Three-Dimensional Multicomponent Lattice Abelian-Higgs Models in the Presence of Frustrated Gauge Interactions

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Three-Dimensional Multicomponent Lattice

Abelian-Higgs Models in the Presence of Frustrated

Gauge Interactions

Candidate: Fabio Sciaulino

Supervisor: Prof. Ettore Vicari

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Introduction

A physical system is said to be frustrated if not all contributions to its potential energy can be simultaneously minimized. In statistical and condensed matter physics, systems with frustrated interactions show peculiar behaviours, which may substantially differ from those of unfrustrated systems. Frustrated behaviour happens frequently for systems of magnetic moments, namely if the minimization of all interaction energies poses incompatible constraints on the system’s configuration. Frustration can arise from the geometry of the underlying lattice and/or from the nature of the interactions, as it will be in our case. The simplest and most popular example of frustration is the antiferromagnetic Ising model on a triangle, where the configuration with all spins lined up in the same direction is does not minimize the interaction energy and the system is forced to choose one of 6 equivalent configuration of lower energy when the tempera-ture is lowered. One important effect of frustration is to weaken the effects of the spontaneous symmetry-breaking ordering that usually occurs at lower temperatures. Sometimes frustration even removes ordering altogether. This often leads to unusual low temperature phases for those kinds of systems. Consequence of the absence of ordering can produce exotic and highly non-trivial phase-diagrams with a rich phenomenology. For this reason, in recent years frustrated systems have been much studied as a source of discoveries of new phases of matter, usually char-acterized by non-trivial topological properties. Examples are: spin liquids with fractionalized degrees of freedom, skyrmion lattices with emergent artifcial electrodynamics, fractionalized Fermi liquids, etc [8].

Many results in the field of phase transitions and critical phenomena have been obtained through the use of numerical simulations of lattice statistical models, which very often provide the only means possible to explore this kind of systems. Models of complex scalar matter fields coupled to gauge fields have been much studied in statistical and condensed matter physics, since they are believed to describe several interesting systems, such as superconductors and superfluids, quantum SU(N) antiferromagnets, unconventional quantum phase transitions, etc. In this the-sis, we study issues related to the effects of frustration in three-dimensional lattice scalar models with U(1) gauge-invariant interactions, in particular, in the limit of fully frustrated gauge in-teractions on a cubic geometry. We are interested in studying the phase diagram of frustrated 3D N-component lattice Abelian Higgs models. The compact Abelian Higgs model is one lat-tice formulation of scalar electrodynamics. The model has already been studied extensively in non frustrated regimes ( [2] [1]) and it displays continuous phase transitions belonging to the Heisenberg O(3) universality class (and to the O(4) universality class in the regime of infinite gauge coupling). We will show how, as the effects of frustration increase, the behaviour of the system deviates from the non-frustrated case and, in the limit of full frustration, belongs to a new universality class altogether.

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Chapter 2

The Model

2.1 The lattice model

For our lattice implementation of the three-dimensional AH model we start by associating a complex N-component vector zx, with the sites x of a cubic lattice. We then define the gauge

invariant lattice Action:

Sinv= −J X x ,µ Re(¯zx · zx +ˆµ) + X x V (¯zx· zx) (2.1) V (X) = rX +1 2uX 2 _(2.2)

The first sum is over all lattice links and the second one is over the lattice sites. ˆµ = ˆ1, ˆ2, ˆ3 are unit vectors along the lattice directions.

We consider the unit-lenght limit of the site variables, which is formally obtained by setting r = −u, and taking the limit u → ∞ in the potential (2.2), so that z satisfies:

¯ zx · zx = 1 (2.3) and (2.1) simplifies to Sinv= −J X x ,µ Re(¯zx · zx +ˆµ) (2.4)

We then proceed by gauging some of the degrees of freedom. We associate a U(1) complex variable λx ,µ (with |λx ,µ| = 1) to the links, each link connecting the site x to the site x + ˆµ,

and extend the (2.1) Action to ensure U(1) gauge invariance. We then add a kinetic term for the gauge variables in the wilson form [11]. We obtain the model defined by:

S = Sz+ Sg (2.5) where: Sz= −J N X x ,µ 2Re(¯zx · λx ,µzx + ˆµ), (2.6)

where the sum is over all links of the lattice and J plays the role of the inverse of the temperature T.

The pure gauge term is given by: Sg = −κ X x ,µ6=ν 2Re(λx ,µλx +µ,ν¯λx +ˆν,µ¯λx ,ν) ≡ −κ X x ,µ6=ν 2Re(Πx ,µν), (2.7)

where the sum is over all the lattice plaquettes and Πx ,µν is the field strenght associated with

each plaquette. κ plays the role of the inverse of the gauge coupling. The partition function of the system is then given by:

Z = X

{z },{λ}

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As we wrote in the introduction, we say that a system is frustrated if not all contributions to its potential energy can be simultaneously minimized. In our case frustration is not generated by the particular geometry of the lattice, but from the gauge interactions. The interactions between the z fields on the lattice sites are mediated by the λ fields. The value of the minimum value of the Re(¯zx · λx ,µzx + ˆµ) term varies depending on the orientation of λx ,µ. If Re(λx ,µ) > 0

the interaction is ferromagnetic and therefore it minimizes the interaction energy by having parallel z neighbors. If Re(λx ,µ) < 0the interaction is anti-ferromagnetic and it minimizes the

interaction energy by favoring anti-parallel z neighbors. For negative values of κ the average value of the plaquette product W ≡ hRe(Πx ,µν)i < 0. A negative value of the plaquette product

gives rise to frustrated interactions so that it cannot minimize all the Re(¯zx · λx ,µzx + ˆµ)terms

around the plaquette. To have a negative value of the plaquette product Re(Πx ,µν), it is not

possible to have all interactions ferromagnetic or all anti-ferromagnetic at the same time. As κ → −∞, i.e. in the fully frustrated regime, W → −1 . In this limit, due to the gauge invariance we can choose the frustrated configuration for the λx,µ field with some freedom, as long as it

preserves the original properties of unitarity and minimizes the Action by giving W = −1. The configuration we have chosen is:

λx,1= 1 ∀x; λx,2= (−1)x1 ∀x; λx,3= (−1)x1+x2 ∀x (2.9)

Where th xi are the i-th components of the x vector.

2.2 N -component Abelian Higgs models

Scalar electrodynamics, or Abelian-Higgs (AH) model, is a paradigmatic model il wich a N-component complex scalar field Φ is minimally coupled to the electromagnetic field Aµ.

The continuum Lagrangian of the system is:

L = |DµΦ|2+ r|Φ|2+ u(|Φ|2)2+

1 4g2FµνF

µν _(2.10)

where Fµν≡ ∂µAν− ∂νAµ and Dµ≡ ∂µ+ igAµ.

The system is characterized by a U(N) global symmetry and a U(1) local gauge symmetry. Under a U(1) transformation the fields transform as:

Φ(x) → U (x)Φ(x); Aµ(x) → Aµ(x) −

1

g∂µΛ(x) (2.11) where:

U ≡ eiΛ(x) _{Λ(x) ∈ R, ∀x} (2.12) We assume that the field belongs to the fundamental representation of the U(1) group, i.e. , it has charge 1.

The lattice model described in the previous section is a possible discretization of the Abelian Higgs model.

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Chapter 3

Phase transitions, critical

phenomena and scaling

3.1 Phase transitions and critical behaviour

If we take a homogeneous system in thermal equilibrium, we can measure its macroscopic prop-erties like: magnetization, energy density, magnetic susceptivity, specific heat, etc. The value of these macroscopic quantities is obtained by averages over the collective behaviour of the microscopic degrees of freedom which fluctuate in a way that is determined by the equilib-rium state of the system, which in turn is determined by external parameters like temperature, pressure, external magnetic fields etc. One such quantity of great importance in the study of thermodinamic systems is the correlation lenght of the system. It is the distance at which the fluctuations of the microscopic degrees of freedom are significantly correlated to each other. If two parts of the object are further from each other than the correlation lenght of the material, then the fluctuations of their degrees of freedom are effectively disconnected from each other. Many systems show abrupt changes in their macroscopic behaviour as external quantities like temperature, magnetic field, etc. are smoothly varied. The set of external parameters that correspond to these phenomena happen are called transition points and they mark a phase transition from a state of matter to another. They can be represented as points on the phase diagram of the system.

