Subjects Of Scientific Research Methods Training Program International Physics.pdf

An Ising model is introduced and used to investigate theproperties of a two dimensional ferromagnet with respect toits magnetization and energy at varying temperatures.. The relative imp

VIETNAM NATIONAL UNIVERSITY HANOI UNIVERSITY OF SCIENCE DEPARTMENT OF PHYSICS

ⴰⴰ oOo ⴰⴰ

STUDENT: LE MINH QUAN

MONTE CARLO SIMULATION FOR MAGNETIC MATERIALS

SUBJECTS OF SCIENTIFIC RESEARCH METHODS TRAINING PROGRAM: INTERNATIONAL PHYSICS

SUPERVISORS: Assoc Prof Dr BACH HUONG GIANG

PG NGUYEN HAI PHONG

HANOI, 01/2023

VIETNAM NATIONAL UNIVERSITYHANOI UNIVERSITY OF SCIENCEDEPARTMENT OF PHYSICS

SUPERVISORS: Assoc Prof Dr BACH HUONG GIANG

PG NGUYEN HAI PHONG

STUDENT: LE MINH QUAN

CLASS: K65 INTERNATIONAL PHYSICS

HANOI, 01/2023

First of all, I would like to give my thanks to Assoc Prof.Bach Huong Giang for guiding and helping me in the process ofresearching and finishing this topic And a special thanks must beextended to the Postgraduate Nguyen Hai Phong who was alwaysavailable for invaluable discussion, advice and assistance Myclassmate Nguyen Dinh Tien also helped me in sharing hisexpertise with regards this report With their assistances, I foundmyself making a lot of progress in my reading and researchingskills This will be an indispensable baggage in the path of myeducation and research

Regards, Hanoi, January 13, 2023

Student

Le Minh Quan

Table of Contents

1 THEORY 5

1.1 General Introduction 5

1.2 Aims of achievement 6

1.3 Background 6

1.4 Model 8

1.5 Computational problems 11

1.6 Sampling and Averaging 12

1.7 Monte Carlo method 16

1.8 Calculation of observables – the quantities have to measure in a system of magnetic material 16

1.9 Metropolis Algorithm 17

1.10 Run program 20

2 RESULTS 21

2.1 Energy 21

2.2 Magnetization 22

2.3 Heat capacity 23

2.4 Susceptibility 24

3 Conclusion 26

4 References: 28

1 THEORY

1.1 General Introduction

Magnetic material has been found for the mileniums,and by around 300 - 200 B.C, humans had created one of thefirst devices to apply magnetism – the compass With themagnetic properties of materials, human history has seenmany major turning points Humans have applied materials

to form magnetic components used in many devices for dailylife such as fans, televisions, refrigerators, etc

To meet the increasing application of magneticmaterials in life Human have developed the discipline ofmagnetism to study the properties and phenomena ofmagnetic materials Among the phenomena of magneticmaterials, there is a phenomenon that has received muchattention from the scientific community, which is the phasetransition of magnetic materials From the classical Langevintheory (1905) on the research work on paramagnetism anddiamagnetism, or the model of molecular field theory ofWeiss (1907) explaining the phenomenon of ferromagnetismare famous theories, typical of magnetic material research.And in 1925, Ernst Ising published his doctoral thesis in theform of a scientific report

1.2 Aims of achievement

This discussion serves as an introduction to the use ofMonte Carlo simulations as a useful way to evaluate theobservables of a ferromagnet Key background is givenabout the relevance and effectiveness of this stochasticapproach and in particular the applicability of theMetropolis-Hastings algorithm Importantly the potentiallydevastating effects of spontaneous magnetization arehighlighted and a means to avert this is examined

An Ising model is introduced and used to investigate theproperties of a two dimensional ferromagnet with respect toits magnetization and energy at varying temperatures Theobservables are calculated and a phase transition at a criticaltemperature is also illustrated and evaluated Lastly a finitesize scaling analysis is undertaken to determine the criticalexponents and the Curie temperature is calculated using aratio of cumulants with differing lattice sizes The resultsobtained from the simulation are compared to exactcalculations to endorse the validity of this numerical process

In this research experiment, I relied on J Kotze’s articlebecause I read and realized it is very detailed From thereason why we have to do this work, find the solutions ofobservables, to the way to solve the problem

