### Facolt`

### a di scienze matematiche, fisiche e

### naturali

### Corso di laurea magistrale in Fisica Teorica

### Anno Accademico 2017/2018

### Scaling behaviour of Ising

### systems at first-order transitions

### Candidato

### Pierpaolo Fontana

### Relatore

### Prof. Ettore Vicari

## Contents

Introduction 3

1 Phase transitions and critical phenomena 8

1.1 Classification and properties of phase transitions . . . 8

1.1.1 Phase diagram for a ferromagnet . . . 9

1.2 Landau theory for FOTs . . . 11

1.3 Renormalisation group theory . . . 14

1.4 Finite-size scaling theory . . . 17

1.5 Finite-size scaling for first-order phase transitions . . . 19

1.5.1 Double gaussian approximation . . . 19

1.5.2 Surface effects in the coexistence region . . . 22

2 The Ising model 26 2.1 General definition of the model . . . 26

2.2 Statistical mechanics of the model . . . 27

2.3 The two-dimensional Ising model . . . 29

2.3.1 Thermodynamic functions . . . 30

2.3.2 Boundary conditions for spin systems . . . 32

3 Monte Carlo methods 34 3.1 General information on MC methods . . . 34

3.1.1 Applications in statistical physics: the importance sampling 35 3.2 Dynamic Monte Carlo: Markov chains . . . 37

3.2.1 Properties of the transition matrix W . . . 38

3.2.2 Equilibrium condition: the detailed balance principle . . . 39

3.3 The Metropolis algorithm . . . 40

3.3.1 The algorithm for the 2D Ising model . . . 41

3.4 Statistical analysis of Monte Carlo data . . . 42

3.4.1 Equilibration of Markov chains . . . 43

3.4.2 Correlated data: the integrated autocorrelation time . . . 43

3.4.3 Binning analysis . . . 45

3.5.1 The Γ-method . . . 46

3.5.2 The binning method . . . 47

4 Dynamic finite-size scaling: results for OBC 50 4.1 Details of the dynamics . . . 50

4.2 Dynamic scaling behaviour in the coexistence region . . . 51

4.2.1 General arguments for PBC . . . 51

4.2.2 General arguments for OBC . . . 54

4.2.3 Coarse-grained flip dynamics . . . 55

4.3 Data and results in the coexistence region . . . 58

4.4 Conclusions . . . 61

5 Finite-size scaling: results for EFBC 63 5.1 General background and definitions . . . 63

5.1.1 Details of the dynamics . . . 64

5.2 Square lattice: anomalous FSS . . . 64

5.2.1 FSS for general boundary conditions . . . 65

5.2.2 Data and results . . . 67

5.3 Anisotropic lattice: data and results . . . 71

5.4 Conclusions . . . 73

6 Finite-size scaling: results for OFBC 74 6.1 General arguments and definitions . . . 74

6.1.1 Details of the dynamics . . . 75

6.2 Scaling of the magnetization . . . 76

6.3 The equilibrium dynamic exponent and its computation . . . 78

6.3.1 Estimator of τ : the binning method . . . 79

6.3.2 Data and results: value of z . . . 80

6.4 Conclusions . . . 81

Conclusions 83 A Test of the 2D Ising model MC code 85 A.1 Thermalization . . . 85

A.2 Trend of thermodynamic quantities . . . 86

A.3 FSS analysis: critical exponents . . . 89

## Introduction

Phase transitions are common and familiar phenomena in nature: in the mod-ern classification they are divided into two wide blocks, i.e. discontinuous or first-order transitions (FOTs) and continuous transitions. The first ones are characterized by the presence of a latent heat, and in general the first deriva-tives of the free energy show jump singularities at the transition point; for the second ones there is no latent heat, but in general higher derivatives of the free energy are divergent or discontinuous. To continuous transitions are associated critical points in a phase diagram.

A phenomenological theory that can be used to describe phase transitions is the Landau theory: this is a useful theory since it introduces the concept of the or-der parameter, a thermodinamic variable distinguishing the different phases of the system. The free energy can be expanded in powers of the order parameter, taking into account also the symmetries of the model that we are describing. Nevertheless, continuous phase transitions and critical phenomena find a correct explanation within the renormalization group (RG) theory: at the base of this approach there is the fact that continuous phase transitions are characterized by a divergent correlation length associated to the fluctuation of the order pa-rameter. While critical phenomena are described in a convincing way with the RG theory, the case of FOTs is different and in a way more complicated, since there is no systematic and well-established theory that describes them.

However, all the previous properties and singular behaviours of phase transi-tions can occur only in the thermodynamic limit, i.e. when the volume of the considered system tends to infinity: clearly this is not a realistic situation, since physical experiments deal with finite systems, whose properties near a phase transition are characterized by a finite-size scaling (FSS) behaviour. Therefore the understanding of finite-size effects at phase transitions has great phenomeno-logical importance, because it allows us to interpret in the right way experiments and also numerical investigations of finite-systems at the transition point. In the case of a continuous phase transition this FSS behaviour is characterized by power laws with universal critical exponents, in the sense that they depend only on global features of the system (e.g. symmetries of the hamiltonian,

boundary conditions. For a FOT the FSS behaviour may instead depend on the boundary conditions and on the geometry [34]: in the case of periodic boundary conditions (PBC), which is the most studied in literature, the finite-size effects are characterized by a power-law behaviour, whose exponents are related to the space dimension of the system. These effects are different if general boundary conditions are considered: it is then interesting to show how the scaling be-haviour of a finite system is modified in correspondence of different boundary conditions, both for equilibrium and off-equilibrium properties.

In this work we investigate how the behaviour of the 2D Ising model, presenting a field-driven FOT, changes in correspondence of various boundary conditions by using Monte Carlo (MC) simulations and a FSS analysis in the case of a purely relaxational dynamics.

In the case of PBC a universal dynamic FSS theory (DFSS) has been developed
for the dynamic behaviour of coexisting phases, and it has been tested in the
case of the 2D Ising model [29] on a square lattice: for this system, and in
general for a system with a discrete order parameter, the time scale τ (L) of the
relaxational dynamics is very large, i.e. τ (L) ∼ eσL _{where σ is related to the}

interface free energy, due to an exponentially large tunneling time between the coexisting phases at the transition. In the thesis this DFSS theory is extended to the case of open boundary conditions (OBC): the main point is the correct identification of the dynamic time scale of the system. In particular, we show that the time scale τ (L) is again exponential in terms of L but with a different coefficient σ0, due to the fact that the typical configurations in the coexistence region are different compared to the PBC case. We thus verify the behaviour of this time scale and estimate the coefficient σ0by means of numerical simulations with the Metropolis algorithm, which provides a purely relaxational dynamics. In the second part of the thesis we study the equilibrium scaling properties in the case of equal fixed boundary conditions (EFBC) and opposite fixed bound-ary conditions (OFBC). We consider a square lattice of linear dimension L with EFBC and observe the equilibrium scaling properties of the magnetization. We show that an anomalous FSS emerges in the region characterized by the physical coexistence of the two phases, i.e. the magnetization scales with an anomalous scaling variable r1 = hL, with ≈ 1.7. Furthermore, to test also the

depen-dence on the lattice geometry, we do the same analysis for an anisotropic 2L × L lattice with EFBC along the horizontal direction and PBC along the vertical one: in this case the magnetization scales with r1= hL2.

We then consider a square lattice with PBC along the vertical direction and OFBC along the horizontal one, i.e. the spins on the boundary of the lattice in-teract with a column of fixed +1 on the right side and −1 on the left side: in this case the typical configurations at the transition are characterized by an interface separating the phases. We want to estimate the equilibrium dynamic exponent

z of the purely relaxational dynamics: indeed for these boundary conditions the
dynamics is no more exponentially slow as for the PBC, but is characterized
by a power-law behaviour τ (L) ∼ Lz_{. Numerical simulations associated to a}

purely relaxational dynamics allow us to observe this power-law behaviour and estimate the exponent z. It is also observed the scaling of the average mag-netization as a function of r1 = hL2, underlining the connection between the

scaling function of the magnetization and the position of the interface within the lattice.

