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Journal of Mathematical Physics

On a two-parameter generalization of Hecke algebras and the XY chain in a magnetic field

Paul Martin

Published Nov 1, 1992• 2 citations

Abstract

The structure and representation theory of a two-parameter generalization of Hecke algebras, denoted HL(q,η), which plays an important role in the construction of the L-site XY chain in a magnetic field is investigated. In particular, the structure of the XY chain representation of the algebra, on C2⊗L, and hence the structure of the centralizer algebra EndHL(q,η)(C2⊗L) for the XY chain representation, is analyzed. It is proved that the full algebra never reduces to the ordinary (one-parameter) Hecke algebra HL(q) in any specialization of the parameters, but contains the Hecke algebra as a quotient in a certain specialization of one of the parameters. This corrects an earlier claim of Hinrichsen and Rittenberg [Phys. Lett. B 275, 350 (1992)]. Bounds are established on the generic dimension of the centralizer algebra 4 ≤ dim(EndHL(q,η)(C⊗L2)) ≤ 8 (the upper bound is realized in every known case). Hence, it is argued that the structure of the centralizer algebra is independent of L, which is consistent with a result of Hinrichsen and Rittenberg, but that it is isomorphic to two commuting copies (rather than one copy) of the Pauli matrices. This contrasts sharply with the dimension of EndHL(q)(C⊗L2), which grows without bound as L increases.

DOI: 10.1063/1.529843

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The structure and representation theory of a two-parameter generalization of Hecke algebras, denoted HL(q,η), which plays an important role in the construction of the L-site XY chain in a magnetic field is investigated. In particular, the structure of the XY chain representation of the algebra, on C2⊗L, and hence the structure of the centralizer algebra EndHL(q,η)(C2⊗L) for the XY chain representation, is analyzed. It is proved that the full algebra never reduces to the ordinary (one-parameter) Hecke algebra HL(q) in any specialization of the parameters, but contains the Hecke algebra as a quotient in a certain specialization of one of the parameters. This corrects an earlier claim of Hinrichsen and Rittenberg [Phys. Lett. B 275, 350 (1992)]. Bounds are established on the generic dimension of the centralizer algebra 4 ≤ dim(EndHL(q,η)(C⊗L2)) ≤ 8 (the upper bound is realized in every known case). Hence, it is argued that the structure of the centralizer algebra is independent of L, which is consistent with a result of Hinrichsen and Rittenberg, but that it is isomorphic to two commuting copies (rather than one copy) of the Pauli matrices. This contrasts sharply with the dimension of EndHL(q)(C⊗L2), which grows without bound as L increases.

We show that the XY quantum chain in a magnetic field is invariant under a two parameter deformation of the

SU(1/1)

superalgebra. One is led to an extension of the braid group and the Hecke algebras which reduce to the known ones when the two parameter coincide. The physical significance of the two parameters is discussed. When both are equal to one, one gets a Pokrovski-Talapov phase transition. We also show that the representation theory of the quantum superalgebras indicates how to take the appropriate thermodynamical limits.

We consider a series of N-state L(≥N) site quantum spin chains, characterised by the ordered partition of N into 2 parts, N=P+M. These (P/M) chains are invariant under an action of UqSU(P/M), and are built from a representation of the Hecke algebra HL-1(q). We establish that the intersection of the spectra of a (P/M) and (P'/M') chain of fixed length L is the spectrum of the (min(P,P')/min(M,M')) chain of that length. We establish that the spectrum of the (P/M) chain breaks into blocks corresponding to irreducible representations of HL-1(q) (or equivalently irreducible representations of UqSU(P/M)) characterised by Young diagrams with no rectangular subdiagrams of dimension (P+1)×(M+1) (height × width resp.). We give the corresponding quotient relations for the Hecke algebra. We discuss several implications of these results.

With appropriate boundary conditions the anisotropic XY chain in a magnetic field is known to be invariant under quantum group transformations. We generalize this model defining a class of integrable chains with several fermionic degrees of freedom per site. In order to maintain the quantum group symmetry a general condition on the parameters of these systems is derived. It is shown that the corresponding quantum algebra is a multi-parameter deformation of the Clifford algebra. Discussing a special physical example we observe a new type of zero mode.

Models generalizing the su(2) XX spin-chain were recently introduced. These XXC models also have an underlying su(2) structure. Their construction method is shown to generalize to the chains based on the fundamental representations of the A_m Lie algebras. Integrability of the new models is shown in the context of the quantum inverse scattering method. Their R-matrix is found and shown to yield a representation of the Hecke algebra. The diagonalization of the transfer matrices is carried out using the algebraic Bethe Ansatz. I comment on eventual generalizations and possible links to reaction-diffusion processes.

Models generalizing the su(2) XX spin-chain were recently introduced. These XXC models also have an underlying su(2) structure. Their construction method is shown to generalize to the chains based on the fundamental representations of the Am Lie algebras. Integrability of the new models is shown in the context of the quantum inverse scattering method. Their R-matrix is found and shown to yield a representation of the Hecke algebra. The diagonalization of the transfer matrices is carried out using the algebraic Bethe Ansatz. I comment on eventual generalizations and possible links to reaction-diffusion processes.