Functional relations and the Yang-Baxter algebra

Functional equations methods are a fundamental part of the theory of Exactly Solvable Models in Statistical Mechanics and they are intimately connected with Baxter's concept of commuting transfer matrices. This concept has culminated in the celebrated Yang-Baxter equation which plays a fundamental role for the construction of quantum integrable systems and also for obtaining their exact solution. Here I shall discuss a proposal that has been put forward in the past years, in which the Yang-Baxter algebra is viewed as a source of functional equations describing quantities of physical interest. For instance, this method has been successfully applied for the description of the spectrum of open spin chains, partition functions of elliptic models with domain wall boundaries and scalar product of Bethe vectors. Further applications of this method are also discussed.Comment: 23 pages. Contribution to the proceedings of the ISQS2

Six-vertex model and non-linear differential equations I. Spectral problem

In this work we relate the spectral problem of the toroidal six-vertex model's transfer matrix with the theory of integrable non-linear differential equations. More precisely, we establish an analogy between the Classical Inverse Scattering Method and previously proposed functional equations originating from the Yang-Baxter algebra. The latter equations are then regarded as an Auxiliary Linear Problem allowing us to show that the six-vertex model's spectrum solves Riccati-type non-linear differential equations. Generating functions of conserved quantities are expressed in terms of determinants and we also discuss a relation between our Riccati equations and a stationary Schr\"odinger equation.Comment: 42 pages, 3 figure

Off-shell scalar products for the XXZXXZ spin chain with open boundaries

In this work we study scalar products of Bethe vectors associated with the $XXZ$ spin chain with open boundary conditions. The scalar products are obtained as solutions of a system of functional equations. The description of scalar products through functional relations follows from a particular map having the reflection algebra as its domain and a function space as the codomain. Within this approach we find a multiple contour integral representation for the scalar products in which the homogeneous limit can be obtained trivially.Comment: 33 pages. v2: figure and references added, presentation slightly modifie

Twisted Heisenberg chain and the six-vertex model with DWBC

In this work we establish a relation between the six-vertex model with Domain Wall Boundary Conditions (DWBC) and the $XXZ$ spin chain with anti-periodic twisted boundaries. More precisely, we demonstrate a formal relation between the zeroes of the partition function of the six-vertex model with DWBC and the zeroes of the transfer matrix eigenvalues associated with the six-vertex model with a particular non-diagonal boundary twist.Comment: 17 pages. v2: 20 pages, presentation modified, details of derivations added, references added. v3: typos fixed, minor changes, accepted for publication in JSTA