Physical review. B, Condensed matter

Low-temperature behavior of the S=(1/2) ferromagnetic Heisenberg chain

P. Schlottmann

Published Apr 1, 1986• 58 citations

Abstract

A detailed account of the results of the numerical solution of the thermodynamic Bethe-ansatz equations for the isotropic ferromagnetic S=(1/2) Heisenberg chain is presented. The extrapolation procedure used in approximating the infinite set of coupled nonlinear integral equations is discussed. The data for the truncated integral equations are analyzed in terms of a finite-string-size scaling. Analytic expressions for the free energy and susceptibility for T\ensuremath{\rightarrow}0 are obtained. All the results are consistent with entropy S\ensuremath{\sim}(T/\ensuremath{\Vert}J\ensuremath{\Vert}${)}^{1/2}$ and \ensuremath{\chi}\ensuremath{\sim} \ensuremath{\Vert}J\ensuremath{\Vert} / ${T}^{2}$ [${\mathrm{scrL}}^{\mathrm{\ensuremath{-}}1}$+(lnscrL)/${\mathrm{scrL}}^{2}$+...]+O (${T}^{\mathrm{\ensuremath{-}}3/2}$) , where scrL=ln(\ensuremath{\Vert}J\ensuremath{\Vert}/T), suggesting the existence of a marginal variable. The logarithmic corrections reflect the analogy to the Kondo problem.

DOI: 10.1103/physrevb.33.4880

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Critical behavior of the isotropic ferromagnetic Heisenberg chain with arbitrary spinS

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Critical Behavior of Spin-1/2 One-Dimensional Heisenberg Ferromagnet at Low Temperatures

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Critical Behavior of the Isotropic Ferromagnetic Quantum Heisenberg Chain

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

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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

Physical review. B, Condensed matter

Low-temperature behavior of the S=(1/2) ferromagnetic Heisenberg chain

P. Schlottmann

Published Apr 1, 1986• 58 citations

Publisher

Abstract

{)}^{1/2}

and \ensuremath{\chi}\ensuremath{\sim} \ensuremath{\Vert}J\ensuremath{\Vert} /

{T}^{2}

[

{\mathrm{scrL}}^{\mathrm{\ensuremath{-}}1}

+(lnscrL)/

{\mathrm{scrL}}^{2}

+...]+O (

{T}^{\mathrm{\ensuremath{-}}3/2}

) , where scrL=ln(\ensuremath{\Vert}J\ensuremath{\Vert}/T), suggesting the existence of a marginal variable. The logarithmic corrections reflect the analogy to the Kondo problem.

DOI: 10.1103/physrevb.33.4880

Related papers

Critical behavior of the isotropic ferromagnetic Heisenberg chain with arbitrary spinS

Kong-Ju-Bock Lee, P. Schlottmann

We have extracted the leading low-temperature contributions to the specific heat and the magnetic susceptibility from the thermodynamic Bethe-ansatz equations of the SU(2) invariant ferromagnetic Heisenberg chain of arbitrary spin S. This extends previous analytic results for S=1/2 to other spin values. The critical exponent for the specific heat is \ensuremath{\alpha}=-1/2, that for the susceptibility is \ensuremath{\gamma}=2, and that for the correlation length is \ensuremath{\nu}=1. The susceptibility and the correlation length exhibit logarithmic corrections, reflecting the analogy to the Kondo problem. These logarithmic corrections are quenched by relatively small magnetic fields. In large fields the energy required to flip a spin gives rise to an exponential activation.We have extracted the leading low-temperature contributions to the specific heat and the magnetic susceptibility from the thermodynamic Bethe-ansatz equations of the SU(2) invariant ferromagnetic Heisenberg chain of arbitrary spin S. This extends previous analytic results for S=1/2 to other spin values. The critical exponent for the specific heat is \ensuremath{\alpha}=-1/2, that for the susceptibility is \ensuremath{\gamma}=2, and that for the correlation length is \ensuremath{\nu}=1. The susceptibility and the correlation length exhibit logarithmic corrections, reflecting the analogy to the Kondo problem. These logarithmic corrections are quenched by relatively small magnetic fields. In large fields the energy required to flip a spin gives rise to an exponential activation.

