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

(1)

Università di Pisa

Dipartimento di Fisica E. Fermi

Corso di Laurea Magistrale in Fisica

Curriculum Fisica Teorica

Investigation of the topological

properties of the CP

N −1

model

using Monte Carlo simulations

Candidate:

Supervisor:

(2)

(3)

Abstract

Our purpose is to study the topological properties of the CPN −1 _model

us-ing numerical Monte Carlo simulations. In particular, our goal is to calculate the first three coefficients of the θ-expansion of the vacuum energy density f(θ), parametrized by the three coefficients χ, b2 and b4, and to compare them with

the analytical results obtained within the large-N scheme.

After a quick review of the definition and of the properties of the continuum CPN −1 model and of its lattice counterpart, we first studied and implemented a Monte Carlo simulation which employs the usual local over-heat-bath algorithm to simulate the theory, so that we could check its characteristics and limitations. In particular, we found that the algorithm suffers a severe Critical Slowing Down (CSD) when approaching the continuum limit [18]; namely, the machine time needed to generate two decorrelated configurations exponentially grows as the lattice spacing tends to zero. Besides, we also found that the CSD worsens expo-nentially at fixed lattice spacing as N gets larger. This facts prevents a precise numerical verification of the analytical large-N predictions using the local algo-rithm.

Therefore, we studied and implemented an improvement of this algorithm in or-der to dampen the effects of the CSD: the simulated tempering [4, 11, 29, 41]. The main idea this algorithm is based on is to make simulations with a dynamical coupling constant, id est, (thanks to the asymptotic freedom of the theory) with a dynamical lattice spacing instead of a fixed one, so that finer lattice spacings can benefit from the correct functioning of the local algorithm for larger ones. We made a systematic study of the performances of the simulated tempering and compared its efficiency with the one of the local algorithm, finding a significant gain in the autocorrelation time and in the measure accuracy of topological ob-servables.

This algorithm, along with the method of the imaginary-θ fit developed recently in the field of lattice QCD [2, 3, 7, 8, 9, 30] and applied for the first time to the case of the CPN −1 model in this work, allowed us to measure χ, b

2 and b4 for

various values of N and to make a comparison between Monte Carlo results and large-N analytical predictions.

(4)

(5)

4.5 Extrapolation of the continuum limit . . . 79 4.6 Extrapolation of the large-N limit . . . 83 Conclusions and future perspective 89 A Details on the sampling of the distribution of the paths 93 The Von Neumann algorithm . . . 93 Sampling of the U field distribution . . . 94 Sampling of the z field distribution . . . 95 B The jackknife and blocking method 97 C Derivation of the standard deviation of correlated data 101

Bibliography 103

(7)

Introduction

Quantum Chromodynamics (QCD) is the physical theory that describes strong interactions and is part of the Standard Model of elementary particles. The theory describes the strong force inside the general framework of Quantum Field Theo-ries (QFT) as the result of an interaction between quarks, elementary constituents of hadrons, and gluons, massless vector bosons mediators of the interaction. In particular, it is mathematically formulated as a non-abelian gauge theory, with gauge group SU(3), where quark fields are minimally coupled with gluon ones. After the formulation of the theory, QCD was successfully compared with ex-perimental evidence; indeed, it correctly described many important phenomena involving strong interactions, such as the formation of hadron jets in high-energy collisions or the run of the Drell ratio with the energy. First results were obtained using one of the main tools available in QFT that allows analytic calculations: perturbation theory. This approach is based on the assumption that the strong coupling constant g is small and that all the quantities of interest can be ex-panded in a perturbative series of powers of g.

The reason why perturbative QCD describes so well all these phenomena is that the theory enjoys a peculiar property: asymptotic freedom; this means that the in-teraction between quarks mediated by gluons becomes weak at high energies. This property of QCD was theoretically discovered using renormalization group and experimentally verified with high precision for the first time in the deep-inelastic scattering experiment and constitutes a peculiar difference between strong inter-actions and electroweak ones.

Asymptotic freedom is the key to understand in which regime perturbative expan-sion makes sense and has a physical interpretation: only high-energies processes, where g is small, can be described by perturbative QCD. The low-energy regime, instead, is characterized by a coupling constant of order 1 (or larger) and is there-fore dominated by non-perturbative effects; here perturbation theory fails and a different approach is needed to study the behaviour of the theory.

Non-perturbative QCD has been the core of numerous researches since, from an experimental point of view, many interesting phenomena involving strong in-teractions take place in the low-energy regime, such as the formation of bound states of quarks (hadrons) or quark confinement. Besides, many interesting prop-erties of QCD are purely non-perturbative and cannot emerge in the perturbative formulation of the theory.

(8)

Introduction

One of the main non-perturbative characteristics enjoyed by QCD is the existence of a non-trivial topological structure, which enters many interesting aspects in hadron physics. For example, the solution of the U(1)a anomaly, which emerges

in the study of the effective Lagrangian of the low-energy modes in QCD, was found only when topology was taken into account.

In the chiral limit (realized when the masses of the lightest quarks go to zero), the QCD Lagrangian acquires an extra axial U(1) global symmetry (obtained when the right and the left components of the quarks fields are rotated with opposite phases) which is not realized in the real world. The first attempts to interpret it as a broken symmetry resulted in a failure. Indeed, the candidate for the role of pseudo-Goldstone boson of U(1)a was the η0 meson but, as Weinberg showed

in reference [43], if U(1)a were spontaneously broken then m (theo) η0 <

√

3mπ0, a

bound which is violated by the experimental value of the η0 _mass.

Actually, the U(1)a symmetry is anomalous and its anomaly (namely, the

vari-ation of the quantum QCD action under U(1)a rotations) is proportional to the

topological charge density q, a quantity related to the topology of gluon field configurations. Since the total topological charge Q can be expressed as the in-tegral of a total divergence, the U(1)a anomaly was thought to be negligible in

first studies. However, thanks to the non-trivial topological structure of QCD, that is not true. Indeed, there is a class of exact solutions of the classical equa-tions of motion (id est a class of configuraequa-tions that minimizes the action) with definite, non-zero topological charge Q ∈ Z, called instantons. The existence of such solutions makes the contribution of the anomaly in the quantum theory non-vanishing. Besides, instantonic field configurations are very different from those of the vacuum and are purely non-perturbative objects which cannot be studied with usual techniques of small-coupling expansion around vacuum configurations. This fact explains why instantonic contribution does not emerge in perturbation theory.

The discovery of instantons made clear that the nature of the violation of the Weinberg’s limit was topological and that topology was a determinant aspect of the dynamics of low-energy modes in QCD. Indeed, Witten and Veneziano related the theoretical value of the mass of the η0 _{meson to the topological susceptibility}

χ [40, 45, 46] (a quantity proportional to the vacuum expectation value of Q2₎

in the large number of colours limit (which is an approximation of the physical case, where Nc = 3). The existence of instantons also pointed out another

is-sue: if one introduced in the QCD Lagrangian an additional term proportional to the topological charge density, its contribution in the quantum theory would be non-zero thanks to the existence of configurations with non-trivial topology. This term, referred to as topological term or as θ-term (from the name of the constant that appears in front of it) is particularly interesting since it introduces in QCD an explicit breaking of the CP -symmetry, absent in the original theory, referred to as strong-CP violation. So far no violation of the CP -symmetry for strong interactions has yet been observed experimentally, thus θ is believed to be zero (recent measurements of the neutron electric dipole suggest an upper bound

(9)

Introduction of 10−9 _{- 10}−10_{), however, in principle, θ could assume any value in [0, 2π). This}

fine-tuning problem is still an open issue and many solutions have been proposed during last years, one of the main being the axion (see references [31, 44]). The axion is an hypothetical particle, theorized by Peccei and Quinn, that couples with quarks and that, once included in the theory, provides a mechanism that sets dynamically θ = 0. In the context of axion physics the study of the θ-dependence of QCD is extremely relevant since the profile of the axion potential depends explicitly on θ and on the topological susceptibility of the pure-gauge theory.

Since QCD is an extremely complicated theory, no analytic calculation is pos-sible to explore the non-perturbative regime in the full theory. This has led to the study of simpler models in lower dimensions that allowed analytic calcula-tions and that, at the same time, maintained many of the peculiar features of QCD. It is in this context that the CPN −1 _{model has emerged and gained much}

importance.

