Magnetic Monopoles in Spin Ice

Master of Science Thesis

Axel Nordstr¨om

Supervisor: Patrik Henelius

Department of Theoretical Physics, School of Engineering Sciences Royal Institute of Technology, SE-106 91 Stockholm, Sweden

Stockholm, Sweden 2014 Typeset in LATEX

Examensarbete inom ämnet teoretisk fysik för avläggande av civilingenjörsexamen inom utbildningstprogrammet Teknisk fysik. Graduation thesis on the subject Theoretical Physics for the degree of Master of Science in Engineering from the School of Engineering Sciences.

TRITA-FYS 2014:26 ISSN 0280-316X ISRN KTH/FYS/–14:26–SE

In this thesis, we investigate the behaviour of magnetic monopoles in spin ice when an external magnetic field is applied. We find that steady state direct currents of magnetic monopoles cannot be maintained for long and consider the possibility of alternating magnetic currents by investigating the alternating current susceptibility using both analytical and Monte Carlo techniques. Moreover, we look at the transition that occurs when a magnetic field is applied in a 111 direction. We show that the transition is a continuous crossover rather thanh a phasei transition in the nearest neighbour model and we study the behaviour of the system during the crossover, especially at the critical field where a temperature independent state appears. Using Monte Carlo methods and analytical methods based on the Bethe approximation, we find that the mean monopole density is 0.4 monopoles per tetrahedron in the temperature independent state at the critical field.

Keywords: spin ice, magnetic monopoles, phase transitions.

iii iv Preface

This thesis is the result of my degree project at the Department of Theoretical Physics at the Royal Institute of Technology (KTH) during the spring semester of 2014. The work concerns magnetic monopoles in frustrated pyrochlore magnets – spin ice – and their statics and dynamics in an applied magnetic ﬁeld.

Overview

The thesis is divided into six chapters and three appendices. Chapter 1 briefly introduces the subject of magnetism and spin ice, whereas we in Chapter 2 go into some more detail on the generalities of spin ice, introducing the pyrochlore lattice and the model used. Chapter 3 concerns Monte Carlo simulations and we introduce the Metropolis algorithm used in all simulations throughout this thesis. In Chapters 4 and 5, the main results of the work are presented. In Chapter 4, we investigate the dynamics of magnetic monopoles in an applied magnetic field and in Chapter 5 we address the potential phase transition when a magnetic field is applied to spin ice in a 111 direction. Finally, the results are summarised in Chapter 6. Theh firsti appendix contains detailed information regarding the simulations performed throughout the thesis. The other two appendices contain brief reviews of rare earth magnetism and complexity classes in computer science, respectively. Knowledge of these topics is not required in order to understand the results presented in the thesis, but may help putting models and problems into context.

Conventions

It is conventional in the ﬁeld of frustrated magnetism to use the same unit for energy and temperature since it makes comparison of energy scales convenient.Thus we put kB = 1 throughout the thesis and measure energy in kelvins. Otherwise we use standard SI units.

v vi Acknowledgements

First and foremost, I would like to thank my supervisor Assoc. Prof. Patrik Henelius for giving me the opportunity to work with this thesis, for valuable guid- ance and discussions as well as for helping me with practical matters surrounding the thesis. I would also like to thank Mikael Twengstr¨omwith whom I have shared oﬃce this past semester and with whom I have had many fruitful discussions on coding, simulations and physics. Many thanks to my colleagues at the Department of Theoretical Physics at KTH for welcoming me among them. Finally I want to thank my friends and family for their encouragement and support.

vii viii Contents

Abstract ...... iii

Preface v

Acknowledgements vii

Contents ix

1. Introduction 1

2. Background theory 3 2.1. Magnetism ...... 3 2.1.1. Para-, ferro- and antiferromagnetism ...... 3 2.1.2. Frustration ...... 4 2.2. Spin ice ...... 5 2.2.1. The pyrochlore lattice and the ice rules ...... 5 2.2.2. Residual entropy ...... 8 2.2.3. Magnetic monopoles ...... 8 2.3. Modelling spin ice ...... 9 2.3.1. The nearest neighbour model ...... 9 2.3.2. The dipole model ...... 10 2.3.3. Eﬀect of magnetic ﬁelds ...... 10

3. Monte Carlo simulations 13 3.1. Pseudo-random numbers ...... 13 3.2. Monte Carlo algorithms ...... 13 3.2.1. Requirements on Monte Carlo algorithms ...... 13 3.2.2. Single spin ﬂips ...... 14 3.2.3. The Metropolis algorithm ...... 14 3.2.4. Limitations of Metropolis dynamics ...... 15

ix x Contents

4. Magnetricity 17 4.1. Magnetic monopole current ...... 17 4.2. Expression for the monopole current ...... 18 4.2.1. Magnetisation ...... 21 4.2.2. AC susceptibility ...... 22 4.2.3. Analogy to an RL-circuit ...... 22 4.3. Spin-lattice relaxation ...... 23 4.3.1. Monopole current ...... 25 4.3.2. Interpretation of χS ...... 26 4.4. Numerical determination of the susceptibility ...... 26 4.5. Predicting the monopole current ...... 28

5. Phase transitions in a 111 magnetic field 31 5.1. Introduction to phaseh transitionsi ...... 31 5.2. Kagoméice ...... 32 5.2.1. Value of the critical field ...... 33 5.2.2. Magnetisation plateau ...... 36 5.2.3. Phase diagram of the nearest neighbour model ...... 37 5.2.4. Low temperature simulations ...... 40 5.2.5. Phase transition or continuous crossover? ...... 41 5.3. Temperature independent state at Hc ...... 42 5.3.1. Monopole density for larger systems ...... 44 5.3.2. Bethe approximation ...... 46 5.3.3. Onset of the temperature dependence ...... 49 5.4. Phase transitions in the dipole model ...... 50

6. Conclusions 53 6.1. Magnetricity ...... 53 6.2. Phase transitions ...... 54

A. Simulations 55 A.1. Details ...... 55 A.2. Simulation parameters ...... 55

B. Rare-earth magnetism 59 B.1. Generalities ...... 59 B.2. Hund’s rules ...... 59

C. Complexity classes 61 C.1. P, NP and #P problems ...... 61 C.2. NP- and #P-complete problems ...... 61 C.3. The P versus NP problem ...... 62

Bibliography 63 Chapter 1

Introduction

The phenomenon of magnetism has been known since about 500 BC, possibly even longer. Since the 12th century, ferromagnets have been used as compasses for navigation and much modern technology, such as hard disks and techniques for medical imaging, relies heavily on magnetism. Although magnetism has been used in technology for nearly a millennium, it was not until the 20th century that the microscopic mechanisms of magnetism were properly understood and the discoveries of other types of magnetism than the usual ferromagnetism were made. To this day, magnetism remains a much studied phenomenon and the research into magnetic systems is vast to say the least. One speciﬁc kind of systems that have acquired must attention over the past few decades are so-called frustrated magnetic systems. In these systems, it is impossible to minimise the energy of all interactions simultaneously and the ground state is massively degenerate (see Section 2.1.2). When Harris et al. [1] performed susceptibility measurements and neutron scat- tering on the rare earth pyrochlore compound Ho2Ti2O7 in 1997, it turned out that this material was not only frustrated, but also had a net ferromagnetic nearest neighbour interaction. No such system had previously been observed. Due to an analogy to the frustrated structure of water ice, the family of frustrated pyrochlore magnets to which Ho2Ti2O7 belongs was dubbed ”spin ice”. Since ideal frustrated systems retain their degeneracy even when their temperature reach absolute zero, they seem to violate the third law of thermodynamics – which states that the entropy of a perfect crystal, at absolute zero kelvin, is exactly equal to zero. It is, however, widely believed that real frustrated systems have a phase transition to an ordered state at low temperatures, due to small perturba- tions in the system starting to play an important role. In many cases, though, the phase transition would occur at a temperature below what is experimentally attainable today. Electricity and magnetism are intimately related via Maxwell’s equations. How- ever, whereas there exist free electric charges which are sources of the electric ﬁeld,

1 2 CHAPTER 1. INTRODUCTION magnetic ﬁelds seem only to be generated by magnetic multipoles. The question thus arises: do magnetic charges, or monopoles, exist? The existence of magnetic monopoles was proposed by Paul Dirac in 1931 [2], but no experiment has been able to detect any magnetic monopoles to this day. However, in 2008 it was suggested that elementary excitations in spin ice materials behave much like classical analogues of Dirac’s magnetic monopoles [3]. This discovery raises the questions of whether or not it is possible to create currents of magnetic monopoles, as is possible with electric charges, and if it even would be possible to construct magnetic equivalents of electronic circuits in condensed matter systems despite the apparent rareness or lack of natural magnetic monopoles. Chapter 2

Background theory

Here we introduce diﬀerent kinds of magnetism and how these can give rise to frustration. We also introduce the spin ice compounds, the pyrochlore lattice and present the models used to describe spin ice.

2.1 Magnetism

2.1.1 Para-, ferro- and antiferromagnetism

Magnetism occurs when the intrinsic magnetic moments of elementary particles, spins, interact – either with a magnetic field or with each other. If the spins do not interact among themselves, but only with external magnetic fields, we are dealing with paramagnetism. The spins tend to align with the magnetic field according to M = χH, (2.1) where M is the sample magnetisation, H is the external magnetic field and χ is the susceptibility. In purely paramagnetic materials, the susceptibility obeys Curie’s law [4] 1 χ , (2.2) ∝ T where T is the temperature. If we consider a simple exchange interaction between nearest neighbour spins, we get two different cases: ferromagnetism and antiferromagnetism. In a ferromagnet, the spins tend to align parallel to each other below the Curie temperature

3 4 CHAPTER 2. BACKGROUND THEORY

TC . Above this temperature, the behaviour is essentially paramagnetic. However, Equation 2.2 must be substituted for the Curie-Weiss law [4] 1 χ . (2.3) ∝ T TC − In antiferromagnets, the spins instead align anti-parallel to each other below the N´eeltemperature TN . As in the ferromagnet, the antiferromagnet becomes paramagnetic above TN , but obeys 1 χ , (2.4) ∝ T + θ where θ is a material dependent parameter in the order of TN [4]. Equations 2.2, 2.3 and 2.4 can thus be written as ∼

Θ = 0 paramagnet 1  CW χ where ΘCW > 0 ferromagnet (2.5) ∼ T ΘCW − ΘCW < 0 antiferromagnet, and thus we can determine what kind of interactions dominate at low temperature by measuring the susceptibility as a function of temperature in the paramagnetic phase and determining the Curie-Weiss temperature, ΘCW [5].

2.1.2 Frustration When a system cannot minimise the energy of all its interactions simultaneously, we say that the system is frustrated. This implies that the ground state of frustrated systems will be degenerate, since there are several ways to minimise the total energy of the system. The link between frustration and degeneracy is so fundamental that the appearance of degeneracy sometimes is taken as a deﬁnition of frustration [6].

Random frustration One way to achieve frustration in a magnetic system is by introducing randomness in the system. If spins interact ferromagnetically with some neighbours, but antiferromagnetically with others, it may not be possible to satisfy every bond on the lattice, possibly giving rise to frustration. Random frustration is for example present in spin glasses [5].

Geometrical frustration Geometrical frustration arises when all bonds between spins cannot be satisﬁed because of the lattice geometry. A simple example is the Ising antiferromagnet on a triangular lattice shown to the left in Figure 2.1. In this case, each spin wants both its neighbours to anti-align, which is not possible for all three spins on a single triangle. 2.2. SPIN ICE 5 ?

Figure 2.1: Examples of geometrical frustration. To the left is a triangular unit cell with antiferromagnetic interactions making one spin point up and one down. The last spin can be either up or down, being unable to satisfy both of its bonds. To the right is a primitive unit cell of the pyrochlore lattice. Here the spins interact ferromagnetically resulting in the frustrated ground state with two spins pointing into the tetrahedron and two spins pointing out of it.

In spin ice, the frustration arises from the geometry of the pyrochlore lattice (see Section 2.2.1). Essentially, four spins located on the vertices of a tetrahedron are forced to point either into or out of the tetrahedron. The spins interact ferromagnetically meaning that if a spin points into the tetrahedron, it wants all other spins to point out of the tetrahedron, or vice versa. This is obviously not possible to satisfy for all four spins in the tetrahedron, and the compromise leading to the lowest energy of the system is two spins pointing into and two spins pointing out of the tetrahedron, as in the right panel of Figure 2.1.

