• quantum chromodynamics
• QCD
• Monte Carlo simulations
• lattice quantum field theory
• lattice QCD
• CPN model
• theta-dependence
• topology

In this work we studied the CP^N-1 model in two-dimensions using numerical Monte Carlo (MC) simulations. This model has been extensively studied in the past because it presents several analogies with the theory that describes Quantum Chromodynamics (QCD), the physical theory that explains strong interactions.
The QCD theory is part of the Standard Model of particles and describes strong forces inside the general framework of Quantum Field Theories (QFT), in terms of fundamental interactions between quarks and gluons. To study the behavior of the theory at high energy, it is possible to use perturbative expansion since the theory is asymptotically free, i.e. the interaction between quarks and gluons becomes weak at high energies. However, this method can not be used at low energies because we no longer have an asymptotically free theory. But the interest in studying this regime is still very high, since other properties, such as the quarks confinement, come out. Another peculiar non-perturbative property of QCD is the existence of a non-trivial topological structure. Indeed, we can introduce a non-perturbative term in the QCD Lagrangian, called topological term or theta-term since it depends on a theta parameter and on the topological charge Q, which is closely related to many physical aspects of the theory, such as the explanation of the U(1)_a anomaly and the physics of the eta' meson. These contributions can not be explained by the perturbation theory but we can explain them thanks a particular class of configurations called instantons. The problem is that the introduction of theta-term in the theory entails a serious consequence because the term breaks the charge-parity symmetry, which is not observed in nature. This issue is known as "strong CP problem". Thus, the non-perturbative QCD has been the focus of many studies in the last few years.
It is in this context that the CP^N-1 model came out, because this two-dimensional QFT was considered as a toy model to study the non-perturbative regime better and more easily. Indeed, it has several properties in common with QCD, including: confinement, asymptotic freedom, the existence of a non-trivial topological structure and the presence of instantons for each N. The ease in the study of this model is due to the fact that, unlike the QCD, it is possible to carry out a 1/N expansion keeping the coupling constant fixed. This allows us to do an analytical study of the theory even in the non-perturbative regime. Thus, the study of the CP^N-1 model has led to a better comprehension of the mechanisms that underlie non-perturbative QCD, such as confinement or the theta-dependence of the theory.
With the advent of supercomputers, the most used method to study the non-perturbative QCD was the numerical MC simulations of lattice QCD, but also in this context the CP^N-1 model has acquired a notable importance. Indeed, MC simulations of QCD have several computational problems that are also present in the MC simulations of CP^N-1. The latter, being easier to study numerically, has made it a very useful model for studying new algorithms to solve these problems.
Our purpose is to use some of these computational solutions of MC simulations to further study the topological properties of the CP^N-1 model. To do this, we will study a function that is related to topology: the vacuum free energy density f(theta). The theta parameter must be small in order to motivate the CP-invariance presents in nature for QCD and this means that we can study the development of f(theta) around theta=0. We will focus on the study of its first three terms of the theta-expansion, parameterized by the coefficients chi, b_2 and b_4, and we will do for both low-N and large-N limit. The study of these parameters, being they proportional to the cumulants of the topological charge distribution, gives us information on the topology as N varies and this is interesting because there are analytical predictions in the large-N limit that are not completely confirmed and also predictions about the existence of a divergence in the instanton density in the limit of N->2, where the coefficients of the theta-expansion are not completely verified numerically. For these reasons, we considered both small and large values of N, in particular we have analyzed N=2, 3, 4, 6, 31, 41 and 51.
To did it, we had to solve some technical problems. The first is that theta-term in the action is a purely imaginary term and this means that it is impossible to define a probability through the exponential of the action. To solve this problem we applied the imaginary-theta method, i.e. we performed an analytic continuation in the complex plane for theta that allowed us to have a purely real action. The second computational problem is present in MC simulations of CP^N-1 with big values of N but also in the lattice QCD ones, and is called "critical slowing down", i.e. the local algorithms, usually employed in MC simulations, experience an exponential growth of the CPU-time needed to generate two decorrelated measures of topological observables (called autocorrelation time) when approaching the continuum limit ( when the lattice spacing a->0). To avoid this problem, we adopted a recent algorithm proposed by Martin Hasenbusch, which has been adapted for our purposes.
At first we studied the low-N limit, adopting the usual local over-heat-bath algorithms to simulate the theory, and we confirmed for small-N values the presence of a divergence for the topological susceptibility, but for b_2 and b_4 we obtained finite values that tend to values compatible with the ones predicted from the Dilute Instantons Gas Approxiamtion model.
Then we studied the large-N one adopting a mixture of local over-heat-bath algorithms and Hasenbusch algorithm. The results in the large-N limit confirmed the analytical prediction for the leading order (LO) and for the next-to-leading order (NLO) in 1/N expansion for the topological susceptibility chi and also gave a prediction to the NNLO. Moreover, we had a confirmation for the LO of b_2 and a prediction for the NLO. In general, we can say that the convergence to the large-N limit is very slow and affected by large corrections.