Variational Approach to Quantum Statistical Mechanics of Nonlinear Systems with Application to Sine-Gordon Chains

Abstract

The path-integral method is used for determination of the quantum corrections to the free energy of nonlinear systems. All quantum effects of the harmonic part of the potential are considered and a variational principle is used to account for the quantum corrections due to the anharmonic part. Correct renormalized frequencies are obtained at any temperature and an effective potential to be inserted in the configurational integral is found. A new general expression for the partition function at any temperature in the low-coupling limit is obtained.PACS numbers: 05.30.Ch, 75.10.Jm Methods of reducing quantum statistical calculations to classical ones 1,2 are finding nowadays increasing applications to fluid 3 and solid 4 models. In one-dimensional magnetic chains, classical models are unsatisfactory 5 (see, e.g., the CsNiF 3 case) and quantum corrections have been taken into account 6,7 in the low-coupling limit, restricted to the noninteracting soliton approximation, which fails in the range of temperature where the peak of the specific heat occurs. 8 Different expansions have also been proposed. 9, 10 We use here a path-integral approach which improves upon previous variational treatments 11 by considering all the quantum effects of the harmonic part of the potential, while the variational principle, in the first cumulant approximation, is used to account for the renormalized quantum corrections, both of potential and of frequencies, due to the anharmonic part. Besides unifying the previous methods, 6,9,10 our approach gives a correct effective potential for all temperatures in the low-coupling limit and can be applied to systems where the energy scales of linear and nonlinear excitations are well separated. Relegating the mathematical details of the theory to an extended paper, we present in this Letter a brief account of the formalism and a concrete application to the sine-Gordon chain, including a numerical evaluation of the specific heat throughout the temperature range, thus clarifying some controversial issues on soliton behavior in planar magnetic chains.The path-integral form of the partition function Z = e ~& F isThe functional integral is evaluated over all the closed paths and the potential V is taken in the discrete form K=(m/2) Z x&jXjg Z U{x,).(2)We assume that Bjj = B i + nj + n contains all the harmonic interaction [£/"(0) = 0]. The anharmonic part U(x) has been taken to be local for simplicity, since the extension to a general U(x) is straightforward. We defineThe functions W, w,-, and w u must be chosen in order to minimize the right-hand side of the first-order cumulant inequalitywhere F 0 and the average are defined with use of V 0 instead of V. It is worthwhile to note that w u (y) are not taken equal to djj V(y) = mB^•+gd iJ U"(y), but are determined by the variational principle. As will be shown in the following, this is a crucial point to avoid unphysical results at lowest temperatures and just this full application of the variational principle leads to the correct frequency renormaliza...

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