The CPN −1 model is a two-dimensional quantum field theory that constitutes

a useful toy model for the study of non-perturbative QCD. The theory shares many fundamental properties with QCD such as confinement, gauge invariance, existence of a non-trivial topological structure and of a θ-term, semiclassical in-stantonic solutions, dynamical mass generation and asymptotic freedom. How-ever, unlike QCD, the model allows an analytic investigation of the behaviour of the theory even in the non-perturbative regime thanks to the possibility of performing a systematic 1/N expansion in the large-N limit (which is similar to the infinite number of colours limit in QCD).

The study of the CPN −1 allows therefore to check non-trivial non-perturbative

scenarios by means of analytic calculations, such as the structure of the θ-vacua, the mass gap formation or the θ-dependence of the vacuum energy. However, the importance of CPN −1 does not finish here, since the theory plays a key role also

for what concerns numerical simulations.

Indeed, 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. For example, the study of the masses of quarks and hadrons, the evaluation of Standard Model matrix elements, the behaviour of QCD in the presence of strong background magnetic fields, the thermody-namics of QCD at finite temperature and density, the measure of the topological susceptibility, the study of the θ-dependence of the theory or the verification of the large-Nc predictions have been extensively studied through MC simulations

and MC results have been successfully compared with experimental evidence, corroborating the importance of the Monte Carlo approach to non-perturbative QCD.

Monte Carlo lattice QCD simulations are, however, extremely computationally expensive; therefore, many efforts were put in the study and in the development of more efficient algorithms in order to solve the several highly non-trivial

(10)

com-Introduction

putational challenges that lattice QCD presents and that slow down progresses in this field. Thus, the CPN −1 model, being simpler than QCD but showing many

of its peculiar properties, has been the object of many numerical studies, mostly aiming to test new algorithms that could solve such computational problems and to verify numerically the consistency of large-N predictions. Moreover, the fact that the model is exactly solvable in the large-N limit offers the possibility of comparing MC results with analytic calculations.

One of the main open computational problems that affects lattice QCD simula-tions is the critical slowing down (CSD) of topological modes. This problem is due both to the non-trivial topological properties of the theory and to the local structure of the algorithms typically used in MC simulations. To understand the nature of the problem it is necessary to give to the reader a brief overview on how a Monte Carlo simulation of a quantum field theory works.

Simulations in QFT are based on a non-perturbative regularization of the Feyn-man’s path integral, which is at the bases of the definition of every field theory. Feynman’s formulation of quantum mechanics assumes that a particle doesn’t fol-low a particular path when travelling from a point A to a point B of space-time (unlike in classical mechanics, where the particle follows the path that minimizes the action) but can follow any path from A to B. Given a certain observable O, every possible path gives a contribution to it, thus, we have to integrate over all the possible trajectories that the particle can follow to find the expectation value of the observable hOi. In QFT trajectories become the configurations of the fields associated to the particles of the theory and are specified giving the value of the field for every point of the space-time. Every path has a certain weight in the integration that gives hOi. If we make an analytic continuation of the theory from Minkowski to Euclidean time, performing the Wick rotation t → it ≡ tE,

the weight of a trajectory can be expressed as P ∝ e−S_{, which can be interpreted,}

if the Euclidean action S is real, as the probability of following a certain path. Since, as it is well-known, the path-integral formulation of quantum mechanics is ill-defined and leads to divergences, a regularization of the path integral is needed to a correct definition of the theory. A non-perturbative regularization that turns to be very suitable for numerical simulations is the lattice regularization, which consists in substituting the continuum space-time with a finite-size, discrete lat-tice R. This way, one introduces in the theory a length scale a, the latlat-tice spacing of R, whose inverse sets an energy scale which acts as an ultraviolet cut-off for the path integral, making the theory convergent. The lattice theory, besides, has a finite number of degrees of freedom, which is a necessary condition to perform simulations. The limit in which the regulator is removed coincides with the con-tinuum limit in which the lattice tends to the concon-tinuum space-time. This limit is achieved when a → 0.

In QFT, the idea MC simulations are based on is to treat the trajectory φ as a random variable with probability distribution P [φ] ∝ e−S[φ]_{. First of all, a}

(11)

Introduction sample of the paths {φ1, ..., φk} with respect to the distribution P is taken; then,

the observable of interest O is measured on the lattice on the sampled config-urations. Finally, once the sample {O1, ..., Ok} has been obtained, the sample

mean ¯O tends, for the central limit theorem, to the expectation value hOi when k → ∞.

In lattice field theory, simulations make use of Markov chains to sample the prob-ability distribution of the trajectories. This means that the algorithm generates a stochastic path in the space of configurations of the theory (the Markov chain) that visits different states during its evolution. The dynamics of evolution of the chain must be chosen so that, asymptotically, the frequency of visit of a certain state is proportional to the probability P of occupying that state.

A Markov chain is practically achieved updating, at each Monte Carlo step, the current field configuration so that it evolves into another configuration of the space. Since trajectories are defined only on the sites of the lattice, the sim-plest update one can adopt is a local one, which stochastically transforms the configuration site by site, keeping the value of the field in the others sites fixed. Local algorithms are ideal to apply to lattice QCD but tend to be less and less efficient as one approaches the continuum limit because of the non-trivial topolog-ical structure of the theory. Indeed, in continuum QCD, gluon field configurations are divided in homotopy classes characterized by their topological charge, which represents the number of windings of the path. This is similar to what happens for the free particle on the ring, whose trajectories are characterized by the num-ber of times they wrap around the centre of the ring. Therefore, in QCD, a path having charge Q cannot be deformed continuously in one having a different charge Q0, just like in the case of the particle on the ring. This implies that the space of configurations is divided into topological sectors separated by infinitely-high potential barriers.

On the lattice homotopy classes do not exist and potential barriers between dif-ferent topological sectors have finite height; hence, it is possible to deform locally (id est updating site by site) a path with charge Q into one with charge Q0_.

However, tunnelling from a topological sector to another becomes more and more difficult as the lattice spacing tends to zero since the height of the barriers tends to infinity in the continuum limit, because of the existence of topological sec-tors in the continuum theory. Hence, when approaching the continuum limit, it becomes more and more difficult to change the topological charge of a path and this causes the trajectory to remain trapped in a certain topological sector, preventing the correct sampling of the probability distribution of the paths and causing the freezing of the Monte Carlo evolution of the topological charge dur-ing the simulation. In other words, as the continuum limit is approached, the machine time needed to generate a trajectory with charge Q0 _{starting from one}

with charge Q grows exponentially (as showed in references [18, 28, 42]). This problem is referred to as critical slowing down of topological modes and is one of the main, still-open, computational problems that affects simulations of lattice QCD. Naturally, since the critical slowing down is due to the use of a local

(12)

algo-Introduction

rithm in the presence of non-trivial topological properties, also lattice CPN −1_MC

simulations suffer from it. This issue has been studied extensively in literature, both in the former (see references [4, 11, 25]) and in the latter case (see references [13, 18, 22, 27, 33, 35, 41]).

Another open computational problem in lattice QCD is the sign problem. Previ-ously it was stated that, thanks to the analytic continuation in Euclidean time, it is possible to express the probability distribution of the paths as P ∝ e−S_.

However, P can be interpreted as an actual distribution if and only if S is real. Otherwise, if P is a complex quantity, e−S _{cannot be interpreted as a probability}

distribution and a MC simulation of the theory is impossible. In lattice QCD this happens, for example, when a θ-term is added to the theory. Indeed, the topo-logical term is always purely imaginary, thus, if θ 6= 0, the weight P is complex. For the same reason, the CPN −1 _{action becomes complex if a θ-term is added}

and the simulation of the model is impossible if θ 6= 0.

The goal of our studies is to develop strategies to deal with the critical slow-ing down of topological modes and with the sign problem in order to simulate the CPN −1 model in the presence of the θ-term and investigate the topological prop-erties of the theory. In particular, we aim to study the θ-dependence of the theory, a consequence of topology, measuring the first three coefficients, usually referred to as χ, b2 and b4, of the Taylor expansion of the vacuum energy density f(θ).

