We extend basic properties of two dimensional integrable models within the Algebraic Bethe Ansatz approach to 2+1 dimensions and formulate the sufficient conditions for the commutativity of transfer matrices of different spectral parameters, in analogy with Yang-Baxter or tetrahedron equations. The basic ingredient of our models is the R-matrix, which describes the scattering of a pair of particles over another pair of particles, the quark-anti-quark (meson) scattering on another quark-anti-quark state. We show that the Kitaev model belongs to this class of models and its R-matrix fulfills well-defined equations for integrability.We extend basic properties of two dimensional integrable models within the Algebraic Bethe Ansatz approach to 2+1 dimensions and formulate the sufficient conditions for the commutativity of transfer matrices of different spectral parameters, in analogy with Yang-Baxter or tetrahedron equations. The basic ingredient of our models is the R-matrix, which describes the scattering of a pair of particles over another pair of particles, the quark-anti-quark (meson) scattering on another quark-anti-quark state. We show that the Kitaev model belongs to this class of models and its R-matrix fulfills well-defined equations for integrability.

arXiv:1502.04055Feb 13, 20151 citation

Aspects of Integrable Models

Pranav Diwakar

The objective of this thesis is to study the isotropic XXX-1/2 spin chain model using the Algebraic Bethe Ansatz. To this end, we discuss the concept of integrability as well as the Lax operator and R-matrix, which help generate as many commuting operators in involution as there are degrees of freedom. We establish that the spin chain Hamiltonian belongs to this set and provide a definition of a state vector whose parameters, the Bethe roots, are constrained by a set of equations called the Bethe Ansatz Equations. We show that there is a one-to-one correspondence between the Bethe roots and the eigenfunctions of the system. Next, we proceed to study the nature of the low-lying excitations of both the ferromagnetic and antiferromagnetic model in the thermodynamic limit N → ∞ and show that the Bethe roots can be grouped into complexes or strings, which behave like bound states. We see that integrability is directly related to diffractionless scattering, which is obeyed by systems whose scattering matrices satisfy the Yang-Baxter Equation. In order to provide a more physical interpretation, we calculate the scattering matrix of the two-body problem for a system that satisfies the Yang-Baxter Equation and obtain exchange relations that are identical to those obtained using the Algebraic Bethe Ansatz for the XXX-1/2 spin chain model. Finally, we calculate the scattering matrix for a two-body problem interacting with a delta potential and show that this is the same as what we derived using the Coordinate Bethe Ansatz.The objective of this thesis is to study the isotropic XXX-1/2 spin chain model using the Algebraic Bethe Ansatz. To this end, we discuss the concept of integrability as well as the Lax operator and R-matrix, which help generate as many commuting operators in involution as there are degrees of freedom. We establish that the spin chain Hamiltonian belongs to this set and provide a definition of a state vector whose parameters, the Bethe roots, are constrained by a set of equations called the Bethe Ansatz Equations. We show that there is a one-to-one correspondence between the Bethe roots and the eigenfunctions of the system. Next, we proceed to study the nature of the low-lying excitations of both the ferromagnetic and antiferromagnetic model in the thermodynamic limit N → ∞ and show that the Bethe roots can be grouped into complexes or strings, which behave like bound states. We see that integrability is directly related to diffractionless scattering, which is obeyed by systems whose scattering matrices satisfy the Yang-Baxter Equation. In order to provide a more physical interpretation, we calculate the scattering matrix of the two-body problem for a system that satisfies the Yang-Baxter Equation and obtain exchange relations that are identical to those obtained using the Algebraic Bethe Ansatz for the XXX-1/2 spin chain model. Finally, we calculate the scattering matrix for a two-body problem interacting with a delta potential and show that this is the same as what we derived using the Coordinate Bethe Ansatz.

Oct 18, 20180 citations

Spectra of Transfer Matrices of Exactly Solvable Models with D(D3)Symmetry

Christopher S. Campbell

The Quantum Inverse Scattering Method, formalised in the late 70s by Faddeev, Sklyanin and Takhtadzhyan, is a method for constructing integrable quantum systems given a solution to the Yang-Baxter equation. Once constructed it is often possible to find the energy spectrum of the associated hamiltonian using the method of mutually commuting transfer matrices, and the algebraic Bethe ansatz. If the algebraic Bethe ansatz fails we are still often able to solve the system using functional relation methods. In 2006 Dancer, Isaac and Links found several solutions to the Yang-Baxter equation arising from the Drinfeld double of the dihedral group $D(D_3)$ . The associated hamiltonian describes a system of interacting, non-abelian anyons. In this dissertation we take these solutions to the Yang-Baxter equation and, using a combination of algebraic Bethe ansatz and functional relations, we find the eigenvalues of the anyonic hamiltonian. The solution we find is dependent on two distinct sets of solutions to the Bethe equations, something that is rather uncommon, though not unprecedented. Further results are also obtained for other integrable models arising from the same set of solutions to the Yang-Baxter equation.The Quantum Inverse Scattering Method, formalised in the late 70s by Faddeev, Sklyanin and Takhtadzhyan, is a method for constructing integrable quantum systems given a solution to the Yang-Baxter equation. Once constructed it is often possible to find the energy spectrum of the associated hamiltonian using the method of mutually commuting transfer matrices, and the algebraic Bethe ansatz. If the algebraic Bethe ansatz fails we are still often able to solve the system using functional relation methods. In 2006 Dancer, Isaac and Links found several solutions to the Yang-Baxter equation arising from the Drinfeld double of the dihedral group $D(D_3)$. The associated hamiltonian describes a system of interacting, non-abelian anyons. In this dissertation we take these solutions to the Yang-Baxter equation and, using a combination of algebraic Bethe ansatz and functional relations, we find the eigenvalues of the anyonic hamiltonian. The solution we find is dependent on two distinct sets of solutions to the Bethe equations, something that is rather uncommon, though not unprecedented. Further results are also obtained for other integrable models arising from the same set of solutions to the Yang-Baxter equation.

