This thesis presents a method to simulate scalar electrodynamics on the lattice in a dual representation which avoids the sign problem arising at finite density in conventional simulations. We first introduce the model as a classical field theory and canonically quantize it paying special attention to the role of conserved charges. We then derive the path integral formulation of the grand canonical partition function, and formulate a lattice regulated version of the model.
The dual representation used in the simulations is based on well known high temperature expansion techniques. We discuss these methods in order to give the reader a general picture of their applicability to spin models and abelian lattice field theory. The existing literature on simulations of lattice models in dual representations is also reviewed. We find that, besides solving some sign problems, a dual representation often alleviates the inefficiency of Monte Carlo simulations near phase transitions. This is partly due to the availability of efficient update algorithms, such as the worm algorithm.
Using the expansion techniques introduced earlier we derive a dual formulation of lattice regulated scalar electrodynamics. We show that the dynamical variables of the dual model can be intuitively interpreted as field strengths and current densities. The dual representation of other observables, such as general correlation functions, is also discussed.
Finally, we present an algorithm to simulate the dual model. This algorithm is based on simple local updates combined with a worm algorithm that updates the current densities. By comparing simulation results at vanishing density with a conventional simulation we show that the algorithm is working correctly. We find that the worm algorithm behaves very differently in different phases of the system, and argue that this phenomenon is directly linked with the presence or absence of long range order. We also perform simulations at nonzero chemical potential where the system exhibits the silver blaze phenomenon and a transition to a finite density phase.
