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Physical review. B, Condensed matter

Thermal Bethe-ansatz study of the correlation length of the one-dimensionalS=1/2 Heisenberg antiferromagnet

K. Nomura, Miki Yamada

Published Apr 1, 1991• 30 citations
Publisher

Abstract

We investigate the correlation length of the one-dimensional S=1/2 Heisenberg antiferromagnet by the thermal Bethe-ansatz method proposed by Koma. The numerical result shows that the correlation length diverges as \ensuremath{\xi}\ensuremath{\sim}${\mathit{T}}^{\mathrm{\ensuremath{-}}1}$ at low temperatures with logarithmic corrections. This logarithmic correction can be explained by conformal field theory.

DOI
10.1103/physrevb.43.8217

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We investigate the correlation length of the one-dimensional S=1/2 Heisenberg antiferromagnet by the thermal Bethe-ansatz method proposed by Koma. The numerical result shows that the correlation length diverges as \ensuremath{\xi}\ensuremath{\sim}T\ensuremath−1{\mathit{T}}^{\mathrm{\ensuremath{-}}1}T\ensuremath−1 at low temperatures with logarithmic corrections. This logarithmic correction can be explained by conformal field theory.
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 T\sqrt{T}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.
We have calculated the spin correlation length of the half-filled one-dimensional Hubbard model down to the temperature of T =0.01×(transfer integral) by the Bethe-Ansatz method for the quantum transfer matrix. Its temperature dependence shows the crossover from (2 log π T ) -1 at high temperatures to a T -1 with the log corrections at low temperatures, the low- T behavior being consistent with the prediction by the conformal field theory. When the Coulomb repulsion U is small, we have found at low temperatures another crossover in the proportional constant a from the value for the free electron system to that for the antiferromagnetic Heisenberg model. This crossover is due to the disappearance of charge fluctuation. A similar crossover has been observed for the linear coefficient of specific heat, C / T .
We have calculated the spin correlation length of the half-filled one-dimensional Hubbard model down to the temperature of T =0.01×(transfer integral) by the Bethe-Ansatz method for the quantum transfer matrix. Its temperature dependence shows the crossover from (2 log π T ) -1 at high temperatures to a T -1 with the log corrections at low temperatures, the low- T behavior being consistent with the prediction by the conformal field theory. When the Coulomb repulsion U is small, we have found at low temperatures another crossover in the proportional constant a from the value for the free electron system to that for the antiferromagnetic Heisenberg model. This crossover is due to the disappearance of charge fluctuation. A similar crossover has been observed for the linear coefficient of specific heat, C / T .
The one-dimensional spin-12\frac{1}{2}21​ Heisenberg antiferromagnet is considered using a simple quasiparticle picture: a weakly interacting Fermi gas of kinks. Using this picture, the low-temperature heat capacity, and the magnetic susceptibility with logarithmic field and temperature corrections are derived. The results obtained are in agreement with previous Bethe ansatz and conformal field-theory calculations. We demonstrate from comparison with previous calculations that the magnetic moment of the kink gkink=2gelectron.{g}_{\mathrm{kink}}=\sqrt{2}{g}_{\mathrm{electron}}.gkink​=2​gelectron​.
We construct a new mean-field theory for a quantum (spin-1/2) Heisenberg antiferromagnet in one (1D) and two (2D) dimensions using a Hartree-Fock decoupling of the four-point correlation functions. We show that the solution to the self-consistency equations based on two-point correlation functions does not produce any unphysical finite-temperature phase transition, in accord with the Mermin-Wagner theorem, unlike the common approach based on the mean-field equation for the order parameter. The next-neighbor spin-spin correlation functions, calculated within this approach, reproduce closely the strong renormalization by quantum fluctuations obtained via a Bethe ansatz in 1D and a small renormalization of the classical antiferromagnetic state in 2D. The heat capacity approximates with reasonable accuracy the full Bethe ansatz result at all temperatures in 1D. In 2D, we obtain a reduction of the peak height in the heat capacity at a finite temperature that is accessible by high-order 1/T1/T1/T expansions.