arXiv subjects
Publications and source records attributed to Christian Korff.
We present in detail a recent conjecture on self-adjoint representations of the Temperley-Lieb algebra for particular values on the unit circle. The formulation in terms of graphical calculus is emphasized and discussed for several examples. The role of PT (parity and time reversal) invariance is highlighted as it might prove important for generalising the construction to other cases.
This is a continuation of a previous joint work with Robert Weston on the quantum group invariant XXZ spin-chain (math-ph/0703085). The previous results on quasi-Hermiticity of this integrable model are briefly reviewed and then connected with a new construction of an inner product with respect to which the Hamiltonian and the representation of the Temperley-Lieb algebra become Hermitian. The approach is purely algebraic, one starts with the definition of a positive functional over the Temperley-Lieb algebra whose values can be computed graphically. Employing the Gel'fand-Naimark-Segal (GNS) construction for C*-algebras a self-adjoint representation of the Temperley-Lieb algebra is constructed when the deformation parameter q lies in a special section of the unit circle. The main conjecture of the paper is the unitary equivalence of this GNS representation with the representation obtained in the previous paper employing the ideas of PT-symmetry and quasi-Hermiticity. An explicit example is presented.
We investigate the PT-symmetry of the quantum group invariant XXZ chain. We show that the PT-operator commutes with the quantum group action and also discuss the transformation properties of the Bethe wavefunction. We exploit the fact that the Hamiltonian is an element of the Temperley-Lieb algebra in order to give an explicit and exact construction of an operator that ensures quasi-Hermiticity of the model. This construction relys on earlier ideas related to quantum group reduction. We then employ this result in connection with the quantum analogue of Schur-Weyl duality to introduce a dual pair of C-operators, both of which have closed algebraic expressions. These are novel, exact results connecting the research areas of integrable lattice systems and non-Hermitian Hamiltonians.
Baxter's Q-operator for the quantum transfer matrix of the XXZ spin-chain is constructed employing the representation theory of quantum groups. The spectrum of this Q-operator is discussed and novel functional relations which describe the finite temperature regime of the XXZ spin-chain are derived. For non-vanishing magnetic field the previously known Bethe ansatz equations can be replaced by a system of quadratic equations which is an important advantage for numerical studies. For vanishing magnetic field and rational coupling values it is argued that the quantum transfer matrix exhibits a loop algebra symmetry closely related to the one of the classical six-vertex transfer matrix at roots of unity.
Taking the isotropic limit in a recent representation theoretic construction of Baxter's Q-operators for the XXZ model with quasi-periodic boundary conditions we obtain new results for the XXX model. We show that quasi-periodic boundary conditions are needed to ensure convergence of the Q-operator construction and derive a quantum Wronskian relation which implies two different sets of Bethe ansatz equations, one above the other below the "equator" of total spin zero. We discuss the limit to periodic boundary conditions at the end and explain how this construction might be useful in the context of correlation functions on the infinite lattice. We also identify a special subclass of solutions to the quantum Wronskian for chains up to a length of 10 sites and possibly higher.
We apply a recently introduced reduction procedure based on the embedding of non-crystallographic Coxeter groups into crystallographic ones to Calogero-Moser systems. For rational potentials the familiar generalized Calogero Hamiltonian is recovered. For the Hamiltonians of trigonometric, hyperbolic and elliptic type, we obtain novel integrable dynamical systems with a second potential term which is rescaled by the golden ratio. We explicitly show for the simplest of these non-crystallographic models how the corresponding classical equations of motion can be derived from a Lie algebraic Lax pair based on the larger, crystallographic Coxeter group.
An explicit construction for Q-operators of the finite XXZ spin-chain with twisted boundary conditions is presented. The massless and the massive regime is considered as well as the root of unity case. It is explained how these results yield an alternative expression for the trace function employed in the description of the correlation functions of the inhomogeneous XXZ model on the infinite lattice by Boos, Jimbo, Miwa, Smirnov and Takeyama(hep-th/0412191).
