Kurt Langfeld
University of Leeds, Mathematics, Faculty Member
- University of Liverpool, Mathematical Sciences, Faculty Memberadd
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We develop a free surface tracking solver for numerical simulation of unsteady irrotational fully non-linear water waves in a freely available open-source computational fluid dynamics toolbox OpenFOAM R ©-Ext, which is community-driven... more
We develop a free surface tracking solver for numerical simulation of unsteady irrotational fully non-linear water waves in a freely available open-source computational fluid dynamics toolbox OpenFOAM R ©-Ext, which is community-driven release of OpenFOAM R ©. The solver is based on the solution of the Laplacian of the velocity potential with moving free surface. The free surface is tracked by solving the kinematic boundary condition based on the normal flux out of the surface. We also develop the necessary boundary conditions for the realistic wave generation at inlet and the absorption boundary condition at the outlet boundary. To avoid numerical instability, a 5-point smoothing technique is used to smooth the free surface elevation. Solution of Laplace’s equation for the velocity potential, the non-linear free surface boundary conditions, the wave generation and the absorption boundary conditions are all not part of the standard OpenFOAM R © distribution. The potential flow solve...
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In this paper, we develop the Virtual Source Method for simulation of incompressible and irrotational fluid flows. The method is based upon the integral equations derived by using Green’s identity with Laplace’s equation for the velocity... more
In this paper, we develop the Virtual Source Method for simulation of incompressible and irrotational fluid flows. The method is based upon the integral equations derived by using Green’s identity with Laplace’s equation for the velocity potential. The velocity potential within the fluid domain is completely determined by the potential on a virtual boundary located above the fluid. This avoids the need to evaluate singular integrals. Furthermore, the solution method developed here is meshless in space in that discretisation is in terms of the spectral components of the solution along this virtual boundary. These are determined by specifying non-linear boundary conditions on the velocity potential on the air/water surface using Bernoulli’s equation. A fourth-order Runge-Kutta procedure is used to update the spectral components in time. The method is used to model high-amplitude standing waves and sloshing. Results are compared with theory where applicable and some interesting physica...
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Although Monte Carlo calculations using Importance Sampling have matured into the most widely employed method for determining first principle results in QCD, they spectacularly fail for theories with a sign problem or for which certain... more
Although Monte Carlo calculations using Importance Sampling have matured into the most widely employed method for determining first principle results in QCD, they spectacularly fail for theories with a sign problem or for which certain rare configurations play an important role. Non-Markovian Random walks, based upon iterative refinements of the density-of-states, overcome such overlap problems. I will review the Linear Logarithmic Relaxation (LLR) method and, in particular, focus onto ergodicity and exponential error suppression. Applications include the high-state Potts model, SU(2) and SU(3) Yang-Mills theories as well as a quantum field theory with a strong sign problem: QCD at finite densities of heavy quarks.
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Quantum field theories at finite matter densities generically possess a partition function that is exponentially suppressed with the volume compared to that of the phase quenched analogue. The smallness arises from an almost uniform... more
Quantum field theories at finite matter densities generically possess a partition function that is exponentially suppressed with the volume compared to that of the phase quenched analogue. The smallness arises from an almost uniform distribution for the phase of the fermion determinant. Large cancellations upon integration is the origin of a poor signal to noise ratio. We study three alternatives for this integration: the Gaussian approximation, the "telegraphic" approximation, and a novel expansion in terms of theory-dependent moments and universal coefficients. We have tested the methods for QCD at finite densities of heavy quarks. We find that for two of the approximations the results are extremely close - if not identical - to the full answer in the strong sign problem regime.
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Improved actions in SU(2) and SU(3) lattice gauge theories are investigated with an emphasis on asymptotic scaling. A new scheme for tadpole improvement is proposed. The standard but heuristic tadpole improvement emerges from a mean field... more
Improved actions in SU(2) and SU(3) lattice gauge theories are investigated with an emphasis on asymptotic scaling. A new scheme for tadpole improvement is proposed. The standard but heuristic tadpole improvement emerges from a mean field approximation from the new approach. Scaling is investigated by means of the large distance static quark potential. Both, the generic and the new tadpole scheme yield significant improvements on asymptotic scaling when compared with loop improved actions. A study of the rotational symmetry breaking terms, however, reveals that only the new improvement scheme efficiently eliminates the leading irrelevant term from the action.
