Junichi Tsuda - Tokyo Institute of Technology #
Exactness of the mean-field theory for the random-field spin system with 
long-range interactions #
We study the spin system with long-range interactions 
and show the exactness of the mean-field theory under certain mild 
conditions [1].
This is a generalization of the result of Mori [2] for the non-random 
and spin-glass cases.
To treat random fields, we evoke the self-averaging property of a 
function of random fields,
which is our new contribution to the framework established in [2].
The result is that the mean-field theory gives the exact expression of 
the canonical free energy
for systems with power-decaying interactions if the power is smaller 
than the spatial dimension.
This is a highly non-trivial result because each site is given a very 
different environment
by site-dependent random fields
whereas the mean-field theory highlights the averages of physical 
quantities over many sites.
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[1] J. Phys. Soc. Jpn. (to be published)
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[2] T. Mori, "Instability of the mean-field states and generalization of 
phase separation in long range interacting systems", Phys. Rev. E 84, 
031128 (2011)