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arXiv:cond-mat/0201096 · Europhysics Letters (EPL)

Comment on “Bounding and approximating parabolas for the spectrum of Heisenberg spin systems” by H.-J. Schmidt, J. Schnack and M. Luban

O. Waldmann

Published Feb 1, 2002• 8 citations

Abstract

Recently, Schmidt et al. proved that the energy spectrum of a Heisenberg spin system (HSS) is bounded by two parabolas, i.e. lines which depend on the total spin quantum number S as S(S+1). The prove holds for homonuclear HSSs which fulfill a weak homogenity condition. Moreover, the extremal values of the exact spectrum of various HSS which were studied numerically were found to lie on approximate parabolas, named rotational bands, which could be obtained by a shift of the boundary parabolas. In view of this, it has been claimed that the rotational band structure (RBS) of the energy spectrum is a general behavior of HSSs. Furthermore, since the approximate parabolas are very close to the true boundaries of the spectrum for the examples discussed, it has been claimed that the methods allow to predict the detailed shape of the spectrum and related properties for a general HSS. In this comment I will show by means of examples that the RBS hypothesis is not valid for general HSSs. In particular, weak homogenity is neither a necessary nor a sufficient condition for a HSS to exhibit a spectrum with RBS.

DOI: 10.1209/epl/i2002-00506-9

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We prove that for a wide class of quantum spin systems with isotropic Heisenberg coupling the energy eigenvalues which belong to a total spin quantum number S have upper and lower bounds depending at most quadratically on S . The only assumption adopted is that the mean coupling strength of any spin with respect to its neighbours is constant for all N spins. The coefficients of the bounding parabolas are given in terms of special eigenvalues of the N × N coupling matrix which are usually easily evaluated. In addition, we show that the bounding parabolas, if properly shifted, provide very good approximations of the true boundaries of the spectrum. We present numerical examples of frustrated rings, a cube, and an icosahedron.

PACS. 75.10.Jm – Quantized spin models. PACS. 75.10.Hk – Classical spin models. PACS. 75.50.Xx – Molecular magnets. Abstract. – We prove that for a wide class of quantum spin systems with isotropic Heisenberg coupling the energy eigenvalues which belong to a total spin quantum number S have upper and lower bounds depending at most quadratically on S. The only assumption adopted is that the mean coupling strength of any spin w. r. t. its neighbours is constant for all N spins. The coefficients of the bounding parabolas are given in terms of special eigenvalues of the N × N coupling matrix which are usually easily evaluated. In addition we show that the bounding parabolas, if properly shifted, provide very good approximations of the true boundaries of the spectrum. We present numerical examples of frustrated rings, a cube, and an icosahedron. Introduction. – The study of finite spin systems is not only interesting in its own right but has also important applications for understanding so-called “molecular magnets”. The synthesis of these systems has undergone rapid progress in recent years building on successes in coordination and polyoxometalate chemistry [1–4]. Each of the identical molecular units can

The spin pair-correlation function of finite bipartite antiferromagnetic Heisenberg quantum spin rings was studied numerically by means of exact diagonalization techniques. For large spins, the spectrum of the spin pair-correlation function is of a remarkably simple structure: It consists of few characteristic peaks at low, and a broad featureless signal at high temperatures. This arises as the energy spectrum exhibits a set of parallel rotational bands at low energies emerging into a quasi-continuum of states, with transitions from the lowest rotational band to the quasi-continuum being highly suppressed. The energies of the rotational bands can be accurately described by a generalized dispersion relation that depends on the total spin quantum number and, as usual, on the shift quantum number. These regularities are better fulfilled the larger the spin length but the smaller the ring size. It will be shown that all these features are associated with the underlying sublattice structure of the spin rings, and it is argued that they are valid for a more general class of finite Heisenberg systems than rings.

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