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Journal of Physics A Mathematical and Theoretical

Exact solutions of a multi-component anyon model withSU(N) invariance

Yi-Luo Yao, Jun-Peng Cao, Guang-Liang Li, Heng Fan

Published Jan 4, 2012• 4 citations

Abstract

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.

DOI: 10.1088/1751-8113/45/4/045207

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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.

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.

An exact solvable model of the coupled supersymmetric t - J chains interacting via the spin exchange interaction is proposed and solved by the means of the nested Bethe ansatz. It is shown, that in addition to the direct spin-exchange interaction an effective interaction between fermions of different chains called as "an effective attractive hard-core interaction" is realized in the model. The critical exponents describing asymptotic behavior of the correlation functions are calculated using the Bethe ansatz equations and the finite-state scaling analysis in conformal field theory. It is shown that the two-component Tomonaga-Luttinger liquid with the dominating correlations of fermion pairs is realized in the low-density region. This non-traditional behavior of the correlation functions is realized due to the effective attractive hard-core interaction between particles.

Abstract The contribution of anyonic degrees of freedom emerging in the non-Abelian spin sector of a one-dimensional system of interacting fermions carrying both spin and SU ( N f ) orbital degrees of freedom to the thermodynamic properties of the latter is studied based on the exact solution of the model. For sufficiently small temperatures and magnetic fields the anyons appear as zero energy modes localized at the massive kink excitations (Tsvelik 2014 Phys. Rev. Lett . 113 066401). From their quantum dimension they are identified as spin- anyons. The density of kinks (and anyons) can be controlled by an external magnetic field leading to the formation of a collective state of these anyons described by a parafermion conformal field theory for large fields. Based on the numerical analysis of the thermodynamic Bethe ansatz equations we propose a phase diagram for the anyonic modes.

The author considers a gas of fermions with parabolic dispersion and N spin-components (or spin S, N=2S+1) with SU(N) symmetry in one dimension interacting via a delta -function potential. The model is integrable and its solution has been obtained by Sutherland in terms of N nested Bethe ansatze. He analyses the discrete Bethe ansatz equations and classifies their solutions according to the string hypothesis. The thermodynamic Bethe-ansatz equations are derived for arbitrary band-filling for both repulsive and attractive interaction in terms of the thermodynamic energy potentials for the classes of eigenstates of the Hamiltonian. These equations are then discussed in several limiting cases, e.g. S=1/2, the ground state (T to 0), for vanishing interaction strength, for strong coupling, in the high-temperature limit, and the large-N limit.