Indeed, these coefficients are related to the cumulants of the topological charge and they are non-vanishing because of the non-trivial topological structure of the theory. In particular, the n-th cumulant of Q is proportional to the n-th deriva-tive of f. Besides, we want to study the large-N behaviour of these coefficients in order to make a comparison between lattice measures and analytic predictions. In chapter 1 we present and define the continuum CPN −1 _{model, focusing on}

its topological properties and on its affinities with QCD. Besides, we summarise the principal analytic results obtained in the large-N limit, in particular the be-haviour of the coefficients of the Taylor expansion of f(θ).

In chapter 2 we present the lattice formulation of the CPN −1 theory and a brief

overview of the most important local algorithms employed in Monte Carlo simu-lations of lattice field theories. Then, we present the usual local over-heat-bath algorithm employed in literature to simulate the model at θ = 0, where the sign problem is absent. Finally, we discuss the state of the art concerning the study of the behaviour of the critical slowing down with the lattice spacing and with N. In particular, past studies (for example [18]) have shown that the CSD worsens exponentially both when N grows at fixed lattice spacing and when the lattice spacing goes to 0 at fixed N. Therefore, simulating large values of N with a local algorithm is prohibitive. This aspect, along with the fact that all topological-related quantities vanish for N → ∞, makes the measure of the θ-dependence of the vacuum energy for large values of N extremely difficult.

In chapter 3 we present and discuss the new numerical strategies we studied and 10

(13)

Introduction adopted to deal with the sign problem and with the critical slowing down of topo-logical modes.

As for the former, we applied to the CPN −1 _{a solution proposed and studied in}

lattice QCD (see references [2, 3, 7, 8, 9, 30]) and never tested for this theory: the imaginary-θ method. This method consists in analytically continuing the theory for imaginary values of θ in order to make the topological term real. This way, the total action becomes a real quantity and it is possible to interpret e−S _{as a}

probability distribution. This allows to simulate the theory in the presence of the topological term, avoiding the limitation to the θ = 0 case that past studies suffered from (for example, references [13, 14, 41]). Naturally, this approach relies on the assumption that the θ-dependence of the theory is analytic, an hypothesis proven to be true in the large-N limit (see references [8, 34]).

Instead, concerning the critical slowing down of topological modes, we imple-mented the simulated tempering method: an algorithm that has been first pro-posed in the field of statistical mechanics simulations but that has been applied in a wide variety of physical problems, such as molecular dynamics, study of protein folding or quantum field theories, thanks to its general formulation and to the loose hypothesis it is based on. This method was applied just once to the CPN −1

model (see reference [41]) but without any systematic study of its properties. The simulated tempering algorithm is used when the statistical system that has to be simulated presents a many-vacua potential. At low temperatures, fluctua-tions are damped; therefore, the Markov chain that explores the phase space of the system can remain trapped in a local minima of the energy if fluctuations are not large enough to make the system tunnel out of it. This causes a critical slowing down of the algorithm when approaching the zero-temperature limit and a freezing of the evolution of the chain since it cannot visit other vacua of the po-tential. The idea the simulated tempering algorithm is based on is to change the temperature during the simulation so that the system, once heated, can escape from the local minima in which it is entrapped, exploiting the large fluctuations that occur at high temperatures. In the case of the CPN −1 _{model, the}

param-eter which is dynamically changed during the simulation is the lattice coupling constant g. Indeed, thanks to the asymptotic freedom of the theory, a decrease of the coupling results in a decrease of the lattice spacing a too. Thus, when g is large, the theory is more far from the continuum limit and the algorithm can change the topological charge of the configurations more easily since, for higher values of a, the free-energy barriers separating the topological sectors are lower. We also studied a possible extension to the original idea of the simulated temper-ing, making a two-dimensional parameter exploration and letting also θ change during the simulation. Indeed, the mean value of the topological charge depends on θ, thus, varying this parameter during the simulation constitutes a further stimulation to explore different topological sectors.

In chapter 4 we apply the algorithms described in chapter 3 and present the re-sults we have obtained.

(14)

au-Introduction

tocorrelation time of the lattice topological susceptibility (namely, the machine time needed to generate two decorrelated measures of this observable), the pa-rameter N and the lattice spacing. This ansatz is based on the picture of barrier crossings described above for the spread of topological information and has been successfully compared with the lattice data collected.

Then, we present a systematic study of the characteristics of the simulated tem-pering and we compare its efficiency with the one of the local algorithm. Besides, we also make a comparison between the performances of the single simulated tempering and the double one.

Lastly, we show how we obtained lattice measures of χ, b2 and b4 using the

imaginary-θ fit, a method which is based on the global fit of the θ-dependence of the cumulants of the topological charge, and how we extrapolated their contin-uum limit. Then, we compare the numerically-extrapolated large-N limit with the analytic results.

In the final chapter we draw our conclusions, summarizing the original results obtained in this work and discussing their possible future applications to the physical case of lattice QCD.

(15)

Chapter 1 The continuum CP

N −1

model

In this chapter we define the Euclidean CPN −1 _{quantum model in the}

con-tinuum using the path integral formulation after having introduced its Lagrangian and discussed its main properties, with particular stress on the topological term and on the analogies with QCD. Then, we define the free energy density and we discuss its θ-dependence, an aspect which is strictly connected with the non-trivial topological properties of the theory. Lastly, we summarise the most important an-alytic results obtained in the large-N limit.

1.1 The Lagrangian of the model

The CPN −1_{model is a two-dimensional quantum field theory whose Euclidean}

Lagrangian is

L0(x) =

1

λ2D¯µz(x)D¯ µz(x), (1.1)

where z(x) is a N-component complex scalar field that satisfies the constraint ¯ z(x)z(x) = N X k=1 |zk(x)|2 = 1, (1.2)

λ is the dimensionless coupling constant of the theory and

Dµ= ∂µ+ iAµ (1.3)

is the U(1) covariant derivative.

The field Aµis a background real abelian gauge field that has been introduced to

have a manifest U(1) gauge invariance. This means that, if one makes the local field transformation

z(x) → z0(x) ≡ eiΩ(x)z(x), Aµ(x) → A0µ(x) ≡ Aµ(x) − ∂µΩ(x),

the Lagrangian (1.1) is invariant.

Indeed, under a gauge transformation, the derivative of z(x) gets an extra term: ∂µz(x) → ∂µz0(x) = eiΩ(x)[∂µz(x) + iz(x)∂µ.Ω(x)]

(16)

Chapter 1. The continuum CPN −1 _model

This term, with our definition of Dµ, is exactly cancelled by the variation of

Aµ(x).

Therefore, Dµz(x)transforms as z(x); indeed, one has:

Dµz(x) → Dµ0z 0

(x) = ∂µz0(x) + iA0µ(x)z 0

(x)

= eiΩ(x)∂µz(x) + ieiΩ(x)z(x)∂µΩ(x) − ieiΩ(x)z(x)∂µΩ(x) + ieiΩ(x)Aµ(x)z(x)

= eiΩ(x)Dµz(x).

From this fact it trivially follows the invariance of (1.1) under gauge transforma-tions.

Note that in Lagrangian (1.1) all indices are low since we are defining the theory in a Euclidean space-time. Thus, being the metric g = 1, there is no distinction between covariant and contravariant vectors. Obviously, the Euclidean theory has to be intended as a theory defined on a Minkowski space-time and analytically extended in imaginary time via the Wick rotation t → it ≡ x2 ∈ R. Indeed, one

of the effects of this transformation is to change the metric from η to 1.

The name CPN −1_{derives from the name of the manifold in which the z field lives.}

Indeed, CPN −1 _{is a complex projective manifold that is obtained by projecting}

the complex N-dimensional sphere SN

C = {z ∈ C

N_|¯_{zz = 1}} _{with respect to the}

equivalence relation z1 ∼ z2 ⇐⇒ ∃ α ∈ R | z1 = eiαz2 (note that the ∼ relation

is equivalent to a gauge transformation).

Since Aµ is a background field, id est it has no kinetic term, it is possible to

integrate it out and to express it in terms of the z field. To do so, let us first rewrite Lagrangian (1.1) expliciting its quadratic dependence from Aµ. Indeed,

using the fact that

∂µ(¯zz) = ∂µ1 = 0 = ∂µzz + ¯¯ z∂µz,

it is possible to rewrite (1.1) as L0 =

1

λ2 [∂µz∂¯ µz − 2iAµ(¯z∂µz) + AµAµ] .