2.2 Spin ice

2.2.1 The pyrochlore lattice and the ice rules Spin ice is the name given to a family of magnetically frustrated pyrochlore oxides 4+ 4+ such as Ho2Ti2O7, Dy2Ti2O7, Ho2Sn2O7 and Dy2Sn2O7. The Ti and Sn ions are non-magnetic, but the Ho3+ and Dy3+ ions are both rare earth elements with large magnetic moments in the order of 10µB [7]. Here, µB is the Bohr magneton, e deﬁned as µB = ~ , where e is the elementary charge, is the reduced Planck’s 2me ~ constant and me is the electron mass. It is the large magnetic moment of the ions, the large crystal ﬁeld splitting of the compounds and the fact that the magnetic ions reside on a pyrochlore lattice that give rise to the interesting properties of spin ice. The pyrochlore lattice is a face-centered cubic (fcc) Bravais lattice with a four site basis. The atoms are oriented such that each primitive unit cell forms a tetrahedron by itself, and when several unit cells are combined we get a lattice of vertex- 6 CHAPTER 2. BACKGROUND THEORY

[001]

[010]

[100]

Figure 2.2: The pyrochlore lattice. The magnetic ions reside on the vertices of vertex-sharing tetrahedra. sharing tetrahedra as in Figure 2.2. Each tetrahedron’s center occupies a site on a diamond lattice [3]. The diamond lattice dual to the pyrochlore lattice is drawn in blue in Figure 2.3. Due to holmium and dysprosium being rare earth elements, the unpaired electrons contributing to their magnetic moment are of 4f type. The wavefunctions of these electrons do not stretch as far from the nucleus as for example 5s and 5p electrons [8] which are also present in Ho3+ and Dy3+. The relatively confined electronic wavefunctions make the exchange interaction relatively small, in the order of a few kelvins, whereas the crystal field splitting is around 200 to 300 K [7]. The large crystal field completely eclipses the exchange interaction and forces the spins to point along their local 111 axes, that is, into or out of a tetrahedron. h i However, the exchange interaction still has a role to play. Whereas the crystal 2.2. SPIN ICE 7

Figure 2.3: The tetrahedra of the pyrochlore lattice form a diamond lattice. The diamond lattice sites are marked with blue spheres.

ﬁeld pins the spins to the local 111 axes, it is the spin-spin interactions (mainly nearest neighbour interactions, suchh i as exchange) that determines whether a spin should point into or out of a tetrahedron. With the net nearest neighbour interaction being ferromagnetic [9], the energy of one spin pointing into a tetrahedron is minimised if the other spins of the tetrahedron point outwards. Thus it is impossible to minimise the energy of all nearest neighbour interactions simultaneously and the system is frustrated. The ground state is when two spins point into and two spins point out of each tetrahedron – the so-called ice rules or two-in-two-out rule [1] – and is massively degenerate. The net nearest neighbour interaction is known to be ferromagnetic since measurements of the Curie-Weiss temperature gave a positive ΘCW +1.9 K which implies ferromagnetism as noted in Section 2.1.1. Spin ice was the≈ ﬁrst ferromagnetic system to show frustration – all previously known frustrated magnetic systems had been antiferromagnetic, such as the triangular Ising antiferromagnet mentioned previously [1]. 8 CHAPTER 2. BACKGROUND THEORY

The name spin ice and the ice rules come from an analogy to water ice. In water ice, the oxygen atoms reside on a diamond lattice with four hydrogen atoms tetrahedrally coordinated [10]. The hydrogen atoms are thus situated on the pyrochlore lattice, dual to the diamond lattice. In the ground state, two of these hydrogen atoms are covalently bound to the oxygen atom and two form hydrogen bonds – analogous to the two-in-two-out rules of spin ice. Thus arises geometrical frustration which gives rise to a large degeneracy of the water ice ground state.

2.2.2 Residual entropy Due to the degeneracy of the spin ice ground state, spin ice compounds – just like water ice – possess a zero-point entropy. This entropy was estimated for water ice by Pauling in 1935 [10] by estimating the number of possible ground states, Ω. For each tetrahedron there are four spins that can point either in or out, giving rise to 24 = 16 unique conﬁgurations. The ones obeying the ice rules, i.e. two-in- 4 two-out, can be calculated as 2 = 6. If we now take N tetrahedra, we have 2N spins and thus 22N possible conﬁgurations, but since only a fraction 6/16 of the states per tetrahedron obey the ice rules, we obtain

6 N 3N Ω = 22N = , (2.6) 16 2 which gives the entropy 3 S = ln Ω = N ln . (2.7) 2 Experiments have conﬁrmed that spin ice compounds possess a residual entropy close to Pauling’s estimate [11, 12].

2.2.3 Magnetic monopoles As stated above, the ground state of spin ice compounds is a state obeying the ice rules two-in-two-out. The elementary excitation occurs when a single spin is flipped, creating one tetrahedron with three-in-one-out and one tetrahedron with one-in-three-out. However, this excited state is also degenerate, and we can con- tinue flipping certain spins without increasing the energy of the system. In this way we can separate the three-in-one-out and the one-in-three-out tetrahedra by a macroscopic distance as in Figure 2.4. In 2008, Castelnovo, Moessner and Sondhi [3] argued that these elementary excitation can be viewed as magnetic monopoles, carrying a magnetic charge and being under the influence of a magnetic Coulomb potential. Since only a finite amount of energy is required to separate the monopoles infinitely, the monopoles are deconfined, unlike similar excitations in ordered ferromagnets [3]. Whereas these excitations effectively can be seen as magnetic monopoles, they are not equivalent to the magnetic monopoles suggested by Dirac in 1931 [2]. The 2.3. MODELLING SPIN ICE 9

Figure 2.4: Magnetic monopoles in spin ice. A chain of flipped spins (in white) gives rise to two defects of three-in-one-out and one-in-three-out respectively. These monopoles are deconfined and can be separated indefinitely at a finite energy cost. so-called Dirac string connecting two magnetic monopoles must be unobservable for Dirac monopoles, but in the case of spin ice, the Dirac string is a chain of flipped spins (in white in Figure 2.4) connecting the magnetically charged tetrahedra and thus it is actually observable [3]. Furthermore, the monopole charge is not quantised in spin ice. The charge is defined as qm = 2µ/ad, where µ is the rare earth ion ± magnetic moment and ad is the diamond lattice parameter. Thus, it is possible to change the monopole charge by applying pressure to the sample – i.e. changing the lattice parameter. Dirac monopoles, on the other hand, must be quantised in units of h/µ0e where h is Planck’s constant, µ0 is the vacuum permeability and e is the elementary charge [2, 3]. Furthermore, magnetic monopoles in spin ice always appear in pairs of one positive and one negative monopole.

2.3 Modelling spin ice

2.3.1 The nearest neighbour model

The simplest way to describe the magnetic properties of spin ice materials is to employ the nearest neighbour model. This is essentially the Ising model [13] on a 10 CHAPTER 2. BACKGROUND THEORY pyrochlore lattice. The Hamiltonian of the Ising model is given by X H = J Si Sj, (2.8) − · hi,ji where i, j indicates summation over nearest neighbour spins Si and Sj. The spins areh vectorsi of unit length directed in the 111 directions. J is the interaction constant. If J > 0, the interaction is ferromagnetic,h i whereas if J < 0 it is antiferromagnetic. For the two most common spin ice compounds, Ho2Ti2O7 and Dy2Ti2O7, J equals 5.4 K and 3.3 K [7], respectively. The nearest neighbour model is used in all simulations throughout this thesis and is – despite its simplicity – surprisingly successful in reproducing the qualitative behaviour of real spin ice compounds.

2.3.2 The dipole model As one might suspect, it turns out that the nearest neighbour model is a simpliﬁ- cation of reality. Due to the large magnetic moments of the Ho3+ and Dy3+ ions, long range dipole interactions are bound to play an important role in the behaviour of spin ice [9]. When accounting for dipole-dipole interactions, the Hamiltonian is given by [9]

X 3 X Si Sj 3 (Si rij)(Sj rij) H = J Si Sj + Drnn · 3 · 5 · , (2.9) − · rij − rij hi,ji j>i | | | | where D is an interaction constant for the dipole-dipole interaction, rnn is the nearest neighbour distance and rij is the vector pointing from site i to site j. Using the above model, it actually turns out that J < 0 – i.e. the exchange interaction is antiferromagnetic. D is, however, positive and large enough to make the net nearest neighbour interaction ferromagnetic [9]. −3 For simulations, the long range interactions due to the rij dependence of the dipole term are problematic. Attempts have been made to| truncate| the interaction after the 5th or 12th nearest neighbour [14, 15] but in these cases even the nearest neighbour model yields better results due to screening of medium- to long-range interactions in the real system. In order to properly account for the long range interactions, one often turns to Ewald summation techniques [9].

2.3.3 Effect of magnetic fields Applying a magnetic field to the sample is equivalent to adding a Zeeman term to the Hamiltonians of the respective models in Equations 2.8 and 2.9 given by [8] X HZ = µ0H µSi, (2.10) − · i 2.3. MODELLING SPIN ICE 11

where µ0 is the vacuum permeability, µ is the magnetic moment of the ions and H is the applied magnetic ﬁeld. The spins will thus seek to align with the magnetic ﬁeld in order to minimise their free energy. 12 Chapter 3

Monte Carlo simulations

This chapter is concerned with Monte Carlo algorithms. We introduce some general requirements on Monte Carlo algorithms and present the single spin ﬂip Metropolis algorithm which is used in the simulations throughout the thesis.

3.1 Pseudo-random numbers

Monte Carlo simulations are based on random numbers and are therefore highly dependent on our ability to generate large amounts of random numbers on a computer. Unfortunately, computers are by construction deterministic in their nature, i.e. they are unable to generate pure random numbers. However, by making use of clever algorithms we can generate simulated random numbers – pseudo-random numbers [16]. Hence, when referring to random numbers throughout this thesis, we actually mean pseudo-random numbers.

3.2 Monte Carlo algorithms 3.2.1 Requirements on Monte Carlo algorithms In order to construct a Monte Carlo algorithm that properly describes reality we generally impose two requirements on the algorithm: it must be ergodic and it must obey detailed balance. Ergodicity is fulﬁlled if the system – being in any allowed state – can reach all other states in a ﬁnite time [16]. That is, all states of our system must be connected in phase space – we do not allow separate ”islands” of states. Ergodicity ensures that we sample a fraction of phase space that is representative of the entire phase space. The condition of detailed balance comes from the interpretation of Monte Carlo sampling as Markov chains. A Markov chain is a chain of stochastic events that is

13 14 CHAPTER 3. MONTE CARLO SIMULATIONS memoryless. This means that the transition probability from state n to state m at time t, Wn→m only depends on the current state of the system, n(t), and not on any previous states, n(t0 < t) [16]. The probability of being in state n at time t, Pn(t), obeys the master equation

dPn(t) X = [P (t)W P (t)W ] . (3.1) dt m m→n n n→m m −

The ﬁrst term in the brackets of Equation 3.1 accounts for the change in Pn(t) due to transitions from a state m to the state n, whereas the last term represents the change due to transitions from n to m, hence the minus sign. Finally, we sum over all possible states in order to account for all possible transitions in the system1. dPn(t) In the steady state – equilibrium – the derivative, dt , must be zero. The simplest way to satisfy this is to say that all terms in the sum must equal zero, which leads to the condition for detailed balance [16]

Pn(t)Wn→m = Pm(t)Wm→n. (3.2)

Imposing detailed balance when choosing transition rates for a Monte Carlo algorithm ensures that we are sampling the equilibrium properties of the system. Detailed balance is thus a sufficient condition to fulfil the steady state master equation. However, it is not a necessary condition – it is possible to fulfil the steady state master equation by mixing different terms in the sum of Equation 3.1, although this is considerably more complicated and generally, detailed balance is preferred.

3.2.2 Single spin flips Throughout our simulations we have used single spin flip methods. This means that the accepted moves are the flipping of single spins with a probability 0 < W 1. Since this probability is strictly non-zero, single spin flip algorithms are trivially≤ ergodic.

