Abstract
The coordinate Bethe ansatz solutions of the XXZ model for a one-dimensional spin-1/2 chain are analyzed with focus on the statistical properties of the constituent quasiparticles. Emphasis is given to the special cases known as XX, XXX, and Ising models, where considerable simplications occur. The XXZ spectrum can be generated from separate pseudovacua as congur ations of sets of quasiparticles with different exclusion statistics. These sets are complementary in the sense that the pseudovacuum of one set contains the maximum number of particles from the other set. The Bethe ansatz string solutions of the XXX model evolve differently in the planar and axial regimes. In the Ising limit they become ferromagnetic domains with integer-valued exclusion statistics. In the XX limit they brake apart into hard-core bosons with (effectively) fermionic statistics. Two sets of quasiparticles with spin 1/2 and fractional statistics are distinguished, where one set (spinons) generates the XXZ spectrum from the unique, critical ground state realized in the planar regime, and the other set (solitons) generates the same spectrum from the twofold, antiferromagnetically ordered ground state realized in the axial regime. In the Ising limit, the solitons become antiferromagnetic domain walls.
Highlights
Quantum spin chains are physically realized in quasi-one-dimensional magnetic compounds
The XXZ spectrum can be generated from different pseudovacua by the systematic creation of quasiparticles with different exclusion statistics
In the following we describe three ways of generating the spectrum of the Ising chain, each producing a distinct set of quasiparticles with different exclusion statistics
Summary
Quantum spin chains are physically realized in quasi-one-dimensional magnetic compounds. The XXZ spectrum can be generated from different pseudovacua by the systematic creation of quasiparticles with different exclusion statistics In most cases these pseudovacua are states of lowest energy (physical vacua). The boxes in the top row represent quasiparticles whose names are derived from jargon used to describe coordinate Bethe ansatz solutions These quasiparticles are characterized by strings of complex momenta with common real part and different imaginary parts [7]. Whereas the string particles have integer-valued exclusion statistics, the spinons and solitons are realizations of fractional statistics [18] They are both semions but with different pseudovacua. When a fermion is removed from the intact sea, kc slightly increases to generate not just one but two vacancies in the interval |k| < kc of the band These two holes are identified with a pair of spin-up spinons. We discuss the quasiparticle composition of the XXZ spectrum in the XXX limit (section 3.1), the Ising limit (section 3.2), and the XX limit (section 3.3)
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