Narrow Results

The gradient flow scheme has emerged as a prominent nonperturbative renormalization scheme on the lattice, where flow time is introduced to define the renormalization scale. In this study, we perturbatively compute the gradient flow coupling for the SU(N) Yang–Mills theory in the large-N limit in terms of the lattice bare coupling up to three-loop order. This is achieved by combining the twisted Eguchi–Kawai model with the numerical stochastic perturbation theory. We analyze the flow time dependence of the perturbative coefficients to determine the perturbative beta function coefficients, successfully computing the one-loop coefficient in the large-N limit using three matrix sizes $N=289,441,529$. However, the higher-order coefficients are affected by large statistical errors. We also explore the potential for reducing these statistical errors through variance reduction combined with the large-N factorization property of the SU(N) Yang–Mills theory, and estimate the required number of samples for the precise determination of the higher-order coefficients.

Simulation of quantum field theories and fundamental interactions is one of the most challenging tasks in modern particle physics. Classical computers generally fail to reproduce accurate results when it comes to strongly coupled theories such as QCD. Recent developments in quantum technologies open up the possibility of simulating such physical regimes by using quantum computers. In this paper, we study the quantum resource related to the simulability of a quantum theory, i.e. non-stabilizerness for Lattice Gauge Theory (LGT) with discrete symmetry gauge groups. We show that enforcing gauge constraints for ${\mathbb{Z}}_l$ LGTs has no cost in terms of this resource and discuss the relation between non-abelianity of the gauge group with the average non-stabilizerness of the gauge invariant Hilbert space.

Higher gauge theory is a higher order version of gauge theory that makes possible the definition of 2-dimensional holonomy along surfaces embedded in a manifold where a gauge 2-connection is present. In this paper, we study Hamiltonian models for discrete higher gauge theory on a lattice decomposition of a manifold. We show that a construction for higher lattice gauge theory is well-defined, including in particular a Hamiltonian for topological phases of matter in $3+1$ dimensions. Our construction builds upon the Kitaev quantum double model, replacing the finite gauge connection with a finite gauge 2-group 2-connection. Our Hamiltonian higher lattice gauge theory model is defined on spatial manifolds of arbitrary dimension presented by slightly combinatorialized CW-decompositions (2-lattice decompositions), whose 1-cells and 2-cells carry discrete 1-dimensional and 2-dimensional holonomy data. We prove that the ground-state degeneracy of Hamiltonian higher lattice gauge theory is a topological invariant of manifolds, coinciding with the number of homotopy classes of maps from the manifold to the classifying space of the underlying gauge 2-group.

The operators of our Hamiltonian model are closely related to discrete 2-dimensional holonomy operators for discretized 2-connections on manifolds with a 2-lattice decomposition. We therefore address the definition of discrete 2-dimensional holonomy for surfaces embedded in 2-lattices. Several results concerning the well-definedness of discrete 2-dimensional holonomy, and its construction in a combinatorial and algebraic topological setting are presented.

In this paper, we study Feynman checkers, one of the most elementary models of electron motion. It is also known as a one-dimensional quantum walk or an Ising model at an imaginary temperature. We add the simplest non-trivial electromagnetic field and find the limits of the resulting model for small lattice step and large time, analogous to the results by Narlikar from 1972 and Grimmet–Jason–Scudo from the 2000s. It turns out that the limits in the model with the added field are obtained from the ones without field by mass renormalization. Also, we find an exact solution of the resulting model.

We discuss hybrid Monte Carlo algorithms for odd-flavor lattice QCD simulations. The algorithms include a polynomial approximation, which enables us to simulate odd-flavor QCD in the framework of the hybrid Monte Carlo algorithm. In order to make the algorithms exact, the correction factor to the polynomial approximation is also included in an economical, stochastic way. We test the algorithms for n_f = 1, 1 + 1 and 2 + 1 flavors and compare results with other algorithms.

We investigated a critical short-time relaxation in a lattice gauge theory. A systematic procedure of estimating critical point based on the "short-time scaling" is formulated. It is applied to the (2+1)-dimensional SU(2) lattice gauge theory at finite temperature to deduce its critical point. Finally, we studied the short-time relaxation behavior at the critical temperature starting either from "cold" and "hot" initial configuration, and calculated the dynamic critical exponents θ and z, as well as the static exponents β/ν.

We examine SU(2) gauge theory in 3 + 1 dimensions at finite temperature in the vicinity of critical point. For various lattice sizes in time direction (N_τ = 1, 2, 4, 8) we extract high precision values of the inverse critical coupling and critical values of the 4th order cumulant of Polyakov loops (Binder cumulant). We check the universality class of the theory by comparing the cumulant values to that of the 3D Ising model and find very good agreement. The Polyakov loop correlators for the indicated lattices are also measured and the string tension values extracted. The high precision values of critical coupling and string tension allow us to study the scaling of dimensionless ratio. The violation of scaling by < 10% is observed as the coupling is varied from weak to strong coupling regime.

Fields exhibit a variety of topological properties, like different topological charges, when field space in the continuum is composed by more than one topological sector. Lattice treatments usually encounter difficulties describing those properties. In this work, we show that by augmenting the usual lattice fields to include extra variables describing local topological information (more precisely, regarding homotopy), the topology of the space of fields in the continuum is faithfully reproduced in the lattice. We apply this extended lattice formulation to some simple models with nontrivial topological charges, and we study their properties both analytically and via Monte Carlo simulations.