3.2.3 The Metropolis algorithm The algorithm of choice for our simulations of nearest neighbour interactions in spin ice is the Metropolis algorithm [17]. The essential part of the algorithm is to choose the transition rate according to

e−β∆E if ∆E > 0 W = (3.3) n→m 1 otherwise,

−1 where ∆E = Em En is the change in energy due to the spin ﬂip and β = T is the inverse thermal− energy. While we for standard non-importance sampling Monte

1Note that for n = m, the two terms cancel so we have no problems with transitions to the same state. 3.2. MONTE CARLO ALGORITHMS 15

Carlo would weight the contribution from each conﬁguration by its Boltzmann factor, choosing the transition rate according to Equation 3.3 introduces an importance sampling and we instead weight all states equally, but sample the state with larger Boltzmann factor more often. Equation 3.3 also guarantees that detailed balance is obeyed. Since the probability of being in a state n is given by 1 P (t) = e−βEn , (3.4) n Z where Z is the partition function [16], Equation 3.2 reduces to 1 1 e−βEn e−β(Em−En) = e−βEm , (3.5) Z Z if Em En, or ≥ 1 1 e−βEn = e−βEm e−β(En−Em), (3.6) Z Z if Em En, both of which are trivially true, and thus detailed balance is obeyed. Schematically,≤ we can describe a Monte Carlo simulation of an Ising-like model utilising the Metropolis algorithm as follows:

1. Set up the lattice and assign each spin a random direction. 2. Perform a number of Monte Carlo steps to equilibrate the system.

3. Perform a number of Monte Carlo steps during which the quantities of interest are calculated.

A Monte Carlo step means 1. Select a random spin from your lattice.

2. Calculate the transition rate, Wn→m, according to Equation 3.3. 3. Generate a uniformly distributed random number 0 < r < 1.

4. If Wn→m > r, the spin ﬂip is accepted. Else it is rejected and nothing happens. 5. For a system of N spins, repeat steps 1 to 4 N times.

3.2.4 Limitations of Metropolis dynamics A rule of thumb for optimising Monte Carlo algorithms is to have an acceptance rate of about 50% for the moves [18] – in our case the single spin flips. However, in the 16 CHAPTER 3. MONTE CARLO SIMULATIONS low temperature limit, β , the Metropolis transition rate goes to zero, meaning that barely any moves will→ ∞ be accepted making the algorithm very inefficient in sampling a representative part of the system states. In the limit of high temperatures, β 0, the transition rate rather goes to 1. This means that every spin flip will be→ allowed, which risks giving rise to larger correlations in time than should be the case. If one is to simulate high temperatures one may instead choose the Glauber algorithm, which instead uses the transition rate [16] 1 1 W = 1 + S tanh βE , (3.7) n→m 2 2 i n where Si = 1 is the spin being flipped and En is the energy of the original configuration.± The Glauber transition rate tends to 1/2 for large temperatures, avoiding the problem of the Metropolis algorithm at high temperatures. In other aspects, the Metropolis and Glauber algorithms do not differ significantly and they are both special cases of a more general transition rate [16]. Chapter 4

Magnetricity

In this chapter we treat direct and alternating currents of magnetic monopoles. We derive expressions for the monopole current and the AC susceptibility in the nearest neighbour model and calculate the AC susceptibility numerically when the field is applied in a 001 direction. Finally, we present a scheme to find the magnetisation and monopoleh currenti for an arbitrary time dependent magnetic field without using Monte Carlo simulations.

4.1 Magnetic monopole current

Due to the symmetry of Maxwell’s equations with respect to electric and magnetic fields, one may na¨ıvely expect magnetic monopoles in spin ice to behave like electric charges. This might lead us to believe that it is possible to drive a current of magnetic monopoles through a spin ice material. If we were to apply a constant magnetic field to a sample with periodic boundary conditions, we could get a steady state current through the sample – a magnetic equivalent of electricity, known to some as ”magnetricity” [19]. However, as can be seen from the Monte Carlo simulations of the monopole current presented in Figure 4.1, it turns out that the monopole current that emerges upon applying a magnetic field decays to zero exponentially with a characteristic relaxation time τ, both for periodic and open boundary conditions. It is important to keep in mind that while we in some respects indeed can view the elementary excitations in spin ice as magnetic monopoles, these excitations are not free particles. Every time a monopole passes through a plane, a spin in that plane is flipped in the direction of the current [20] as in Figure 4.2. Thus all monopole current contributes to magnetising the sample according to [21]

∂M j = , (4.1) mp ∂t

17 18 CHAPTER 4. MAGNETRICITY

0.012

0.008

0.004 Monopole current [a.u.]

0.000

0 20 40 60 80 100 Monte Carlo time

Figure 4.1: Monte Carlo simulation of the current of magnetic monopoles that appears upon applying a constant magnetic ﬁeld at t = 0 at a temperature T/J = 0.2. Note that both in the case of open boundary conditions (red) and periodic boundary conditions (blue), the current decays to zero. For precise simulation parameters, please refer to Appendix A.

where jmp is the monopole current and M is the magnetisation of the sample. If we were to have a steady state current, the magnetisation of the sample would tend to infinity, which is obviously unphysical. A persistent direct current is thus out of the question. However, alternating currents with a large enough frequency, ω such that ωτ 2π, should remain relatively unaffected by the relaxation of the monopole current, provided that the system is able to respond swiftly enough to a change in the magnetic field.

4.2 Expression for the monopole current

Let us consider a system of non-interacting magnetic monopoles, equivalent to a nearest neighbour spin ice model. In the general case, we can write the entropy of a system slightly out of equilibrium as a quadratic form

1 X TS( αj ) = TS( 0 ) gikαi αk (4.2) { } { } − 2 · i,k 4.2. EXPRESSION FOR THE MONOPOLE CURRENT 19

Figure 4.2: An excited spin ice state projected on the (100) plane with a magnetic field applied along the [001] direction. Spins aligned with the field are black and spins anti-aligned are white. When a positive monopole (blue) moves in the direction of the field, a spin is aligned with the field in the plane marked by the red rectangle, contributing to increasing the magnetisation of the sample. Eventually, the marked plane will be fully polarised and no positive monopoles can pass in the field direction. This effectively makes any direct monopole current transient.

where gik is a positive definite form [22] – that is, the eigenvalues of gik are all positive. In our case, the thermodynamic quantities, αj, are the magnetic field α1 = H and the magnetisation α2 = M. The form of Equation 4.2 is due to the fact that the entropy has a maximum at thermal equilibrium, meaning that the first derivatives with respect to αj must be zero and the second derivatives must be negative. Introducing the magnetic field and magnetisation into Equation 4.2, we get

2 2 TS(H, M) = TS(0, 0) C1H C2H M C3M , (4.3) − − · − where C1,C2,C3 are suitable constants. On diﬀerential form, taking dM as the diﬀerential, this becomes

T dS = C2H dM 2C3M dM. (4.4) − · − · As stated above, we require the entropy to have a maximum at equilibrium. Thus

T dS = (C2Heq + 2C3Meq) dM (4.5) − · must be zero for any inﬁnitesimal change in magnetisation, dM,

C2Heq + 2C3Meq = 0. (4.6) 20 CHAPTER 4. MAGNETRICITY

Since we know that at equilibrium, Meq = χT Heq, where χT is the isothermal susceptibility, we get the relation 2C 1 3 = (4.7) C2 −χT for the constants. Let us now take the time derivative of the entropy in order to arrive at an expression for the entropy production [21]

∂S ∂M −1 T = C2 H χ M . (4.8) ∂t − ∂t · − T Returning to the formulation of the entropy in terms of the general state variables, αj [22], we deﬁne the generalised currents as the time derivative of the state ∂αi variables, Ji , and the generalised forces as the linear combination of the ≡ ∂t P state variables, Xi k gikαk. If we now take the time derivative of Equation 4.2, we get ≡ −

∂S 1 X ∂αk 1 X ∂αi X ∂αi T = gikαi gik αk = gik αk. (4.9) ∂t −2 · ∂t − 2 ∂t · − ∂t · i,k i,k i,k This means that we can always write the entropy production as a product of generalised currents and forces in the following manner ∂S X T = Ji Xi. (4.10) ∂t · i ∂M −1 In Equation 4.8, we identify J1 = jmp = ∂t and X1 = C2 H χT M . We will now make the assumption that the generalised forces− and currents obey the phenomenological equations [22] X Ji = LikXk (4.11) i where Lik are phenomenological constants. Since we only have one current, jmp, Equation 4.11 becomes −1 jmp = L11C2 H χ M . (4.12) − T Introducing the monopole mobility, u, the monopole concentration, c, and the monopole charge qm = 2µ/ad, where µ is the magnetic moment of the ions and ad is the lattice parameter of the diamond lattice dual to the pyrochlore lattice, we can use these to express the constants in Equation 4.12 as L11C2 = ucqmµ0 κ [23] and the ﬁnal expression for the monopole current in the case of non-interacting≡ monopoles reads −1 jmp = κ H χ M . (4.13) − T The ﬁrst term of Equation 4.13 looks rather familiar if we make an analogy with the transport of electric charges. The Drude model of electric transport tells 4.2. EXPRESSION FOR THE MONOPOLE CURRENT 21

us that the electric current density, jel, is proportional to the applied electric field, E, by the following relation jel = neueqeE (4.14) where ne is the carrier concentration, ue is the carrier mobility and qe is the charge of the carrier. Exchanging electric field for magnetic field, E µ0H, electric → current for magnetic current, jel jmp and electric carrier concentration, mobility → and charge for their magnetic equivalents, ne c, ue u, qe qm, Equation 4.14 is equivalent to the first term of Equation 4.13.→ → → The origin of the second term in Equation 4.13 cannot be inferred from an electric equivalent as easily as the first term. Whereas the magnetic Drude force works to increase the current, this entropic force works to reduce the current, thus effectively making any direct current of magnetic monopoles transient (see Section 4.2.1). The entropic contribution arises from the attempt to minimise the free energy of the system. Since the Helmholtz free energy is given by F = U TS, where U is the internal energy, T is the temperature and S is the entropy,− an increase in entropy serves to reduce the free energy of the system. When the system is magnetised, the internal energy is reduced because of the Zeeman term of the Hamiltonian (Equation 2.10). However, the system also gets more ordered with increasing magnetisation, reducing the entropy. In the fully polarised state, all the residual entropy of the system has vanished – the ground state lacks degeneracy. Equation 4.13 can thus be seen as the competition between internal energy and entropy.

4.2.1 Magnetisation

∂M Since jmp = ∂t , Equation 4.13 is actually a ﬁrst order linear diﬀerential equation for the magnetisation ∂M = κ H(t) χ−1M(t) , (4.15) ∂t − T with the general solution

t Z 0 M(t) = e−κt/χT eκt /χT κH(t0, )dt0 + M(0) (4.16) 0 which gives the current

t κ Z 0 −κt/χT κt /χT 0 0 jmp = κH(t) e e κH(t )dt + M(0) . (4.17) − χT 0

If we look at the special case of a constant magnetic ﬁeld H0 applied to a sample at thermal equilibrium, i.e. M(0) = 0, at t = 0, we get −κt/χT M(t) = χT H0 1 e (4.18) − 22 CHAPTER 4. MAGNETRICITY

−κt/χT jmp = κH0e . (4.19) We see that the monopole current relaxes and the magnetisation rises exponentially with a relaxation time τ χT /κ. ≡ 4.2.2 AC susceptibility In order to investigate the behaviour of spin ice in an alternating magnetic ﬁeld, we take interest in the AC susceptibility, χ(ω), deﬁned via

M(ω) = χ(ω)H(ω). (4.20)

To obtain an expression for χ(ω), we transform Equation 4.15 to Fourier space

iωM(ω) = κ H(ω) χ−1M(ω) . (4.21) − T Using M(ω) = χ(ω)H(ω) and rearranging terms, we get the following expression for the AC susceptibility [21] χ χ(ω) = T (4.22) 1 + iωτ where the relaxation time again is given by τ χT /κ. Extracting the real (in- phase) and imaginary (out-of-phase) parts of χ(≡ω) = χ0(ω) iχ00(ω), we have − χ χ0(ω) = T (4.23) 1 + ω2τ 2 χ ωτ χ00(ω) = T . (4.24) 1 + ω2τ 2 This form of the susceptibility is known as the Debye model, which was originally proposed for modelling the permittivity of dielectrics.

