Ising spin glass: developments and challenges

Alexander K. Hartmann

Alba Theumann, Vieira Gusmão

Jing-Huei Chen, T. C. Lubensky

A Landau-Ginzburg phenomenological free energy for the Edwards and Anderson spin-glass model when there is competition between spin-glass and ferromagnetic ordering is developed. This free energy obtained with the use of the $n\ensuremath{\rightarrow}0$ replication procedure is analyzed using mean-field theory and the $\ensuremath{\epsilon}$ expansion. Critical exponents for the ferromagnetic-spin-glass multicritical point are calculated in $6\ensuremath{-}\ensuremath{\epsilon}$ dimensions. For Ising systems, $\ensuremath{\nu}=\frac{1}{2}+(\frac{2}{3})\ensuremath{\epsilon}$ and $\ensuremath{\varphi}=1+(\frac{1}{2})\ensuremath{\epsilon}$. For $\mathrm{XY}$ and Heisenberg systems, these exponents are complex. This result is not fully understood. The Harris-Plischke-Zuckermann model for amorphous magnetism is shown to have an Ising-like spin-glass fixed point in high enough dimension.

E. Nogueira, S. Coutinho, Fernando Dantas Nobre +1 more

M. V. Feigel’man, L. B. Ioffe

The analytical theory of slow relaxation of the Ising and vector 3-D spin glasses with the finite range interaction near the freezing transition point Tf has been developed. In the Ising case, the maximum relaxation time tmax increases as tmax ∼ ( T - Tf)-6 exp[C(T - T f)-1/4]. For the Heisenberg spins tmax ∼ (T - T f)-10.5. It is shown that the inclusion of weak random-anisotropy exchange leads to the Ising asymptotics for t max in the Heisenberg spin glass.

Hamid Khoshbakht, Martin Weigel

The Ising spin glass in two dimensions is a model system with extraordinarily rich behavior. Although it does not show a spin-glass phase at finite temperatures, it exhibits many of the salient features of fully fledged spin glass systems. The authors use recently developed mappings to graph-theoretic problems together with highly efficient implementations of combinatorial optimization algorithms to determine exact ground states for systems with up to $10,000\ifmmode\times\else\texttimes\fi{}10,000$ spins. These and further results from new algorithms for fully periodic lattices and for systems with degeneracies allow for the determination of critical exponents with unprecedented accuracy.