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qKZ equation and ground state of the O(1) loop model with open boundary conditions

Luigi Cantini A

Published Mar 30, 2009• 13 citations
OpenAlex

Abstract

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.

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qKZ equations and ground state of the O(1) loop model with open boundary conditions

Luigi Cantini A

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 $q^{1/2}= e^{-2πi/3}$ 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.

arXiv:0903.5050•Mar 30, 2009•0 citations

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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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The exact finite size ground state of the O(n= 1) loop model with open boundaries

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Exact expressions for correlations in the ground state of the dense O(1) loop model

Saibal Mitra, Bernard Nienhuis, Jan de Gier +1 more

Conjectures for analytical expressions for correlations in the dense O (1) loop model on semi-infinite square lattices are given. We have obtained these results for four types of boundary conditions. Periodic and reflecting boundary conditions have been considered before. We give many new conjectures for these two cases and review some of the existing results. We also consider boundaries on which loops can end. We call such boundaries 'open'. We have obtained expressions for correlations when both boundaries are open, and one is open and the other one is reflecting. Also, we formulate a conjecture relating the ground state of the model with open boundaries to fully packed loop models on a finite square grid. We also review earlier obtained results about this relation for the three other types of boundary conditions. Finally, we construct a mapping between the ground state of the dense O (1) loop model and the XXZ spin chain for the different types of boundary conditions.

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Bethe ansatz for the Temperley–Lieb loop model with open boundaries

Jan de Gier, Pavel Pyatov

We diagonalize the Hamiltonian of the Temperley–Lieb loop model with open boundaries using a coordinate Bethe ansatz calculation. We find that in the ground-state sector of the loop Hamiltonian, but not in other sectors, a certain constraint on the parameters has to be satisfied. This constraint has a natural interpretation in the Temperley–Lieb algebra with boundary generators. The spectrum of the loop model contains that of the quantum spin-1/2 XXZ chain with non-diagonal boundary conditions. We hence derive a recently conjectured solution of the complete spectrum of this XXZ chain. We furthermore point out a connection with recent results for the two-boundary sine–Gordon model.

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A Positive integral property on the ground state of the two-boundary Temperley--Lieb Hamiltonian

Keiichi Shigechi

We study the two-boundary Temperley--Lieb $O(n)$ 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 $U_q(\mathfrak{sl}_2)$. 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.

arXiv:1412.7617•Dec 24, 2014•2 citations
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.
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.
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.
Conjectures for analytical expressions for correlations in the dense O (1) loop model on semi-infinite square lattices are given. We have obtained these results for four types of boundary conditions. Periodic and reflecting boundary conditions have been considered before. We give many new conjectures for these two cases and review some of the existing results. We also consider boundaries on which loops can end. We call such boundaries 'open'. We have obtained expressions for correlations when both boundaries are open, and one is open and the other one is reflecting. Also, we formulate a conjecture relating the ground state of the model with open boundaries to fully packed loop models on a finite square grid. We also review earlier obtained results about this relation for the three other types of boundary conditions. Finally, we construct a mapping between the ground state of the dense O (1) loop model and the XXZ spin chain for the different types of boundary conditions.
We diagonalize the Hamiltonian of the Temperley–Lieb loop model with open boundaries using a coordinate Bethe ansatz calculation. We find that in the ground-state sector of the loop Hamiltonian, but not in other sectors, a certain constraint on the parameters has to be satisfied. This constraint has a natural interpretation in the Temperley–Lieb algebra with boundary generators. The spectrum of the loop model contains that of the quantum spin-1/2 XXZ chain with non-diagonal boundary conditions. We hence derive a recently conjectured solution of the complete spectrum of this XXZ chain. We furthermore point out a connection with recent results for the two-boundary sine–Gordon model.
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.