Journal of Physics A Mathematical and General

The general Uq(sl(2)) invariant XXZ integrable quantum spin chain

P. P. Kulish, Evgeny Konstantinovich Sklyanin

Published Apr 21, 1991• 159 citations

Abstract

The boundary conditions are constructed for the general XXZ quantum integrable spin chain such that the resulting system is invariant under the quantum algebra U q (sl(2)) action. The result generalizes that due to Pasquier and Saleur (1990), obtained for spin-1/2 and spin-1.

DOI: 10.1088/0305-4470/24/8/009

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$$ {U}_{\mathfrak{q}}{\mathfrak{sl}}_2 $$-invariant non-compact boundary conditions for the XXZ spin chain

Dmitry Chernyak, Azat M. Gainutdinov, Hubert Saleur

A bstract We introduce new $$ {U}_{\mathfrak{q}}{\mathfrak{sl}}_2 $$ U q sl 2 -invariant boundary conditions for the open XXZ spin chain. For generic values of $$ \mathfrak{q} $$ q we couple the bulk Hamiltonian to an infinite-dimensional Verma module on one or both boundaries of the spin chain, and for $$ \mathfrak{q}={e}^{\frac{i\pi}{p}} $$ q = e iπ p a 2 p -th root of unity — to its p -dimensional analogue. Both cases are parametrised by a continuous “spin” α ∈ ℂ. To motivate our construction, we first specialise to $$ \mathfrak{q} $$ q = i , where we obtain a modified XX Hamiltonian with unrolled quantum group symmetry, whose spectrum and scaling limit is computed explicitly using free fermions. In the continuum, this model is identified with the ( η, ξ ) ghost CFT on the upper-half plane with a continuum of conformally invariant boundary conditions on the real axis. The different sectors of the Hamiltonian are identified with irreducible Virasoro representations. Going back to generic q we investigate the algebraic properties of the underlying lattice algebras. We show that if $$ \mathfrak{q} $$ q α ∉ ± $$ \mathfrak{q} $$ q ℤ , the new boundary coupling provides a faithful representation of the blob algebra which is Schur-Weyl dual to $$ {U}_{\mathfrak{q}}{\mathfrak{sl}}_2 $$ U q sl 2 . Then, modifying the boundary conditions on both the left and the right, we obtain a representation of the universal two-boundary Temperley-Lieb algebra. The generators and parameters of these representations are computed explicitly in terms of $$ \mathfrak{q} $$ q and α . Finally, we conjecture the general form of the Schur-Weyl duality in this case. This paper is the first in a series where we will study, at all values of the parameters, the spectrum and its continuum limit, the representation content of the relevant lattice algebras and the fusion properties of these new spin chains.

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A quantum algebra invariant integrable closed spin 1 chain is introduced and analyzed in detail. The Bethe ansatz equations as well as the energy eigenvalues of the model are obtained. The highest weight property of the Bethe vectors with respect to Uq(sl(2)) is proved.

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Journal of Physics A Mathematical and General

The general Uq(sl(2)) invariant XXZ integrable quantum spin chain

P. P. Kulish, Evgeny Konstantinovich Sklyanin

Published Apr 21, 1991• 159 citations

Publisher

Abstract

DOI: 10.1088/0305-4470/24/8/009

Related papers

INTEGRABILITY OF OPEN SPIN CHAINS WITH QUANTUM ALGEBRA SYMMETRY

Luca Mezincescu, Rafael I. Nepomechie

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Dmitry Chernyak, Azat M. Gainutdinov, Hubert Saleur

