Journal of Physics A Mathematical and General

An integrablesl(2|1) vertex model for the spin quantum Hall critical point

R. M. Gade

Published Sep 28, 1999• 10 citations

Abstract

An integrable vertex model associated to the Lie superalgebra sl (2|1) is constructed for the description of the spin quantum Hall critical phase. The model involves R -matrix solutions of the Yang-Baxter equation with respect to both the vector representation of sl (2|1) and its dual and an inhomogenity in the spectral parameters. On the torus the model can be mapped onto a Chalker-Coddington-type network model.

DOI: 10.1088/0305-4470/32/41/303

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Journal of Physics A Mathematical and General

An integrablesl(2|1) vertex model for the spin quantum Hall critical point

R. M. Gade

Published Sep 28, 1999• 10 citations

Publisher

Abstract

DOI: 10.1088/0305-4470/32/41/303

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In this study, an integrable vertex model based on the quantum affine superalgebra

U_q\bigl(\hat{gl}(2|2)\bigr)

is constructed. The model is characterized by a particular assignment of spectral parameters and lowest as well as highest weight

U_q\bigl(gl(2|2) \bigr)

modules to its lattice links. Solutions of the corresponding intertwining conditions yield the Boltzmann weights. The set of mutually commuting charges contains a quantity equivalent to the super spin chain Hamiltonian proposed for the description of the integer quantum Hall transition.

This work concerns to the studies of boundary integrability of the vertex models from representations of the Temperley-Lieb algebra associated with the quantum group

{\cal U}_{q}[X_{n}]

for the affine Lie algebras

X_{n}

A_{1}^{(1)}

B_{n}^{(1)}

C_{n}^{(1)}

and

D_{n}^{(1)}

. A systematic computation method is used to constructed solutions of the boundary Yang-Baxter equations. We find a

2n^{2}+1

free parameter solution for $A_{1}^{(1)} $ spin-

(n-1/2)

and $ C_{n}^{(1)}$ vertex models. It turns that for $A_{1}^{(1)} $ spin-

n

, $ B_{n}^{(1)}$ and

D_{n}^{(1)}

vertex models, the solution has

2n^{2}+2n+1

free parameters.

A new type of Yang-Baxter equation (YBE) for

R

-operators depending on three spectral parameters is constructed from the star-triangle relation for the chiral Potts model. As the

Z_N

symmetric generalization to the Ising model, its Boltzmann weights are known to depend on two variables describing a curve with genus larger than one for

N>2

, except for the self-dual point corresponding to the Fateev-Zamolodchikov chain. Combined with the fact that quantum Hamiltonians of edge-type models such as the Ising model contain both nearest-neighbor interaction and onsite potential terms, this leads naturally to an additional spectral parameter in the associated

R

-operator. The construction extends the edge-vertex correspondence of solvable lattice models, and provides a bridge between the Bazhanov-Stroganov four-parameter

R

-matrix---realized as an intertwiner of cyclic representations of

U_q(\mathfrak{sl}_2)

at a root of unity---and Shastry's two-parameter

R

-operator obtained from the decorated YBE.