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SU(2) lattice gauge theory is investigated where the traces of the Wilson lines at any lattice point and along each direction is constrained to zero. Hence, each of the lattice configurations possesses a vanishing density of heavy (anti-)... more
SU(2) lattice gauge theory is investigated where the traces of the Wilson lines at any lattice point and along each direction is constrained to zero. Hence, each of the lattice configurations possesses a vanishing density of heavy (anti-) quarks. The results are compared with those of pure SU(2) gauge theory which can be interpreted as the grand canonical realization of the heavy quark theory where only the ensemble average of the heavy quark density vanishes. The static quark anti-quark potential of the constrained theory is obtained from (spatially smeared) Wilson loops at zero temperature. We find that the potential coincides with that of pure SU(2) gauge theory (without constraints). Hence, the familiar "running" of the lattice spacing with beta is recovered.
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Z3 gauge theory with dynamical (bosonic) matter is studied in 4 dimensions with a finite chemical potential. This theory could be viewed as an effective theory describing the centre vortex picture of QCD colour confinement, but it is... more
Z3 gauge theory with dynamical (bosonic) matter is studied in 4 dimensions with a finite chemical potential. This theory could be viewed as an effective theory describing the centre vortex picture of QCD colour confinement, but it is studied here with local interactions as theory in its own right. It is shown that the sign-problem can be solved by dualisation. The dual theory is derived: the pure gauge sector is a theory of closed membranes with Nambu-Goto action, and matter is described by open branes bounded by closed matter loops. The brane theory is simulated with Monte-Carlo techniques. Some evidence is found that the theory possesses a weakly-renormalisable phase with the scale set by a mass gap. Deconfinement at low temperatures and finite chemical potentials appears as a percolation transition for matter loops.
In Wang-Landau type algorithms, Monte-Carlo updates are performed with respect to the density of states, which is iteratively refined during simulations. The partition function and thermodynamic observables are then obtained by standard... more
In Wang-Landau type algorithms, Monte-Carlo updates are performed with respect to the density of states, which is iteratively refined during simulations. The partition function and thermodynamic observables are then obtained by standard integration. In this work, our recently introduced method in this class (the LLR approach) is analysed and further developed. Our approach is a histogram free method particularly suited for systems with continuous degrees of freedom giving rise to a continuum density of states, as it is commonly found in Lattice Gauge Theories and in some Statistical Mechanics systems. We show that the method possesses an exponential error suppression that allows us to estimate the density of states over several orders of magnitude with nearly-constant relative precision. We explain how ergodicity issues can be avoided and how expectation values of arbitrary observables can be obtained within this framework. We then demonstrate the method using Compact U(1) Lattice G...
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Quantum field theories (QFTs) at finite densities of matter generically involve complex actions. Standard Monte-Carlo simulations based upon importance sampling, which have been producing quantitative first principle results in particle... more
Quantum field theories (QFTs) at finite densities of matter generically involve complex actions. Standard Monte-Carlo simulations based upon importance sampling, which have been producing quantitative first principle results in particle physics for almost fourty years, cannot be applied in this case. Various strategies to overcome this so-called Sign Problem or Complex Action Problem were proposed during the last thirty years. We here review the sign problem in lattice field theories, focussing on two more recent methods: Dualization to world-line type of representations and the density-of-states approach.
Although Monte Carlo calculations using Importance Sampling have matured into the most widely employed method for determining first principle results in QCD, they spectacularly fail for theories with a sign problem or for which certain... more
Although Monte Carlo calculations using Importance Sampling have matured into the most widely employed method for determining first principle results in QCD, they spectacularly fail for theories with a sign problem or for which certain rare configurations play an important role. Non-Markovian Random walks, based upon iterative refinements of the density-of-states, overcome such overlap problems. I will review the Linear Logarithmic Relaxation (LLR) method and, in particular, focus onto ergodicity and exponential error suppression. Applications include the high-state Potts model, SU(2) and SU(3) Yang-Mills theories as well as a quantum field theory with a strong sign problem: QCD at finite densities of heavy quarks.