The effects discussed in the thesis are general and should be observable in var-ious physical situations, where a FOT is approached by varying the external parameters of a finite system. Indeed, the Ising model is useful not only for the description of magnetic systems, but also for a large class of systems showing an order-disorder phase transitions: other examples of such statistical systems are binary alloys and lattice gases. The first are important for the modeling of the order-disorder phase transtions in solids: an example could be the β-CuZn bi-nary alloy, which is a body-centered cubic lattice made up of Cu and Zn atoms. Here the mechanism at the base of the transition is the diffusion process of atoms between the various lattice sites, and such a process is activated by the temperature. In general, a binary alloy is composed by two kinds of atoms, A and B, with positions confined on the sites of a given lattice, with the constraint that each lattice site has only one atom. Assuming only nearest-neighbor inter-actions, the model for binary alloys can be reduced to the Ising model. For what concerns the lattice gas, it is a collection of atoms arranged into discrete cells, forming a lattice of given geometry: each lattice site can be occupied at most by one atom. Assuming again only nearest-neighbor interactions, this model is equivalent to a gas in which the atoms are located on the lattice sites and interact through a given two-body potential. This model in the two-dimensional case could be used to investigate hydrogen adsorbed on metal surfaces, and it can be easily mapped into the Ising model. These examples show that the Ising model can be used to describe several systems in statistical mechanics, but the importance of our studies is not restricted only to this field.

For instance, the QCD phase diagram at finite temperature is expected to have a FOT line, separating the quark-gluon plasma and the hadronic phase [13]. The diagram is showed in fig. 1: considering the left region of the diagram, the quark-gluon plasma is the high-T phase, while the hadronic matter is the low-T phase. The FOT line separating these two phases ends at the critical point E, characterized by a finite value µ = µE.

Heavy-ion collision experiments are trying to find evidence for such a line [30], in order to investigate the features of the hadronic matter’s phase diagram. One of the problem in identifying its signature is the presence of space-time inhomo-geneities in the plasma generated in the collisions: this plasma is expected to be

Figure 1: QCD phase diagram for ms 6= 0 in the plane µ − T , where µ is the

baryon chemical potential. The FOT line in the phase diagram, separating the quark-gluon plasma (QGP) and the hadron gas, is object of current studies in heavy-ion collision experiments.

confined in a small spatial region with a size of a few femtometers (fm), and to hadronize within a time interval of a few fm/c. In the corresponding region of the phase diagram, as the plasma cools down the system should cross the FOT line (fig. 1). As the size and time scales are finite, the singularities at the FOT must be rounded: thus for a correct interpretation of the experimental results is important to clearly understand the effects of space-time inhomogeneities at FOTs. Since, contrary to what usually happens for a typical order-disorder transition, the low-T and high-T phases are the disordered and ordered one, respectively, the dynamics of a quark-gluon plasma corresponds to that of a finite-size statistical system slowly heated across a FOT. Moreover, the hadron gas surrounds the quark-gluon plasma, so the appropriate boundary conditions in the equivalent statistical system must favor one of the phases of the system (in this specific case the disordered one). We clearly understand here that the PBC are not suitable to describe this physical situation, as opposed to other boundaries favoring one specific phase.

The thesis is organised as follows:

• in Chapter 1 the general properties of classical phase transitions are ex-plained, focusing on the Landau theory and on the RG approach. Partic-ular attention is given to FOTs and FSS analysis;

• in Chapter 2 we give some details about the Ising model in d = 2 dimen-sions and the definitions of the various boundary conditions;

explaining the Markov chains, the Metropolis-Hastings algorithm and the statistical analysis of MC data;

• in Chapters 4, 5 and 6 we present our results for the different boundary conditions considered;

• in Appendix A we report the MC code test in the case of PBC, done as a preliminary excercise for the thesis.

### Chapter 1

## Phase transitions and

## critical phenomena

In this chapter the general properties and classification of phase transitions are introduced in a general way. The phase diagram of the uniaxial ferromagnet is explained and the relative critical exponents are defined. Then we discuss first-order phase transitions, marking the fundamental differences as compared with the continuous transitions. Finally finite-size scaling theory is explained.

### 1.1

### Classification and properties of phase

### tran-sitions

A phase transition is a discontinuous change in the properties of a system due to a smooth variation of the external conditions determining its state, such as the temperature or the pressure. The points at which this abrupt change happens are called critical points: they mark the transition of the system from one state of matter to another.

Phase transitions can occur basically in two possible ways:

1. The two or more states of the system on either side of the critical point also coexist exactly at the critical point and they are distinct from each other. However, slightly away from the critical point there is a unique phase with properties connected to one of the previous coexistent phases. In this case we expect to find discontinuities in various thermodynamic quantities as we pass through the critical point. These transitions are called discontin-uous phase transitions or first-order phase transitions (FOTs);

2. The two or more phases on either side of the critical point become iden-tical as it is approached. In this case the difference in the various

ther-modynamic quantities between the phases (e.g. the energy density of the phases) go to zero smoothly. These are continuous phase transitions or second-order phase transitions.

This is the modern classification of phase transitions, as opposed to the histor-ical thermodynamic classification of Ehrenfest: in the latter a phase transition is of ith order when the derivatives of the Helmoltz free energy F with respect to the parameters are continuous up to the (i − 1)th order and some of the ith derivatives are discontinuous at the transition. This is an ambiguous clas-sification since in many transitions the derivatives of F are divergent (and not discontinuous) at the transition, as described in the next sections.

An important quantity that we have to bear in mind is the correlation length ξ, defined physically as the distance over which the fluctuations of the microscopic degrees of freedom are significantly correlated with each other. For FOTs this length is generally finite at both side of transition, while for continuous transi-tions is divergent, ξ → ∞.

It is worth to point out that when ξ → ∞ many properties of the system turn out to be largely independent of the microscopic details of the interaction be-tween the individual atoms (or molecules). These systems fall into one of a relatively small number of different classes, each characterised only by global features (e.g. symmetries of the hamiltonian, number of spatial dimensions). This phenomenon is known as universality of critical phenomena and finds an explanation in the framework of the renormalization group (RG).

Furthermore, we introduce the critical exponents (or critical indices), since ξ and other thermodynamic quantities exhibit power-law dependence on the pa-rameters specifying the distance from the critical point. These exponents are pure numbers and depend only on the universality class of the model considered. We define

t ≡ T − Tc Tc

(1.1) as the reduced temperature, representing the deviation from the critical point. Another important quantity is the order parameter : in general it is an observable equal to zero in one phase and different from zero in the other one (if we have two competing phase).

### 1.1.1

### Phase diagram for a ferromagnet

If a ferromagnetic substance is considered, we observe that there are two impor-tant external parameters: the temperature T and the magnetic field h. If we suppose that the local magnetization is constrained to lie parallel (anti-parallel) to a particular axis, the phase diagram is pretty simple, as shown in fig. 1.1. The relevant theromdynamic quantities are analytic functions of T , h except

Figure 1.1: Phase diagram of a uniaxial ferromagnet. Figure is taken from [5]. M = M (h) is discontinuous. This is a feature of a FOT. As T → Tc this

discon-tinuity in the magnetization approaches zero and ξ → ∞, a sign of a continuous transition. The point (T, h) = (Tc, 0) is called the critical end point : the FOT

becomes continuous; the line h = 0, T < Tc is known as the coexistence line.

For T ≥ Tc is M = 0 for h = 0, while for T < Tc there is a spontaneous

magne-tization different from zero: in particular, the two limits h → 0+ _{and h → 0}−

give two different possible values for the magnetization: lim

h→0+M (T, h) = M0, _{h→0}lim−M (T, h) = −M0, (1.2)

with M0 > 0. Which one of these values the system choose depends on its

previous history (spontaneous symmetry breaking). It is clear from these con-siderations that the order parameter for this transition is the magnetization M . Finally we give the definitions of the principal critical exponents for the case of ferromagnet:

• For T ∼ Tc the specific heat C(T, h = 0) ∼ A|t|−α. The exponent α is a

universal quantity while the amplitude A is not; • For T ∼ Tc, T ≤ Tc is M (T, h = 0) ∝ |t|β;

• For T ∼ Tc the magnetic susceptibility is χ ≡ (∂M/∂h)|h=0∝ |t|−γ;

• For h ∼ 0 is M (Tc, h) ∝ |h|1/δ;

• For T ∼ Tc is ξ(T, h = 0) ∝ |t|−ν;

• For T = Tc and h = 0 is G(ri, rj) ∝ |ri− rj|2−d−η, where G(ri, rj) is the

two point correlation function.