Phase transitions are classified in two ways, depending on the presence or absence of differ-ent coexisting phases at the transition point. In the case of two or more coexisting phases at the critical point we have that each of the phases is continuously connected to a pure phase far away from the transition point. In this situation, we expect to observe discontinuities in the behaviour of various thermodinamical quantities as we go through the critical point during a phase transition. These transitions are called discontinuous or first-order transitions (because the discontinuities are found in the first derivatives of the free energy of the system). Some examples of first-order transitions are the melting of a three-dimensional solid or the condensa-tion of a gas into a liquid. The correlacondensa-tion length at such first-order transicondensa-tions is generally finite. The other possibility is that of a continuous transition, where the correlation lenght effec-tively becomes infinite. This means that fluctuations become correlated at all distance scales and therefore the whole system, assumes a unique critical phase. During continuous transitions, therefore the various phases on either side of the transition point, also known as critical point for continuous transitions, must become identical in its proximity. Also, during a continuous transition, other thermodinamic quantities have a smooth behaviour during the transition. Ex-amples of continuous transitions occur at the Curie temperature in a ferromagnet, and at the liquid-gas critical point in a fluid. As an example we are going to look at the behaviour of uniaxial ferromagnets.

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For ferromagnets the relevant external parameters controlling the macroscopic behaviour of the system: the temperature T and the applied magnetic field H. All the thermodinamic quantities of the systems are smooth functions of T and H except on the line H = 0, T ≤ Tc,

where Tc is referred to as Curie temperature. On the line T ≤ Tc the magnetisation M as a

function of H is discontinuous. This discontinuity is characteristic of a first-order transition, with a finite correlation lenght. As T approaches Tc from below, the discontinuity approaches

zero and the correlation lenght at the transition diverges. The point H = 0, T = Tc is called

critical end-point and marks the point at wich the first-order transition becomes a continuous transition.

For T < Tc, the magnetization exhibits different values depending on how the limit on H → 0

is taken, giving rise to two possible values for the magnetization M. (M0for H → 0+, −M0for

H → 0−). This phenomenon is referred to as spontaneous symmetry breaking and it means that even though the action of the system is invariant (symmetric) under the reversal of all the spin variables, the thermodinamical equilibrium state is not symmetrical under the transformation. The symmetry is broken because the system chooses a direction to align the spins and a nonzero magnetisation arises. In this situation, the magnetisation M is called the order parameter for the transition, and its value measures the amount of magnetic order in the system. It is usu-ally possible for systems in critical regime, to identify one or more order parameters, and, by studying their behaviourm characterize the transition.

3.2 Critical Exponents

Although systems with large correlation lengths might appear to be very complex, they also exhibit important simplifications. One such simplification is the phenomenon of universality. Many properties of a system close to a continuous phase transition turn out to be largely independent of the microscopic details of the interactions between the individual atoms and molecules. Instead, they fall into one of a relatively small number of different classes, each characterised only by global features such as the symmetries of the underlying action, the num-ber of spatial dimensions of the system, the range of the interaction and so on. These are called universality classes. Typically, close to a critical point, the correlation length and the other thermodynamic quantities exhibit power-law dependences on the parameters specifying the distance away from the critical point. These powers, or critical exponents, are pure numbers, usually not integers or even simple rational numbers, which depend solely on the universality class.

To show what the critical exponents look like we again look at the behaviour of ferromag-nets. It is useful to define dimensionless measures for the deviation from the critical point:

t ≡ (T − Tc)/Tc; h ≡

H kbTc

(3.1) The exponents we are interested in are:

α: The specific heat in the absence of external magnetic field goes as:

C v A|t|−α+ Creg (3.2)

Where Creg is the contribution regular in t.

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These definitions can be extended to the case of general continuous phase transitions, paying close attention to which quantities serve the function of the order parameter in every specific situation and which external parameters regulate the transition.

3.3 The Renormalization group

The basic idea behind the Renormalization group is that, in critical regimes, only the long range fluctuations determine the equilibrium dynamics of the thermodinamic system. This is because during continuous transitions the correlations between the fluctuations of the degrees of freedom of the system become very large, i.e. the correlation lenght discussed before becomes infinite. This phenomenon makes it possible to simplify in various ways the study of critical systems. All renormalization group studies have in common the idea of re-expressing the parameters which define a problem in terms of some other, perhaps simpler, set, while keeping unchanged those physical aspects of the problem which are of interest. This may happen, for example, through some kind of coarse-graining of the short-distance degrees of freedom, because only the long-distance physics is of interest. These kinds of methods all define transformations in the system’s parameter space, which in turn, describe what is called Renormalization Group Flow. From the study of the properties of these transformations and their effects on the partition function and the free energy of the system, many important aspects of the critical behaviour can be explained. For example, the critical exponents discussed in the previous section can be derived in the RG framework by studying the linearizations of the transformation laws near the fixed points. Another aspect of great importance, especially in the case of lattice simulations, is the dependence of the critical behaviour of a given system on its particular geometrical limitations. This phenomena are labeled Finite-size Scaling effects.

3.3.1 Finite-size Scaling

Consider a system of linear size L, for example a cubic lattice of volume L3_{. For this system}

there are three relevant lenght-scales: its linear size L in units of the lattice spacing a and its correlation lenght ξ and both L and ξ are implicitly expressed in units of a. In the case in which this system undergoes a continuous transition there are several remarks to be made on the effect of the finite size of the system. In critical regime, it is usual to express the behaviour of the thermodinamic quantities in a more convenient way:

ξ v |t|−ν (3.6)

χ v |t|−γv ξγ/ν (3.7) C v |t|−αv ξα/ν (3.8) In the case of a finite system, this kind of behaviour is only approximately manifested. There is a range of temperatures in which ξ starts to grow. In this regime we have that ξ ≈ L and we can use this relation to derive some important relations. In this regime peaks that increase with the size L are observed in C and χ . During a continuous transition we can assume that the only relevant quantity is the ratio ξ/L. We can write an expression for the various thermodinamic quantities, for example the susceptivity χ, in terms of the two relevant lenghts L and ξ:

χ(J, L) = Lγ/νφ(L, ξ) (3.9) where φ describes the behaviour of χ near the peak. We then write:

χ(J, L) = Lγ/νφ(ξ/L) = Lγ/νφ((J − J˜ c)−νL−1) = Lγ/νΦ((J − Jc)L1/ν) (3.10)

where φ, ˜φ and Φ are universal functions that only depend on the universality class of the transition.

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3.4 Gauge-invariant Landau-Ginzburg-Wilson framework

The phase diagram of the system may be investigated through the Landau-Ginzburg-Wilson (LGW) field-theoretical approach. In this framework the critical features are uniquely specified by the nature of the order parameter associated with the critical modes, by the symmetries of the model, and by the symmetries of the phases coexisting at the transition, the so-called symmetry-breaking pattern. In the presence of gauge symmetries, the traditional LGW approach starts by considering a gauge-invariant order parameter, effectively integrating out the gauge degrees of freedom, and by constructing a LGW field theory that is invariant under the global symmetries of the original model. The order parameter of the transition for the AH model is identified with the gauge invariant local composite site variable:

Qab_x = ¯z_xaz_xa− 1 Nδ

ab_, _(3.11)

which is a hermitian and traceless N × N matrix. Under the global U(N) transformations it transforms as:

Qx → U†QxU (3.12)

The corresponding order-parameter field in the LGW theory is hence a traceless hermitian matrix field Ψab_(x)_{which can be formally defined as the average of Q}ab

x over a large but finite

lattice domain. The LGW theory is then obtained by considering the most general fourth-order polynomial in Ψ consistent with the U(N) global symmetry:

HLGW =T r(∂µΨ)2+ rT rΨ2

+ wT rΨ3+ u(T rΨ2)2+ vT rΨ4. (3.13) A continuous transition is possible if the renormalization group (RG) flow computed in the LGW theory has a stable fixed point. For N = 2, the cubic term in eq. (3.13) vanishes and the quartic terms are equivalent. Therefore the O(3)-symmetric vector LGW theory is recovered, giving some hints that the transition may be continuous and belonging to the Heisenberg universality class. This has already been proven for κ ≥ 0 in [1] [2]. We will see that this general assumption from LGW theory is not applicaple in the case of fully frustrated regime, as the system’s universality class in that regime is different from O(3).

