Hierarchical spin glasses in a magnetic field: A renormalization-group study

By using renormalization-group (RG) methods, we study a non-mean-field model of a spin glass built on a hierarchical lattice, the hierarchical Edwards-Anderson model in a magnetic field. We investigate the spin-glass transition in a field by studying the existence of a stable critical RG fixed point (FP) with perturbation theory. In the parameter region where the model has a mean-field behavior—corresponding to $d\ge 4$ for a $d$-dimensional Ising model—we find a stable FP that corresponds to a spin-glass transition in a field. In the non-mean-field parameter region the FP above is unstable, and we determine exactly all other FPs: to our knowledge, this is the first time that all perturbative FPs for the full set of RG equations of a spin glass in a field have been characterized in the non-mean-field region. We find that all potentially stable FPs in the non-mean-field region have a nonzero imaginary part: this constitutes, to the best of our knowledge, the first demonstration for a spin glass in a field that there is no perturbative FP corresponding to a spin-glass transition in the non-mean-field region. Finally, we discuss the possible interpretations of this result, such as the absence of a phase transition in a field or the existence of a transition associated with a nonperturbative FP.

Figure 1

Hierarchical Edwards-Anderson model in a magnetic field with $k=3$. Dots represents spins $S_1,...,S_8$ from left to right, arcs represent interactions between spins below them, and magnetic fields are not shown. The red path starts from the two circled spins $S_1,S_3$ and it goes from the bottom to the top of the hierarchical tree until a common arc between the two spins is found: since this requires ascending two hierarchical levels, the hierarchical distance between $S_1$ and $S_3$ is $d_{13}=2$.