• There is an exponent related to the time-dependent properties of the sys-tem close to the critical point: the typical relaxation time τ diverges according to τ ∝ ξz.

An important result of universality is that, if the phase diagram of the liquid/gas transition is considered, we observe that near the critical point the trend is more or less the same of the ferromagnetic phase diagram: we can then define critical exponents in analogy to those for the ferromagnet, and they are identical to the exponents previously listed.

Since in this work of thesis we will consider only FOTs, in the next sections the relative phenomenological theory and the renormalisation group approach are explained.

### 1.2

### Landau theory for FOTs

In this section we present an elementary thermodynamic discussion of phase transitions introducing the Landau theory, which will clarify the role of the order parameter previously introduced [19].

We will denote with φ the order parameter of the system (the magnetization in the example of the uniaxial ferromagnet), so that φ = 0 in the disordered phase and φ 6= 0 in the ordered one. Transitions are usually classified according to whether the order parameter is a scalar quantity or has vector (or tensor) character. In the case of scalar order parameter, the Landau theory is based on the expansion of the free energy F in terms of φ:

βF {φ(x)} = βF0+
Z
ddx r
2φ
2_{+}u
4φ
4
− hφ + [R∇φ]
2
2d
, (1.3)
with β ≡ (kBT )−1, u, R > 0 and h ≡ βH is the reduced external field. The fact

that u > 0 is essential if F has to be bounded from below. In this expansion φ is small and slowly varying in space. The form of F is consistent with the symmetry against the change of sign of the order parameter, i.e. φ → −φ, in the case of h = 0. We can redefine r in term of the distance from the critical temperature:

r = r0(T − Tc), with r0 > 0. (1.4)

Denoting with V the volume of the system, in the homogeneous case (∇φ = 0) we find for h = 0 that

∂F
∂φ = βV (rφ0+ uφ
3
0) = 0 ⇒
_{φ}
0= 0, (T ≥ Tc)
φ0= ±
q
r0
u(Tc− T ), (T < Tc)
(1.5)
Thus eq. (1.3) yields a continuous transition when T is varied (temperature
driven), while for T < Tc a FOT occurs as a function of h, since the order

parameter jumps from (−r/u)1/2 _{to −(−r/u)}1/2 _{as h change sign. This is an}

example of field driven FOT. For temperature driven FOTs, they can occur in several ways in this framework:

Figure 1.2: (a): case of continuous transition; (b): case of FOT. In both cases a symmetry around φ = 0 is assumed. Figure is taken from [19].

1. If u < 0 a term vφ6_{must be included in the expansion of F , with v > 0. In}

this case F has three minima for T0< T < T1, and the critical temperature

is reached when the three minima are equally deep (see fig. 1.2). T0 and

T1 can be found from r = r0(T − T0):

Tc= T0+

3u2

32r0_{v}, T1= T0+

u2
8r0_{v}.

In this case φ jumps from φ0= ±(3u/4v)1/2 to φ = 0 discontinuously.

2. If there is no symmetry of F against the change of sign of the order
parameter a term wφ3/3 will be present in the expansion. For u > 0 the
free energy F may have two minima, and also in this case the transition
occurs when they are equally deep (fig. 1.3). As in the previous case we
can find:
Tc= T0+
8w2
81r0_{u}, T1= T0+
w2
4r0_{u},

and φ jumps from φ0= −9r/w to φ = 0 discontinuously.

If we consider for eq. (1.3) the variation of φ with the field h for T < Tc, where

Tcis the critical temperature of the continuous transition, we obtain a trend like

that reported in fig. 1.3 (on the right). The order parameter is discontinuous as a function of the external field h, the stability limit (or spinodal ) φsis

φs= φ0 √ 3 = ± r − r 3u, the critical field hc is

hc= ±β −2r 3 3/2 (√u)−1.

Figure 1.3: (a) Free energy as a function of φ in the presence of a cubic term and (b) FOT due to variation of h for T < Tc. Figure is taken from [19].

Furthermore, from general thermodynamic principles we observe that, in ther-mal equilibrium, thermodynamic potentials are convex functions of the respec-tive variables, thus F {φ} should be a convex function of the order parameter: this exludes the possibility of having multiple minima. This means that, if we consider free energies like those reported in fig. 1.2, for states with φ ∈ (−φ0, φ0)

the thermal equilibrium state is a mixed-phase state, not a pure state.

If we denote with χT the susceptibility of the order parameter fluctuations,

states with χT > 0 (i.e. (∂2F /∂φ2)T > 0) are called metastable states while

states with χT < 0 (i.e. (∂2F /∂φ2)T < 0) are called unstable states.

All previous statements are correct if the order parameter is a scalar quantity, but they could be generalized also for vector (or tensor) order parameters. Some general symmetry conditions exist for which Landau theory allows continuous phase transitions, that can be presented in group theoretical language (see ref. [23] for a rigorous treatment) for a general order parameter φ.

Despite that, if these conditions are satisfied a transition could be discontinuous: an example is the case of Ginzburg-Landau hamiltonian, for an n-component order parameter φ, with a cubic anisotropy term that breaks the O(n) symme-try. It can be shown that for n ≥ 4 there are no stable fixed points, and the first-order character of the transition is due precisely to the large number of the order parameter components. Transitions like this are called FOTs induced by fluctuations and they are associated to runaway trajectories in the renormali-sation group flow.

A last remark should be done about the free Landau energy F {φ}: plots of F {φ} reported in fig. 1.2 and 1.3 are not permissible as bulk free energies of a macroscopic system (i.e. global free energies), but they can be interpreted as the result of coarse graining.

Starting from eq. (1.3), we want to eliminate short-wavelength fluctuations. This may be done by dividing the system into cells of linear dimension L and taking φ(x) = 1 Ld X i∈Ld si (1.6)

as the order parameter. In the previous expression, si is the n-component spin

at site i in a lattice model and x is the centre of gravity of the d-dimensional
cell Ld_{. A coarse-grained hamiltonian may be defined as}

e−βHcg{φ(x)}_{= Tr}

siP (φ(x), si)e

−βH{si}_{,} _{(1.7)}

where P is a projection operator, its meaning is related to the fact that we perform the trace keeping a particular configuration of φ(x) fixed. This coarse-grained hamiltonian should have the Landau form if we are close to a continuous transition and the dimension L is much smaller than the correlation length of the system, L ξ. If the latter condition does not hold, the hamiltonian may not have an analytic expansion in terms of φ(x). However, we suppose that L ξ and we consider Hcg= F {φ}.

### 1.3

### Renormalisation group theory

The Landau theory neglects statistical fluctuations; this can be easily seen from the free energy F :

F = −ln Z β = −β

−1Z _{d{φ}e}−βHcg{φ(x)}_{= −β}−1

Z

d{φ}g{φ}, (1.8) where Z is the partition function. If g{φ} is sharply peaked at the minimum of the free Landau energy we may replace it by a delta function: in this way it is F which has to be minimised in order to describe thermal equilibrium.

