Baxterization, dynamical systems, and the symmetries of integrability

We resolve the `baxterization' problem with the help of the automorphism group of the Yang-Baxter (resp. star-triangle, tetrahedron, \dots) equations. This infinite group of symmetries is realized as a non-linear (birational) Coxeter group acting on matrices, and exists as such, {\em beyond the narrow context of strict integrability}. It yields among other things an unexpected elliptic parametrization of the non-integrable sixteen-vertex model. It provides us with a class of discrete dynamical systems, and we address some related problems, such as characterizing the complexity of iterations.Comment: 25 pages, Latex file (epsf style). WARNING: Postscript figures are BIG (600kB compressed, 4.3MB uncompressed). If necessary request hardcopy to [email protected] and give your postal mail addres

Nonlinear stabilitty for steady vortex pairs

In this article, we prove nonlinear orbital stability for steadily translating vortex pairs, a family of nonlinear waves that are exact solutions of the incompressible, two-dimensional Euler equations. We use an adaptation of Kelvin's variational principle, maximizing kinetic energy penalised by a multiple of momentum among mirror-symmetric isovortical rearrangements. This formulation has the advantage that the functional to be maximized and the constraint set are both invariant under the flow of the time-dependent Euler equations, and this observation is used strongly in the analysis. Previous work on existence yields a wide class of examples to which our result applies.Comment: 25 page

Determinant Representations for Correlation Functions of Spin-1/2 XXX and XXZ Heisenberg Magnets

We consider correlation functions of the spin-\half XXX and XXZ Heisenberg chains in a magnetic field. Starting from the algebraic Bethe Ansatz we derive representations for various correlation functions in terms of determinants of Fredholm integral operators.Comment: 23 pages, TeX, BONN-TH-94-14, revised version: typos correcte

Exact solution of new integrable nineteen-vertex models and quantum spin-1 chains

New exactly solvable nineteen vertex models and related quantum spin-1 chains are solved. Partition functions, excitation energies, correlation lengths, and critical exponents are calculated. It is argued that one of the non-critical Hamiltonians is a realization of an integrable Haldane system. The finite-size spectra of the critical Hamiltonians deviate in their structure from standard predictions by conformal invariance.Comment: 16 pages, to appear in Z. Phys. B, preprint Cologne-94-474

Schur Polynomials and the Yang-Baxter equation

We show that within the six-vertex model there is a parametrized Yang-Baxter equation with nonabelian parameter group GL(2)xGL(1) at the center of the disordered regime. As an application we rederive deformations of the Weyl character formule of Tokuyama and of Hamel and King.Comment: Revised introduction; slightly changed reference

Exact Results on Potts Model Partition Functions in a Generalized External Field and Weighted-Set Graph Colorings

We present exact results on the partition function of the $q$-state Potts model on various families of graphs $G$ in a generalized external magnetic field that favors or disfavors spin values in a subset $I_s = \{1,...,s\}$ of the total set of possible spin values, $Z(G,q,s,v,w)$, where $v$ and $w$ are temperature- and field-dependent Boltzmann variables. We remark on differences in thermodynamic behavior between our model with a generalized external magnetic field and the Potts model with a conventional magnetic field that favors or disfavors a single spin value. Exact results are also given for the interesting special case of the zero-temperature Potts antiferromagnet, corresponding to a set-weighted chromatic polynomial $Ph(G,q,s,w)$ that counts the number of colorings of the vertices of $G$ subject to the condition that colors of adjacent vertices are different, with a weighting $w$ that favors or disfavors colors in the interval $I_s$. We derive powerful new upper and lower bounds on $Z(G,q,s,v,w)$ for the ferromagnetic case in terms of zero-field Potts partition functions with certain transformed arguments. We also prove general inequalities for $Z(G,q,s,v,w)$ on different families of tree graphs. As part of our analysis, we elucidate how the field-dependent Potts partition function and weighted-set chromatic polynomial distinguish, respectively, between Tutte-equivalent and chromatically equivalent pairs of graphs.Comment: 39 pages, 1 figur

