数理解析研究所講究録

Root of unity symmetries in the 8 and 6 vertex models(Solvable Lattice Models 2004 : Recent Progress on Solvable Lattice Models )

K. Fabricius, Barry M. McCoy

Published Apr 1, 2006• 0 citations

Abstract

We review the recently discovered symmetries of the 8 and 6 vertex models which exist at roots of unity and present their relation with representation theory of affine Lie algebras, Drinfeld polynomials and Bethe vectors.

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Root of unity symmetries in the 8 and 6 vertex models

K. Fabricius, Barry M. McCoy

arXiv:cond-mat/0411419To appear in the proceedings ofNov 16, 20046 citations

Loop symmetry of integrable vertex models at roots of unity

Korff, C, Barry M. McCoy

Jan 1, 200113 citations

Evaluation Parameters and Bethe Roots for the Six-Vertex Model at Roots of Unity

K. Fabricius, Barry M. McCoy

arXiv:cond-mat/0108057Birkhäuser Boston eBooksJan 1, 200243 citations

Auxiliary matrices for the six-vertex model atqN= 1: II. Bethe roots, complete strings and the Drinfeld polynomial

Christian Korff

arXiv:math-ph/0305035Journal of Physics A Mathematical and GeneralDec 15, 200317 citations

Representation Theory and Baxter's TQ equation for the six-vertex model. A pedagogical overview

Christian Korff

A summary of the construction procedure of generalized versions of Baxter's Q-operator is given. Illustrated by several figures and diagrams the use of representation theory is explained step-by-step avoiding technical details. The relation with the infinite-dimensional non-abelian symmetries of the six-vertex model at roots of unity is discussed and parallels with the eight-vertex case outlined.

arXiv:cond-mat/0411758ENLIGHTEN (Jurnal Bimbingan dan Konseling Islam)Nov 30, 20043 citations

BETHE ANSATZ AND QUANTUM GROUPS

Hector J. de Vega

arXiv:hep-th/9308008Jun 1, 19930 citations

数理解析研究所講究録

Root of unity symmetries in the 8 and 6 vertex models(Solvable Lattice Models 2004 : Recent Progress on Solvable Lattice Models )

K. Fabricius, Barry M. McCoy

Published Apr 1, 2006• 0 citations

OpenAlex

Abstract

Related papers

Root of unity symmetries in the 8 and 6 vertex models

K. Fabricius, Barry M. McCoy

arXiv:cond-mat/0411419To appear in the proceedings ofNov 16, 20046 citations

Loop symmetry of integrable vertex models at roots of unity

Korff, C, Barry M. McCoy

It has been recently discovered in the context of the six vertex or XXZ model in the fundamental representation that new symmetries arise when the anisotropy parameter $(q+q^{-1})/2$ is evaluated at roots of unity $q^{N}=1$ . These new symmetries have been linked to an $U(A^{(1)}_1)$ invariance of the transfer matrix and the corresponding spin-chain Hamiltonian.In this paper these results are generalized for odd primitive roots of unity to all vertex models associated with trigonometric solutions of the Yang-Baxter equation by invoking representation independent methods which only take the algebraic structure of the underlying quantum groups $U_q(\\hat g)$ into account. Here $\\hat g$ is an arbitrary Kac-Moody algebra. Employing the notion of the boost operator it is then found that the Hamiltonian and the transfer matrix of the integrable model are invariant under the action of $U(\\hat{g})$ . For the simplest case $\\hat g=A_1^{(1)}$ the discussion is also extended to even primitive roots of unity.It has been recently discovered in the context of the six vertex or XXZ model in the fundamental representation that new symmetries arise when the anisotropy parameter $(q+q^{-1})/2$ is evaluated at roots of unity $q^{N}=1$. These new symmetries have been linked to an $U(A^{(1)}_1)$ invariance of the transfer matrix and the corresponding spin-chain Hamiltonian.In this paper these results are generalized for odd primitive roots of unity to all vertex models associated with trigonometric solutions of the Yang-Baxter equation by invoking representation independent methods which only take the algebraic structure of the underlying quantum groups $U_q(\\hat g)$ into account. Here $\\hat g$ is an arbitrary Kac-Moody algebra. Employing the notion of the boost operator it is then found that the Hamiltonian and the transfer matrix of the integrable model are invariant under the action of $U(\\hat{g})$. For the simplest case $\\hat g=A_1^{(1)}$ the discussion is also extended to even primitive roots of unity.