The gaussian integration of Aµ in the path integral (see subsection 1.3.1) results

in replacing Aµ in the Lagrangian by the solution of its classical equation of

motion, which reads: ∂L0 ∂(∂νAµ) = 0 = ∂L0 ∂Aµ = 2Aµ− 2i (¯z∂µz) . This implies: Aµ = i¯z∂µz. (1.4)

Thus, substituting the explicit expression of Aµ in terms of z, Lagrangian (1.1)

becomes

L0 =

1

λ2 [∂µz∂¯ µz + (¯z∂µz) (¯z∂µz)] , (1.5)

where now ¯z∂µz acts as a composite gauge field. In the following, however, it is

more convenient to treat A and z as independent fields in order to maintain a 14

(17)

1.2. The topological term manifest U(1) gauge invariance.

The action of the theory reads S0[¯z, z, A] = Z R2 L0(x)d2x = 1 λ2S˜0,

where we have defined ˜S0 ≡ λ2S0 for later convenience.

1.2 The topological term

Just like in QCD, one may also add to (1.1) a topological term Lθ = −i

θ

4πεµνFµν ≡ −iθq(x), (1.6) where θ ∈ [0, 2π) is a real, dimensionless parameter, εµν is the totally

antisym-metric tensor (ε12 = 1), Fµν ≡ ∂µAν − ∂νAµ is the field strength and q(x) is the

topological charge density, which reads q(x) = 1

4πεµνFµν(x).

Indeed, the topological term enjoys all the characteristics of Lagrangian (1.1): it is renormalizable, manifestly gauge-invariant (Fµν is gauge-invariant and εµν is

an invariant tensor), manifestly Lorentz-invariant and has only two derivatives in the z field. However, unlike L0, it explicitly breaks the P and the T symmetries

and, therefore, also breaks CP (just like the topological term in QCD). In order to clearly see why, it is useful to write explicitly the θ-term:

Lθ = −i θ 2πεµν∂µAν = −i θ 2π ∂1A2− ∂2A1 , (1.7) where 1 and 2 indicate, respectively, the space and the time components of ∂µ

and Aµ.

Parity P reverses the space direction x1 → −x1, thus, ∂2 and A2 are left

un-changed (since the time component of the gauge field transforms as a scalar under parity) while ∂1 → −∂1 and A1 → −A1 (since the spatial component of

the gauge field transforms as a vector under parity); this implies P Lθ = −Lθ.

Time reversal T , instead, leaves A1 and ∂1 unchanged but changes the signs of

∂2 and A2, therefore, T Lθ = −Lθ.

The topological term can be rewritten more intuitively noticing that

B = εµν∂µAν (1.8)

is the magnetic field orthogonal to the (x1, x2) plane. The topological term,

therefore, simply reads:

Lθ = −i

θ

(18)

The total topological charge is defined as the integral of the density q: Q = Z R2 q(x)d2x = 1 4πεµν Z R2 Fµν(x)d2x. (1.10)

Using equation (1.8), one can rewrite Q using the magnetic flux through the (x1, x2) plane: Q = 1 2π Z R2 B(x)d2x = 1 2πΦ(B) = 1 2πR→∞lim I S1_(R) Aµtµdl. (1.11)

The last equality represents the line integral of the vector potential and can be obtained applying the Stokes’ theorem to the integral that defines Φ(B) (tµis the

tangent versor of the integration curve and dl is its infinitesimal length).

The contribute of the topological term to the action can be expressed in terms of the topological charge as

Sθ = Z R2 Lθ(x)d2x = −iθQ = −i θ 2πΦ(B). The total action of the theory therefore reads:

S ≡ S0+ Sθ = S0− iθQ. (1.12)

It is possible to show that for every configuration of the fields the topological charge satisfies Q = n ∈ Z. This number represents how many times the Aµ field

configuration wraps around S1 _{making a full rotation around the manifold and}

is called winding number. It is impossible to continuously deform a trajectory with topological charge Q = n into one having charge Q = n0 _{since a continuous}

transformation cannot change the number of winding of the configuration around the manifold; paths are therefore divided into separated homotopy classes called topological sectors and each class is characterized by the value of the topologi-cal charge. The existence of trajectories with definite topologitopologi-cal charge makes the contribution of the θ-term non-vanishing in the quantum theory (see subsec-tion 1.3.1 for further details) even if it is a boundary term (as previously shown in equation (1.11) using Stokes’s theorem) .

Inside every topological sector there is a class of paths that minimize the action S0 (id est that solve exactly the classical equations of motion at θ = 0) called

instantons. The existence of such solutions is possible thanks to the non-trivial topological properties of the CPN −1 _{model. In particular, the action of an}

in-stanton of a generic sector reads

S₀(min)= 2π|n|

λ2 , (1.13)

where n is the winding number of the class. The peculiar 1/λ2 _{dependence of}

the action is the hallmark of instantons and is exhibited by QCD instantonic configurations too.

(19)

1.3. Path integral formulation of quantum CPN −1 _model

The form of the CPN −1 _{instantonic solutions is known exactly for every value of}

N. Two examples, derived in reference [49], for N = 2 with opposite, unitary topological charge are:

z1(x) = 1 p1 + |x|2; z2(x) = x1± ix2 p1 + |x|2.

The − sign stands for the instanton with charge n = 1 while the + sign for the anti-instanton with charge n = −1. We have indicated the components of x as (x1, x2) and used the notation |x|2 = x21 + x22. Note that |z1|2 + |z2|2 = 1 as

it should. These configurations minimize the action S0 inside their respective

topological sectors.

To show that both have unitary charge one must first derive the vector potential using equation (1.4), obtaining:

A(±)_µ = ± εµνxν 1 + |x|2.

Now, let us evaluate the magnetic flux using the fact that εµνxν = −Rtµand that

dl = Rdφ, since integration is carried over a ring of fixed radius R. One has: Φ(B(±)) = ± lim R→∞ Z 2π 0 −Rtµ 1 + R2tµRdφ = ± lim R→∞ −R2 1 + R2 Z 2π 0 dφ = ±2π lim R→∞ −R2 1 + R2 = ∓2π.

Thus, the topological charge reads Q± =

1 2πΦ(B

(±)_{) =} 1

2π(∓2π) = ∓1, as previously stated. It is interesting to notice that A(±)

µ is directed along the

az-imuthal versor and, in particular, it is proportional to it (since ±εµνxν = ∓Rtµ);

thus, when φ makes a full rotation around S1_{, the vector potential makes a}

com-plete rotation around the manifold too. In particular, A(+)

µ wraps clockwise one

time (therefore Q+ = −1) while A (−)

µ wraps counter-clockwise one time (therefore

Q−= +1).

1.3 _{Path integral formulation of quantum CP}

N −1

model

1.3.1 Partition function

The formulation of the quantum CPN −1 model via the Feynman’s path

(20)

time, the zero-temperature partition function of the CPN −1 _{model reads:}

Z(θ) = I

[d¯z dz dA]e−S[¯z,z,A]= I

[d¯z dz dA]e−S0[¯z,z,A]+iθQ[A]_, (1.14)

where the functional measure is defined as [d¯z dz dA] ≡ Y x∈R2 " _N Y i=1 d¯zi(x) dzi(x) 2 Y µ=1 dAµ(x) # .

The circle means that the integration is carried over all periodic paths with z(x2 =

−∞) = z(x2 = +∞) and Aµ(x2 = −∞) = Aµ(x2 = +∞).

This partition function is evaluated at zero temperature since, in the path integral representation, temperature T = ∆t−1_{, where ∆t is the temporal extension of the}

integral domain in the definition of the action. Indeed, for a generic field theory with Hamiltonian H, the partition function at temperature T reads:

Z(T ) = Tr{e−H/T} = ∞ X n=0 e−En/T ₌ I [dφ] exp ( − Z _2T1 − 1 2T dx2 Z +∞ −∞ dx1 L [φ(x1, x2)] ) . Since, in our case, the x2-integral is extended to the whole real axis, it follows

that partition function (1.14) is evaluated at T = 0.