The Ginzburg criterion provides an argument to understand if it is legitimate to neglect these fluctuations. We consider the fluctuations δφ = φ − φ0 of the

order parameter in the temperature region T < Tc: they will make a small

contribution to the free energy if h(δφ)2_{i}

T ,L φ20. This last relation may be

written as
h(δφ)2_{i}
T ,L=
1
L2d
X
i,j∈Ld
hsisjiT − φ20 ∼
1
Ld
Z
Ω
ddx hφ(0)φ(x)iT − φ20 , (1.9)

where Ω is the sphere of volume Ld. Recalling the static limit of the fluctuation-dissipation relation, we can write the correlation function in terms of the

sus-ceptibility χ(~k)

χ(~k)β−1= Z

Ω

ddxei~k·~x hφ(0)φ(x)iT − φ20 . (1.10)

Considering x . ξ, we obtain hφ(0)φ(x)iT− φ20∝ R−2x2−d. From eq. (1.10) we

may write h(δφ)2iT ,L∝ R−2L2−d. Choosing L ∼ ξ and computing χ in φ = φ0

we obtain

1 R2ξ2−dφ20∝ Rdt

4−d

2 _{,} _{(1.11)}

where t is the reduced temperature. This is the Ginzburg criterion: we observe that for d < 4 and R < ∞ fluctuations of the order parameter become impor-tant, since near Tceq. (1.11) does not hold. This means that the Landau theory

is wasteful.

The renormalisation group (RG) developed by Wilson and Kogut gives a cor-rect description of critical phenomena. The idea is to construct an iteration method where short-wavelength fluctuations are integrated out step by step, as anticipated in the previous section.

The general RG transformation can be written as

H(k){φ(k)} = Rk[H{φ}], (1.12)
where R is the operator that reduces the degrees of freedom by a factor bd_{, where}

b is the scale factor. H(k){φ(k)_{} is the hamiltonian obtained after k applications}

of R, and the corresponding rescaling of the order parameter is necessary if we want to keep the partition function invariant under the transformation. It is also useful to think of the couplings in the hamiltonian H{φ} as forming a vector {k} = (k1, . . . , kn), and think about the RG transformations as acting

on the space of all possible couplings:

{k0} = R{k}. (1.13)

This change of {k} induces a RG flow on the coupling constants space. The values of couplings under the RG flow are called running couplings.

If we are in the region T > Tc, after a sufficient number of iterations k we will

have that bkL > ξ: this means that φ(k), the average of the spin in a block of
size bk_{L, is decoupled from the other blocks. When k → ∞ the hamiltonian}

H reaches the infinite temperature fixed-point: the spin field is distributed according to a gaussian distribution.

If we are at T = Tc there is a non-trivial fixed point, represented by H∗, at

which H converges when k → ∞. If we assume that R is differentiable at the fixed point, we can linearize the RG equations near H∗: the linear operator associated to R is represented by L. We also define the eigenoperators Qi and

the eigenvalues Λi such that LQi= ΛiQi. Thus we can write: H = H∗+X j hjQj, (1.14) H0= LH = H∗+X j h0jQj = H∗+ X j hjΛjQj, (1.15)

where h0_{j} = Λjhj = bλjhj, with the exponents λj independent from the choice

of the block size. The hj are the scaling fields and we can see from the relation

h0_{j}= bλj_{h}

j that they transform multiplicatively near the fixed point; physically,

they are linear combinations of the couplings deviations from the fixed point. The exponents λj are called RG eigenvalues. Scaling fields are classified

accord-ing to the sign of these exponents: for λi> 0 the scaling field hi is relevant, for

λi= 0 is marginal and for λi< 0 is irrelevant.

We can consider the RG transformation of the reduced free energy density de-fined as a function of the scaling fields, i.e. f ({h}). Since the RG transformation preserves the partition function and the couplings flow according to eq. (1.13), the law of transformation is

f ({h}) = g({h}) + b−df ({h0}), (1.16)
where g{h} is an analytic function of the couplings, even at the critical point.
It is clear that the free energy transforms inhomogeneously under RG. Despite
that, the singular part of the free energy density transforms homogeneously
under the RG, indeed f ({h0}) = bd_{f ({H}). In terms of the scaling fields we}

have:

f (h1, . . . , hn) = b−df (bλ1h1, . . . , bλnhn). (1.17)

If we relate h1and h2to the reduced temperature and to the external magnetic

field (k1,2 are constant values)

h1= k1t, h2= k2h, (1.18)

and choose bλ1_{t = 1 the previous expression becomes}

f (t, h, h3, . . . , hn) = td/λ1f (k1, k2htλ1/λ2, h3tλ1/λ3, . . . , hntλ1/λn). (1.19)

Thus the RG eigenvalues are related to the critical exponent defined in the pre-vious sections, in particular λ1= ν−1 and λ2= βδ/ν.

In the case of the ferromagnetic uniaxial model considered in this chapter there are two parameters that must be adjusted to bring the system to its critical point, and they are the temperature and the magnetic field. This is the reason why we expect the fixed point corresponding to this universality class to have

only two relevant scaling fields, as shown before. These fields are respectively the thermal scaling field and the magnetic scaling field.

At last, if we are in the region T < Tc the RG transformation brings the

sys-tem to the zero sys-temperature strong-coupling fixed point: we still have a fixed point but with only one relevant RG eigenvalue, λ2= d. A fixed point with an

eigenvalue λ = d is called discontinuity fixed point [21].

A FOT could be described also following the method reported in ref. [22], where in the RG approach is obtained as a limiting case of a continuous transi-tion. Considering a phase diagram associated to an ”ordinary” field-driven FOT in which the system switches from one noncritical phase to another noncritical phase, like that reported in fig. 1.1, the magnetization should follow the power law

M − Mt∼ ±D | h − ht|1/δ,

where htis the transition field and Mtis the mean magnetization on the phase

boundary (both quantities are zero for the phase diagram considered). A
dis-continuity in the order parameter M is obtained choosing δ → ∞, thus this
is a description of a FOT. If we would represent this transition by a RG fixed
point, we have the recursion relation for the field h0 = bλ_{h and f (h}0_{) = b}d_{f (h).}

Choosing b|h|1/λ = 1 we obtain f (h) ∼ D±|h|1/δ with D± ∼ f (±1). But the

order parameter M is proportional to ∂f /∂h at fixed temperature, and that leads to the relation:

d λ = 1 −

1

δ. (1.20)

In the limit δ → ∞ this relation implies λ = d, reproducing the original FOT.

### 1.4

### Finite-size scaling theory

We have seen that discontinuities or divergences in thermodynamic quantities are features of phase transitions; however, these singular behaviours occur when the thermodynamic limit is considered, i.e. when all the dimensions of the sys-tem under consideration tend to infinity. If some or all of these dimensions are finite we expect that, at the transition, thermodynamic quantities will be both rounded and shifted.

The finite-size scaling (FSS) method is a powerful tool used to extrapolate the information available from a finite (or partially infinite) system to the thermody-namic limit. The starting point for the method’s explanation is the identification of the length scales involved in a finite-size system; they are basically three: the correlation length ξ, the linear dimension of the system L and the microscopic length a (i.e. the parameter that governs the range of the interactions). In principle, all the thermodynamic quantities may depend on the dimensionless

ratios ξ/a, L/a. The finite-size scaling hypothesis assumes that, close to the
critical point, the length a drops out: this means that we can consider only the
ratio ξ/L in the description of the system. For example, in the case of the
uni-axial ferromagnet described in the subsec. 1.1.1 the susceptibility behaves like
χ ∼ |t|−γ (for L → ∞); since ξ ∼ |t|−ν we have χ ∼ ξγ/ν_{. In a finite geometry}

characterized by the length L we thus expect that

χ = ξγ/νφ(ξ/L), (1.21) where φ(x) is a scaling function, with the property φ(x → 0) = const., that contains the precise way in which χ gets cut off close to the critical temperature. Eq. (1.21) contains all the information we need about our system’s behaviour with varying size L. Nevertheless, this form is not very useful since it still contains ξ, the correlation length in the infinite system, which we don’t know. It is convenient to reorganize the equation a little, using the power-law behaviour of ξ as a function of t:

χ = Lγ/νφ(Le 1/νt), (1.22) where eφ is another scaling function. From this last equation is clear that all the L-dependence of χ is already showed explicitly: eφ does not contain any extra hidden dependence on L which is not accounted for. Thus, assuming the validity of FSS hypothesis, studies of finite systems with different L can give information about the critical exponent γ of the infinite system. This method can be easily extended to the other thermodynamic quantities associated to the uniaxial ferromagnet.

