Non-analytic quantum oscillator image of complete replica symmetry breaking

Abstract

We describe the effect of replica symmetry breaking in the field distribution function P(h) of the T = 0 Sherrington–Kirkpatrick (SK) model as the difference between a split Gaussian and the first excited state ψ₁ of a weakly anharmonic oscillator with non-analytic shift by means of the analogy P(h) ↔ |ψ₁(x)|. New numerical calculations of the leading 100 orders of replica symmetry breaking (RSB) were performed in order to obtain P(h), employing the exact mapping between the density of states ρ(E) of the fermionic SK model and P(h) of the standard model, as derived by Perez-Castillo and Sherrington. Fast convergence towards a fixed point function ρ(E) for infinite steps of RSB is observed. A surprisingly small number of harmonic oscillator wavefunctions suffices to represent this fixed point function. This allows us to determine an anharmonic potential V(x) with non-analytic shift, whose first excited state represents ρ(E) and hence P(h). The harmonic potential with unconventional shift V ₂(x) ∼ (|x| − x ₀)² = (x − x ₀ sign(x))² already yields a very good approximation, since anharmonic couplings of V(x) − V ₂(x) ∼ |x| ^m , m > 2, decay rapidly with increasing m. We compare the pseudo-gap-forming effect of replica symmetry breaking, hosted by the fermionic SK model, with the analogous effect in the Coulomb glass as designed by Davies, Lee and Rice, and described by Müller and Pankov.

Keywords:

• pseudo-gap
• replica symmetry breaking
• magnetic order
• numerical modelling
• quantum mechanical calculation
• disordered systems
• localised states
• critical phenomena

Acknowledgements

We wish to thank the DFG for support under Op28/7-2. One of us is indebted to A. Crisanti, C. De Dominicis, and T. Sarlat for useful remarks, and for hospitality extended to one of us (R.O.) at the CEA Saclay, where part of this work was initiated. We are also indebted to David Sherrington for his long-term interest in our research on low T and T = 0-RSB, and for his constant emphasis on the importance of P(h).