A Q-operator for the quantum transfer matrix

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.Comment: 20 pages, v2: some minor style improvement

Workshop island 3: algebraic aspects of integrability. Introduction to an additional volume of selected papers arising from the conference on algebraic aspects of integrable systems, Island 3, Islay 2007

As did the very first ISLAND workshop, ISLAND 3 took place on the Hebridean island of Islay, providing a beautiful and serene surrounding for the meeting which ran for over four days. Building on the success of the previous meetings, ISLAND 3 saw the largest number (so far) of participants coming from countries all over the world. A complete list can be found below

The twisted XXZ chain at roots of unity revisited

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.Comment: 10 pages, 2 figures. One footnote and some comments in the conclusions adde

Turning the Quantum Group Invariant XXZ Spin-Chain Hermitian: A Conjecture on the Invariant Product

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.Comment: 12 page

Auxiliary matrices on both sides of the equator

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.Comment: 22 pages; 2nd version: one paragraph added in the conclusion and some typos correcte

Auxiliary matrices for the six-vertex model and the algebraic Bethe ansatz

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.Comment: 32 page

Pile-soil interaction and settlement effects induced by deep excavations

Deep excavations may cause settlement and damage to adjacent buildings, even if they are found on piles. The corresponding pile deformations are determined by axial and lateral effects. This paper describes an analytical model relating axial pile deformation to the vertical soil displacement resulting from the deep excavation and also suggests ways to determine the pile response to lateral displacements. The axial pile-soil interaction is clearly different for end-bearing and friction piles. Common generalizations that end-bearing piles settle the same as the soil settlement at the base level and friction piles with the ground surface settlement present lower and upper bounds, which are only valid for certain idealized cases. The settlement of piles with a large component of shaft friction is determined mainly by the actual load on the pile relative to the pile ultimate capacity. The lateral pile response is governed mainly by the relative stiffness of the pile to the soil. The proposed model was validated with measurements of the North South Line project in Amsterdam.This paper is based on the first author’s Ph.D. study at Cambridge University in cooperation with the Netherlands Centre of Underground Construction

PT Symmetry of the non-Hermitian XX Spin-Chain: Non-local Bulk Interaction from Complex Boundary Fields

The XX spin-chain with non-Hermitian diagonal boundary conditions is shown to be quasi-Hermitian for special values of the boundary parameters. This is proved by explicit construction of a new inner product employing a "quasi-fermion" algebra in momentum space where creation and annihilation operators are not related via Hermitian conjugation. For a special example, when the boundary fields lie on the imaginary axis, we show the spectral equivalence of the quasi-Hermitian XX spin-chain with a non-local fermion model, where long range hopping of the particles occurs as the non-Hermitian boundary fields increase in strength. The corresponding Hamiltonian interpolates between the open XX and the quantum group invariant XXZ model at the free fermion point. For an even number of sites the former is known to be related to a CFT with central charge c=1, while the latter has been connected to a logarithmic CFT with central charge c=-2. We discuss the underlying algebraic structures and show that for an odd number of sites the superalgebra symmetry U(gl(1|1)) can be extended from the unit circle along the imaginary axis. We relate the vanishing of one of its central elements to the appearance of Jordan blocks in the Hamiltonian.Comment: 37 pages, 5 figure