Improved bounds on the number of ternary square-free words

Improved upper and lower bounds on the number of square-free ternary words are obtained. The upper bound is based on the enumeration of square-free ternary words up to length 110. The lower bound is derived by constructing generalised Brinkhuis triples. The problem of finding such triples can essentially be reduced to a combinatorial problem, which can efficiently be treated by computer. In particular, it is shown that the number of square-free ternary words of length n grows at least as 65^(n/40), replacing the previous best lower bound of 2^(n/17).Comment: 17 pages, AMS LaTeX. Paper has been completely rewritten and comprises new results on both lower and upper bounds. The Mathematica program mentioned in the article can be downloaded at http://mcs.open.ac.uk/ugg2/wordcomb/brinkhuistriples.

N=2 central charge superspace and a minimal supergravity multiplet

We extend the notion of central charge superspace to the case of local supersymmetry. Gauged central charge transformations are identified as diffeomorphisms at the same footing as space-time diffeomorphisms and local supersymmetry transformations. Given the general structure we then proceed to the description of a particular vector-tensor supergravity multiplet of 24+24 components, identified by means of rather radical constraints

Probing Phases and Quantum Criticality using Deviations from the Local Fluctuation-Dissipation Theorem

Introduction Cold atomic gases in optical lattices are emerging as excellent laboratories for testing models of strongly interacting particles in condensed matter physics. Currently, one of the major open questions is how to obtain the finite temperature phase diagram of a given quantum Hamiltonian directly from experiments. Previous work in this direction required quantum Monte Carlo simulations to directly model the experimental situation in order to extract quantitative information, clearly defeating the purpose of an optical lattice emulator. Here we propose a new method that utilizes deviations from a local fluctuation dissipation theorem to construct a finite temperature phase diagram, for the first time, from local observables accessible by in situ experimental observations. Our approach extends the utility of the fluctuation-dissipation theorem from thermometry to the identification of quantum phases, associated energy scales and the quantum critical region. We test our ideas using state-of-the-art large-scale quantum Monte Carlo simulations of the two-dimensional Bose Hubbard model.Comment: 7 pages; 4 figures; also see supplementary material of 7 pages with 3 figure

Spectral weight redistribution in strongly correlated bosons in optical lattices

We calculate the single-particle spectral function for the one-band Bose-Hubbard model within the random phase approximation (RPA). In the strongly correlated superfluid, in addition to the gapless phonon excitations, we find extra gapped modes which become particularly relevant near the superfluid-Mott quantum phase transition (QPT). The strength in one of the gapped modes, a precursor of the Mott phase, grows as the QPT is approached and evolves into a hole (particle) excitation in the Mott insulator depending on whether the chemical potential is above (below) the tip of the lobe. The sound velocity of the Goldstone modes remains finite when the transition is approached at a constant density, otherwise, it vanishes at the transition. It agrees well with Bogoliubov theory except close to the transition. We also calculate the spatial correlations for bosons in an inhomogeneous trapping potential creating alternating shells of Mott insulator and superfluid. Finally, we discuss the capability of the RPA approximation to correctly account for quantum fluctuations in the vicinity of the QPT.Comment: 14 pages, 12 figure

Tomographic RF Spectroscopy of a Trapped Fermi Gas at Unitarity

We present spatially resolved radio-frequency spectroscopy of a trapped Fermi gas with resonant interactions and observe a spectral gap at low temperatures. The spatial distribution of the spectral response of the trapped gas is obtained using in situ phase-contrast imaging and 3D image reconstruction. At the lowest temperature, the homogeneous rf spectrum shows an asymmetric excitation line shape with a peak at 0.48(4)$\epsilon_F$ with respect to the free atomic line, where $\epsilon_F$ is the local Fermi energy

Non-universal behavior for aperiodic interactions within a mean-field approximation