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Chapter 4

Monte-Carlo Simulations

4.1 Monte-Carlo methods

Monte-Carlo methods are a very powerful tool for the numerical solution of a wide range of problems in physics and elsewhere. They are often used in problems where there is the need to evaluate quantities of the form:

hF i = Z

Dxp(x)F (x) (4.1)

where x ≡ {x1_{, x}2_{, . . . , x}n_}_{are a set of stochastic variables with probability distribution Dxp(x) ≡}

dx1_{. . . dx}n_p(x1_{, . . . , x}n₎_{and F (x) is a function of the stochastic variables. In short, MC}

meth-ods are used to compute expectation values of functions of the stochastic variables. The basic approach is as follows:

• With a suitable algorithm, the stochastic variables are sampled N times from the distri-bution p(x)Dx, producing x1, . . . , xN.

• From the samplings f1= F (x1), . . . , fN = F (xN)are computed and the sample mean:

¯ f = 1 N N X i=1 fi (4.2) is computed.

• The fi are stochastic variables, as it is ¯f. To understand the distribution of ¯f various

techniques are employed, paying careful attention to the presence of correlations in the sample data. Error estimates in the case of correlations to be discussed more thoroughly in Section 4.3.

For the sake of simplicity, the following discussion is based on the hypotesis of statis-tically independent data sets.

• In the case of statistically independent data sets and for large enough values of N, the Central Limit Theorem holds, therefore the distribution of ¯f is given by:

P ( ¯f ) =√ 1 2π ˜σ2e

−f −hF i¯

2˜σ2 (4.3)

where ˜σ = σ/N and σ2 _{≡ hF}2_{i − hF i}2 _{is the true variance of F . This is true regardless}

of the original distribution of the stochastic variables, granted that it has a well defined variance.

• As a consequence ¯f has a 68% probability of being within one standard deviation, i.e. ˜σ, from hF i. Therefore, by measuring the sample mean, the uncertainty on the estimate of hF idecreases with N as 1/√N.

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4.1.1 Application to physical problems

Equilibrium systems with many degrees of freedom are governed by the laws of statistical mechanics. All the physical quantities are calculated in terms of averages with respect to the Gibbs distribution, G(H) = e−βH_.

This statistical formulation lends itself very well to the study through the application of Monte-Carlo methods, and in fact, MC methods have proved time and time again extremely useful tools, often the only ones available, in the investigation of such systems, which are very hard to approach analitically.

4.2 Monte-Carlo algorithms

4.2.1 Markov chains

A Markov process is a stochastic evolution process, defined on a set of stochastic variables (the system) as a function of a series of "steps" (MC time), with the property that the state of the system is uniquely determined by the immediately previous state of the system, in other words, the system has no memory of its past. When the time variable is discrete, we talk of Markov Chains. In addition, time independence is assumed, meaning that the evolution process is independent during the stochastic process. All the properties of a Markov chain can therefore be represented using a transition matrix W acting on the system’s states, and, for each couple of states a and b, having probabililty Wba of going from a to b. In order for a

Markov chain to be useful it has to satisfy several requests, we only list the most important ones. Let the space of all possible states that one system can assume be called configuration space and be denoted by Ω.

• Normalization: All the elements in the matrix must satisfy the relations: 0 ≤ Wab≤ 1 ∀a, b ∈ Ω;

X

a∈Ω

Wab= 1 ∀b ∈ Ω (4.4)

as a consequence, the probability of reaching a starting from b in k steps: P (b → a, k) = X

c1,...,ck−1

Wack−1. . . Wc2c1Wc1b= W

k

ab (4.5)

• Ergodicity: A Markov chain is called ergodic if, for any couple of states a,b ∈ ω, exists ksuch that (Wk₎

ab6= 0

• Aperiodicity: Lets consider a system is in some state a, and all integer n such that Wn

aa6= 0. The greatest common divisor daof all n’s is called the period of a. If da= 1 ∀a

the Markov chain is called aperiodic.

A Markov chain which is both ergodic and aperiodic is called regular.

The close resemblance to the evolution of dynamical systems make Markov chains very use-ful to simulate thermodinamical systems for which the macroscopic variables are distributed stochastically. The idea is to devise a stochastic dynamical process that behaves in a way that averages taken over MC time coincide with averages over a given probability distribution.

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4.2.2 Metropolis algorithm

Let a and b be two possible states of the system. The Metropolis algorithm can be summarized in three steps that describe how the system moves from state a to state b:

• Starting from a, a trial state ˜b is selected/generated according to a tentative transition probability A˜_ba(A must be properly normalized).

• If p˜b > pa then b = b˜b.

• Metropolis step: If p˜_b< pa then b = b˜_b with probability p˜_b/pa and a otherwise.

In our case the probabilities of the various states are calculated using the Gibbs distribution.

4.2.3 Over-relaxation algorithm

A cheap technique that is often used in tandem with stochastic algorithms like the Metropolis is the over-relaxation algorithm. It is based on the assumption that the local distribution of the field Φ(x) in the lattice site x0 is a Gaussian centered in G, a sort of mean field centered

on x0. At any given time, the field in x0 is not exactly at the maximum but somewhere near

it. It makes sense then to move the value of the field to the value symmetric with respect to the maximum, in other words:

Φ = 2G − Φ(x0) (4.6)

which has the same probability of the original value of Φ(x0). The cheapness of the

over-relaxation algorithm comes at the price that it is deterministic, therefore it cannot be used on its own and must be deployed in tandem with a stochastic algorithm like the Metropolis.

4.3 Error estimates

In the following we present some of the sources of errors that are characteristic of MC simulations and describe the techniques that have been used to estimate the uncertainties in the data collected from the simulations.

4.3.1 Autocorrelations and data blocking techniques

One of the main things to take into account when using Markov chain based Monte Carlo sim-ulations is that consecutive draws of a Markov chain are not statistically independent. Each draw has some memory of the previous configuration. This fact means that standard error estimation techniques cannot be used or that, at least, they must be adjusted accordingly. One rather simple method to estimate the statistical error taking into account autocorrela-tions is that of dividing the sample into blocks of smaller size of adjacent draws and doing the standard computation on the averages of the smaller blocks. By varying the dimension of the blocks we can control how much the consecutive averages are correlated to each other. Once the blocks are big enough that their dimension is comparable to the autocorrelation time of the Markov chain we can treat their averages as statistically independent and therefore use them in place of the original draws to compute the quantities of interest.

Another method that can be used in parallel with the blocking of the data is simply to only do the measuraments once every a certain number of updates, so that the system has time to decorrelate and the successive measuraments fluctuate more according to the distribution that we want to sample.

In our case, at the start of every run of the simulation, the system is updated many times (≈ 105_{) in order to reach thermal equilibrim. Once thermal equilibrium is reached, the updates}

continue and measuraments are made every 10 updates of the system. Every time 103

measur-aments are reached, they are averaged and stored as a data point from the simulation. This means that we get one data point for every 104 _{updates of the system in thermal equilibrium.}

After the simulation is over, the error estimates on the data are carried out using the previ-ously described data blocking techniques to take into account the residual correlations, plus the bootstrap technique described in the following section.