May 1, 20100 citations

INTRODUCTION TO QUANTUM INTEGRABILITY

Anastasia Doikou, STEFANO EVANGELISTI, Giovanni Feverati +1 more

In this paper, we review the basic concepts regarding quantum integrability. Special emphasis is given on the algebraic content of integrable models. The associated algebras are essentially described by the Yang–Baxter and boundary Yang–Baxter equations depending on the choice of boundary conditions. The relation between the aforementioned equations and the braid group is briefly discussed. A short review on quantum groups as well as the quantum inverse scattering method (algebraic Bethe ansatz) is also presented.In this paper, we review the basic concepts regarding quantum integrability. Special emphasis is given on the algebraic content of integrable models. The associated algebras are essentially described by the Yang–Baxter and boundary Yang–Baxter equations depending on the choice of boundary conditions. The relation between the aforementioned equations and the braid group is briefly discussed. A short review on quantum groups as well as the quantum inverse scattering method (algebraic Bethe ansatz) is also presented.

arXiv:0912.3350International Journal of Modern Physics AJul 8, 201020 citations

Introduction to Quantum Integrability

Anastasia Doikou, Stefano Evangelistib, Giovanni Feverati +1 more

In this article we review the basic concepts regarding quantum integrability. Special emphasis is given on the algebraic content of integrable models. The associated algebras are essentially described by the Yang-Baxter and boundary Yang-Baxter equations depending on the choice of boundary conditions. The relation between the aforementioned equations and the braid group is briefly discussed. A short review on quantum groups as well as the quantum inverse scattering method (algebraic Bethe ansatz) is also presented.In this article we review the basic concepts regarding quantum integrability. Special emphasis is given on the algebraic content of integrable models. The associated algebras are essentially described by the Yang-Baxter and boundary Yang-Baxter equations depending on the choice of boundary conditions. The relation between the aforementioned equations and the braid group is briefly discussed. A short review on quantum groups as well as the quantum inverse scattering method (algebraic Bethe ansatz) is also presented.

Aug 14, 201626 citations

Integrability of planar-algebraic models

Xavier Poncini, Jørgen Rasmussen

Abstract The quantum inverse scattering method is a scheme for solving integrable models in 1 + 1 dimensions, building on an R-matrix that satisfies the Yang–Baxter equation (YBE) and in terms of which one constructs a commuting family of transfer matrices. In the standard formulation, this R-matrix acts on a tensor product of vector spaces. Here, we relax this tensorial property and develop a framework for describing and analysing integrable models based on planar algebras, allowing non-separable R-operators satisfying generalised YBEs. We also re-evaluate the notion of integrals of motion and characterise when an (algebraic) transfer operator is polynomial in a single integral of motion. We refer to such models as polynomially integrable. In an eight-vertex model, we demonstrate that the corresponding transfer operator is polynomial in the natural Hamiltonian. In the Temperley–Lieb loop model with loop fugacity β ∈ C , we likewise find that, for all but finitely many β-values, the transfer operator is polynomial in the usual Hamiltonian element of the Temperley–Lieb algebra T L n ( β ) , at least for n ⩽ 17 . Moreover, we find that this model admits a second canonical Hamiltonian, and that this Hamiltonian also acts as a polynomial integrability generator for small n and all but finitely many β-values.Abstract The quantum inverse scattering method is a scheme for solving integrable models in 1 + 1 dimensions, building on an R-matrix that satisfies the Yang–Baxter equation (YBE) and in terms of which one constructs a commuting family of transfer matrices. In the standard formulation, this R-matrix acts on a tensor product of vector spaces. Here, we relax this tensorial property and develop a framework for describing and analysing integrable models based on planar algebras, allowing non-separable R-operators satisfying generalised YBEs. We also re-evaluate the notion of integrals of motion and characterise when an (algebraic) transfer operator is polynomial in a single integral of motion. We refer to such models as polynomially integrable. In an eight-vertex model, we demonstrate that the corresponding transfer operator is polynomial in the natural Hamiltonian. In the Temperley–Lieb loop model with loop fugacity β ∈ C , we likewise find that, for all but finitely many β-values, the transfer operator is polynomial in the usual Hamiltonian element of the Temperley–Lieb algebra T L n ( β ) , at least for n ⩽ 17 . Moreover, we find that this model admits a second canonical Hamiltonian, and that this Hamiltonian also acts as a polynomial integrability generator for small n and all but finitely many β-values.

arXiv:2206.14462Journal of Statistical Mechanics Theory and ExperimentJul 1, 20231 citation

D(D_3)

. The associated hamiltonian describes a system of interacting, non-abelian anyons. In this dissertation we take these solutions to the Yang-Baxter equation and, using a combination of algebraic Bethe ansatz and functional relations, we find the eigenvalues of the anyonic hamiltonian. The solution we find is dependent on two distinct sets of solutions to the Bethe equations, something that is rather uncommon, though not unprecedented. Further results are also obtained for other integrable models arising from the same set of solutions to the Yang-Baxter equation.