1.3 Background

In most ordinary materials the associated magneticdipoles of the atoms have a random orientation In effect this

non-specific distribution results in no overall macroscopicmagnetic moment However in certain cases, such as iron, amagnetic moment is produced as a result of a preferredalignment of the atomic spins

This phenomenon is based on two fundamentalprinciples, namely energy minimization and entropymaximization These are competing principles and areimportant in moderating the overall effect Temperature isthe mediator between these opposing elements andultimately determines which will be more dominant The relative importance of the energy minimization andentropy maximization is governed in nature by a specificprobability

(α)

kbT

(1)which is illustrated in figure below

Figure 1: Boltzmann probability distribution as a landscapefor varying Energy (E) and Temperature (T )

1.4 Model

To understand this theory, we need to clearly understandthe spin and magnetic moment These factors prove to beunnecessary complications

So the central idea to a model is to simplify thecomplexity of the problem to such a degree that it ismathematically tractable to deal with while retaining theessential physics of the system The Ising Model does thisvery effectively and even allows for a good conceptualunderstanding

In this report, I will present the Ising 2D model This isthe mathematical model of ferromagnet in statisticalmechanic

Figure 2: A lattice illustration of model Ising 2D

An Ising model describes the interaction between spinsplaced in the juntions of a lattice d-dimensions With asystem of spin S = ½, the quantum values of it on the z-axisare just one of 2 states “up” or “down” (+1/2 or -1/2) This isone of the simplest statistical model to describe the phasetransition of the ferromagnet

We will consider a 2D squared lattice have size L andthe total spins N = L x L Thus, a simplest classical Isingmodel is decribed by a Hamiltonian in the form (2) below:

si and s : value of spin at position i and jj

The reason why we set the J larger of smaller than 0dues to the direction of spin i and spin j, when spin i and jhave the same direction, assume positive and negative spinhave value +1 and -1, it means that energy of the system issum of 4 value of summation of multiplication between spin

i and j and equal 4, with the minus sign of Hamiltonian,energy of system equal −4, so we set J>0 for parallel link tolet energy smallest, similar for the case spin i have oppositedirection with spin j, so J<0 for anti-parallel link to satisfythe minimization energy

For simple, I will stipulate that J = +1 (take the example

of ferromagnet), spins +1 for spin “up” and -1 for spin

“down” and J/k is taken to be unity Thus, we get theb

relative position of the nearest neighbor of spins

Figure 3: Nearest neighbor coupling

With dark dot is the position spin s considering,i

surrounding spins are the nearest neighbor s of spin s Wej i

can see that for a spin, there’re 4 nearest neighborsurrounding it But what if the spin is in the edge?

The answer is to maximize the interaction, those spinsare made to interact with the spins at the geometric oppositeedges of the lattice This is referred to as periodic boundarycondition (pbc) and can be visualized better if we considerthe 2D lattice being folded into a donut-shaped 3D withspins being on the surface of this topological structure

Figure 4: An illustration of a donut-shaped 3D which isrepresentative of a 2D lattice with periodic boundary

conditions

1.5 Computational problems

With the help of Ising model, we can anticipate thesolutions of observables For a possible state α, theBoltzmann distribution function, equation (1) will be:

For example, for any fix state α, the magnetization will

be the different between the summation of spins “up” minussummation of spins “down”

M ( α )= Nup( α )− Ndown( α ) (3)while energy of lattice will be the Hamiltonian above,equation (2)

Ei=Hi=−J ∑

j nn

sisj

The expected value M and E is given by

⟨ M ⟩= ∑α M ( α ) P( α ) (4)

⟨ E ⟩= ∑α E( α ) P( α ) (5)The problem of these calculations is, when we use them,

in N spins lattice there are 2 different states That means theN

computations become so difficult with N large

It may seem a natural suggestion to use a computersimulation to do these calculations but by examiningequations (4) and (5) more closely it becomes apparent thatusing this procedure would waste as much computing effort

on calculating an improbable result as it does on a veryprobable result Thus a better numerical alternative would be

to use a simulation to generate data over the ‘representativestates’ These representative states constitute the appropriateproportions of different states This is a form of biasedsampling which essentially boils down to satisfying thecondition

GENERATED FREQUENCY ≡ ACTUAL PROBABILITY

(computer) (theory)

With that condition, we now need to examine how to accomplish the objective

1.6 Sampling and Averaging

The thermal average for an observable A(x) is defined in thecanonical ensemble

Figure 5: Metropolis Flowchart

• In the first step the lattice is INITIALIZED to a starting configuration This may either be a homogeneous or random configuration A random configuration has a benefit in that it uses less computing time to reach a equilibrated configuration with the associated heat bath

• In the following PROCESS the random number generator is used to select a position on the lattice by producing a uniformly generated number between 1 andN.