4.2.3 Analogy to an RL-circuit Equation 4.18 for the magnetisation resembles the expression for the current in a −t/τ RL-circuit biased with a constant voltage switched on at t = 0, i(t) = i0(1 e ). Thus we turn to a model of a RL-circuit biased with an alternating voltage− in order to see if we can model spin ice by a simple circuit model. The current-voltage relationship in this circuit is v = Z(ω)i (4.25) where i is the current, v is the voltage and Z(ω) is the complex impedance. If we now make the substitution v(ω) H(ω) I(ω) → M(ω) (4.26) Z−1(ω) → χ(ω) → 4.3. SPIN-LATTICE RELAXATION 23 we get back M(ω) = χ(ω)H(ω). This leads us to investigate whether or not we can model spin ice susceptibility data with the expression for the impedance of the RL-circuit. The expression for the impedance is

Z(ω) = R + iωL (4.27) where R is the DC resistance and L is the circuit inductance. Introducing the time constant of the circuit as τ = L/R, we can write

Z(ω) = R (1 + iωτ) . (4.28)

The inverse impedance is now given by

1 1 1 iωτ = − . (4.29) Z(ω) R 1 + (ωτ)2

If we identify the inverse resistance with the isothermal susceptibility, χT , we get the following expressions for the real and imaginary parts of the susceptibility χ χ0(ω) = T (4.30) 1 + ω2τ 2 χ ωτ χ00(ω) = T (4.31) 1 + ω2τ 2 which are identical to Equations 4.23 and 4.24.

4.3 Spin-lattice relaxation

A refinement of the above model can be made by considering the coupling between the lattice vibrations and the spins [24]. When an external magnetic field is applied to the system, this results in an almost instantaneous magnetisation [23, 25], resulting in a shift in the population of spin up and spin down states. However, this shift requires the release of energy. In a thermally isolated system, there is no way to release this energy and thus, instead of shifting the populations, the temperature of the spin system would increase from T1 to T2 such that H/T1 = (H + ∆H)/T2, where H is the initial external field and ∆H is the change in the field [24]. Now, the spins are not thermally isolated, but the heat transport from the spin system to the lattice is slower than the magnetisation of the sample, giving rise to a slightly higher temperature of the spin system. We model the spin-lattice relaxation by assuming that the exchange of heat is proportional to the temperature difference between the spin system and the lattice [23–25], i.e. dQ = α (T Tl) = αθ, (4.32) dt − − − 24 CHAPTER 4. MAGNETRICITY

where T is the temperature of the spin system, Tl is the temperature of the lattice and α is a proportionality constant. Let us now express the entropy in terms of temperature and magnetisation ∂S ∂S T dS = T dT + T dM. (4.33) ∂T M ∂M T We now make use of the identity dQ ∂U ∂S CM = = = T (4.34) dT M ∂T M ∂T M and the Maxwell relation ∂H ∂S = (4.35) ∂T M − ∂M T and write Equation 4.33 as ∂H ∂H T dS = CM dT T dM = CM dT T χdH. (4.36) − ∂T M − ∂T M For a reversible process, dQ = T dS. Using Equation 4.32, we get ∂H αθdt = CM dT T χdH. (4.37) − − ∂T M iωt Let now the magnetic field vary sinusoidally, H = H0e . As a consequence, the iωt temperature difference will vary in the same manner: θ = θ0e . Dividing Equation 4.37 by dt and inserting the explicit time dependences of the state variables, we arrive at ∂H αθ0 = iω CM θ0 T χH0 . (4.38) − − ∂T M In order to get an expression for the susceptibility, χ, we set out to eliminate θ0 in Equation 4.38. This is done by writing the magnetisation in terms of the magnetic field and the temperature ∂M ∂M dM = dH + dT (4.39) ∂H T ∂T H and differentiate with respect to time ∂M ∂M χH0 = H0 + θ0. (4.40) ∂H T ∂T H

Eliminating θ0 from Equation 4.38, we get ∂M ∂H ∂M (α + iωCM ) χ = iωT χ (4.41) − ∂H T ∂T M ∂T H 4.3. SPIN-LATTICE RELAXATION 25 where we can use the thermodynamic relation [24]

∂H ∂M CH CM = T (4.42) − − ∂T M ∂T H resulting in

∂M (α + iωCM + iω (CH CM )) χ = (α + iωCM ) (4.43) − ∂H T

∂M and ﬁnally, with ∂H T = χT ,

α + iωCM χ(ω) = χT . (4.44) α + iωCH

If we take the relaxation time to be τ = CH /α [24] and the adiabatic susceptibility to be χS = χT CM /CH [23], we can write the real and imaginary parts of the susceptibility, χ(ω) = χ0(ω) iχ00(ω), as [23] − χT χS χ0(ω) = χ + − (4.45) S 1 + ω2τ 2

(χT χS)ωτ χ00(ω) = − . (4.46) 1 + ω2τ 2 Note that the above expressions are identical to those of Section 4.2.2 in the limit χS 0. → 4.3.1 Monopole current To determine the monopole current in the case when spin-lattice relaxation is taken into account, we can make use of the inverse procedure of Section 4.2.2. First, by using M(ω) = χ(ω)H(ω), we write Equation 4.44 as

χT −1 iωM(ω) = H(ω) χ M(ω) + iωχSH(ω) (4.47) τ − T and take the inverse Fourier transform to get

∂M χT −1 ∂H = H(t) χ M(t) + χS (4.48) ∂t τ − T ∂t

∂M or equivalently for the monopole current, jmp = , and introducing κ χT /τ ∂t ≡

−1 ∂H jmp = κ H(t) χ M(t) + χS (4.49) − T ∂t which is identical to Equation 4.13 apart from the last term on the right. 26 CHAPTER 4. MAGNETRICITY

4.3.2 Interpretation of χS

The introduction of the adiabatic susceptibility, χS, above may require some clari- ﬁcation. From Equation 4.44 we see that at low frequencies the susceptibility will take the isothermal value χT . This essentially means that at low frequencies, the spin system and the lattice are in equilibrium with each other – they have the same temperature. In the other extreme of very high frequencies the susceptibility takes the adiabatic value χS. This means that at high frequencies, there is no heat exchange between the spin system and the lattice [24], ¯dQ = 0, because the variation of the magnetic ﬁeld is much faster than the spin lattice relaxation. The absence of heat exchange implies an adiabatic process, why the susceptibility takes the adiabatic value, χS. Experiments indicate that the adiabatic susceptibility in spin ice is proportional to the concentration of magnetic monopoles, c [26],

cµ q2 χ = 0 m (4.50) S K where K 0.06 N m−1 is a force constant. In the≈ approximation done in Section 4.2 we neglected the adiabatic susceptibility. This approximation was said to be valid only for small fields, as it is based on a Taylor expansion of the entropy. At small fields and low temperature, the concentration of monopoles, c, will be vanishingly small, which justifies neglecting χS.

4.4 Numerical determination of the susceptibility

From our Monte Carlo simulations, we get the magnetisation, M, as a function of time for a given ﬁeld. If we take a sinusoidally varying magnetic ﬁeld, H(ω) = H0 sin ω, as input in the simulations we can calculate the two components of the AC susceptibility for a given input frequency, ω, as [27, 28]

NMC 0 2 X χ (ω)H0 = M(tMC ) sin(ωtMC ) (4.51) NMC tMC =1

NMC 00 2 X χ (ω)H0 = M(tMC ) cos(ωtMC ) (4.52) −NMC tMC =1 where NMC is the total number of Monte Carlo steps in the simulation. The resulting data for a field parallel to a 001 direction is presented in Figures 4.3 and 4.4 for the real and imaginary parts,h respectively.i The data is fitted to Equations 4.45 and 4.46 and fits rather well. The slight discrepancy at lower temperatures may be due to the applied magnetic field being to large compared to the temperature for linear response theory – Equation 4.20 – to be valid. 4.4. NUMERICAL DETERMINATION OF THE SUSCEPTIBILITY 27

2.5

2.0

1.5 ) [a.u.] ω ( ′

χ 1.0

0.5

0.0 4 3 2 1 0 10− 10− 10− 10− 10 1 Frequency [(Monte Carlo time)− ]

Figure 4.3: Real part of the AC susceptibility as a function of frequency. From blue to red: T/J = 0.20, 0.22, 0.25, 0.27, 0.30, 0.32, 0.35, 0.37, 0.40. The dotted lines are ﬁts to the real part data, whereas the dashed lines are ﬁtted to the imaginary part. Please refer to Appendix A for further details.

1.2

1.0

0.8

) [a.u.] 0.6 ω ( ′′ χ 0.4

0.2

0.0 4 3 2 1 0 10− 10− 10− 10− 10 1 Frequency [(Monte Carlo time)− ]

Figure 4.4: Imaginary part of the AC susceptibility as a function of frequency. From blue to red: T/J = 0.20, 0.22, 0.25, 0.27, 0.30, 0.32, 0.35, 0.37, 0.40. The dotted lines are ﬁts to the real part data, whereas the dashed lines are ﬁtted to the imaginary part. Please refer to Appendix A for further details. 28 CHAPTER 4. MAGNETRICITY

4.5 Predicting the monopole current

Knowing how the magnetisation is related to the applied magnetic ﬁeld and how to extract the monopole current from the magnetisation, we set out to calculate the magnetisation and monopole current for an arbitrary magnetic ﬁeld without having to run lengthy Monte Carlo simulations, according to the following scheme

−1 ∂ F χ(ω) F ∂t H(t) H(ω) M(ω) M(t) jmp(t). (4.53) −→ −−−→ −−→ −−→ Since we know the susceptibility in frequency domain from Equations 4.45 and 4.46, we start by transforming the magnetic ﬁeld, H(t), to frequency domain using a discrete Fourier transform

N−1 X iωm∆t H(ω) = F(H(t)) = H(m∆t) exp . (4.54) N m=0 −

n Here, m∆t = t and ω = ∆t where n is an integer between 0 and N 1. Using Equations 4.45 and 4.46 we get the magnetisation in frequency domain,− which we can transform back into the time domain to get the sought M(t). Finally, we use Equation 4.1 to calculate the monopole current. Examples of results using this method are shown in Figures 4.5 and 4.6 for applied ﬁelds of the form H sin ω1t sin ω2t and a series of Heaviside pulses, respectively. ∝ 4.5. PREDICTING THE MONOPOLE CURRENT 29

1.5

1.0 ) [a.u.] t

( 0.5 mp j 0.0 ) and t (

M 0.5

), − t (

H 1.0 −

1.5 − 0 20 40 60 80 100 Time [a.u.]

Figure 4.5: Magnetic ﬁeld (red), magnetisation (blue) and magnetic monopole current (green) calculated using the method of Section 4.5. The applied ﬁeld is of the form H sin ω1t sin ω2t with ω1 = 0.4 and ω2 = 0.1. Input parameters where ∝ arbitrarily set to χT = 1 and τ = 10.

1.6 1.4 1.2 ) [a.u.] t

( 1.0 mp j 0.8

) and 0.6 t (

M 0.4 ), t ( 0.2 H 0.0 0.2 − 0 20 40 60 80 100 Time [a.u.]

Figure 4.6: Magnetic ﬁeld (red), magnetisation (blue) and magnetic monopole current (green) calculated using the method of Section 4.5. The applied ﬁeld is a series of Heaviside pulses. Input parameters where arbitrarily set to χT = 1 and τ = 10. 30 Chapter 5

Phase transitions in a 111 magnetic ﬁeld h i

In this chapter, we investigate what happens when a spin ice compound is exposed to a magnetic ﬁeld in a 111 direction. We are mainly concerned with what happens at the transition fromh thei two-in-two-out Kagom´eice state to the fully polarised state.