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Recently, a new and efficient algorithm (the LLR method) has been proposed for computing densities of states in statistical systems and gauge theories. In this talk, we explore whether this novel density of states method can be applied to... more
Recently, a new and efficient algorithm (the LLR method) has been proposed for computing densities of states in statistical systems and gauge theories. In this talk, we explore whether this novel density of states method can be applied to numerical computations of observables in systems for which the action is complex. To this purpose, we introduce a generalised density of states, in terms of which integrals of oscillating observables can be determined semi-analytically, and we define a strategy to compute it with the LLR method. As a case study, we apply these ideas to the Z(3) spin model at finite density, finding a remarkable agreement of our results for the phase twist with those obtained with the worm algorithm for all explored chemical potentials, including values for which there are cancellations over sixteen orders of magnitude. These findings open new perspectives for dealing with the sign problem on physically more relevant systems.
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We present a novel algorithm to compute the density of states, which is proven to converge to the correct result. The algorithm is very general and can be applied to a wide range of models, in the frameworks of Statistical Mechanics and... more
We present a novel algorithm to compute the density of states, which is proven to converge to the correct result. The algorithm is very general and can be applied to a wide range of models, in the frameworks of Statistical Mechanics and Lattice Gauge Theory. All the thermal or quantum expectation values can then be obtained by a simple integration of the density of states. As an application, a numerical study of 4d U(1) compact lattice gauge theory is presented.
QCD at finite densities of heavy quarks is investigated using the density-of-states method. The phase factor expectation value of the quark determinant is calculated to unprecedented precision as a function of the chemical potential.... more
QCD at finite densities of heavy quarks is investigated using the density-of-states method. The phase factor expectation value of the quark determinant is calculated to unprecedented precision as a function of the chemical potential. Results are validated using those from a reweighting approach where the latter can produce a significant signal-to-noise ratio. We confirm the particle-hole symmetry at low temperatures, find a strong sign problem at intermediate values of the chemical potential, and an inverse Silver Blaze feature for chemical potentials close to the onset value: here, the phase quenched theory underestimates the density of the full theory.
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We consider two very different models of the flux tube linking two heavy quarks: a string linking the matter fields and a Coulombic description of two separately gauge invariant charges. We compare how close they are to the unknown true... more
We consider two very different models of the flux tube linking two heavy quarks: a string linking the matter fields and a Coulombic description of two separately gauge invariant charges. We compare how close they are to the unknown true ground state in compact U(1) and the SU(2) Higgs model. Simulations in compact U(1) show that the string description is better in the confined phase but the Coulombic description is best in the deconfined phase; the last result is shown to agree with analytical calculations. Surprisingly in the non-abelian theory the Coulombic description is better in both the Higgs and confined phases. This indicates a significant difference in the width of the flux tubes in the two theories.
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Topological configurations, monopoles and vortices, successfully describe quark confinement and the spontaneous breakdown of chiral symmetry. Despite their infinite action, these configurations are relevant due to a subtle cancellation... more
Topological configurations, monopoles and vortices, successfully describe quark confinement and the spontaneous breakdown of chiral symmetry. Despite their infinite action, these configurations are relevant due to a subtle cancellation between action and entropy. A natural explanation for this intrinsic fine-tuning is that smooth low action configurations exist which confine and which appear as singular topological objects in certain gauges. To reveal these confining semi-classical configurations, a new cooling method is proposed which largely reduces the action while preserv-ing the asymptotic quark-antiquark potential. First numerical results for a SU(2) gauge theory show that confining configurations with an average plaquette as high as 0.95 do exist.
