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Register a new userMonte Carlo Methods in Quantum Field Theory
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I. Montvay
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In these lecture notes some applications of Monte Carlo integration methods in Quantum Field Theory - in particular in Quantum Chromodynamics - are introduced and discussed.
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In Hybrid Monte Carlo(HMC) simulations for full QCD, the gauge fields evolve smoothly as a function of Molecular Dynamics (MD) time. Thus we investigate improved methods of estimating the trial solutions to the Dirac propagator as superpositions of the solutions in the recent past. So far our best extrapolation method reduces the number of Conjugate Gradient iterations per unit MD time by about a factor of 4. Further improvements should be forthcoming as we further exploit the information of past trajectories.
The extreme computational costs of calculating the sign of the Wilson matrix within the overlap operator have so far prevented four dimensional dynamical overlap simulations on realistic lattice sizes, because the computational power required to invert the overlap operator, the time consuming part of the Hybrid Monte Carlo algorithm, is too high. In this series of papers we introduced the optimal approximation of the sign function and have been developing preconditioning and relaxation techniques which reduce the time needed for the inversion of the overlap operator by over a factor of four, bringing the simulation of dynamical overlap fermions on medium-size lattices within the range of Teraflop-computers. In this paper we adapt the HMC algorithm to overlap fermions. We approximate the matrix sign function using the Zolotarev rational approximation, treating the smallest eigenvalues of the Wilson operator exactly within the fermionic force. We then derive the fermionic force for the overlap operator, elaborating on the problem of Dirac delta-function terms from zero crossings of eigenvalues of the Wilson operator. The crossing scheme proposed shows energy violations which are better than O($Deltatau^2$) and thus are comparable with the violations of the standard leapfrog algorithm over the course of a trajectory. We explicitly prove that our algorithm satisfies reversibility and area conservation. Finally, we test our algorithm on small $4^4$, $6^4$, and $8^4$ lattices at large masses.
We perform quantum Monte Carlo simulations in the background of a classical black hole. The lattice discretized path integral is numerically calculated in the Schwarzschild metric and in its approximated metric. We study spontaneous symmetry breaking of a real scalar field theory. We observe inhomogeneous symmetry breaking induced by inhomogeneous gravitational field.
We study constraint effective potentials for various strongly interacting $phi^4$ theories. Renormalization group (RG) equations for these quantities are discussed and a heuristic development of a commonly used RG approximation is presented which stresses the relationships among the loop expansion, the Schwinger-Dyson method and the renormalization group approach. We extend the standard RG treatment to account explicitly for finite lattice effects. Constraint effective potentials are then evaluated using Monte Carlo (MC) techniques and careful comparisons are made with RG calculations. Explicit treatment of finite lattice effects is found to be essential in achieving quantitative agreement with the MC effective potentials. Excellent agreement is demonstrated for $d=3$ and $d=4$, O(1) and O(2) cases in both symmetric and broken phases.
We propose the clock Monte Carlo technique for sampling each successive chain step in constant time. It is built on a recently proposed factorized transition filter and its core features include its O(1) computational complexity and its generality. We elaborate how it leads to the clock factorized Metropolis (clock FMet) method, and discuss its application in other update schemes. By grouping interaction terms into boxes of tunable sizes, we further formulate a variant of the clock FMet algorithm, with the limiting case of a single box reducing to the standard Metropolis method. A theoretical analysis shows that an overall acceleration of ${rm O}(N^kappa)$ ($0 ! leq ! kappa ! leq ! 1$) can be achieved compared to the Metropolis method, where $N$ is the system size and the $kappa$ value depends on the nature of the energy extensivity. As a systematic test, we simulate long-range O$(n)$ spin models in a wide parameter regime: for $n ! = ! 1,2,3$, with disordered algebraically decaying or oscillatory Ruderman-Kittel-Kasuya-Yoshida-type interactions and with and without external fields, and in spatial dimensions from $d ! = ! 1, 2, 3$ to mean-field. The O(1) computational complexity is demonstrated, and the expected acceleration is confirmed. Its flexibility and its independence from the interaction range guarantee that the clock method would find decisive applications in systems with many interaction terms.
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