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Dispersion relations and speeds of sound in special sectors for the integrable chain with a.pdf

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Dispersion relations and speeds of sound in special sectors for the integrable chain with a

a r X i v : c o n d - m a t / 9 8 0 3 2 1 0 v 1 1 7 M a r 1 9 9 8 Dispersion relations and speeds of sound in special sectors for the integrable chain with alternating spins B - D Do?rfel? and St Mei?ner§ Institut fu?r Physik, Humboldt-Universita?t , Theorie der Elementarteilchen Invalidenstra?e 110, 10115 Berlin, Germany Abstract. Based on our previous analysis [1] of the anisotropic integrable chain consisting of spins s = 1 2 and s = 1 we compare the dispersion relations for the sectors with infinite Fermi zones. Further we calculate the speeds of sound for regions close to sector borders, where the Fermi radii either vanish or diverge, and compare the results. PACS numbers: 75.10 JM, 75.40 Fa ? E-mail: doerfel@qft2.physik.hu-berlin.de § E-mail: meissner@qft2.physik.hu-berlin.de Dispersion relations and speeds of sound 2 1. Introduction The integrable spin chain XXZ(1 2 , 1) constructed in 1992 by de Vega and Woynarovich [2] shows a rich physical structure with different ground states depending on the anisotropy parameter and the two coupling constants. In our previous papers [1], [3], [4] and [5] we have analyzed this structure and calculated several important quantities. In this paper we deal mainly with dispersion relations and speeds of sound. The paper is organized as follows. After having reviewed the definitions in section 2 we found it necessary to remind the reader results of our last paper [1] concerning the phase diagram. In the same section the dispersion relations for all sectors with infinite Fermi zones only are compared. In section 4 we present the calculations for the speeds of sound obtained either by power expansion or Wiener-Hopf technique for small or large Fermi radii respectively. Our conclusions are contained in section 5. Some useful definitions are compiled in an appendix. 2. Description of the model We refer the reader to papers [2] and [3] for the basics of the model. Our Hamiltonian of a spin chain of length 2N is given by H(γ) =