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arXiv:1506.00822 · Journal of Physics A Mathematical and Theoretical

Non-Abelian ${SU}{(3)}_{k}$ anyons: inversion identities for higher rank face models

Holger Frahm, Nikos Karaiskos

Published Oct 29, 2015• 7 citations
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Abstract

The spectral problem for an integrable system of particles satisfying the fusion rules of is expressed in terms of exact inversion identities satisfied by the commuting transfer matrices of the integrable fused interaction round a face model of Jimbo, Miwa and Okado. The identities are proven using local properties of the Boltzmann weights, in particular the Yang–Baxter equation and unitarity. They are closely related to the consistency conditions for the construction of eigenvalues obtained in the separation of variables approach to integrable vertex models.

DOI
10.1088/1751-8113/48/48/484001

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Non-Abelian SU(3)k{SU}{(3)}_{k}SU(3)k​ anyons: inversion identities for higher rank face models
The spectral problem for an integrable system of particles satisfying the fusion rules of is expressed in terms of exact inversion identities satisfied by the commuting transfer matrices of the integrable fused interaction round a face model of Jimbo, Miwa and Okado. The identities are proven using local properties of the Boltzmann weights, in particular the Yang–Baxter equation and unitarity. They are closely related to the consistency conditions for the construction of eigenvalues obtained in the separation of variables approach to integrable vertex models.
Using the properties of the local Boltzmann weights of integrable interaction-round-a-face (IRF or face) models we express local operators in terms of generalized transfer matrices. This allows for the derivation of discrete functional equations for the reduced density matrices in inhomogeneous generalizations of these models. We apply these equations to study the density matrices for IRF models of various solid-on-solid type and quantum chains of non-Abelian su(²)³ or Fibonacci anyons. Similar as in the six vertex model we find that reduced density matrices for a sequence of consecutive sites can be’factorized’, i.e. expressed in terms of nearest-neighbour correlators with coefficients which are independent of the model parameters. Explicit expressions are provided for correlation functions on up to three neighbouring sites.
Using the properties of the local Boltzmann weights of integrable interaction-round-a-face (IRF or face) models we express local operators in terms of generalized transfer matrices. This allows for the derivation of discrete functional equations for the reduced density matrices in inhomogeneous generalizations of these models. We apply these equations to study the density matrices for IRF models of various solid-on-solid type and quantum chains of non-Abelian \mathbold{su(2)_3} 𝐬 𝐮 ( 2 ) 3 or Fibonacci anyons. Similar as in the six vertex model we find that reduced density matrices for a sequence of consecutive sites can be ‘factorized’, i.e. expressed in terms of nearest-neighbour correlators with coefficients which are independent of the model parameters. Explicit expressions are provided for correlation functions on up to three neighbouring sites.
Chains of interacting non-Abelian anyons with local interactions invariant under the action of the Drinfeld double of the dihedral group D 3 are constructed. Formulated as a spin chain the Hamiltonians are generated from commuting transfer matrices of an integrable vertex model for periodic and braided as well as open boundaries. A different anyonic model with the same local Hamiltonian is obtained within the fusion path formulation. This model is shown to be related to an integrable fusion interaction round the face model. Bulk and surface properties of the anyon chain are computed from the Bethe equations for the spin chain. The low-energy effective theories and operator content of the models (in both the spin chain and fusion path formulation) are identified from analytical and numerical studies of the finite-size spectra. For all boundary conditions considered the continuum theory is found to be a product of two conformal field theories. Depending on the coupling constants the factors can be a Z 4 parafermion or a minimal model.
We derive exact inversion identities satisfied by the transfer matrix of inhomogeneous interaction-round-a-face (IRF) models with arbitrary boundary conditions using the underlying integrable structure and crossing properties of the local Boltzmann weights. For the critical restricted solid-on-solid (RSOS) models these identities together with some information on the analytical properties of the transfer matrix determine the spectrum completely and allow to derive the Bethe equations for both periodic and general open boundary conditions.
Starting from the fusion rules for the algebra SO(5)2 we construct one-dimensional lattice models of interacting anyons with commuting transfer matrices of ‘interactions round the face’ (IRF) type. The conserved topological charges of the anyon chain are recovered from the transfer matrices in the limit of large spectral parameter. The properties of the models in the thermodynamic limit and the low energy excitations are studied using Bethe ansatz methods. Two of the anyon models are critical at zero temperature. From the analysis of the finite size spectrum we find that they are effectively described by rational conformal field theories invariant under extensions of the Virasoro algebra, namely WB2 and WD5, respectively. The latter contains primaries with half and quarter spin. The modular partition function and fusion rules are derived and found to be consistent with the results for the lattice model.
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.