arXiv:0912.0348

On domain wall boundary conditions for the XXZ spin Hamiltonian

Domenico Orlando, Susanne Reffert, Nicolai Reshetikhin

Published Dec 2, 2009• 6 citations

Abstract

In this note, we derive the spectrum of the infinite quantum XXZ spin chain with domain wall boundary conditions. The eigenstates are constructed as limits of Bethe states for the finite XXZ spin chain with quantum sl(2) invariant boundary conditions.

DOI: 10.48550/arxiv.0912.0348

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arXiv:0912.0348

On domain wall boundary conditions for the XXZ spin Hamiltonian

Domenico Orlando, Susanne Reffert, Nicolai Reshetikhin

Published Dec 2, 2009• 6 citations

arXiv HTML PDF

Abstract

DOI: 10.48550/arxiv.0912.0348

Related papers

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arXiv:hep-th/9912252Journal of Physics A Mathematical and GeneralMar 23, 200039 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.