## Tesi etd-09262017-185353 |

Thesis type

Tesi di laurea magistrale

Author

BONANNO, CLAUDIO

URN

etd-09262017-185353

Title

Investigation of the topological properties of the CP^(N-1) model using Monte Carlo simulations

Struttura

FISICA

Corso di studi

FISICA

Supervisors

**relatore**Prof. D'Elia, Massimo

Parole chiave

- CPN model
- theta-dependence
- QCD
- lattice QCD
- quantum chromodynamics
- critical slowing down
- lattice quantum field theory
- topology
- Monte Carlo simulations

Data inizio appello

18/10/2017;

Consultabilità

Completa

Riassunto analitico

Quantum Chromodynamics (QCD) is the physical theory that describes strong interactions. The theory is part of the Standard Model (SM) of particles and describes strong forces inside the general framework of Quantum Field Theories in terms of fundamental interactions between quarks and gluons. Unlike Quantum Electrodynamics, QCD is an asymptotic-free theory, id est, quarks and gluons weakly couple at high energies. This allows, in this regime, to use the perturbative expansion to study the behaviour of the theory. However, many other peculiar properties enjoyed by QCD emerge in the low-energy regime, where non-perturbative effects are dominant, for example quarks confinement. Another important non-perturbative property of QCD is the existence of a non-trivial topological structure. In fact, one can introduce a non-perturbative contribution in the QCD Lagrangian, called topological term (depending on the parameter theta and on the topological charge Q), which is strictly connected with many interesting physical aspects such as the explanation of the U(1)_a anomaly or the physics of the eta' meson and whose contribution does not emerge in perturbation theory. Besides, there are some open problems concerning beyond-SM physics which are connected with the theta-dependence of QCD, such as the strong-CP violation or the dynamics of the axion. Therefore, non-perturbative QCD has been the core of numerous researches in the latest years.

It is in this context that the CP^(N-1) model has emerged and has gained a great importance. In fact, this two-dimensional quantum field theory is an interesting toy model that shares many fundamental properties with QCD such as confinement, asymptotic freedom, theta-term and topological properties. Therefore, it has been used as a theoretical laboratory to check non-trivial non-perturbative scenarios by means of analytical calculations. In fact, unlike in QCD, in the CP^(N-1) model it is possible to make a systematic 1/N expansion keeping the coupling constant fixed (which is analogous to the large number of colours limit in QCD) which allows an analytical investigation 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 (a consequence of topology).

The CP^(N-1) model is also interesting for numerical studies. In fact, thanks to the development of supercomputers, numerical Monte Carlo (MC) simulations of lattice QCD have become one of the main tools used to study non-perturbative QCD. Since they present many non-trivial computational challenges, which are suffered by MC simulations of CP^(N-1) model too, the latter has been the object of many numerical studies. Their goal is to test new algorithms that could solve such computational problems and to verify large-N predictions obtained by analytical studies.

One of the main computational problems that is still unsolved is the critical slowing down (CSD), namely, the local algorithms usually employed in MC simulations of lattice QCD experience an exponential growth of the machine time needed to generate two decorrelated measures of topological observables (called autocorrelation time) when approaching the continuum limit (lattice spacing a -> 0). This has the practical consequence of preventing to get closer to the continuum limit beyond a certain lattice spacing threshold. The CSD is a consequence of the existence of a non-trivial topological structure in the theory; for this reason this problem affects MC simulations of lattice CP^(N-1) too.

The main goal of our studies is to investigate the topological properties of the CP^(N-1) model using numerical Monte Carlo simulations; in particular, we aim to measure the first three terms of the theta-expansion of the vacuum free energy density f(theta), parametrized by the coefficients chi, b_2 and b_4, and to compare them with the large-N analytical results. To do so, it is of utmost importance to study a new algorithm to deal with the CSD. In fact, the slowing down worsens exponentially as N grows and this fact has prevented precise numerical verifications of the large-N limit in past studies.

At first, we have implemented a Monte Carlo simulation which employs the usual local over-heat-bath algorithm in order to check its characteristics and limitations. In particular, we studied the behaviour of the autocorrelation time of topological observables with N and with the lattice spacing. This study was pursued including larger values of N compared to past studies and exploring a different ansatz, obtaining a better mathematical modelling of the CSD.

Then, we have studied and implemented an improvement of the local algorithm in order to dampen the effects of the CSD: the simulated tempering. The main idea this algorithm is based on is to make simulations with a dynamical coupling constant g instead of a fixed one, as it is usually done. Changing the lattice coupling constant results in a change of the lattice spacing thanks to the asymptotic freedom of the theory (g -> 0 ==> a -> 0). When the system is far from the continuum limit, the algorithm correctly decorrelates the configurations and the change of g also allows finer lattice spacings to exploit it. The simulated tempering algorithm is based on rather loose hypothesis and can be easily formulated for a generic quantum field theory; thus, it is easily extendible to the case of lattice QCD or to other theories with a topological structure.

