Phase Transitions and Vortex Structures in Multicomponent Superconductors

Phase transitions and vortex structures in multicomponent superconductors

KARL SELLIN

Licentiate thesis in Theoretical Physics Stockholm, Sweden 2015 TRITA-FYS 2015:78 ISSN 0280-316X KTH Teoretisk fysik ISRN KTH/FYS/--1578:SE AlbaNova universitetscentrum ISBN 978-91-7595-758-6 SE-106 91 Stockholm Sweden

Akademisk avhandling som med tillstånd av Kungl Tekniska högskolan framlägges till oentlig granskning för avläggande av Teknologie licentiatexamen i fysik med inriktning teoretisk fysik 7 december 2015 kl 10:00 i sal FB42, AlbaNova Univer- sitetscentrum.

Tryck: Universitetsservice US AB 3

Sammanfattning

Teoretiska aspekter angående erkomponentsupraledning och system med konkurrerande interaktioner studeras med hjälp av Monte Carlo-metoder. Motiverat av de senaste experimentella och teoretiska resultaten av kom- plexa strukturformationer av virvlar i erkomponentsystem, studeras möjliga virvelstrukturformationer på grund av virvelinteraktioner som inte är rent attraktiva eller repellerande. Virvelstrukturer såsom kluster, superkluster, hi- erarkiska strukturer, remsor, gossamer-mönster, glasiga faser, samt schack- brädesmönster och ringar demonstreras vara möjliga. Ordningen hos den supraledande fasövergången studeras för London- supraledare med era komponenter. Fasövergången visas vara antingen av första ordningen eller kontinuerlig beroende på styrkan hos en symmetribry- tande Josephsoninteraktion mellan komponenterna. Det föreslås att fasövergån- gen är av första ordningen på grund av en fasseparation av virvlar, som i sin tur är på grund av en uktuationsinducerad attraktiv växelverkan mellan virvellinjer. 4

Abstract

Theoretical aspects of multicomponent superconductivity and systems with competing interactions are studied using Monte Carlo techniques. Motivated by recent experimental and theoretical results of complex structure formation of vortices in multicomponent systems, possible vortex structure formations due to vortex interactions that are not purely attractive or repulsive are considered. Vortex structures such as clusters, superclusters, hierarchical structure formation, stripes, gossamer patters, glassy phases, as well as checkerboard lattices and loops are demonstrated to be possible. The order of the superconducting phase transition is considered for multicomponent lattice London superconductors. The phase transition is demonstrated to be either rst-order or continuous depending on the strength of a symmetry-breaking Josephson intercomponent interaction. It is argued that the rst-order phase transition is caused by a vortex phase separation due to a uctuation-induced attractive interaction between vortex lines. Acknowledgements

The numerical data on which this thesis relies is a result of the more than 700 processor years that I have consumed from resources provided by the Swedish Na- tional Infrastructure for Computing (SNIC) at National Supercomputer Center at Linköping, Sweden. I want to thank my supervisor Egor Babaev for giving me the chance to do this research, which has been a pleasure, as it should be. Thanks to Mats Wallin, Jack Lidmar and Hannes Meier for helpful discussions and advice on computational physics. Thanks also to past and present group members: Chris Varney who worked with me on my rst paper, Johan Carlström, Julien Garaud and Mihail Silaev for many scientic discussions, and Daniel Weston for being an excellent 'bollplank' when encountering something non-sensible. I thank also collectively the PhD and master students and postdocs who have passed through the department for their insightful analyses, substantiated opinions and polite discussions during lunch. Thanks to my parents for all support and for letting me keep the top notch 200 MHz family computer in my room as a child, thanks to Viktor for growing up with me and for our friendship, thanks to my aunt Birgitta for taking me grocery shopping when I was a starved student and thanks to Maja for being a big sister and sorry for being a little brother. And thanks to Hanna for her love and support that has been crucial and for having the courage of living in the same apartment as me. I also send thoughts to my late grandfather Sten Forsmark, a physics teacher, who did his best to spark my interest for physics but never had the pleasure to see it develop.

5 Contents

Contents 6

I Background and selected results 1

1 Physical and computational framework 3 1.1 Averages in statistical physics ...... 3 1.2 Monte Carlo integration ...... 4 1.3 The Metropolis Monte Carlo method ...... 5 1.4 Finite-size scaling and parallel tempering ...... 6 1.5 Phase transitions: denition and classication ...... 7

2 Superconductivity: an overview 9 2.1 Vortices and dierent types of superconductors ...... 9 2.2 The order of the superconducting phase transition ...... 11

3 Vortex structure formation 13 3.1 Hierarchical structure formation ...... 14 3.2 Stripe phases due to non-pairwise interaction ...... 14 3.3 Vortices with dipolar forces ...... 14

4 Phase transitions in multicomponent London superconductors 25 4.1 Two-component London superconductivity ...... 25

4.2 Derivation of the Josephson length ξJ ...... 26 4.3 Discretization of the two-band London free energy ...... 27 4.4 Two-component lattice London superconductivity ...... 29

Bibliography 35

II Scientic Papers 41

6 Part I

Background and selected results

Chapter 1

Physical and computational framework

We begin by outlining the principles upon which the work in this thesis relies, namely equilibrium statistical physics and Monte Carlo methods.

1.1 Averages in statistical physics

In equilibrium statistical physics, one attempts to understand the properties of physical systems by computing and analyzing expectation values of an observable

O = O[ θi ] where θi denotes the degrees of freedom of the system. Given a Hamiltonian{ } H = H[{ θ } ] the ensemble average of O is given by { i}

1 βH O = Tr Oe− , (1.1) h i Z 1 βH where β = (kBT )− , Z = Tr e− is the partition function, and Tr = d θi is the trace operator which consists of integration (and/or summation) over the{ range} of values the degrees of freedom takes. R The pen-and-paper approach to computing expressions like Eq. (1.1) only rarely leads to exact solutions, see e.g. [1], and otherwise the strategy has historically been to look for approximations which ease the mathematical complexity of the integrals while still preserving the essential physics. However, with the advent of computers came the possibility of the numerical computation of expressions like (1.1) [2]. Since the trace operation in (1.1) amounts to high dimensional integration for any realistic system, straightforward integration by numerical discretization may very well not be a feasible strategy, when Monte Carlo integration techniques are. The Monte Carlo techniques not only have the advantage of practicality, but also the principal advantage of being an exact method, which by the law of large numbers will converge towards the right answer [3].

3 4 CHAPTER 1. PHYSICAL AND COMPUTATIONAL FRAMEWORK

Figure 1.1: Calculation of π with a Monte Carlo method, illustrating the principle of Monte Carlo integration. Red dots are hits and blue misses.

