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arXiv:math-ph/0504032 · Journal of Physics A Mathematical and General

Inhomogeneous loop models with open boundaries

Philippe Di Francesco

Published Jun 22, 2005• 28 citations
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Abstract

We consider the crossing and non-crossing O(1) dense loop models on a semi-infinite strip, with inhomogeneities (spectral parameters) that preserve the integrability. We compute the components of the ground state vector and obtain a closed expression for their sum, in the form of Pfaffian and determinantal formulas.
DOI
10.1088/0305-4470/38/27/001

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Abstract We consider the integrable dilute Temperley–Lieb (dTL) O( n = 1) loop model on a semi-infinite strip of finite width L . In the analogy with the Temperley–Lieb (TL) O( n = 1) loop model the ground state eigenvector of the transfer matrix is studied by means of a set of q -difference equations, sometimes called the q KZ equations. We compute some ground state components of the transfer matrix of the dTL model, and show that all ground state components can be recovered for arbitrary L using the q KZ equation and certain recurrence relation. The computations are done for generic open boundary conditions.
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We consider the qKZ equations based on the two boundaries Temperley Lieb algebra. We construct their solution in the case s=q−3/2s=q^{-3/2}s=q−3/2 using a recursion relation. At the combinatorial point q1/2=e−2πi/3q^{1/2}= e^{-2πi/3}q1/2=e−2πi/3 the solution reduces to the ground state of the dense O(1) loop model on a strip with open boundary conditions. We present an alternative construction of such ground state based on the knowledge of the ground state of the same model with mixed boundary conditions and prove that the sum rule as of its components is given by the product of four symplectic characters.
We consider the qKZ equations based on the two boundaries Temperley Lieb algebra. We construct their solution in the case s = q −3/2 using a recursion relation. At the combinatorial point q 1/2 = e −2�i/3 the solution reduces to the ground state of the dense O(1) loop model on a strip with open boundary conditions. We present an alternative construction of such ground state based on the knowledge of the ground state of the same model with mixed boundary conditions and prove that the sum rule as of its components is given by the product of four symplectic characters.