THE FREE-ENERGY OF SOLITONS COUPLED TO FERMIONS - ABSENCE OF A GRADIENT EXPANSION AT ZERO TEMPERATURE DUE TO BOUND-STATES

Polyacetylene, regarded as one of the simplest systems bearing soliton s which are coupled to fermions, is investigated at zero temperature. With lambda-denoting the soliton size in units of the coherence length , solitons with lambda >> 1 are considered. It is found, from the exac t results of Nakahara et al (for a hyperbolic tangent soliton profile) , that the soliton creation energy does not possess an expansion in po wers of 1/lambda, as would be expected of a gradient expansion. Rather , the leading terms in a large-lambda expansion are found to be a-lamb da + b-lambda-1/2 with constant a and b. Calculations are presented fo r general soliton profiles. They show that the term linear in lambda i s the soliton creation energy to zeroth order in gradients. The term b -lambda-1/2 is shown to arise purely from bound states of the fermions (electrons) trapped by the soliton. The evaluation of the coefficient b requires the extraction of the finite difference between a divergen t sum and integral, a procedure employed in the Casimir effect.