Math. Model. Nat. Phenom.
Volume 9, Number 5, 2014Spectral problems
|44 - 72
|17 July 2014
The Infinite XXZ Quantum Spin Chain Revisited: Structure of Low Lying Spectral Bands and Gaps
1 School of Mathematics, Statistics and
Actuarial Science University of Kent Canterbury, Kent
2 Department of Mathematics, University of Alabama at Birmingham Birmingham, AL 35294, USA
Corresponding author. E-mail: firstname.lastname@example.org
We study the structure of the spectrum of the infinite XXZ quantum spin chain, an anisotropic version of the Heisenberg model. The XXZ chain Hamiltonian preserves the number of down spins (or particle number), allowing to represent it as a direct sum of N-particle interacting discrete Schrödinger-type operators restricted to the fermionic subspace. In the Ising phase of the model we use this representation to give a detailed determination of the band and gap structure of the spectrum at low energy. In particular, we show that at sufficiently strong anisotropy the so-called droplet bands are separated from higher spectral bands uniformly in the particle number. Our presentation of all necessary background is self-contained and can serve as an introduction to the mathematical theory of the Heisenberg and XXZ quantum spin chains.
Mathematics Subject Classification: 82B20 / 82B44
Key words: XXZ spin chain / Heisenberg model / droplet spectrum
© EDP Sciences, 2014
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