Stiffness of the Heisenberg Spin-Glass Model at Zero- and Finite-Temperatures in Three Dimensions

We examine the stiffness of the Heisenberg spin-glass (SG) model at both zero temperature (T=0) and finite temperatures (T ≠ 0) in three dimensions. We calculate the excess energies at T=0 which are gained by rotating and reversing all the spins on one surface of the lattice, and find that they increase with the lattice size L. We also calculate the excess free-energies at T ≠ 0 which correspond to these excess energies, and find that they increase with L at low temperatures, while they decrease with increasing L at high temperatures. These results strongly suggest the occurrence of the SG phase at low temperatures. The SG phase transition temperature is estimated to be TSG ~ 0.19J from the lattice size dependences of these excess free-energies.