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 ($T \ne 0$) 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 $T \ne 0$ 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 $T_{\rm SG} \sim 0.19J$ from the lattice size dependences of these excess free-energies.