• hep-lat
• Lattice QCD
• Monte Carlo
• QCD
• Topology

Among the wide range of their potential uses, Markov Chain Monte Carlo algorithms are an essential tool for computing expectation values of physical observables in systems at thermodynamic equilibrium.
Furthermore, the same calculation method is not only applicable to classical thermodynamic systems. Indeed, phenomenological predictions of Quantum Mechanics and Quantum Field Theories can be computed in the same way once their path integral formulation has been discretized.

In particular, one of the fields of research that has taken more advantage of this method is Quantum Chromodynamics (QCD), which is the Quantum Field Theory of the Standard Model that describes the strong interactions between quark and gluons. The coupling strength of QCD depends on the scale of energy, and the strong-coupling regime cannot be studied with perturbation theory. Lattice QCD, instead, provides a non-perturbative method to compute theory prediction in such a regime: QCD can be discretized on a finite lattice, and, performing simulations at different lattice spacings, the continuum limit can then be extrapolated.

However, at small values of lattice spacing, there is a problem that affects Lattice QCD simulations and undermines the accuracy and precision of results. Continuum QCD has a non-trivial topology, and gluon fields configurations are partitioned in homotopy classes, identified by their integer topological charge. On a discrete lattice, there is still a finite probability of tunneling between configurations with a different topological charge, but it decreases rapidly and vanishes when the continuum limit is approached, causing a loss of ergodicity.

This problem is called topological freezing, and can affect the correctness of any Lattice QCD extrapolation, especially for those quantities that depend directly on the topological charge. In particular, this is the case of the mass of the Axion, a hypothetical particle that rises in the context of the Peccei-Quinn mechanism. It is expected that the Axion is weakly coupled with the Standard Model interactions, and, for this reason, it is a promising candidate to be a constituent of Dark Matter. An accurate Lattice QCD extrapolation would indicate the point of the energy spectrum in which the Axion should be, and its existence could then be verified experimentally.

The topological freezing is an instance of critical slowing down, which is also a well-known problem in Condensed Matter Physics, since it characterizes the divergence of the relaxation time of materials in critical conditions.

The source of the slowing down is the increasing of the correlation length, and hence a partitioning of the system in clusters that have the correlation length as their typical size. If the time evolution is determined by a local interaction, it is improbable that all the microscopic components of a cluster behave coherently to change the macroscopic properties of the entire cluster.

In Markov chain Monte Carlo simulations, the problem is analogous: using the fact that the energy of the whole system is the sum of all the microscopic contributions, microscopic parts can be updated independently, generating Markov chains of phase space configurations that are distributed according to the Boltzmann distribution. In presence of a critical increasing of the correlation length, the autocorrelation time of the Markov chain increases dramatically, and the effective number of independent configurations becomes too small to perform accurate and precise measures.

For spin systems and hard sphere models simulations, the critical slowing down can be tempered very effectively with the implementation of cluster algorithms. The idea is to create more specialized algorithms that can exploit the properties of the system to update efficiently big clusters of elements, without compromising the Boltzmann distribution.

However, no cluster algorithms for Lattice QCD have been found, and only strategies that dampen the topological freezing have been proposed. It is then difficult to prove whether the results they lead to are actually correct.

A possible way to evaluate all these strategies is to implement them in a simplified model, called toy model, that shares many properties with QCD. If an exact solution is available for the toy model, then the results obtained with these strategies can be evaluated, and their possible biases can be measured.

The simplest model with a non-trivial topology, in which a topological charge can be defined, is a quantum mechanical particle that can move on a circumference. The topological charge here is equal to the winding number, i. e. the number of times a path configuration wraps around the circle.

Many of the most common strategies used to damped the topological freezing have been implemented and benchmarked in such a toy model.

However, even though the topological charge of the quantum particle shares many characteristics with the one defined in QCD, many more similarities could be found if another gauge field theory is considered as a toy model, and the U(1) pure gauge theory in two dimensions is, among them, the one that has the most straightforward implementation. It is also possible to include fermions into it, and such a theory is known as the Schwinger model. It is a widely used toy model since it exhibits fermions confinement, with a potential similar to that of quarks in QCD.

In this work, a new cluster algorithm for the two dimensional U(1) pure gauge theory is presented. It constructs a closed path of lattice links and reverses the sign of all gauge fields configurations inside the region surrounded by the path, accepting or rejecting the operation with a Metropolis algorithm. A chain of local gauge transformations along the path can make the action variation of the entire move to be the same as with a local update, making the acceptance of the algorithm reasonably high and independent of the cluster size, which is also possible to be set arbitrarily.

The results of this algorithm have shown that the cluster algorithm allows to perform measures in a region of parameter space that is impossible to study with a local algorithm.

This algorithm is statistically correct, and mean values obtained with it are unbiased. For this reason, it could be used to benchmark the strategies usually implemented in QCD, or to study the behaviour of the topological charge in a lattice gauge theory very close to the continuum limit.

The only other cluster algorithm for this theory that is present in literature was published in 1992 by Robert Sinclair. His algorithm uses the local gauge invariance to map the system into a one dimensional XY model, and updates it using the Wolff algorithm. Although it is a valid solution, its implementation is more complicated, it does not leave the choice of cluster size and the updating is not specifically focused in tunnelling between topological sectors, which is the only main problem of the local algorithm.

The algorithm proposed in this work, on the other hand, is specifically tailored for this theory, and for this reason, it is more adaptable, easier to implement and debug, and closer to the underlying Physics.