Progress of Theoretical Physics

Numerical Calculation of Thermodynamic Quantities of Spin-1/2 Anisotropic Heisenberg Ring

Minoru Takahashi

Published May 1, 1974• 38 citations

Abstract

Thermodynamic quantities such as specific heat and magnetic susceptibility are calculated numerically for the one-dimensional spin-1/2 Heisenberg-Ising model at Δ= 0, ±0.5, ±1/√2. We use a set of nonlinear integral equations which is derived from the Bethe ansatz and from some assumptions on the distribution of quasi-momenta. In the same way we calculate the specific heat of the one-dimensional X-Y-Z model in zero field, putting various values into the coupling constants (Jx, Jy, Jz).

DOI: 10.1143/ptp.51.1348

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Progress of Theoretical Physics

Numerical Calculation of Thermodynamic Quantities of Spin-1/2 Anisotropic Heisenberg Ring

Minoru Takahashi

Published May 1, 1974• 38 citations

Publisher

Abstract

DOI: 10.1143/ptp.51.1348

Related papers

Low-Temperature Specific Heat of Spin-1/2 Anisotropic Heisenberg Ring

Makoto Konagai Takahashi

Low-temperature specific heat of Heisenberg-Ising ring at | Δ| ≪ 1 is calculated by the method of non-linear integral equations. Specific heat in constant magnetic field (CH) is proportional to temperature (T) at |2µ0H| ≪ (1+Δ)J. In particular lim H →0 lim T →0CH/T is 2θ/(3Jsin θ) where θ= cos -1Δ, ≤θ≪ π. It is conjectured that lim T→0 lim H→0CH/T = lim H→0 lim T→0CH/T from the result of numerical calculation. Low-temperature specific heat of the one-dimensional X-Y-Z model is proportional to T-3/2 exp (-α/T) in the case of Jz > Jy > 0, Jy ≧Jx > -Jy.Low-temperature specific heat of Heisenberg-Ising ring at | Δ| ≪ 1 is calculated by the method of non-linear integral equations. Specific heat in constant magnetic field (CH) is proportional to temperature (T) at |2µ0H| ≪ (1+Δ)J. In particular lim H →0 lim T →0CH/T is 2θ/(3Jsin θ) where θ= cos -1Δ, ≤θ≪ π. It is conjectured that lim T→0 lim H→0CH/T = lim H→0 lim T→0CH/T from the result of numerical calculation. Low-temperature specific heat of the one-dimensional X-Y-Z model is proportional to T-3/2 exp (-α/T) in the case of Jz > Jy > 0, Jy ≧Jx > -Jy.

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The thermodynamics of the one-dimensional Heisenberg-Ising model for |Δ| ≪ 1 as well as of the X-Y-Z model is reduced to a set of non-linear integral equations under some plausible assumptions. It is remarkable that the number of unknown functions involved in them becomes finite when π/ cos -1Δ is a rational number for the Heisenberg-Ising model and when Kl /ζ is a rational number for the X-Y-Z model (where coupling constants Jx, Jy and Jz are parametrized by ζ, l, and Jz as Jx = Jz cn(2ζ, l) and Jy = Jz dn(2ζ, l); 1 ≥l ≥0, Kl ≥2ζ≥0 and Kl is the complete elliptic integral of the first kind of modulus l). The validity of our theory has been confirmed by the high-temperature expansion of the free energy through the second term for a general value of Δ and through the fourth them for Δ= ½.The thermodynamics of the one-dimensional Heisenberg-Ising model for |Δ| ≪ 1 as well as of the X-Y-Z model is reduced to a set of non-linear integral equations under some plausible assumptions. It is remarkable that the number of unknown functions involved in them becomes finite when π/ cos -1Δ is a rational number for the Heisenberg-Ising model and when Kl /ζ is a rational number for the X-Y-Z model (where coupling constants Jx, Jy and Jz are parametrized by ζ, l, and Jz as Jx = Jz cn(2ζ, l) and Jy = Jz dn(2ζ, l); 1 ≥l ≥0, Kl ≥2ζ≥0 and Kl is the complete elliptic integral of the first kind of modulus l). The validity of our theory has been confirmed by the high-temperature expansion of the free energy through the second term for a general value of Δ and through the fourth them for Δ= ½.

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\sqrt{T/|J|}

about T =0. The critical exponents are a =-1/2 and y =2. The coefficient of T 3/2 term in the free energy agrees with the spin wave theory with the accuracy of 0.1%. The coefficient of T -2 term in the zero-field susceptibility agrees with Fisher's solution of the classical Heisenberg model with the accuracy of 0.3%. The contribution from the second and higher order terms of the expansions in these thermodynamic quantities is not negligible in the region T /| J |>0.002.

{\mathit{S}}_{0}^{\mathit{z}}

{\mathit{S}}_{\mathit{n}}^{\mathit{z}}

〉 and 〈

{\mathit{S}}_{0}^{+}

${\mathit{S}}_{\mathit{n}}^{\mathrm{\ensuremath{-}}}$〉 under the constraint of 〈

{\mathit{S}}^{\mathit{z}}

〉=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,

1, and

3/2

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.