Document 
Eur. Phys. J. B 33, 439-445 (2003)
DOI: 10.1140/epjb/e2003-00184-5
Reduction of spin glasses applied to the Migdal-Kadanoff hierarchical lattice
S. BoettcherPhysics Department, Emory University, Atlanta, Georgia 30322, USA
(Received 20 February 2003 / Received in final form 4 April 2003 Published online 3 July 2003)
Abstract
A reduction procedure to obtain ground states of spin glasses on
sparse graphs is developed and tested on the hierarchical lattice
associated with the Migdal-Kadanoff approximation for low-dimensional
lattices. While more generally applicable, these rules here lead to a
complete reduction of the lattice. The stiffness exponent governing
the scaling of the defect energy
with system size
L,
, is obtained as
y3=0.25546(3) by
reducing the equivalent of lattices up to
L=2100 in
d=3, and as
y4=0.76382(4) for up to
L=235 in
d=4. The reduction rules
allow the exact determination of the ground state energy, entropy, and
also provide an approximation to the overlap distribution. With these
methods, some well-know and some new features of diluted hierarchical
lattices are calculated.
05.50.+q - Lattice theory and statistics (Ising, Potts, etc.).
75.10.Nr - Spin-glass and other random models.
02.60.Pn - Numerical optimization.
© EDP Sciences, Società Italiana di Fisica, Springer-Verlag 2003