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4.3.2 Resampling techniques: Bootstrap

In situation where the observables of interest are functionals of the entire sample, error estima-tion can become problematic because the funcestima-tional is calculated using the entire sample and therefore one needs more samples in order to estimate the degree of fluctuation to calculate the error associated to the measurament.

Usually, this procedure is too computationally expensive to be viable and therefore, more in-genious methods have been developed to tackle the problem. The one we have used is the Bootstrap.

As an example, let’s consider an observable ON over a sample of N draws xi, i = 1, ..., N,

so that ON = ON(x1, ..., xN). For N → ∞, ON → O(P (x))(where P (x) is the distribution we

are sampling). We are interested in computing the quantity: σON =

q hO2

Nis− hONi2s (4.7)

where with h·i we mean an average over the probability distribution P (x1, ..., xN).

The idea of the Bootstrap is like follows. Since the N elements of the sample are extracted from the same distribution P (x), they can be used to make new "fake" samples of N elements generated by randomly extracting elements from the original sample. We write x(j)

i to indicate

the i-th element from the j-th "fake" sample. The new samples are fake in the sense that they are not drawn from the original distribution function P (x), but just by a finite sample of it, however they are very cheaply produced, since no new Monte-Carlo generation is needed. For every sample, we then compute the new estimate of ON using:

O_N(j)= ON(x (j) 1 , ..., x

(j)

N ). (4.8)

Then, if we have M samples we measure the variance of O(j)

N on them and take it as an estimate

of σON, i.e. σON u v u u u t 1 M M X j=1 (O(j)_N )2₋   1 M M X j=1 O_N(j)   2 (4.9)

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4.4 Implementation

We consider cubic lattices of linear size L with periodic boundary conditions and the simulations are carried out by fixating the value of κ and studying the behaviour of the system as a function of J. The cubic lattices are rendered by complex arrays with 4 indices. The first index is used to address the components of the fields zxand λx,µ, while the other three identify the coordinates

on the lattice. The linearity of the Action (2.5) with respect to the lattice variables allows to employ an overrelaxed algorithm for the updating of the lattice configurations. The updating of the system therefore consists of a stochastic mixing of microcanonical (overrelaxed) and standard Metropolis updates. The process is carried out in a way that it randomly chooses either a standard Metropolis update, which is slower but ensures ergodycity, or a micrcanonical move, which is more efficient than the Metropolis one but does not change the energy of the system. On average, the rate is of 3-4 microcanonical updates for every Metropolis proposal. The Metropolis update is tuned so that the acceptance is 1/3.

4.4.1 Updating of the z

x

field

For every site x of the system, our algorithm associates a complex 2-dimensional unit-vector with components denoted as za

x, with a = 1, 2. For every site x, we construct f defined as:

f_za =X

µ

(λx,µzxa_{+ ˆ}_µ+ ¯λ_{x− ˆ}_µ,µz_{x− ˆ}a _µ) (4.10)

we also, using Action (2.5), calculate the nearest neighbor interaction energy: Zint= X a 2Re( ¯za xf a z) (4.11)

and then the algorithm randomly chooses either the Metropolis update (typically once every 4 updates), or the micro-canonical update (over-relaxation algorithm, typically 3 out of every 4 updates).

• Metropolis update: from the vector zx we create a vector of the form:

Zcp(x) =     Re(z1x) Im(zx1) Re(z2 x) Im(z2 x)     (4.12)

We then take two random components i1 and i2, from Zcp(x)and a randomly generated

phase φ and use them to generate two new components for Zcp(x):

[Zcp(x)]i1 → [Zcp(x)]i1cos φ − [Zcp(x)]i2sin φ (4.13)

[Zcp(x)]i2 → [Zcp(x)]i1sin φ + [Zcp(x)]i2cos φ (4.14)

We then build a new trial complex vector ztfrom the new Zcpthat is going to be used as

a candidate for the Metropolis step: zt= [Zcp(x)]1+ i[Zcp(x)]2 [Zcp(x)]3+ i[Zcp(x)]4 (4.15) and calculate: ZT = X a 2Re(¯za_tf_za) (4.16) Then we perform the Metropolis step with:

pa= e−N J Zint p˜_b= e−N J ZT (4.17)

• Micro-canonical update: using the quantities fz and Zint defined in (4.10) and (4.11)

we perform the substitution:

z_xa→ Zint fa z |fz|2 − za x (4.18)

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4.4.2 Updating of the λ

x,µ

field

Our algorithm associates a unit-lenght complex number λx,µ to every link in the µ direction

connecting site x to site x + ˆµ. For every link (with direction µ) we construct: fλ= fλ,z+

κ

N Jfλ,µ (4.19)

where fλ,z is the contribution from the z field and fλ,µis the contribution from the plaquettes.

They are defined by:

fλ,z =

X

a

¯

za_{x+ ˆ}_µz_xa (4.20) and, for a link in the µ direction:

fλ,µ=

X

ν6=µ

(λx,νλx+ˆν,µλ¯x+ ˆµ,ν+ ¯λx−ˆν,νλx−ˆν,µλx− ˆµ−ˆν,ν) (4.21)

The fλ,µterms are the sum over all three-link products that form a closed loop with our starting

link λx,µ. We then use fλ to compute:

Λint= 2Re(λx,µfλ) (4.22)

• Metropolis update: We randomly generate a phase φ and compute:

λt= eiφλx,µ (4.23)

and:

ΛT = 2Re(λtfλ) (4.24)

Then we perform the Metropolis step with:

pa= e−N J Λint p˜_b = e−N J ΛT (4.25)

• Micro-canonical update: using the quantities fλ and Λintdefined in (4.20) and (4.22)

we perform the substitution:

λx,µ→ Λint

fλ

|fλ|2

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Chapter 5

Observables

5.1 Lattice Observables

The energy density and specific heat are computed, using the definitions: E = 1 N V hHi, C = 1 N2_V(hH 2_{i − hHi}2 ), (5.1)

where V ≡ L3_{. With h·i we mean averaging over a Monte-Carlo simulation.}

We also consider averages over plaquettes:

W = Re(Πx ,µν) ≡ Re(λx ,µλx +µ,νλ¯x +ˆν,µ¯λx ,ν), (5.2)

Correlations of the hermitian gauge-invariant operator (3.11) are also considered. The associ-ated two-point correlation function is defined as:

G(x − y ) = hT rQxQyi (5.3)

Where the translational invariance of the system has been taken into account. The susceptibility and the correlation lenght are then defined as:

χ =X x G(x ), G(x ) ≡ G(x − 0), (5.4) and ξ2≡ 1 4 sin2(π/L) ˜ G(0) − ˜G(pm) ˜ G(p_m) , (5.5)

where ˜G(p) = Σxeip·xG(x ), and pm= (2π/L, 0, 0)is the minimum nonzero lattice momentum.

The Binder parameter is defined as: U = hµ 2 2i hµ2i2 , µ2= 1 V2 X x ,y T rQxQy. (5.6)

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5.1.1 Continuous transitions

To analyze the continuous transitions we study the FSS behaviour of the Binder parameter U and of the quantity:

Rξ ≡ ξ/L. (5.7)

At continuous transitions the FSS limit is obtained by taking J → Jcand L → ∞ while keeping

X ≡ (J − Jc)L1/ν (5.8)

fixed.

Here Jc is the inverse critical temperature and ν is the correlation lenght critical exponent.

RG invariant quantities such as Rξ and U are expected to behave like:

R(J, L) = fR(X) + O(L−ω) (5.9)

where ω is the leading scaling correction exponent. The ω is dependent on the particular properties of the system and regulates the behaviour of the deviations from the asymptotic limit (L → ∞). fR(X) is a universal function apart from a normalization of its argument and

only depends on the shape of the lattice and on the boundary conditions. In particular, the quantity R∗ _{≡ f}

R(0) is universal. If the scaling function of a RG invariant quantity R1 is

monotonic we may also write:

R2(R1, L) = FR(R1) + G(R1)L−ω (5.10)

Where F (x) is a universal scaling function and G(x) is universal apart from a model dependent normalization.

