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Physical review. B, Condensed matter

Classical statistical mechanics of the sine-Gordon and φ4 chains. Static properties

T. Schneider, E. Stoll

Published Dec 1, 1980• 175 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.22.5317

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We present and discuss the classical statistical mechanics of the static properties associated with a discretized sine-Gordon and \ensuremathφ4{\ensuremath{\varphi}}^{4}\ensuremathφ4 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.
The quantum thermodynamics of the nonintegrable ${\ensuremath{\varphi}}^{4}$ one-dimensional chain is studied by means of a classical effective potential, which includes in a fully quantum way the linear modes of the field. In contrast to the case of the sine-Gordon chain, integrable in the continuum limit, exact results are not available, so that approximate quantum calculations appear to be useful. This effective potential is determined by a variational approach developed in previous papers, which is based on the path-integral formulation of statistical mechanics. The temperature renormalization is studied, in the limit of low temperature, by means of a self-consistent saddle-point method, both for the vacuum and the one-kink sectors, and the results of the semiclassical approximation are recovered. Important results are obtained by a new low-coupling expansion for the effective potential. Its range of validity in temperature is much wider than the range of previous high-temperature expansions. The results for the nonlinear contributions to internal energy and specific heat, obtained by means of original transfer-matrix computations, are finally presented and discussed.
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 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.
Making use of thermal-Green's-function technique, we study the quantum-statistical mechanics of a sine-Gordon system in 1 + 1 dimensions. In the weak-coupling limit, the temperature dependences of the soliton energy, Es{E}_{s}Es​, the soliton inertial mass, and the soliton density are determined. At high temperatures (T>mT>mT>m, where mmm is the mass of the fundamental field), Es{E}_{s}Es​ decreases monotonically as the temperature increases, and Es{E}_{s}Es​ jumps to zero around T=TcrT={T}_{\mathrm{cr}}T=Tcr​ ($\ensuremath{\equiv}{e}^{\ensuremath{-}1}{{E}_{s}}^{0}$), where Es0{{E}_{s}}^{0}Es​0 is the soliton energy at T=0T=0T=0 K. The soliton density agrees with the classical statistical-mechanics results for ${T}_{\mathrm{cr}}>T\ensuremath{\gg}m$, if Es{E}_{s}Es​ in the classical theory is replaced by the temperature-dependent one of the present theory.
We solve the finite-temperature Bethe-ansatz (BA) integral equations with coupling constant $\ensuremath{\mu}=\ensuremath{\pi}(1\ensuremath{-}{n}^{\ensuremath{-}1})+0$, in the semiclassical limit $n=\mathrm{integer}\ensuremath{\rightarrow}\ensuremath{\infty}$. After eliminating long strings and then an exact bosonization, transforming $n\ensuremath{-}2$ fermion strings into a single boson, the BA integral equations are reduced to two, coupling bosons and two fermions (kinks and antikinks). In the classical limit these iterate exactly to the sine-Gordon free energy found by the transfer integral method and by other methods recently reported. We suggest that our result can be extended to general values of $\ensuremath{\mu}$.
This paper reviews soliton theories for classical statistical mechanics of the massive-Thirring-sine-Gordon model. These theories attempt not only to reproduce the classical transfer matrix method result but also to provide thermodynamic information about exact nonlinear excitations of the model. Three theories are discussed. Firstly, the soliton–gas phenomenology which is successful in the Toda lattice is shown to be not even approximately correct for the sine-Gordon model. This breakdown is due to the incorrect counting of the number of degrees of freedom. Secondly, the soliton–phonon theory which reproduces the WKB result is also shown to be not exact. This non-exactness is ascribed to the fact that the nonlinear phonons are not exact elementary excitations in the model. Thirdly, a non-gas phenomenological soliton–breather theory is derived as a classical limit of quantum statistical mechanics of the model. The exactness of the last theory is demonstrated through an accurate numerical analysis of its basic equations.