Conjectured Exact Locations of Dynamical Transition Points for the +-J Ising Spin Glass Model

Details Conjectured Exact Locations of Dynamical Transition Points for the +-J Ising Spin Glass Model Conjectured Exact Locations of Dynamical Transition Points for the +-J Ising Spin Glass Model

2010-04-05

arxiv.org/abs/1004.0654v6

Physics

Condensed Matter

Disordered Systems and Neural Networks

8 pages, 1 figure. v6: minor corrections/additions

Scientific paper

The conjectured exact locations of the dynamical transition points for the +-J Ising spin glass model are theoretically shown based on a percolation theory and conjectures. The dynamical transition is a transition for the time evolution of the distance between two spin configurations. The distance is called the damage or the Hamming distance. The conjectured exact locations of the dynamical transition points are obtained by using the values of the threshold fractions of the random bond percolation problem. The present results are obtained as locations of points on the Nishimori line which is a special line on the phase diagram. We obtain TD = 2/ ln (z / z - 2) and pD = z / 2 (z - 1) for the Bethe lattice, TD -> infinity and pD -> 1 / 2 for the infinite-range model, TD = 2 / ln 3 and pD = 3 / 4 for the square lattice, TD ~ 3.9347 and pD ~ 0.62441 for the simple cubic lattice, TD ~ 6.191 and pD ~ 0.5801 for the 4-dimensional hypercubic lattice, and TD = 2 / ln [1 + 2 sin(pi / 18) / 1 - 2 sin(pi / 18)] and pD = [1 + 2 sin(pi / 18)] / 2 for the triangular lattice, when J / kB = 1, where z is the coordination number, J is the strength of the exchange interaction between spins, kB is the Boltzmann constant, TD is the temperature at the dynamical transition point, and pD is the probability, that the interaction is ferromagnetic, at the dynamical transition point.

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