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Journal of Physics A Mathematical and General

Thermodynamics of 1D soliton-bearing Hamiltonians: transfer integral structure and correlation functions

A. R. Bishop

Published Apr 1, 1981• 14 citations
Publisher

Abstract

Static correlation functions for the sine-Gordon chain and related one-dimensional models are described in terms of the transfer operator formalism. The characteristic signatures of kink-solitons within this formalism are discussed and classes of 'kink-sensitive' and 'kink-insensitive' correlations are emphasised. Special attention is given to the integrated structure factor integral S(q, omega )d Omega and the consistency of several calculations and models of S(q, omega is addressed. Possible applications to the interpretation of scattering experiments in quasi-one-dimensional ferro- and antiferromagnetic materials such as CsNiF 3 and (CD 3 ) 4 NMnCl 3 are discussed.

DOI
10.1088/0305-4470/14/4/017

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Static correlation functions for the sine-Gordon chain and related one-dimensional models are described in terms of the transfer operator formalism. The characteristic signatures of kink-solitons within this formalism are discussed and classes of 'kink-sensitive' and 'kink-insensitive' correlations are emphasised. Special attention is given to the integrated structure factor integral S(q, omega )d Omega and the consistency of several calculations and models of S(q, omega is addressed. Possible applications to the interpretation of scattering experiments in quasi-one-dimensional ferro- and antiferromagnetic materials such as CsNiF 3 and (CD 3 ) 4 NMnCl 3 are discussed.
We present and discuss the classical statistical mechanics of the static properties associated with a discretized sine-Gordon and ${\ensuremath{\varphi}}^{4}$ system by a numerical solution of the transfer-integral equation. The approximate replacement of the transfer-integral equation by a pseudo-Schr\"odinger equation is also considered to clarify the range of validity of this approach and of the associated WKB approximation. On this basis as well as low- and high-temperature perturbative expansions, it becomes possible to identify those thermodynamic properties and static form factors which are sensitive to or even dominated by the kinks. It is found that clear-cut kink effects are not only restricted to particular properties and to low and intermediate temperatures, but also, for the case of static form factors, to very small wave numbers.
We study the classical thermodynamics of a (1+1)(1+1)(1+1)-dimensional double-well sinh-Gordon theory. Remarkably, the Schr\"odinger-like equation resulting from the transfer integral method is quasi-exactly solvable at several temperatures. This allows exact calculation of the partition function and some correlation functions above and below the short-range order (``kink'') transition, in striking agreement with high resolution Langevin simulations. Interesting connections with the Landau-Ginzburg and double sine-Gordon models are also established.
We study the quantum-statistical mechanics, at low temperatures, of a one-dimensional antiferromagnetic Heisenberg model with two anisotropies. In the weak-coupling limit we determine the temperature dependences of the soliton energy and the soliton density. We have found that the leading correction to the sine-Gordon (SG) expression for the soliton density and the quantum soliton energy comes from the out-of-plane magnon mode, not present in the pure SG model. We also show that when an external magnetic field is applied, the chain supports a new type of kink, where the sublattices rotate in opposite directions.
A number of one-dimensional quantum systems like interacting fermions or spin chains can be described in terms of a generalized quantum sine-Gordon Hamiltonian. The transfer-matrix formulation of critical two-dimensional models also frequently leads to sine-Gordon Hamiltonians. We compute here the different correlation functions of such a model using a real-space renormalization technique. Our renormalization calculation gives us the exponents of the correlation functions as well as the corrections to the usual power-law behavior due to marginally irrelevant operators. Except on the critical line, where logarithmic corrections exist at all length scales, the correlation functions decay like power laws, with amplitude corrections from the marginal operators. The logarithmic corrections always appear at short length scale. The complete crossover between the two behaviors is given by our equations. The implication of our calculation for physical systems like the one-dimensional electron gas, quantum spin chains, and two-dimensional statistical systems is discussed.
A variational approach based on the path-integral formulation of the statistical mechanics is applied to calculate the partition function and the related thermodynamic quantities of one-dimensional kink bearing fields. This is done by determining an effective potential which includes in a complete quantum way the linear modes of the field, while treating variationally the nonlinear excitations. The treatment is applied both to integrable systems, like Sine-Gordon and to non integrable ones, like φ 4 and Double Sine-Gordon. The temperature renormalization can be separately studied both for the vacuum and the one-soliton sector in order to find the low temperature properties in self consistent one-loop approximation. Moreover a low-coupling expansion is given and its range of applicability is found to be much wider than the Wigner expansion. Comparisons with quantum Monte-Carlo results and exact results obtained by Bethe Ansatz are presented.
The thermodynamic properties of soliton-bearing systems can be obtained from the statistics of their soliton components, which reflect restrictions on phase space due to interactions. Going beyond the established agreement of the free energy with transfer-integral results, we have successfully tested first derivatives (energy, number density) and second derivatives (specific heat, compressibility) for a one-component soliton gas (corresponding to the case of the low-temperature sine-Gordon chain). A complete description of the equilibrium statistical properties of the soliton gas is given in terms of the averages of occupation numbers and their fluctuations. The latter are characterized by the absence of off-diagonal correlations.