Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:0901.2961 · Journal of Statistical Mechanics Theory and Experiment

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

Jan de Gier, Anita Ponsaing, Keiichi Shigechi

Published Apr 16, 2009• 16 citations
arXivHTMLPDF

Abstract

We explicitly describe certain components of the finite size ground state of the inhomogeneous transfer matrix of the O( n = 1) loop model on a strip with non-trivial boundaries on both sides. In addition we compute explicitly the ground state normalization which is given as a product of four symplectic characters.
DOI
10.1088/1742-5468/2009/04/p04010

Related papers

qKZ equation 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 −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.

•Mar 30, 2009•13 citations

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

Inhomogeneous loop models with open boundaries

Philippe Di Francesco

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.

arXiv:math-ph/0504032•Journal of Physics A Mathematical and General•Jun 22, 2005•28 citations

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

Alexandr Garbali, Bernard Nienhuis

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.

arXiv:1411.7020•Journal of Statistical Mechanics Theory and Experiment•Apr 1, 2017•9 citations

Logarithmic corrections to finite-size scaling in strips

John Cardy

The corrections to the finite-size scaling behaviour of the eigenvalues of the transfer matrix of a critical theory defined on an infinitely long strip of finite width, which occur when the Hamiltonian contains a marginal operator, are computed using conformal invariance. They show a calculable universal logarithmic character. For the four-state Potts model they agree with numerical data.

Journal of Physics A Mathematical and General•Dec 1, 1986•185 citations

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.

arXiv:cond-mat/0401245•Journal of Statistical Mechanics Theory and Experiment•Oct 1, 2004•63 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.