We develop an approach for constructing the Baxter Q-operators for generic sl(N) spin chains. The key element of our approach is the possibility to represent a solution of the the Yang Baxter equation in the factorized form. We prove that such a representation holds for a generic sl(N) invariant R-operator and find the explicit expression for the factorizing operators. Taking trace of monodromy matrices constructed of the factorizing operators one defines a family of commuting (Baxter) operators on the quantum space of the model. We show that a generic transfer matrix factorizes into the product of N Baxter Q-operators and discuss an application of this representation for a derivation of functional relations for transfer matrices.
We construct the Baxter operator $\boldsymbol{ \texttt{Q} }(λ)$ for the $q$-Toda chain and the Toda$_2$ chain (the Toda chain in the second Hamiltonian structure). Our construction builds on the relation between the Baxter operator and Bäcklund transformations that were unravelled in {\cite{GaPa92}}. We construct a number of quantum intertwiners ensuring the commutativity of $\boldsymbol{ \texttt{Q} }(λ)$ with the transfer matrix of the models and the one of $\boldsymbol{ \texttt{Q} }$'s between each other. Most importantly, $\boldsymbol{ \texttt{Q} }(λ)$ is modular invariant in the sense of F…
We construct a basis of solutions of the scalar $\boldsymbol{ \texttt{t} }- \boldsymbol{ \texttt{Q} }$ equation describing the spectrum of the $q$-Toda and Toda$_2$ chains by using auxiliary non-linear integral equations. Our construction allows us to provide quantisation conditions for the spectra of these models in the form of thermodynamic Bethe Ansatz-like equations.
Baxter's Q-operators and operatorial Backlund flow for quantum (super)-spin chains
本文提出了描述非齐次有理gl（K）嵌套过程的运算Bäcklund流的一般形式中的Baxter的TQ关系的运算形式|M）具有扭曲周期边界条件的量子（超）自旋链。明确定义了所有嵌套层次上的Q-算子和T-算子的全集。这些结果是基于作者之一和P.Vieira [V.Kazakov and P. Vieira，JHEP 0810（2008）050 [arXiv：0711.2470]]发现的群特征标及其群余导数关于扭曲矩阵的恒等式的推广。我们的形式主义，基于这个新的“主”身份，允许一个系统的和相当直接的推导整个套嵌套的Bethe anomaly方程的频谱的量子可积自旋链，从R-矩阵。
In this paper we continue our classification of regular solutions of the Yang-Baxter equation using the method based on the spin chain boost operator developed in \cite{deLeeuw:2019zsi}. We provide details on how to find all non-difference form solutions and apply our method to spin chains with local Hilbert space of dimensions two, three and four. We classify all $16\times 16$ solutions which exhibit $\mathfrak{su}(2)\oplus \mathfrak{su}(2)$ symmetry, which include the one-dimensional Hubbard model and the $S$-matrix of the ${\rm AdS}_5 \times {\rm S}^5$ superstring sigma model. In all cases …
We diagonalize Q-operators for rational homogeneous sl(2)-invariant Heisenberg spin chains using the algebraic Bethe ansatz. After deriving the fundamental commutation relations relevant for this case from the Yang-Baxter equation we demonstrate that the Q-operators act diagonally on the Bethe vectors if the Bethe equations are satisfied. In this way we provide a direct proof that the eigenvalues of the Q-operators studied here are given by Baxter's Q-functions.
Supersymmetric generalization of q-deformed long-range spin chains of Haldane-Shastry type and trigonometric GL(N|M) solution of associative Yang-Baxter equation
本文用两类三角超对称 R 矩阵构造对合差分算子，证明分次 R 矩阵满足结合 Yang-Baxter 方程，并借助 Polychronakos 冻结技巧得到 q 形变、可含各向异性的新型长程自旋链。
BC Toda chain I: reflection operator and eigenfunctions
Beijing Institute of Mathematical Sciences and Applications（北京数学科学与应用研究院）；Steklov Mathematical Institute（斯捷克洛夫数学研究所）；Department of Mathematics, Technion（以色列理工学院数学系）；National Research University Higher School of Economics（国立高等经济大学）
This article, based on the author's PhD thesis, reviews recent advancements in the field of quantum integrability, in particular the separation of variables (SoV) program for high-rank integrable spin chains and the boost mechanism for solving the Yang-Baxter equation. We begin with a general overview of quantum integrable systems with special emphasis on their description in terms of quantum algebras. We then provide a detailed account of the Yangian of $\mathfrak{gl}(n)$ in particular the Bethe algebra, fusion, and T- and Q-systems. We then introduce the notion of separation of variables in …
Beijing Institute of Mathematical Sciences and Applications（北京国际数学研究中心）；Steklov Mathematical Institute（斯捷克洛夫数学研究所）；Department of Mathematics, Technion（以色列理工学院数学系）；Skolkovo Institute of Science and Technology（斯科尔科沃科学技术研究所）
We classify all fundamental integrable spin chains with two-dimensional local Hilbert space which have regular R-matrices of difference form. This means that the R-matrix underlying the integrable structures is of the form R(u,v)=R(u-v) and reduces to the permutation operator at some particular point. We find a total of 14 independent solutions, 8 of which correspond to well-known eight or lower vertex models. The remaining 6 models appear to be new and some have peculiar properties such as not being diagonalizable or being nilpotent. Furthermore, for even R-matrices, we find a bijection betwe…