Eq.(5.10) is especially convenient because it allows a direct check of universality without being affected from any model-dependent normalizations. As Rξ is monotonically increasing as a

function of X Eq. (5.10) implies that:

U = F (Rξ) + G(Rξ)L−ω (5.11)

where F (x) and G(X) are universal scaling functions as in eq.(5.10). We will use Eq. (5.11), to perform the check of universality. Then we should expect that if two models belong to the same universaliy class, the data for both should collapse onto the same curve as L increases.

In order to estimate the exponent η, we analyze the scaling behaviour of the susceptivity χ. It scales as:

χ(J, L) v L2−η[fχ(X) + O(L−ω)] (5.12)

or equivalently, as:

χ(Rξ, L) v L2−η[Fχ(Rξ) + O(L−ω)] (5.13)

The specific heat at the transition behaves as, [6],:

C(J, L) ≈ Creg(J ) + Lα/νfC(X) (5.14)

where Creg denotes the regular background, which is an analytical function of J. This

contri-bution is dominant if α < 0 or equivalently ν > 2/3. When α > 0 we may write:

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5.1.2 First-order transitions

To study first order transitions we use the standard phenomenological theory from [9] and [10], which is based on the assumption that at a first-order transition, the statistical distribution of the energy density and magnetization is well approximated by a double gaussian distribution. At first-order transitions the probability distributions of the energy and of the magnetization are expected to show a double peak for large values of L or, in alternative, at reasonably low Lwhen the transition happens to be strong enough. Therefore, two peaks in the distributions are often taken as an indication of a first-order transition. However, the the observation of two maxima in the distribution of the energy is not sufficient to conclude that the transition is of first-order. For example, in the 3D Ising model the distribution of the magnetization has two maxima. To show the nature of the transition we also consider the specific heat C and the Binder parameter U. Both quantities are expected to increase linearly with the volume in a first order transition [9]. If E+ and E− are the values of the energy corresponding to the two

maxima of the energy-density distribution, the latent heat ∆h is given by:

∆h= E(J → Jc+) − E(J → Jc−). (5.16)

The latent heat can be estimated from histograms of the energy density distribution near the transition temperature Jc. An alternative estimate of the latent heat can be obtained from the

behaviour of the specific heat C as a function of the volume of the system V . According to the standard phenomenological theory [9], for a lattice of size L exists a value Jmax,C of J where

Ctakes its maximum value Cmax(L), which asymptotically increases as [9]:

Cmax(V ) = V 1 4∆ 2 h+ O(1/V ) (5.17) Jmax,C(V ) − Jc≈ cV−1 (5.18) where V = Ld _{and ∆}

h is the latent heat .

The Binder parameter U can also be used to characterize a first-order transition. As dis-cussed in [10], the distribution of the order parameter is also expected to show two peaks at µ+ 2 and µ− 2, µ − 2 < µ + 2 with µ −

2 → 0as L → ∞ since there is no spontaneous magnetization in the

high-temperature phase. Therefore, the behavior of the Binder parameter U(J, L) is expected to show a maximum Umax(L)at fixed L (for sufficiently large L) at J = Jmax,U(L) < Jc with:

Umaxv aV + O(1) (5.19)

Jmax,U(V ) − Jc≈ bV−1 (5.20)

The previous relations are valid in the asymptotic limit and, for weak transitions, require data on large lattices.

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Chapter 6

Numerical Results: Exploring the

κ < 0 region

In this section we present all the numerical results of the MC simulations of the AH model for N = 2and κ < 0. In the next chapter, we discuss the overall κ behaviour of the system and our findings.

It must be stressed that all numerical results for this work were obtained by using a laptop computer. This is why we only show data for lattices with maximum values of L = 14. Nev-ertheless we believe that the results obtained are significant. In all of our plots, errorbars are present, even if in some cases they are not visible due to being smaller than the markers used for the data points.

6.1 Comparison with the O(3) model

To make a comparison with the Heisenberg O(3) model is important to identify the correct operators in both models. For the AH lattice model the quantity we are going to consider is the local operator Qab

x defined in (3.11). For the corresponding operator in the O(3) model we

do as in previous works [1] [2], using the explicit relation between the CP1 _{model (κ = 0) and}

AH model. This relation implies that the parameter U and the correlation lenght correspond to the O(3) vector Binder parameter and correlation lenght (computed from correlations of the fundamental spin variable sx).

We now report some results for the features and exponents of the Heisenberg universality class. From [7] [4], we have:

νh= 0.71164(10), ηh= 0.03784(5), ωh= 0.759(2) (6.1)

We report here an accurate expression (deviations are assumed to be well below 0.5%) of the universal curve U = F (Rξ)for the Heisenberg universality class [3]:

F (x) = 5/3 + x(3.0263535 + 23.139470x)(1 − e−15x) − 47.838890x2

+ 58.489668x3− 67.020681x4_{+ 38.408855x}5_{− 8.8557348x}6 (6.2)

We also report accurate results for RG invariant quantities, Rξ and U for systems in the

Heisenberg 3D universality class. For cubic systems with periodic boundary conditions we have [5]:

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6.2 Continuous O(3) Transitions

6.2.1 Data for κ = −0.5

First, as we are studying a system in a partially frustrated regime, we show data for the plaquettes’ expectation value W as a function of J. As visible from the plot in Figure 6.1, we see that for the most part, as expected from a frustrated system, W < 0. As expected from Action (2.5), for higher values of J, W → 1 in order to minimize the interaction energy. In Figure 6.2 we show the results for Rξ = ξ/L. The data corresponding to different values of L

have a crossing point, which provides us a first estimate of the critical temperature, Jc ≈ 1.1055.

The slopes of the curves are related to the critical exponent ν. Their behavior is consistent with the Heisenberg value. An accurate estimate of the critical point is obtained by assuming the Heisenberg critical exponents and fitting the data to Eq. (5.9). The points used for the fit are data belonging to a small interval around Jc, where the behaviour of Rξ can be approximated

well by a linear fit. In particular, we fit Rξ using the ansatz:

Rξ = R∗ξ+ c1X, X = (J − Jc)L1/νh (6.4)

using the known estimates of Rξ and νh. We obtain Jc= 1.055(1)and c1= 0.299(1).

Figure 6.1: Plot of W versus J for the AH lattice model with κ = −0.5

Figure 6.2: Plot of Rξ versus J for the AH lattice model with κ = −0.5, the data for different

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A consequence of Finte-Size Scaling is that, taking into account the distances from the critical temperature (J −Jc)and the size of the system L by using the variable X = (J −Jc)L1/ν,

the data for various values of L automatically collapses on the appropriate universal curve. As a confirmation of this we plot the same data from Figure 6.2 as a function of X = (J − Jc)1/ν,

as shown in figure 6.3, where we have used the Heisenberg value νh = 0.7117, and observe a

very good collapse of the data on the universal curve. Likewise, in figure 6.4 we show data for the magnetic susceptivity χ against Rξ. We plot χL−2+ηh vs. Rξ using the 3D Heisenberg

value ηh= 0.0378and observe the collapse of the points on the universal scaling curve.

Figure 6.3: Scaling plot for Rξ vs. X ≡ (J − Jc)L1/ν for the AH lattice model, for κ = −0.5,

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Next, we check the behaviour of the Binder parameter U, shown in Figure 6.5. We show data for U vs. Rξ together with the universal O(3) curve from Eq.(6.2) . We can see that

all the points collapse on or very close to the Heisenberg curve, a further confirmation of the system belonging to the Heisenberg universality class. There are small deviations, as shown in Figure 6.6. These are the non asymptotic scaling corrections from Eq. 5.11. They scale as L−ωh, where ω

h is the leading scaling correction for the O(3) model. We plot (U − UO(3))Lωh

vs. Rξ. The uncertainties of the data points in the plot are compatible with the collapse onto

the universal scaling curve, following Eq. (5.11). We will see, in the next subsection, that the shape of the scaling curves of Figures 6.6 and 6.12 are equal apart from a overall normalization, like expected from FSS.

