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Journal of the Physical Society of Japan

Thermal Bethe Ansatz Study of Correlation Length of Spin-1/2 Heisenberg Ferromagnetic Chain

Miki Yamada

Published Mar 15, 1990• 21 citations
Publisher

Abstract

The correlation length of the Heisenberg ferromagnetic chain with S =1/2 at sufficiently low temperatures (1/β J 10 -4 ) is obtained numerically (relative 10 -5 ) by the thermal Bethe Ansatz method developed recently by Koma. The numerical result shows directly that the critical exponent ν of the correlation length ξ defined as ξ∼ T -ν is ν=1 and ξ is expanded in \(\sqrt{T}\) near T =0. These results coincide with the new spin wave theory in case of no spontaneous magnetization developed by Takahashi quantitatively with the accuracy of 2×10 -3 % and 7×10 -2 % as for the first and second order terms of a low-temperature expansion respectively.

DOI
10.1143/jpsj.59.848

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The correlation length of the Heisenberg ferromagnetic chain with S =1/2 at sufficiently low temperatures (1/β J 10 -4 ) is obtained numerically (relative 10 -5 ) by the thermal Bethe Ansatz method developed recently by Koma. The numerical result shows directly that the critical exponent ν of the correlation length ξ defined as ξ∼ T -ν is ν=1 and ξ is expanded in (\sqrt{T}) near T =0. These results coincide with the new spin wave theory in case of no spontaneous magnetization developed by Takahashi quantitatively with the accuracy of 2×10 -3 % and 7×10 -2 % as for the first and second order terms of a low-temperature expansion respectively.
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 renormalization-group recursion relations for the spin-S nearest-neighbor quantum Heisenberg ferromagnet recently derived by Chakravarty and the present author are analyzed in one dimension. It is found that the susecptibility \ensuremath{\chi} is given at low temperatures T by T\ensuremath{\chi}=${C}_{\ensuremath{\chi}}^{(0)}$(JS2{\mathrm{JS}}^{2}JS2/T) [1+${C}_{\ensuremath{\chi}}^{(1)}$\ensuremath{\eta}+O(${\ensuremath{\eta}}^{2}$)], where J is the exchange coupling and the dimensionless expansion parameter \ensuremath{\eta} is given by \ensuremath{\eta}=${\ensuremath{\pi}}^{\mathrm{\ensuremath{-}}1}$(T/JS3{\mathrm{JS}}^{3}JS3)1/2{)}^{1/2})1/2. For the correlation length \ensuremath{\xi} we find (\ensuremath{\xi}/a)=${C}_{\ensuremath{\xi}}^{(0)}$(JS2{\mathrm{JS}}^{2}JS2/T) [1+${C}_{\ensuremath{\xi}}^{(1)}$\ensuremath{\eta}+O(${\ensuremath{\eta}}^{2}$)], where a is the lattice spacing. The values of the coefficients ${C}_{\ensuremath{\chi}}^{(j)}$ and ${C}_{\ensuremath{\xi}}^{(j)}$ (j=0,1), have been determined by fitting the above expressions with Monte Carlo data obtained for the S=1/2 Heisenberg chain at temperatures as low as T/J=0.01.
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 T/∣J∣\sqrt{T/|J|}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.
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.
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.