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arXiv:cond-mat/9401074 · Journal of the Physical Society of Japan

The Free Energy and the Scaling Function of the Ferromagnetic Heisenberg Chain in a Magnetic Field

Hiroaki Nakamura, Minoru Takahashi

Published Jul 15, 1994• 20 citations
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Abstract

A nonlinear susceptibilities (the third derivative of a magnetization $m_S$ by a magnetic field $h$ ) of the $S$=1/2 ferromagnetic Heisenberg chain and the classical Heisenberg chain are calculated at low temperatures $T.$ In both chains the nonlinear susceptibilities diverge as $T^{-6}$ and a linear susceptibilities diverge as $T^{-2}.$ The arbitrary spin $S$ Heisenberg ferromagnet $[$ ${\cal H} = \sum_{i=1}^{N} \{ - J{\bf S}_{i} {\bf S}_{i+1} - (h/S) S_{i}^{z} \}$ $(J>0),$ $]$ has a scaling relation between $m_S,$ $h$ and $T:$ $m_S(T,h) = F( S^2 Jh/T^2).$ The scaling function $F(x)$=(2$x$/3)-(44$x^{3}$/135) + O($x^{5}$) is common to all values of spin $S.$

DOI
10.1143/jpsj.63.2563

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The finite-size scaling function of the nonlinear susceptibility of the ferromagnetic Heisenberg chain is given explicitly. It is conjectured that the scaling function is universal for any values of SSS. The conjecture is based on the exact solution of the nonlinear susceptibility for S=∞S=\inftyS=∞, and numerical calculations for S=1/2S=1/2S=1/2 and S=1S=1S=1.
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.
The thermodynamic Bethe-Ansatz equations for the isotropic ferromagnetic S=12S=\frac{1}{2}S=21​ Heisenberg chain have been solved numerically. At low temperatures, I find a power-law dependence on TTT 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 finite-size scaling function of the magnetization of the ferromagnetic Heisenberg chain is argued to be universal with respect to the magnitude of the spin. The finite-size scaling function is given explicitly by an analytical calculation in the classical limit S=∞.S=\infty.S=∞. The universality is checked for S=1/2S=1/2S=1/2 and 111 by means of numerical calculations. Critical exponents are obtained as well. It is concluded that this universal scaling function originates in the universal behavior of the correlation function.
Using Bethe ansatz integral equations, we calculate the free energy and susceptibility of spin-12\frac{1}{2}21​ 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 T/∣J∣\sqrt{T/|J|}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.