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American Journal of Physics

Green's Function Treatment of Short Linear Heisenberg Chains

Henry Unruh

Published Jun 1, 1972• 1 citation
Publisher

Abstract

The Callen and Tyablikov decoupling schemes used in the Green's function method are tested numerically for short Heisenberg linear chains which are first subjected to a constraint. The constraint is imposed to insure that internal excitations alone contribute to the calculated thermal averages so that in this sense these averages correspond to thermodynamic quantities. It is found that improved agreement with the exact results occurs if the Tyablikov decoupling is corrected by a term linear in the temperature.

DOI
10.1119/1.1986685

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The Callen and Tyablikov decoupling schemes used in the Green's function method are tested numerically for short Heisenberg linear chains which are first subjected to a constraint. The constraint is imposed to insure that internal excitations alone contribute to the calculated thermal averages so that in this sense these averages correspond to thermodynamic quantities. It is found that improved agreement with the exact results occurs if the Tyablikov decoupling is corrected by a term linear in the temperature.
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The one-dimensional spin-1/2 Ising-Heisenberg ferromagnets are studied by the Green's function method at finite temperatures. The Green's function is decoupled in terms of the correlation functions of 〈S0z{\mathit{S}}_{0}^{\mathit{z}}S0z​Snz{\mathit{S}}_{\mathit{n}}^{\mathit{z}}Snz​〉 and 〈S0+{\mathit{S}}_{0}^{+}S0+​${\mathit{S}}_{\mathit{n}}^{\mathrm{\ensuremath{-}}}$〉 under the constraint of 〈Sz{\mathit{S}}^{\mathit{z}}Sz〉=0, from which a set of self-consistent equations are derived and solved numerically. The thermodynamic properties such as internal energy, specific heat, and zero-field parallel susceptibility are calculated. Our results predict the correct behavior for the correlation functions and thermodynamic quantities in the entire temperature region, in agreement both with the exact numerical results on the finite chain and the experimental data.
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.
An approximate calculation of the excitation spectrum of the Heisenberg linear chain is performed by the Green's-function technique. Apart from the usual transverse Green's function, a longitudinal Green's function is also found useful. The decoupling procedure is not based on the two-sublattice picture, but utilizes the knowledge of known exact results about the ground state of the linear chain and the insight obtained from a model with explicitly known ground state. The equations are solved in a self-consistent manner, and the spectrum, except near the edge of the Brillouin zone, compares well the exact des Cloizeaux-Pearson result. We also provide an expression for the intensity for neutron-scattering studies; this can be checked against measurements on CuCl2{\mathrm{Cl}}_{2}Cl2​\ifmmode\cdot\else\textperiodcentered\fi{}2N(C5{\mathrm{C}}_{5}C5​D5{\mathrm{D}}_{5}D5​) (CPC).
The Green's function formalism proposed by Kondo and Yamaji for low-dimensional spin systems with S =1/2 has been extended to the cases of S >1/2. We apply this formalism to the study of one- and two-dimensional Heisenberg ferromagnets over the whole temperature range. It is shown that our theory can reproduce the correct results obtained by the high-temperature expansion method. On the other hand, the results at low temperatures are similar to those of the modified spin-wave theory, which is considered to predict the low-temperature properties of such systems rather correctly. The gross behaviors of the calculated thermodynamic quantities of ferromagnetic Heisenberg chains agree with those obtained by the exact diagonalization method.
A new method to obtain the knowledge for the ground state energy as well as one for the elementary excitation energy in an interacting many particle system is considered from the intuitive approach in the double time Green function theory.Using this method of approach the statistical diagonalization which may include properly the effects of temperature and other external fields can be performed when the Green functions are solved by an appropriate approximation, and the physical meaning of the results is considered.The application to the Heisenberg antiferromagnet is shown to lead the temperature and external magnetic field dependent results in the approximation of Tyablikov decoupling.