arXiv:cond-mat/9404034 · Journal of Physics A Mathematical and General

Four-state models and Clifford algebras

Haye Hinrichsen B

Published Aug 21, 1994• 7 citations

Abstract

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.

DOI: 10.1088/0305-4470/27/16/008

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arXiv:cond-mat/9404034 · Journal of Physics A Mathematical and General

Four-state models and Clifford algebras

Haye Hinrichsen B

Published Aug 21, 1994• 7 citations

arXiv HTML PDF

Abstract

DOI: 10.1088/0305-4470/27/16/008

Related papers

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Holger Frahm, Sascha Gehrmann, Rafael I. Nepomechie +1 more

A bstract We consider the quantum integrable spin chain models associated with the Jimbo R-matrix based on the quantum affine algebra ${D}_{n+1}^{(2)}$ D n + 1 2 , subject to quantum-group-invariant boundary conditions parameterized by two discrete variables p = 0, . . . , n and ε = 0, 1. We develop the analytical Bethe ansatz for the previously unexplored case ε = 1 with any n , and use it to investigate the effects of different boundary conditions on the finite-size spectrum of the quantum spin chain based on the rank-2 algebra ${D}_3^{(2)}$ D 3 2 . Previous work on this model with periodic boundary conditions has shown that it is critical for the range of anisotropy parameters 0 < γ < π /4, where its scaling limit is described by a non-compact CFT with continuous degrees of freedom related to two copies of the 2D black hole sigma model. The scaling limit of the model with quantum-group-invariant boundary conditions depends on the parameter ε : similarly as in the rank-1 ${D}_2^{(2)}$ D 2 2 chain, we find that the symmetry of the lattice model is spontaneously broken, and the spectrum of conformal weights has both discrete and continuous components, for ε = 1. For p = 1, the latter coincides with that of the ${D}_2^{(2)}$ D 2 2 chain, which should correspond to a non-compact brane related to one black hole CFT in the presence of boundaries. For ε = 0, the spectrum of conformal weights is purely discrete.A bstract We consider the quantum integrable spin chain models associated with the Jimbo R-matrix based on the quantum affine algebra $$ {D}_{n+1}^{(2)} $$ D n + 1 2 , subject to quantum-group-invariant boundary conditions parameterized by two discrete variables p = 0, . . . , n and ε = 0, 1. We develop the analytical Bethe ansatz for the previously unexplored case ε = 1 with any n , and use it to investigate the effects of different boundary conditions on the finite-size spectrum of the quantum spin chain based on the rank-2 algebra $$ {D}_3^{(2)} $$ D 3 2 . Previous work on this model with periodic boundary conditions has shown that it is critical for the range of anisotropy parameters 0 < γ < π /4, where its scaling limit is described by a non-compact CFT with continuous degrees of freedom related to two copies of the 2D black hole sigma model. The scaling limit of the model with quantum-group-invariant boundary conditions depends on the parameter ε : similarly as in the rank-1 $$ {D}_2^{(2)} $$ D 2 2 chain, we find that the symmetry of the lattice model is spontaneously broken, and the spectrum of conformal weights has both discrete and continuous components, for ε = 1. For p = 1, the latter coincides with that of the $$ {D}_2^{(2)} $$ D 2 2 chain, which should correspond to a non-compact brane related to one black hole CFT in the presence of boundaries. For ε = 0, the spectrum of conformal weights is purely discrete.

arXiv:2509.00610Journal of High Energy PhysicsDec 16, 20250 citations

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.

Quantum-group-invariant

{D}_{n+1}^{(2)}

models: Bethe ansatz and finite-size spectrum

A bstract We consider the quantum integrable spin chain models associated with the Jimbo R-matrix based on the quantum affine algebra

{D}_{n+1}^{(2)}

D n + 1 2 , subject to quantum-group-invariant boundary conditions parameterized by two discrete variables p = 0, . . . , n and ε = 0, 1. We develop the analytical Bethe ansatz for the previously unexplored case ε = 1 with any n , and use it to investigate the effects of different boundary conditions on the finite-size spectrum of the quantum spin chain based on the rank-2 algebra

{D}_3^{(2)}

D 3 2 . Previous work on this model with periodic boundary conditions has shown that it is critical for the range of anisotropy parameters 0 < γ < π /4, where its scaling limit is described by a non-compact CFT with continuous degrees of freedom related to two copies of the 2D black hole sigma model. The scaling limit of the model with quantum-group-invariant boundary conditions depends on the parameter ε : similarly as in the rank-1

{D}_2^{(2)}

D 2 2 chain, we find that the symmetry of the lattice model is spontaneously broken, and the spectrum of conformal weights has both discrete and continuous components, for ε = 1. For p = 1, the latter coincides with that of the

{D}_2^{(2)}

D 2 2 chain, which should correspond to a non-compact brane related to one black hole CFT in the presence of boundaries. For ε = 0, the spectrum of conformal weights is purely discrete.