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arXiv:1008.0918

q-Deformed Onsager symmetry in boundary integrable models related to twisted U$_{q^{1/2}}(\hat{sl_2})$ symmetry

Kozo Koizumi

Published Aug 5, 2010• 0 citations
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Abstract

We consider an unified model, called ancestor model, associated with twisted trigonometric $R$ matrix which model leads to several descendant integrable lattice models related to the U$_{q^{1/2}}(\hat{sl_2})$ symmetry. Boundary operators compatible with integrability are introduced to this model. Reflection and dual reflection equations to ensure integrability of the system are shown to be same as the untwisted case. It follows that underlying symmetry of the ancestor model with integrable boundaries is identified with the q-deformed analogue of Onsager's symmetry. The transfer matrix and its related mutually commuting quantities are expressed in terms of an abelian subalgebra in the $q$-Onsager algebra. It is illustrated that the generalized McCoy-Wu model with general open boundaries enjoys this symmetry.

DOI
10.48550/arxiv.1008.0918

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q-Deformed Onsager symmetry in boundary integrable models related to twisted Uq1/2(sl2^)_{q^{1/2}}(\hat{sl_2})q1/2​(sl2​^​) symmetry
We consider an unified model, called ancestor model, associated with twisted trigonometric RRR matrix which model leads to several descendant integrable lattice models related to the Uq1/2(sl2^)_{q^{1/2}}(\hat{sl_2})q1/2​(sl2​^​) symmetry. Boundary operators compatible with integrability are introduced to this model. Reflection and dual reflection equations to ensure integrability of the system are shown to be same as the untwisted case. It follows that underlying symmetry of the ancestor model with integrable boundaries is identified with the q-deformed analogue of Onsager's symmetry. The transfer matrix and its related mutually commuting quantities are expressed in terms of an abelian subalgebra in the qqq-Onsager algebra. It is illustrated that the generalized McCoy-Wu model with general open boundaries enjoys this symmetry.
The XXZ open spin chain with general integrable boundary conditions is shown to possess a q-deformed analogue of the Onsager's algebra as fundamental non-abelian symmetry which ensures the integrability of the model. This symmetry implies the existence of a finite set of independent mutually commuting nonlocal operators which form an abelian subalgebra. The transfer matrix and local conserved quantities, for instance the Hamiltonian, are expressed in terms of these nonlocal operators. It follows that Onsager's original approach of the planar Ising model can be extended to the XXZ open spin chain.
The XXZ open spin chain with general integrable boundary conditions is shown to possess a q -deformed analogue of the Onsager's algebra as fundamental non-Abelian symmetry which ensures the integrability of the model. This symmetry implies the existence of a finite set of independent mutually commuting nonlocal operators which form an Abelian subalgebra. The transfer matrix and local conserved quantities, for instance the Hamiltonian, are expressed in terms of these nonlocal operators. It follows that Onsager's original approach of the planar Ising model can be extended to the XXZ open spin chain.
It is well known that integrable models associated to rational R matrices give rise to certain non-Abelian symmetries known as Yangians. Analogously boundary symmetries arise when general but still integrable boundary conditions are implemented, as originally argued by Delius, Mackay, and Short from the field theory point of view, in the context of the principal chiral model on the half-line. In the present study we deal with a discrete quantum mechanical system with boundaries, that is the N site gl(n) open quantum spin chain. In particular, the open spin chain with two distinct types of boundary condition known as soliton preserving and soliton nonpreserving is considered. For both types of boundaries we present a unified framework for deriving the corresponding boundary nonlocal charges directly at the quantum level. The nonlocal charges are simply coproduct realizations of particular boundary quantum algebras called boundary or twisted Yangians, depending on the choice of boundary conditions. Finally, with the help of linear intertwining relations between the solutions of the reflection equation and the generators of the boundary or twisted Yangians we are able to exhibit the exact symmetry of the open spin chain, namely we show that a number of the boundary nonlocal charges are in fact conserved quantities.
It is known that integrable models associated to rational R matrices give rise to certain non-abelian symmetries known as Yangians. Analogously ‘boundary ’ symmetries arise when general but still integrable boundary conditions are implemented, as originally argued by Delius, Mackay and Short from the field theory point of view, in the context of the principal chiral model on the half line. In the present study we deal with a discrete quantum mechanical system with boundaries, that is the N site gl(n) open quantum spin chain. In particular, the open spin chain with two distinct types of boundary condition known as soliton preserving and soliton non-preserving is considered. For both types of boundaries we present a unified framework for deriving the corresponding boundary non-local charges directly at the quantum level. The non-local charges are simply coproduct realizations of particular boundary quantum algebras called ‘boundary ’ or twisted Yangians, depending on the choice of boundary conditions. Finally, with the help of linear intertwining relations between the solutions of the reflection equation and the generators of the boundary or twisted Yangians we are able to exhibit the symmetry of the open spin chain, namely we show that a number of the boundary non-local charges are in fact conserved quantities.
The symmetry of the Hamiltonian describing the asymmetric twin model was partially studied in earlier works, and our aim here is to generalize these results for the open transfer matrix.In this spirit we first prove, that the so called boundary quantum algebra provides a symmetry for any generic -independent of the choice of model -open transfer matrix with a trivial left boundary.In addition it is shown that the boundary quantum algebra is the centralizer of the B type Hecke algebra.We then focus on the asymmetric twin representation of the boundary Temperley-Lieb algebra.More precisely, by exploiting exchange relations dictated by the reflection equation we show that the transfer matrix with trivial boundary conditions enjoys the recognized U q (sl 2 ) ⊗ U i (sl 2 ) symmetry.When a nondiagonal boundary is implemented the symmetry as expected is reduced, however again certain familiar boundary non-local charges turn out to commute with the corresponding transfer matrix.
Conventional quantum integrability is encoded in a commuting algebra of transfer matrices. By contrast, several models possess additional non-commuting conserved charges with important physical consequences, yet the nature of the corresponding symmetry has remained elusive. Focusing on the XXZ spin chain at roots of unity, we show that the non-Abelian analogue of the commuting transfer-matrix algebra is governed by quadratic relations following from a new class of unbalanced Yang--Baxter/RLL relations. This quadratic algebra is shown to encode precisely the Onsager algebra, for which we construct explicit matrix-product representations of both its generators and its duality defect line. The latter obeys ZN\mathbb{Z}_NZN​ Tambara--Yamagami fusion rules, thereby providing a lattice realization of the topological defect lines of the compactified boson conformal field theory. Our results identify non-Abelian transfer-matrix algebras as a microscopic origin for Onsager symmetry and dualities in lattice models.