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arXiv:cond-mat/9707222 · Physical Review Letters

Exact Thermodynamics of the Double sinh-Gordon Theory in 1+1 Dimensions

Avinash Khare, Salman Habib, Avadh Saxena

Published Nov 17, 1997• 44 citations
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Abstract

We study the classical thermodynamics of a $(1+1)$-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.

DOI
10.1103/physrevlett.79.3797

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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.
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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 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 $\ensuremath{\mu}$.
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.
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.
We have developed a quantum-mechanical approach to study the critical properties of the quantum sine-Gordon equation describing a massless scalar field in 1+1 space-time dimensions with interaction density proportional to cos(${\mathrm{\ensuremath{\beta}}}_{0}$\ensuremath{\varphi}). The main feature of the present theory is to find an appropriate starting point for perturbation expansion, which can directly treat the infrared divergences encountered in the conventional perturbation theory. Exact critical conditions for the phase transition of the model are derived; and the quantum sine-Gordon Hamiltonian is found to correspond exactly to a free-field model. Near the critical regime, a gap will open near zero momentum in the excitation spectrum, and the corresponding ground-state wave function is a pairing quasiparticle state, analogous to the BCS superconducting state. It is proved that this ground state will possess quasi-long-range and quasi-off-diagonal long-range correlations. Comparisons with the Kosterlitz-Thouless classical theory of the two-dimensional Coulomb gas are also made.