Figure 6.5: We show the Binder parameter U versus Rξ for the AH lattice model, for κ = −0.5.

The green dot-dashed line is the Heisenberg curve for the O(3) vector model.

Figure 6.6: Plot of the deviation of the data from the universal O(3) curve. Here we have used the Heisenberg value ωh= 0.76

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6.2.2 Data for κ = −0.7

As for the case with κ = −0.5 we show data for the expectation value of the plaquettes, W , for the κ = −0.7 system. By comparing with Figure 6.1 we can easily see that frustration is stronger overall, as W < 0 even for greater values of J, as expected from the Action (2.5). In Figure 6.8 we show the results for Rξ = ξ/L vs. J. The data corresponding to different

values of L have a crossing point, which provides us an estimate of the critical temperature, Jc= 1.1177(3).

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We now study the FSS behaviour of the κ = −0.7 system. Here too we use the Heisenberg values for the critical exponents. In Figure 6.9 we show data for Rξ vs. X = (J − Jc)L1/νh,

where we have used the Heisenberg value νh = 0.7117. We observe a good collapse of the

points onto the Heisenberg universal curve as confirmation of the system belonghing to the Heisenberg universality class. In Figure 6.10 we show the FSS behaviour of the magnetic susceptivity χL−2+ηh, with η

h= 0.0378, as a function of Rξ. As expected, we see good collapse

of the points onto the Heisenberg universal curve.

Figure 6.9: Scaling plot for Rξ vs. X ≡ (J − Jc)L1/ν for the AH lattice model, for κ = −0.7,

using the 3D Heisenberg exponent νh = 0.7117 and Jc = 1.177. The points collapse onto the

Heisenberg universal curve.

Figure 6.10: Scaling behavior of the susceptibility χ for the AH lattice model: plot of χL−2+ηh

vs. Rξ for κ = −0.7, using the 3D Heisenberg value ηh= 0.0378. The points collapse onto the

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Like for the κ = −0.5 case we check the FSS behaviour of the Binder parameter U. In Figure 6.11 we show the data for U vs. Rξ and we see a more pronounced deviation from the

universal O(3) curve. In Figure 6.12 we plot (U − UO(3))Lωh vs. Rξ. The deviations scale with

L−ωh, with ω

hthe Heisenberg leading scaling correction exponent, like in the case for κ = −0.5.

Compatibly with the uncertainties, we observe collapse on the universal scaling curve. Apart from a normalization factor, the shape of the curve in Figure 6.12 is the same as for the case for κ = −0.5. As we can see, the closer we get to κ ≈ −0.8 the more the deviations get pronounced.

Figure 6.11: We show the Binder parameter U versus Rξfor the AH lattice model, for κ = −0.7.

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6.3 First Order Transitions

In the following we show data from the simulations for κ < −0.7, which show first-order behaviour. In particular, we study the cases for κ = −0.8, κ = −1 and κ = −1.5. To study discontinuous transitions we use the relations for the specific heat and Binder parameter illustrated in section 5.1.2, in particular Eqs. 5.17 to 5.20. We then use the data for various values of L to estimate the asymptotic value of the critical temperature Jc and the latent heat

∆h of the transition.

6.3.1 Data for κ = −0.8

First of all we show the behaviour of the average value of the plaquettes, W . In Figure 6.13 we show data for W as a function of the temperature J. As expected, we get overall more frustrated results than the systems with smaller values of κ. We also see, as a first hint of a discontinuous transition, that W has a sharp drop, which appears to get sharper as we increase the volume.

Figure 6.13: Plot of W versus J for the AH lattice model with κ = −0.8. The sharp drop in W is a clear indicator of a first order transition. The height of the drop is ∆W ≈ 0.1.

Another clue we find for a first order transition is the sudden jump in the energy density of the system near the transition point. In Figure 6.14 we show data for the energy density E near the transition temperature. As we can see from the plot, the transition is already clearly visible for L = 6. As a matter of fact, the transition is very sharp, and it renders very difficult to obtain clean data even for values as little as L = 10. In Figure 6.15 we show data for the specific heat per unit volume C/V vs. J, with peaks increasing with the volume V ≡ L3_,

confirming the first-order transition behaviour. This also shows that the system is still not in asympthotic behaviour. If that was the case, the peak of the specific heat per unit volume C/V would not grow with increasing L. From (5.17) we get:

Cmax(V ) V → 1 4∆ 2 h (V → ∞) (6.5)

The locations of the peaks in the specific heat will be used to estimate the infinite volume value of the critical temperature by fitting the equation:

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Figure 6.14: Plot of the energy density E vs. J for the κ = −0.8 system. The sharp jump in Eis a clear indicator of a first order transition and is correlated to the drop we saw in the plot of W vs J in Fig. 6.13 .

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In Figure 6.16 we show data for the Binder parameter U vs the temperature J. We observe behaviour compatible with Eq. (5.19). The locations of the peaks in the Binder parameter U will be used to estimate the infinite volume value of the critical temperature by fitting the equation:

J_cU(V ) − Jc= bV−1 (6.7)

and we show the results in Figure 6.17. We estimate JC

c = 1.116(3)and JCU = 1.122(2). (Here,

we have used data corresponding to L = 6, 8 and 10 for the fitting. The data for L = 10 could not be used to fit eqs. (5.17) and (5.19) due to very high uncertainty in the measuraments of Cmax(L = 10) and Umax(L = 10), but it had good accuracy in the measurament of Jc(L)

therefore it could be used for the estimate of Jc). The two estimates should ideally converge

to the same value of Jc but, due to systematic biases they converge to slightly different results.

Taking into account the biases we estimate Jc= 1.119(5).

Figure 6.16: Plot of U vs. J for the κ = −0.8 system. As expected from a first order transition, we observe the maximum value of the curve to be proportional to the volume of the system.

Figure 6.17: Parameter fitting for eqs.(5.18) and (5.20) for the κ = −0.8 system. We obtain JC

c = 1.116(3), a = −11.0(1) and JcU = 1.122(1), b = −19.0(1). From this we can estimate

the value of Jc by taking into account the biases of the two fit estimates and averaging the

two values, keeping in mind that the systematical error from the simulation is greater than the statistical error from the estimates. We get: Jc= 1.119(5).

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One defining feature of first order transitions is the coexistence of different phases at the transition point. By studying the distributions of thermodinamical quantities like the energy density E and the quantity µ2 defined in eq. (5.6) near the critical temperature Jc we can

identify peaks in the distributions as signs of the coexistent phases. In Figures 6.18 and 6.19 we show histograms for the energy density and magnetization at the transition point for L = 8 and J = 1.096. It is easy to see the two peaks for the energy distribution and for the magnetization. The presence of the peaks is futher evidence of a first order transition as it shows the coexistence of two different phases. From the histogram is possible to estimate the latent heat by measuring the distance between the peaks in the energy density in Figure 6.18. We get ∆h= 0.108(2).

Figure 6.18: Histogram for the energy density for the κ = −0.8 and L = 8 system. By measuring the distance between the peaks it is possible to estimate the latent heat ∆h≈ 0.10.

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6.3.2 Data for κ = −1

Next, we look at the system with κ = −1. For this system, the first order transition is a little weaker than the κ = −0.8 system, therefore in order to have meaningful insight it is necessary to study the system on larger lattices. As before, we start by checking the plaquette average value. In Figure 6.20 we show data for the average plaquette value W as a function of J. As we can see from the plot, the average value of W is progressively decreasing as κ decreases, the system is getting more frustrated. Like for the previous case, we observe a sharp drop in W , corresponding to the transition point. We also observe that the height of the drop is shrinking. While for κ = −0.8 the drop was ∆W ≈ 0.1, we clearly see that for κ = −1 the drop is ∆W ≈ 0.02, a factor of 5 smaller than the previous case. Next, we look at the behaviour of the energy density and specific heat. In Figure 6.21 we show data for the energy density E vs. J for the κ = −1 system. The sharp jump in the plots marks a first-order transition, like for the κ = −0.8 system. The jumps are less sharp than the κ = −0.8 case, even for greater values of L. This is another indicato of the fact that the transition is weaker for κ = −1 than for κ = −0.8.

