Discontinuities in disordered systems

Abstract

The entropyS T (j) of a two-dimensional Ising spin glass with an independent distribution of the random couplingp(J)=x·δ(J+1)+(1-x)δ(J-j) is discontinuous for temperatureT=0 and rationalj>0 and continuous elsewhere. The integrated density of frequenciesk M (ω2) of an one-dimensional chain of coupled oscillators with an independent distribution of the random massesp(m)=x·δ(m-1)+(1-x)δ(m-M) has the same behaviour, whereω2 corresponds toj andM to 1/T. The discontinuity points for infiniteM are, for sufficiently large but finiteM, special, frequencies, wherek M (ω2) has a Lifshitz singularity.