5.1 Introduction to phase transitions

A phase transition occurs when a system abruptly changes from one thermodynamic phase to another. Examples are the transition from the liquid to the gas phase in water or the transition from a paramagnetic to a polarised state in a ferromagnet. When we study phase transitions, we often take interest in the specific heat, given by [29] ∂Q ∂S Cx = = T (5.1) ∂T x ∂T x where Q is the heat, T is the temperature, S is the entropy and x represents the constraint that x is kept constant when measuring Cx [29]. Generally, phase transitions are associated with a change in entropy – the transition in a ferromagnet involves the disordered, high entropy paramagnetic state and the ordered, low entropy ferromagnetic state. Likewise, gas phases have higher entropy than liquid or solid phases. Because the specific heat is given by the derivative of the entropy, any change in behaviour of the entropy will show quite clearly in the specific heat. In the case of a discontinuity of the entropy, the specific heat will have a very sharp peak (a delta peak in the thermodynamic limit) at the transition, for example. We can split phase transitions into two categories: first order and continuous phase transitions. For a first order transition, a first derivative of the free energy,

31 32 CHAPTER 5. PHASE TRANSITIONS IN A 111 MAGNETIC FIELD h i

F = U TS, is discontinuous. First derivatives of the free energy include, for example,− entropy, S, and magnetisation, M. If the first derivatives are discontinuous, the second derivatives are singular at the transition. Hence the specific heat, Cx, and the susceptibility, χ, should exhibit sharp peaks at a first order phase transition. The examples of phase transitions above are all examples of first order transitions [29]. On the other hand, a phase transition is continuous if the first derivatives are continuous, but some higher derivative is discontinuous. For example, in a second order (continuous) transition, the specific heat or the susceptibility could exhibit discontinuities, but not singularities, at the phase transition. An example of a second order phase transition is the transition to the superconducting state in a superconductor [29]. All transitions between thermodynamic states are not phase transitions, however. As mentioned above, a phase transition occurs abruptly. As an example, we take a paramagnet. If we apply a magnetic field, we will start aligning the spins with the field, going from a disordered state at zero field, to a fully polarised state at high fields. Whereas these states may not technically be distinct thermodynamic phases, they are very different in terms of the values of thermodynamic quantities, such as magnetisation and entropy. The transition does, however, not occur abruptly, but rather slowly over an extended range of values of the magnetic field. In this and similar cases we rather talk about continuous crossovers than phase transitions.

5.2 Kagom´eice

When viewed from the 111 directions, the pyrochlore lattice can be seen as alternating layers of triangularh i lattices and Kagomélattices as in Figure 5.1. The Kagomélattice is a two dimensional lattice formed by six-pointed stars. If we were to apply a magnetic field in the [111] direction, the spins on the triangular sublattices would be more strongly coupled to the field than those on the Kagomélattices, since the local Ising axis of the triangular lattice spins is [111], whereas the Kagomé spins have [1¯11],¯ [11¯ 1]¯ and [11¯1]¯ as local Ising axes. If the field is of moderate strength, it will be able to polarise the triangular lattice spins, but not the Kagoméspins. The Kagoméspins will still adhere to the ice rules, so we still have two spins pointing into and two spins pointing out of every tetrahedron as in the left panel of Figure 5.2. However, the entropy is reduced, since the triangular lattice spins are forced to be aligned with the field and thus the only degrees of freedom that remain are the spin directions in the Kagoméplanes. This state of polarised triangular lattice spins is called Kagoméice and is illustrated in Figure 5.3. When the field is high enough, even the weakly coupled Kagoméspins will align with the field as in the right panel of Figure 5.2. In this fully polarised state, the sample turns into a crystal of monopoles. Every diamond lattice site is occupied 5.2. KAGOME´ ICE 33

Figure 5.1: The pyrochlore lattice can be seen as alternating layers of triangular (blue) and Kagomé(red) lattices stacked in a 111 direction. h i by a single monopole, together forming a regular pattern of monopoles as in Figure 5.4. Thus, at some point, we go from a state of zero monopoles at moderate fields to a state of 100% occupation of monopoles at higher fields. We will in the following investigate this potential phase transition.

5.2.1 Value of the critical ﬁeld

The value of the critical field, Hc, can easily be calculated by considering the nearest neighbour spin ice Hamiltonian under the influence of a magnetic field in a 111 direction h i X X H = J Si Sj µ0H µSi. (5.2) − · − · hi,ji i

Let us explicitly evaluate Equation 5.2 for the Kagom´eice state and for the fully polarised state. 34 CHAPTER 5. PHASE TRANSITIONS IN A 111 MAGNETIC FIELD h i

Figure 5.2: Left: A tetrahedron in the Kagoméice state. The [111] spin is aligned with the field, but the other spins make sure that the tetrahedron as a whole obeys the ice rules. Right: The fully polarised state. All spins are aligned with the field and the ice rules are broken.

Kagoméstate In the Kagoméstate, to the left in Figure 5.2, the ice rules are obeyed and thus the first term of the Hamiltonian is minimised. Each tetrahedron contains six unique nearest neighbour interactions between the spins. Since the ice rules are obeyed, four of these interactions will give the energy J/3 and two will give +J/3 (because − the scalar product Si Sj = 1/3). In a system of N particles, we will have N/2 tetrahedra and hence the· nearest− neighbour interaction energy in the Kagoméstate will be NJ EJ,kag = . (5.3) − 3 From the second term we will get three different kind of contributions. One quarter of the particles will be completely aligned with the field, giving an energy of µµ0H each. Another quarter will be directed partly against the field giving the − energy µµ0H/3 (since H e ¯¯ = H(1, 1, 1)/√3 ( 1, 1, 1)/√3 = H/3). Finally, · [111] · − − − half the spins will be partly aligned with the field giving an energy of µµ0H/3 each. Summing these contributions for all N particles, we get −

Nµµ0H EH,kag = . (5.4) − 3

Fully polarised state In the fully polarised state – to the right in Figure 5.2 – we have three satisﬁed and three unsatisﬁed nearest neighbour interactions. Thus

EJ,pol = 0. (5.5)

Since the ﬁnal N/4 spins are now also aligned with the magnetic ﬁeld, we have N/4 spins giving a contribution of µµ0H each and 3N/4 spins giving µµ0H/3. − − 5.2. KAGOME´ ICE 35

Figure 5.3: The Kagoméice state. The spins on the triangular sublattices (not shown) are aligned with the field (directed out of the paper), whereas the spins on the Kagomésublattices are free to point into or out of a tetrahedron, as long as the ice rules are obeyed.

Figure 5.4: The fully polarised state. All spins are aligned with the magnetic ﬁeld (directed out of the paper) and the ice rules are no longer obeyed. Thus, all diamond lattice sites contain either a positive or negative magnetic monopole, forming a ”monopole crystal”. 36 CHAPTER 5. PHASE TRANSITIONS IN A 111 MAGNETIC FIELD h i

0.5 ] µ

0.4

0.3

0.2 Magnetisation per spin [ 0.1

0.0 0 2 4 6 8 10 Magnetic ﬁeld [J/µµ0]

Figure 5.5: Simulation of the magnetisation as a function of applied magnetic ﬁeld in the 111 direction for a system of 3 3 3 cubic unit cells. From blue to red: T/Jh =i 0.10, 0.15, 0.20, 0.25, 0.30, 0.35×, 0.40.× For further details, refer to Appendix A.

Thus the magnetic ﬁeld part of the Hamiltonian gives

Nµµ0H EH,pol = . (5.6) − 2 The above gives us a diﬀerence in energy between the fully polarised state and the Kagom´estate of

NJ Nµµ0H ∆E = EJ,pol + EH,pol EJ,kag EH,kag = . (5.7) − − 3 − 6 If this energy difference is positive, the Kagoméstate is energetically favourable, whereas the fully polarised state is favourable if ∆E is negative. The transition, ∆E = 0, occurs exactly for the critical field 2J Hc = . (5.8) µµ0

5.2.2 Magnetisation plateau The presence of the Kagoméice state makes the magnetisation curve behave differ- ently from the case when a magnetic field is applied in a 001 direction, as studied h i 5.2. KAGOME´ ICE 37

1.0

0.8

0.6

0.4

Monopoles per0 tetrahedron .2

0.0 0 2 4 6 8 10 Magnetic ﬁeld [J/µµ0]

Figure 5.6: Simulation of the monopole density as a function of applied magnetic field in the 111 direction for a system of 3 3 3 cubic unit cells. From blue to red: T/Jh =i 0.10, 0.15, 0.20, 0.25, 0.30, 0.35×, 0.40.× For further details, refer to Appendix A. in Chapter 4. Instead of simply rising and flattening out at the maximum value with increasing field, we now get a plateau in the magnetisation curve as in Figure 5.5. First, the strongly coupled triangular spins are polarised and the Kagoméice plateau is reached. Then, when the critical field is approached, the Kagoméspins start aligning and the maximum value can be reached. Evidently, the shape of the plateau is temperature dependent due to the increased fluctuations at higher temperatures being able to overcome the ice rules, as well as the magnetic field. As mentioned above, when going from the Kagoméstate to the fully polarised state, the monopole density goes from essentially 0% to 100% – one monopole occupying every tetrahedron. This is seen in Figure 5.6.

5.2.3 Phase diagram of the nearest neighbour model Based on Monte Carlo simulations and calculations of the specific heat, we can determine the phase diagram of nearest neighbour spin ice. The phase diagram is shown for a system of 3 3 3 cubic unit cells in Figure 5.7. From the specific heat× data× in Figure 5.8 we can distinguish three different peaks in the specific heat, although the first peak (at low fields) and second peak merges at higher temperatures. The first peak corresponds to the green line in the phase 38 CHAPTER 5. PHASE TRANSITIONS IN A 111 MAGNETIC FIELD h i

5 ] 0 4 Fully polarised state J/µµ

Magnetic ﬁeld [ Coexistence state 1 Kagome ice state 0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 Temperature [J]

Figure 5.7: Phase diagram of nearest neighbour spin ice containing 432 spins (3 3 3 cubic unit cells) in a 111 magnetic ﬁeld. Based on peaks in the speciﬁc heat,× the× error bars represent theh widthsi of the peaks.

diagram of Figure 5.7, the second peak corresponds to the blue line and the high ﬁeld peak corresponds to the red line.

Above the red line, the system is in the fully polarised state. All tetrahedra are occupied by monopoles and the magnetisation is saturated at its maximum value. Below the blue line, but above the green line, we have the Kagoméice state, responsible for the magnetisation plateau of Figure 5.5. Between the red and blue lines there is a coexistence state where we have a transition between the Kagomé and fully polarised states, where the magnetisation starts rising from the plateau value to the saturated value, and the monopole density increases from essentially 0% to 100%. Below the green line – at low fields – the paramagnetic state of spin ice is present.

It is interesting to note that at Hc, the specific heat goes to zero for low enough temperatures. Normally, the specific heat is large when fluctuations in the system are large since we can calculate the specific heat as the variance of the energy. However, whereas we indeed have a lot of fluctuations at Hc, the fluctuations are between states of the exact same energy and thus the variance of the energy, and the specific heat, is zero. 5.2. KAGOME´ ICE 39

0.40

0.35

0.30

0.25

0.20

0.15 Speciﬁc heat [a.u.] 0.10

0.05

0.00 0 1 2 3 4 5 6 7 8 Magnetic ﬁeld [J/µµ0]

Figure 5.8: Specific heat versus applied magnetic field in the 111 direction. From blue to red: T/J = 0.10, 0.15, 0.20, 0.25, 0.30, 0.35, 0.40. Threeh differenti peaks are distinguishable for low temperatures. Please refer to Appendix A for details.

0.5

0.4

0.3

0.2 Susceptibility [a.u.]

0.1

0.0 0 1 2 3 4 5 6 7 8 Magnetic ﬁeld [J/µµ0]

Figure 5.9: Susceptibility versus applied magnetic ﬁeld in the 111 direction. From blue to red: T/J = 0.10, 0.15, 0.20, 0.25, 0.30, 0.35, 0.40. Theh peaki approaches the critical ﬁeld as the temperature is lowered. Please refer to Appendix A for details. 40 CHAPTER 5. PHASE TRANSITIONS IN A 111 MAGNETIC FIELD h i

0.25

0.20

0.15

0.10 Speciﬁc heat [a.u.]

0.05

0.00 1.990 1.995 2.000 2.005 2.010 Magnetic ﬁeld [J/µµ0]

Figure 5.10: Speciﬁc heat for 16 spins at T/J = 1 10−3 (red), T/J = 5 10−4 (green) −4 · · and T/J = 1 10 (blue) in the vicinity of Hc = 2J/µµ0. Refer to Appendix A for details. ·

We can also look at the susceptibility in Figure 5.9. This quantity can be calculated from the ﬂuctuations in the magnetisations as

2 2 χ(T ) = µ0β M M (5.9) − h i where β = T −1 is the inverse thermal energy. Unlike the specific heat, the susceptibility does not go to zero at the critical field, but rather exhibits a peak that approaches the critical field as the temperature is lowered. This is because the fluctuations in the magnetisation are indeed largest near the critical field where the Kagoméice state, the fully polarised state and all partly polarised states have the same energy and are thus equally probable.