Figure 6.20: Plot of W versus J for the AH lattice model with κ = −1. The sharp drop in W is a clear indicator of a first order transition. The height of the drop is ∆W ≈ 0.02.

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We then take a look at the specific heat C and study its behaviour to characterize the transition. As shown from Figure 6.22, the specific heat per unit volume C/V shows peaks in presence of the finite size critical temperature JC

c (L),like is expected. The behaviour of the

peaks is closer to asymptotic behaviour than the case for κ = −0.8 as the peaks do not increas as much with L (Eq. (5.17)). The data in the plot were used to estimate the infinite volume value of the critical temperature and the latent heat ∆h. Next, we look at the behaviour of the

Binder parameter U. In Figure 6.23 we can appreciate the Binder parameter U as a function of J. It shows maxima Umax(L)proportional to the volume V as expected from Eq. (5.19). The

peaks are in correspondece of the effctive finite size critical value of the transition temperature JU

c (L). The positions of the maxima were used to estimate the infinite volume of the critical

temperature Jc.

Figure 6.22: Data for the specific heat as a function of J, as we can see the maximum value of the specific heat increases with L and so does the value of Jc. We observe that the peaks’

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Like for the κ = −0.8 case, we study the distribution of the energy density and of the parameter µ2, to be interpreted as the magnetization of the system, near the critical

tempera-ture. In Figure 6.24, we have the histogram of the energy density distribution at the transition temperature. The histogram has two peaks which corroborate the observation that the tran-sition is first order. From the energy density histogram we are able to estimate ∆h ≈ 0.05 in

accordance with the fit estimates from the data shown in Figure 6.27. In Figure 6.25 we show the magnetization histogram for the κ = −1 system, noting the presence of two peaks in the distribution, also a symptom of a first order transition.

Figure 6.24: Histogram for the energy density for the κ = −1 and L = 12 system. It is easy to see the two peaks for the energy distribution. From the histogram it is possible to estimate the latent heat ∆h≈ 0.05.

Figure 6.25: (Histogram for the magnetization for the κ = −1 and L = 12 system. The presence of peaks is futher evidence of a first order transition.

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We then proceed to estimate the infinite volume value of the critical temperature Jc and

latent heat ∆h using the data from the specific heat C and Binder parameter U. Due to the

fact that we only have three points at our disposal, we choose to use linear fits as its the only viable option. We start with the critical temperature estimates, and proceed like in the case for κ = −0.8. We fit equations:

J_cC(V ) − Jc= aV−1, JcU(V ) − Jc = bV−1 (6.8)

In Figure 6.26 we show the parameter fitting for equations (6.8). Taking into account the systematic biases of the two estimates, we estimate Jc= 0.955(5).

Figure 6.26: Parameter fitting for eqs.(6.8) for the κ = −1 system. We obtain JC

c = 0.9556(2),

a = 18.1(1)and J_cU = 0.9545(2), b = −28.88(6). From this we can estimate the value of Jc by

taking into account the biases of the two fit estimates and averaging the two values, keeping in mind that the systematical error from the simulation is greater than the statistical error from the estimates. We get: Jc= 0.955(5).

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We then used the data from the specific heat C and Binder parameter U to fit equations: Cmax(V ) = V 1 4∆ 2 h+ c(1/V ) (6.9) and: Umax= aV + b (6.10)

For the latent heat we estimate ∆h = 0.054(1). The estimate is compatible with the estimate

from the energy density histogram in Figure 6.24. Results of the fits shown with the data points in Figures 6.27 and 6.28.

Figure 6.27: Parameter fitting for eq.(5.17). We obtain: ∆h= 0.054(1), c = −0.25(5), in good

accordance with the estimate from the energy histogram.

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6.3.3 Data for κ = −1.5

The last partially frustrated system we study is the κ = −1.5 system. In Figure 6.29 we show data for the average plaquette value W as a function of J. As expected from the previous data for the systems for κ = −0.8, −1, the system is more frustrated as a consequence of the greater value of κ. The overall range of values in which W varies is narrower, in fact we get ∆W ≈ 0.01, about a factor of two smaller than the κ = −1 case. The overall value of W is, as expected, closer to -1. In this regime the transition is weaker than the previous systems, as confirmed by the drop in W being less pronounced. We then take a look at the energy density as a function of temperature. In Figure 6.30 we show data for the energy density E vs. J. Here too we observe a milder transition than for the previous cases.

Figure 6.29: Plot of W versus J for the AH lattice model with κ = −1.5. The sharp drop in W is a clear indicator of a first order transition. The height of the drop is ∆W ≈ 0.01.

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We then look at the specific heat C and Binder parameter U. In Figure 6.31 we show data for C vs. J and observe increasing peaks of C at the effective value of the transition temperature Jc(L). Here too we observe that the transition is weaker than the ones for κ ≈ −0.8 and κ = −1,

in fact, the peaks of C and U grow weakly with the volume. In Figure 6.32 we show data for the Binder parameter U vs. J. We observe peaks in correspondence of the transition temperature, as expected from a first order transition. Like before, we are going to use the positions and values of the peaks to estimate the latent heat ∆h and the infinite volume value of the critical

temperature Jc.

Figure 6.31: Plot of the specific heat C as a function of J. As shown in the following (Figure 6.12) we estimate Deltah= 0.0113

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We then proceed to estimate the infinite volume value of the critical temperature Jc and

latent heat ∆h using the data from the specific heat C and Binder parameter U. Due to the

fact that we only have three points at our disposal, we choose to use linear fits as its the only viable option. We start with the critical temperature estimates, and proceed like in the case for κ = −0.8. We fit equations:

J_cC(V ) − Jc= aV−1, JcU(V ) − Jc = bV−1 (6.11)

In Figure 6.33 we show the parameter fitting for equations 6.11. Taking into account the sys-tematic biases of the two estimates we estimate Jc= 0.860(4).

We then used the data from the specific heat C and Binder parameter U to fit equations: Cmax(V ) = V 1 4∆ 2 h+ c(1/V ) (6.12) and: Umax= aV + b (6.13)

For the latent heat we estimate ∆h= 0.0113(1). The estimate is compatible with the estimate

from the energy density histogram in Figure 6.24. Results of the fits shown with the data points in Figures 6.27 and 6.28.

Figure 6.33: Parameter fitting for eqs.(6.11) for the κ = −1.5 system. We obtain JC

c = 0.864(3),

a = 6.9(1)and JU

c = 0.857(3), b = −15(1). From this we can estimate the value of Jc. By

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In Figures 6.34 and 6.35 we show parameter fitting for Eqs. (6.12) and (6.13) and obtain an estimate for the latent heat ∆h= 0.0113(1).

Figure 6.34: Parameter fitting for eq.(6.12). We obtain: ∆h= 0.0113(1), c = −0.080(1)

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6.4 What happens when κ → −∞?.

In this section we are going to discuss the behaviour of the system in the regime of |κ| 1, i.e in the regime of full frustration. In this regime, the gauge field is forced to assume a fixed configuration (a frustrated configuration) in order to minimize the Action (2.5), i.e. W = −1, leading to a system evolving through the interaction of the zx field with the frustrated λx,µ

field. As a further proof of this behaviour we ran simulations for large negative values of κ and L = 10 to show the behaviour of W as we approach the fully frustrated regime. We report the data for the average plaquette value W . The data show that the plaquettes’ expectation value for more negative values of κ tends to −1 and is largely independent of the temperature (Figure 6.36 (Top)). In Figure 6.36 (Bottom) we show data for W − 1 as a function of |κ−1_|_as

well as the line:

W + 1 = aκ−1, a = 0.17 (6.14) and we see that W + 1 → 0 as κ → −∞.

