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arXiv:0707.2899 · Physical Review B

Thermodynamics of low-dimensional spin- 12 Heisenberg ferromagnets in an external magnetic field within a Green function formalism

Tatiana N. Antsygina, M. I. Poltavskaya, Igor Poltavsky, Konstantin A. Chishko

Published Jan 9, 2008• 36 citations
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Abstract

The thermodynamics of low-dimensional spin-$\frac{1}{2}$ Heisenberg ferromagnets (HFMs) in an external magnetic field is investigated within a second-order two-time Green function formalism in the wide temperature and field ranges. A crucial point of the proposed scheme is a proper account of the analytical properties for the approximate transverse commutator Green function obtained as a result of the decoupling procedure. A good quantitative description of the correlation functions, magnetization, susceptibility, and heat capacity of the HFMs on a chain and square and triangular lattices is found for both infinite- and finite-sized systems. The dependences of the thermodynamic functions of two-dimensional HFMs on the cluster size are studied. The obtained results agree well with the corresponding data found by Bethe ansatz, exact diagonalization, high temperature series expansions, and quantum Monte Carlo simulations.

DOI
10.1103/physrevb.77.024407

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The thermodynamics of low-dimensional spin-12\frac{1}{2}21​ Heisenberg ferromagnets (HFMs) in an external magnetic field is investigated within a second-order two-time Green function formalism in the wide temperature and field ranges. A crucial point of the proposed scheme is a proper account of the analytical properties for the approximate transverse commutator Green function obtained as a result of the decoupling procedure. A good quantitative description of the correlation functions, magnetization, susceptibility, and heat capacity of the HFMs on a chain and square and triangular lattices is found for both infinite- and finite-sized systems. The dependences of the thermodynamic functions of two-dimensional HFMs on the cluster size are studied. The obtained results agree well with the corresponding data found by Bethe ansatz, exact diagonalization, high temperature series expansions, and quantum Monte Carlo simulations.
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 calculate the low-temperature thermodynamic quantities (magnetization, correlation functions, transverse and longitudinal correlation lengths, spin susceptibility, and specific heat) of the frustrated one-dimensional spin-half J1{J}_{1}J1​-J2{J}_{2}J2​ Heisenberg ferromagnet, i.e., for J2<0.25∣J1∣{J}_{2}<0.25|{J}_{1}|J2​<0.25∣J1​∣, in an external magnetic field using a second-order Green-function formalism and full diagonalization of finite systems. We determine power-law relations for the field dependence of the position and the height of the maximum of the uniform susceptibility. Considering the specific heat at low magnetic fields, two maxima in its temperature dependence are found.
The one-dimensional Heisenberg model with S = 1/2 is treated with the use of the two-time Green's functions. The hierarchy of the equations of motion of the Green'f functions is decoupled at a stage one-step further than Tyablikov's decoupling. The thermal average of the spin component, ≪ Sz >, is set to zero, becaure the long-range order does not exist in one dimension. Instead, our Green's functions are expressed in terms of the correlation functions cn ≡4≪ S0zSnz >. The Green's function is essentially of the form representing undamped spin waves, whose spactrum depends on c1, c2 and one more parameter. They are determined by the requirement that c1 and c2 should be self-consistent and that c0 should be unity. The self-consistency equations have been solved analytically at high- and low-temperature limits, and also solved numerically in the whole range of the temperature. Thermodynamic quantities have been calculated using these solutions. It has turned out that the theory gives the correct high-temperature expansion for the thermodynamic quantities and the correlation functions. The latter is expressed by cn = (J / 4 kBT )n. In the case of ferromagnetic coupling, the correlation function cn at T = 0 is equal to 1/3 for all n's. This is what is expected from the correct ground state of the ferromagnetic Heisenberg system. The spin-wave spectrum at T = 0 also agrees with the correct one. At T ≪J/ kB, we find that the specific heat goes as T1/2 and the susceptibility goes as T-2. The gross feature of the temperature-dependence of the thermodynamic quantities agrees with Bonner and Fisher. In the case of the antiferromagnetic Heisenberg model, we find c1 = -0.55407 and c2 = 0.16100 at T = 0, which are fairly close to the exact values. The thermodynamic quantities are also in gross agreement with Bonner and Fisher.
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.
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 energy, the specific heat, and the magnetic susceptibility of the bond-diluted S=(1/2) Heisenberg ferromagnet on a two-dimensional square lattice are studied using a Monte Carlo sampling of the terms contributing to the partition function. The full lattice is studied in detail for sizes up to 20\ifmmode\times\else\texttimes\fi{}20, and evidence is presented to suggest that the low-temperature correlation length behaves as \ensuremath{\xi}\ensuremath{\sim}${e}^{\ensuremath{\nu}\ensuremath{\beta}J}$ with \ensuremath{\nu}\ensuremath{\sim}1.91\ifmmode\pm\else\textpm\fi{}0.06. The temperature dependence of the susceptibility, however, is not quite a simple exponential in \ensuremath{\beta}J. Knowledge of the correlation length and finite-size scaling allows the determination of the prefactor. Our conclusion is that the low-temperature susceptibility behaves as ${T}^{\mathrm{\ensuremath{-}}1}$${e}^{2\ensuremath{\nu}\ensuremath{\beta}J}$. Square lattices of size ranging from 6\ifmmode\times\else\texttimes\fi{}6 to 12\ifmmode\times\else\texttimes\fi{}12 are studied for kB{k}_{B}kB​T/J ranging from 0.1 to 3 and for several values of the bond probability p. Not surprisingly, the most drastic thermodynamic behavior near the critical bond-percolation probability is the sharp increase in the low-temperature susceptibility for p>pc{p}_{c}pc​. This corresponds to much larger correlation effects in the percolating lattice than below the percolation threshold. We also compare the thermodynamics at p=pc{p}_{c}pc​ to the scaling theories of Lubensky and Stanley et al. which suggest that the correlations at the percolation threshold are predominantly one dimensional.