Baxter Q-operators of the XXZ chain and R-matrix factorization

Derkachov, S. E.; Karakhanyan, D.; Kirschner, R.

doi:10.1016/j.nuclphysb.2005.12.015

Cited by 19 publications

(23 citation statements)

References 38 publications

(48 reference statements)

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“…However, as explained in the Introduction there is an alternative construction of "factorized" Q-operators [37,38] based on the factorization property of the U q (sl(2)) L-operator. This approach works well for the infinite-dimensional representations (or cyclic case q N = 1), but its restriction to a finite-dimensional case requires a regularization.…”

Section: Resultsmentioning

confidence: 99%

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On the Yang–Baxter equation for the six-vertex model

Mangazeev

2014

Nuclear Physics B

109

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In this paper we review the theory of the Yang-Baxter equation related to the 6-vertex model and its higher spin generalizations. We employ a 3D approach to the problem. Starting with the 3D R-matrix, we consider a two-layer projection of the corresponding 3D lattice model. As a result, we obtain a new expression for the higher spin R-matrix associated with the affine quantum algebra U q ( sl (2)). In the simplest case of the spin s = 1/2 this R-matrix naturally reduces to the R-matrix of the 6-vertex model. Taking a special limit in our construction we also obtain new formulas for the Q-operators acting in the representation space of arbitrary (half-)integer spin. Remarkably, this construction can be naturally extended to any complex values of spin s. We also give all functional equations satisfied by the transfer-matrices and Q-operators. and the general scheme of quantum SoV has been developed [29][30][31][32]. The integral Q-operator for the case of the XXX chain was first calculated in [33]. It is worth noting that taking the limit N → ∞ [34] in the Bazhanov and Stroganov construction [22] one can recover the results of [27] and [33].The main difference of the above approach from the original Baxter method is that the "quantum" representation space is infinite-dimensional. It has the structure of a tensor product of Verma modules with the basis chosen as multi-variable polynomials p(x 1 , . . . , x M ), where M is the size of the system. The Q-operators appear as integral operators with an explicit action on such a polynomial basis. A detailed construction can be found in [35,36] for the XXX case and its generalization to the XXZ case in [37,38]. The non-compact case and applications of the Q-operators to Liouville theory are discussed in [39][40][41]. It is worth mentioning that the representation of the Q-operator by an integral operator is known only for the XXX case [33]. The proper deformation of such integral operator for the case of the six-vertex model is still a challenging problem.Another problem arises when spins take (half-) integer values. In this case the quantum space becomes reducible and the action of the Q-operator on the polynomial basis becomes singular. This difficulty can be overcome by expanding near the limit 2s → Z + as shown in [35,36]. However, a removal of such a regularization is technically challenging and it is desirable to have an alternative approach which is free from this difficulty.In 1997 Bazhanov, Lukyanov and Zamolodchikov (BLZ) suggested another method to derive the Q-operators related to the affine algebra U q ( sl(2)) [42,43]. Based on the universal R-matrix theory [44] they showed that the Q-operators can be constructed as special monodromy operators with the auxiliary space being an infinite-dimensional representation of the q-oscillator algebra. Although their original approach was developed in the context of quantum field theory, the results of [42,43] can be easily adjusted to the spin s = 1/2 XXZ chain [45]. However, the derivation of the local Q-operators from...

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Section: Resultsmentioning

confidence: 99%

“…The Q-operators appear as integral operators with an explicit action on such a polynomial basis. A detailed construction can be found in [35,36] for the XXX case and its generalization to the XXZ case in [37,38]. The non-compact case and applications of the Q-operators to Liouville theory are discussed in [39][40][41].…”

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confidence: 99%

On the Yang–Baxter equation for the six-vertex model

Mangazeev

2014

Nuclear Physics B

109

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“…For the simpler models associated with the quantum group U q (g), there have been many papers which extend, generalize, and comment on the T -Q relation. We would like to refer some of these literatures, including Sklyanin's separation variable method, written by E.Sklyanin [14,15,16], by V.Kuzunetsov, V.Mangazeev and E.Sklyanin [17], by V.Pasquier and M.Gaudin [18], by S.Derkachov [19] by S.Derkachov, G.Karakhanyan and A.Mansahov [20,21] by S.Derkachov, G.Karakhanyan and R.Kirschner, [22] by S.Derkachov and A.Mansahov [23], by A.Belisty, S.Derkachov, G.Korchemesky and A.Manasahov [24], by C.Korff [25,26], by A.Bytsko and J.Teschner [27], by V.Bazhanov, S.Lukyanov and Al.Zamolodchikov [28,29,30,31], by M.Rossi and R.Weston [32], by P.Dorey and R.Tateo [33], by V.Bazhanov, A.Hibberd and S.Khoroshkin [34], by P.Kulish and Z.Zeitlin [35], by A.Antonov and B.Feigin [36], by I.Krichever, O.Lipan, P.Wiegmann and A.Zabrodin [38], by V.Bazhanov and N.Reshetikhin [39], by A.Kuniba, T.Nakanishi and J.Suzuki [40], by H.Boos, M.Jimbo, T.Miwa, F.Smirnov and Y.Takeyama [41,42], by A.Chervov and G.Falqui [43]. Each paper added to our understanding of the great Baxter's original paper [1].…”

Section: Introductionmentioning

confidence: 99%

Baxter'sQ-operator for theW-algebraW_N

Kojima¹

2008

J. Phys. A: Math. Theor.

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“…As we will see in the text this actually requires the introduction of quasiperiodic boundary conditions. Previous constructions of Q-operators for the XXX spinchain [25,26,27] have been for periodic boundary conditions only, where Q has been represented as an integral kernel (see also [28] for a related XXZ construction).…”

Section: Introductionmentioning

confidence: 99%

AQ-operator for the twisted XXX model

Korff¹

2006

J. Phys. A: Math. Gen.

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Taking the isotropic limit ∆ → 1 in a recent representation theoretic construction of Baxter's Q-operators for the XXZ model with quasi-periodic boundary conditions we obtain new results for the XXX model. We show that quasi-periodic boundary conditions are needed to ensure convergence of the Q-operator construction and derive a quantum Wronskian relation which implies two different sets of Bethe ansatz equations, one above the other below the "equator" of total spin S z = 0. We discuss the limit to periodic boundary conditions at the end and explain how this construction might be useful in the context of correlation functions on the infinite lattice. We also identify a special subclass of solutions to the quantum Wronskian for chains up to a length of 10 sites and possibly higher.

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Baxter Q-operators of the XXZ chain and R-matrix factorization

Cited by 19 publications

References 38 publications

On the Yang–Baxter equation for the six-vertex model

On the Yang–Baxter equation for the six-vertex model

Baxter'sQ-operator for theW-algebraW_N

AQ-operator for the twisted XXX model

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Baxter Q-operators of the XXZ chain and R-matrix factorization

Cited by 19 publications

References 38 publications

On the Yang–Baxter equation for the six-vertex model

On the Yang–Baxter equation for the six-vertex model

Baxter'sQ-operator for theW-algebraWN

AQ-operator for the twisted XXX model

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Baxter'sQ-operator for theW-algebraW_N