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arXiv:cond-mat/9701197 · Physical Review Letters

Correlation Lengths in Quantum Spin Ladders

Olav F. Syljuåsen, Sudip Chakravarty, M. Greven

Published May 26, 1997• 24 citations
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Abstract

Analytic expressions for the temperature dependences of the correlation length are given for antiferromagnetic spin- $\frac{1}{2}$ Heisenberg ladders using a finite-size nonlinear \ensuremath{\sigma} model approach. These calculations rely on identifying three successive crossover regimes as a function of temperature. In each of these regimes, precise and controlled approximations are formulated. The analytical results are found to be in excellent agreement with Monte Carlo simulations for the Heisenberg Hamiltonian.

DOI
10.1103/physrevlett.78.4115

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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 Heisenberg spin ladder is studied in the semiclassical limit, via a mapping to the nonlinear \ensuremath{\sigma} model. Different treatments are needed if the interchain coupling K is small, intermediate, or large. For intermediate coupling a single nonlinear \ensuremath{\sigma} model is used for the ladder. It predicts a spin gap for all nonzero values of K if the sum s+s\ifmmode \tilde{}\else \~{}\fi{} of the spins of the two chains is an integer, and no gap otherwise. For small K, a better treatment proceeds by coupling two nonlinear \ensuremath{\sigma} models, one for each chain. For integer s=s\ifmmode \tilde{}\else \~{}\fi{}, the saddle point approximation predicts a sharp drop in the gap as K increases from zero. A Monte Carlo simulation of a spin-1 ladder is presented which supports the analytical results.
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​.