Using the fact that the topological charge Q takes only integer values, one can rewrite partition function (1.14) as a sum on topological sectors:

Z(θ) = ∞ X n=−∞ eiθn I Q[A]=n [d¯z dz dA]e−S0[¯z,z,A] _≡ ∞ X n=−∞ eiθnZn. (1.15)

This expression shows that the contribution of the n-th topological sector to the total partition function is weighted with the complex factor eiθn_{; therefore,}

the contribution of the topological term is non-vanishing and non-trivial in the quantum theory despite being it a total derivative. Besides, in this expression, it is manifest the key role played by instantons; indeed, they give the largest contribution to Zn since they minimize the action S0 (so they maximize e−S0).

Lastly, this expression makes manifest that the partition function is 2π-periodic in θ, since

θ → θ + 2π =⇒ Z(θ) → Z(θ). (1.16) This is a general feature of partition functions of field theories with topological terms; indeed, it holds in QCD too. Moreover, this fact also explains why one can limit to take θ ∈ [0, 2π).

The partition function enters the definition of the vacuum expectation value of an observable O[¯z, z, A], which reads

h0| O |0i ≡ hOi (θ) = 1 Z(θ)

I

[d¯z dz dA]e−S[¯z,z,A]O[¯z, z, A] =

I

[d¯z dz dA]P [¯z, z, A]O[¯z, z, A], 18

(21)

where P is the complex weight of the paths and is defined as P [¯z, z, A] ≡ 1

Z(θ)e

−S[¯z,z,A] ₌ 1

Z(θ)e

−S0[¯z,z,A]+iθQ[A]_. (1.17)

The weight P is a complex quantity since the θ-term is purely imaginary, thus, it cannot be interpreted as a probability distribution. This, as we shall see in the next chapter, represents a well-known problem in Monte Carlo simulations, known as sign problem. We shall discuss this issue in detail later, when we will introduce MC simulations of lattice CPN −1 _{in chapter 2.}

From the zero-temperature partition function one can also define the zero-temperature free energy F , which is the main quantity we will study on the lattice. The zero-temperature free energy reads:

F (θ) ≡ − log Z(θ). (1.18) The zero-temperature free energy coincides with the vacuum energy E0, since

Z(T ) ∼

T →0 e −E0/T_,

hence, this implies that

F (T = 0) = lim

T →0−T log Z(T ) = E0.

The non-trivial dependence of the partition function implies a non-trivial θ-dependence of the free energy. Naturally, this aspect is strictly connected with the topological properties of the theory and will be discussed in detail later, in subsection 1.3.2.

The partition function can be evaluated perturbatively if λ 1 using the saddle-point expansion technique since the action has the form

S0 =

1

λ2S˜0. (1.19)

In this scheme, the topological term plays no role since the contribution of in-stantonic configurations to the partition function is, using equation (1.13),

e−S0(min ) =

e−2π/λ2

|n|

, (1.20) id est, their contribution is exponentially suppressed in the λ → 0 limit and is thus negligible in perturbation theory.

The perturbative definition of Z is ill-defined since leads to divergent Feynman diagrams, thus, the theory needs to be renormalized. To do so, it is usually chosen to regularize the model introducing an ultra-violet cut-off ΛU V in the

integrals on the momenta that emerge in the loop expansion of the partition function Z. In this scheme, calling µ the energy scale at which one defines the renormalized theory, the equation that connects the renormalized coupling of the

(22)

physical theory and the bare coupling of the regularized theory is, at one-loop order: 4π λ2 B(ΛU V) = N log Λ 2 U V µ2 + 4π λ2 R(µ) , (1.21) which is a particular case of the general formula, arguable with renormalization-group arguments, λR= λR(λB, ΛU V, µ) = λR λB, µ ΛU V . (1.22)

Indeed, the theory does not contain any dimensional parameter, thus, the renormalization-group flow from the physical theory to the bare one must depend only on the

dimensionless scale µ/ΛU V.

From equation (1.21), one obtains that µ

ΛU V

→ ∞ ⇒ λR(µ) → 0 (1.23)

if one wants to keep λB(ΛU V)fixed; id est, the interaction mediated by the gauge

field becomes irrelevant at high energies. This property holds for every value of N, is addressed to as asymptotic freedom and is one of the most important characteristics that the CPN −1 _{model shares with QCD.}

Asymptotic freedom can also be argued by computing the β function of the theory from equation (1.21), id est evaluating:

β(λR) ≡ µ dλ2 R(µ) dµ λB(ΛU V) = dλ 2 R(µ) d log µ λB(ΛU V) . (1.24) The β function expresses how the renormalized coupling λRvaries with µ at fixed

bare coupling λB. One gets:

β(λR) = − N λ4 R 2π + O(λ 6 R). (1.25)

It is a general fact that, if the coefficient of the leading order of the λR-expansion

of the β function is negative, than the theory is asymptotically free. This is the case of (1.25). Indeed, its solution is

1 λ2 R(µ) − 1 λ2 R(µ0) = N 2πlog µ µ0 , (1.26) where µ0 is a reference scale. From this expression is clear that, if µ/µ0 → ∞,

than λR(µ)/λR(µ0) → 0.

Asymptotic freedom clarifies when the perturbative expansion has a physical meaning: only the high-energy regime can be described in perturbation theory, the low-energy regime is, instead, a non-perturbative domain where λRis of order

1 (or larger) and where perturbative expansion breaks down. Thus, in this regime, a different approach, such as the large-N expansion discussed in section 1.4, is needed to investigate the behaviour of the theory.

(23)

1.3.2 The θ-dependence of the free energy

As already pointed out in subsection 1.3.1, the free energy F depends non-trivially on θ when a topological term is added to the action, just like in QCD. In the present subsection, we will define the Taylor expansion of F and connect the coefficient of the expansion to the topological charge Q.

First of all, let us prove that the free energy F is an even function of θ. This can be easily shown performing the change of variable Aµ → P Aµ inside the

functional integral that defines the partition function. Indeed, P Q = −Q and [d(P A)] = [dA], since P A2 = A2 and P A1 = −A1 (the − sign is absorbed when

adjusting the extremes of integration). Hence, one has: e−F (β,θ) = I [d¯z dz d(P A)]e−S0[¯z,z,P A]+iθQ[P A] = I [d¯z dz dA]e−S0[¯z,z,A]−iθQ[A]_{= e}−F (β,−θ)_.

The next thing to do is to define the free energy density. If the space-time R2 is

restricted to a region Ω with finite volume V , it is a well known and general fact that the free energy is extensive in the volume. Therefore, it is more useful to study its density, defined as:

f (θ) ≡ 1 V h

F (θ) − F (0)i, (1.27) where we have subtracted the value it assumes at θ = 0 in order to set f(θ = 0) = 0. Indeed, f(0) is an unobservable quantity that plays no role in our discussion, therefore, it can be eliminated without any loss of generality.

Assuming that f is an analytic function of θ (assumption that holds at least in the large-N limit, as shown in reference [34]), one can expand it in Taylor series:

f (θ) = ∞ X n=1 a2n θ2n (2n)!. (1.28) Collecting a global factor a2θ2/2, one gets:

f (θ) = 1 2a2θ 2 _{1 +} ∞ X n=1 2 (2n + 2)! a2n+2 a2 θ2n ! ≡ 1 2a2θ 2 _{1 +} ∞ X n=1 b2nθ2n ! , (1.29) where the coefficients

b2n≡ 2 (2n + 2)! a2n+2 a2 (1.30) have been introduced to have a simpler parametrization.

The a2n coefficients are related to the cumulants of the topological charge Q.