1.2 Monte Carlo integration

Monte Carlo methods rely on random numbers1. A classic and visual example of straightforward Monte Carlo integration is the 'hit-and-miss' calculation of the integral 1 √ 2 , i.e. the area of a quarter unit circle. Such a calculation 0 dx 1 x = π/4 consists of repeated− generation of sets of two random numbers x and y between zero andR one, and if y < √1 x2 the iteration counts as a hit. The ratio of the number of hits to the total number− of iterations will tend to the area under the curve y = √1 x2 and thus the integral. The method is exact in the sense that there is no discretization− of the domain, and convergence is guaranteed by the law of large numbers [3]. A realization of such a calculation is presented in Fig. 1.1. The hit-and-miss method can be naturally extended to integrals with higher dimensionality. For example, for a system with degrees of freedom N we can θi i=1 calculate the integral { }

βH[ θi ] Z = d θ e− { } , (1.2) { i} Z (where we denote d θi as the integration of the range of values the degrees of freedom take) by generating{ } random congurations N , and generating a θi i=1 random number withinR the range of the integrand, and counting{ } the number of hits and misses. Note however that in the example of Fig. 1.1 there is no substantial

1Unless special hardware is used, a computer generates random numbers by some mathematical formula, making them not truly random but pseudorandom. 1.3. THE METROPOLIS MONTE CARLO METHOD 5 dierence between the sizes of the areas under and above the curve. If one were to compute an integral with a big such dierence, the straightforward hit-and- miss method would not work well in practice as most of the iterations would be either hits or misses and the convergence would be slow. A solution is to use importance sampling (as opposed to the 'simple sampling' used in Fig. 1.1), where the contributing terms are drawn from a non-uniform distribution. For details, refer to [3, 4, 5].

1.3 The Metropolis Monte Carlo method

The principle of importance sampling in statistical physics as suggested by Metropo- lis [2] is as follows. In Fig. 1.1 we generated numbers x and y from a uniform distribution. In the evaluation of something like Eq. (1.1), we could instead imagine generating the congurations N not uniformly, but in such a way that µ θi i=1 the number of hits and misses are≡ roughly { } equal. The estimated expectation value after M generated congurations µn can be written

M O e βHµn n=1 µn − states are generated with uniform probability. O M , µn h i ≈ βHµn P n=1 e− (1.3) If we insteadP draw congurations with probabilities according to their Boltzmann weights, the estimator is simply an unweighted arithmetic average

M 1 βHµ O O , states µ are generated with probability e− n . (1.4) h i ≈ M µn n n=1 X The remaining question is then how one generates congurations with probability according to their Boltzmann weights. In practice, the importance sampling discussed above is obtained with a Markov Chain through the Metropolis method [2]. Metropolis Monte Carlo works by proposing a new conguration j from an old conguration i with a proposal distribution g(i j), which is accepted with a specied probability A(i j) = β∆E → → min(1, e− ) where ∆E = Ej Ei. To see how this works, consider the time evolution of the probability of being− in state i, dp i = [p P (j i) p P (i j)] , (1.5) dt j → − i → j X where P (i j) is the probability of transitioning from i to j. The rst and second terms on the→ right-hand side correspond to the rate of transitioning into and out of the state i respectively. In equilibrium the time derivative is zero and Eq. (1.5) is fullled if (but not only if)

p P (j i) = p P (i j). (1.6) j → i → 6 CHAPTER 1. PHYSICAL AND COMPUTATIONAL FRAMEWORK

The condition of Eq. (1.6) is called detailed balance and is a sucient but not necessary condition for Metropolis Monte Carlo given that states are generated ergodically2. In a simulation, the probability of transitioning from i to j is the product of the proposal and acceptance probabilities, that is P (i j) = g(i j)A(i j) which after insertion into the balance equation (1.6) gives→ → →

β(Ei Ej ) βEi p g(j i) min(1, e− − ) e− i = → = , (1.7) β(E E ) βEj pj g(i j) min(1, e j i ) e → − − − assuming that the proposal distribution is symmetric. The Metropolis Monte Carlo method thus generates a chain of congurations where each conguration appears with a probability proportional to its Boltzmann weight. The estimation of thermal averages of the sort (1.1) can thus be done via arithmetic averaging of the form (1.4) by starting with some initial conguration µ0 and 3 generating a chain µ1,. . . µM via the Metropolis scheme .

1.4 Finite-size scaling and parallel tempering

At rst glance the computational principles outlined above may seem straightforward, but there is in practice a range of diculties that arise when computing averages with the Metropolis method. Here a few that are relevant for this thesis will be briey discussed, and the reader is referred to the literature for further details. Of course, only systems of nite size are accessible for numerical studies. How- ever, by calculating the same quantities for various system sizes, one can extrap- olate to the thermodynamic limit. For example, scaling of the maximum of the heat capacity and behavior of the energy histogram can determine whether a phase transition is rst order or not [7, 8, 9]. Since the Metropolis Monte Carlo method is a Markov chain process generated congurations will be correlated. Such correlations can be quantied in terms of a characteristic correlation time [4], which can be taken as a safe sampling frequency in order to sample uncorrelated data. However, close to critical points the correlation time can diverge, which is called critical slowing down. There are various model-specic algorithms that circumvent this via global updates. Using the parallel tempering algorithm [10] is another strategy. With parallel tempering, systems with dierent temperatures are simulated in parallel for a range of temperatures, as illustrated in Fig. 1.2. Each processor

2See [3, 6]. Ergodicity means that each state is reachable from every other state in a nite time, however a non-ergodic calculation may still yield useful information within an ergodic class [3]. 3 There are two practical caveats to this: i) if the initial state µ0 is not an equilibrium state the system must be equilibrated before sampling occurs and ii) sampling should be done with uncorrelated samples. See [4] for a valuable reference of how Monte Carlo works in practice. 1.5. PHASE TRANSITIONS: DEFINITION AND CLASSIFICATION 7

Figure 1.2: Illustration of parallel tempering where several replicas of a system are evolved rightwards for dierent temperatures T1, T2,. . . and exchange of such replicas are possible (illustrated by crossing lines).

performs ordinary Monte Carlo updates and to this is added the trial move of swapping systems between temperatures, leading to reduced correlations between states and that the system can tunnel out of metastable states. For strict detailed balance to be satised the swap moves should occur at random, but for reasons of computational eciency the swap moves can be attempted with xed frequency which satises the balance condition [6, 11].

