Extracting small signals from highly oscillating data : the sign problem in particle systems
2021114 pages
Thesis: PhD - U. Grenoble Alpes
- Published: 2021
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Abstract: (dart-europe.org)
Monte Carlo simulations are the numerical method of choice for the study of lattice field theories in a non-perturbative framework. Over the years, Monte Carlo methods in Lattice Field Theories have reached a level of maturity such that in several QCD applications they provide the most reliable predictions for the low-energy behaviour of the theory. However, for many interesting theories, a complex-valued action prevents the use of standard sampling techniques. This is generally known as the sign problem and is present if the Boltzmann weight associated with the field configurations is either non-positive or non-real. A class of models exhibiting this property are finite density theories. These are going to be our primary interest.In this work, we shall focus on the density of states approach to the sign problem. This is a numerical technique that enables us to use standard Monte Carlo techniques to evaluate the density of states relating to the imaginary part of the action. By doing this, the sign problem is reduced to a simpler one-dimensional oscillatory integral, amenable to standard deterministic quadrature methods.At the core of our implementation of the density of states method is the LLR algorithm. We will present the general implementation and recent developments regarding on the control of possible sources of bias. Then we will extend the current formulation to allow for the evaluation of generic observables. Both these topics will be supported by results from numerical studies of the relativistic Bose gas at finite density.Lastly, we will discuss the problem of applying this approach to fermionic models where the sign problem is generated by the complex-valued fermionic determinant. As a specific application, we will study the Thirring model in two different representations discussing the related challenges.- Densité d'états
- Méthodes Monte Carlo
- Problème de signe
- Gaz de Bose relativiste
- Modèle Thirring
- Density of states
- Monte Carlo methods
- Sign problem
- Relativistic Bose gas
- Thirring model
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