An heuristic derivation of the FSS hypothesis may be given with the help of the RG ideas. We have only to extend the arguments already explained in the case of a finite geometry. We will assume that the system does not have boundaries (e.g. choosing periodic boundary conditions). After a first rescaling of scale b, the coarse-grained system has a size L/b. Since the RG transformation is local, it will be identical in the infinite system, if the ratio L/a is large. Thus eq. (1.16) may be generalized as

f ({h}, L−1) = g({h}) + b−df ({h0}, bL−1). (1.23)
Therefore L−1could be considered as a scaling variable with RG eigenvalue λ =
1, and this is the reason for the FSS hypothesis. In fact, defining ∆f = f (L−1_{)−}

f (0) in order to eliminate g and iterating the procedure log(L/L0)/ log(b) times

we obtain ∆f ({h}, L−1) = L L0 −d ∆f L L0 λ1 h1, . . . , L0 ! . (1.24)

We assume also that the irrelevant scaling variables, i.e. terms like (L/L0)λjhj

for j > 2, are small, so we may ignore them: this is valid as long as the free energy is an analytic function in such variables. It follows that the quantity

χ = ∂
2_{(∆f )}
∂h2
_{h=0}
(1.25)
scales according to:

∆χ = Lγ/ν∆ eφ(L1/νt), with ν = 1 λ1

. (1.26)

From this last equation, scaling laws like eq.s (1.21) and (1.22) follow immedi-ately.

If f is not an analytic function of the irrelevant variables, the terms (L/L0)λjhj

can not be neglected. This happens for d > 4 in the case of Ising-like systems: using field-theoretical RG arguments it can be shown that there is a violation of FSS for d > 4, while for d ≤ 4 FSS holds [25].

### 1.5

### Finite-size scaling for first-order phase

### tran-sitions

We have seen in the previous section that the FSS theory is a powerful tool for the studies of continuous transitions in the presence of finite systems. In this section we want to explain the FSS in the case of FOTs, in particular by discussing the finite-size rounding in the case of a field-driven FOT by means of the thermodynamic fluctuation theory.

### 1.5.1

### Double gaussian approximation

Our starting point is again the uniaxial ferromagnet: in an infinite system the order parameter of the transition φ (i.e. the magnetization in this case) jumps from φ0 to −φ0(fig. 1.3) when h → 0+. In a system with all linear dimensions

L 9 ∞ no singularities occur, thus the variation of φ with h is perfectly smooth: rather than an infinitely steep variation with h, φ has a large but finite slope. FSS theory of FOTs may be used in order to locate (in practice) phase bound-aries and estimate jumps of the order parameter.

We explain the method in the case of a field-driven FOT for a ferromagnetic system below the critical temperature (T < Tc): in ref. [26] is analysed the

behaviour of the system for h = 0 and finite size L with periodic boundary conditions; in particular is supposed that the probability distribution of the magnetization PL(φ) is a sum of two gaussians, centered at +φ

(L)

0 and −φ (L) 0

(the values of the order parameter for a finite system):
PL(φ) =
Ld/2
2p2πkBT χ(L)
e−(φ−φ(L)0 )
2_{L}d_{/(2k}
BT χ(L))_{+ e}−(φ+φ(L)0 )
2_{L}d_{/(2k}
BT χ(L))_{.}
(1.27)
This approximation should be accurate when L ξ for h → 0±: if this condition
holds we can make the substitutions χ(L)→ χ(∞), φ

(L)

0 → φ0. We can generalize

this form of PL(φ) for h 6= 0:

PL(φ) = A

e−[(φ−φ0)2−2χφh]Ld/(2kBT χ)_{+ e}−[(φ+φ0)2−2χφh]Ld/(2kBT χ)

, (1.28) where A is a renormalization constant, whose value is

A = L d/2 2 e− χh2 Ld 2kB T √ 2πkBT χ coshhφ0L d kBT −1 . (1.29)

We can obtain the same result starting from the free energy density f described in terms of the Landau theory (away from the critical region):

f = f0+
r
2φ
2_{+}u
4φ
4_{− hφ −→ P}
L(φ) ∝ e−βf L
d
, (1.30)
with β = (kBT )−1. We can easily identify φ0as the minimum of f with respect

to φ in the case of h = 0, and linearization around this value of the order parameter for h 6= 0 yields χ = (r + 3uφ20)−1. Rewriting the free energy in

terms of χ, φ0 we have f = ef0+ (φ20− φ2)2 8φ2 0χ − hφ, (1.31) near φ = ±φ0is f ∼ ef0+ (φ ∓ φ0)2 2χ − hφ. (1.32)

This means that near φ = ±φ0, the peaks of the probability distribution, the

formulation in terms of f and the double gaussian approximation are equivalent. In between the two peaks eq. (1.28) underestimates the real PL(φ), since in

this regime the latter will be dominated by configurations corresponding to the phase’s coexistence in the system: there will be interfacial contributions to the free energy.

We can rewrite eq. (1.28) by completing the squares in the exponential terms, obtaining finally

PL(φ) ∝ eβhφ0L

d

For h 6= 0 the probability distribution is again a sum of two gaussians but now centered around the shifted values ±φ0+ hχ. We can also see that the weights

of the two peaks are different for these values. From PL(φ) we can obtain the

Figure 1.4: Variation of hφi for uniaxial ferromagnet vs h (in the figure φ ≡ M ). Figure is taken from [24].

physical quantities of interest; for example: hφi =

Z 1

−1

φPL(φ)dφ = hχ + φ0tanh βhφ0Ld. (1.34)

We can see that

hφi =

_{hχ + φ}

0, for βhφ0Ld 1

βhφ2_{0}Ld, for βhφ0Ld 1

(1.35) These results are summarized in fig. 1.4. The width over which the transition is rounded is of the order of

δh ∼ (βφ0Ld)−1. (1.36)

For what concerns the magnetic susceptibility of the finite system χ(L)= ∂hφi ∂h = χ + βφ2 0Ld cosh2βhφ0Ld . (1.37)

It is clear that, instead of a δ-function singularity, we have at h = 0 a smooth
peak of height ∝ Ld_{. In all these expressions we observe that the only scaling}

power of L that appears is Ld_{. In this specific case there is no shift of the}

transition, due to the simmetry h → −h. The shift of the transition may be analysed in the case of boundary conditions that break this inversion symmetry

(e.g. fixed): in this case what is important is the contribution due to surface effects.

However, finite-size effects depend strongly on the geometry of the system: if
the system’s geometry is totally finite, as in the case analysed in this section,
the rounding of a field-driven FOT takes place on a scale proportional to the
inverse of the total volume, i.e. Ld _{(see eq. (1.36)); if the system is infinite in}

one direction, e.g. forming a cylinder of cross-sectional area S, the rounding in h is exponential in S. In ref. [27] is analysed in detail the crossover in the sharpness of a FOT that takes place when the shape of a system goes over from a totally finite geometry to a cylindrical geometry with one infinite dimension.

### 1.5.2

### Surface effects in the coexistence region

In the previous subsection we said that eq. (1.28) is reasonable only near the peaks φ = ±φ0, while it gives an underestimate of the real value of PL(φ)

between them. The reason is that in the regime −φ0 φ φ0 the

proba-bility distribution is dominated by configurations corresponding to two-phase coexistence within the system. The correct value of PL(φ) for φ ≈ 0 can be

computed considering interface contributions in the free energy. We are going to consider for the moment finite blocks in d dimension with periodic or open boundary conditions (PBC/OBC). Their free energy can be written in terms of the partition function as

ZL= e−βFL(h)= Tr(e−βH) =

Z +∞

−∞

dφe−β(FL(φ)−hLdφ)_{,} _{(1.38)}

where we are considering h 6= 0 and FL is defined by eq. (1.7). Thus the

probability distribution is written as PL(φ) =

e−βFL(φ)

ZL

. (1.39)

The free energy FL, depending on the boundary conditions, has its minima

where the corresponding probability distribution is maximal, i.e. at φ = ±φmax

near φ = φ0. The free energy cost of having states with φ ≈ 0 is due to interface

contributions.

