Physical Review A

Low-Temperature Thermodynamics of the |Δ|≥1 Heisenberg-Ising Ring

James D. Johnson, Barry M. McCoy

Published Oct 1, 1972• 149 citations

Abstract

We use the equations proposed by Gaudin for the free energy of the Heisenberg-Ising ring for $|\ensuremath{\Delta}|\ensuremath{\ge}1$ to obtain the first temperature-dependent term in a systematic low-temperature expansion of the free energy.

DOI: 10.1103/physreva.6.1613

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Finite-temperature excitations and thermodynamics of the Δ>~1 Heisenberg-Ising ring

James D. Johnson

Physical Review AApr 1, 197435 citations

Thermodynamics of the Heisenberg-Ising Ring for Δ>~1

M. Gaudin

The thermodynamics of the Heisenberg-Ising ring is reduced to the solution of a system of recurrent nonlinear integral equations.Received 9 April 1971DOI:https://doi.org/10.1103/PhysRevLett.26.1301©1971 American Physical Society

Physical Review LettersMay 24, 1971290 citations

One-Dimensional Anisotropic Heisenberg Model at Finite Temperatures

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Journal of the Physical Society of JapanDec 15, 19971 citation

Spin-1/2 One-Dimensional Heisenberg Ferromagnet at Low-Temperature

Minoru Takahashi, Miki Yamada

Using Bethe ansatz integral equations, we calculate the free energy and susceptibility of spin-$\frac{1}{2}$ one-dimensional Heisenberg ferromagnet $\mathcal{H}{=}J\Sigma(\mbi{S}_{\text{i}}\mbi{S}_{\text{i}+1}-\frac{1}{4})$, ( J <0) at T ≥0.004| J |. We find that the free energy and susceptibility are expanded by $\sqrt{T/|J|}$ at low temperature: $f&=|J|\left\{-1.042\left(\frac{T}{|J|}\right)^{3/2}+1.00\left(\frac{T}{|J|}\right)^{2}-0.9\left(\frac{T}{|J|}\right)^{5/2}+O\left(\frac{T}{|J|}\right)^{3}\right\}\\ \chi&=|J|^{-1}\left\{0.1667\left(\frac{|J|}{T}\right)^{2}+0.581\left(\frac{|J|}{T}\right)^{3/2}+0.68\left(\frac{|J|}{T}\right)^{1}+O\left(\frac{|J|}{T}\right)^{1/2}\right\}.$ The first term of free energy coincides with the spin wave calculation. The first term of susceptibility coincides with Fisher's solution of classical Heisenberg Ferromagnet. We conclude α=-0.5 and γ=2.

Journal of the Physical Society of JapanAug 15, 1985104 citations

Thermodynamics of the Heisenberg-Ising Ring for Δ ≥ 1

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Physical Review A

Low-Temperature Thermodynamics of the |Δ|≥1 Heisenberg-Ising Ring

James D. Johnson, Barry M. McCoy

Published Oct 1, 1972• 149 citations

Publisher

Abstract

We use the equations proposed by Gaudin for the free energy of the Heisenberg-Ising ring for

|\ensuremath{\Delta}|\ensuremath{\ge}1

to obtain the first temperature-dependent term in a systematic low-temperature expansion of the free energy.

DOI: 10.1103/physreva.6.1613

Related papers

Finite-temperature excitations and thermodynamics of the Δ>~1 Heisenberg-Ising ring

James D. Johnson

We prove the existence and discuss the uniqueness of the solution of the equations proposed by Gaudin for the free energy of the $\ensuremath{\Delta}\ensuremath{\ge}1$ Heisenberg-Ising ring. A numerical technique for solving the equations is outlined, and some results of the numerical work are displayed. Comparisons are made with the previous numerical study of Bonner and Fisher with favorable results, giving support to Gaudin's formulation. The finite-temperature excitations of the system are derived, and some sample dispersion curves are given.We prove the existence and discuss the uniqueness of the solution of the equations proposed by Gaudin for the free energy of the $\ensuremath{\Delta}\ensuremath{\ge}1$ Heisenberg-Ising ring. A numerical technique for solving the equations is outlined, and some results of the numerical work are displayed. Comparisons are made with the previous numerical study of Bonner and Fisher with favorable results, giving support to Gaudin's formulation. The finite-temperature excitations of the system are derived, and some sample dispersion curves are given.

Physical Review AApr 1, 197435 citations

Thermodynamics of the Heisenberg-Ising Ring for Δ>~1

M. Gaudin

Physical Review LettersMay 24, 1971290 citations

One-Dimensional Anisotropic Heisenberg Model at Finite Temperatures

Minoru Takahashi, Masuo Suzuki

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.

