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arXiv:cond-mat/9803074 · Physical review. B, Condensed matter

One-dimensional XXZ model for particles obeying fractional statistics

Luigi Amico A, Andreas Osterloh, Ulrich Eckern

Published Jul 15, 1998• 44 citations
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Abstract

We define one-dimensional particles as non-Abelian representations of the symmetric group ${S}_{N}.$ The exact solution of an $\mathrm{XXZ}$ type Hamiltonian built up with such particles is achieved using the coordinate Bethe ansatz. The Bethe equations show that fractional statistics effectively account for coupling an external gauge field to an integer statistics' system.

DOI
10.1103/physrevb.58.r1703

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We define one-dimensional particles as non-Abelian representations of the symmetric group SN.{S}_{N}.SN​. The exact solution of an XXZ\mathrm{XXZ}XXZ type Hamiltonian built up with such particles is achieved using the coordinate Bethe ansatz. The Bethe equations show that fractional statistics effectively account for coupling an external gauge field to an integer statistics' system.
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 simplifications occur.The XXZ spectrum can be generated from separate pseudovacua as configurations 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.
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 simplifications occur. The XXZ spectrum can be generated from separate pseudovacua as configurations 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.
We present a comprehensive analysis of the exact Bethe ansatz solution for the onedimensional spinor Bose gas. The Bethe ansatz equations and the thermodynamic Bethe ansatz equations are derived, before investigating properties of these equations in limiting regimes. We explore the difference between spinless and spinor particles in the onedimensional Bose gas. For the spinor Bose gas, we present a new approximation to the thermodynamic Bethe ansatz equations which allows them to be solved in the strong coupling regime. We derive the first order correction to the solutions of the thermodynamic Bethe ansatz equations in a low temperature regime, where spin effects are most evident. We then derive thermodynamic quantities for this model, adding a first order correction to the thermodynamics of the spinless system. ¶ The subject of fractional exclusion statistics is presented, and its significance with regards to the spinless Bose gas is discussed. We also discuss its significance with regards to the spinor Bose gas, and find a regime in which non-mutual statistics may be applied to the spinor Bose gas to obtain expansions of thermodynamic quantities. These expressions are seen to improve upon previous analytical results for the model.