Let us analyse firstly the case of a finite block embedded in a larger system: for large enough blocks the free energy FL(φ ≈ 0) will be dominated by

configura-tions characterized by the minimum interface area, i.e. a spherical domain (see fig. 1.5). Denoting with R the radius of this domain, we can give an estimate of it by putting:

Ld 2 = VdR

Figure 1.5: Typical configuration of 2D block for L ξ, φ ≈ 0 in the case where the block is a subsystem of a large system. Red areas indicates domain of positive φ, blue areas have negative φ.

where Ld _{is the volume of the whole system and V}

d is the volume of the

d-dimensional unit sphere. The surface area of this domain is estimated as A = Sd

|φ − φ0|

2φ0Vd

d−1_{d}

Ld−1, (1.41)

and this time Sdis the surface area of the d-dimensional unit sphere. By means

of A we can estimate the free energy cost of the domain (for h = 0) as FL(φ ≈ 0) − FL(φ = φmax) = Afs= Sd

|φ − φ0|

2φ0Vd

d−1d

Ld−1fs (1.42)

where fs is the surface tension: physically this is the difference in free energy

between the state characterized by the coexistence of the phases (φ ≈ 0) and the one characterized by the presence of a single phase with order parameter φ = φmax.

For PBC the minimum free energy excess is again needed for the spherical domain configurations as long as

∆FL≡ FL(φ ≈ 0) − FL(φ = φmax) (1.43)

is less than ∆F_{L}p, i.e. the cost of the configuration characterized by a rectangular
domain extending throughout the block (fig. 1.6a): we are introducing here an
additional index p to stress that PBC are considered. If we call f_{s}p the surface
tension in this case, we can easily observe that

(a) Configuration for PBC. (b) Configuration for OBC.

Figure 1.6: Typical configuration of 2D block for L ξ, φ ≈ 0 in the case of PBC and OBC. Red areas indicates domain of positive φ, blue areas have negative φ.

Imposing ∆FL = ∆F_{L}p we find that there is a critical value φpcrit of the order

parameter given by
φp_{crit}= φ0
"
1 − 2Vd
2
Sd
d−1d #
. (1.45)

We can conclude that for φ > φp_{crit} eq. (1.42) holds again, while in the opposite
case φ < φp_{crit}we have to replace it with eq. (1.44).

The case of OBC is similar to the previous one but this time the typical config-uration in the coexistence region is given by the creation of just one wall (fig. 1.6b). Introducing again an additional index o to distinguish the two cases, we observe that Fo L(φ ≈ 0) − F o L(φ = φmax) = Ld−1fso, (1.46) where fo

s is the surface tension. Again imposing ∆FL= ∆FLowe find the critical

value of φ, i.e.
φo_{crit}= φ0
"
1 − 2Vd
1
Sd
d−1d #
. (1.47)

For φ > φo_{crit}the minimum free energy excess is needed for the spherical domain,
while for φ < φo

criteq. (1.46) holds. There are two final important remarks that

must be discussed:

• In eq. (1.46) it is assumed that the free surface contributions to Fo L(φ ≈ 0)

and Fo

L(φ = φmax) are essentially the same and cancel out;

• We have considered fs, fsp and fso as three different surface tensions. In

with the other, but in principle it is not an allowed operation. We can justify this fact by saying that we expect

lim L→∞fs= limL→∞f p s = lim L→∞f o s = Fs, (1.48)

where Fsis the usual interface tension between two phases of opposite φ

separated by an infinitely large interface. We are actually assuming that in the thermodynamic limit L → ∞ these three surface tensions converge to the same value.

### Chapter 2

## The Ising model

In this chapter we introduce the Ising model, showing its general features in d dimensions. Then we focus on the 2D case, giving also various definitions for the boundary conditions that will be studied in the next chapters.

### 2.1

### General definition of the model

The electrons responsible for magnetic behaviour in many magnetic materials are localized principally near the atoms of a lattice, and the force which tends to orient the spins is a short-range force (exchange interaction).

This consideration gives consistency to the Ising model, introduced by the physi-cist Ernst Ising, which is a simple mathematical model used in statistical me-chanics to simulate a domain in a ferromagnetic substance. It consists of an N -points lattice of discrete spin variables si, where the index i = 0, . . . , N

la-bels the position of each site, which can take only two possible values: si= +1

or si = −1. In the first case, the ith site is said to have spin up (↑), in the

second case it is said to have spin down (↓).

The mutual interaction energy of any couple of lattice spins can be written as −Jijsisj: it is −Jij if the spins have the same sign (↑↑, ↓↓) or +Jij if they have

opposite sign (↑↓, ↓↑). Furthermore, in the presence of an external magnetic field h the interaction energy is −hsi: again, it is −h or +h according to the

sign of the spin si (↑ or ↓ respectively).

Throughout this work we will consider only the case where the coupling Jij

vanishes unless the positions i and j are nearest neighbors on the lattice and does not depend on i and j: this is called the isotropic coupling case, and we can write Jij = J for all i, j ∈ Λ. Furthermore, we will consider (in 2D case)

mainly the square lattice, where the spins are situated at the intersection of a square grid, although there are various 2D lattices other than this one (tri-angular, hexagonal and so on). The motivation is that most of the physical

properties of other types of lattices reveal no new phenomena with respect to the square lattice.

We can formalize these considerations by saying that for a set of lattice sites Λ = {i = 1, . . . , N }, forming a d-dimensional lattice, the energy of a spin con-figuration s = (si)i∈Λ is given by the hamiltonian

H(s) = −JX hi,ji sisj− h X i si, (2.1)

where si ∈ {−1, 1} and the symbol hi, ji denotes a nearest-neighbor pair of

spins.

For h = 0 the hamiltonian is invariant when all the spins change sign simulta-neously: si → −si, this is known as Z2 inversion symmetry. The spontaneous

symmetry breaking associated to the Z2inversion symmetry is a sign of the

pres-ence of a phase transition in the Ising model, depending on the dimensionality: for d = 1 and T > 0 no phase transition is observed while for d ≥ 2 and T < Tc

the spontaneous magnetization is different from zero. For this reason the av-erage magnetization is identified as the order parameter of the ferromagnetic transition.

Exact solutions of the model exist in the specific case of d = 1 and d = 2, they can be explained with the help of the transfer-matrix method.

For d = 3 the critical point can be described by a conformal field theory, as
evidenced by Monte Carlo simulations and theoretical arguments, while near
d = 4 the critical behavior of the model is understood to correspond to the
renormalization behavior of the scalar φ4_{-theory (see ref. [2] for a detailed }

ex-planation). For d > 4 the Ising model’s phase transition is described with the help of mean field theory.

### 2.2

### Statistical mechanics of the model

We summarize here the relevant statistical quantities for the Ising model in d dimension. In statistical mechanics a general system is described by the density matrix, which determines the probability of the occurrence of a given configuration. In the following we will consider the canonical ensamble, where the energy H(s) of the configuration s determines its occurence probability: in this case, a spin configuration s occurs with probability P (s) ∝ e−βH(s), where β = (kBT )−1, T is the temperature and kB is the Boltzmann constant. The

partition function is defined as

Z(β) =X

s∈Ω

where the sum is extended over all s and Ω is the set of all possible spin config-urations (this can be easily generalized also if we consider a continuous model instead of a discrete one, replacing the sum with an integral over the lattice spins).

We define also the expectation values of observables O(s) in the canonical en-samble hOi(β) = 1 Z(β) X s O(s)e−βH(s). (2.3) Other thermodynamic quantities are derived from the partition function Z(β):

• Helmholtz free energy:

F (β) = −1

βlog Z(β) (2.4)

and the associated free energy density: f (β) = 1

VF (β)

• expectation value of the energy at equilibrium (internal energy): U (β) = hHi(β) = − 1 Z(β) ∂ ∂β X s e−βH(s)= − ∂ ∂βlog Z(β), (2.5) that can be written also as a function of the free energy:

U (β) = F (β) − T ∂ ∂TF (β)

• the specific heat capacity CV, i.e. the increase in internal energy caused

by an infinitesimal increase in temperature CV =

∂U (β) ∂T = −β

2∂U (β)

∂β . (2.6)

Using for U (β) the eq. (2.5) we can write it in terms of a second-order derivative of the partition function

CV = β2

∂2

∂β2log Z (2.7)

• The average magnetization hM i is computed by coupling the spins to a homogeneous external magnetic field and differentiating the correspongind h-dependent free energy density with respect to h:

m ≡ * 1 V X i si + = − ∂ ∂hf (β, h) (2.8)

• The n-point correlation functions:

G(n)(i1, . . . , in) = hsi1· · · sini, i1, . . . , in∈ Λ, (2.9)

as a specific case we mention the 2-point correlation function

G(2)(i, j) ≡ G(i, j) = hsisji. (2.10)

With eq. (2.10) we can know how much the spins of the sites i and j are correlated.

