Journal of the Physical Society of Japan

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

Minoru Takahashi, Miki Yamada

Published Aug 15, 1985• 104 citations

Abstract

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.

DOI: 10.1143/jpsj.54.2808

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Journal of the Physical Society of Japan

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

Minoru Takahashi, Miki Yamada

Published Aug 15, 1985• 104 citations

Publisher

Abstract

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.

DOI: 10.1143/jpsj.54.2808

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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.

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.

{\mathit{J}}_{0}

{\mathit{r}}_{\mathit{i}\mathit{j}}^{\mathit{p}}

)

{\mathbf{S}}_{\mathit{i}}

\ensuremath{\cdot}

{\mathbf{S}}_{\mathit{j}}

]. It is shown that for dp2d there exists a phase transition at finite temperatures; the critical temperature is estimated. The susceptibility and the specific heat are obtained at low temperatures. Our results for d=1, p=2, and S=1/2 agree with the exact solution of the Haldane-Shastry model. For d=1, we find a scaling relation \ensuremath{\alpha}+1=${\ensuremath{\gamma}}^{\mathrm{\ensuremath{-}}1}$ when 2p3.

{\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.

{)}^{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.