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

Journal of Physics A Mathematical and General

Central charges of the 6- and 19-vertex models with twisted boundary conditions

Andreas Klümper, Murray T. Batchelor, Paul A. Pearce

Published Jul 7, 1991• 260 citations
Publisher

Abstract

A new and general analytic method for calculating finite-size corrections and central charges is applied to the 6- and 19-vertex models and their related spin-1/2 and spin-1 XXZ chains with twisted boundary conditions. Nonlinear integral equations are derived from which the central charge c can be extracted in terms of Rogers dilogarithms. For twist angle phi , the central charge is c=3S/S+1 (1-4(S-1) phi 2 / pi ( pi -2S gamma )) where gamma is the crossing parameter or chain anisotropy and spin S=1/2 or 1. For periodic boundary conditions ( phi =0) this reduces to the known results c=1 and c=3/2, respectively.

DOI
10.1088/0305-4470/24/13/025

Related papers

Groundstate finite-size corrections and dilogarithm identities for the twisted A 1 ( 1 ) , A 2 ( 1 ) and A 2 ( 2 ) models

Alexi Morin-Duchesne, Andreas Klümper, Paul A. Pearce

Abstract We consider the Y -systems satisfied by the A 1 ( 1 ) , A 2 ( 1 ) , A 2 ( 2 ) vertex and loop models at roots of unity with twisted boundary conditions on the cylinder. The vertex models are the 6-, 15- and Izergin–Korepin 19-vertex models respectively. The corresponding loop models are the dense, fully packed and dilute Temperley–Lieb loop models respectively. For all three models, our focus is on roots of unity values of e i λ with the crossing parameter λ corresponding to the principal and dual series of these models. Converting the known functional equations to nonlinear integral equations in the form of thermodynamic Bethe ansatz equations, we solve the Y -systems for the finite-size 1 N corrections to the groundstate eigenvalue following the methods of Klümper and Pearce. The resulting expressions for c − 24Δ, where c is the central charge and Δ is the conformal weight associated with the groundstate, are simplified using various dilogarithm identities. Our analytic results are in agreement with previous results obtained by different methods.

arXiv:2006.08233•Journal of Statistical Mechanics Theory and Experiment•Mar 1, 2021•1 citation

Groundstate finite-size corrections and dilogarithm identities for the twisted \n\t A1(1)\n\t , \n\t A2(1)\n\t and \n\t A2(2)\n\t models

Alexi Morin-Duchesne, Andreas Klümper, Paul A. Pearce

We consider the Y -systems satisfied by the A(1)1, A(1)2, A(2)2 vertex and loop models at roots of unity with twisted boundary conditions on the cylinder. The vertex models are the 6-, 15- and Izergin-Korepin 19-vertex models respectively. The corresponding loop models are the dense, fully packed and dilute Temperley-Lieb loop models respectively. For all three models, our focus is on roots of unity values of eiλ with the crossing parameter λ corresponding to the principal and dual series of these models. Converting the known functional equations to nonlinear integral equations in the form of thermodynamic Bethe ansatz equations, we solve the Y -systems for the finite-size 1/N corrections to the groundstate eigenvalue following the methods of Klümper and Pearce. The resulting expressions for c. 24Δ, where c is the central charge and Δ is the conformal weight associated with the groundstate, are simplified using various dilogarithm identities. Our analytic results are in agreement with previous results obtained by different methods.

•Jan 1, 2021•2 citations

Conformal spectrum of the six-vertex model

Andreas Klümper, T Wehner, J. Zittartz

By way of a generalization of analytical methods recently introduced, the conformal spectrum of the six-vertex model with twisted boundary conditions is analysed. Nonlinear integral equations are derived, from which all scaling dimensions and the spins can be extracted analytically in terms of Rogers dilogarithms. The scaling dimensions of low-lying excitations are given by the so-called Gaussian dimensions.

Journal of Physics A Mathematical and General•Jun 21, 1993•50 citations

Excited States in the Twisted XXZ Spin Chain

Tarek Nassar, Olav Tirkkonen

We compute the finite size spectrum for the spin 1/2 XXZ chain with twisted boundary conditions, for anisotropy in the regime 0 < γ < π/2, and arbitrary twist θ. The string hypothesis is employed for treating complex excitations. The Bethe Ansatz equtions are solved within a coupled non-linear integral equation approach, with one equation for each type of string. The root-of-unity quantum group invariant periodic chain reduces to the XXZ1/2 chain with a set of twist boundary conditions (π/γ ∈ ZZ, θ an integer multiple of γ). For this model, the restricted Hilbert space corresponds to an unitary conformal field theory, and we recover all primary states in the Kač table in terms of states with specific twist and strings.

