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arXiv:1412.7617

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

Keiichi Shigechi

Published Dec 24, 2014• 2 citations
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Abstract

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.

DOI
10.48550/arxiv.1412.7617

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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 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.
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.
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 give an exact spectral equivalence between the quantum group invariant XXZ chain with arbitrary left boundary term and the same XXZ chain with purely diagonal boundary terms. This equivalence, and a further one with a link pattern Hamiltonian, can be understood as arising from different representations of the one-boundary Temperley–Lieb algebra. For a system of size L these representations are all of dimension 2 L and, for generic points of the algebra, equivalent. However, at exceptional points they can possess different indecomposable structures. We study a centralizer of the one-boundary Temperley–Lieb algebra in the 'non-diagonal' spin- representation and find its eigenvalues and eigenvectors. In the exceptional cases this centralizer becomes indecomposable. We show how to get a truncated space of 'good' states. The indecomposable part of the centralizer leads to degeneracies in the three mentioned Hamiltonians.
We discuss generalizations of the Temperley–Lieb algebra in the Potts and XXZ models. These can be used to describe the addition of integrable boundary terms of different types. We use the Temperley–Lieb algebra and its one-boundary, two-boundary, and periodic extensions to classify different integrable boundary terms in the two-, three-, and four-state Potts models. The representations always lie at critical points where the algebras becomes non-semisimple and possess indecomposable representations. In the one-boundary case we show how to use representation theory to extract the Potts spectrum from an XXZ model with particular boundary terms and hence obtain the finite size scaling of the Potts models. In the two-boundary case we find that the Potts spectrum can be obtained by combining several XXZ models with different boundary terms. As in the Temperley–Lieb case, there is a direct correspondence between representations of the lattice algebra and those in the continuum conformal field theory.