1.5 Phase transitions: denition and classication

1 The thermodynamical bulk free-energy density f = β− ln(Z)/V depends on the coupling parameters of the Hamiltonian, and thermodynamic functions are calculated via dierentiation of the free energy. Such thermodynamic functions may exhibit discontinuities at certain sets of coupling parameters, for which the free energy is not analytic. Such sets dene phase boundaries which separate phases (i.e. regions of analyticity). The crossing of a phase boundary is a phase transition. For further details of mathematical aspects of phase transitions, refer to [12], and for a more physical discussion refer to [13]. By the Ehrenfest classication [14] phase transitions are classied by the order of the derivative of the free energy which has a discontinuity at the critical point. If a rst order derivative (entropy, internal energy) is discontinuous, then the phase transition is of rst order. If a second order derivative (heat capacity) is discontinuous, the phase transition is of second order, and so on. The phase transitions of dierent order may furthermore be grouped into dierent types depending on the behavior of higher order derivatives, see [15]. The Ehrenfest classication is however discouraged [13, 16], for example there may divergences rather than discontinuities in thermodynamic functions (as for the 2D Ising model in no external

eld where the heat capacity diverges logarithmically at Tc). In the classical work of Landau, Lifshitz et al. [17] phase transitions are grouped 8 CHAPTER 1. PHYSICAL AND COMPUTATIONAL FRAMEWORK into those of the 'rst kind' and those of the 'second kind', which can be a source of terminological confusion with the Ehrenfest classication. See [18] for a historical assessment of the terminology and classication of phase transitions. The modern classication of phase transitions is binary where a phase transition is either 'rst- order' (there is a latent heat) or 'continuous' (no latent heat) [16, 13]. Chapter 2

Superconductivity: an overview

In this chapter a brief overview of superconductivity is given, with emphasis on the aspects that are relevant to this thesis. The key properties of superconducting materials are the vanishing of electrical resistivity [19] and magnetic elds [20] (the Meissner eect). Insight to the microscopic mechanism was established with the BCS-theory [21] (for a more modern approach to the microscopic theory, see [22]). More useful to our purposes is Ginzburg-Landau theory [23] which was for- mulated before the BCS-theory but can be derived from BCS-theory close to the critical point [24]. Within the framework of this theory, an order parameter eld ψ determines the free-energy density

1 1 1 f = α ψ 2 + β ψ 4 + ( + ieA)ψ 2 + ( A)2, (2.1) | | 2 | | 2 | ∇ | 2 ∇ × where the coecients α and β may depend on temperature, A is the magnetic vector potential and e the charge coupling. Writing ψ = ψ eiθ, it can be noted that this theory has a local U(1)-symmetry, as Eq. (2.1)| is| invariant under the transformations θ θ + φ, A A φ/e where φ is a spatially dependent and single-valued function.→ It can→ also be− ∇ noted that two characteristic length scales are contained in this theory, the magnetic penetration depth λ and the coherence length ξ [25], corresponding to the decay of the magnetic eld in a Meissner domain and the correlation of the order parameter eld respectively. The limit of constant density ψ but varying phase θ is referred to as the London limit and will be used extensively| | in this thesis.

2.1 Vortices and dierent types of superconductors

Abrikosov [26] found when that when superconductors with κ λ/ξ > 1/√2 in a suciently strong applied magnetic eld, the eld penetrates the≡ sample with spatial periodicity. At the points of magnetic maximum, the order parameter vanishes and following a contour around such a point causes the phase to wind by 2π. This

9 10 CHAPTER 2. SUPERCONDUCTIVITY: AN OVERVIEW

−M −M

H H

H H H

1 2 Figure 2.1: Schematic behavior of the magnetization M in type-1 (left) and type-2 (right) superconductors, in an applied magnetic eld H.

Figure 2.2: Illustration of a vortex in a superconducting slab which allows magnetic eld lines (red) to penetrate.

state is called the Abrikosov vortex state. For superconductors with κ < 1/√2 such a state does not occur. Superconductors have therefore been sorted into two kinds, those with attractively interacting vortices (type-1: κ < 1/√2) and those with repulsively interaction vortices (type-2: κ > 1/√2) [25]. In the Meissner state an external magnetic eld H is canceled by the magnetization M. However, this screening is only possible up to a critical point, see Fig. 2.1. For type-1 superconductors, there is a single critical eld Hc, and the superconductor is in its Meissner state for H < Hc and in its normal state for H > Hc. For type-2 superconductors there are two critical elds, Hc1 and Hc2, where H < Hc1 is the Meissner state, Hc1 < H < Hc2 is the Abrikosov vortex state (Hc1 corresponds to the entrance of the rst vortex), and H > Hc2 is the normal state. In the Abrikosov vortex state magnetic ux tubes penetrate the superconductor as is illustrated in Fig. 2.2 and minimize their mutual repulsion energy by forming a triangular lattice. In recent years it has been shown that multicomponent superconductors can have length scales such that they are not classiable as type-1 nor as type-2 [27].

For a two-component superconductor which has two coherence lengths ξ1 and ξ2, the case with κ1 < 1/√2 and κ2 > 1/√2 was labeled type-1.5 superconductivity [28] and generated further theoretical and experimental activity [29, 30, 31, 32, 2.2. THE ORDER OF THE SUPERCONDUCTING PHASE TRANSITION 11

33, 34, 35, 36, 37, 38]. In such systems the vortex pairwise interactions can be short-range repulsive and long-range attractive, leading to vortex clustering and a 'semi-Meissner' state, where regions of a sample are in a vortex state, and other regions are in a Meissner state. Such systems may also have strong non-pairwise interactions due to the non-linearity of the theory, further complicating the vortex structures into lamentary shapes [33, 39, 40].

