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Journal of the Physical Society of Japan

Low Temperature Thermodynamics of Magnetic Chains

Naobumi Honda, Harukazu Igarashi

Published Sep 15, 1984• 2 citations
Publisher

Abstract

A full description of the previously proposed method of calculating low temperature specific heat and susceptibility of quantum spin chains are given. By the use of internal energy as the expansion variable the domain of convergence of the high temperature series is remarkably extended. The low temperature properties are obtained by continuing it in a natural way to polynomials in internal energy, whose form is suggested by the spin wave theory. The method is checked in the S =1/2 XY model and is then applied to the Heisenberg model of S =1/2, 1, 3/2 and 2. The results are satisfactory in ferromagnets and reasonable in the S =1/2 antiferromagnet. In antiferromagnets of S ≧1 the results suggest inadequacy of our polynomial, which is, presumably, due to the existence of an energy gap.

DOI
10.1143/jpsj.53.2930

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A full description of the previously proposed method of calculating low temperature specific heat and susceptibility of quantum spin chains are given. By the use of internal energy as the expansion variable the domain of convergence of the high temperature series is remarkably extended. The low temperature properties are obtained by continuing it in a natural way to polynomials in internal energy, whose form is suggested by the spin wave theory. The method is checked in the S =1/2 XY model and is then applied to the Heisenberg model of S =1/2, 1, 3/2 and 2. The results are satisfactory in ferromagnets and reasonable in the S =1/2 antiferromagnet. In antiferromagnets of S ≧1 the results suggest inadequacy of our polynomial, which is, presumably, due to the existence of an energy gap.
An improved Padé method is applied to the entropy and the reduced susceptibility χ R (=χ· k B T / N ( g µ) 2 ) expanded in powers of the internal energy in order to calculate the specific heat c ( T ) for S ≤2 and the zero-field susceptibility for S =1/2 of a quantum Heisenberg spin chain, respectively. The exponents α( c ( T )∼ T α ) obtained for an antiferromagnetic exchange and 1≤ S ≤2 are 1.6, 1.4 and 1.3, which are much different from the value (α=1) given by the spin-wave theory (SW). It is also shown that the polynomials with exponents given by SW cannot fit in with the Padé approximants for these cases. This disagreement with the behaviors of SW is ascribed, at least for S =1, to the nonzero gap found by Botet and Jullien.
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.
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.
We have implemented three approaches to describe the thermodynamic properties of ferrimagnetic (S=5∕2S=5∕2S=5∕2, s=2s=2s=2) spin chains. The application of cumulant expansion has been generalized to the ferrimagnetic chain in the presence of an external magnetic field. Using cumulants, we have obtained the field-dependent effective Hamiltonian in terms of the classical variables up to the second order of quantum corrections. Thermodynamic functions, the internal energy, the specific heat, and the magnetic susceptibility are obtained from the effective Hamiltonian. We have also examined the modified spin-wave theory to derive the same physical properties. Finally, we have studied our model using quantum Monte Carlo simulation to obtain accurate results. The comparison of the above results and also the high temperature series expansion shows that cumulant expansion gives good results for moderate and high temperature regions while the modified spin wave theory is good for low temperatures. Moreover, the convergence regions of the cumulant expansion and the modified spin-wave theory overlap each other which propose these two as a set of complement methods to get the thermodynamic properties of spin models.
We explain how and why all thermodynamic properties of spin systems can be computed in one and two dimensions in the whole range of temperatures overcoming the divergence towards zero temperature of the standard high-temperature series expansions (HTEs). The method relies on an approximation of the entropy versus energy (microcanonical potential function) on the whole range of energies. The success is related to the intrinsic physical constraints on the entropy function and a careful treatment of the boundary behaviors. This method is benchmarked against two one-dimensional solvable models: the Ising model in longitudinal field and the XY model in a transverse field. With ten terms in the HTE, we find a spin susceptibility within a few percent of the exact results over the entire range of temperatures. The method is then applied to two two-dimensional models: the supposedly gapped Heisenberg model and the J(1)-J(2)-J(d) model on the kagome lattice.
The low-energy structures of the quantum ferrimagnetic Heisenberg chain consisting of (5/2, 1/2, 1/2) trimers are investigated theoretically. The results of the linear spin-wave theory are compared with those from the numerical exact diagonalization calculation and the matrix product method for the lowest optical mode. The temperature behaviour of thermodynamical properties such as magnetic susceptibility, specific heat and entropy are analysed in the framework of the modified spin-wave theory. The results of the calculations are used to explain the experimental data obtained for the molecule-based heterospin magnets [Mn(hfac) 2 BNO R ] (R = H, F, Cl, Br) with one-dimensional chain structure.