The form factor program: a review and new results − the nested SU(N) off-shell Bethe ansatz

The purpose of the ''bootstrap program'' for integrable quantum field theories in 1+1 dimensions is to construct explicitly a model in terms of its Wightman functions. In this article, this program is mainly illustrated in terms of the sinh-Gordon model and the SU(N) Gross-Neveu model. The nested off-shell Bethe ansatz for an SU(N) factorizing S-matrix is constructed. We review some previous results on sinh-Gordon form factors and the quantum operator field equation. The problem of how to sum over intermediate states is considered in the short distance limit of the two point Wightman function for the sinh-Gordon model

A class of ansatz wave functions for 1D spin systems and their relation to DMRG

We investigate the density matrix renormalization group (DMRG) discovered by White and show that in the case where the renormalization eventually converges to a fixed point the DMRG ground state can be simply written as a ``matrix product'' form. This ground state can also be rederived through a simple variational ansatz making no reference to the DMRG construction. We also show how to construct the ``matrix product'' states and how to calculate their properties, including the excitation spectrum. This paper provides details of many results announced in an earlier letter.Comment: RevTeX, 49 pages including 4 figures (macro included). Uuencoded with uufiles. A complete postscript file is available at http://fy.chalmers.se/~tfksr/prb.dmrg.p

Superconducting correlations in metallic nanoparticles: exact solution of the BCS model by the algebraic Bethe ansatz

Superconducting pairing of electrons in nanoscale metallic particles with discrete energy levels and a fixed number of electrons is described by the reduced BCS model Hamiltonian. We show that this model is integrable by the algebraic Bethe ansatz. The eigenstates, spectrum, conserved operators, integrals of motion, and norms of wave functions are obtained. Furthermore, the quantum inverse problem is solved, meaning that form factors and correlation functions can be explicitly evaluated. Closed form expressions are given for the form factors that describe superconducting pairing.Comment: revised version, 5 pages, revtex, no figure

Separation of variables for the quantum SL(2,R) spin chain

We construct representation of the Separated Variables (SoV) for the quantum SL(2,R) Heisenberg closed spin chain and obtain the integral representation for the eigenfunctions of the model. We calculate explicitly the Sklyanin measure defining the scalar product in the SoV representation and demonstrate that the language of Feynman diagrams is extremely useful in establishing various properties of the model. The kernel of the unitary transformation to the SoV representation is described by the same "pyramid diagram" as appeared before in the SoV representation for the SL(2,C) spin magnet. We argue that this kernel is given by the product of the Baxter Q-operators projected onto a special reference state.Comment: 26 pages, Latex style, 9 figures. References corrected, minor stylistic changes, version to be publishe

ZnZ_n elliptic Gaudin model with open boundaries

The $Z_n$ elliptic Gaudin model with integrable boundaries specified by generic non-diagonal K-matrices with $n+1$ free boundary parameters is studied. The commuting families of Gaudin operators are diagonalized by the algebraic Bethe ansatz method. The eigenvalues and the corresponding Bethe ansatz equations are obtained.Comment: 21 pages, Latex fil

On the 3-particle scattering continuum in quasi one dimensional integer spin Heisenberg magnets

We analyse the three-particle scattering continuum in quasi one dimensional integer spin Heisenberg antiferromagnets within a low-energy effective field theory framework. We exactly determine the zero temperature dynamical structure factor in the O(3) nonlinear sigma model and in Tsvelik's Majorana fermion theory. We study the effects of interchain coupling in a Random Phase Approximation. We discuss the application of our results to recent neutron-scattering experiments on the Haldane-gap material ${\rm CsNiCl_3}$.Comment: 8 pages of revtex, 5 figures, small changes, to appear in PR

Limits of Gaudin algebras, quantization of bending flows, Jucys--Murphy elements and Gelfand--Tsetlin bases

Gaudin algebras form a family of maximal commutative subalgebras in the tensor product of $n$ copies of the universal enveloping algebra U(\g) of a semisimple Lie algebra \g. This family is parameterized by collections of pairwise distinct complex numbers $z_1,...,z_n$. We obtain some new commutative subalgebras in U(\g)^{\otimes n} as limit cases of Gaudin subalgebras. These commutative subalgebras turn to be related to the hamiltonians of bending flows and to the Gelfand--Tsetlin bases. We use this to prove the simplicity of spectrum in the Gaudin model for some new cases.Comment: 11 pages, references adde

Causality and dispersion relations and the role of the S-matrix in the ongoing research

The adaptation of the Kramers-Kronig dispersion relations to the causal localization structure of QFT led to an important project in particle physics, the only one with a successful closure. The same cannot be said about the subsequent attempts to formulate particle physics as a pure S-matrix project. The feasibility of a pure S-matrix approach are critically analyzed and their serious shortcomings are highlighted. Whereas the conceptual/mathematical demands of renormalized perturbation theory are modest and misunderstandings could easily be corrected, the correct understanding about the origin of the crossing property requires the use of the mathematical theory of modular localization and its relation to the thermal KMS condition. These new concepts, which combine localization, vacuum polarization and thermal properties under the roof of modular theory, will be explained and their potential use in a new constructive (nonperturbative) approach to QFT will be indicated. The S-matrix still plays a predominant role but, different from Heisenberg's and Mandelstam's proposals, the new project is not a pure S-matrix approach. The S-matrix plays a new role as a "relative modular invariant"..Comment: 47 pages expansion of arguments and addition of references, corrections of misprints and bad formulation

Haldane-Gapped Spin Chains as Luttinger Liquids: Correlation Functions at Finite Field

We study the behavior of Heisenberg, antiferromagnetic, integer-spin chains in the presence of a magnetic field exceeding the attendant spin gap. For temperatures much smaller than the gap, the spin chains exhibit Luttinger liquid behavior. We compute exactly both the corresponding Luttinger parameter and the Fermi velocity as a function of magnetic field. This enables the computation of a number of correlators from which we derive the spin conductance, the expected form of the dynamic structure factor relevant to inelastic neutron scattering experiments, and NMR relaxation rates. We also comment upon the robustness of the magnetically induced gapless phase both to finite temperature and finite couplings between neighbouring chains.Comment: 32 pages, 8 figures; published version includes additions discussing the robustness of the magnetically induced gapless phase to ordering between chains as well as the relationship between the spin-1 chains and spin-1/2 ladders in the presence of a magnetic fiel

Universality class of S=1/2 quantum spin ladder system with the four spin exchange

We study s=1/2 Heisenberg spin ladder with the four spin exchange. Combining numerical results with the conformal field theory(CFT), we find a phase transition with central charge c=3/2. Since this system has an SU(2) symmetry, we can conclude that this critical theory is described by k=2 SU(2) Wess-Zumino-Witten model with Z$_2$ symmetry breaking