We propose affine Toda field theories related to the non-crystallographic Coxeter groups H_2, H_3 and H_4. The classical mass spectrum, the classical three-point couplings and the one-loop corrections to the mass renormalisation are determined. The construction is carried out by means of a reduction procedure from crystallographic to non-crystallographic Coxeter groups. The embedding structure explains for various affine Toda field theories that their particles can be organised in pairs, such that their relative masses differ by the golden ratio.
A summary of the construction procedure of generalized versions of Baxter's Q-operator is given. Illustrated by several figures and diagrams the use of representation theory is explained step-by-step avoiding technical details. The relation with the infinite-dimensional non-abelian symmetries of the six-vertex model at roots of unity is discussed and parallels with the eight-vertex case outlined.
Baxter's TQ-equation is solved for the six-vertex model using the representation theory of quantum groups at roots of unity. A novel simplified construction of the Q-operator is given depending on a new free parameter. Specializing this general construction to even roots of unity and lattices with an odd number of columns two linearly independent operator solutions of Baxter's TQ equation are obtained as special limits.
We establish that by parameterizing the configuration space of a one-dimensional quantum system by polynomial invariants of q-deformed Coxeter groups it is possible to construct exactly solvable models of Calogero type. We adopt the previously introduced notion of solvability which consists of relating the Hamiltonian to finite dimensional representation spaces of a Lie algebra. We present explicitly the $G_2^q $-case for which we construct the potentials by means of suitable gauge transformations.
The spectra of previously constructed auxiliary matrices for the six-vertex model at roots of unity are investigated for spin-chains of even and odd length. The two cases show remarkable differences. In particular, it is shown that for even roots of unity and an odd number of sites the eigenvalues contain two linear independent solutions to Baxter's TQ-equation corresponding to the Bethe ansatz equations above and below the equator. In contrast, one finds for even spin-chains only one linear independent solution and complete strings. The other main result is the proof of a previous conjecture on the degeneracies of the six-vertex model at roots of unity. The proof rests on the derivation of a functional equation for the auxiliary matrices which is closely related to a functional equation for the eight-vertex model conjectured by Fabricius and McCoy.
We connect two alternative concepts of solving integrable models, Baxter's method of auxiliary matrices (or Q-operators) and the algebraic Bethe ansatz. The main steps of the calculation are performed in a general setting and a formula for the Bethe eigenvalues of the Q-operator is derived. A proof is given for states which contain up to three Bethe roots. Further evidence is provided by relating the findings to the six-vertex fusion hierarchy. For the XXZ spin-chain we analyze the cases when the deformation parameter of the underlying quantum group is evaluated both at and away from a root of unity.
The spectra of recently constructed auxiliary matrices for the six-vertex model respectively the spin s=1/2 Heisenberg chain at roots of unity q^N=1 are investigated. Two conjectures are formulated both of which are proven for N=3 and are verified numerically for several examples with N>3. The first conjecture identifies an abelian subset of auxiliary matrices whose eigenvalues are polynomials in the spectral variable. The zeroes of these polynomials are shown to fall into two sets. One consists of the solutions to the Bethe ansatz equations which determine the eigenvalues of the six-vertex transfer matrix. The other set of zeroes contains the complete strings which encode the information on the degeneracies of the model due to the loop symmetry sl_2 present at roots of 1. The second conjecture then states a polynomial identity which relates the complete string centres to the Bethe roots allowing one to determine the dimension of the degenerate eigenspaces. Its proof for N=3 involves the derivation of a new functional equation for the auxiliary matrices and the six-vertex transfer matrix. Moreover, it is demonstrated in several explicit examples that the complete strings coincide with the classical analogue of the Drinfeld polynomial. The latter is used to classify the finite-dimensional irreducible representations of the loop algebra sl_2. This suggests that the constructed auxiliary matrices not only enable one to solve the six-vertex model but also completely characterize the decomposition of its eigenspaces into irreducible representations of the underlying loop symmetry.
