24 research outputs found
On the efficient numerical solution of lattice systems with low-order couplings
We apply the Quasi Monte Carlo (QMC) and recursive numerical integration
methods to evaluate the Euclidean, discretized time path-integral for the
quantum mechanical anharmonic oscillator and a topological quantum mechanical
rotor model. For the anharmonic oscillator both methods outperform standard
Markov Chain Monte Carlo methods and show a significantly improved error
scaling. For the quantum mechanical rotor we could, however, not find a
successful way employing QMC. On the other hand, the recursive numerical
integration method works extremely well for this model and shows an at least
exponentially fast error scaling
Lattice meets lattice: Application of lattice cubature to models in lattice gauge theory
High dimensional integrals are abundant in many fields of research including
quantum physics. The aim of this paper is to develop efficient recursive
strategies to tackle a class of high dimensional integrals having a special
product structure with low order couplings, motivated by models in lattice
gauge theory from quantum field theory. A novel element of this work is the
potential benefit in using lattice cubature rules. The group structure within
lattice rules combined with the special structure in the physics integrands may
allow efficient computations based on Fast Fourier Transforms. Applications to
the quantum mechanical rotor and compact lattice gauge theory in two and
three dimensions are considered
Overcoming the sign problem in 1-dimensional QCD by new integration rules with polynomial exactness
In this paper we describe a new integration method for the groups and , for which we verified numerically that it is polynomially exact for . The method is applied to the example of 1-dimensional QCD with a chemical potential. We explore, in particular, regions of the parameter space in which the sign problem appears due the presence of the chemical potential. While Markov Chain Monte Carlo fails in this region, our new integration method still provides results for the chiral condensate on arbitrary precision, demonstrating clearly that it overcomes the sign problem. Furthermore, we demonstrate that our new method leads to orders of magnitude reduced errors also in other regions of parameter space
Overcoming the sign problem in one-dimensional QCD by new integration rules with polynomial exactness
In this paper we describe a new integration method for the groups U(N) and SU(N), for which we verified numerically that it is polynomially exact for N≤3. The method is applied to the example of one-dimensional QCD with a chemical potential. We explore, in particular, regions of the parameter space in which the sign problem appears due the presence of the chemical potential. While Markov chain Monte Carlo fails in this region, our new integration method still provides results for the chiral condensate on arbitrary precision, demonstrating clearly that it overcomes the sign problem. Furthermore, we demonstrate that also in other regions of parameter space our new method leads to errors which are reduced by orders of magnitude
Lattice field computations via recursive numerical integration
We investigate the application of efficient recursive numerical integration strategies to models in lattice gauge theory from quantum field theory. Given the coupling structure of the physics problems and the group structure within lattice cubature rules for numerical integration, we show how to approach these problems efficiently by means of Fast Fourier Transform techniques. In particular, we consider applications to the quantum mechanical rotor and compact lattice gauge theory, where the physical dimensions are two and three. This proceedings article reviews our results presented in J. Comput. Phys 443 (2021) 110527