A bstract We introduce new ${U}_{\mathfrak{q}}{\mathfrak{sl}}_2$ U q sl 2 -invariant boundary conditions for the open XXZ spin chain. For generic values of $\mathfrak{q}$ q we couple the bulk Hamiltonian to an infinite-dimensional Verma module on one or both boundaries of the spin chain, and for $\mathfrak{q}={e}^{\frac{i\pi}{p}}$ q = e iπ p a 2 p -th root of unity — to its p -dimensional analogue. Both cases are parametrised by a continuous “spin” α ∈ ℂ. To motivate our construction, we first specialise to $\mathfrak{q}$ q = i , where we obtain a modified XX Hamiltonian with unrolled quantum group symmetry, whose spectrum and scaling limit is computed explicitly using free fermions. In the continuum, this model is identified with the ( η, ξ ) ghost CFT on the upper-half plane with a continuum of conformally invariant boundary conditions on the real axis. The different sectors of the Hamiltonian are identified with irreducible Virasoro representations. Going back to generic q we investigate the algebraic properties of the underlying lattice algebras. We show that if $\mathfrak{q}$ q α ∉ ± $\mathfrak{q}$ q ℤ , the new boundary coupling provides a faithful representation of the blob algebra which is Schur-Weyl dual to ${U}_{\mathfrak{q}}{\mathfrak{sl}}_2$ U q sl 2 . Then, modifying the boundary conditions on both the left and the right, we obtain a representation of the universal two-boundary Temperley-Lieb algebra. The generators and parameters of these representations are computed explicitly in terms of $\mathfrak{q}$ q and α . Finally, we conjecture the general form of the Schur-Weyl duality in this case. This paper is the first in a series where we will study, at all values of the parameters, the spectrum and its continuum limit, the representation content of the relevant lattice algebras and the fusion properties of these new spin chains.A bstract We introduce new $$ {U}_{\mathfrak{q}}{\mathfrak{sl}}_2 $$ U q sl 2 -invariant boundary conditions for the open XXZ spin chain. For generic values of $$ \mathfrak{q} $$ q we couple the bulk Hamiltonian to an infinite-dimensional Verma module on one or both boundaries of the spin chain, and for $$ \mathfrak{q}={e}^{\frac{i\pi}{p}} $$ q = e iπ p a 2 p -th root of unity — to its p -dimensional analogue. Both cases are parametrised by a continuous “spin” α ∈ ℂ. To motivate our construction, we first specialise to $$ \mathfrak{q} $$ q = i , where we obtain a modified XX Hamiltonian with unrolled quantum group symmetry, whose spectrum and scaling limit is computed explicitly using free fermions. In the continuum, this model is identified with the ( η, ξ ) ghost CFT on the upper-half plane with a continuum of conformally invariant boundary conditions on the real axis. The different sectors of the Hamiltonian are identified with irreducible Virasoro representations. Going back to generic q we investigate the algebraic properties of the underlying lattice algebras. We show that if $$ \mathfrak{q} $$ q α ∉ ± $$ \mathfrak{q} $$ q ℤ , the new boundary coupling provides a faithful representation of the blob algebra which is Schur-Weyl dual to $$ {U}_{\mathfrak{q}}{\mathfrak{sl}}_2 $$ U q sl 2 . Then, modifying the boundary conditions on both the left and the right, we obtain a representation of the universal two-boundary Temperley-Lieb algebra. The generators and parameters of these representations are computed explicitly in terms of $$ \mathfrak{q} $$ q and α . Finally, we conjecture the general form of the Schur-Weyl duality in this case. This paper is the first in a series where we will study, at all values of the parameters, the spectrum and its continuum limit, the representation content of the relevant lattice algebras and the fusion properties of these new spin chains.

