Search for a spin glass phase of the edge-cubic spin model on 3D lattice

Discreteness is important for the existence of finite temperature 

phase transition, not only for regular magnetic system but also 

for random system called spin glass (SG). This is exemplified by 

the presence of three dimensional (3D) SG phase for the Ising model 

and not for the Heisenberg model.  The SG  phase  is characterized by

the low-temperature frozen random spin orientation; considered

as a temporally ordered phase rather than spatially ordered. The present 

study probes one of discrete counterparts of the Heisenberg model, i.e.,

the edge-cubic  model where spins are allowed to have 12 possible

orientations, namely to the middle point of the edges of the cubic.

We search for the existence of SG phase on the square and cubic lattices 

with random mixture of ferromagnetic and anti-ferromagnetic interactions.

In previous study, the 2D ferromagnetic version of this model was reported to indicate two consecutive phase transitions, respectively associated 

with the broken of $O_h$ and $C_{3h}$ symmetries.  In searching for the spin glass phase, we use Replica Exchange algorithm of Monte Carlo method. We estimate the critical temperature and exponents of the existing SG phase transition.