Vertex Models and Spin Chains in Formulas and Pictures

Abstract

We systematise and develop a graphical approach to the investigations of quantum integrable vertex statistical models and the corresponding quantum spin chains. The graphical forms of the unitarity and various crossing relations are introduced. Their explicit analytical forms for the case of integrable systems associated with the quantum loop algebra U q (L(sl l+1 )) are given. The commutativity conditions for the transfer operators of lattices with a boundary are derived by the graphical method. Our consideration reveals useful advantages of the graphical approach for certain problems in the theory of quantum integrable systems.

$B_k$ Spin Vertex Models and Quantum Algebras

“…Our aim here is to study the n-CB algebra for vertex models. For recent works on vertex models, see [13,14,15,16]. We establish that the same algebra is obeyed by vertex models.…”

Reduced qKZ equation and genuine qKZ equation

“…We actively use the graphical approach developed in the paper [24]. This approach has already proved to be useful in deriving the reduced qKZ equation and has confirmed its capabilities in our consideration.…”

“…Following Kac, we denote by L(g) the loop algebra of g and by L(g) its standard central extension. It is natural to denote by h the arising extension of h. For details see the monograph by Kac [27] and, in the form adopted for our purposes, the paper [24]. Let be a complex number such that q = exp is not a root of unity.…”

“…. l], and q x , x ∈ h, subjected to the corresponding defining relations, see, for example, the paper [24]. The quantum loop algebra U q (L(g)) is a Hopf algebra with the comultiplication Δ, antipode S, and counit ε.…”

On SO$(N)$ spin vertex models

“…Well known among these are the six, eight and nineteen vertex models [3,4]. For recent works on vertex models see, e.g., [5,6,7,8,9]. Our purpose here is to introduce vertex models based on the algebra D k and the spin representation.…”