5.2.4 Low temperature simulations Since the Metropolis transition rate, Equation 3.3, goes to zero as we approach absolute zero temperature, fluctuations are strongly suppressed and we would expect that simulations at low temperatures would take too long to yield accurate results to be feasible. Indeed, at low temperatures and low fields, we have problems reproducing the expected behaviour of the magnetisation. Data below T/J = 0.05 exhibits discontinuities, rather than rising continuously, in the range of fields corresponding to the transition from the spin ice state to the Kagoméice state. The 5.2. KAGOME´ ICE 41

100

1 10−

2 10−

3 10− Acceptance rate

4 10−

5 10−1.985 1.990 1.995 2.000 2.005 2.010 2.015 Magnetic ﬁeld [J/µµ0]

Figure 5.11: Acceptance rate for the Metropolis algorithm for 16 spins at T/J = 10−3 close to the transition value of the magnetic field. Refer to Appendix A for more details. problem is that in order for the spins to start aligning with the field, the ice rules must at some point be broken. However, with the high energy cost associated with breaking the ice rules and the low temperature of the system, spin flips are extremely rare. However, as can be seen in Figure 5.10, we can study the transition region, −4 close to Hc = 2J/µµ0 even at temperatures as low as 10 J. The reason for this is that close to the transition, the exchange and Zeeman contributions to the energy almost cancel each other, making the energy comparable to the thermal energy. This effectively raises the Metropolis transition rate by several orders of magnitude, peaking at about 30% as seen in Figure 5.11. This allows for fluctuations and sampling of a representative part of phase space near the transition.

5.2.5 Phase transition or continuous crossover?

The shape of the specific heat in Figures 5.8 and 5.10 is not compatible with a continuous phase transition, where the specific heat should exhibit discontinuities [29]. If we are dealing with a first order transition, then the peaks in the specific heat should be singularities in the thermodynamic limit – when the system size tends to infinity. It is possible that the apparent smoothness of the peaks in Figures 5.8 42 CHAPTER 5. PHASE TRANSITIONS IN A 111 MAGNETIC FIELD h i

0.25

0.20

0.15

0.10 Speciﬁc heat [a.u.]

0.05

0.00 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 Magnetic ﬁeld [J/µµ0]

Figure 5.12: Specific heat at T/J = 0.1 for different system sizes: 16 spins, 1 1 1 cubic unit cells, in blue; 128 spins, 2 2 2 cubic unit cells, in green; and 432× spins,× 3 3 3 cubic unit cells, in red.× The× peaks are identical for all three system sizes,× showing× no trace of finite size scaling. For detailed simulation parameters, see Appendix A. and 5.10 is due to finite size effects. If so, the maximum value of the specific heat peaks should scale with the system size according to [30]

Cpeak a + bN (5.10) ∼ where a and b are constants and N is the number of particles in the system. In Figure 5.12, we have conducted simulations for different system sizes. The specific heat peaks fall precisely on top of each other, indicating that this is not a phase transition, but rather a continuous crossover. Indeed, it has been mathemat- ically proven that no phase transition is possible at finite temperature in nearest neighbour spin ice in a 111 magnetic field [3,31]. However, calculations based on the Bethe approximationh showi that there may be a first order phase transition at T = 0, H = 2J/µµ0 [32], which is the point that the two peaks in the specific heat in Figure 5.10 seem to tend to for low temperatures.

5.3 Temperature independent state at Hc The magnetisation and monopole density data in Figures 5.5 and 5.6 show an interesting feature at the critical ﬁeld, Hc = 2J/µµ0, where the magnetisation and 5.3. TEMPERATURE INDEPENDENT STATE AT HC 43

0.400

0.398

0.396

0.394

0.392

Monopoles per0 tetrahedron .390

0.388 0 50 100 150 200 250 300 350 400 450 Number of spins

Figure 5.13: Size dependence of the monopole density at the critical field. The markers are larger than the extension of the errorbars. For detailed simulation parameters, refer to Appendix A monopole density curves intersect for all temperatures. This implies that the state at Hc is temperature independent (for low enough temperatures). As we calculated in Section 5.2.1, the difference in energy is ∆E = 0 meaning that ∆E/T = 0 for all finite temperatures. Thus, the system is free to fluctuate between the Kagomé ice state and any partly polarised states and the fluctuations will be completely temperature independent. However, if the temperature is to high, we will start getting double defects and fluctuations to states where two or more spins in a tetrahedron are anti-aligned with the field, in which case the energy difference is non-zero and the temperature dependence returns. The critical point in Figures 5.5 and 5.6 has another curious property as well. Na¨ıvely, we might guess that the monopole density should be 50% at the critical field and, hence, the magnetisation equal to the mean of the Kagoméice value (1/3 µ per spin) and the fully polarised value (1/2 µ per spin) – that is 5/12 µ per spin. However, the monopole density turns out to be about 0.4 monopoles per tetrahedron instead of 0.5. Running more accurate Monte Carlo simulations for different system sizes, we see that the monopole density is relatively size independent for systems of some hundred spins, see Figure 5.13. For a single unit cell, the monopole density is slightly lower than for the larger systems. The value of the monopole density for the 3 3 3 system (432 spins) presented in Figure 5.13 is 0.3995 1 10−4. However,× it is× possible that this value will change slightly in the thermodynamic± · 44 CHAPTER 5. PHASE TRANSITIONS IN A 111 MAGNETIC FIELD h i limit since the 3 3 3 system still is quite small. The reason for× the× observed value of the monopole density is simply that the number of microscopic states with a small number of monopoles is larger than the number of states with many monopoles. The microcanonical mean number of monopoles per tetrahedron is given by

n 1 PNt nΩ(n) x = h i = n=0 (5.11) N N PNt h i t t n=0 Ω(n) where x is the number of monopoles per tetrahedron, n is the total number of monopoles in the system, Nt is the number of tetrahedra in the system and Ω(n) is the number of microscopic states with n monopoles. We can calculate this mean exactly for a single cubic unit cell of 16 spins. This gives a total of 108 states being either Kagoméstates or partly (fully) polarised. The total number of monopoles in these 108 states is 336. With the unit cell containing 8 tetrahedra, this yields a monopole density of x 0.3889. It is also possible to count every state for a system of two unit cells.h Ini ≈ this case, the monopole density is x 0.3965. h i ≈ 5.3.1 Monopole density for larger systems Since the number of states grows exponentially with system size, it is not possible to use the above method and calculate the microcanonical mean for any larger systems. Thus we seek an exact expression for the monopole density as a function of system size, or at least an algorithm that can calculate the monopole density fast enough to be feasible for larger systems. We start by mapping the problem of counting the number of states with a certain number of monopoles to an equivalent problem. If we start with the Kagomélattice in the fully polarised state, this is a state of Nt monopoles, where Nt is the number of tetrahedra on the lattice. In order to reach a state of Nt 2 monopoles, we − flip a single spin. To get Nt 4 monopoles, we flip two spins, etc. However, no two spins that are nearest neighbours− may be flipped at the same time, since this would create monopoles of the ”wrong” sign – that is, a one-in-three-out monopole in a tetrahedron with the spin coupled strongly to the field pointing outwards or a three-in-one-out monopole in a tetrahedron with the strongly coupled spin pointing inwards. The problem is thus reduced to placing spin-flip-quasi-particles on the Kagomélattice with nearest neighbour exclusion. This system on different lattices, known as the hard core lattice gas, has been studied extensively with Monte Carlo techniques – mainly focusing on phase transitions in the model. However, no exact results for the number of states are available. The hard core lattice gas with nearest neighbour exclusion on a Kagomélattice can, in turn, be mapped to another problem: the monomer-dimer problem on a honeycomb lattice. If we take the Kagomélattice and substitute every tetrahedron for a vertex and every site for an edge as in Figure 5.14, we end up with a honeycomb lattice. A spin-flip-quasi-particle is now represented by placing a dimer on an edge, 5.3. TEMPERATURE INDEPENDENT STATE AT HC 45

Figure 5.14: Left: mapping every tetrahedron to a vertex and every Kagom´esite (black) to an edge, we can transform the hard core lattice gas with nearest neighbour exclusion to the monomer-dimer problem on the honeycomb lattice (red). Right: a honeycomb lattice completely covered with dimers (thick lines). occupying the two vertices that the edge connects. A vertex can either be occupied by a single dimer or unoccupied (or equivalently occupied by a monomer). The problem is thus: can we count the ways to place a certain number of monomers and dimers on the honeycomb lattice? Monomer-dimer problems have been of interest to scientists since 1937 when Fowler and Rushbrooke published a paper on liquid mixtures of molecules of different sizes [33]. In 1961, Fisher, Kasteleyn and Temperley [34–36] independently published an exact solution to the dimer covering problem – the special case when all sites are covered by dimers, corresponding to the Kagom´eice state in our problem. The general monomer-dimer problem has proven more elusive, though. In 1987, Jerrum proved that the general monomer-dimer problem on a two dimensional lattice is ”#P-complete” and thus computationally intractable [37]. This does not mean that there cannot be an analytical expression for the number of states or a fast algorithm for counting them, but it is very improbable that either of these exist [37]. Finding such a solution or algorithm that works for the general monomer- dimer problem on any two dimensional lattice would mean solving one of the most important problems in theoretical computer science – the P versus NP problem. The P versus NP problem is one of the seven Millennium Problems proposed by the Clay Mathematics Institute and the reward for solving it is $1 million [38, 39]. A brief explanation of complexity classes, #P-complete problems and the P versus 46 CHAPTER 5. PHASE TRANSITIONS IN A 111 MAGNETIC FIELD h i

Figure 5.15: The Husimi cactus (black). Every triangle shares corners with two other triangles. Unlike in the Kagom´elattice, two branches of the cactus never meet. The Cayley tree dual to the Husimi cactus is drawn in red.

NP problem is given in Appendix C. Solving the P versus NP problem is beyond the scope of this thesis.

5.3.2 Bethe approximation Whereas solving the general monomer-dimer problem will not be accomplished here, it may be possible to solve the problem in certain special cases by exploiting symmetries. In spite of extensive research, no one has found such a solution for the honeycomb lattice to the best of our knowledge. Thus we turn to an approximate model which can be solved exactly. Instead of considering the hard core lattice gas with nearest neighbour exclusion on the Kagomélattice, we consider the same problem on a Husimi cactus. The Husimi cactus is shown in Figure 5.15. It is a graph consisting of triangular units connected to each others corners. Unlike in the Kagomélattice, no two branches of the cactus are connected – the only way to move between two different branches is to go via the central triangle. This allows for computing thermodynamic properties via recursion relations, which simplifies the problem [5]. 5.3. TEMPERATURE INDEPENDENT STATE AT HC 47

Like in the previous section, we can map the hard core lattice gas with nearest neighbour exclusion to the monomer-dimer problem on the dual graph. The graph dual to the Husimi cactus is a Cayley tree with coordination number 3, drawn in red in Figure 5.15. The Cayley tree will have a very large surface and to avoid surface eﬀects, we will only consider the region far from the surface called the Bethe lattice [5, 40]. The problem of calculating the entropy for polymers of length M on a Bethe lattice of coordination z was solved by Stilck and de Oliveira [40]. They considered the entropy per vertex, deﬁned by

s(ρ) = lim N −1 ln Ω(ρ), (5.12) N→∞ where ρ is the fraction of the vertices that are covered by a polymer, N is the number of vertices and Ω(ρ) is the number of states corresponding to a given ρ. They found the general expression

ρ 2ρ 1 s(ρ) = (1 ρ) ln (1 ρ) ln + 1 ρ (ln z 1) + − − − − M M − M − 2 1 1 + 1 ρ ln 1 + 1 ρ+ (5.13) − M − z − M z 1 2 1 + 1 ρ ln 1 1 ρ , 2 − − M − z − M which in our case of dimers (M = 2) on a Bethe lattice of coordination z = 3 reduces to ρ ρ s(ρ) = (1 ρ) ln (1 ρ) ln + 2 3 − − − − (5.14) 3 ρ ρ + 1 ln 1 . 2 − 3 − 3 From the deﬁnition of the entropy per vertex, Equation 5.12, we can write the number of states in the thermodynamic limit as

Ω(ρ) = eNs(ρ). (5.15)

This means that we can calculate the microcanonical mean number of monopoles, x = 1 ρ , as h i h − i R 1 (1 ρ) eNs(ρ)dρ x = lim 0 − (5.16) h i N→∞ R 1 Ns(ρ) 0 e dρ with s(ρ) as given in Equation 5.14. The integrals in Equation 5.16 cannot be evaluated analytically, but we can investigate the behaviour for increasing values of N numerically. In Figure 5.16, x is plotted for 1 N 1000. Clearly the monopole density tends to 0.4 for largeh i value of N. Indeed,≤ if we≤ look more precisely at larger systems as in Table 5.1, a system of N = 105 sites we get x = 0.4000024. h i 48 CHAPTER 5. PHASE TRANSITIONS IN A 111 MAGNETIC FIELD h i Vertices, N Monopole density, x h i 101 0.4234153 102 0.4024327 103 0.4002440 104 0.4000244 105 0.4000024

Table 5.1: Monopole densities for diﬀerent system sizes in the Bethe approximation.

