Modern Physics Letters B

A Note on the Symmetry Algebra of An Asymmetric XXZ Quantum Spin Chain

M. A. Rego-Monteiro

Published Jan 10, 1997• 0 citations

Abstract

We show that an interesting open asymmetric XXZ quantum spin chain with surface fields is invariant under a two-parametric generalization of the slq(2) algebra with deformation parameters being a particular root of unit.

DOI: 10.1142/s0217984997000049

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

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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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Modern Physics Letters B

A Note on the Symmetry Algebra of An Asymmetric XXZ Quantum Spin Chain

M. A. Rego-Monteiro

Published Jan 10, 1997• 0 citations

Publisher

Abstract

DOI: 10.1142/s0217984997000049

Related papers

XXZ Model and the Structure of the Symmetric Algebra

Ma Z, Baolin Hou

In this paper, an XXZ model of the Heisenberg spin chain with the symmetry of the quantum SL_q(2) enveloping algebra is discussed. In terms of the structure of the Temperley-Lieb algebra, in this model we analyse the finite-dimensional representations of the quantum SL_q(2) with q~p=±1 in some detail, including the condition of appearance of the reducible but indecomposable representations (type Ⅰ representation), their properties, and the completeset of the states related to a definite energy where the new states are obtained by an appropriare limit process.In this paper, an XXZ model of the Heisenberg spin chain with the symmetry of the quantum SL_q(2) enveloping algebra is discussed. In terms of the structure of the Temperley-Lieb algebra, in this model we analyse the finite-dimensional representations of the quantum SL_q(2) with q~p=±1 in some detail, including the condition of appearance of the reducible but indecomposable representations (type Ⅰ representation), their properties, and the completeset of the states related to a definite energy where the new states are obtained by an appropriare limit process.

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We construct an open quantum spin chain from the “twisted” [Formula: see text]R matrix in the fundamental representation which has the quantum algebra symmetry Uq[ su (2)]. This anisotropic spin-1 chain is different from the Uq[ su (2)]-invariant chain constructed from the “untwisted” [Formula: see text] spin-1 R matrix (namely, the spin-1 XXZ chain of Fateev-Zamolodchikov with boundary terms) but, nevertheless, is also completely integrable. We discuss the general case of an R matrix of the type g(k), where k∈{1, 2, 3}, and g is any simple Lie algebra.We construct an open quantum spin chain from the “twisted” [Formula: see text]R matrix in the fundamental representation which has the quantum algebra symmetry Uq[ su (2)]. This anisotropic spin-1 chain is different from the Uq[ su (2)]-invariant chain constructed from the “untwisted” [Formula: see text] spin-1 R matrix (namely, the spin-1 XXZ chain of Fateev-Zamolodchikov with boundary terms) but, nevertheless, is also completely integrable. We discuss the general case of an R matrix of the type g(k), where k∈{1, 2, 3}, and g is any simple Lie algebra.

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

Journal of High Energy PhysicsNov 4, 20226 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.