The cumulants kn of a random variable X are defined in terms of the moments

µn ≡ hXni and in terms of the generating function F:

F (θ) ≡ heiθX_{i =} ∞ X n=0 in n!µnθ n_. _(1.31)

(24)

In particular, cumulants are defined as the coefficients of the θ-expansion of log F: log F (θ) = log heiθXi =

∞ X n=1 in n!knθ n_. _(1.32)

Using the expression of the θ-expansion of F in equation (1.31) it is easy to express the n-th cumulant kn in terms of the first n moments µ1, ..., µn. The

expression of the first four cumulants reads: k1 = µ1, k2 = µ2− µ21, k3 = µ3 − 3µ2µ1+ 2µ31, k4 = µ4− 4µ3µ1− 3µ22+ 12µ2µ21− 6µ 4 1. (1.33) Using the definitions of cumulants and observing that every derivative with re-spect to θ of the partition function brings down a factor iQ inside the path integral, it is easy to show that:

dm_{f (θ)} dθm = − 1 V dm dθmlog Z(θ) = −im V km(Q)(θ). (1.34) For example, for m = 1 one gets

df (θ) dθ = −1 V Z0(θ) Z(θ) = −i V hQi (θ) = −i V k1(Q)(θ). (1.35) Using equation (1.34) and evaluating it for θ = 0, one obtains the expression of the a2n coefficients: a2n= d2n_f dθ2n(θ = 0) = −i2n V k2n(Q)(θ = 0) = (−1)n+1 V k2n(Q)(θ = 0). (1.36) Hence, the expression of the b2n coefficients is:

b2n = 2 (2n + 2)! a2n+2 a2 = (−1)n 2 (2n + 2)! k2n+2(Q)(θ = 0) k2(Q)(θ = 0) . (1.37) The coefficient a2 is known as topological susceptibility and is usually indicated

with the Greek letter χ. Using the fact that hQi (θ = 0) = 0, one can express the topological susceptibility as:

χ ≡ a2 =

1 V hQ

2_{i (θ = 0).} _(1.38)

Therefore, one can express the free energy density in terms of the topological susceptibility as: f (θ) = 1 2χθ 2 _{1 +} ∞ X n=1 b2nθ2n ! . (1.39) 22

(25)

1.4. Large-N expansion Combining equations (1.34) and (1.39), one obtains the θ-expansion of the cumu-lants of Q in terms of χ and of the b2n coefficients, which will be extremely useful

later. The expression of the first four cumulants up to b4 is:

k1(Q) V = iχθ1 + 2b2θ 2_{+ 3b} 4θ4+ O(θ5), k2(Q) V = χ1 + 6b2θ 2_{+ 15b} 4θ4+ O(θ5), k3(Q) V = −iχ12b2θ + 60b4θ 3_{+ O(θ}4_), k4(Q) V = −χ12b2 + 180b4θ 2_{+ O(θ}3_). (1.40)

1.4 Large-N expansion

As discussed in subsection 1.3.1, the perturbative expansion of the theory is allowed only in the high-energy regime. Indeed, thanks to the asymptotic freedom, it is in this regime that the coupling constant of the theory λ 1. However, many interesting physical properties of the CPN −1 model emerge only

in the low-energy regime, where non-perturbative effects are dominant. The CPN −1 offers an analytic way to investigate the non-perturbative behaviour of the theory: the large-N limit.

The large-N limit is obtained when the number of bosons of the theory N → ∞ and the coupling constant λ → 0 as 1/√N, so that

g ≡ N λ2 (1.41) is kept fixed for every value of N. The quantity g is called t’Hooft coupling. The large-N limit is equivalent to the large number of colours limit proposed by Witten and Veneziano to explain the U(1)a anomaly, but, unlike in QCD, in the

CPN −1 model analytic calculations is possible in this scheme.

Using the reparametrization of the coupling presented in equation (1.41), the total action becomes

S = N g

˜

S0− iθQ, (1.42)

thus, it is possible to perform a systematic 1/N analytic expansion of the partition function around the large-N saddle point for arbitrary value of g, which allows a non-perturbative renormalization of the theory (see references [28, 36]).

The regularization usually chosen in the large-N limit is the same adopted in the perturbative case, obtained by introducing an ultra-violet cut-off ΛU V. At

leading order in 1/N, one gets that the renormalized coupling and the bare one are related by 4π gB(ΛU V) = log Λ 2 U V µ2 + 4π gR(µ) . (1.43)

(26)

This equation leads to the β function β(gR) ≡ µ dgR(µ) dµ gB(ΛU V) = −g 2 R 2π + O 1 N , (1.44) which is equal to the one found with perturbation theory in subsection 1.3.1. This assures that asymptotic freedom holds in the large-N limit too (which is consistent with the perturbative result, which holds for every N) and that it is not only a perturbative property of the theory.

The solution of the equation for the running coupling is, has already seen in subsection 1.3.1, 2π gR(µ) − 2π gR(µ0) = log µ µ0 , (1.45) therefore, one gets:

µ0e−2π/gR(µ0) = µe−2π/gR(µ) ≡ Λ0. (1.46)

This fact is referred to as dimensional transmutation and means that the renor-malized theory generates dynamically an energy scale Λ0, despite the process of

regularization started from a bare theory which did not contain any dimensional scale. Besides, Λ0 is independent on the energy scale µ at which the theory is

renormalized, as shown in equation (1.46). Dimensional transmutation is very important since it is connected to the mechanism of dynamical mass generation for the z field and is similar to what happens in pure-gauge QCD for gluons. However, in the CPN −1 _{theory, the dynamical mass acquired by the z field can}

be computed analytically in the large-N scheme.

The mass term for the z field can be obtained imposing the constraint ¯zz = 1 adding to the total Lagrangian a Lagrange multiplier of the type Lc = σ(¯zz−1)(σ

is an auxiliary field that, if integrated away supposing z unconstrained, imposes the desired constraint). Indeed, quantum corrections make σ acquire a non-zero vacuum expectation value hσi ≡ M2 _{6= 0}_{. If one then defines the fluctuations of}

σas S ≡ σ − M2 and substitutes this expression into Lc, obtains a mass term for

the z fields of the type M2_zz_¯ _.

At first order in 1/N, the mass gap reads M = µe−2π/gR(µ)_{+ O} 1 N = Λ0+ O 1 N . (1.47) Note that M depends on the coupling constant gR non-perturbatively and that,

at leading order in 1/N, the mass gap does not depend on θ. Indeed, in reference [42], it is reported that

M = M (N = ∞) + M2θ2+ O(θ4), (1.48)

where M2 is suppressed as O(1/N2). Quantum corrections generate also a

ki-netic term for the gauge field, which, therefore, becomes a dynamical field and generates a confining potential for the z field, as it is well-known happening for an abelian gauge field in two space-time dimensions.

(27)

1.4. Large-N expansion From the dynamically acquired mass one can define a length scale called correla-tion length and defined as

ξc ≡ M−1. (1.49)

This name is due to the fact that the two-point connected correlation function of the projector

P (x) ≡ z(x) ⊗ ¯z(x) (1.50) decays exponentially with ξc as characteristic length scale:

G(x) ≡ hTr [P (x)P (0)]i − 1

N |x|→∞∼ e −|x|_ξc

. (1.51) In the context of large-N expansion, however, it is more useful to define a length scale from the second moment of G(x):

ξ2 ≡ 1 R G(x)d2_x

Z

G(x)|x|2d2x. (1.52) Indeed, it can be shown that ξ allows an expansion in powers of 1/N while ξc

introduces non-trivial power corrections since, in the large-N limit, the two length scales are connected by the relation

ξ ξc = r 2 3+ O N−23 .

Therefore, ξ turns to be more suitable for a large-N expansion.

For what concerns the θ-expansion of the free energy f, one obtains, for N → ∞, ξ2f (θ) = 1 4πNθ 2_{+ O} 1 N2 , hence ξ2χ = 1 2πN + O 1 N2 . (1.53) Rather cumbersome calculations show that the next-to-leading-order term of the 1/N expansion of ξ2_χ is ξ2χ = 1 2πN + e2 2πN2 + O 1 N3 (1.54) where e2 ≈ −0.38088 (see references [22, 42]).

The leading large-N behaviour of the b2n coefficients is b2n = O(N−2n). In

par-ticular, the explicit expression of the leading term of the 1/N expansion of the first two coefficients is:

b2 = − 27 5N2 + O 1 N3 , (1.55) b4 = − 25338 175N4 + O 1 N5 . (1.56) In general, both χ and the b2n coefficients vanish for N → ∞; besides b2n < 0for

(28)

(29)

Chapter 2 Lattice formulation of the model

In this chapter we will define the lattice CPN −1 _{model after a general}

discus-sion about how a discretization of a continuum quantum field theory is achieved. In particular, we will discuss all the possible formulations of the topological charge on the lattice and the lattice version of the quantities we are going to measure during the simulations. Then, we will discuss how the continuum limit is prac-tically achieved on the lattice using renormalization-group arguments. Lastly, we will discuss the main local algorithms usually employed in lattice field theories’ simulations and all the problems and limitations that they present.