0.49 0.48 0.47 0.46 0.45 0.44 0.43

Monopoles per0 tetrahedron .42 0.41 0.40 100 101 102 103 Number of vertices

Figure 5.16: The monopole density in the Bethe approximation as a function of the number of vertices on the Bethe lattice.

After completing the above calculations, we discovered that the same problem had been treated by Isakov et al. [41] using similar methods. In their work, the Monte Carlo data and the Bethe approximation also agrees very well. It is clear that the Monte Carlo and Bethe approximation results approach each other as the system size is increased. However, determining whether the actual value of the mean monopole density at the critical ﬁeld in the thermodynamic limit is exactly 0.4 (as the Bethe approximation suggests) or if it is slightly smaller (as simulations for small systems imply) would require accurate Monte Carlo simulations for signiﬁcantly larger systems. 5.3. TEMPERATURE INDEPENDENT STATE AT HC 49

8000

7000 500 400 6000 300 200 100 5000 0 16 14 − − 4000

3000 Number of states 2000

1000

0 20 15 10 5 0 5 10 15 20 − − − − Energy [J]

Figure 5.17: Histogram of the energies of all possible states in a single cubic unit cell at the critical field. The inset shows the ground state and the two first excited 4 states. The first excited state is situated at an energy of 3 J above the ground state.

5.3.3 Onset of the temperature dependence

As mentioned previously, the temperature independence of the state at the critical field is due to all relevant states having the same energy. However, all possible states do not have the same energy. At higher temperatures, excited states will start playing a role and the energy increases. A histogram of the energy of all possible states at Hc in a single cubic unit cell is plotted in Figure 5.17. The inset shows the ground state and the first two excited states. We see that the energy gap 4 from the ground state to the first excited state is 3 J. This agrees well with the observed temperature dependence at higher temperatures as shown in Figure 5.18. When the Boltzmann factor for the first excited state is added to the ground state energy, the increase in energy occurs simultaneously for the different curves in Figure 5.18. Obviously, only the ground state and the first excited state cannot alone account for the behaviour of the energy. This is why the green curve of Figure 5.18 increases more slowly than the red and blue curves. A realisation of the (multiply degenerate) first excited state is shown in Fig- ure 5.19. From a Kagoméice state, two spins are flipped (marked in red in Fig- ure 5.19). Since these two spins projections on the [111] axis will sum to zero both before and after the flip, the difference in the magnetisation contribution to the energy will be zero. All tetrahedra except for the three depicted will remain un- 50 CHAPTER 5. PHASE TRANSITIONS IN A 111 MAGNETIC FIELD h i

14.8 − 15.8 15.0 − −

15.2 15.9 − −

15.4 − 16.0

Energy [J] − 0.1 0.2 0.3 0.4 15.6 −

15.8 −

16.0 3 2 1 0 − 10− 10− 10− 10 Temperature [J]

Figure 5.18: The energy for a single cubic unit cell (blue) and a 2 2 2 unit cell system (red) at the critical field as a function of temperature. The× green× curve is the ground state energy plus the Boltzmann weight corresponding to the energy 4 3 J. The inset shows a magnification of the area where the energy starts to rise above the ground state level. changed, and in the middle tetrahedron of Figure 5.19, the ice rules are preserved. However, the ice rules are broken in the left and right tetrahedra, each contributing 2 4 3 J to the energy, thus giving an energy of 3 J above the ground state for the first excited state.

5.4 Phase transitions in the dipole model

Experiments [42, 43] have revealed an actual first order phase transition in real spin ice materials in a magnetic field in the 111 direction at low temperatures. The phase transition is similar to a liquid gas transitionh i and terminates in a critical point at µ0Hc = 0.929 T, Tc = 0.36 K in Dy2Ti2O7 [26]. At temperatures above Tc, the phase transition turns into a continuous crossover [44] similar to the case of the nearest neighbour model outlined above – that is, we pass through the coexistence state when going from the Kagoméice state to the fully polarised state. This phase transition is apparently absent in the nearest neighbour model. How- ever, taking dipolar interactions into account, as described briefly in Section 2.3.2, it is possible to reproduce the phase transition in simulations [3]. 5.4. PHASE TRANSITIONS IN THE DIPOLE MODEL 51

Figure 5.19: Upper panel: one of the many degenerate ground states at the critical field in the 111 direction. Lower panel: the first excited state. Two spins are flipped (markedh i in red), rendering the magnetisation contribution to the energy unchanged as well as the ice rule in the middle tetrahedron. In the left and right 2 tetrahedra, the ice rules are broken giving a contribution of 3 J to the energy for each tetrahedron. 52 Chapter 6

Conclusions

In this ﬁnal chapter, we present the conclusions of the work and summarise the results presented in the thesis.

6.1 Magnetricity

The main result on the subject of magnetic monopole currents is that any direct current must be transient due to the magnetic monopoles not being free particles. Two equations were presented, one that states the relation between the monopole current and the magnetisation ∂M j = (6.1) mp ∂t and one that gives the monopole current in terms of the applied magnetic field −1 jmp = κ H χ M . (6.2) − T The latter of these can be viewed as the competition between a magnetic Drude force (first term) and an entropic force (second term). Whereas steady state direct currents proved to be impossible, we concluded that alternating currents are possible provided that the frequency is large enough for the current not to have time to vanish. However, the frequency cannot be so high that the system is unable to respond to the changing magnetic field, i.e. the AC susceptibility must be non-zero. In Chapter 4 we derived expressions for the AC susceptibility and compared this with simulation data. We found that the obtained Debye model, especially when corrected for spin-lattice relaxation, is able to describe nearest neighbour spin ice in a time dependent magnetic field. We also found that we can determine the magnetisation and monopole current for an arbitrary time dependent applied magnetic field by using Fourier transforms, ∂M the expression for the AC susceptibility and jmp = ∂t . This has the advantage of not requiring lengthy Monte Carlo simulations if we want to look at spin ice behaviour in different time dependent magnetic fields.

53 54 CHAPTER 6. CONCLUSIONS

6.2 Phase transitions

As predicted by analytical calculations, simulations show no sign of an actual phase transition in nearest neighbour spin ice when a field is applied in a 111 direction, but rather a continuous crossover from a state of no monopoles – theh Kagoméicei state – to a state of monopoles on every tetrahedron – the fully polarised state. At the critical field, Hc = 2J/µµ0, we found a temperature independent state (for low temperatures) due to the sudden increase of the gap between the ground state and the first excited state. This state turned out to have a mean monopole density of 0.3995 1 10−4 monopoles per tetrahedron according to Monte Carlo simulations. Exact± evaluation· of the microcanonical mean value on a 32 spin system gave a monopole density of 0.3965. Finding an exact algorithm that can evaluate the microcanonical mean fast enough to be feasible for larger systems seems to be an intractable problem. How- ever, by using the Bethe approximation in and known results [40], we evaluated the monopole density on the Husimi cactus which turned out to converge towards 0.4. This value is in agreement with both exact calculations for small systems and Monte Carlo data for somewhat larger systems. Appendix A

Simulations

In this appendix we present the details of the Monte Carlo simulations performed throughout the thesis.

A.1 Details

The Monte Carlo simulations were written in C++ and were based on the single spin ﬂip Metropolis algorithm described in Chapter 3. The model implemented was the nearest neighbour model with an applied magnetic ﬁeld, described in Section 2.3. The code is based on the cubic unit cell of the pyrochlore lattice – making the smallest possible system one of 16 spins – and it is possible to specify the number of unit cells in the [100], [010] and [001] directions individually. The random numbers used in the simulations were produced by the built in random number generator in C++, std::rand(), since this was considered a good enough random number generator for our purposes. Some data analysis and plotting was done in Python using the numpy and matplotlib libraries.

A.2 Simulation parameters

On the next pages there is a table where the input parameters of all simulations referred to are presented. System size is given in terms of the number of unit cells in each direction and temperature and magnetic field are given in dimensionless units. The field can vary either sinusoidally (for the AC susceptibility simulations) or be constant and can either be applied before equilibration ( t) or after equilibration (t 0). One Monte Carlo step corresponds to one spin flip∀ attempt per lattice site. ≥

55 56 APPENDIX A. SIMULATIONS

Figure Size Temperature, Field mag- Field Field T/J nitude direc- time de- µµ0H/J tion pendence 4.1 40 40 40 0.2 0.1 001 Constant; × × h i t 0 4.3 15 15 15 0.20, 0.22, 0.25, 0.1 001 sin≥ωt; t × × 0.27, 0.30, 0.32, h i ∀ 0.35, 0.37, 0.40 4.4 15 15 15 0.20, 0.22, 0.25, 0.1 001 sin ωt; t × × 0.27, 0.30, 0.32, h i ∀ 0.35, 0.37, 0.40 5.5 3 3 3 0.10, 0.15, 0.20, 0 to 10; 111 Constant; × × 0.25, 0.30, 0.35, 10,000 steps h i t 0.40 ∀ 5.6 3 3 3 0.10, 0.15, 0.20, 0 to 10; 111 Constant; × × 0.25, 0.30, 0.35, 10,000 steps h i t 0.40 ∀ 5.8 3 3 3 0.10, 0.15, 0.20, 0 to 10; 111 Constant; × × 0.25, 0.30, 0.35, 10,000 steps h i t 0.40 ∀ 5.9 3 3 3 0.10, 0.15, 0.20, 0 to 10; 111 Constant; × × 0.25, 0.30, 0.35, 10,000 steps h i t 0.40 ∀ 5.10 1 1 1 0.001 1.9 to 2.1; 111 Constant; × × 1000 steps h i t 5.10 1 1 1 0.0005 1.9 to 2.1; 111 Constant;∀ × × 1000 steps h i t 5.10 1 1 1 0.0001 1.85 to 2.15; 111 Constant;∀ × × 1000 steps h i t 5.11 1 1 1 0.001 1.9 to 2.1; 111 Constant;∀ × × 1000 steps h i t 5.12 1 1 1, 0.1 0 to 10; 111 Constant;∀ 2 × 2 × 2, 10,000 steps h i t 3 ×3 ×3 ∀ 5.13 1 ×1 ×1 to 0.1 2 111 Constant; 3 × 3× 3 h i t 5.18 1 × 1× 1, 0.0001 to 0.1 2 111 Constant;∀ 2 ×2 ×2 h i t × × ∀ A.2. SIMULATION PARAMETERS 57

Equilibration Sampling Boundary Comments steps steps conditions 1000 100 open and Relaxation of the monopole periodic current 1,000 1,000 periodic Real part of the AC susceptibility.

1,000 1,000 periodic Imaginary part of the AC susceptibility.

