Reduction of Spin Glasses applied to the Migdal-Kadanoff Hierarchical Lattice

Details Reduction of Spin Glasses applied to the Migdal-Kadanoff Hierarchical Lattice Reduction of Spin Glasses applied to the Migdal-Kadanoff Hierarchical Lattice

2003-02-20

arxiv.org/abs/cond-mat/0302424v3

Eur.Phys.J. B33 (2003) 439-445

Physics

Condensed Matter

Disordered Systems and Neural Networks

7 pages, RevTex, 6 figures (postscript), added results for d=4, some corrections; final version, as to appear in EPJB

Scientific paper

10.1140/epjb/e2003-00184-5

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 $\Delta E$ with system size $L$, $\sigma(\Delta E)\sim L^y$, is obtained as $y_3=0.25546(3)$ by reducing the equivalent of lattices up to $L=2^{100}$ in $d=3$, and as $y_4=0.76382(4)$ for up to $L=2^{35}$ 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.

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