•Jan 1, 1997•4 citations

Central charge for the integrable higher-spin XXZ model

Henrik Johannesson

The central charge for the integrable higher-spin XXZ model is derived, exploiting the exact Bethe ansatz solution by Babujian and Tsvelick (1986). For integer values of v, where gamma = pi /v is the anisotropy parameter (0< gamma < pi /2S), we find c=3S/(S+1), with S the magnitude of the spin. Hence, contrary to expectation, the U(1)-invariant critical point of the theory does not renormalise onto a free massless scalar field when S>1/2.

Journal of Physics A Mathematical and General•Jun 7, 1988•13 citations

An analytic treatment of finite-size corrections in the spin-1 antiferromagnetic XXZ chain

Andreas Klümper, Murray T. Batchelor

It is known that the standard method for calculating finite-size corrections in Bethe ansatz solvable systems is not applicable to the Takhtajan-Babujian model and its anisotropic XXZ generalisation. The authors develop a new analytic method explicitly avoiding root densities and associated problems. Nonlinear integral equations are derived whose solutions yield the correct central charges c=1 and c=3/2 for the spin-1/2 and spin-1 XXZ chains, respectively. In the spin-1 case the authors obtain as a by-product the finite-size deviation of the Bethe ansatz roots from the 2-string formation.

Journal of Physics A Mathematical and General•Mar 7, 1990•91 citations
A new and general analytic method for calculating finite-size corrections and central charges is applied to the 6- and 19-vertex models and their related spin-1/2 and spin-1 XXZ chains with twisted boundary conditions. Nonlinear integral equations are derived from which the central charge c can be extracted in terms of Rogers dilogarithms. For twist angle phi , the central charge is c=3S/S+1 (1-4(S-1) phi 2 / pi ( pi -2S gamma )) where gamma is the crossing parameter or chain anisotropy and spin S=1/2 or 1. For periodic boundary conditions ( phi =0) this reduces to the known results c=1 and c=3/2, respectively.
Abstract We consider the Y -systems satisfied by the A 1 ( 1 ) , A 2 ( 1 ) , A 2 ( 2 ) vertex and loop models at roots of unity with twisted boundary conditions on the cylinder. The vertex models are the 6-, 15- and Izergin–Korepin 19-vertex models respectively. The corresponding loop models are the dense, fully packed and dilute Temperley–Lieb loop models respectively. For all three models, our focus is on roots of unity values of e i λ with the crossing parameter λ corresponding to the principal and dual series of these models. Converting the known functional equations to nonlinear integral equations in the form of thermodynamic Bethe ansatz equations, we solve the Y -systems for the finite-size 1 N corrections to the groundstate eigenvalue following the methods of Klümper and Pearce. The resulting expressions for c − 24Δ, where c is the central charge and Δ is the conformal weight associated with the groundstate, are simplified using various dilogarithm identities. Our analytic results are in agreement with previous results obtained by different methods.
We consider the Y -systems satisfied by the A(1)1, A(1)2, A(2)2 vertex and loop models at roots of unity with twisted boundary conditions on the cylinder. The vertex models are the 6-, 15- and Izergin-Korepin 19-vertex models respectively. The corresponding loop models are the dense, fully packed and dilute Temperley-Lieb loop models respectively. For all three models, our focus is on roots of unity values of eiλ with the crossing parameter λ corresponding to the principal and dual series of these models. Converting the known functional equations to nonlinear integral equations in the form of thermodynamic Bethe ansatz equations, we solve the Y -systems for the finite-size 1/N corrections to the groundstate eigenvalue following the methods of Klümper and Pearce. The resulting expressions for c. 24Δ, where c is the central charge and Δ is the conformal weight associated with the groundstate, are simplified using various dilogarithm identities. Our analytic results are in agreement with previous results obtained by different methods.
We compute the finite size spectrum for the spin 1/2 XXZ chain with twisted boundary conditions, for anisotropy in the regime 0 < γ < π/2, and arbitrary twist θ. The string hypothesis is employed for treating complex excitations. The Bethe Ansatz equtions are solved within a coupled non-linear integral equation approach, with one equation for each type of string. The root-of-unity quantum group invariant periodic chain reduces to the XXZ1/2 chain with a set of twist boundary conditions (π/γ ∈ ZZ, θ an integer multiple of γ). For this model, the restricted Hilbert space corresponds to an unitary conformal field theory, and we recover all primary states in the Kač table in terms of states with specific twist and strings.
It is known that the standard method for calculating finite-size corrections in Bethe ansatz solvable systems is not applicable to the Takhtajan-Babujian model and its anisotropic XXZ generalisation. The authors develop a new analytic method explicitly avoiding root densities and associated problems. Nonlinear integral equations are derived whose solutions yield the correct central charges c=1 and c=3/2 for the spin-1/2 and spin-1 XXZ chains, respectively. In the spin-1 case the authors obtain as a by-product the finite-size deviation of the Bethe ansatz roots from the 2-string formation.