The Hubbard model with smooth boundary conditions

We apply recently developed smooth boundary conditions to the quantum Monte Carlo simulation of the two-dimensional Hubbard model. At half-filling, where there is no sign problem, we show that the thermodynamic limit is reached more rapidly with smooth rather than with periodic or open boundary conditions. Away from half-filling, where ordinarily the simulation cannot be carried out at low temperatures due to the existence of the sign problem, we show that smooth boundary conditions allow us to reach significantly lower temperatures. We examine pairing correlation functions away from half-filling in order to determine the possible existence of a superconducting state. On a $10\times 10$ lattice for $U=4$, at a filling of $\langle n \rangle = 0.87$ and an inverse temperature of $\beta=10$, we did find enhancement of the $d$-wave correlations with respect to the non-interacting case, a possible sign of $d$-wave superconductivity.Comment: 16 pages RevTeX, 9 postscript figures included (Figure 1 will be faxed on request

Revealing Topological Structure in the SU(2) Vacuum

In this paper we derive a simple parametrization of the cycling method developed by us in our earlier work. The new method, called renormalization group (RG) mapping, consists of a series of carefully tuned APE-smearing steps. We study the relation between cycling and RG mapping. We also investigate in detail how smooth instantons and instanton-anti-instanton pairs behave under the RG mapping transformation. We use the RG mapping technique to study the topological susceptibility and instanton size distribution of SU(2) gauge theory. We find scaling in both quantities in a wide range of coupling values. Our result for the topological susceptibility, chi^1/4=220(6) MeV, agrees with our earlier results.Comment: 28 pages, LaTeX, 7 eps figure

Boosting the accuracy of SPH techniques: Newtonian and special-relativistic tests

We study the impact of different discretization choices on the accuracy of SPH and we explore them in a large number of Newtonian and special-relativistic benchmark tests. As a first improvement, we explore a gradient prescription that requires the (analytical) inversion of a small matrix. For a regular particle distribution this improves gradient accuracies by approximately ten orders of magnitude and the SPH formulations with this gradient outperform the standard approach in all benchmark tests. Second, we demonstrate that a simple change of the kernel function can substantially increase the accuracy of an SPH scheme. While the "standard" cubic spline kernel generally performs poorly, the best overall performance is found for a high-order Wendland kernel which allows for only very little velocity noise and enforces a very regular particle distribution, even in highly dynamical tests. Third, we explore new SPH volume elements that enhance the treatment of fluid instabilities and, last, but not least, we design new dissipation triggers. They switch on near shocks and in regions where the flow --without dissipation-- starts to become noisy. The resulting new SPH formulation yields excellent results even in challenging tests where standard techniques fail completely.Comment: accepted for publication in MNRA

Cooling, Physical Scales and Topology

We develop a cooling method controlled by a physical cooling radius that defines a scale below which fluctuations are smoothed out while leaving physics unchanged at all larger scales. We apply this method to study topological properties of lattice gauge theories, in particular the behaviour of instantons, dislocations and instanton--anti-instanton pairs. Monte Carlo results for the SU(2) topology are presented. We find that the method provides a means to prevent instanton--anti-instanton annihilation under cooling. While the instanton sizes are largely independent from the smoothing scale, the density and pair separations are determined by the particular choice made for this quantity. We discuss the questions this raises for the "physicality" of these concepts.Comment: 25 pages, 8 figures, minor corrections, references adde

Smoothed Particle Magnetohydrodynamics III. Multidimensional tests and the div B = 0 constraint

In two previous papers (Price & Monaghan 2004a,b) (papers I,II) we have described an algorithm for solving the equations of Magnetohydrodynamics (MHD) using the Smoothed Particle Hydrodynamics (SPH) method. The algorithm uses dissipative terms in order to capture shocks and has been tested on a wide range of one dimensional problems in both adiabatic and isothermal MHD. In this paper we investigate multidimensional aspects of the algorithm, refining many of the aspects considered in papers I and II and paying particular attention to the code's ability to maintain the div B = 0 constraint associated with the magnetic field. In particular we implement a hyperbolic divergence cleaning method recently proposed by Dedner et al. (2002) in combination with the consistent formulation of the MHD equations in the presence of non-zero magnetic divergence derived in papers I and II. Various projection methods for maintaining the divergence-free condition are also examined. Finally the algorithm is tested against a wide range of multidimensional problems used to test recent grid-based MHD codes. A particular finding of these tests is that in SPMHD the magnitude of the divergence error is dependent on the number of neighbours used to calculate a particle's properties and only weakly dependent on the total number of particles. Whilst many improvements could still be made to the algorithm, our results suggest that the method is ripe for application to problems of current theoretical interest, such as that of star formation.Comment: Here is the latest offering in my quest for a decent SPMHD algorithm. 26 pages, 15 figures, accepted for publication in MNRAS. Version with high res figures available from http://www.astro.ex.ac.uk/people/dprice/pubs/spmhd/spmhdpaper3.pd

The QCD vacuum

We review issues involved in understanding the vacuum, long-distance and low-energy structure of non-Abelian gauge theories and QCD. The emphasis will be on the role played by instantons.Comment: 12p with 7 figs. Review presented at Lattice'97, Edinburgh, 22-26 July, 199

Comparison of different lattice definitions of the topological charge

We present a comparison of different definitions of the topological charge on the lattice, using a small-volume ensemble with 2 flavours of dynamical twisted mass fermions. The investigated definitions are: index of the overlap Dirac operator, spectral projectors, spectral flow of the Hermitian Wilson-Dirac operator and field theoretic with different kinds of smoothing of gauge fields (HYP and APE smearings, gradient flow, cooling). We also show some results on the topological susceptibility.Comment: 7 pages, 2 figures, presented at the 32nd International Symposium on Lattice Field Theory (Lattice 2014), 23-28 June 2014, Columbia University, New York, NY, US