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Physical review. B, Condensed matter

Finite-lattice expansion for quantum spin chains

Rajamani Narayanan, Rajiv R. P. Singh

Published Dec 1, 1990• 7 citations
Publisher

Abstract

We investigate the thermal properties of s=1/2 and 1 antiferromagnetic Heisenberg spin chains by an expansion in finite lattices. The expansion is based on expressing the quantity of interest, in the thermodynamic limit, as a sum over contributions from different length scales. The first few terms in the expansion, obtained by diagonalizing chains of length up to 13 for s=1/2 and length up to 9 for s=1, provide an accurate estimate for the internal energy and specific heat for T/J\ensuremath{\ge}0.25 and 0.35, respectively. At lower temperatures, by studying the exponential decay of the various terms in the expansion, we estimate the correlation length \ensuremath{\xi}. We get good estimates for \ensuremath{\xi} down to T/J=0.06 for s=1/2 and T/J=0.02 for s=1. We find that for s=1 \ensuremath{\xi} increases much more slowly than for s=1/2 and is consistent with a value of \ensuremath{\xi}\ensuremath{\approxeq}7 at T=0.

DOI
10.1103/physrevb.42.10305

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We investigate the thermal properties of s=1/2 and 1 antiferromagnetic Heisenberg spin chains by an expansion in finite lattices. The expansion is based on expressing the quantity of interest, in the thermodynamic limit, as a sum over contributions from different length scales. The first few terms in the expansion, obtained by diagonalizing chains of length up to 13 for s=1/2 and length up to 9 for s=1, provide an accurate estimate for the internal energy and specific heat for T/J\ensuremath{\ge}0.25 and 0.35, respectively. At lower temperatures, by studying the exponential decay of the various terms in the expansion, we estimate the correlation length \ensuremath{\xi}. We get good estimates for \ensuremath{\xi} down to T/J=0.06 for s=1/2 and T/J=0.02 for s=1. We find that for s=1 \ensuremath{\xi} increases much more slowly than for s=1/2 and is consistent with a value of \ensuremath{\xi}\ensuremath{\approxeq}7 at T=0.
Thermodynamic properties of the quantum Heisenberg spin chains with S=1/2,S=1/2,S=1/2, 1, and 3/23/23/2 are investigated using the transfer-matrix renormalization-group method. The temperature dependence of the magnetization, susceptibility, specific heat, spin-spin correlation length, and several other physical quantities in a zero or finite applied field are calculated and compared. Our data agree well with the Bethe ansatz, exact diagonalization, and quantum Monte Carlo results and provide further insight into the quantum effects in the antiferromagnetic Heisenberg spin chains.
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.
Using the semiclassical method called pure-quantum self-consistent harmonic approximation (PQSCHA), we evaluate thermodynamic quantities of one-dimensional Heisenberg ferromagnets and antiferromagnets. Since the PQSCHA reduces their evaluation to classical-like calculations, we take advantage of Fisher's exact solution [M. E. Fisher, Am J. Phys. 32, 343 (1964)] to get all results in an almost fully analytical way. Explicitly considered here are the specific heat, the correlation length, and the susceptibility. Good agreement with available numerical data and Monte Carlo simulations is found for S>1S>1S>1 ferromagnets and antiferromagnets; for the latter it is seen that topological terms and the related Haldane gap are relevant only for the lowest spin values and temperatures.
Thermodynamic properties of S=2 antiferromagnetic Heisenberg chains are studied not only under the periodic boundary condition but also under the open one employing a quantum Monte Carlo method. Temperature and size dependences of the energy, the specific heat, and the magnetic susceptibility are calculated and edge effects on them are investigated in detail. The specific heat shows a well-pronounced Schottky anomaly but the maximum is located at a temperature much larger than the Haldane gap of the system. As temperature goes to zero, the magnetic susceptibility vanishes for the periodic chains, while it diverges for the open chains. The edge contribution in the open-chain susceptibility is attributed to the two S=1 effective spins localized at the chain ends at low temperatures, while to an S=2 free spin at high temperatures. This is a visualization of a quantum-classical crossover and an evidence that the present model is a Haldane system. High-temperature behaviors of the thermodynamic quantities are also discussed with the help of a series-expansion method. \textcopyright{} 1996 The American Physical Society.
Energy spectra and thermodynamic quantities (partition function, entropy, specific heat, susceptibility) of ferrimagnetic Heisenberg chains, made up of two spin sublattices [(1/2-S], are discussed from the exact results computed for finite rings. Extrapolation of the theoretical data versus the length of finite rings N is discussed for S ranging from 1 to (5/2 and compared with previous reported findings for the [(1/2-\ensuremath{\infty}] spin chain. In the low-temperature region, similarities between the ferrimagnetic chains [(1/2-S] and the regular ferromagnetic ones (with spin S-(1/2) are used for the determination of the limiting curves.