This paper is divided into three parts: an introduction to the Hybrid Monte Carlo (HMC) algorithm; a detailed discussion of its implementation for Quantum Chromodynamics (QCD) in CMIS on the Connection Machine (CM-2); and a summary of recent analytic results on the behavior of HMC for free field theory. We present a detailed description of the architecture of the CM-2 and explain how it can execute the Conjugate Gradient (CG) solver for HMC at 6.35 GFlops/s.

Performance of the Intel iPSC/860 parallel processor for Quantum Chromodynamics codes with dynamical fermions is described. After reviewing the hardware and software environments provided by the manufacturer, the data structures appropriate for the QCD code are described. Techniques for maximum performance are briefly discussed. We achieve a speed of 10–15 Mŕlops per node depending upon how many lattice sites are located on each node.

In this talk we present results of a recent investigation of the topological vacuum structure in SU(2) lattice gauge theory with Wilson fermions. An effective action given by the hopping parameter expansion up to the 6-th order is used. To check the reliability of this approximation we compare it with the outcomes of an exact algorithm. The topological charge is measured according to a geometric prescription.

In the first part of the talk, a short review of baryon number violation in the electroweak standard model is given. I concentrate on perturbative estimates for the electroweak phase transition. A strong first order phase transition could be relevant both for a possible generation of the baryon asymmetry of the universe, and the survival of this asymmetry afterwards. In the second part of the talk, some lattice results for the electroweak phase transition are presented. They tend to indicate a transition more strongly first order than predicted by perturbation theory. A definite condusion cannot be drawn, however, because of severe finite size effects.

The standard PTMG method, invented and tested for the inversion of the fermion matrix (Dirac-operator) in the case of U(1) and SU(2) lattice gauge theories with staggered fermions, has now been generalized to SU(3). A crucial ingredient of the new algorithm is the way SU(3) link elements are averaged in the generation of the coarse grid link elements. Numerical results are presented for the two-dimensional case. For 128 × 128 lattices and physically relevant values of the coupling constant β and of the quark mass m_q, the new algorithm is an order of magnitude faster in CPU time than its leading competitor, the Conjugate-Gradient algorithm.

This meeting produces another evidence that present parallel computers are (a) real instruments of computational physics, (b) largely in the hands of still-pioneers, (c) efficiently promoted by basic research groups with large-scale computational needs. Progress in parallel computing is carried by two types of such groups, that either follow the build-it-yourself or the early-use strategies. In this contribution, we describe, as an example to the second approach, the Wuppertal university pilot project in applied parallel computing. We report in particular about one of our key applications in theoretical particle physics on the Connection Machine CM-2: a high statistics computer experiment to determine the static quark-antiquark potential from quenched quantum chromodynamics.

A detailed investigation of the temperature dependence of the spatial string tension σ_s in SU(2) gauge theory is presented. A sustained performance of 3 GFLOPS on a 64K Connection Machine CM-2 equivalent has been achieved. Scaling of σ_s between β = 2.5115 and β = 2.74, on large lattices, is demonstrated. Below the critical temperature, T_c, σ_s remains constant. For temperatures larger than 2T_c the temperature dependence can be parametrized by σ_s(T) = (0.369 ± 0.014)²g⁴(T)T², where g(T) is a 2-loop running coupling constant with the scale parameter determined as Λ_T = (0.076 ± 0.013)T_c.

We show numerically that the lowest eigenmodes of the two-dimensional Laplace operator with SU(2) gauge couplings and periodic boundary conditions are strongly localized. A connection is drawn to the Anderson localization problem. A new multigrid algorithm, capable of dealing with these modes, shows no critical slowing down for this problem independent of the disorder.

We present an improvement of global Metropolis updating steps, the instanton hits, used in a hybrid Monte Carlo simulation of the two-flavor Schwinger model with staggered fermions. These hits are designed to change the topological sector of the gauge field. In order to match these hits to an unquenched simulation with pseudofermions, the approximate zero mode structure of the lattice Dirac operator has to be considered explicitly.

The principle of indirect elimination states that an algorithm for solving discretized differential equations can be used to identify its own bad-converging modes. When the number of bad-converging modes of the algorithm is not too large, the modes thus identified can be used to strongly improve the convergence. The method presented here is applicable to any linear algorithm like relaxation or multigrid. An example from theoretical physics, the Dirac equation in the presence of almost-zero-modes arising from instantons, is studied. Using the principle, bad-converging modes are removed efficiently. It is sketched how the method can be used for a Conjugate Gradient algorithm. Applied locally, the principle is one of the main ingredients of the Iteratively Smoothing Unigrid algorithm.

We compare different conjugate gradient–like matrix inversion methods (CG, BiCGstab1 and BiCGstab2) employing for this purpose the compact lattice quantum electrodynamics (QED) with Wilson fermions. The main goals of this investigation are the CPU time efficiency of the methods as well as the influence of machine precision on the reliability of (physical) results especially close to the 'critical' line κ_c (β).

We present a new multigrid method called neural multigrid which is based on joining multigrid ideas with concepts from neural nets. The main idea is to use the Greenbaum criterion as a cost functional for the neural net. The algorithm is able to learn efficient interpolation operators in the case of the ordered Laplace equation with only a very small critical slowing down and with a surprisingly small amount of work comparable to that of a Conjugate Gradient solver

In the case of the two-dimensional Laplace equation with SU(2) gauge fields at β = 0 the learning exhibits critical slowing down with an exponent of about z≈0.4. The algorithm is able to find quite good interpolation operators in this case as well. Thereby it is proven that a practical true multigrid algorithm exists even for a gauge theory. An improved algorithm using dynamical blocks that will hopefully overcome the critical slowing down completely is sketched.