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Modern Physics Letters B

ABOUT THE CORRELATION LENGTHS OF QUANTUM HEISENBERG FERRO AND ANTIFERROMAGNETS

Th. Jolicœur, J. C. Le Guillou

Published Apr 10, 1991• 3 citations
Publisher

Abstract

The mass gap for the two-dimensional O (3) non-linear sigma model has been recently extracted from the Bethe Ansatz solution. Combining this result with the Renormalization Group treatment of the Quantum Heisenberg Ferromagnet as well as the Quantum non-linear sigma model we present new formulas for the correlation length of the square lattice quantum ferro and antiferromagnet.

DOI
10.1142/s021798499100071x

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The mass gap for the two-dimensional O (3) non-linear sigma model has been recently extracted from the Bethe Ansatz solution. Combining this result with the Renormalization Group treatment of the Quantum Heisenberg Ferromagnet as well as the Quantum non-linear sigma model we present new formulas for the correlation length of the square lattice quantum ferro and antiferromagnet.
The low-temperature behavior of quantum Heisenberg models can be obtained by combining a renormalization-group analysis of the quantum models with a Monte Carlo simulation of the corresponding classical model, where the ``spins'' are represented by unit vectors defined on a lattice. In this approach renormalization-group equations for the quantum models lead to an effective classical O(3) nonlinear \ensuremath{\sigma} model in appropriate regimes of the parameter space. In the present work we apply this technique, which was recently used by Chakravarty, Halperin, and Nelson to discuss the low-temperature behavior of quantum antiferromagnets, to ferromagnets and calculate the correlation length and the susceptibility of a two-dimensional quantum ferromagnet at low temperatures. The results obtained with this method are then compared with those obtained from a direct simulation of the spin S=1/2 quantum ferromagnet. Excellent agreement between the two approaches leads us to believe that the analytic expressions obtained in the renormalization-group method are accurate at low temperatures. This strengthens the validity of the procedure adopted for antiferromagnets as well, where no such simulation results are yet known. The present paper also further substantiates recent work on the properties of solid 3He^{3}\mathrm{He}3He in two dimensions.
The scaling function for the spin correlation length \ensuremath{\xi} of the two-dimensional (2D) quantum nonlinear \ensuremath{\sigma} model is obtained by Monte Carlo simulations. This is used to extract the temperature dependence of the spin correlation length, which sheds light on the nature of the ground state of low-dimensional quantum antiferromagnets. A crossover from quantum critical to renormalized classical behavior of the correlation length is found. Comparison is made with numerical results on the 2D square lattice Heisenberg model, as well as with neutron-scattering and nuclear magnetic relaxation experiments.
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.
Using an extended version of Handscomb's Monte Carlo method we study the spin-(1/2 quantum antiferromagnetic Heisenberg model on square lattices of various sizes. The calculated correlation length associated with antiferromagnetic order grows very rapidly as the temperature is lowered suggesting an essential singularity. Our results are consistent with those recently observed by neutron scattering done on Lo2{\mathrm{Lo}}_{2}Lo2​CuO4{\mathrm{CuO}}_{4}CuO4​.
In a recent preprint (cond-mat/9905415), Fujimoto has used the Bethe ansatz to compute the finite temperature, zero frequency Drude weight of spin transport in the quantum O(3) non-linear sigma model in a magnetic field H≠0H \neq 0H=0. We show here that, contrary to his claims, the results are in accord with earlier semiclassical results (Sachdev and Damle, cond-mat/9610115). We also comment on his 1/N expansion, and show that it does not properly describe the long-time correlations.