If G(i, j) is positive these spins have the tendency to align, and if this condition holds for an arbitrary distance |i − j| between any two lattice spins the system is said to be spontaneously magnetized.

### 2.3

### The two-dimensional Ising model

A square lattice of linear dimension L is considered, so there are N = L2 spins organised in L rows and L columns, as shown in fig. 2.1a. In this section periodic boundary conditions (PBC) are considered, since they endow the lattice with the topology of a torus, as in fig. 2.1b: these are the best boundary conditions that can be choosen in order to simulate an infinite lattice.

(a) An example of 2D lattice configuration.

(b) Topology of the lattice with PBC.

Figure 2.1: An example of configuration is reported on the left figure. If we consider PBC the lattice has the topology of a torus, as showed in the right figure.

An exact solution of the model was found by Onsager [14], in the isotropic coupling case for h = 0. In his original paper he showed that the partition function’s computation can be reduced to an eigenvalue problem. The fact that, under particular conditions, the partition function may be written in terms of a trace of a matrix is at the base of the transfer-matrix method, a powerful

the evaluation its eigenvalues allows us to write the explicit solution.

This method is simple and elegant for the Ising chain (d = 1), but highly non-trivial fo d = 2. For that reason is important to develop some approximate methods that permit us to analyse the most relevant physical aspects and to extract an estimate of critical exponents or other thermodynamic quantities.

### 2.3.1

### Thermodynamic functions

Following the transfer-matrix method cited previously it is possible to compute
all the thermodynamic quantities of interest1. The Helmoltz free energy density
is
f (β) = − log 2 cosh 2βJ − 1
2π
Z π
0
dφ log1
2
1 +
q
1 − K2_{sin}2_{φ}
, (2.11)
where
K ≡ 2
cosh 2βJ sinh 2βJ. (2.12)
The internal energy density is

u(β) = −J coth 2βJ
1 + 2
π 2 tanh
2_{2βJ − 1 K}
1(K)
, (2.13)
where K1(K) is the complete elliptic integral of the first kind, defined as:

K1(K) ≡
Z π_{2}
0
dφ
p
1 − K2_{sin}2_{φ}.

The specific heat is obtained differentiating the internal energy with respect to
the temperature
CV(β)
kB
= 4
π(βJ coth 2βJ )
2_{K}
1(K) − E1(K) − (1 − tanh22βJ )K0 , (2.14)
where K0 _{is defined as}
K0 ≡π
2 + (2 tanh
2
2βJ − 1)K1(K)

and E1(K) is the complete elliptic integral of the second kind, defined as:

E1(K) ≡
Z π_{2}
0
dφ
q
1 − K2_{sin}2_{φ.}

We observe that K1 is singular for K = 1 and the phase transition occurs at

1_{Here we do not report all the calculations in details since they are not relevant for our}

Figure 2.2: Specific heat and internal energy density as functions of K ≡ βJ . Figure is taken from [8].

this point. Therefore all thermodynamic quantities have a singularity of some kind at a critical temperature Tc such that:

2 tanh22βcJ = 1 =⇒ βcJ =

log (1 +√2) 2

Approximately is βcJ ∼ 0.4406. It can be also seen that near the critical

tem-perature the specific heat approaches infinity logarithmically, while the internal energy is continuous: the phase transition at T = Tc does not present latent

heat, then is not of the first order but continuous.

In order to call this phenomenon a phase transition we must evaluate the spon-taneous magnetization, but in this framework this is not possible since the Onsager’s solution holds for h = 0: the presence of an external field makes more difficult the diagonalization of the transfer-matrix. A closed expression is obtained in ref. [17], again using the matrix method. Although the method is very long and non-trivial, the final result is pretty simple:

m(β) =1 − sinh−4_{2βJ}18_{,} _{T < T}

c (2.15)

While is m = 0 for T > Tc. Thus for T < Tc and h → 0 the system is

spontaneously magnetized, and we can conclude that the model undergoes a continuous transition, known as paramagnetic-ferromagnetic transition, for h = 0 and β = βc. The phase diagram relative to the 2D Ising model is the same

analysed in Chapter 1, fig. 1.1: by considering a specific model we are specifying the hamiltonian H, but the properties of the phase transition have been already discussed in Chapter 1.

### 2.3.2

### Boundary conditions for spin systems

All the previous considerations on the Ising model are valid for L → ∞. In a finite square of size L the behaviour of the system depends on the bound-ary conditions considered: to be precise, the presence of physical boundaries contributes to the observable properties of a finite system, due to modified in-terparticle interactions close to (and across) the boundary. All these effects break isotropy and traslational invariance, both built in the theory previously explained.

The net result is that the average values of physical observables may differ near the surface and in the bulk of the system. If we suppose that surface contri-butions are small compared to the extensive properties of the (finite) system, we can write in principle the thermodynamic functions separating bulk terms (proportional to the volume) from surface terms (proportional to the area of the surface).

We have seen in Section 2.1 and 2.2 that we can write H(s) and P (s) given a specific s = (si)i∈Λ in the finite space of all the allowed spin configurations Ω;

this could be done keeping fixed the spin configuration in the infinite
comple-ment Ωc_{of Ω. The fixation of the spin configuration in Ω}c_{is a formal expression}

for the boundary conditions (BC) of the system: they define the interaction of the spins in Ω with a specified configuration sc= (sj)j∈Λc in Ωc.

In general two types of BC can be considered: macroscopic BC, characterized by an average fixed value of the magnetization at the boundaries, and microscopic BC, characterized by fixed spin variables at the boundaries. We label different BC with an index ρ; the most common types of BC are the following:

• Periodic boundary conditions (PBC, ρ = p): often used when the bound-ary effects have to be minimized, e.g. in the studies of bulk critical phe-nomena and FSS in the absence of boundary effects. The configuration sc

is represented by repeated copies of s shifted along each coordinate axis. Formally, in the general d-dimensional case it is

s(r1+ m1L, . . . , rd+ mdL) = s(r1, . . . , rd), (2.16)

with m1, . . . , md ∈ Zd. In the case of nearest neighbor interactions and

d = 2 geometry L × L it is sufficient to specify the spin configurations in the layers of coordinate (r1= 0, r2), (r1= L + 1, r2) for all r2∈ Λ, i.e.

s(0, r2) = s(L, r2), s(L + 1, r2) = s(1, r2). (2.17)

The same may be done also for r2, setting PBC along both directions of

the lattice.

outside the physical lattice there is nothing for the spins to interact with. In the case of nearest neighbor interactions for d = 2 they are simply

s(0, r2) = s(L + 1, r2) = 0. (2.18)

The same may be done also for r2, setting OBC along both directions.

These are known in literature also as zero Dirichlet BC.

• Equal fixed boundary conditions (EFBC, ρ = +, −): it is a generalization
of the previous BC. For OBC the value of the spin in Ωc_{can be only s}

c = 0

while in this case it takes values from a finite set, i.e. sc= {−sm, . . . , sm}.

We will consider only the case sm= ±1, the two possible cases are ρ =

(+), (−):

s(0, r2) = s(L + 1, r2) = +1, (2.19)

s(0, r2) = s(L + 1, r2) = −1. (2.20)

The same may be done also for r2, setting EQFB along both directions.

