One-boundary Temperley–Lieb algebras in the XXZ and loop models

Nichols, A.; Rittenberg, V.; Gier, Jan de

doi:10.1088/1742-5468/2005/03/p03003

Cited by 72 publications

(173 citation statements)

References 39 publications

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“…However at exceptional points (2.3) the 1BTL algebra gives rise to degeneracies in H 1T . As the spectrum of H 1T and H d is identical this implies degeneracies in H d at these points [6]. If one examines the coefficients of σ z 1 and σ z L in (2.17) there is no obvious difference between the parameters ω − and δ − .…”

Section: Temperley Lieb Algebras and Xxz Chains With Boundary Termsmentioning

confidence: 99%

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Magic in the spectra of the XXZ quantum chain with boundaries at and

et al. 2005

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We show that from the spectra of the Uq(sl(2)) symmetric XXZ spin-1/2 finite quantum chain at ∆ = −1/2 (q = e πi/3 ) one can obtain the spectra of certain XXZ quantum chains with diagonal and non-diagonal boundary conditions. Similar observations are made for ∆ = 0 (q = e πi/2 ). In the finite-size scaling limit the relations among the various spectra are the result of identities satisfied by known character functions. For the finite chains the origin of the remarkable spectral identities can be found in the representation theory of one and two boundaries Temperley-Lieb algebras at exceptional points. Inspired by these observations we have discovered other spectral identities between chains with different boundary conditions.

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Section: Temperley Lieb Algebras and Xxz Chains With Boundary Termsmentioning

confidence: 99%

“…These representations are equivalent except at the exceptional points of the 1BTL algebra. At these exceptional points, although the spectra remain the same, degeneracies appear and the three Hamiltonians have different Jordan cell structure and therefore describe different physical problems [6].…”

Section: Introductionmentioning

confidence: 99%

Magic in the spectra of the XXZ quantum chain with boundaries at and

et al. 2005

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“…Their proposal is based on a previous work [25] and possibly overlaps, for rational θ, with the results of [26]. The claim of [18] is that there is a continuum of boundary conditions characterized by a real variable y.…”

Section: Introductionmentioning

confidence: 99%

Boundary loop models and 2D quantum gravity

Kostov

2007

J. Stat. Mech.

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We study the O(n) loop model on a dynamically triangulated disk, with a new type of boundary conditions, discovered recently by Jacobsen and Saleur. The partition function of the model is that of a gas of self and mutually avoiding loops covering the disk. The Jacobsen-Saleur (JS) boundary condition prescribes that the loops that do not touch the boundary have fugacity n ∈ [−2, 2], while the loops touching at least once the boundary are given different fugacity y. The class of JS boundary conditions, labeled by the real number y, contains the Neumann (y = n) and Dirichlet (y = 1) boundary conditions as particular cases. Here we consider the dense phase of the loop gas, where we compute the two-point boundary correlators of the L-leg operators with mixed Neumann-JS boundary condition. The result coincides with the boundary two-point function in Liouville theory, derived by Fateev, Zamolodchikov and Zamolodchikov. The Liouville charge of the boundary operators match, by the KPZ correspondence, with the L-leg boundary exponents conjectured by JS.

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“…Moreover, it has also provided valuable insight into important universality class in condensed matter physics [3] and cold atom systems [4]. In the past several decades, the integrable quantum spin chains with U(1)-symmetry (with periodic boundary or with diagonal open boundaries [5]) and with some constrained open boundaries [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21] have been extensively studied by various Bethe ansatz methods for a finite lattice and by the vertex operator method [22] in an infinite or a half-infinite lattice [23][24][25][26][27][28].…”

Section: Introductionmentioning

confidence: 99%

A representation basis for the quantum integrable spin chain associated with the su(3) algebra

Hao

Cao

et al. 2016

J. High Energ. Phys.

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An orthogonal basis of the Hilbert space for the quantum spin chain associated with the su(3) algebra is introduced. Such kind of basis could be treated as a nested generalization of separation of variables (SoV) basis for high-rank quantum integrable models. It is found that all the monodromy-matrix elements acting on a basis vector take simple forms. With the help of the basis, we construct eigenstates of the su(3) inhomogeneous spin torus (the trigonometric su(3) spin chain with antiperiodic boundary condition) from its spectrum obtained via the off-diagonal Bethe Ansatz (ODBA). Based on small sites (i.e. N = 2) check, it is conjectured that the homogeneous limit of the eigenstates exists, which gives rise to the corresponding eigenstates of the homogenous model.

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One-boundary Temperley–Lieb algebras in the XXZ and loop models

Cited by 72 publications

References 39 publications

Magic in the spectra of the XXZ quantum chain with boundaries at and

Magic in the spectra of the XXZ quantum chain with boundaries at and

Boundary loop models and 2D quantum gravity

A representation basis for the quantum integrable spin chain associated with the su(3) algebra

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