 01231 Pietro Caputo, Fabio Martinelli
 Asymmetric diffusion and the energy gap above the 111 ground state of
the quantum XXZ model
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Jun 26, 01

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Abstract. We consider the anisotropic three dimensional XXZ Heisenberg ferromagnet
in a cylinder with axis along the $111$ direction and boundary
conditions that induce ground states describing an interface orthogonal
to the cylinder axis. Let $L$ be the linear size of the basis of the
cylinder. Because of the breaking of the continuous symmetry around the
$\hat z$ axis, the Goldstone theorem implies that the spectral gap above
such ground states must tend to zero as $L\to \infty$. In \cite{BCNS} it
was proved that, by perturbing in a subcylinder with basis of linear
size $R\ll L$ the interface ground state, it is possible to construct
excited states whose energy gap shrinks as $R^{2}$. Here we prove that,
uniformly in the height of the cylinder and in the location of the
interface, the energy gap above the interface ground state is bounded
from below by $\text{const.}L^{2}$. We prove the result by first
mapping the problem into an asymmetric simple exclusion process
on $\Z^3$ and then by adapting to the latter the recursive analysis to
estimate from below the spectral gap of the associated Markov
generator developed in \cite{CancMart}.
Along the way we improve some bounds on the equivalence
of ensembles already discussed in \cite{BCNS} and we establish an upper
bound on the density of states close to the bottom of the spectrum.
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