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The spectral gap of the ferromagnetic $$XXZ$$ chain. (English) Zbl 0880.60103
Summary: We prove that the spectral gap of the spin-$${1\over 2}$$ ferromagnetic $$XXZ$$ chain with Hamiltonian $$H= -\sum_x S^{(1)}_x S^{(1)}_{x+1}+ S^{(2)}_x S^{(2)}_{x+1}+\Delta S^{(3)}_x S^{(3)}_{x+1}$$, is given by $$\Delta-1$$ for all $$\Delta\geq 1$$. This is the gap in the spectrum of the infinite chain in any of its ground states, the translation invariant ones as well as the kink ground states, which contain an interface between an up and a down region. In particular, this shows that the lowest magnon energy is not affected by the presence of a domain wall. This surprising fact is a consequence of the $$\text{SU}_q(2)$$ quantum group symmetry of the model.

##### MSC:
 60K40 Other physical applications of random processes 82B10 Quantum equilibrium statistical mechanics (general) 82B24 Interface problems; diffusion-limited aggregation arising in equilibrium statistical mechanics 81R50 Quantum groups and related algebraic methods applied to problems in quantum theory 82B23 Exactly solvable models; Bethe ansatz
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