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arXiv:0807.3197 · Journal of Physics A Mathematical and Theoretical

The quantum inverse scattering method with anyonic grading

Murray T. Batchelor, Angela Foerster, Xi‐Wen Guan, Jon Links, Huan-Qiang Zhou

Published Oct 16, 2008• 17 citations
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Abstract

We formulate the quantum inverse scattering method for the case of anyonic grading. This provides a general framework for constructing integrable models describing interacting hard-core anyons. Through this method we reconstruct the known integrable model of hard core anyons associated with the XXX model, and as a new application we construct the anyonic t − J model. The energy spectrum for each model is derived by means of a generalization of the algebraic Bethe ansatz. The grading parameters implementing the anyonic signature give rise to sector-dependent phase factors in the Bethe ansatz equations.

DOI
10.1088/1751-8113/41/46/465201

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We formulate the quantum inverse scattering method for the case of anyonic grading. This provides a general framework for constructing integrable models describing interacting hard-core anyons. Through this method we reconstruct the known integrable model of hard core anyons associated with the XXX model, and as a new application we construct the anyonic t − J model. The energy spectrum for each model is derived by means of a generalization of the algebraic Bethe ansatz. The grading parameters implementing the anyonic signature give rise to sector-dependent phase factors in the Bethe ansatz equations.
The Quantum Inverse Scattering Method, formalised in the late 70s by Faddeev, Sklyanin and Takhtadzhyan, is a method for constructing integrable quantum systems given a solution to the Yang-Baxter equation. Once constructed it is often possible to find the energy spectrum of the associated hamiltonian using the method of mutually commuting transfer matrices, and the algebraic Bethe ansatz. If the algebraic Bethe ansatz fails we are still often able to solve the system using functional relation methods. In 2006 Dancer, Isaac and Links found several solutions to the Yang-Baxter equation arising from the Drinfeld double of the dihedral group D(D3)D(D_3)D(D3​). The associated hamiltonian describes a system of interacting, non-abelian anyons. In this dissertation we take these solutions to the Yang-Baxter equation and, using a combination of algebraic Bethe ansatz and functional relations, we find the eigenvalues of the anyonic hamiltonian. The solution we find is dependent on two distinct sets of solutions to the Bethe equations, something that is rather uncommon, though not unprecedented. Further results are also obtained for other integrable models arising from the same set of solutions to the Yang-Baxter equation.
We construct a multi-component interacting hard-core anyons model with SU(N) invariance and obtain the exact solutions of the systems by using the nested algebraic Bethe ansatz method with anyonic grading. We find that the grading parameters enter the Bethe ansatz equations as the gauge potentials, and the magnetic flux-like effects are different in the different rapidity sectors. Our results cover the interacting multi-component bosonic or multi-component fermionic models with SU(N) symmetry. In the system, the effective interactions among each component can be tuned by the grading parameters which do not break the integrability, while the integrability of conventional multi-component bosons or fermions requires that all the interactions are equal. The results will be useful to theories and experiments of the cold atomic systems.