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arXiv:solv-int/9711006

Comment on ``Equal-time temperature correlators of the one-dimensional Heisenberg XY chain'', preprint solv-int/9710028

Kirill Ilinski, A. S. Stepanenko

Published Nov 6, 1997• 1 citation
arXivHTMLPDF

Abstract

In the comment we give references to our papers where the problem was solved for more general case of time-dependent finite temperature correlators.
DOI
10.48550/arxiv.solv-int/9711006

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This thesis consists in a theoretical study of some one-dimensional models. Special attention is given to simple but non-trivial soluble models. The study starts with a resume of the Green function method which is the formalism used throughout the work. A discussion of coupled electron-phonon and spin-phonon systems is presented and a decoupling scheme introduced. The approximation gives the transition temperature for one-dimensional systems exactly, namely T = 0, and consequently represents a great improvement on the mean-field approximation. The impure XY-chain in a transverse field is studied and the formal solution presented. The static and time-dependent correlations are discussed, and the effect of the boundary term reviewed in details. The one-impurity case is solved for the isotropic chain, and it is shown how to derive from it the open chain result in the thermodynamic limit. The heat-flux operator is obtained for the impure isotropic chain by solving the continuity equation for energy density. The dynamics of the one-dimensional transverse Ising model is discussed within several approximations. The limitations of each approximation are examined in detail and the time-dependent correlations are calculated using the most successful approximation. It is also shown that this approximation is only valid in the high temperature limit in which case no critical behaviour is to be observed. The one-impurity solution of the isotropic impure XY-chain is extended to many impurities in two different cases. In the first case the specific heat of the dilute chain is calculated in the framework of the average t-matrix approximation, and the results compared with exact numerical calculations for finite chains. In the second case the specific heat and thermal conductivity of the isotropic XY-chain in a random transverse field are calculated using the coherent-potential approximation (CPA). The results are used to explain the low temperature thermal properties of praseodymium ethyl sulphate. Finally, two interacting systems, an electron-phonon system and a spin-phonon system, are studied after a brief discussion of the so-called Peierls instability. The first model is solved exactly in the framework of the approximation discussed at the beginning of the work. It presents a giant Kohn anomaly at zero temperature which drives a Peierls transition. The renormalized modes are discussed, and the real and imaginary parts of the phonon propagator are presented for wave vector π⁄a and several coupling constants and temperatures. The spin-phonon system is also studied in the framework of the approximation discussed initially in the work, and it is chosen in such a way that the relevant properties are given in terms of Green functions calculated in the transverse Ising model. The results obtained for the dynamics of this model are used, and since they are valid only in the high temperature limit, the discussion is restricted to this temperature region. Since this region is far away from the critical one, no critical behaviour is observed, and the study is restricted to a simple discussion of modes. The real and imaginary parts of the renormalized phonon propagator are presented for the wave vector π⁄a, high temperature and various coupling constants.
The one-dimensional spin-1 Heisenberg antiferromagnetic chains are studied by a generalized two-time Green's-function method at finite temperature. 〈Siz{\mathit{S}}_{\mathit{i}}^{\mathit{z}}Siz​〉 is set to zero at each site to guarantee the lack of the long-range order in the one-dimensional Heisenberg system. The Green's functions are decoupled in terms of the correlation functions of spin operators. A set of self-consistent equations of the correlation functions are derived and solved numerically. Correlation functions of spin operators, thermodynamic properties such as the internal energy, and specific heat in the entire temperature region are obtained. The Haldane gap in the excitation spectrum appears naturally in the analytical result. When k is zero, there is a gap 2\ensuremath{\Delta}, which is 1.0J and close to the theoretical result of White and Huse. And there is a broad maximum in the temperature dependence of the specific heat. The properties of the thermodynamic quantities are consistent with the numerical results on the finite chains. \textcopyright{} 1996 The American Physical Society.