Bounding and approximating parabolas for the spectrum of Heisenberg spin systems

Details Bounding and approximating parabolas for the spectrum of Heisenberg spin systems Bounding and approximating parabolas for the spectrum of Heisenberg spin systems

2001-01-16

arxiv.org/abs/cond-mat/0101228v1

Europhysics Letters 55 (2001) 105-111

Physics

Condensed Matter

Statistical Mechanics

8 pages, 3 figures. Submitted to Europhysics Letters

Scientific paper

10.1209/epl/i2001-00388-9

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 times 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.

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