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By using the method of center projection the center vortex part of the gauge field is isolated and its propagator is evaluated in the center Landau gauge, which minimizes the open 3-dimensional Dirac volumes of non-trivial center links... more
By using the method of center projection the center vortex part of the gauge field is isolated and its propagator is evaluated in the center Landau gauge, which minimizes the open 3-dimensional Dirac volumes of non-trivial center links bounded by the closed 2-dimensional center vortex surfaces. The center field propagator is found to dominate the gluon propagator (in Landau gauge) in the low momentum regime and to give rise to an OPE correction to the latter of √ σ/p 3.The screening mass of the center vortex field vanishes above the critical temperature of the deconfinement phase transition, which naturally explains the second order nature of this transition consistently with the vortex picture. Finally, the ghost propagator of maximal center gauge is found to be infrared finite and thus shows no signal of confinement. PoS(LAT2005)321
By using the method of center projection the center vortex part of the gauge field is isolated and its propagator is evaluated in the center Landau gauge, which minimizes the open 3-dimensional Dirac volumes of non-trivial center links... more
By using the method of center projection the center vortex part of the gauge field is isolated and its propagator is evaluated in the center Landau gauge, which minimizes the open 3-dimensional Dirac volumes of non-trivial center links bounded by the closed 2-dimensional center vortex surfaces. The center field propagator is found to dominate the gluon propagator (in Landau gauge) in the low momentum regime and to give rise to an OPE correction to the latter of $\sqrt{\sigma} /p^3 $.The screening mass of the center vortex field vanishes above the critical temperature of the deconfinement phase transition, which naturally explains the second order nature of this transition consistently with the vortex picture. Finally, the ghost propagator of maximal center gauge is found to be infrared finite and thus shows no signal of confinement.
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The coupled system of renormalized Dyson-Schwinger equations for the electron self-energy and the photon propagator are supplied with the tree level vertex as Ansatz for the renormalized three point function. The system is investigated... more
The coupled system of renormalized Dyson-Schwinger equations for the electron self-energy and the photon propagator are supplied with the tree level vertex as Ansatz for the renormalized three point function. The system is investigated numerically. In the case of a massive electron, the theory is “weakly renormalizable”, i.e. cutoff independent for values of the cutoff below an upper limit. In this regime of cutoff independence, the quenched approximation yields good results for the electron self-energy. In the chiral limit, a logarithmic cutoff dependence of the electron self-energy is found. The question, whether a regime of cutoff independence with a spontaneously broken chiral symmetry exists in strongly coupled QED, remains open.
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A systematic approach to the non-perturbative regime of Yang-Mills theory is provided by computer simulations of lattice gauge theory. Regularization is introduced by replacing continuous spacetime by a hypercubic lattice with lattice... more
A systematic approach to the non-perturbative regime of Yang-Mills theory is provided by computer simulations of lattice gauge theory. Regularization is introduced by replacing continuous spacetime by a hypercubic lattice with lattice spacing a. Improved actions for those lattice gauge simulations have attracted much interest in the recent past, since they allow for simulations on coarse lattices without an overwhelming impact of discretization errors. For example if the infra red regime of QCD Green functions is addressed (see e.g. [1, 2]), the low lying momenta are of order 1/Na, where N the number of lattice points in one direction. For a reasonable amount of lattice points, large lattice spacings are highly desirable for these purposes. Using the standard Wilson action, large lattice spacings give rise to sizable violations of rotational symmetry. These induce a systematic error to the lattice data and severely limit their significance. The benefit of improved actions is that th...
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The vortex theory which emerges from SU(2) lattice gauge theory by center projection is briefly reviewed. In this vortex picture, quark confinement is due to percolating (closed) vortices which are randomly linked to the Wilson loop. The... more
The vortex theory which emerges from SU(2) lattice gauge theory by center projection is briefly reviewed. In this vortex picture, quark confinement is due to percolating (closed) vortices which are randomly linked to the Wilson loop. The deconfinement phase transition appears as a de-percolation phase transition.
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The pandemic of Severe Acute Respiratory Syndrome Coronavirus 2 (SARS-CoV-2) suggests a novel type of disease spread dynamics. We here study the case where infected agents recover and only develop immunity if they are continuously... more
The pandemic of Severe Acute Respiratory Syndrome Coronavirus 2 (SARS-CoV-2) suggests a novel type of disease spread dynamics. We here study the case where infected agents recover and only develop immunity if they are continuously infected for some time τ. For large τ, the disease model is described by a statistical field theory. Hence, the phases of the underlying field theory characterise the disease dynamics: (i) a pandemic phase and (ii) a response regime. The statistical field theory provides an upper bound of the peak rate of infected agents. An effective control strategy needs to aim to keep the disease in the response regime (no ‘second’ wave). The model is tested at the quantitative level using an idealised disease network. The model excellently describes the epidemic spread of the SARS-CoV-2 outbreak in the city of Wuhan, China. We find that only 30% of the recovered agents have developed immunity.