2.2 The order of the superconducting phase transition

In early heat capacity measurements a discontinuity at the superconducting critical point was observed as well as an exponential behavior for low temperatures, see for example [41]. BCS-theory successfully reproduced both of these features1, suggesting that the superconducting phase transition is of second order. However, the mean-eld BCS-calculation does not account for thermodynamic contributions from thermally induced vortex excitations. Fluctuations in the gauge eld in a Ginzburg-Landau setting can drive the phase transition to rst-order for very type-1 superconductors [43]. There are calculations taking vortices into account at the level of Ginzburg-Landau theory (which is accu- rate at criticality), concluding that the phase transition is rst-order for very type-1 superconductors, and continuous for very type-2 superconductors, with a tricritical point slightly below the mean-eld critical value of κ = 1/√2 [44]. A tricritical point is indeed found by Monte Carlo [45] and renormalization group [46] calculations. The extension of the continuous phase transition slightly into κ < 1/√2 regime was also found in large-scale Monte Carlo calculations [47]. By appealing to universality and symmetry, we may consider the results of the three-dimensional globally U(1)-symmetric xy-model

H = J cos(θ θ ), (2.2) − i+ˆµ − i iµ X (where i is a site index and µ = x, y, z and J is a coupling strength) to be informative for the problem of the order of the superconducting phase transition. The 3Dxy- model indeed describes the continuous superuid λ-transition ([48], see also helium microgravity measurements [49, 50]) and is furthermore dual to the lattice London superconductor [51] (see also [52, 53])

1 H = J cos(θ θ A ) + λ2[∆ A]2 , (2.3) − i+ˆµ − i − iµ 2 × iµ iµ X where the second term in the bracket is a discrete curl. In the duality mapping the temperature axis is reversed and the London superconductor is said to be in the inverted 3Dxy universality class, which was conrmed with Monte Carlo

1See Eqs. (3.47), (3.54) in [21] as well as [25] p. 64-66 and [42] p. 128-130. 12 CHAPTER 2. SUPERCONDUCTIVITY: AN OVERVIEW calculation [54]. See also [55, 56, 57, 58] for further numerical studies of lattice London superconductivity. The order of the superconducting phase transition may be dierent in multicomponent cases. An U(1) U(1)-superconductor can have a rst-order phase transition [59, 60, 61, 62]. The rst-order× phase transition can be interpreted as an eect of the intercomponent vortex interactions, Monte Carlo calculations for eective 3D vortex-loop models with non-monotonic interactions of the type-1.5 kind indeed demonstrate a rst-order phase transition [63]. Paper 4 of this thesis considers the order the phase transition of a two-component model where the U(1) U(1)- symmetry has been broken to a global U(1)-symmetry by an intercomponent× phase coupling term. Chapter 3

Vortex structure formation

The triangular Abrikosov vortex lattice has been detected experimentally for various compounds and with various experimental techniques, see e.g. [64, 65, 66, 67]. However, more recent measurements have displayed complex vortex structures with vortex stripes, clusters and voids [28, 29, 37, 68, 69]. Although eects of pinning1 may certainly be important for understanding such results, eects of a vortex interaction which is not purely repulsive can not be ruled out, which motivates theoretical considerations of vortex structure formation. A recent proposal [70] of using vortex arrays as a magnetic lattice for ultracold atoms further motivates studies of vortex structure formation as it could provide experimental control of the magnetic lattice [71]. One theoretical approach to vortex structure formation is to introduce vortices into a domain and numerically minimize the free energy in a Ginzburg-Landau setting, see for example [33]. Another approach, which will be used here, is to assume forms of vortex interactions found elsewhere [33, 39] and considering vortices to be point-particles in two dimensions. The properties of a system of such point- particles can be determined with Monte Carlo or molecular dynamics (for vortex molecular dynamics type simulations, see e.g. [71, 72, 73, 74]). In section 3.1 paper 1 is discussed as well as results of hierarchical structure formation that were not included in the paper. In section 3.2 is discussed paper 2, which describes stripe phases that arise due to non-pairwise interactions. In section 3.3 is discussed paper 3 where vortices with dipolar interactions are considered, the derivation of the vortex interactions in the London limit from a Ginzburg-Landau theory is included, and results that were not included in the paper are discussed.

1Regions of the superconductor which attracts vortices, e.g. impurities can cause vortex pinning.

13 14 CHAPTER 3. VORTEX STRUCTURE FORMATION

3.1 Hierarchical structure formation

In paper 1, the eects of pairwise vortex interactions with several repulsive and/or attractive length scales are analyzed with both Monte Carlo and molecular dynamics. Such interactions are argued to be possible by fabrication of layered superconducting structures with each layer having a dierent penetration depth, and/or layers of insulating materials of varying thickness. It is demonstrated that a variety of structures like giant clusters, ring clusters, voids, stripes and clusters of clusters (so-called superclusters) are possible depending on the shape of the interaction potential and density. A molecular dynamics follow-up paper demonstrated that honeycomb and kagome lattices are also possible [71] and argued that it could provide control of a magnetic lattice for ultracold atomic experiments. An illustrative example obtained with Monte Carlo of hierarchical structure formation due to multi-scale repulsion is shown in Fig. 3.1. The melting of such a superclustering phase is considered in Fig. 3.2 where it is seen that the melting occur in two steps, corresponding to melting of the subclusters followed by melting of the superclusters. In Fig. 3.1 there are three repulsive scales and thus there can be clusters of clusters, but by adding more repulsive scales it is possible to have more steps of clustering of clusters, as shown in Fig. 3.3 where a four-step potential yields clusters of clusters of clusters.

3.2 Stripe phases due to non-pairwise interaction

In paper 2, an eective vortex model with short-range repulsive and long-range attractive pairwise interaction and a repulsive three-body interaction following the forms of calculated interactions from Ginzburg-Landau theory [33, 39] is studied. A variety of phases like stripes, 'gossamer' (web like) patterns and glassy phases arise. See Fig. 3.4 for an illustrative example of the stripe phase.

3.3 Vortices with dipolar forces

In paper 3, Ginzburg-Landau theory calculations and point-particle Monte Carlo is combined to study two-band systems where vortices core-split giving rise to eective dipolar-like interactions. The paper considers a two-component London theory with a dissipationless drag term (3.1b) which gives rise to skyrmions (to be discussed below). For a multicomponent system, there may be current-current (Andreev- Bashkin) interactions of the form v v [75], and we may consider the following i · j 3.3. VORTICES WITH DIPOLAR FORCES 15

3.5 3.0 2.5 2.0 u(r) 1.5 1.0 0.5 0.0 r 7 ρ = 0.06 6 ρ = 0.11 5 ρ = 0.29 g(r) 4 ρ = 0.51 3 ρ = 0.69 2 ρ = 1.09 1 0 0 5 10 15 20 r (a)

(b) (c) Figure 3.1: An illustrative example of hierarchical structure formation for 2000 parti- (r 1)2 cles. In a) is plotted the (toy) interaction potential u(r) = (K0(R) + e− − + 1) 1/(10 r) · e− − (if r < 10 and zero otherwise) and the the radial distribution function for various densities ρ = N/L2. In b) and c) are shown the hierarchical patterns for two densities. 16 CHAPTER 3. VORTEX STRUCTURE FORMATION

U 55

2 4 6 8 10 12 14 T Figure 3.2: Melting of the superclustering phase for a three-step potential u(r) where u = for r < 0.25, u = 2.25 for 0.25 < r < 1.75, u = 0.8 for 1.75 < r < 10 and u = 0∞for r > 10, here with 2000 particles for a density ρ = N/L2 = 0.51. U is the internal energy and the inset show snapshots of the structure formation at the marked temperatures.