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6.4.1 Data for κ = −∞

As it will be clear from the data in the following, the nature of the transition for the κ = −∞ regime appears to be continuous. In order to estimate the critical exponents for the transitions we check the behaviour of the quantities Rξ, C and χ and look for behaviour compatible with

the equations of Section 5.1.1. In the following we show data from simulations of the frustrated system for various values of L. We first look at the behaviour of Rξ in order to estimate the

critical temperature Jc and the correlation lenght exponent ν. In Figure 6.38 we show the plot

for the quantity Rξ vs. J for the κ = −∞ system. The data shows a crossing point that can

be used to estimate the critical temperature Jc = 0.761(1). We used data belonging to a small

interval around Jc so that the behaviour of Rξ is linear enough in X to fit the equations:

Rξ= a + cX, X = (J − Jc)L1/ν (6.15)

In table 6.1 we show the fit results for the estimates of the exponent ν. From the table we see that the measuraments are stable when we vary |XM AX|and we have χ2/n ≈ 1, where n is the

number of data points used for the fit. The χ2 _{values are reasonable and therefore we estimate}

the exponent value to be ν = 0.570(5). In figure 6.37 we show the data points near Jc used for

the estimates in Table 6.1.

|XM AX| Lmin Data points χ2 ν a c

1.443 8 44 45.99 0.569(5) 0.1819(2) 0.029(3) 1.445 10 33 38.10 0.569(5) 0.1820(2) 0.029(3) 1.178 8 40 42.47 0.570(5) 0.1818(2) 0.028(3) 1.181 10 30 35.61 0.569(5) 0.1819(2) 0.028(3) 0.914 8 36 38.83 0.571(5) 0.1817(2) 0.028(3) 0.917 10 27 32.97 0.570(5) 0.1819(2) 0.028(3) 0.927 8 32 31.76 0.569(5) 0.1816(2) 0.029(3) 0.928 10 24 26,67 0.569(5) 0.1817(2) 0.029(3)

Table 6.1: Data from the fit estimates for Eq. (6.15).

Figure 6.37: Plot of the data points for Rξ vs J near the critical temperature used for the fit

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Next, we check the FSS behaviour of Rξ using the estimate for the critical exponent ν. We

first show in Figure 6.38 the data for Rξ as a function of J and then we show the same data

as a function of the scaling variable X = (J − Jc)L1/ν. If our estimate is correct, we should

see the points for various values of L collapse onto a single curve as predicted from Eq. (5.9). The curve is, modulo normalization, the universal scaling curve for the system’s universality class. In Figure 6.39 we show R vs. X, with ν = 0.57, the collapse of the data points provides confirmation for the estimate for the value of the ν exponent.

Figure 6.38: Plot of Rξ vs J in a larger range of J. We can see the crossing point signaling the

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We then look at the behaviour of the specific heat C in order to estimate the critical exponent α. We expect to see peaks in the specific heat near the critical temperature Jc, growing with

Las:

Cmax≈ a + bLα/ν (6.16)

. In Figure 6.40 we show the plot for the specific heat C vs J for the κ = −∞ system. As expected from FSS, the peaks of the curves increase with the size L of the lattice as predicted from Eq. (6.16). The data shown in Figure 6.40 has been used to estimate the ratio α/ν by fitting the equations:

C(J, L) ≈ (a + bX + cX2)Lα/ν, X = (J − Jc)L1/ν (6.17)

The reason we used a non-linear parametrization this time comes from the fact that, due to the shape of the curve near the critical temperature, just a linear expansion in X would not lead to a reasonable estimate of the exponent. In Table 6.2 we show fit results for the estimate of the ratio α/ν. From the table we see that measuraments are stable as we vary |XM AX and we

have χ2_/n _{reasonably close to 1, where, again, n is the number of data points used for the fit}

estimates. The χ2 _{values are reasonable and we estimate the ratio α/ν = 0.370(5) and, using}

the previous estimate for the ν exponent we get α = 0.21(1).

|XM AX| Lmin Data points χ2 α/ν a b c

5.25 8 36 50.08 0.365(5) 0.066(3) 0.0020(2) -0.00058(3) 5.24 10 27 40.11 0.364(5) 0.066(3) 0.0020(1) -0.00057(3) 5.25 8 33 17.98 0.368(5) 0.066(3) 0.0022(1) -0.00071(4) 5.25 10 24 11.97 0.367(5) 0.066(3) 0.0022(1) -0.00069(4) 3.97 8 28 13.19 0.376(4) 0.065(2) 0.0021(1) -0.00075(4) 3.97 10 21 7.96 0.374(4) 0.065(2) 0.0021(1) -0.00073(5) 2.68 8 20 7.39 0.379(4) 0.065(2) 0.0023(2) -0.0010(1) 2.68 10 15 3.85 0.377(4) 0.065(2) 0.0023(2) -0.0010(1)

Figure 6.40: Plot of C vs. J for the κ = −∞ system. The value of the peaks increases as Lα/ν_.

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We then check the FSS behaviour of the specific heat C using the estimate for the critical exponents ν and α. Like for Rξ, if the estimates are correct, we expect the data points for

various values of L to collapse onto a single curve. In this case, in addition to plotting the data as a function of the scaling variable X, like in (5.14), we can plot the data in terms of the RG invariant quantity Rξ, like in (5.15), with the advantage of being independent of normalizations.

In Figures 6.41 and 6.42 we show plots for CL−α/ν _{vs. X and vs. R}

ξ respectively, having used

the values ν = 0.57 and α = 0.21. We observe a good collapse of the data points onto the universal scaling curve.

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Finally, we study the behaviour of the magnetic susceptivity χ in order to estimate the critical exponent η. In Figure 6.43 we show the plot for the magnetic susceptivity χ vs. J for the κ = −∞ system. The data has been used to estimate the critical exponent η by fitting Eq. (5.12) using the ansatz:

χ(J, L) = (a + bX + cX2)L2−η, X = (J − Jc)L1/ν (6.18)

As for the specific heat, we chose to use a quadratic curve to better approximate the behaviour of the data for not too small values of X. In Table 6.3 we show fit results for the estimate of the exponent η. From the table we see reasonable values of the χ2 _{but we see fluctuations in}

the estimate of η as we vary |XM AX|. This leads to greater uncertainty in the estimate. We

get η = 1.2(1)

|XM AX| Lmin Data points χ2 η a b c

3.97 8 32 35.46 1.28(1) 0.43(1) 0.089(2) 0.0076(6) 3.97 10 24 24.92 1.25(1) 0.40(1) 0.082(2) 0.0072(6) 2.68 8 24 26.90 1.26(1) 0.422(5) 0.088(3) 0.007(1) 2.68 10 18 16.18 1.23(1) 0.38(2) 0.080(3) 0.007(1) 1.40 8 16 16.74 1.26(1) 0.42(2) 0.084(6) 0.003(2) 1.40 10 12 9.55 1.21(1) 0.37(2) 0.078(6) 0.005(3)

Figure 6.43: Plot of the magnetic susceptivity χ vs. J for the κ = −∞ system. The data have been used for the estimates in Table 6.3

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As for Rξ and the specific heat C, we look at the FSS behaviour of the magnetic susceptivity

χ. As usual, if the estimates are correct, the data points for various values of L should collapse onto a single curve. Like for the specific heat, we plot the data both as a function of the scaling variable X, like in (5.12), and as a function of Rξ, like in (5.13). In Figures 6.44 and 6.45 we

show data for χL−2+η _{vs. X and vs. R}

ξ respectively, having used the value η = 1.25. We

observe good collapse of the points onto the universal scaling curve.

Figure 6.44: Plot of χL−2+η _{vs. X, where we have used η = 1.25. We observe the collapse of}

Three-Dimensional Multicomponent Lattice Abelian-Higgs Models in the Presence of Frustrated Gauge Interactions