##
Lower Critical Dimension of Ising Spin Glasses

A.K. Hartmann
and
A.P. Young

cond-mat/0105310.
**Abstract**

Exact ground states of two-dimensional Ising spin glasses with Gaussian
and bimodal (+/- J) distributions of the disorder are calculated using a
``matching''algorithm, which allows
large system
sizes of up to N=480^2 spins to be investigated. We study domain walls
induced by
two rather different types of boundary-condition changes,
and, in each case, analyze the system-size dependence of
an appropriately defined
``defect energy'', which we
denote by Delta E.
For Gaussian disorder, we find a power-law behavior
Delta E ~ L^theta, with
theta=-0.266(2) and theta=-0.282(2) for the two types of
boundary condition changes. These results are in reasonable agreement with
each
other, allowing for small systematic effects.
They also agree well with earlier
work on smaller sizes. The
negative value indicates that
two dimensions is below the
lower critical dimension d_c.
For the +/- J
model, we obtain a different result, namely
the domain-wall energy saturates at a nonzero value for L -> infinity,
so \theta = 0,
indicating that the lower critical dimension for the +/- J model
is exactly d_c=2.

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