London theory:

1 f = ψ 2( θ(a) + eA)2 (3.1a) 2 | a| ∇ a=1,2 X 1 2 + ν ψ 2 θ(1) + eA + ψ 2 θ(2) + eA (3.1b) 2 | 1| ∇ | 2| ∇ 1 + ( A)2 , (3.1c) 2 ∇ × where (3.1b) corresponds to the Andreev-Bashkin intercomponent drag term with strength determined by ν, and the other two terms are the usual kinetic and magnetic energy densities.

Skyrmions

A skyrmion is a topological object in which a three-dimensional unit texture n assumes all possible values on the unit two-sphere2. Such objects can occur for in- stance in magnets, see Fig. 3.5 for a vortex skyrmion. While covering the plane, the

2For two-dimensional cases, as here, the objects are sometimes referred to as baby skyrmions [76]. 3.3. VORTICES WITH DIPOLAR FORCES 17

500

400

300

200

100

0 0 100 200 300 400 500 Figure 3.3: Clusters of clusters of clusters for an interaction with four repulsive scales. Here the potential u(r) is a four-step potential with u = for r < 0.25, u = 2.25 for 0.25 < r < 1.75, u = 0.8 for 1.75 < r < 10, u = 0.3 for∞10 < r < 80, and u = 0 for r > 80, and there are 5000 particles in a square box with side length of 500.

35 35 30 30 25 25 20 20 15 15 10 10 5 5 0 0 0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35

(a) (b) Figure 3.4: Structure formation for a short-range repulsive long-range attractive two- body interaction and a repulsive three-body interaction. In a) the three-body interaction is weak and the vortices form clusters. In b) the three-body interaction is strong and the vortices form stripes. 18 CHAPTER 3. VORTEX STRUCTURE FORMATION

Figure 3.5: A vortex skyrmion. Blue arrows point down, and red arrows point up. number of wrappings around the unit sphere may be quantied by the topological skyrmion number (Chern number)

1 ∂n ∂n Q = dxdyn , (3.2) 4π · ∂x × ∂y Z To get a feeling of how the skyrmion number works write

sin(θ(r)) cos(Φ(ϕ)) n = sin(θ(r)) sin(Φ(ϕ)) , (3.3)  cos(θ(r))    where r and ϕ are the coordinates of a cylindrical system, Φ and θ are the azimuthal and polar angles of n (see [77]). Insertion into (3.2) yields after some computation

2π 1 ∞ dθ(r) dΦ(ϕ) Q = dr dϕ sin(θ(r)) (3.4a) 4π dr dϕ Z0 Z0 1 r= ϕ=2π = [ cos(θ(r))] ∞ [Φ(ϕ)] . (3.4b) 4π − r=0 ϕ=0 Since the skyrmion texture is such that n switches pole from r = 0 to r = we r= ∞ have [ cos(θ(r))] ∞ = 2, giving − r=0 [Φ(ϕ)]ϕ=2π Q = ϕ=0 m, (3.5) 2π ≡ 3.3. VORTICES WITH DIPOLAR FORCES 19 which we identify as the phase winding number m (vorticity). Skyrmions may be identied in two-component superconductors when composite vortices core-split. This is understood through the mapping of the density-varying

GL theory to an easy-plane non-linear sigma model n = Ψ†σΨ/Ψ†Ψ where Ψ = T (ψ1, ψ2) and σ = (σ1, σ2, σ3) with σi the Pauli matrices. For details, refer to paper 3. The texture of n for a composite vortex with increasing core-splitting is schematically shown in Fig. 3.6.

Derivation of vortex interaction potentials From Eq. (3.1) one can derive interaction potentials of vortices in the London limit. First, rewrite the free energy (3.1) into the form of neutral (3.6a) and charged modes (3.6b):

1 ψ 2 ψ 2 f = | 1| | 2| ( (θ(1) θ(2)))2 (3.6a) 2 ψ 2 + ψ 2 ∇ − | 1| | 2| 2 1 1 2 (1) 2 (2) 2 2 + 2 2 + ν ψ1 θ + ψ2 θ + e( ψ1 + ψ2 )A 2 ψ1 + ψ2 | | ∇ | | ∇ | | | | | | | | (3.6b) 1 + ( A)2 . (3.6c) 2 ∇ × The free energy associated with excitations in A is 1 1 f = (keA)2 + ( A)2 , (3.7) 2 2 ∇ ×

2 2 2 g 2gg0 2 rdr (ke) g + + + (g0) = drL[r, g, g0], (3.9) r2 r Z Z which presents a variational problem to nd the g(r) that minimizes the total free- energy functional. The variational terms are

∂L 2 1 = 2 (ke) r + g + 2g0, (3.10a) ∂g r ∂ ∂L = 2g0 + 2(rg0)0, (3.10b) ∂r ∂g0 20 CHAPTER 3. VORTEX STRUCTURE FORMATION

Figure 3.6: Schematic texture of n = Ψ†σΨ/Ψ†Ψ for a composite vortex with core- splitting increasing downwards in the gure, with no core-splitting in the upper-most plot. Red arrows point up, blue arrows point down, and green arrows point in-plane. 3.3. VORTICES WITH DIPOLAR FORCES 21

1.0 1.620 − 0.9

0.8 1.625 − 0.7

0.6 u 1.630 − M 0.5

0.4 1.635 − 0.3

1.640 0.2 − 0.1 0.002 0.004 0.006 0.008 0.010 T

2 2 Figure 3.7: Melting of a dipolar lattice for ψ1 + ψ2 = 4.125, m = 2.1429, e = 0.8, ν = 0.025, N = 198 and density ρ = 0.04|. Red| and| | blue dots correspond to vortices from dierent bands. On the left axis is the internal energy per particle and the quantity M on the right (red) axis is a measure of the alignment of the dipoles.