Progress of Theoretical PhysicsDec 1, 1972403 citations

Low–Temperature Expansions of the XXZ Heisenberg Model

Hideyuki Mizuta, Miki Wadati

Thermodynamic quantities of solvable models such as one–dimensional Heisenberg model can be obtained through the Finite temperature Baxter formula. For the low–temperature expansions, the estimate of the finite size corrections by use of the Euler–Maclaurin formula is essential. There has been a question on the convergence of such calculations. In this paper, we confirm that the Euler–Maclaurin formula gives the correct low–temperature expansion of the free energy for the XXZ Heisenberg model.Thermodynamic quantities of solvable models such as one–dimensional Heisenberg model can be obtained through the Finite temperature Baxter formula. For the low–temperature expansions, the estimate of the finite size corrections by use of the Euler–Maclaurin formula is essential. There has been a question on the convergence of such calculations. In this paper, we confirm that the Euler–Maclaurin formula gives the correct low–temperature expansion of the free energy for the XXZ Heisenberg model.

Journal of the Physical Society of JapanDec 15, 19971 citation

Spin-1/2 One-Dimensional Heisenberg Ferromagnet at Low-Temperature

Minoru Takahashi, Miki Yamada

Using Bethe ansatz integral equations, we calculate the free energy and susceptibility of spin- $\frac{1}{2}$ one-dimensional Heisenberg ferromagnet $\mathcal{H}{=}J\Sigma(\mbi{S}_{\text{i}}\mbi{S}_{\text{i}+1}-\frac{1}{4})$, ( J <0) at T ≥0.004| J |. We find that the free energy and susceptibility are expanded by $\sqrt{T/|J|}$ at low temperature: $f&=|J|\left\{-1.042\left(\frac{T}{|J|}\right)^{3/2}+1.00\left(\frac{T}{|J|}\right)^{2}-0.9\left(\frac{T}{|J|}\right)^{5/2}+O\left(\frac{T}{|J|}\right)^{3}\right\}\\ \chi&=|J|^{-1}\left\{0.1667\left(\frac{|J|}{T}\right)^{2}+0.581\left(\frac{|J|}{T}\right)^{3/2}+0.68\left(\frac{|J|}{T}\right)^{1}+O\left(\frac{|J|}{T}\right)^{1/2}\right\}.$ The first term of free energy coincides with the spin wave calculation. The first term of susceptibility coincides with Fisher's solution of classical Heisenberg Ferromagnet. We conclude α=-0.5 and γ=2.Using Bethe ansatz integral equations, we calculate the free energy and susceptibility of spin-$\frac{1}{2}$ one-dimensional Heisenberg ferromagnet $\mathcal{H}{=}J\Sigma(\mbi{S}_{\text{i}}\mbi{S}_{\text{i}+1}-\frac{1}{4})$, ( J <0) at T ≥0.004| J |. We find that the free energy and susceptibility are expanded by $\sqrt{T/|J|}$ at low temperature: $f&=|J|\left\{-1.042\left(\frac{T}{|J|}\right)^{3/2}+1.00\left(\frac{T}{|J|}\right)^{2}-0.9\left(\frac{T}{|J|}\right)^{5/2}+O\left(\frac{T}{|J|}\right)^{3}\right\}\\ \chi&=|J|^{-1}\left\{0.1667\left(\frac{|J|}{T}\right)^{2}+0.581\left(\frac{|J|}{T}\right)^{3/2}+0.68\left(\frac{|J|}{T}\right)^{1}+O\left(\frac{|J|}{T}\right)^{1/2}\right\}.$ The first term of free energy coincides with the spin wave calculation. The first term of susceptibility coincides with Fisher's solution of classical Heisenberg Ferromagnet. We conclude α=-0.5 and γ=2.

Journal of the Physical Society of JapanAug 15, 1985104 citations

Thermodynamics of the Heisenberg-Ising Ring for Δ ≥ 1

M. Gaudin

EDP Sciences eBooksJan 1, 19960 citations

We use the equations proposed by Gaudin for the free energy of the Heisenberg-Ising ring for

|\ensuremath{\Delta}|\ensuremath{\ge}1

to obtain the first temperature-dependent term in a systematic low-temperature expansion of the free energy.

Using Bethe ansatz integral equations, we calculate the free energy and susceptibility of spin-

\frac{1}{2}

one-dimensional Heisenberg ferromagnet $\mathcal{H}{=}J\Sigma(\mbi{S}_{\text{i}}\mbi{S}_{\text{i}+1}-\frac{1}{4})$, ( J <0) at T ≥0.004| J |. We find that the free energy and susceptibility are expanded by

\sqrt{T/|J|}

at low temperature: $f&=|J|\left\{-1.042\left(\frac{T}{|J|}\right)^{3/2}+1.00\left(\frac{T}{|J|}\right)^{2}-0.9\left(\frac{T}{|J|}\right)^{5/2}+O\left(\frac{T}{|J|}\right)^{3}\right\}\\ \chi&=|J|^{-1}\left\{0.1667\left(\frac{|J|}{T}\right)^{2}+0.581\left(\frac{|J|}{T}\right)^{3/2}+0.68\left(\frac{|J|}{T}\right)^{1}+O\left(\frac{|J|}{T}\right)^{1/2}\right\}.$ The first term of free energy coincides with the spin wave calculation. The first term of susceptibility coincides with Fisher's solution of classical Heisenberg Ferromagnet. We conclude α=-0.5 and γ=2.