The symmetries of the twisted XXZ spin-chain (alias the twisted six-vertex model) at roots of unity are investigated. It is shown that when the twist parameter is chosen to depend on the total spin an infinite-dimensional non-abelian symmetry algebra can be explicitly constructed for all spin sectors. This symmetry algebra is identified to be the upper or lower Borel subalgebra of the sl_2 loop algebra. The proof uses only the intertwining property of the six-vertex monodromy matrix and the familiar relations of the six-vertex Yang-Baxter algebra.
The construction of auxiliary matrices for the six-vertex model at a root of unity is investigated from a quantum group theoretic point of view. Employing the concept of intertwiners associated with the quantum loop algebra $U_q(\tilde{sl}_2)$ at $q^N=1$ a three parameter family of auxiliary matrices is constructed. The elements of this family satisfy a functional relation with the transfer matrix allowing one to solve the eigenvalue problem of the model and to derive the Bethe ansatz equations. This functional relation is obtained from the decomposition of a tensor product of evaluation representations and involves auxiliary matrices with different parameters. Because of this dependence on additional parameters the auxiliary matrices break in general the finite symmetries of the six-vertex model, such as spin-reversal or spin conservation. More importantly, they also lift the extra degeneracies of the transfer matrix due to the loop symmetry present at rational coupling values. The extra parameters in the auxiliary matrices are shown to be directly related to the elements in the enlarged center of the quantum loop algebra $U_q(\tilde{sl}_2)$ at $q^N=1$. This connection provides a geometric interpretation of the enhanced symmetry of the six-vertex model at rational coupling. The parameters labelling the auxiliary matrices can be interpreted as coordinates on a three-dimensional complex hypersurface which remains invariant under the action of an infinite-dimensional group of analytic transformations, called the quantum coadjoint action.
We consider integrable vertex models whose Boltzmann weights (R-matrices) are trigonometric solutions to the graded Yang-Baxter equation. As is well known the latter can be generically constructed from quantum affine superalgebras $U_{q}(\hat g)$. These algebras do not form a symmetry algebra of the model for generic values of the deformation parameter $q$ when periodic boundary conditions are imposed. If $q$ is evaluated at a root of unity we demonstrate that in certain commensurate sectors one can construct non-abelian subalgebras which are translation invariant and supercommute with the transfer matrix and therefore with all charges of the model. In the line of argument we introduce the restricted quantum superalgebra $U^{res}_q(\hat g)$ and investigate its root of unity limit. We prove several new formulas involving supercommutators of arbitrary powers of the Chevalley-Serre generators and derive higher order quantum Serre relations as well as an analogue of Lustzig's quantum Frobenius theorem for superalgebras.
The leading excitations of the dilute $A_L$ model in regime 2 are considered using analytic arguments. The model can be identified with the integrable $ϕ_{1,2}$ perturbation of the unitary minimal series $M_{L,L+1}$. It is demonstrated that the excitation spectrum of the transfer matrix satisfies the same functional equations in terms of elliptic functions as the exact S-matrices of the $ϕ_{1,2}$ perturbation do in terms of trigonometric functions. In particular, the bootstrap equation corresponding to a self-fusing process is recovered. For the special cases $L=3,4,6$ corresponding to the Ising model in a magnetic field, and the leading thermal perturbations of the tricritical Ising and three-state Potts model, as well as for the unrestricted model, $L=\infty$, we relate the structure of the Bethe roots to the Lie algebras $E_{8,7,6}$ and $D_4$ using Coxeter geometry. In these cases Coxeter geometry also allows for a single formula in generic Lie algebraic terms describing all four cases. For general $L$ we calculate the spectral gaps associated with the leading excitation which allows us to compute universal amplitude ratios characteristic of the universality class. The ratios are of field theoretic importance as they enter the bulk vacuum expectation value of the energy momentum tensor associated with the corresponding integrable quantum field theories.