Jan 1, 200113 citations

Evaluation Parameters and Bethe Roots for the Six-Vertex Model at Roots of Unity

K. Fabricius, Barry M. McCoy

We propose an expression for the current form of the lowering operator of the sl 2 loop algebra symmetry of the six-vertex model (XXZ spin chain) at roots of unity. This operator has poles which correspond to the evaluation parameters of representation theory which are given as the roots of the Drinfeld polynomial. We explicitly compute these polynomials in terms of the Bethe roots which characterize the highest weight states for all values of S z . From these polynomials we find that the Bethe roots satisfy sum rules for each value of S z .We propose an expression for the current form of the lowering operator of the sl 2 loop algebra symmetry of the six-vertex model (XXZ spin chain) at roots of unity. This operator has poles which correspond to the evaluation parameters of representation theory which are given as the roots of the Drinfeld polynomial. We explicitly compute these polynomials in terms of the Bethe roots which characterize the highest weight states for all values of S z . From these polynomials we find that the Bethe roots satisfy sum rules for each value of S z .

arXiv:cond-mat/0108057Birkhäuser Boston eBooksJan 1, 200243 citations

Auxiliary matrices for the six-vertex model atqN= 1: II. Bethe roots, complete strings and the Drinfeld polynomial

Christian Korff

The spectra of recently constructed auxiliary matrices for the six-vertex model respectively the spin s = 1/2 Heisenberg chain at roots of unity are investigated. Two conjectures are formulated both of which are proved for N = 3 and are verified numerically for several examples with N > 3. The first conjecture identifies an Abelian subset of auxiliary matrices whose eigenvalues are polynomials in the spectral variable. The zeros of these polynomials are shown to fall into two sets. One consists of the solutions to the Bethe ansatz equations which determine the eigenvalues of the six-vertex transfer matrix. The other set of zeros contains the complete strings which encode the information on the degeneracies of the model due to the loop symmetry present at roots of 1. The second conjecture then states a polynomial identity which relates the complete string centres to the Bethe roots allowing one to determine the dimension of the degenerate eigenspaces. Its proof for N = 3 involves the derivation of a new functional equation for the auxiliary matrices and the six-vertex transfer matrix. Moreover, it is demonstrated in several explicit examples that the complete strings coincide with the classical analogue of the Drinfeld polynomial. The latter is used to classify the finite-dimensional irreducible representations of the loop algebra . This suggests that the constructed auxiliary matrices not only enable one to solve the six-vertex model but also completely characterize the decomposition of its eigenspaces into irreducible representations of the underlying loop symmetry.The spectra of recently constructed auxiliary matrices for the six-vertex model respectively the spin s = 1/2 Heisenberg chain at roots of unity are investigated. Two conjectures are formulated both of which are proved for N = 3 and are verified numerically for several examples with N > 3. The first conjecture identifies an Abelian subset of auxiliary matrices whose eigenvalues are polynomials in the spectral variable. The zeros of these polynomials are shown to fall into two sets. One consists of the solutions to the Bethe ansatz equations which determine the eigenvalues of the six-vertex transfer matrix. The other set of zeros contains the complete strings which encode the information on the degeneracies of the model due to the loop symmetry present at roots of 1. The second conjecture then states a polynomial identity which relates the complete string centres to the Bethe roots allowing one to determine the dimension of the degenerate eigenspaces. Its proof for N = 3 involves the derivation of a new functional equation for the auxiliary matrices and the six-vertex transfer matrix. Moreover, it is demonstrated in several explicit examples that the complete strings coincide with the classical analogue of the Drinfeld polynomial. The latter is used to classify the finite-dimensional irreducible representations of the loop algebra . This suggests that the constructed auxiliary matrices not only enable one to solve the six-vertex model but also completely characterize the decomposition of its eigenspaces into irreducible representations of the underlying loop symmetry.

arXiv:math-ph/0305035Journal of Physics A Mathematical and GeneralDec 15, 200317 citations

Representation Theory and Baxter's TQ equation for the six-vertex model. A pedagogical overview

