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

Gier, Jan de; Ponsaing, Anita; Shigechi, Keiichi

doi:10.1088/1742-5468/2009/04/p04010

Cited by 15 publications

(34 citation statements)

References 45 publications

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“…This is a consequence of the stochasticity of the transfer matrix. Analogous situation happens in the dense TL loop model at n = 1 [26,6]. There are in total 2 L different choices of the occupancy for the link patterns in LP L , hence Z L = 2 L ψ e .…”

Section: The First Recurrence Relationmentioning

confidence: 86%

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

Garbali¹,

Nienhuis²

2017

J. Stat. Mech.

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This is the second part of our study of the ground state eigenvector of the transfer matrix of the dilute Temperley-Lieb loop model with the loop weight n = 1 on a semi infinite strip of width L [14]. We focus here on the computation of the normalization (otherwise called the sum rule) Z L of the ground state eigenvector, which is also the partition function of the critical site percolation model. The normalization Z L is a symmetric polynomial in the inhomogeneities of the lattice z 1 , .., z L . This polynomial satisfies several recurrence relations which we solve independently in terms of Jacobi-Trudi like determinants. Thus we provide a few determinant expressions for the normalization Z L .

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Section: The First Recurrence Relationmentioning

confidence: 86%

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

Garbali¹,

Nienhuis²

2017

J. Stat. Mech.

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“…Using this conjecture all other elements follow from the qKZ equations. This was the basis of the algorithm for the computation of the ground state components and the sum rules for the TL O(1) loop models with various boundary conditions [12,11,28,7,3] and for the dTL O(1) model with periodic boundary conditions [10]. We extend these results to the dTL O(1) model with open boundary conditions.…”

Section: The Dilute O(n) Loop Modelmentioning

confidence: 98%

“…The action of ρ (i) on the link patterns should respect the occupation, i.e., the top left j should coincide with the occupations of the edges j and j + 1 (respectively) in the link pattern, otherwise the action gives zero. We notice that ρ (2) , ρ (4) , ρ (7) and ρ (9) act as projectors on the different occupancies of two edges, ρ (5) and ρ (6) act locally by interchanging an occupied edge with an unoccupied edge, the operator ρ (1) also acts locally by inserting a little arch at the position j, j + 1 of a link pattern. The two remaining operators ρ (3) and ρ (8) , change the global picture, because they connect two existing paths, or close one.…”

Section: Link Pattern Basis and The R-matrixmentioning

confidence: 99%

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

Garbali¹,

Nienhuis²

2017

J. Stat. Mech.

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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 qKZ 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 qKZ equation and certain recurrence relation. The computations are done for generic open boundary conditions.

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“…. It is in this form that the reflection quantum KZ equations often appear in the literature on spin chains, see, e.g., [12,22,24,54,77].…”

Section: Proof Consider the Induced C(t )#W -Modulementioning

confidence: 99%

Koornwinder polynomials and the XXZ spin chain

Stokman

Vlaar

2015

Journal of Approximation Theory

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Abstract. Nonsymmetric Koornwinder polynomials are multivariable extensions of nonsymmetric Askey-Wilson polynomials. They naturally arise in the representation theory of (double) affine Hecke algebras. In this paper we discuss how nonsymmetric Koornwinder polynomials naturally arise in the theory of the Heisenberg XXZ spin-1 2 chain with general reflecting boundary conditions. A central role in this story is played by an explicit two-parameter family of spin representations of the two-boundary Temperley-Lieb algebra. These spin representations have three different appearances. Their original definition relates them directly to the XXZ spin chain, in the form of matchmaker representations they relate to Temperley-Lieb loop models in statistical physics, while their realization as principal series representations leads to the link with nonsymmetric Koornwinder polynomials. The nonsymmetric difference Cherednik-Matsuo correspondence allows to construct for special parameter values Laurent-polynomial solutions of the associated reflection quantum KZ equations in terms of nonsymmetric Koornwinder polynomials. We discuss these aspects in detail by revisiting and extending work of De Gier, Kasatani, Nichols, Cherednik, the first author and many others.

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

Cited by 15 publications

References 45 publications

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

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

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

Koornwinder polynomials and the XXZ spin chain

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