• A DECISION is then made whether the change in energy of flipping that particular spin selected is lower than zero This is in accordance with the principle of energy minimization

– If the change in energy is lower than zero then a PROCESS is invoked to flip the spin at the selected site and the associated change in the observables that want to be monitored are stored.– If the change in energy is higher than zero then aDECISION has to be used to establish if the spin

is going to be flipped, regardless of the higher energy consideration A random number is generated between 0 and 1 and then weighed against the Boltzmann Probability factor If the random number is less than the associated probability, e-βδH, then the spin is flipped (This would allow for the spin to be flipped as a result

of energy absorbed from the heat bath, as in keeping with the principle of entropy

maximization) else it is left unchanged in its original configuration

• The above steps are repeated N times and checked at this point in a DECISION to determine if the loop is

completed The steps referred to here do not include the initialization which is only required once in the beginning of the algorithm

• Once the N steps are completed a PROCESS is used to add all the progressive changes in the lattice

configuration together with the original configuration inorder to produce a new lattice configuration

• All these steps are, in turn, contained within a Monte Carlo loop A DECISION is used to see if these steps are completed

• Once the Monte Carlo loop is completed the program is left with, what amounts to, the sum of all the generated lattices within the N loops A PROCESS is thus employed to average the accumulated change in observables over the number of spins and the number ofMonte Carlo steps

• Lastly this data can be OUTPUT to a file or plot

order to produce these observables for a range oftemperature.Take the temperature T start from 1 to 5 withintervals of 0.1

Consider lattice sizes N = 2 × 2 ; 4 × 4 ; 8 × 8 ; 16 × 16.Run 5.000.000 Monte Carlo steps, thow away 3.000.000steps, each 100.000 steps write out 1 time

The steep gradient in the larger lattices points towards apossible phase transition but isn’t clearly illustrated

The energy per spin for higher temperatures is relativelyhigh This is in keeping with our expectation of having arandom configuration while it stabilizes to a E/N = −2J = −2

at low temperatures This indicates that the spins are allaligned in parallel In the region of low temperature T < 2,energy of the lattice keeps stable ,and rapidly increases whentemperature increase T > 2

2.2 Magnetization

Graph 2: The differing results of the Magnetization for

varying lattice sizes, L × L

In the region T < 2, Average Magnetization get thelargest value and approximately 1 (this mean the spins areparallel aligned)

When the Temperature increases T > 2, the spinsfluctuate lead to rapidly reduced the magnetization of thelattice

Decrease in magnetization of the larger lattices is fasterthan the smaller ones evidently due to the size of the lattices.With the small lattices, change of the spins not as clear as the

larger ones, leads to the decrease of magnetization slowerthan the larger lattices.

In the explaination of this phenomenon, because themagnetization is determined by excess of the spins, in thelow temperature, spins are parallel align with the samedirection to satisfy the minimization energy, so we have thelargest magnetization due to the equation (3) and (4) Andwith the temperature increasing, the spins are unstable, theyflip continously and nonstop This destroy the internalmagnetization of sample, lead to the magnetization very low,can be considered as insignificant

2.3 Heat capacity

Graph 3: The differing results of Specific Heat Capacity

for varying lattice sizes, L × L

The side graph shows the heat capacity as a function oftemperature

There’s a peak for each increasing size of lattice, for thelarger size, the peak appears more clearly

We concluded previously that a divergence would occur

at a phase transition and thus should be looking for such adivergence on the graph It is however clear that there is nosuch divergence but merely a progressive steepening of thepeak as the lattice size increases The point at which the plot

is peaked will be noted as a possible point of divergence

2.4 Susceptibility

With the susceptibility of sample, we will get a problemthat is our simulating susceptibility give out the data thus notexactly equivalent to the theoretical susceptibility, χ So, todistinguish these 2 quantities, we call the simulationsusceptibility is χ’

The scaling characteristic of this susceptibility is,however, equivalent to the theoretical value and only varies

by a constant factor above the Curie temperature