140

40 120 35

100 30

80 25

20 60

15 40 10

20 5

0 0 0 20 40 60 80 100 120 140 0 5 10 15 20 25 30 35 40

(a) ρ = 0.01 (b) ρ = 0.1 Figure 3.8: Loops and checkerboard lattices occur for low and high density ρ = N/L2 respectively, with ψ 2 + ψ 2 = 2, m = 1, e = 0.6, ν = 2 and N = 198. | 1| | 2| 22 CHAPTER 3. VORTEX STRUCTURE FORMATION which with the Euler-Lagrange equation gives

2 1 (ke) r + r− g (rg0)0 = 0. (3.11) − The physical solution is a rst order modied Bessel function of the second kind g(r) K (ker), (3.12) ∝ 1 from which we can identify the penetration depth λ = 1/ke, 1 λ = . (3.13) e ( ψ 2 + ψ 2) (1 + ν ( ψ 2 + ψ 2)) | 1| | 2| | 1| | 2| Integrating the magneticp eld over the surface of the system at a radius R yields a ux φ

φ = A dl (3.14a) · I R = constant 2πRK , (3.14b) · 1 λ where we can determine the constant by assumption of large λ and limx 0 xK1(x) = 1. Using (d/dx)xK (x) = xK (x) we then have → 1 − 0 φ r B(r) = K . (3.15) 2πλ2 0 λ Assuming an a-type vortex at ra and a b-type vortex at rb we get the energy contribution

charged 1 φ r r φ r r E = dS a K | − a| + b K | − b| (3.16a) ab 2 2πλ2 0 λ 2πλ2 0 λ × Z (φ δ(r r ) + φ δ(r r )) (3.16b) × a − a b − b φ φ r r = a b K | a − b| + self-energy terms. (3.16c) 2πλ2 0 λ Each vortex carries a fraction of the ux quantum φ0 = 2π/e which is given by 2 2 2 φa = φ0 ψa /( ψ1 + ψ2 ). Writing r = ra rb the interactions due to the charged mode| | take| | the| form| | − |

charged 2 2 1 r (3.17a) E12 = 2π ψ1 ψ2 2 2 + ν K0 , | | | | ψ1 + ψ2 λ | | | | charged 4 1 r (3.17b) Eaa = 2π ψa 2 2 + ν K0 . | | ψ1 + ψ2 λ | | | | The interaction energy due to the neutral mode is obtained by assuming that θ(a) = θ where θ is the polar angle, which gives 1 θ(a) = θ = θˆ = zˆ ln( r r ). (3.18) ∇ ∇ ρ × ∇ | − a| 3.3. VORTICES WITH DIPOLAR FORCES 23

2.06 −

2.08 −

2.10 −

2.12 u −

2.14 −

2.16 −

2.18 −

0.02 0.04 0.06 0.08 0.10 T

2 2 Figure 3.9: Melting of the loop phase, with ψ1 + ψ2 = 4.125, m = 2.1429, e = 0.8, ν = 1.0, N = 198 and ρ = 0.001.| In the| highest| | temperature phase the vortices are still bound in pairs, and for even higher temperatures this pairing is also broken.

The interaction energy due to the neutral mode for two vortices a = b is calculated with the divergence theorem3 and using 2 ln r r = 2πδ( r r6 ) as follows ∇ | − a| | − a|

2 2 neutral 1 ψ1 ψ2 (1) (2) 2 (3.19a) Ea=b = | | | | dS( (θ θ )) 6 2 ψ 2 + ψ 2 ∇ − | 1| | 2| Z ψ 2 ψ 2 = | 1| | 2| dS θ(1) θ(2) + self-energy terms (3.19b) − ψ 2 + ψ 2 ∇ · ∇ | 1| | 2| Z ψ 2 ψ 2 = | 1| | 2| dSθ(1) 2θ(2) (3.19c) ψ 2 + ψ 2 · ∇ | 1| | 2| Z ψ 2 ψ 2 = | 1| | 2| dS ln( r r ) 2πδ(r r ) (3.19d) ψ 2 + ψ 2 | − 1| · − 2 | 1| | 2| Z ψ 2 ψ 2 = 2π | 1| | 2| ln( r r ). (3.19e) ψ 2 + ψ 2 | 1 − 2| | 1| | 2|

For two vortices 1 and 2, in the same band , we have (a) (a) (a) while a θ = θ1 + θ2 θ(b) = 0 and the calculation is identical to the one above in Eq. (3.19) but with the

3R H with (1) (2). S dS∇ · F = C nˆds · F = 0 F = θ ∇θ 24 CHAPTER 3. VORTEX STRUCTURE FORMATION sign reversed. The interaction energies due to the neutral mode are thus

2 2 neutral ψ1 ψ2 (3.20a) E12 = 2π | 2| | | 2 ln(r), ψ1 + ψ2 | | 2| | 2 neutral ψ ψ E = 2π | 1| | 2| ln(r). (3.20b) aa − ψ 2 + ψ 2 | 1| | 2| with r = r r . Putting together (3.17) and (3.20) gives | 1 − 2| r r E = ln + wK , (3.21a) 12 R 0 λ r r E = ln + mwK , (3.21b) 11 − R 0 λ r w r E = ln + K , (3.21c) 22 − R m 0 λ 2 2 2 2 where w = 1 + ν( ψ1 + ψ2 ) and m = ψ1 / ψ2 . The potentials| (3.21)| allow| | for a variety| of| structures| | like checkerboard lattices, octagonal structures and rings, which also arise in the Ginzburg-Landau setting. For small core-splitting, the vortices form a dipolar lattice as shown in Fig. 3.7. When the core-splitting is large, loops and strings are found at low densities, and square lattices for high densities, see Figs. 3.8 and 3.9. Chapter 4

Phase transitions in multicomponent London superconductors

The thermal phase transitions of a multicomponent London superconductor are studied in paper 4. The main result is that the phase transition can be rst-order for small Josephson couplings and continuous for large Josephson couplings, and the scaling arguments for this is contained in the paper. The following sections contain details of the derivation of length scales and discretization of the model, and is concluded with results from Monte Carlo calculations.

4.1 Two-component London superconductivity

The limit of constant amplitudes ψa in the Ginzburg-Landau free energy (2.1) is called the London limit, where we| may| write for two components a = 1, 2 that iθ(a)(r) 2 ψa(r) = ψa e with ψa = αa/βa = a positive constant. The gradient | | | | − (a) terms of the Ginzburg-Landau free energy then becomes ψa = ψai θ so that 2 (a) 2 2 (a) ∇ 2 ∇ ( + ieA)ψa = ψai θ + ieAψa = ψa ( θ + eA) . The free-energy density| ∇ is then| ∇ | | ∇

1 f = ψ 2( θ(a) + A)2 (4.1a) 2| a| ∇ a=1,2 X 1 + ( A)2 (4.1b) 2e2 ∇ × + η ψ ψ cos θ(1) θ(2) , (4.1c) | 1|| 2| − where the last term is the Josephson interaction. In Eq. 4.1 two characteristic length scales can be identied, namely the Joseph- son length ξJ , and the magnetic penetration depth λ. We shall below derive formulas for these scales in terms of the dening parameters of the model (the

