number of
examples.