Spin Glasses by Marco Baity Jesi

Author:Marco Baity Jesi
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

3.9 An ex post Interpretation

We can reinterpret the results on the Heisenberg model with random anisotropic exchange interaction from an RG perspective. It was already established that in the isotropic limit there is a phase transition at . There are controversies on whether , but this is unimportant to us, because it is generally accepted that , though it was not verified until [BJ14d]. Therefore in the present section we mention the critical temperature as .

One of our main questions was whether the universality class changes when . Since in nature anisotropies are always present, though weak, the problem was initially tackled by studying low random anisotropies in [MM11]. To the light of the remarks of Sect. 1.​3.​5.​1, it was expectable that the numerical results be of hard interpretation. In fact when starting the RG flow from a small anisotropy, the system will initially feel strong effects from the FP. Furthermore, if the flow does not start close to , the numerical simulations will only feel at first the effects of the FP, and then those of the or FP (recall Fig. 1.​7 and discussion).

Of the three options that in Sect. 1.​3.​5.​1 are suggested to get away from this hard regime, we are able to adopt two, increasing drastically both the anisotropy and the lattice sizes, and finished obtaining also a better estimate of the critical temperature. The result is depicted in Fig. 3.6. Starting the flow from a large anisotropy leads the system far from the zone where echoes of the transition are strong, and simulating on larger lattices is equivalent to taking more RG steps, toward the Ising FP. Furthermore, large lattices gave us a better estimate of the critical temperature, so our movement in the phase diagram sped towards the Ising FP in an effective way. In terms of Fig. 3.6 we moved from the outer part of the Heisenberg fixed point influence (drawn in orange, smaller lattices), to the zone where the Ising behavior is strong, blue zone (), so we were able to measure an Ising behavior.

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