Christian Korff

arXiv:cond-mat/0411758ENLIGHTEN (Jurnal Bimbingan dan Konseling Islam)Nov 30, 20043 citations

BETHE ANSATZ AND QUANTUM GROUPS

Hector J. de Vega

The formulation and resolution of integrable lattice statistical models in a quantum group covariant way is the subject of this review. The Bethe Ansatz turns to be remarkably useful to implement quantum group symmetries and to provide quantum group representations even when $q$ is a root of unity. We start by solving the six-vertex model with fixed boundary conditions (FBC) that guarantee exact $SU(2)_q$ invariance on the lattice. The algebra of the Yang-Baxter (YB) and $SU(2)_q$ generators turns to close. The infinite spectral parameter limit of the YB generators yields {\bf cleanly} the $SU(2)_q$ generators. The Bethe Ansatz states constructed for FBC are shown to be {\bf highest weights} of $SU(2)_q$ . The higher level Bethe Ansatz equations (BAE, describing the physical excitations) are explicitly derived for FBC. We then solve the RSOS( $p$ ) models on the light--cone lattice with fixed boundary conditions by disentangling the type II representations of $SU(2)_q$ , at $q=e^{iπ/p}$ , from the full SOS spectrum obtained through Algebraic Bethe Ansatz. The RSOS states are those with quantum spin $JThe formulation and resolution of integrable lattice statistical models in a quantum group covariant way is the subject of this review. The Bethe Ansatz turns to be remarkably useful to implement quantum group symmetries and to provide quantum group representations even when $q$ is a root of unity. We start by solving the six-vertex model with fixed boundary conditions (FBC) that guarantee exact $SU(2)_q$ invariance on the lattice. The algebra of the Yang-Baxter (YB) and $SU(2)_q$ generators turns to close. The infinite spectral parameter limit of the YB generators yields {\bf cleanly} the $SU(2)_q$ generators. The Bethe Ansatz states constructed for FBC are shown to be {\bf highest weights} of $SU(2)_q$. The higher level Bethe Ansatz equations (BAE, describing the physical excitations) are explicitly derived for FBC. We then solve the RSOS($p$) models on the light--cone lattice with fixed boundary conditions by disentangling the type II representations of $SU(2)_q$, at $q=e^{iπ/p}$, from the full SOS spectrum obtained through Algebraic Bethe Ansatz. The RSOS states are those with quantum spin $J

arXiv:hep-th/9308008Jun 1, 19930 citations

It has been recently discovered in the context of the six vertex or XXZ model in the fundamental representation that new symmetries arise when the anisotropy parameter

(q+q^{-1})/2

is evaluated at roots of unity

q^{N}=1

. These new symmetries have been linked to an

U(A^{(1)}_1)

invariance of the transfer matrix and the corresponding spin-chain Hamiltonian.In this paper these results are generalized for odd primitive roots of unity to all vertex models associated with trigonometric solutions of the Yang-Baxter equation by invoking representation independent methods which only take the algebraic structure of the underlying quantum groups

U_q(\\hat g)

into account. Here

\\hat g

is an arbitrary Kac-Moody algebra. Employing the notion of the boost operator it is then found that the Hamiltonian and the transfer matrix of the integrable model are invariant under the action of

U(\\hat{g})

. For the simplest case

\\hat g=A_1^{(1)}

the discussion is also extended to even primitive roots of unity.

q

is a root of unity. We start by solving the six-vertex model with fixed boundary conditions (FBC) that guarantee exact

SU(2)_q

invariance on the lattice. The algebra of the Yang-Baxter (YB) and

SU(2)_q

generators turns to close. The infinite spectral parameter limit of the YB generators yields {\bf cleanly} the

SU(2)_q

generators. The Bethe Ansatz states constructed for FBC are shown to be {\bf highest weights} of

SU(2)_q

. The higher level Bethe Ansatz equations (BAE, describing the physical excitations) are explicitly derived for FBC. We then solve the RSOS(

p

) models on the light--cone lattice with fixed boundary conditions by disentangling the type II representations of

SU(2)_q

, at

q=e^{iπ/p}

, from the full SOS spectrum obtained through Algebraic Bethe Ansatz. The RSOS states are those with quantum spin $J