Root multiplicities and number of nonzero coefficients of a polynomial

It is known that the weight (that is, the number of nonzero coefficients) of a univariate polynomial over a field of characteristic zero is larger than the multiplicity of any of its nonzero roots. We extend this result to an appropriate statement in positive characteristic. Furthermore, we present a new proof of the original result, which produces also the exact number of monic polynomials of a given degree for which the bound is attained. A similar argument allows us to determine the number of monic polynomials of a given degree, multiplicity of a given nonzero root, and number of nonzero coefficients, over a finite field of characteristic larger than the degree.Comment: 6 pages. Minor change from previous version: added Example 6, illustrating the difficulties arising when one tries to relax the hypothesis n<p of Theorem

Comprehensive quantum Monte Carlo study of the quantum critical points in planar dimerized/quadrumerized Heisenberg models

We study two planar square lattice Heisenberg models with explicit dimerization or quadrumerization of the couplings in the form of ladder and plaquette arrangements. We investigate the quantum critical points of those models by means of (stochastic series expansion) quantum Monte Carlo simulations as a function of the coupling ratio $\alpha = J^\prime/J$. The critical point of the order-disorder quantum phase transition in the ladder model is determined as $\alpha_\mathrm{c} = 1.9096(2)$ improving on previous studies. For the plaquette model we obtain $\alpha_\mathrm{c} = 1.8230(2)$ establishing a first benchmark for this model from quantum Monte Carlo simulations. Based on those values we give further convincing evidence that the models are in the three-dimensional (3D) classical Heisenberg universality class. The results of this contribution shall be useful as references for future investigations on planar Heisenberg models such as concerning the influence of non-magnetic impurities at the quantum critical point.Comment: 10+ pages, 7 figures, 4 table

Evidence of columnar order in the fully frustrated transverse field Ising model on the square lattice

Using extensive classical and quantum Monte Carlo simulations, we investigate the ground-state phase diagram of the fully frustrated transverse field Ising model on the square lattice. We show that pure columnar order develops in the low-field phase above a surprisingly large length scale, below which an effective U(1) symmetry is present. The same conclusion applies to the Quantum Dimer Model with purely kinetic energy, to which the model reduces in the zero-field limit, as well as to the stacked classical version of the model. By contrast, the 2D classical version of the model is shown to develop plaquette order. Semiclassical arguments show that the transition from plaquette to columnar order is a consequence of quantum fluctuations.Comment: 5 pages (including Supplemental Material), 5 figure

Casimir-Lifshitz Force Out of Thermal Equilibrium and Asymptotic Nonadditivity

We investigate the force acting between two parallel plates held at different temperatures. The force reproduces, as limiting cases, the well-known Casimir-Lifshitz surface-surface force at thermal equilibrium and the surface-atom force out of thermal equilibrium recently derived by M. Antezza et al., Phys. Rev. Lett. 95, 113202 (2005). The asymptotic behavior of the force at large distances is explicitly discussed. In particular when one of the two bodies is a rarefied gas the force is not additive, being proportional to the square root of the density. Nontrivial crossover regions at large distances are also identified

Moving forward with combinatorial interaction testing

Combinatorial interaction testing (CIT) is an efficient and effective method of detecting failures that are caused by the interactions of various system input parameters. In this paper, we discuss CIT, point out some of the difficulties of applying it in practice, and highlight some recent advances that have improved CIT’s applicability to modern systems. We also provide a roadmap for future research and directions; one that we hope will lead to new CIT research and to higher quality testing of industrial systems

Seeing zeros of random polynomials: quantized vortices in the ideal Bose gas

We propose a physical system allowing one to experimentally observe the distribution of the complex zeros of a random polynomial. We consider a degenerate, rotating, quasi-ideal atomic Bose gas prepared in the lowest Landau level. Thermal fluctuations provide the randomness of the bosonic field and of the locations of the vortex cores. These vortices can be mapped to zeros of random polynomials, and observed in the density profile of the gas.Comment: 4 page

Optimized auxiliary oscillators for the simulation of general open quantum systems

A method for the systematic construction of few-body damped harmonic oscillator networks accurately reproducing the effect of general bosonic environments in open quantum systems is presented. Under the sole assumptions of a Gaussian environment and regardless of the system coupled to it, an algorithm to determine the parameters of an equivalent set of interacting damped oscillators obeying a Markovian quantum master equation is introduced. By choosing a suitable coupling to the system and minimizing an appropriate distance between the two-time correlation function of this effective bath and that of the target environment, the error induced in the reduced dynamics of the system is brought under rigorous control. The interactions among the effective modes provide remarkable flexibility in replicating non-Markovian effects on the system even with a small number of oscillators, and the resulting Lindblad equation may therefore be integrated at a very reasonable computational cost using standard methods for Markovian problems, even in strongly non-perturbative coupling regimes and at arbitrary temperatures including zero. We apply the method to an exactly solvable problem in order to demonstrate its accuracy, and present a study based on current research in the context of coherent transport in biological aggregates as a more realistic example of its use; performance and versatility are highlighted, and theoretical and numerical advantages over existing methods, as well as possible future improvements, are discussed.Comment: 23 + 9 pages, 11 + 2 figures. No changes from previous version except publication info and updated author affiliation

Collapse and revival in inter-band oscillations of a two-band Bose-Hubbard model

We study the effect of a many-body interaction on inter-band oscillations in a two-band Bose-Hubbard model with external Stark force. Weak and strong inter-band oscillations are observed, where the latter arise from a resonant coupling of the bands. These oscillations collapse and revive due to a weak two-body interaction between the atoms. Effective models for oscillations in and out of resonance are introduced that provide predictions for the system's behaviour, particularly for the time-scales for the collapse and revival of the resonant inter-band oscillations.Comment: 10 pages, 5 figure

Effect of the Casimir-Polder force on the collective oscillations of a trapped Bose-Einstein condensate

We calculate the effect of the interaction between an optically active material and a Bose-Einstein condensate on the collective oscillations of the condensate. We provide explicit expressions for the frequency shift of the center of mass oscillation in terms of the potential generated by the substrate and of the density profile of the gas. The form of the potential is discussed in details and various regimes (van der Waals-London, Casimir-Polder and thermal regimes) are identified as a function of the distance of atoms from the surface. Numerical results for the frequency shifts are given for the case of a sapphire dielectric substrate interacting with a harmonically trapped condensate of $^{87}$Rb atoms. We find that at distances of $4-8 \mu m$, where thermal effects become visible, the relative frequency shifts produced by the substrate are of the order $10^{-4}$ and hence accessible experimentally. The effects of non linearities due to the finite amplitude of the oscillation are explicitly discussed. Predictions are also given for the radial breathing mode.Comment: 28 pages, 10 figures. Submitted to PR