Still many open questions exist related to the spin glass phase transition.
Being the renormalization group controlled by a zero temperature fixed point, it seems very reasonable to study the spin glass transition directly at zero temperature, e.g. varying the external field.
In this project, we focus on the study of long-range spin glass models that undergo a transition at zero temperature varying the external field.
In order to have a spin glass phase transition at zero temperature, we focus on models with m-components variables that are known to undergo a phase transition at a finite field, and on models defined on sparse random graphs.
We plan to study these models via the cavity method and via numerical simulations. For Ising spin glass models we aim at understanding in detail how the replica symmetry breaking transition arises. While for models with continuous variables, i.e. with an O(m) symmetry and m=2 (XY) or m=3 (Heisenberg), we aim at understanding the properties of the low-energy excitations via the study of the Hessian spectrum at the lowest energy minima.
The spin glass transition is possibly the most important and ubiquitous in strongly disordered models. Its comprehension is still not fully detailed but in a very small number of fully-connected models. The scope of the present project is to deepen our understanding of the spin glass transition in some fundamental models.
Apart from the obvious advance in knowledge, we believe the success of the present project may have also an impact on other close-by fields. For example, the vector spin glass models may be very good mean-field models for the glass transition in case we would find the features of realistic glass formers.
Having at hand a solvable mean-field model is always very desirable and usually boosts a deep understanding of the physical problem (the glass transition in this case).