25 CHAPTER 4. PHASE TRANSITIONS IN MULTICOMPONENT LONDON 26 SUPERCONDUCTORS ground-state amplitudes ψ1 , ψ2 , charge coupling e and Josephson coupling η), and illuminate our model| by| writing| | it in terms of the two length scales 1 f = ψ 2( θ(a) + A)2 (4.2a) 2 | a| ∇ a=1,2 X 1 + λ2( ψ 2 + ψ 2)( A)2 (4.2b) 2 | 1| | 2| ∇ × 1 ψ 2 ψ 2 + | 1| | 2| cos(θ(1) θ(2)), (4.2c) ξ2 ψ 2 + ψ 2 − J | 1| | 2| where λ is the penetration depth and ξJ the Josephson length. The penetration depth λ is obtained by letting ν = 0 in Eq. (3.13) giving 1 λ = . (4.3) e ψ 2 + ψ 2 | 1| | 2| The Josephson length ξJ is derived below.p

4.2 Derivation of the Josephson length ξJ We follow [78] and imagine that we are in the ground state where there are no phase gradients and the phases are locked. Assume that η is positive and write θ(1) θ(2) = π + δ(r), (4.4) − where the total free energy due to the phase dierence is minimized when δ(r) = 0 everywhere. We have (θ(1) θ(2)) = δ(r) and cos(θ(1) θ(2)) = cos(π + δ(r)) = cos(δ(r)). Imagine that∇ the− phase dierence∇ is perturbed− slightly from its ground − state value at x = 0. By enforcing that δ(0) = δ0, the response of the system can be found by calculating the prole δ(x) which minimizes the total free energy. The gradient term opposes any spatial changes in the phase dierence and the phase locking term favors immediate nullication of any nite δ(x). The δ(x) which optimizes these competing tendencies is found by minimizing the functional

2 2 1 ψ1 ψ2 2 dx | | | | (δ0(x)) η ψ ψ cos(δ(x)) = dxL[x, δ, δ0]. (4.5) 2 ψ 2 + ψ 2 − | 1|| 2| Z | 1| | 2| Z 2 2 2 We have that ∂L/∂δ = η ψ1 ψ2 sin(δ) and (d/dx)(∂L/∂δ0) = δ00 ψ1 ψ2 /( ψ1 + ψ 2) which after expanding| || the| sine gives the Euler-Lagrange equation| | | | | | | 2| 2 2 η( ψ1 + ψ2 ) δ00(x) | | | | δ(x) = 0, (4.6) − ψ ψ | 1|| 2| with the physical solution

x δ(x) = δ exp , (4.7) 0 −ξ J 4.3. DISCRETIZATION OF THE TWO-BAND LONDON FREE ENERGY 27 where the characteristic length scale associated with the system's return to the ground state is the Josephson length

ψ ψ ξ = | 1|| 2| . (4.8) J η( ψ 2 + ψ 2) s | 1| | 2|

4.3 Discretization of the two-band London free energy

We begin by describing the notation used in the discretization. We dene a square grid in three dimensions with isotropic grid spacing h. All vectors are Cartesian vectors with C = (Cx,Cy,Cz) = Cxxˆ + Cxyˆ + Cxzˆ. A site is indexed by i so that where means sum over all sites and denotes the discrete H = i(H)i i i (H)i value of H at the site i, the discrete value of the vector C on site i is denoted (C)i, P P the µ-component of a vector C is denoted by [C]µ = Cµ, the µ-component of the vector (C)i is denoted ([C]µ)i = Ciµ so that we can write (C)i = (Cix,Ciy,Ciz) 2 2 and the value of a scalar C on the lattice site i is likewise denoted (C )i. Since a vector squared is the sum of its squared components we can then write C2 = 2 2 2 2 2 , C C = CxCx + CyCy + CzCz = Cx + Cy + Cz = µ=x,y,z Cµ = µ=x,y,z[C]µ and· for brevity denote . Thus we can write the value of the square µ=x,y,z = µ of the vector C at site i as P P P P

2 2 2 2 (4.9) (C )i = [C]µ = ([C]µ)i = Ciµ. µ ! µ µ X i X X On every site we have ve variables: the two phases (1), (2) and the three com- θi θi ponents of the vector (A)i: Aix, Aiy and Aiz. The Josephson term is trivially discretized by

1 1 cos(θ(1) θ(2)) = cos(θ(1) θ(2)) (4.10a) ξ2 − ξ2 i − i J i J 1 h 2 = cos(θ(1) θ(2)), (4.10b) h2 ξ i − i J where the last step is performed for reason that will become apparent below. The gradient term of the kinetic energy density is discretized straightforwardly by the nite dierence approximation

θ(a) θ(a) ([ θ] ) = i+ˆµ − i . (4.11) ∇ µ i h We apply this rewriting on our Hamiltonian terms to get the discrete kinetic energy CHAPTER 4. PHASE TRANSITIONS IN MULTICOMPONENT LONDON 28 SUPERCONDUCTORS of component a on site i as

(a) 2 (a) 2 (4.12a) ( θ + A) = [ θ + A]µ ∇ i ∇ µ ! X i = (([ θ(a)] ) + ([A] ) )2 (4.12b) ∇ µ i µ i µ X 2 θ(a) θ(a) = i+ˆµ − i + A (4.12c) h iµ µ ! X 1 = (θ(a) θ(a) + hA )2, (4.12d) h2 i+ˆµ − i iµ µ X To discretize the magnetic eld energy density term is used the denition of the curl of a vector eld C at some point r as the limit ∇ × 1 ( C) nˆ = lim C dr, (4.13) ∇ × · S 0 area(S) · → I∂S where S is some small surface around r with a unit normal nˆ, and ∂S is the border of S, with the direction of the curve ∂S dened by the right-hand rule. On our square lattice δS is a square plaquette which we denote , and the area 2 is h , and ∂S is the curve connected by the links of the plaquette , which gives 2 ( C) nˆ h− C dr. The unit normal is nˆ = xˆ, nˆ = yˆ or nˆ = zˆ. We ∇ × · ≈ · introduce the notation x0 = y, y0 = z, z0 = x, and x00 = y0 = z etc so that we can write the discrete latticeH curl on site i in the plane with normal µˆ as (refer to Fig. 4.1)