We study the spin-1/2 Ising model on a Bethe lattice in the mean-field limit, with the interaction constants following two deterministic aperiodic sequences: Fibonacci or period-doubling ones. New algorithms of sequence generation were implemented, which were fundamental in obtaining long sequences and, therefore, precise results. We calculate the exact critical temperature for both sequences, as well as the critical exponent $\beta$, $\gamma$ and $\delta$. For the Fibonacci sequence, the exponents are classical, while for the period-doubling one they depend on the ratio between the two exchange constants. The usual relations between critical exponents are satisfied, within error bars, for the period-doubling sequence. Therefore, we show that mean-field-like procedures may lead to nonclassical critical exponents.Comment: 6 pages, 7 figures, to be published in Phys. Rev.

Four types of special functions of G_2 and their discretization

Properties of four infinite families of special functions of two real variables, based on the compact simple Lie group G2, are compared and described. Two of the four families (called here C- and S-functions) are well known, whereas the other two (S^L- and S^S-functions) are not found elsewhere in the literature. It is shown explicitly that all four families have similar properties. In particular, they are orthogonal when integrated over a finite region F of the Euclidean space, and they are discretely orthogonal when their values, sampled at the lattice points F_M \subset F, are added up with a weight function appropriate for each family. Products of ten types among the four families of functions, namely CC, CS, SS, SS^L, CS^S, SS^L, SS^S, S^SS^S, S^LS^S and S^LS^L, are completely decomposable into the finite sum of the functions. Uncommon arithmetic properties of the functions are pointed out and questions about numerous other properties are brought forward.Comment: 18 pages, 4 figures, 4 table

Dilute Birman--Wenzl--Murakami Algebra and Dn+1(2)D^{(2)}_{n+1} models

A ``dilute'' generalisation of the Birman--Wenzl--Murakami algebra is considered. It can be ``Baxterised'' to a solution of the Yang--Baxter algebra. The $D^{(2)}_{n+1}$ vertex models are examples of corresponding solvable lattice models and can be regarded as the dilute version of the $B^{(1)}_{n}$ vertex models.Comment: 11 page

Fluxes and Warping for Gauge Couplings in F-theory

We compute flux-dependent corrections in the four-dimensional F-theory effective action using the M-theory dual description. In M-theory the 7-brane fluxes are encoded by four-form flux and modify the background geometry and Kaluza-Klein reduction ansatz. In particular, the flux sources a warp factor which also depends on the torus directions of the compactification fourfold. This dependence is crucial in the derivation of the four-dimensional action, although the torus fiber is auxiliary in F-theory. In M-theory the 7-branes are described by an infinite array of Taub-NUT spaces. We use the explicit metric on this geometry to derive the locally corrected warp factor and M-theory three-from as closed expressions. We focus on contributions to the 7-brane gauge coupling function from this M-theory back-reaction and show that terms quadratic in the internal seven-brane flux are induced. The real part of the gauge coupling function is modified by the M-theory warp factor while the imaginary part is corrected due to a modified M-theory three-form potential. The obtained contributions match the known weak string coupling result, but also yield additional terms suppressed at weak coupling. This shows that the completion of the M-theory reduction opens the way to compute various corrections in a genuine F-theory setting away from the weak string coupling limit.Comment: 46 page

Non-universality of artificial frustrated spin systems

Magnetic frustration effects in artificial kagome arrays of nanomagnets with out-of-plane magnetization are investigated using Magnetic Force Microscopy and Monte Carlo simulations. Experimental and theoretical results are compared to those found for the artificial kagome spin ice, in which the nanomagnets have in-plane magnetization. In contrast with what has been recently reported, we demonstrate that long range (i.e. beyond nearest-neighbors) dipolar interactions between the nanomagnets cannot be neglected when describing the magnetic configurations observed after demagnetizing the arrays using a field protocol. As a consequence, there are clear limits to any universality in the behavior of these two artificial frustrated spin systems. We provide arguments to explain why these two systems show striking similarities at first sight in the development of pairwise spin correlations.Comment: 7 pages, 6 figure