• Opposite fixed boundary conditions (OFBC, ρ = ±): this is the case of (+) on one side of the lattice and (−) on the other one. These BC play an important role for the creation of interfaces at FOTs, as we will see in the next chapters. Formally:

s(0, r2) = +1, s(L + 1, r2) = −1, (2.21)

while in the other direction PBC, previously defined, can be considered. These are the BC used in the thesis, but also the most common in literature. Other variations with respect to these may be considered, e.g. considering dif-ferent types of BC at the opposite surfaces of the lattice.

As a last remark, we mention that a possible way of creating interfaces be-tween domains of opposite magnetization for contiuous spin’s systems consists in imposing antiperiodic BC (ρ = a):

### Chapter 3

## Monte Carlo methods

In this chapter we first introduce static Monte Carlo (MC) methods in a gen-eral way, then we focus on dynamic MC and Markov chains. Subsequently we describe carefully the Metropolis-Hastings algorithm. In the last part of the chapter the statistical analysis of MC data is explained, giving numerical estimators of the integrated autocorrelation time with two different methods.

### 3.1

### General information on MC methods

MC methods are a wide class of computational algorithms that make use of random numbers to obtain numerical results, with applications in statistical mechanics, quantum mechanics, lattice field theories and ohters.

In most cases a MC method applies to a problem in which we have to compute a quantity like the mean value of a certain observable O, i.e.

hOi = Z

dq1dq2. . . dqkp(q1, . . . , qk)O(q1, . . . , qk) ≡ Z

{dq}p(q)O(q), (3.1)
where q ≡ {q1_{, . . . , q}k_{} is a set of stochastic variables, distributed according}

to the probability distribution {dq}p(q) ≡ dq1_{dq}2_{. . . dq}k_{p(q}1_{, . . . , q}k_{). The MC}

approach to this problem is the following:

1. using an appropriate algorithm we generate a sample of N copies of the stochastic variables, i.e. a set q1, . . . , qN, each one extracted according to

{dq}p(q);

2. on this sample we measure O1 ≡ O(q1), . . . , ON ≡ O(qN) and compute

the sample mean

ON = 1 N N X k=1 Oi. (3.2)

This quantity is itself a stochastic variable. Moreover, in the large-N limit, we know also its probability distribution thanks to the central limit theorem: independently from the initial probability distribution, provided that it has a finite variance, and under the further assumption of stochastic independent variables, ON has a normal distribution

P (ON) =
1
√
2πσN
e−
ON −hOi
2σ2_{N} _{,} _{(3.3)}
where σN ≡ σ/
√

N and σ ≡ hO2_{i − hOi}2_{is the variance of the observable}

O;

3. the variance σN can be considered as an estimate of the standard error,

i.e. hOi = ON ±pσ2N.

Given this brief summary, we observe that MC calculations are to all effect equivalent to integration problems. In the low-dimensional case, MC is in gen-eral a very bad integration method: if we are integrating a generic function f in a certain region of its domain, evaluating it in N points, again thanks to the central limit theorem we can conclude that the error on the estimate decreases as 1/√N , independently on the dimensionality d of the integral. Nevertheless, other methods of direct integration have different properties: the final result is that the error in general scales with N−k/d, where k is an index that labels different direct integration methods (e.g. for the trapezoidal rule is k = 2, for the Simpson rule is k = 4).

Then it is clear that for d = 1 any of these direct integration methods (trape-zoidal, Simpson and ohters) will be better than MC, while for large d, e.g. d ≈ O(10), MC beats any of them. By these considerations we see that for problems with many degrees of freedom, e.g. related to statistical mechanics or quantum field theory, MC methods are far better than the others.

### 3.1.1

### Applications in statistical physics: the importance

### sampling

Statistical mechanics deals with systems with many degrees of freedom. In general, the typical problem is to compute average macroscopic observables of a system for which the hamiltonian is assumed to be known.

To be specific, the thermal average of a generic observable A(~x), where ~x is a vector in phase space that denotes the set of variables describing the considered degree of freedom, in the canonical ensamble is defined as

hA(~x)iT =
R d~x e−βH(~x)_{A(~}_{x)}
R d~x e−βH(~x) =
1
Z
Z
d~x e−βH(~x)A(~x), (3.4)

where Z is the partition function and β = (kBT )−1. The role of the probability

distribution here is played by the normalized Boltzmann factor p(~x) ≡ 1

Ze

−βH(~x)_{,} _{(3.5)}

and it represents the statistical weight with which the configuration ~x occurs in thermal equilibrium.

Using the method explained in the previous section, we obtain that

A(~x) =
N
P
i=1
e−βH( ~xi)_{A( ~}_{x}
i)
N
P
i=1
e−βH( ~xi)
(3.6)

must approximate eq. (3.4). Unless this method is correct, it is totally inef-ficient: from statistical mechanics we know that the configurations which give the relevant contribution to eq. (3.4) have energy E ∈ [E − ∆E, E + ∆E], where E is the average energy at fixed β and ∆E has the property of going to zero as the volume of the system tends towards infinity (self-averaging property). In eq. (3.6) we are trying to estimate the observable A at fixed β using configurations distributed according to the Gibbs measure at β = 0: it is clear that if β is not small the configurations we are using are not dominant in the sum and thus the estimate is completely unreliable.

Therefore what is needed is a more efficient method that samples the config-urations ~xi in the MC average not completely at random, but preferentially

from that region of phase space which is important at given β. This leads to importance sampling: suppose we produce configurations ~xi according to some

probability P ( ~xi). Choosing this set for the estimation of the thermal average

of A, we obtain
A(~x) =
N
P
i=1
e−βH( ~xi)_{A( ~}_{x}
i)/P ( ~xi)
N
P
i=1
e−βH( ~xi)/P ( ~x_{i})
. (3.7)

The simplest and most natural choice for P ( ~xi) is P ( ~xi) ∝ e−βH( ~xi); then the

Boltzmann factor cancels out altogether and eq. (3.7) is a simple arithmetic average, i.e. A(~x) = 1 N N X i=1 A( ~xi). (3.8)

The problem here is to find a procedure which practically realizes the impor-tance sampling: at this point, we do not know how to generate independent configurations from the given probability P ( ~xi). As we shall see, the solution is

### 3.2

### Dynamic Monte Carlo: Markov chains

The general idea is to build a stochastic evolution process that allows us to explore the various states with established frequency. This has to be done in such a way that the final points are distributed with the desired probability distribution: in statistical mechanics this is represented by eq. (3.5).

This stochastic process is realized by means of Markov chains: let us call Ω the configuration space, i.e. the set of all possible states of the system, then a Markov chain with configuration space Ω is a sequence of Ω-valued random variables X1, . . . , XN such that the state Xi+1 depends only on the previous

state Xi.

We observe that in a Markov chain the probability that Xt → Xt+1 depends

explicitly only on Xtand not on Xi with i < t. Therefore, the whole process is

completely defined by the transition probability matrix

W ≡ {Wij}i,j∈Ω, Wij≡ P (j → i), (3.9)

where P (j → i) gives the probability that the system, starting from a state j, arrives in a state i after one discrete time step of the stochastic process. The elements Wij of the transition matrix will be encoded in the algorithm used

to move from one state to the other during the computer implementation of the Markov chain: in particular, this is a square matrix with dimension given by the number of states in Ω. Furthermore, their elements must have the following properties

0 ≤ Wij≤ 1, ∀i, j ∈ Ω;

X

i∈Ω

Wij = 1, ∀j ∈ Ω. (3.10)

The first property states that W is a probability matrix, the second one assures us that the system will surely reach some new state after each step.

Analogously to the statistical mechanics, where one makes use of statistical
ensamble to rewrite time averages of complex systems in terms of averages
over a given probability distribution, in a Markov process we will consider an
ensamble of different stochastic walks; if we denote with π(0)_{j} the distribution of
the initial ensamble, from every state j starts a walk

π(0)_{j} → π_{j}(1)→ π_{j}(2)→ · · · → π_{j}(k)→ π_{j}(k+1). (3.11)
Our goal is to build W in such a way that this walk tends towards an equilibrium
probability distribution π(eq)_{j} = pj. In matrix form we can write

π(k+1)= W π(k)⇒ π_{i}(k)= (Wk)ijπ
(0)