Optical excitations in a one-dimensional Mott insulator

The density-matrix renormalization-group (DMRG) method is used to investigate optical excitations in the Mott insulating phase of a one-dimensional extended Hubbard model. The linear optical conductivity is calculated using the dynamical DMRG method and the nature of the lowest optically excited states is investigated using a symmetrized DMRG approach. The numerical calculations agree perfectly with field-theoretical predictions for a small Mott gap and analytical results for a large Mott gap obtained with a strong-coupling analysis. Is is shown that four types of optical excitations exist in this Mott insulator: pairs of unbound charge excitations, excitons, excitonic strings, and charge-density-wave (CDW) droplets. Each type of excitations dominates the low-energy optical spectrum in some region of the interaction parameter space and corresponds to distinct spectral features: a continuum starting at the Mott gap (unbound charge excitations), a single peak or several isolated peaks below the Mott gap (excitons and excitonic strings, respectively), and a continuum below the Mott gap (CDW droplets).Comment: 12 pages (REVTEX 4), 12 figures (in 14 eps files), 1 tabl

Decoupling of the S=1/2 antiferromagnetic zig-zag ladder with anisotropy

The spin-1/2 antiferromagnetic zig-zag ladder is studied by exact diagonalization of small systems in the regime of weak inter-chain coupling. A gapless phase with quasi long-range spiral correlations has been predicted to occur in this regime if easy-plane (XY) anisotropy is present. We find in general that the finite zig-zag ladder shows three phases: a gapless collinear phase, a dimer phase and a spiral phase. We study the level crossings of the spectrum,the dimer correlation function, the structure factor and the spin stiffness within these phases, as well as at the transition points. As the inter-chain coupling decreases we observe a transition in the anisotropic XY case from a phase with a gap to a gapless phase that is best described by two decoupled antiferromagnetic chains. The isotropic and the anisotropic XY cases are found to be qualitatively the same, however, in the regime of weak inter-chain coupling for the small systems studied here. We attribute this to a finite-size effect in the isotropic zig-zag case that results from exponentially diverging antiferromagnetic correlations in the weak-coupling limit.Comment: to appear in Physical Review

Peierls transition in the presence of finite-frequency phonons in the one-dimensional extended Peierls-Hubbard model at half-filling

We report quantum Monte Carlo (stochastic series expansion) results for the transition from a Mott insulator to a dimerized Peierls insulating state in a half-filled, 1D extended Hubbard model coupled to optical bond phonons. Using electron-electron (e-e) interaction parameters corresponding approximately to polyacetylene, we show that the Mott-Peierls transition occurs at a finite value of the electron-phonon (e-ph) coupling. We discuss several different criteria for detecting the transition and show that they give consistent results. We calculate the critical e-ph coupling as a function of the bare phonon frequency and also investigate the sensitivity of the critical coupling to the strength of the e-e interaction. In the limit of strong e-e couplings, we map the model to a spin-Peierls chain and compare the phase boundary with previous results for the spin-Peierls transition. We point out effects of a nonlinear spin-phonon coupling neglected in the mapping to the spin-Peierls model.Comment: 7 pages, 5 figure

Existence and Nonlinear Stability of Rotating Star Solutions of the Compressible Euler-Poisson Equations

We prove existence of rotating star solutions which are steady-state solutions of the compressible isentropic Euler-Poisson (EP) equations in 3 spatial dimensions, with prescribed angular momentum and total mass. This problem can be formulated as a variational problem of finding a minimizer of an energy functional in a broader class of functions having less symmetry than those functions considered in the classical Auchmuty-Beals paper. We prove the nonlinear dynamical stability of these solutions with perturbations having the same total mass and symmetry as the rotating star solution. We also prove local in time stability of W^{1, \infty}(\RR^3) solutions where the perturbations are entropy-weak solutions of the EP equations. Finally, we give a uniform (in time) a-priori estimate for entropy-weak solutions of the EP equations