Physics – Condensed Matter – Disordered Systems and Neural Networks
Scientific paper
2011-05-16
Physics
Condensed Matter
Disordered Systems and Neural Networks
11 pages, 3 figures, submitted to Phil. Mag., special edition in honour of David Sherrington's 70th birthday
Scientific paper
We describe the effect of replica symmetry breaking in the field distribution function P(h) of the T=0 SK-model as the difference between a split Gaussian and the first excited state $\psi_1$ of a weakly anharmonic oscillator with nonanalytic shift by means of the analogy $P(h)<->|\psi_1(x)|$. 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 density of states $\rho(E)$ 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 $\rho(E)$ for infinite steps of RSB is observed. A surprisingly small number of harmonic oscillator wave-functions suffices to represent this fixed point function. This allows to determine an anharmonic potential V(x) with nonanalytic shift, whose first excited state represents $\rho(E)$ and hence P(h). The harmonic potential with unconventional shift $V_2(x)\sim (|x|-x_0)^2=(x-x_0\,sign(x))^2$ yields already a very good approximation, since anharmonic couplings of $V(x)-V_2(x)\sim |x|^{m}, m>2,$ decay rapidly with increasing m. We compare the pseudogap-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-Rice and described by M\"uller-Pankov.
Oppermann Reinhold
Schenck Hannes
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