The Heisenberg Antiferromagnet and the Lanczos Algorithm Abstract

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BSc Final Year Report - Matthew HolmesPrevious comparisons with exact values, in Figs. 8 & 10, hinted at the trend toward linearity. With nofield, Fig. 8, the ground state increases haphazardly with N. In a field of 10T (Fig. 10), when thevariational estimates first agree for all N, the rise in energies shows more linearity. The field strengthin Fig. 10 is similar to the h=0.9 line in Fig. 13, above. For confidence in the estimations beyond 8spins, the field parameter h is increased for safety. In this way we can be confident that the degree ofaccuracy of the modified Lanczos estimates for small chains, is retained beyond N=8.Limitations of The Modified Lanczos MethodThe Modified Lanczos method is confined to making estimation of the ground state energies, for boththe ferro- and antiferromagnetic cases. It is thus useful only in the zero temperature limit and cannotevaluate excited states of the chain, for which the full algorithm is required.ConclusionBy applying the Heisenberg Hamiltonian to antiferromagnetically coupled spins along a lineardimension, a model of a closed 1D spin chain has been developed. The justification for such a model ismade with reference to experimental observations of quasi-1D chains and a theoretical consideration ofthe nature of antiferromagnetic exchange interactions.The ground state eigenvalues of the Hamiltonian, calculated with the Lanczos method were found to bestrongly dependent on the eigenvalue spacing. Accuracy was found to be significantly increased byusing the modified Lanczos method, a variational technique. To achieve complete accuracy, however,required that the chain be placed in a magnetic field. With a field of ~10T a complete agreementbetween the Lanczos estimates and a direct evaluation with the Jacobi method, was found up to chainlength of 8 spins.Direct evaluation with Jacobi, for a chain of 8 spins was the limit on the author's computer. The fieldhad the effect of pulling apart the eigenvalues and allowing complete accuracy up to a chain of 8 spins.It also permitted larger systems to be treated than the 8 spin limit using the Jacobi method.The ability for accurate eigenvalue estimation with the Lanczos method, and its ability to treat largersystems, has thus been demonstrated.It is suggested that the excited states of the Hamiltonian be investigated using the full Lanczosalgorithm, the modified method being confined to ground state estimation. The ground stateeigenvector might also be used to construct the operators with which to investigate a wider range ofphysical properties than have been considered herein.AcknowledgmentsI would like to thank my project supervisor, Dr Paul Stevenson, for his excellent guidance (and infinitepatience!) throughout. Now is also the time to acknowledge my debt to the sun; for shining throughmy window one day.References[1] Aharoni A., Introduction to the Theory of Ferromagnetism, Oxford (2000).[2] Brussard P. J. & Glaudemans P. W. M., Shell Model Applications in Nuclear Spectroscopy,North-Holland Publishing Company, Amsterdam, (1977).[3] Chatelin F., Eigenvalues of Matrices, Wiley (1993).[4] Cloizeaux J. & Pearson J. J., Phys. Rev. 128, 2131 (1962).[5] Coldea R. et al., Phys. Rev. Lett. 86, 5377 (2001).[6] Dagotto E. & Moreo A., Phys. Rev. D 31, 865 (1985).[7] Dender D. C. et al., Phys. Rev. B 53, 2583 (1996).[8] Ellis T. M. R. et al., Fortan 90 Programming, Addison-Wesley (1994).[9] Hihilashvili R., Derivation of the Heisenberg Hamiltonian, http://phjoan5.technion.ac.il/~riki/,Israel Institute of Technology Dept. of Physics, (2003).[10] Hutchings M. T., Ikeda H., Milne J. M., J. Phys. C 12, L739 (1979)[11] Krupi ka S. et al, Elements of Theoretical Magnetism, Iliffe (1968).[12] Lake B. et al, ISIS Science, www.isis.rl.ac.uk/science/index.htm, Rutherford AppletonLaboratory, (2001).[13] Lieb E. H. et al., Mathematical Physics in One Dimension, Academic Press (1996).16
BSc Final Year Report - Matthew Holmes[14] Malvezzi A. L., Brazilian J. Phys. 33, 55 (2003).[15] Merzbacher E., Quantum Mechanics, Wiley (1998).[16] Nepomechie R. I., A Spin Chain Primer, www.physics.miami.edu/nepomechie/primer.pdf,University of Miami Phys. Dept. (1998).[17] O'Reilly E., Quantum Theory of Solids, Taylor & Francis (2002).[18] Sandvik A. W., 'Computational Studies of Quantum Spin Systems' & 'Numerical Solutions ofthe Schrödinger Equation', PY502 Computational Physics, Dept. of Physics, BostonUniversity (2004).[19] Scaife B. K. P. et al, Studies in Numerical Analysis, Academic Press (1974)[20] Stevenson P. D., University of Surrey Phys. Dept., Private communication (2005).[21] Tyablikov S. V., Methods in The Quantum Theory of Magnetisation, Plenum Press (1967).17
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