2.1 General rules for the discretization of a

quan-tum field theory

2.1.1 Lattice regularization and discretization of space-time

In order to make a Monte Carlo simulation of a quantum field theory, it is necessary to define the theory on a finite-volume, discrete lattice. This fact is indispensable since it provides a non-perturbative regularization of the path integral which is also suitable for a numerical approach, since the number of degrees of freedom of the lattice theory is finite (unlike the continuum theory which has an infinite number of degrees of freedom). In this case, the ultraviolet cut-off ΛU V is provided by the introduction of the lattice spacing a, in particular

ΛU V = 1/a. In the following discussion we will choose a square-lattice R with

V = L2 _{sites and periodic boundary conditions.}

The limit in which the regulator is removed, a → 0, is addressed to as continuum limit since, in this limit, the lattice R tends to a continuous space-time with finite volume VF = a2L2 = a2V. We will discuss in detail how to achieve the

continuum limit of a lattice field theory later, in section 2.5. The first implication of the discretization of the space-time is that the integral is substituted by a finite

(30)

Chapter 2. Lattice formulation of the model sum over the sites of the lattice:

Z

R2

d2x → a2X

x∈R

.

The discretization of the integral will enter the definition of the lattice action of the model.

2.1.2 Discretization of the simple derivative

In order to discretize a field theory on the lattice, it is necessary to define a lattice version of the simple derivative. However, there is not a unique way to do so. In our work, we have chosen the forward derivative, which reads

∂µz(x) →

1

a[z(x + ˆµ) − z(x)] ,

where x ± nˆµ stands for the n-th near neighbour of x in the direction ±ˆµ. It is trivial to show that

1

a[z(x + ˆµ) − z(x)] = ∂µz(x) + O(a). Indeed, using Taylor expansion, one has:

z(x + ˆµ) = z(x) + a∂µz(x) + O(a2).

2.1.3 Gauge connection and discretization of the covariant

derivative

As already seen in chapter 1, z(x) and ∂µz(x) transform differently under

a gauge transformation. This is due to the fact that gauge transformations are local transformations of the fields, thus, z(x+dx) and z(x) transforms differently. Indeed, one has:

z(x + dx) → z0(x + dx) ≡ eiΩ(x+dx)z(x + dx) = eiΩ(x)+i∂µΩ(x)dxµ+O(dx2)_{z(x + dx)}

= ei∂µΩ(x)dxµ+O(dx2)eiΩ(x)z(x + dx) 6= eiΩ(x)_{z(x + dx).}

Therefore, to obtain a gauge-invariant theory, we had to introduce the covariant derivative Dµ. Here, we will present an alternative way of obtaining a

gauge-covariant derivative that is easier to discretize on the lattice. Let us start defining the infinitesimal gauge connection as

U (x ← x + dx) = eiAµ(x)dxµ _{+ O(dx}2_{) = 1 + iA}

µ(x)dxµ+ O(dx2). (2.1)

The gauge connection is associated to a rectilinear path that starts from the point x + dx and finishes in the point x and has the remarkable property that U (x ← x + dx)z(x + dx)transforms as z(x) under a gauge transformation. This

(31)

2.2. Lattice CPN −1 _action

fact has a clear geometrical interpretation: U(x ← x + dx) parallel transports z from the point x + dx to the point x along a rectilinear, infinitesimal, path. Indeed, under a gauge transformation and limiting to first order in dx, one has:

U (x ← x + dx)z(x + dx) → U0(x ← x + dx)z0(x + dx) = 1 + iAµ(x)dxµ− i∂µΩ(x)dxµ+ O(dx2) eiΩ(x+dx)z(x + dx).

Expanding iΩ(x + dx) in the exponential, one gets

1 + iAµ(x)dxµ− i∂µΩ(x)dxµ+ O(dx2) eiΩ(x)1 + i∂µΩ(x)dxµ+ O(dx2) z(x + dx)

= eiΩ(x)U (x ← x + dx)z(x + dx),

as previously stated. Using the gauge connection, it is easy to rewrite the covari-ant derivative of z in an alternative way, which can be easily discretized on the lattice. Indeed, neglecting O(dx2₎ terms, one has:

U (x ← x + dx)z(x + dx) − z(x) = z(x + dx) − z(x) + iAµ(x)z(x)dxµ

= [∂µz(x) + iAµ(x)z(x)] dxµ = Dµz(x)dxµ.

Thus, introducing the lattice gauge connection associated to U(x ← x + dx) Uµ(x) = eiaAµ(x)+ O(a2) = 1 + iaAµ(x) + O(a2), (2.2)

one can write immediately the expression for the lattice covariant derivative: Dµz(x) →

1

a[Uµ(x)z(x + ˆµ) − z(x)].

This definition of lattice covariant derivative transforms correctly under a gauge transformation and has the correct continuum limit, as can be trivially veri-fied expanding Uµ(x) and z(x + ˆµ). The lattice gauge connection Uµ(x) can

be geometrically interpreted as an operator living on the link between the two near-neighbour sites x + ˆµ and x, as shown in Figure 2.1.

x

x + ˆ

µ

U

µ

(x)

Figure 2.1: Geometrical interpretation of the lattice gauge connection.

2.2 _{Lattice CP}

N −1

action

Using the rules defined above, one can easily obtain the lattice action for the CPN −1 model without the θ-term (which will be discussed in the next section):

S₀(L)= −2N g X x∈R 2 X µ=1 {<[ ¯Uµ(x)¯z(x + ˆµ)z(x)] − 1},

(32)

Chapter 2. Lattice formulation of the model which satisfies, for a → 0, S(L)

0 = S0 + O(a). Note that we have directly used

the t’Hooft coupling since we are interested in a numerical study of the large-N limit.

In order to diminish the correction to the continuum limit, one can define an improved version of the lattice action that takes into account higher-order terms in the lattice spacing. The idea is to add to the action appropriate O(a) terms so that they cancel exactly the O(a) terms that emerge from the a-expansion of the action (see references [28, 39]). The tree-level, O(a), Symanzik-improved lattice action for the CPN −1 _is

S₀(L)= −2N g X x∈R 2 X µ=1 c1{<[ ¯Uµ(x)¯z(x + ˆµ)z(x)] − 1} −2N g X x∈R 2 X µ=1 c2{<[ ¯Uµ(x + ˆµ) ¯Uµ(x)¯z(x + 2ˆµ)z(x)] − 1},

with c1 = 4₃ and c2 = −₁₂1. This lattice action contains, besides the

near-neighbour interactions, also terms that couple next-near near-neighbours (these are the O(a) terms) and satisfies S(L)

0 = S0 + O(a2) for a → 0. Dropping the

irrele-vant constants inside the curvy brackets, one obtains the expression: S₀(L)= −2N g X x∈R 2 X µ=1 c1<[ ¯Uµ(x)¯z(x + ˆµ)z(x)] −2N g X x∈R 2 X µ=1 c2<[ ¯Uµ(x + ˆµ) ¯Uµ(x)¯z(x + 2ˆµ)z(x)]. (2.3) For later convenience, it is useful to set β ≡ 1/g and to define EL≡ gS

(L) 0 = 1 βS (L) 0 ,

so that we can rewrite the lattice action as

S₀(L)= βEL. (2.4)

2.3 Lattice topological charge

A straightforward discretization of the topological charge is obtained using the plaquette tensor, defined as

Πµν(x) ≡ Uµ(x)Uν(x + ˆµ) ¯Uµ(x + ˆν) ¯Uν(x). (2.5)

Indeed, substituting (2.2) into the expression of Π, one gets: Πµν(x) = eia{aAµ(x)+aAν(x+ˆµ)−aAν(x)−aAµ(x+ˆν)+O(a

2_)}

= eia2Fµν(x)+O(a3)_{= 1 + ia}2_F

µν(x) + O(a3).