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$U_q\mathfrak{sl}_2$ -invariant non-compact boundary conditions for the XXZ spin chain$U_q\mathfrak{sl}_2$-invariant non-compact boundary conditions for the XXZ spin chain

Dmitry Chernyak, Azat M. Gainutdinov, Hubert Saleur

We introduce new $U_q\mathfrak{sl}_2$ -invariant boundary conditions for the open XXZ spin chain. For generic values of $q$ we couple the bulk Hamiltonian to an infinite-dimensional Verma module on one or both boundaries of the spin chain, and for $q=e^{\frac{iπ}{p}}$ a $2p$ -th root of unity $ - $ to its $p$ -dimensional analogue. Both cases are parametrised by a continuous "spin" $α\in\mathbb{C}$ . To motivate our construction, we first specialise to $q=i$ , where we obtain a modified XX Hamiltonian with unrolled quantum group symmetry, whose spectrum and scaling limit is computed explicitly using free fermions. In the continuum, this model is identified with the $(η,ξ)$ ghost CFT on the upper-half plane with a continuum of conformally invariant boundary conditions on the real axis. The different sectors of the Hamiltonian are identified with irreducible Virasoro representations. Going back to generic $q$ we investigate the algebraic properties of the underlying lattice algebras. We show that if $q^α\notin\pm q^{\mathbb{Z}}$ , the new boundary coupling provides a faithful representation of the blob algebra which is Schur-Weyl dual to $U_q\mathfrak{sl}_2$ . Then, modifying the boundary conditions on both the left and the right, we obtain a representation of the universal two-boundary Temperley-Lieb algebra. The generators and parameters of these representations are computed explicitly in terms of $q$ and $α$ . Finally, we conjecture the general form of the Schur-Weyl duality in this case. This paper is the first in a series where we will study, at all values of the parameters, the spectrum and its continuum limit, the representation content of the relevant lattice algebras and the fusion properties of these new spin chains.We introduce new $U_q\mathfrak{sl}_2$-invariant boundary conditions for the open XXZ spin chain. For generic values of $q$ we couple the bulk Hamiltonian to an infinite-dimensional Verma module on one or both boundaries of the spin chain, and for $q=e^{\frac{iπ}{p}}$ a $2p$-th root of unity $ - $ to its $p$-dimensional analogue. Both cases are parametrised by a continuous "spin" $α\in\mathbb{C}$. To motivate our construction, we first specialise to $q=i$, where we obtain a modified XX Hamiltonian with unrolled quantum group symmetry, whose spectrum and scaling limit is computed explicitly using free fermions. In the continuum, this model is identified with the $(η,ξ)$ ghost CFT on the upper-half plane with a continuum of conformally invariant boundary conditions on the real axis. The different sectors of the Hamiltonian are identified with irreducible Virasoro representations. Going back to generic $q$ we investigate the algebraic properties of the underlying lattice algebras. We show that if $q^α\notin\pm q^{\mathbb{Z}}$, the new boundary coupling provides a faithful representation of the blob algebra which is Schur-Weyl dual to $U_q\mathfrak{sl}_2$. Then, modifying the boundary conditions on both the left and the right, we obtain a representation of the universal two-boundary Temperley-Lieb algebra. The generators and parameters of these representations are computed explicitly in terms of $q$ and $α$. Finally, we conjecture the general form of the Schur-Weyl duality in this case. This paper is the first in a series where we will study, at all values of the parameters, the spectrum and its continuum limit, the representation content of the relevant lattice algebras and the fusion properties of these new spin chains.

arXiv:2207.12772Jul 26, 20220 citations

The spin-S XXZ quantum chain with general toroidal boundary conditions

Francisco Castilho Alcaraz, Marcio Jose Martins

The authors derive the operator content of the spin-S XXZ quantum chain with generalised toroidal boundary conditions compatible with the U(1) symmetry of the model. These results are derived by solving numerically the associated Bethe ansatz equations for finite chains and exploring the consequences of the conformal invariance of the infinite system. They also show that, as in the spin-1/2 case, the conformal anomaly and dimensions of general extended SU(2) algebras can be obtained from these spin-S chains by choosing properly the coupling constant and the boundary condition.The authors derive the operator content of the spin-S XXZ quantum chain with generalised toroidal boundary conditions compatible with the U(1) symmetry of the model. These results are derived by solving numerically the associated Bethe ansatz equations for finite chains and exploring the consequences of the conformal invariance of the infinite system. They also show that, as in the spin-1/2 case, the conformal anomaly and dimensions of general extended SU(2) algebras can be obtained from these spin-S chains by choosing properly the coupling constant and the boundary condition.

