Gaussian Summation: An Exponentially Converging Summation Scheme

Gaussian Quadrature is a well known technique for numerical integration. Recently Gaussian quadrature with respect to discrete measures corresponding to finite sums have found some new interest. In this paper we apply these ideas to infinite sums in general and give an explicit construction for the weights and abscissae of GAUSSIAN SUMMATION formulas. The abscissae of the Gaussian summation have a very interesting asymptotic distribution function with a (cusp) singularity. We apply the Gaussian summation technique to two problems which have been discussed in the literature. We find that the Gaussian summation has an extremely rapid convergence rate for the Hardy-Littlewood sum for a large range of parameters. For functions which are smooth but have a large scale, a, the error of Gaussian Summation shows exponential convergence as a function of summation points. The Gaussian summation achieves a given accuracy with a number of points proportional to the sqrt of the large scale whereas other summation schemes require at least a number of function evaluations proportional to the scale.Comment: 14 pages, 4 figures This manuscript is written by a non-expert (me). Any advice on where to publish these results or any other useful comments is appreciate

Hankel determinants of Dirichlet series

We derive a general expression for the Hankel determinants of a Dirichlet series F(s) and derive the asymptotic behavior for the special case that F(s) is the Riemann zeta function. In this case the Hankel determinant is a discrete analogue of the Selberg integral and can be viewed as a matrix integral with discrete measure. We briefly comment on its relation to Plancherel measures.Comment: 11 pages, 1 figur

What is wrong with paramagnons?

Systems with itinerant fermions close to a zero temperature quantum phase transition like the high temperature superconductors exhibit unusual non-Fermi liquid properties. The interaction of the long-range and low-energy fluctuations of the incipient order with the fermions modify the dynamical properties of the fermions strongly by inducing effective long-range interactions. Close to the transition the interaction of the order parameter fluctuations becomes important. In this paper we discuss the physics of the non-Gaussian order parameter fluctuations on the electronic spectrum and illustrate their effect by considering the charge-density-wave transition and the phase fluctuations in a two-dimensional d-wave superconductor.Comment: 12 pages, 5 figure

Dynamics and Thermodynamics of the Bose-Hubbard model

We report results from a systematic analytic strong-coupling expansion of the Bose-Hubbard model in one and two spatial dimensions. We obtain numerically exact results for the dispersion of single particle and single hole excitations in the Mott insulator. The boundary of the Mott phase can be determined with previously unattainable accuracy in one and two dimensions. In one dimension we observe the occurrence of reentrant behavior from the compressible to the insulating phase in a region close to the critical point which was conjectured in earlier work. Our calculation can be used as a benchmark for the development of new numerical techniques for strongly correlated systems.Comment: RevTex, 4 pages, 4 figures (eps format

The Complexity of Equilibria for Risk-Modeling Valuations

We study the complexity of deciding the existence of mixed equilibria for minimization games where players use valuations other than expectation to evaluate their costs. We consider risk-averse players seeking to minimize the sum ${\mathsf{V}} = {\mathsf{E}} + {\mathsf{R}}$ of expectation ${\mathsf{E}}$ and a risk valuation ${\mathsf{R}}$ of their costs; ${\mathsf{R}}$ is non-negative and vanishes exactly when the cost incurred to a player is constant over all choices of strategies by the other players. In a ${\mathsf{V}}$-equilibrium, no player could unilaterally reduce her cost. Say that ${\mathsf{V}}$ has the Weak-Equilibrium-for-Expectation property if all strategies supported in a player's best-response mixed strategy incur the same conditional expectation of her cost. We introduce ${\mathsf{E}}$-strict concavity and observe that every ${\mathsf{E}}$-strictly concave valuation has the Weak-Equilibrium-for-Expectation property. We focus on a broad class of valuations shown to have the Weak-Equilibrium-for-Expectation property, which we exploit to prove two main complexity results, the first of their kind, for the two simplest cases of the problem: games with two strategies, or games with two players. For each case, we show that deciding the existence of a ${\mathsf{V}}$-equilibrium is strongly ${\mathcal{NP}}$-hard for certain choices of significant valuations (including variance and standard deviation).Comment: 49 page

Mott transition in the Hubbard model on the hyper-kagome lattice

Motivated by recent experiment on the Na$_4$Ir$_3$O$_8$ compound we study the Hubbard model on the "hyper-kagome lattice", which forms a three-dimensional network of corner sharing triangles, using dynamical cluster approximation (DCA) method with $N_c$=12 combined with the continuous-time quantum Monte Carlo (CT QMC) method. The system undergoes a Mott transition if the Hubbard interaction $U/W$ ($W$ is the bandwidth) exceeds the value of 1.2 for $T$=0.1 and displays reentrant behavior due to competition between the magnetic correlation and the kinetic energy of electrons due to the geometrical frustration. We observe a "critical slowing down" of the double occupancy which shows evidence of a continuous transition. The nearest-neighbor and next nearest-neighbor spin-spin correlations indicate a paramagnetic metallic state in the weak-coupling regime and an antiferromagnetic (AF) Mott insulator in the strong-coupling regime within the temperature range which we can access with our numerical tools.Comment: 4 pages and 6 figure

Nonequilibrium dynamical mean-field theory of a strongly correlated system

We present a generalized dynamical mean-field approach for the nonequilibrium physics of a strongly correlated system in the presence of a time-dependent external field. The Keldysh Green's function formalism is used to study the nonequilibrium problem. We derive a closed set of self-consistency equations in the case of a driving field with frequency Omega and wave vector q. We present numerical results for the local frequency-dependent Green's function and the self-energy for different values of the field amplitude in the case of a uniform external field using the iterated perturbation theory. In addition, an expression for the frequency-dependent optical conductivity of the Hubbard model with a driving external field is derived.Comment: 4 pages, 7 figure

Doping on the kagome lattice: A variational Monte-Carlo study of the t−Jt-J model

We study doping on the Kagome lattice by exploring the $t-J$-model with variational Monte-Carlo. We use a number of Gutzwiller projected spin-liquid and valence bond-crystal states and compare their energies at several system-sizes. We find that introducing mobile holes drives the system away from the Spin-Liquid state proposed by Ran et al for the undoped system, towards a uniform state with zero-flux. On top of the uniform-state a VBC of the Hastings-type is formed for low doping. The results are compared to exact diagonalization on small clusters. This agrees well.Comment: 5 pages, 7 figure

A numerical exact solution of the Bose-Hubbard model

In this paper we report results from a systematic strong-coupling expansion of the Bose-Hubbard model in one and two spatial dimensions. We obtain numerically exact results for the structure factor and the spectrum of single particle and single hole excitations in the Mott insulator. This enables the determination of the zero-temperature phase diagram and the location of the critical endpoints of the Mott lobes. In one dimension we confirm the occurrence of reentrance behavior from the compressible to the insulating phase in a region close to the critical point.Comment: revtex, postscript figure

Strong coupling expansion for bosons on the kagome lattice

We use series expansion techniques for analyzing properties of the phase transition between the Mott insulating and superfluid phase for bosons on the kagome lattice, and the multicritical point in the ground-state phase diagram for unit-filling is calculated. It is seen that of the clusters that contribute with non-zero weights to the ground state energy, many contain rings. The decay exponents of ground state correlations are also obtained within the Mott phase. For single-particle excited states, quasiparticle dispersion and effective masses for particles and holes are computed along certain symmetry cuts in the first Brillouin zone. Furthermore at eighth order, the coherence-length critical exponent is found to be comparably close to that of the 3D XY model.Comment: 5 pages, 4 figures: added reference, consolidated figures, included correlator tabl