1 (( C) µˆ) = C dr (4.14a) ∇ × · i h2 · I i 1 i+ˆµ0 1 i+ˆµ0+ˆµ00 = C µˆ0dµ0 + C µˆ00dµ00 (4.14b) h2 · h2 ˆ · Zi Zi+µ0 1 i+ˆµ00 1 i + C µˆ0dµ0 + C µˆ00dµ00 (4.14c) h2 · h2 · Zi+ˆµ0+ˆµ00 Zi+ˆµ00 1 = (hC + hC hC hC ). (4.14d) h2 iµ0 i+ˆµ0,µ00 − i+ˆµ00,µ0 − iµ00 In the last line we can identify the discrete lattice curl with unit grid spacing which we denote

[∆ C] C + C C C . (4.15) × iµ ≡ iµ0 i+ˆµ0,µ00 − i+ˆµ00,µ0 − iµ00 4.4. TWO-COMPONENT LATTICE LONDON SUPERCONDUCTIVITY 29

We then have that 2 and can write the magnetic eld (( A) µˆ)i = [∆ hA]iµ/h energy density as ∇ × · ×

λ2( A)2 = λ2 [ A]2 (4.16a) ∇ × i ∇ × µ µ ! X i = λ2 (( A µˆ) )2 (4.16b) ∇ × · i µ X 1 2 = λ2 [∆ hA] (4.16c) h2 × iµ µ X 1 λ 2 = [∆ hA]2 . (4.16d) h2 h × iµ µ X

Ci-+ˆµ00,µ0 i +µ ˆ0 +µ ˆ00 i +µ ˆ s 00 s

6 6 Ciµ00 Ci+ˆµ0,µ00 µ00 - 6- µ0 i C i +µ ˆ © siµ0 s 0 µ Figure 4.1: Lattice curl with µˆ normal to the plaquette.

4.4 Two-component lattice London superconductivity

Absorb the grid spacing into the length scales by

hA A (4.17a) → λ/h λ (4.17b) → ξ /h ξ , (4.17c) J → J the Hamiltonian at i is then 2 2 (a) (a) 2 (H) =h− ψ (θ θ + A ) /2 (4.18a) i | a| i+ˆµ − i iµ µ a X X 2 2 2 2 2 + h− λ ( ψ + ψ ) [∆ A] /2 (4.18b) | 1| | 2| × iµ µ X 2 2 2 1 ψ1 ψ2 (1) (2) + h− | | | | cos(θ θ ). (4.18c) ξ2 ψ 2 + ψ 2 i − i J | 1| | 2| CHAPTER 4. PHASE TRANSITIONS IN MULTICOMPONENT LONDON 30 SUPERCONDUCTORS We can absorb the grid spacing into the Hamiltonian with

h2(H) (H) , (4.19) i → i and rewriting the kinetic energy term into xy-terms (cos(x) 1 x2/2 so that x2 2 cos(x) + 2), we obtain the nal expression ≈ − ≈ −

H = ψ 2 cos(θ(a) θ(a) + A ) (4.20a) −| a| i+ˆµ − i iµ i " µ a X X X 1 + λ2( ψ 2 + ψ 2) [∆ A]2 + (4.20b) 2 | 1| | 2| × iµ µ X 1 ψ 2 ψ 2 + | 1| | 2| cos(θ(1) θ(2)) . (4.20c) ξ2 ψ 2 + ψ 2 i − i J | 1| | 2| # The form of Eq. (4.20) is suitable for numerical computations. 2 Paper 4 contains a nite-size scaling analysis for the case with twin bands ψ1 = 2 | | ψ2 = 1 which concludes that the phase transition is rst-order for small Josephson couplings,| | and continuous for large Josephson couplings. The reason for the rst- order phase transition is argued to be because of a uctuations-induced attraction between vortex lines, leading to phase separation at the transition. The vortex attraction for a two-dimensional cross-section of vortex lines may be seen from the vortex interaction potentials Eq. (3.21) by letting w = m = 1 and considering the total energy of a pair of composite vortices. Without core-splitting, the composite vortices repel, but the energetic cost of core-splitting is small, and is thus not unlikely to happen at a nite temperature. One may thus nd a situa- tion where two composite vortices form two dipoles which are mutually long-range attractive. This point-particle analysis does not account for a presence of a - nite Josephson coupling. A nite Josephson coupling opposes the core-splitting, as may be seen in Fig. 4.2, which shows snapshots from simulations of the three- dimensional lattice theory (4.20) of vortex phases where an external magnetic eld has been applied1 in the vertical axis of the system. The displayed quantity is the interpolated isosurface of the kinetic energy density. A suciently strong Joseph- son coupling would thus remove the uctuation-induced attraction, making the transition continuous. We include here also results for a nite-size system when varying the dierence 2 in the phase stinesses ψa as well as varying the Josephson interaction η. In Fig. 4.3, one of the phase| stinesses| is kept constant and the other is varied until they are equal. For a big dierence, two peaks are visible in the heat capacity C,

1An external magnetic eld is applied along the z-axis by decomposing the gauge eld into a uctuating part and an applied part, A = Auct +Aapp, where Aapp = 2πfyxˆ so that the corresponding magnetic induction Bapp = ∇×Aapp = −2πfzˆ, where the parameter f determines the ground-state vortex density in the xy-plane. Note that because of periodic boundary conditions, special care must be taken when choosing f and the size of the lattice in the y-direction. 4.4. TWO-COMPONENT LATTICE LONDON SUPERCONDUCTIVITY 31

Figure 4.2: Top and perspective views of low-temperature vortex phases for a two- component system with an applied magnetic eld. The upper row is for weak Josephson interaction, and the lower row is with a strong Josephson interaction. CHAPTER 4. PHASE TRANSITIONS IN MULTICOMPONENT LONDON 32 SUPERCONDUCTORS

8 4 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5

T 2 Figure 4.3: For a big dierence between the phase stinesses ψa there are two visible peaks. For clarity, each curve is displaced vertically from the| underlying| curve. 2 When the dierence is decreased by increasing ψ2 , the peaks merge into one more pronounced peak. Here ψ 2 = 1.0, e = 2.2, η =| 0|and ψ 2 varies. | 1| | 2| which then merges into one pronounced peak when the dierence is small. In Fig. 4.4, the phase stiness dierence is kept constant and the Josephson coupling is increased, causing the two visible peaks to merge into one peak which is about the same height as the previous peaks. 4.4. TWO-COMPONENT LATTICE LONDON SUPERCONDUCTIVITY 33

0.50

0.45

0.40

0.35

C 0.30

0.25

0.20

0.15

0.10

0.05 3 2 1 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0

T Figure 4.4: A Josephson coupling η causes a re-merging of peaks in the heat capacity. Here ψ 2 = 0.8, ψ 2 = 1.2, e = 1, L = 32 and η varies. | 1| | 2|

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