arXiv subjects
Publications and source records attributed to W. Galleas.
In this letter we show the partition function of the 8VSOS model with domain-wall boundaries satisfies the same type of functional equations as its six-vertex model counterpart. We then use these refined functional equations to obtain novel determinantal representations for the aforementioned partition function.
In this work we study an elliptic solid-on-solid model with domain-wall boundaries having the elliptic quantum group $\mathcal{E}_{p, γ}[\widehat{\mathfrak{gl}_2}]$ as its underlying symmetry algebra. We elaborate on results previously presented by the author and extend our analysis to include continuous families of single determinantal representations for the model's partition function. Interestingly, our families of representations are parameterized by two continuous complex variables which can be arbitrarily chosen without affecting the partition function.
In this work we elaborate on a previous result relating the partition function of the six-vertex model with domain-wall boundary conditions to eigenvalues of a transfer matrix. More precisely, we express the aforementioned partition function as a determinant of a matrix with entries being eigenvalues of the anti-periodic six-vertex model's transfer matrix.
This work is concerned with functional properties shared by partition functions of nineteen-vertex models with domain-wall boundary conditions. In particular, we describe both Izergin-Korepin and Fateev-Zamolodchikov models with the aforementioned boundary conditions and show their partition functions are governed by a system of functional equations originated from the associated Yang-Baxter algebra.
This paper is a continuation of our previous work "Six-vertex model and non-linear differential equations I. Spectral problem" in which we have put forward a method for studying the spectrum of the six-vertex model based on non-linear differential equations. Here we intend to elaborate on that approach and also discuss properties of the spectrum unveiled by the aforementioned differential formulation of the transfer matrix's eigenvalue problem. In particular, we intend to demonstrate how this differential approach allows one to study continuous symmetries of the transfer matrix's spectrum through the Lie groups method.
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 Schrödinger equation.
We present families of single determinantal representations of on-shell scalar products of Bethe vectors. Our families of representations are parameterized by a continuous complex variable which can be fixed at convenience. Here we consider Bethe vectors in two versions of the six-vertex model: the case with boundary twists and the case with open boundaries.
We propose a new method to compute connection matrices of quantum Knizhnik-Zamolodchikov equations associated to integrable vertex models with super algebra and Hecke algebra symmetries. The scheme relies on decomposing the underlying spin representation of the affine Hecke algebra in principal series modules and invoking the known solution of the connection problem for quantum affine Knizhnik-Zamolodchikov equations associated to principal series modules. We apply the method to the spin representation underlying the $\mathcal{U}_q\bigl(\hat{\mathfrak{gl}}(2|1)\bigr)$ Perk-Schultz model. We show that the corresponding connection matrices are described by an elliptic solution of a supersymmetric version of the dynamical quantum Yang-Baxter equation with spectral parameter.
In this work we express the partition function of the integrable elliptic solid-on-solid model with domain-wall boundary conditions as a single determinant. This representation appears naturally as the solution of a system of functional equations governing the model's partition function.
This letter is concerned with the analysis of the six-vertex model with domain-wall boundaries in terms of partial differential equations (PDEs). The model's partition function is shown to obey a system of PDEs resembling the celebrated Knizhnik-Zamolodchikov equation. The analysis of our PDEs naturally produces a family of novel determinant representations for the model's partition function.
In this work we obtain hierarchies of partial differential equations describing on-shell scalar products for two types of six-vertex models. More precisely, six-vertex models with two different diagonal boundary conditions are considered: the case with boundary twists and the case with open boundary conditions. Solutions and properties of our partial differential equations are also discussed.
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.
In this work we propose a mechanism for converting the spectral problem of vertex models transfer matrices into the solution of certain linear partial differential equations. This mechanism is illustrated for the $U_q[\widehat{\mathfrak{sl}}(2)]$ invariant six-vertex model and the resulting partial differential equation is studied for particular values of the lattice length.
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.
In this work we investigate the possibility of using the reflection algebra as a source of functional equations. More precisely, we obtain functional relations determining the partition function of the six-vertex model with domain-wall boundary conditions and one reflecting end. The model's partition function is expressed as a multiple-contour integral that allows the homogeneous limit to be obtained straightforwardly. Our functional equations are also shown to give rise to a consistent set of partial differential equations for the partition function.
In this work the scalar product of Bethe vectors for the six-vertex model is studied by means of functional equations. The scalar products are shown to obey a system of functional equations originated from the Yang-Baxter algebra and its solution is given as a multiple contour integral.
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.
Osculating paths are sets of directed lattice paths which are not allowed to cross each other or have common edges, but are allowed to have common vertices. In this work we derive a constant term formula for the number of such lattice paths by solving a set of simultaneous difference equations.