QQ-system and Weyl-type transfer matrices in integrable SO(2r) spin chains

Abstract

We propose the full system of Baxter Q-functions (QQ-system) for the integrable spin chains with the symmetry of the Dr Lie algebra. We use this QQ-system to derive new Weyl-type formulas expressing transfer matrices in all symmetric and antisymmetric (fundamental) representations through r + 1 basic Q-functions. Our functional relations are consistent with the Q-operators proposed recently by one of the authors and verified explicitly on the level of operators at small finite length.

Determinant form of correlators in high rank integrable spin chains via separation of variables

“…Another interesting direction is developing SoV for higher rank spin chains with different symmetry groups such as so(N ) where progress was made recently in [75,76]. Other natural extensions include the super-symmetric case (see [18,77] for related results), boundary problems and q-deformations (see [15,78] for recent work).…”

Algebraic Bethe Ansatz for spinor R-matrices

“…The authors provided explicit examples of spinorspinor R-matrices of low rank and commented on the corresponding cases of the algebraic Bethe ansatz. Similar spectral problems were also addressed by Reshetikhin in [4], De Vega and Karowski in [5], Babujian, Foerster and Karowski in [6,7], Ferrando, Frassek and Kazakov in [8], Liashyk and Pakuliak in [9], and Gerrard together with the author in [10].…”

“…in [17]. Moreover, it would be interesting to construct q-deformed spinor-oscillator R-matrices and investigate the spinor-type QQ-system following the steps of Ferrando, Frassek and Kazakov in [8]. Lastly, we believe this work might help to better undertstand the Bethe ansazt for fishnets and fishchains emerging in the AdS/CFT integrability framework, see [18][19][20][21] and references therein.…”

Bethe Algebra using Pure Spinors

“…For so 2r models, candidates for analogs of Q a can be spotted in the work [8] by G. Ferrando, R. Frassek, and V. Kazakov who considered Q-system on the Weyl orbit (we use the terminology of [5]) and proposed determinant-type formulae for transfer matrices in symmetric powers of the vector representation. These formulae feature only r Q-functions and, by simple inspection, are gl r -invariant, where gl r ⊂ so 2r corresponds to the standard embedding U(r) ⊂ SO(2r) (we do not need to specify real forms and hence select gl r for notation).…”

“…We build on the findings of [5] and [8] to offer and investigate the explicit gl r -covariant description of so 2r Bethe algebra. The key observation is the following: Because Qfunctions are Plücker coordinates, those of them who define maximal isotropic subspaces are components of pure spinors.…”