Journal of Physics A Mathematical and GeneralApr 21, 199026 citations

{U}_{\mathfrak{q}}{\mathfrak{sl}}_2

-invariant non-compact boundary conditions for the XXZ spin chain

A bstract We introduce new

{U}_{\mathfrak{q}}{\mathfrak{sl}}_2

U q sl 2 -invariant boundary conditions for the open XXZ spin chain. For generic values of

\mathfrak{q}

q we couple the bulk Hamiltonian to an infinite-dimensional Verma module on one or both boundaries of the spin chain, and for

\mathfrak{q}={e}^{\frac{i\pi}{p}}

q = e iπ p a 2 p -th root of unity — to its p -dimensional analogue. Both cases are parametrised by a continuous “spin” α ∈ ℂ. To motivate our construction, we first specialise to

\mathfrak{q}

q = i , where we obtain a modified XX Hamiltonian with unrolled quantum group symmetry, whose spectrum and scaling limit is computed explicitly using free fermions. In the continuum, this model is identified with the ( η, ξ ) ghost CFT on the upper-half plane with a continuum of conformally invariant boundary conditions on the real axis. The different sectors of the Hamiltonian are identified with irreducible Virasoro representations. Going back to generic q we investigate the algebraic properties of the underlying lattice algebras. We show that if

\mathfrak{q}

q α ∉ ±

\mathfrak{q}

q ℤ , the new boundary coupling provides a faithful representation of the blob algebra which is Schur-Weyl dual to

{U}_{\mathfrak{q}}{\mathfrak{sl}}_2

U q sl 2 . Then, modifying the boundary conditions on both the left and the right, we obtain a representation of the universal two-boundary Temperley-Lieb algebra. The generators and parameters of these representations are computed explicitly in terms of

\mathfrak{q}

q and α . Finally, we conjecture the general form of the Schur-Weyl duality in this case. This paper is the first in a series where we will study, at all values of the parameters, the spectrum and its continuum limit, the representation content of the relevant lattice algebras and the fusion properties of these new spin chains.

U_q\mathfrak{sl}_2

-invariant non-compact boundary conditions for the XXZ spin chain

We introduce new

U_q\mathfrak{sl}_2

-invariant boundary conditions for the open XXZ spin chain. For generic values of

q

we couple the bulk Hamiltonian to an infinite-dimensional Verma module on one or both boundaries of the spin chain, and for

q=e^{\frac{iπ}{p}}

2p

-th root of unity $ - $ to its

p

-dimensional analogue. Both cases are parametrised by a continuous "spin"

α\in\mathbb{C}

. To motivate our construction, we first specialise to

q=i

, where we obtain a modified XX Hamiltonian with unrolled quantum group symmetry, whose spectrum and scaling limit is computed explicitly using free fermions. In the continuum, this model is identified with the

(η,ξ)

ghost CFT on the upper-half plane with a continuum of conformally invariant boundary conditions on the real axis. The different sectors of the Hamiltonian are identified with irreducible Virasoro representations. Going back to generic

q

we investigate the algebraic properties of the underlying lattice algebras. We show that if

q^α\notin\pm q^{\mathbb{Z}}

, the new boundary coupling provides a faithful representation of the blob algebra which is Schur-Weyl dual to

U_q\mathfrak{sl}_2

. Then, modifying the boundary conditions on both the left and the right, we obtain a representation of the universal two-boundary Temperley-Lieb algebra. The generators and parameters of these representations are computed explicitly in terms of

q

and

α

. Finally, we conjecture the general form of the Schur-Weyl duality in this case. This paper is the first in a series where we will study, at all values of the parameters, the spectrum and its continuum limit, the representation content of the relevant lattice algebras and the fusion properties of these new spin chains.