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arXiv:1411.7020 · Journal of Statistical Mechanics Theory and Experiment

The dilute Temperley–Lieb O( n = 1) loop model on a semi infinite strip: the ground state

Alexandr Garbali, Bernard Nienhuis

Published Apr 1, 2017• 9 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1742-5468/aa6a30

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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 study the two-boundary Temperley--Lieb O(n)O(n)O(n) loop model on Kazhdan--Lusztig bases of type A and B. We obtain explicit expressions of the ground state of the two-boundary Temperley--Lieb Hamiltonian by means of a coideal subalgebra of Uq(sl2)U_q(\mathfrak{sl}_2)Uq​(sl2​). This ground state possesses a positive integral property. We conjecture that some components of the ground state are directly related to an enumeration of binary or permutation matrices.
Abstract We introduce the ghost algebra, a two-boundary generalisation of the Temperley–Lieb (TL) algebra, using a diagrammatic presentation. The existing two-boundary TL algebra has a basis of string diagrams with two boundaries, and the number of strings connected to each boundary must be even; in the ghost algebra, this number may be odd. To preserve associativity while allowing boundary-to-boundary strings to have distinct parameters according to the parity of their endpoints, as seen in the one-boundary TL algebra, we decorate the boundaries with bookkeeping dots called ghosts. We also introduce the dilute ghost algebra, an analogous two-boundary generalisation of the dilute TL algebra. We then present loop models associated with these algebras, and classify solutions to their boundary Yang–Baxter equations, given existing solutions to the Yang–Baxter equations for the TL and dilute TL models. This facilitates the construction of a one-parameter family of commuting transfer tangles, making these models Yang–Baxter integrable.