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We propose two methods of estimating a systematic error in extrapolation to the infinite-size limit in the study of measuring the Haldane gaps of the one-dimensional Heisenberg antiferromagnet with the integer spin up to S ¼ 5. The finite-size gaps obtained by numerical diagonalizations based on Lanczos algorithm are presented for sizes that have not previously been reported. The changes of boundary conditions are also examined. We successfully demonstrate that our methods of extrapolation work well. The Haldane gap for S ¼ 1 is estimated to be 0:4104789 AE 0:0000013. We successfully obtain the gaps up to S ¼ 5, which make us confirm the asymptotic formula of the Haldane gap in S ! 1.

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We investigate the E(g) x in circle e(g) Jahn-Teller system for the purpose of revealing the nature of quantum chaos in crystals. This system simulates the interaction between the nuclear vibrational modes and the electronic motion in non-Kramers doublets for multiplets of transition-metal ions. Inclusion of the anharmonic potential due to the trigonal symmetry in crystals makes the system nonintegrable and chaotic. Besides the quantal analysis of the transition from Poisson to Wigner level statistics with increasing the strength of anharmonicity, we study the effect of chaos on the electronic orbital angular momentum and explore the magnetic g-factor as a function of the system's energy. The regular oscillation of this factor changes to a rapidly decaying irregular oscillation by increasing the anharmonicity (chaoticity).

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