(33)

2.3. Lattice topological charge Thus, at first order in a, the relation between Π and F is

={Πµν(x)} = −<{iΠµν(x)} = a2Fµν(x) + O(a3), (2.6)

and the expression of the lattice topological charge is QL= − 1 4π X x∈R 2 X µ,ν=1 <{iεµνΠµν(x)}. (2.7)

The tensor Πµν(x)can be geometrically interpreted as an operator living on the

plaquette obtained rotating clockwise starting from the site x, as shown in Figure 2.2.

x

x + ˆ

µ

x + ˆ

µ + ˆ

ν

x + ˆ

ν

¯

U

ν

(x)

¯

U

µ

(x + ˆ

ν)

U

ν

(x + ˆ

µ)

U

µ

(x)

Figure 2.2: Geometrical interpretation of the plaquette tensor Πµν(x).

It is worth noticing that, being QL an analytic function of U, this formulation

of the lattice topological charge will not result in an integer value for a generic configuration of the field.

Later it will be useful to use the fact that the plaquette tensor can be written as Πµν(x) = ¯Uµ(x + ˆν)Γ(↓)µν(x).

where Γ(↓)

µν(x)is called staple and is the complement of the plaquette Πµν(x)with

respect to ¯Uµ(x + ˆν)(see Figure 2.3). The full expression of the down-staple reads

Γ(↓)_µν(x) = ¯Uν(x)Uµ(x)Uν(x + ˆµ).

Similarly, one can write the complex conjugate of the plaquette as ¯

Πµν(x) = ¯Uµ(x)Γ(↑)µν(x),

where we have introduced the up-staple (depicted in Figure 2.4) Γ(↑)_µν(x) = ¯Uν(x + ˆµ)Uµ(x + ˆν)Uν(x).

(34)

Chapter 2. Lattice formulation of the model

¯

U

µ

(x + ˆ

ν)

Γ

(↓)µν

(x)

Figure 2.3: Geometrical interpretation of the down-staple Γ(↓) µν(x).

Γ

(↑)µν

(x)

¯

U

µ

(x)

Figure 2.4: Geometrical interpretation of the up-staple Γ(↑) µν(x).

It is also possible to formulate two geometrical definitions of the topological charge on the lattice that will result in an integer value for every field configuration (ex-cept for a set of measure zero) if we assume periodic boundary conditions, both reviewed and discussed in reference [42].

One definition, originally proposed in reference [5], makes use of the projector P defined in chapter 1 and reads:

Qz = 1 2π X x∈R = log Tr[P (x + ˆ1 + ˆ2)P (x + ˆ1)P (x)] + log Tr[P (x + ˆ2)P (x + ˆ1 + ˆ2)P (x)] . (2.8) Another one, presented and studied in reference [13], makes use of the lattice gauge connection Uµ(z) and reads:

QU = 1 2π X x∈R = log[U1(x)U2(x + ˆ1) ¯U1(x + ˆ2) ¯U2(x)] = 1 2π X x∈R = log Π12(x) . (2.9) This definition can be interpreted in terms of magnetic flux; indeed, using equa-tion (2.6), one has:

log [Π12(x)] = ia2F12(x)+O(a2) = ia2B(x)+O(a2) = iΦP[B(x)]+O(a2), (2.10)

(35)

2.4. Lattice correlation length where ΦP[B(x)] = a2B(x) is the magnetic flux through the plaquette Π12(x),

which has a surface S = a2. In the latter geometric definition of the lattice

topological charge, as well as in definition (2.8), the complex logarithm is to be taken in the interval [−π, π] so that it can assume non-zero values. Besides, it is worth noticing that the two geometric definitions of the lattice topological charge are non-analytic functions of the fields. Indeed, they depend on the function f (x) = = log x = arg x. It is thanks to their non-analyticity that they can always return integer values for every configuration.

Lastly, it is important to underline that, despite the fact that integer definitions of the lattice topological charge do exist, homotopy classes on the lattice do not exist. Therefore, it is possible to deform continuously a path with charge Q = n into one with charge Q = n0_{. The notion of topology is, however, recovered in}

the continuum limit since, in this limit, the contribution of discontinuous con-figurations in the path integral is suppressed compared to the one of smooth configurations.

2.4 Lattice correlation length

In this section we will briefly derive the definition of the lattice correlation length proposed in reference [15]. This definition satisfies, in the continuum limit,

a ξL(a) → a→0ξ,

where ξ is the length scale defined in equation (1.52).

Starting from the lattice definition of the two-point correlation function of the projector P (x) (which has been defined in section 1.4)

GL(x) = h|¯z(x)z(0)|2i −

1

N, (2.11) one can define its lattice Fourier transform:

˜

GL(p) =

X

x∈R

GL(x)eip·x. (2.12)

For small momenta, it can be shown that (see reference [15]) ˜ GL(p) ∼ p→0 1 m2 L+ 4 P µsin 2 pµ 2 ,

where mL≡ ξL−1. Therefore, one can express the lattice correlation length as

ξ_L2 = 1 4 sin2 k₂ ˜ GL(0) ˜ GL(k) − 1 ,

where kµ = k(0, 1) = 2π_L(0, 1)is the minimum impulse one can have on a finite

lat-tice. It is worth noticing that ˜GL(0) ≡ χmis referred to as magnetic susceptibility,

(36)

Chapter 2. Lattice formulation of the model

2.5 The continuum limit

At the beginning of this chapter we stated that the continuum limit is achieved when the lattice spacing a tends to 0. However, it is not possible to perform this limit directly since, in the discretized model described so far, there is no dimensional quantity. To solve this issue renormalization group comes in our help. Indeed, in chapter 1 we showed that the renormalized coupling is connected to the bare coupling of the regularized theory by the general equation

gR= gR gB(ΛU V), µ ΛU V = gR(gL(a), aµ), (2.13)

where the second expression derives from the fact that, in the case of lattice regularization, one has gB = gL (which is the lattice coupling constant) and

ΛU V = a−1.

In section 1.4 we discussed the running of the renormalized coupling with the energy scale µ at fixed bare coupling. Now we are interested in the answer to the opposite question: how the bare coupling changes with the ultra-violet cut-off at fixed physics, id est fixing gR(µ)?

Taking the logarithmic derivative of equation (2.13) with respect to the lattice spacing, one obtains

adgR da = 0 = ∂gR ∂gL adgL da + a ∂gR ∂(aµ) d(aµ) da . Defining the lattice beta function as

βL≡ −a

∂gL

∂a (2.14)

and remembering that the β function of the running coupling gR is defined as

β ≡ µdgR dµ gL = µ∂gR ∂µ , (2.15) one obtains 0 = −∂gR ∂gL βL+ β.

Therefore, the lattice beta function reads βL=

∂gR

∂gL

−1

β. (2.16)

In general, the CPN −1 _{model’s β function is a polynomial in g}2 R:

β(gR) = β0gR2 + O(g4R). (2.17)

The bare and renormalized coupling, instead, are connected by an equation of the form

gR(gL) = gL(1 + c1g2L+ O(g 4

L)). (2.18)

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

Università di Pisa

Dipartimento di Fisica E. Fermi

Corso di Laurea Magistrale in Fisica

Curriculum Fisica Teorica

Investigation of the topological

properties of the CP

N −1

model

using Monte Carlo simulations

Candidate:

Supervisor:

Abstract

Contents

Introduction

Chapter 1

The continuum CP

N −1

model

1.1

The Lagrangian of the model

1.2

The topological term

1.3

Path integral formulation of quantum CP

model

1.3.1

Partition function

1.3.2

The θ-dependence of the free energy

1.4

Large-N expansion

Chapter 2

Lattice formulation of the model

2.1

General rules for the discretization of a

quan-tum field theory

2.1.1

Lattice regularization and discretization of space-time

2.1.2

Discretization of the simple derivative

2.1.3

Gauge connection and discretization of the covariant

derivative

x

x + ˆ

µ

U

(x)

2.2

Lattice CP

action

2.3

Lattice topological charge

x

x + ˆ

µ

x + ˆ

µ + ˆ

ν

x + ˆ

ν

¯

U

(x)

¯

U

(x + ˆ

ν)

U

(x + ˆ

µ)

U

(x)

¯

U

(x + ˆ

ν)

Γ

(x)

_{Path integral formulation of quantum CP}

_{Lattice CP}