We made a systematic study of the characteristics and of the performances of the simulated tempering and compared its efficiency with the one of the local algorithm, finding a significant reduction by an order of magnitude in the autocorrelation time and a significant improvement in measurement accuracy of topological observables.

Using this algorithm, along with the imaginary-theta method originally proposed in lattice QCD and first applied to the CP^(N-1) model in this work, we achieved precise measurements of chi, b_2 and b_4 for various values of N. This has allowed a more precise comparison between lattice data and large-N analytical predictions compared to past studies.

Since the use of the simulated tempering method resulted in more efficient simulations of the model and in more precise measures of topological related quantities, we plan, in the next future, to apply this algorithm to the physical case of lattice QCD. In particular, we aim to study the theta-dependence of the theory, which is related to axion physics and to the strong-CP violation problem, and the sector of hadron physics whose dynamics is influenced by topology, such as the eta' meson physics.

It is in this context that the CP^(N-1) model has emerged and has gained a great importance. In fact, this two-dimensional quantum field theory is an interesting toy model that shares many fundamental properties with QCD such as confinement, asymptotic freedom, theta-term and topological properties. Therefore, it has been used as a theoretical laboratory to check non-trivial non-perturbative scenarios by means of analytical calculations. In fact, unlike in QCD, in the CP^(N-1) model it is possible to make a systematic 1/N expansion keeping the coupling constant fixed (which is analogous to the large number of colours limit in QCD) which allows an analytical investigation 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 (a consequence of topology).

The CP^(N-1) model is also interesting for numerical studies. In fact, thanks to the development of supercomputers, numerical Monte Carlo (MC) simulations of lattice QCD have become one of the main tools used to study non-perturbative QCD. Since they present many non-trivial computational challenges, which are suffered by MC simulations of CP^(N-1) model too, the latter has been the object of many numerical studies. Their goal is to test new algorithms that could solve such computational problems and to verify large-N predictions obtained by analytical studies.

One of the main computational problems that is still unsolved is the critical slowing down (CSD), namely, the local algorithms usually employed in MC simulations of lattice QCD experience an exponential growth of the machine time needed to generate two decorrelated measures of topological observables (called autocorrelation time) when approaching the continuum limit (lattice spacing a -> 0). This has the practical consequence of preventing to get closer to the continuum limit beyond a certain lattice spacing threshold. The CSD is a consequence of the existence of a non-trivial topological structure in the theory; for this reason this problem affects MC simulations of lattice CP^(N-1) too.

The main goal of our studies is to investigate the topological properties of the CP^(N-1) model using numerical Monte Carlo simulations; in particular, we aim to measure the first three terms of the theta-expansion of the vacuum free energy density f(theta), parametrized by the coefficients chi, b_2 and b_4, and to compare them with the large-N analytical results. To do so, it is of utmost importance to study a new algorithm to deal with the CSD. In fact, the slowing down worsens exponentially as N grows and this fact has prevented precise numerical verifications of the large-N limit in past studies.

At first, we have implemented a Monte Carlo simulation which employs the usual local over-heat-bath algorithm in order to check its characteristics and limitations. In particular, we studied the behaviour of the autocorrelation time of topological observables with N and with the lattice spacing. This study was pursued including larger values of N compared to past studies and exploring a different ansatz, obtaining a better mathematical modelling of the CSD.

Then, we have studied and implemented an improvement of the local algorithm in order to dampen the effects of the CSD: the simulated tempering. The main idea this algorithm is based on is to make simulations with a dynamical coupling constant g instead of a fixed one, as it is usually done. Changing the lattice coupling constant results in a change of the lattice spacing thanks to the asymptotic freedom of the theory (g -> 0 ==> a -> 0). When the system is far from the continuum limit, the algorithm correctly decorrelates the configurations and the change of g also allows finer lattice spacings to exploit it. The simulated tempering algorithm is based on rather loose hypothesis and can be easily formulated for a generic quantum field theory; thus, it is easily extendible to the case of lattice QCD or to other theories with a topological structure.

We made a systematic study of the characteristics and of the performances of the simulated tempering and compared its efficiency with the one of the local algorithm, finding a significant reduction by an order of magnitude in the autocorrelation time and a significant improvement in measurement accuracy of topological observables.

Using this algorithm, along with the imaginary-theta method originally proposed in lattice QCD and first applied to the CP^(N-1) model in this work, we achieved precise measurements of chi, b_2 and b_4 for various values of N. This has allowed a more precise comparison between lattice data and large-N analytical predictions compared to past studies.

Since the use of the simulated tempering method resulted in more efficient simulations of the model and in more precise measures of topological related quantities, we plan, in the next future, to apply this algorithm to the physical case of lattice QCD. In particular, we aim to study the theta-dependence of the theory, which is related to axion physics and to the strong-CP violation problem, and the sector of hadron physics whose dynamics is influenced by topology, such as the eta' meson physics.

File

Nome file | Dimensione |
---|---|

riassunt...audio.pdf | 110.77 Kb |

Tesi_Bon...audio.pdf | 2.59 Mb |

Contatta l'autore |