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Register a new userNumerical Implementation of Gauge-Fixed Fourier Acceleration
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Authors
Yidi Zhao
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In hybrid Monte Carlo evolution, by imposing a physical gauge condition, simple Fourier acceleration can be used to generate conjugate momenta and potentially reduce critical slowing down. This modified gauge evolution algorithm does not change the gauge-independent properties of the resulting gauge field configurations. We describe this algorithm and it numerical implementation.
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For an asymptotically free theory, a promising strategy for eliminating Critical Slowing Down (CSD) is naive Fourier acceleration. This requires the introduction of gauge-fixing into the action, in order to isolate the asymptotically decoupled Fourier modes. In this article, we present our approach and results from a gauge-fixed Fourier-accelerated hybrid Monte Carlo algorithm, using an action that softly fixes the gauge links to Landau gauge. We compare the autocorrelation times with those of the pure hybrid Monte Carlo algorithm. We work on a small-volume lattice at weak coupling. We present preliminary results and obstacles from working with periodic boundary conditions, and then we present results from using fixed, equilibrated boundary links to avoid $mathbb{Z}_3$ and other topological barriers and to anticipate applying a similar acceleration to many small cells in a large, physically-relevant lattice volume.
The analysis developed by Luscher and Schaefer of the Hybrid Monte Carlo (HMC) algorithm is extended to include Fourier acceleration. We show for the $phi^4$ theory that Fourier acceleration substantially changes the structure of the theory for both the Langevin and HMC algorithms. When expanded in perturbation theory, each five-dimensional auto-correlation function of the fields $phi(x_i, t_i)$, $1le i le N $, corresponding to a specific 4-dimensional Feynman graph separates into two factors: one depending on the Monte-Carlo evolution times $t_i$ and the second depending on the space-time positions $x_i$. This separation implies that only auto-correlation times at the lattice scale appear, eliminating critical slowing down in perturbation theory.
We test a set of lattice gauge actions for QCD that suppress small plaquette values and in this way also suppress transitions between topological sectors. This is well suited for simulations in the epsilon-regime and it is expected to help in numerical simulations with dynamical quarks.
We investigate the continuum limit of a compact formulation of the lattice U(1) gauge theory in 4 dimensions using a nonperturbative gauge-fixed regularization. We find clear evidence of a continuous phase transition in the pure gauge theory for all values of the gauge coupling (with gauge symmetry restored). When probed with quenched staggered fermions with U(1) charge, the theory clearly has a chiral transition for large gauge couplings. We identify the only possible region in the parameter space where a continuum limit with nonperturbative physics may appear.
We study thermodynamics of SU(3) gauge theory at fixed scales on the lattice, where we vary temperature by changing the temporal lattice size N_t=(Ta_t)^{-1}. In the fixed scale approach, finite temperature simulations are performed on common lattice spacings and spatial volumes. Consequently, we can isolate thermal effects in observables from other uncertainties, such as lattice artifact, renormalization factor, and spatial volume effect. Furthermore, in the EOS calculations, the fixed scale approach is able to reduce computational costs for zero temperature subtraction and parameter search to find lines of constant physics, which are demanding in full QCD simulations. As a test of the approach, we study the thermodynamics of the SU(3) gauge theory on isotropic and anisotropic lattices. In addition to the equation of state, we calculate the critical temperature and the static quark free energy at a fixed scale.
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