Local and Nonlocal Continuum Limits of Ising-Type Energies for Spin Systems

We study, through a !-convergence procedure, the discrete to continuum limit of Ising-type energies of the form F"(u) = −Pi,j c" i,juiuj , where u is a spin variable defined on a portion of a cubic lattice "Zd 8 ⌦, ⌦ being a regular bounded open set, and valued in {−1, 1}. If the constants c" i,j are nonnegative and satisfy suitable coercivity and decay assumptions, we show that all possible !-limits of surface scalings of the functionals F" are finite on BV (⌦; {±1}) and of the formR Su'(x, ⌫u) dH11 d−1. If such decay assumptions are violated, we show that we may approximate nonlocal functionals of the form R Su '(⌫u) dHd−1+ K(x, y)g(u(x), u(y)) dxdy. We focus on the approximation of two relevant examples: fractional perimeters and Ohta–Kawasaki-type energies. Eventually, we provide a general criterion for a ferromagnetic behavior of the energies F" even when the constants c" i,j change sign. If such a criterion is satisfied, the ground states of F" are still the uniform states 1 and −1 and the continuum limit of the scaled energies is an integral surface energy of the form above

Low-Dimensional Spin Systems: Hidden Symmetries, Conformal Field Theories and Numerical Checks

We review here some general properties of antiferromagnetic Heisenberg spin chains, emphasizing and discussing the role of hidden symmetries in the classification of the various phases of the models. We present also some recent results that have been obtained with a combined use of Conformal Field Theory and of numerical Density Matrix Renormalization Group techniques.Comment: To be published in the proceedings of the XIII Conference on "Symmetries in Physics", held in Bregenz (Voralberg, Austria), 21-24/7/2003. Plain LaTeX2e, 4 EPS figure

Interplay between the edge-state magnetism and long-range Coulomb interaction in zigzag graphene nanoribbons: quantum Monte Carlo study

We perform projective quantum Monte Carlo simulations of zigzag graphene nanoribbons within a realistic model with long-range Coulomb interactions. Increasing the relative strength of nonlocal interactions with respect to the on-site repulsion does not generate a phase transition but has a number of nontrivial effects. At the single-particle level we observe a marked enhancement of the Fermi velocity at the Dirac points. At the two-particle level, spin- and charge-density-wave fluctuations compete. As a consequence, the edge magnetic moment is reduced but the edge dispersion relation increases in the sense that the single-particle gap at momentum $q=\pi/|{\pmb a}_1|$ grows. We attribute this to nonlocal charge fluctuations which assist the spin fluctuations to generate the aforementioned gap. In contrast, the net result of the interaction-induced renormalization of different energy scales is a constant spin-wave velocity of the edge modes. However, since the particle-hole continuum is shifted to higher energies---due to the renormalization of the Fermi velocity---Landau damping is reduced. As a result, a roughly linear spin-wave-like mode at the edge spreads out through a larger part of the Brillouin zone.Comment: 11 pages, 11 figures, comment about doped nanoribbon

Quantum criticalities in a two-leg antiferromagnetic S=1/2 ladder induced by a staggered magnetic field

We study a two-leg antiferromagnetic spin-1/2 ladder in the presence of a staggered magnetic field. We consider two parameter regimes: strong (weak) coupling along the legs and weak (strong) coupling along the rungs. In both cases, the staggered field drives the Haldane spin-liquid phase of the ladder towards a Gaussian quantum criticality. In a generalized spin ladder with a non-Haldane, spontaneously dimerized phase, the staggered magnetic field induces an Ising quantum critical regime. In the vicinity of the critical lines, we derive low-energy effective field theories and use these descriptions to determine the dynamical response functions, the staggered spin susceptibility and the string order parameter.Comment: 29 pages of revtex, 10 figure

Does the XY Model have an integrable continuum limit?

The quantum field theory describing the massive O(2) nonlinear sigma-model is investigated through two non-perturbative constructions: The form factor bootstrap based on integrability and the lattice formulation as the XY model. The S-matrix, the spin and current two-point functions, as well as the 4-point coupling are computed and critically compared in both constructions. On the bootstrap side a new parafermionic super selection sector is found; in the lattice theory a recent prediction for the (logarithmic) decay of lattice artifacts is probed.Comment: 69 pages, 18 figures. Equation (3.20) correcte

Fermionic SK-models with Hubbard interaction: Magnetism and electronic structure

Models with range-free frustrated Ising spin- and Hubbard interaction are treated exactly by means of the discrete time slicing method. Critical and tricritical points, correlations, and the fermion propagator, are derived as a function of temperature T, chemical potential \mu, Hubbard coupling U, and spin glass energy J. The phase diagram is obtained. Replica symmetry breaking (RSB)-effects are evaluated up to four-step order (4RSB). The use of exact relations together with the 4RSB-solutions allow to model exact solutions by interpolation. For T=0, our numerical results provide strong evidence that the exact density of states in the spin glass pseudogap regime obeys \rho(E)=const |E-E_F| for energies close to the Fermi level. Rapid convergence of \rho'(E_F) under increasing order of RSB is observed. The leading term resembles the Efros-Shklovskii Coulomb pseudogap of localized disordered fermionic systems in 2D. Beyond half filling we obtain a quadratic dependence of the fermion filling factor on the chemical potential. We find a half filling transition between a phase for U>\mu, where the Fermi level lies inside the Hubbard gap, into a phase where \mu(>U) is located at the center of the upper spin glass pseudogap (SG-gap). For \mu>U the Hubbard gap combines with the lower one of two SG-gaps (phase I), while for \mu<U it joins the sole SG-gap of the half-filling regime (phase II). We predict scaling behaviour at the continuous half filling transition. Implications of the half-filling transition between the deeper insulating phase II and phase I for delocalization due to hopping processes in itinerant model extensions are discussed and metal-insulator transition scenarios described.Comment: 29 pages, 26 Figures, 4 jpeg- and 3 gif-Fig-files include

Quantum Field Theory for the Three-Body Constrained Lattice Bose Gas -- Part II: Application to the Many-Body Problem

We analyze the ground state phase diagram of attractive lattice bosons, which are stabilized by a three-body onsite hardcore constraint. A salient feature of this model is an Ising type transition from a conventional atomic superfluid to a dimer superfluid with vanishing atomic condensate. The study builds on an exact mapping of the constrained model to a theory of coupled bosons with polynomial interactions, proposed in a related paper [11]. In this framework, we focus by analytical means on aspects of the phase diagram which are intimately connected to interactions, and are thus not accessible in a mean field plus spin wave approach. First, we determine shifts in the mean field phase border, which are most pronounced in the low density regime. Second, the investigation of the strong coupling limit reveals the existence of a new collective mode, which emerges as a consequence of enhanced symmetries in this regime. Third, we show that the Ising type phase transition, driven first order via the competition of long wavelength modes at generic fillings, terminates into a true Ising quantum critical point in the vicinity of half filling.Comment: 22 pages, 5 figure

Critical properties of the double-frequency sine-Gordon model with applications

We study the properties of the double-frequency sine--Gordon model in the vicinity of the Ising quantum phase transition displayed by this model. Using a mapping onto a generalised lattice quantum Ashkin-Teller model, we obtain critical and nearly-off-critical correlation functions of various operators. We discuss applications of the double-sine-Gordon model to one-dimensional physical systems, like spin chains in a staggered external field and interacting electrons in a staggered potential.Comment: 51 pages, Latex fil