Physical review. B, Condensed matterJul 1, 198720 citations

Critical Behavior of Spin-1/2 One-Dimensional Heisenberg Ferromagnet at Low Temperatures

Miki Yamada, Minoru Takahashi

A linear chain of S =1/2 spins with the isotropic Hamiltonian H = J Σ( S i x S i +1 x + S i y S i +1 y + S i z S i +1 z -1/4), ( J <0) , at low temperatures is studied numerically by solving a set of nonlinear integral equations which are derived from the Bethe Ansatz. We find that the free energy and susceptibility are both expanded in $\sqrt{T/|J|}$ 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.A linear chain of S =1/2 spins with the isotropic Hamiltonian H = J Σ( S i x S i +1 x + S i y S i +1 y + S i z S i +1 z -1/4), ( J <0) , at low temperatures is studied numerically by solving a set of nonlinear integral equations which are derived from the Bethe Ansatz. We find that the free energy and susceptibility are both expanded in $\sqrt{T/|J|}$ 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.

Journal of the Physical Society of JapanJun 15, 198686 citations

Critical Behavior of the Isotropic Ferromagnetic Quantum Heisenberg Chain

P. Schlottmann

The thermodynamic Bethe-Ansatz equations for the isotropic ferromagnetic $S=\frac{1}{2}$ Heisenberg chain have been solved numerically. At low temperatures, I find a power-law dependence on $T$ for the specific heat and the susceptibility with critical exponents $\ensuremath{\alpha}=\ensuremath{-}0.49\ifmmode\pm\else\textpm\fi{}0.02$, $\ensuremath{\gamma}=2.00\ifmmode\pm\else\textpm\fi{}0.02$, and $\ensuremath{\Delta}\ensuremath{\simeq}\ensuremath{\gamma}$. The exponent $\ensuremath{\alpha}$ is compatible with effective spin waves and $\ensuremath{\gamma}$ and $\ensuremath{\Delta}$ are the exponents of the classical chain. Amplitudes and corrections to scaling are obtained and differences with previous results are discussed.The thermodynamic Bethe-Ansatz equations for the isotropic ferromagnetic $S=\frac{1}{2}$ Heisenberg chain have been solved numerically. At low temperatures, I find a power-law dependence on $T$ for the specific heat and the susceptibility with critical exponents $\ensuremath{\alpha}=\ensuremath{-}0.49\ifmmode\pm\else\textpm\fi{}0.02$, $\ensuremath{\gamma}=2.00\ifmmode\pm\else\textpm\fi{}0.02$, and $\ensuremath{\Delta}\ensuremath{\simeq}\ensuremath{\gamma}$. The exponent $\ensuremath{\alpha}$ is compatible with effective spin waves and $\ensuremath{\gamma}$ and $\ensuremath{\Delta}$ are the exponents of the classical chain. Amplitudes and corrections to scaling are obtained and differences with previous results are discussed.

