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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 $d \geq 4$ 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 determined 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 non-perturbative FP.