10,000 10,000 periodic Magnetisation plateau in a 111 ﬁeld. h i 10,000 10,000 periodic Monopole density in a 111 ﬁeld. h i

10,000 10,000 periodic Speciﬁc heat as a function of magnetic ﬁeld.

10,000 10,000 periodic Susceptibility as a function of magnetic ﬁeld.

10,000 10,000 periodic Specific heat at T/J = 1 10−3. · 10,000 10,000 periodic Specific heat at T/J = 5 10−4. · 10,000 10,000 periodic Specific heat at T/J = 1 10−4. · 10,000 10,000 periodic Metropolis acceptance rate.

10,000 10,000 periodic Finite size scaling of the spe- ciﬁc heat.

10,000 10,000 periodic Size dependence of the monopole density. 10,000 10,000 periodic Temperature dependence of the energy at the critical ﬁeld. 58 Appendix B

Rare-earth magnetism

This appendix gives a brief background on rare earth magnetism and motivates why the Ising model of Section 2.3 can be used. Furthermore, we show how the magnetic moment can be calculated using Hund’s rules.

B.1 Generalities

The rare earth elements are the elements of the 4f block of the periodic table of the elements. These elements have partially filled 4f orbitals ranging from lanthanum (La) with no 4f electrons to lutetium (Lu) with the 4f orbital filled. Due to the large orbital angular momentum of the f orbital, it fits as many as 14 electrons and it is possible for some of the elements with partially filled orbitals – such as holmium (Ho) and dysprosium (Dy) – to achieve a very large magnetic moment (both Ho and Dy have magnetic moments of 10µB [8]). The large magnetic moments gives rise to the strong crystal field observed∼ in spin ice compounds. The 4f electronic wavefunctions are confined close to the nucleus, compared to other orbitals present in the rare-earth compounds, such as the 5s and 5p orbitals [8]. This makes the exchange interaction between 4f electrons of different ions relatively weak [7] – which is why the crystal field is able to pin the spins to the local 111 axes in spin ice and we can use Ising spins in our models. h i B.2 Hund’s rules

The total angular momentum of an ion can be calculated using Hund’s rules. Ac- cording to Hund’s rules, the electron states in the partially ﬁlled orbital are chosen in the following manner [8]:

1. Maximise the total spin angular momentum, S.

59 60 APPENDIX B. RARE-EARTH MAGNETISM

2. Maximise the total orbital angular momentum, L, under the condition of S being maximised.

3. Calculate the total angular momentum, J, as J = L + S if the shell is more than half ﬁlled. Otherwise, J = L S . | − | The magnetic moment of the ions can be calculated as

µ = gLµBJ (B.1) where µB is the Bohr magneton, J is the total angular momentum and gL is the Land´efactor given by

3 S(S + 1) L(L + 1) g = + − . (B.2) L 2 2J(J + 1) Appendix C

Complexity classes

Here we give a brief introduction to diﬀerent complexity classes in computer science in order to motivate why the problem of counting the allowed states at Hc in Section 5.3.1 is intractable.

C.1 P, NP and #P problems

Problems in computer science are often categorised based on how long they take to solve on a computer, or rather how the time required to solve it scales with the size of the problem. In Section 5.3.1 we mentioned three different complexity classes without further specification: P, NP and #P. The class P (short for polynomial time) consists of the problems that can be solved by algorithms that scales polynomially with the size of the problem [39]. These problems are considered easy to solve, even though large problems may still be unsolvable with today’s computers. NP problems (non- deterministic polynomial time), however, are considered hard to solve. These are problems for which a given solution can be verified in polynomial time [39]. Find- ing a solution may take super-polynomial time, though. Obviously, the complexity class P is a subset of NP, that is P NP. For NP problems we are tasked⊆ with finding a single solution. If we on the other hand were to take interest in the number of possible solutions to the same NP problem, this problem is said to be in the class #P [37, 45]. Since we must now find all the solutions to the corresponding NP problem, the #P problem is at least as hard as the NP problem.

C.2 NP- and #P-complete problems

A problem is called NP-complete if it is at least as hard as any other problem in NP. Equivalently, a #P-complete problem is at least as hard as any other

61 62 APPENDIX C. COMPLEXITY CLASSES problem in #P. This means that if we can ﬁnd an eﬃcient algorithm that solves a NP-complete problem, we can solve any NP-problem [39].

C.3 The P versus NP problem

As previously stated, P NP. The P versus NP problem is whether or not P=NP, that is can NP problems⊆ not only be veriﬁed in polynomial time, but also solved in polynomial time [39]? This open problem is probably the most important one in theoretical computer science and one of the most important problems in all mathematics [39]. Proving or disproving P=NP would not only have a huge impact on many scientiﬁc disciplines, but also give you a prize of $1 million, being one of the seven Millennium Problems – the seven most important open problems in mathematics, according to the Clay Mathematics Institute [38, 39]. Since NP-complete problems are the hardest problems of the NP class, solving one single such problem in polynomial time would be equivalent to proving P=NP. No such solution exists today, though, and most computer scientist rather believe that P=NP. Numerous attempt at proving and disproving P=NP have been at- tempted6 since the formulation of the problem in 1971, but a solution appears to be quite far in the future [39]. Bibliography

[1] M. J. Harris et al., Geometrical Frustration in the Ferromagnetic Pyrochlore Ho2Ti2O7, Phys. Rev. Lett. 79, 2554 (1997).

[2] P. A. Dirac, Quantised Singularities in the Electromagnetic Field, Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character , 60 (1931).

[3] C. Castelnovo, R. Moessner and S. L. Sondhi, Magnetic Monopoles in Spin Ice, Nature 451, 42 (2008).

[4] C. Kittel, Introduction to Solid State Physics, 6th edition (Wiley, New York, 1986).

[5] L. D. Jaubert, Topological Constraints and Defects in Spin Ice, PhD thesis, Ecole Normale Sup´erieurede Lyon-ENS LYON, 2009.

[6] L. Balents, Spin Liquids in Frustrated Magnets, Nature 464, 199 (2010).

[7] L. D. Jaubert and P. C. Holdsworth, Magnetic Monopole Dynamics in Spin Ice, Journal of Physics: Condensed Matter 23, 164222 (2011).

[8] J. Jensen and A. Mackintosh, Rare Earth Magnetism: Structures and Excita- tions, International Series of Monographs on Physics (Clarendon Press, 1991).

[9] B. C. den Hertog and M. J. P. Gingras, Dipolar Interactions and Origin of Spin Ice in Ising Pyrochlore Magnets, Phys. Rev. Lett. 84, 3430 (2000).

[10] L. Pauling, The Structure and Entropy of Ice and of Other Crystals with Some Randomness of Atomic Arrangement, Journal of the American Chemical Society 57, 2680 (1935).

[11] A. P. Ramirez et al., Zero-Point Entropy in ‘Spin Ice’, Nature 399, 333 (1999).

[12] X. Ke et al., Nonmonotonic Zero-Point Entropy in Diluted Spin Ice, Phys. Rev. Lett. 99, 137203 (2007).

63 64 Bibliography

[13] E. Ising, Beitrag zur Theorie des Ferromagnetismus, Zeitschrift f¨urPhysik 31, 253 (1925). [14] R. Siddharthan et al., Ising Pyrochlore Magnets: Low-Temperature Properties, “Ice Rules,” and Beyond, Phys. Rev. Lett. 83, 1854 (1999). [15] R. Siddharthan, B. S. Shastry and A. P. Ramirez, Spin Ordering and Partial Ordering in Holmium Titanate and Related Systems, Phys. Rev. B 63, 184412 (2001). [16] D. Landau and K. Binder, A Guide to Monte Carlo Simulations in Statistical Physics (Cambridge University Press, 2009). [17] N. Metropolis et al., Equation of State Calculations by Fast Computing Ma- chines, The Journal of Chemical Physics 21, 1087 (1953). [18] S. Koonin, Computational Physics, The Advanced Book Program (Addison- Wesley Publishing Company Incorporated, 1986). [19] S. Bramwell et al., Measurement of the Charge and Current of Magnetic Monopoles in Spin Ice, Nature 461, 956 (2009). [20] O. Tchernyshyov, Magnetism: Freedom for the Poles, Nature 451, 22 (2008). [21] I. Ryzhkin, Magnetic Relaxation in Rare-Earth Oxide Pyrochlores, Journal of Experimental and Theoretical Physics 101, 481 (2005). [22] S. Groot, Thermodynamics of Irreversible Processes, Selected Topics in Mod- ern Physics (North-Holland Pub. Co., 1952). [23] S. T. Bramwell, Generalized Longitudinal Susceptibility for Magnetic Monopoles in Spin Ice, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 370, 5738 (2012). [24] A. Morrish, The Physical Principles of Magnetism, An IEEE Press Classic Reissue (Wiley, 2001). [25] H. Casimir and F. du Pr´e, Note on the Thermodynamic Interpretation of Paramagnetic Relaxation Phenomena, Physica 5, 507 (1938). [26] L. Bovo et al., Brownian Motion and Quantum Dynamics of Magnetic Monopoles in Spin Ice, Nature Communications 4, 1535 (2013).

[27] H. Takatsu et al., AC Susceptibility of the Dipolar Spin Ice Dy2Ti2O7: Exper- iments and Monte Carlo Simulations, Journal of the Physical Society of Japan 82 (2013). [28] M. Picco, F. Ricci-Tersenghi and F. Ritort, Chaotic, Memory, and Cool- ing Rate Eﬀects in Spin Glasses: Evaluation of the Edwards-Anderson Model, Phys. Rev. B 63, 174412 (2001). Bibliography 65

[29] S. Blundell and K. Blundell, Concepts in Thermal Physics (Oxford University Press, 2006).

[30] R. G. Melko and M. J. Gingras, Monte Carlo Studies of the Dipolar Spin Ice Model, Journal of Physics: Condensed Matter 16, R1277 (2004). [31] O. J. Heilmann and E. H. Lieb, Theory of Monomer-Dimer Systems, Commu- nications in Mathematical Physics 25, 190 (1972).

[32] P. Timonin, Spin Ice in a Field: Quasi-Phases and Pseudo-Transitions, Jour- nal of Experimental and Theoretical Physics 113, 251 (2011). [33] R. Fowler and G. Rushbrooke, An Attempt to Extend the Statistical Theory of Perfect Solutions, Transactions of the Faraday Society 33, 1272 (1937).

[34] M. E. Fisher, Statistical Mechanics of Dimers on a Plane Lattice, Physical Review 124, 1664 (1961). [35] P. W. Kasteleyn, The Statistics of Dimers on a Lattice: I. The Number of Dimer Arrangements on a Quadratic Lattice, Physica 27, 1209 (1961). [36] H. Temperley and M. E. Fisher, Dimer Problem in Statistical Mechanics – an Exact Result, Philosophical Magazine 6, 1061 (1961). [37] M. Jerrum, Two-Dimensional Monomer-Dimer Systems are Computationally Intractable, Journal of Statistical Physics 48, 121 (1987), see also erratum. [38] Clay mathematics institute, Millennium Problems, http://www.claymath.org/millenium-problems/p-vs-np-problem, Accessed April 23 2014. [39] L. Fortnow, The Status of the P versus NP Problem, Communications of the ACM 52, 78 (2009). [40] J. F. Stilck and M. J. de Oliveira, Entropy of Flexible Chains Placed on Bethe and Husimi Lattices, Phys. Rev. A 42, 5955 (1990). [41] S. Isakov et al., Magnetization Curve of Spin Ice in a [111] Magnetic Field, Physical Review B 70, 104418 (2004).

[42] Z. Hiroi et al., Speciﬁc Heat of Kagom´eIce in the Pyrochlore Oxide Dy2Ti2O7, Journal of the Physical Society of Japan 72, 411 (2003). [43] T. Sakakibara et al., Observation of a Liquid-Gas-Type Transition in the Py- rochlore Spin Ice Compound Dy2Ti2O7 in a Magnetic Field, Physical review letters 90, 207205 (2003).

[44] R. Higashinaka et al., Low Temperature Speciﬁc Heat of Dy2Ti2O7 in the Kagome Ice State, Journal of the Physical Society of Japan 73, 2845 (2004). 66 Bibliography

[45] L. Fortnow and S. Homer, A Short History of Computational Complexity, Bulletin of the EATCS 80, 95 (2003).