Physical Review LettersMay 13, 198570 citations

Critical behavior of the SU(3) ferromagnetic Heisenberg chain

Young-Ju Jee, Kong-Ju-Bock Lee, P. Schlottmann

We have extracted the leading low-temperature contributions to the specific heat and the magnetic susceptibility from the thermodynamic Bethe-Ansatz equations of the SU(3) invariant ferromagnetic Heisenberg chain of spin 1. The critical exponents for the specific heat, the susceptibility, and the correlation length are $\ensuremath{\alpha}=\ensuremath{-}\frac{1}{2}$, $\ensuremath{\gamma}=2$, and $\ensuremath{\nu}=1$, respectively. The susceptibility and the correlation length exhibit logarithmic corrections, which are quenched by relatively small magnetic fields. In large fields, the energy required to flip a spin gives rise to an exponential activation.We have extracted the leading low-temperature contributions to the specific heat and the magnetic susceptibility from the thermodynamic Bethe-Ansatz equations of the SU(3) invariant ferromagnetic Heisenberg chain of spin 1. The critical exponents for the specific heat, the susceptibility, and the correlation length are $\ensuremath{\alpha}=\ensuremath{-}\frac{1}{2}$, $\ensuremath{\gamma}=2$, and $\ensuremath{\nu}=1$, respectively. The susceptibility and the correlation length exhibit logarithmic corrections, which are quenched by relatively small magnetic fields. In large fields, the energy required to flip a spin gives rise to an exponential activation.

Physical review. B, Condensed matterFeb 1, 198916 citations

Thermodynamics of the Spin- 1/2 Antiferromagnetic Uniform Heisenberg Chain

Andreas Klümper, D. C. Johnston

We present a new application of the traditional thermodynamic Bethe ansatz to the spin-1/2 antiferromagnetic uniform Heisenberg chain and derive exact nonlinear integral equations for just two functions describing the elementary excitations. By using this approach, the magnetic susceptibility chi and specific heat C versus temperature T are calculated to high accuracy for 5x10(-25)</=T/J</=5. The chi(T) data agree very well at low T with the asymptotically exact theoretical low- T prediction of Lukyanov [Nucl. Phys. B522, 533 (1998)]. The unknown coefficients of the second and third lowest-order logarithmic correction terms in Lukyanov's theory for C(T) are estimated from the C(T) data.We present a new application of the traditional thermodynamic Bethe ansatz to the spin-1/2 antiferromagnetic uniform Heisenberg chain and derive exact nonlinear integral equations for just two functions describing the elementary excitations. By using this approach, the magnetic susceptibility chi and specific heat C versus temperature T are calculated to high accuracy for 5x10(-25)</=T/J</=5. The chi(T) data agree very well at low T with the asymptotically exact theoretical low- T prediction of Lukyanov [Nucl. Phys. B522, 533 (1998)]. The unknown coefficients of the second and third lowest-order logarithmic correction terms in Lukyanov's theory for C(T) are estimated from the C(T) data.

arXiv:cond-mat/0002140Physical Review LettersMay 15, 2000125 citations

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

{)}^{1/2}

and \ensuremath{\chi}\ensuremath{\sim} \ensuremath{\Vert}J\ensuremath{\Vert} /

{T}^{2}

[

{\mathrm{scrL}}^{\mathrm{\ensuremath{-}}1}

+(lnscrL)/

{\mathrm{scrL}}^{2}

+...]+O (

{T}^{\mathrm{\ensuremath{-}}3/2}

) , where scrL=ln(\ensuremath{\Vert}J\ensuremath{\Vert}/T), suggesting the existence of a marginal variable. The logarithmic corrections reflect the analogy to the Kondo problem.

\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.

The thermodynamic Bethe-Ansatz equations for the isotropic ferromagnetic

S=\frac{1}{2}

Heisenberg chain have been solved numerically. At low temperatures, I find a power-law dependence on

T

for the specific heat and the susceptibility with critical exponents $\ensuremath{\alpha}=\ensuremath{-}0.49\ifmmode\pm\else\textpm\fi{}0.02$, $\ensuremath{\gamma}=2.00\ifmmode\pm\else\textpm\fi{}0.02$, and $\ensuremath{\Delta}\ensuremath{\simeq}\ensuremath{\gamma}$. The exponent $\ensuremath{\alpha}$ is compatible with effective spin waves and $\ensuremath{\gamma}$ and $\ensuremath{\Delta}$ are the exponents of the classical chain. Amplitudes and corrections to scaling are obtained and differences with previous results are discussed.

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.