New Scaling Method for the Kosterlitz-Thouless Transition

Transcription

1 Master of Science Thesis New Scaling Method for the Kosterlitz-Thouless Transition Hannes Lindström Department of Theoretical Physics, School of Engineering Sciences Royal Institute of Technology, SE Stockholm, Sweden Stockholm, Sweden 2016

2 Typeset in L A TEX Examensarbete inom ämnet teoretisk fysik för avläggande av civilingenjörsexamen inom utbildningsprogrammet 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 2016:76 ISSN X ISRN KTH/FYS/--16:76 SE c Hannes Lindström, December 2016 Printed in Sweden by Universitetsservice US AB, Stockholm December 2016

3 Abstract The Kosterlitz-Thouless transition is studied from the perspective of two different non-neutral 2D Coulomb gas models by Monte Carlo simulations using a Metropolis- Hastings algorithm. A new approach of allowing charge fluctuations allows us to compare the results to scaling formulas linked to the magnetic permeability that are well suited to uncover the critical behavior. We focus on pinpointing the transition temperature T c for the models by least squares optimization of the inverse squared net vorticity 1/m 2 to its expected critical and low temperature forms. The parameters varied include the system size L, the chemical potential µ and a model-specific parameter λ related to the magnetic susceptibility. We find that the method accurately portrays the phase transition and can be used to calculate the Kosterlitz-Thouless temperature and related quantities in a direct manner. The results are roughly the same for the two models and different values of λ, but there are some differences in the efficiency of the simulations. We also outline some future applications of the method and how a connection to experiments can be made. Key words: Kosterlitz-Thouless transition, 2D Coulomb gas, charge fluctuations, magnetic permeability, renormalization group, finite-size scaling, Monte Carlo simulation, Metropolis-Hastings algorithm. iii

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5 Preface This thesis is the result of my Master s degree project at the Department of Theoretical Physics of the Royal Institute of Technology, which lasted throughout the spring and fall of The project concerns Monte Carlo simulations of the nonneutral 2D Coulomb gas near the critical temperature and comparing the results to formulas obtained from a scaling argument. The thesis is divided into five chapters. Chapter 1 provides an overview of the Kosterlitz-Thouless transition in two-dimensional systems and introduces the problem at hand. Chapter 2 is dedicated to deriving the Coulomb gas model from Ginzburg-Landau theory and introducing two proposed non-neutral models. The concepts are more thoroughly explained in Ch. 3, which describes the phase transition and its treatment with renormalization group theory and finite-size scaling analysis. Chapter 4 is about the Monte Carlo method used and also gives an account of the technical details of the simulation. The results of the simulation and data analysis are presented in Ch. 5. Finally, Ch. 6 summarizes the findings, attempts to make some general conclusions and concludes with an outlook for future work. v

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7 Acknowledgements I would like to thank my supervisor Prof. Mats Wallin and additional advisor Assoc. Prof. Jack Lidmar for giving me the opportunity to work on this project. Their constant availability to answer my questions and provide me with feedback has been invaluable. Without their patience and positive attitude, this thesis could not have been finished. My thanks also go out to my friends and family for their support. The years I have spent as a student have been tough at times, but they have always been there to help me pull through. vii

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9 Contents Abstract Preface Contents iii v ix 1 Introduction 1 2 2D Coulomb Gas Ginzburg-Landau Theory Spin Waves Flux Quantization Coulomb Gas Picture Regularization Magnetic Field Fluctuations Model A Model B Kosterlitz-Thouless Transition Nature of the Transition Thermodynamics Renormalization Group Finite-Size Scaling Scaling Formulas Simulation Method Monte Carlo Method Metropolis-Hastings Algorithm Main Program Pre- and Post-Processing Implementation ix

10 x Contents 5 Results Choice of Parameters Potential Configuration Examples Basic Quantities Critical Point and Scaling Properties Efficiency Summary, Conclusions and Outlook 39 Bibliography 40

11 Chapter 1 Introduction One of the great breakthroughs of classical physics was the development of electrostatics by Charles Augustin de Coulomb in the late 18th century. The law of Coulomb allowed physicists for the first time to make accurate predictions concerning the attraction and repulsion of electric charges. Because of the simplicity of the mathematical formulation, it is not surprising that the model would be rediscovered in all kinds of different fields. Indeed, the old gravitation theory of Newton is just another instance of the Coulomb gas. For its historical importance and straightforward interpretation, the model is also well suited to picturesque explanations of more complicated phenomena. A current cause for study of the Coulomb gas is connected to a phase transition known as the Kosterlitz-Thouless (KT) transition, discovered in the early 1970s by Berezinskii [1], Kosterlitz and Thouless [2]. The KT transition can be found in two-dimensional systems containing thermal excitations in the form of vortices, which includes thin films of superconductors and superfluids. It is characterized by a shift between a low temperature phase of bound vortex-antivortex pairs and a high temperature phase that contains free vortices. The analogy to the Coulomb gas model is that vortices are represented by charged particles, with their winding numbers acting as charge quanta [3]. Under certain conditions, these virtual particles interact with one another according to the Poisson equation of electrostatics. The comparatively high complexity of the model lies in part with the statistical treatment of a large number of particles and in part with the logarithmic potential that follows from the restriction to two dimensions. Recent breakthroughs have been made in the construction of thin materials in general, such as the single-layered graphene [4]. In particular, the feasibility of thin superconductors has contributed to the importance of better understanding the KT 1

12 2 Chapter 1. Introduction transition. Not only are there a wide range of possible applications for such materials, but they also serve as an ideal setting for experiments at a single-atom level that can test out theories and simulation methods of condensed matter physics. The great potential of these thin films is reflected in the decision of the Royal Swedish Academy of Sciences to award the Nobel Prize of 2016 to Kosterlitz, Thouless and Haldane, in part due to the discovery of the KT transition. The KT transition has been subject to a great deal of experimental and theoretical studies, but there still remain some open questions. The focus of this thesis lies in testing a new scaling method for the Coulomb gas based on a finite-size scaling argument made by Andersson and Lidmar [5]. Most previous simulation studies examine the neutral Coulomb gas, where the charges only occur in dipole pairs. We here take a different approach and study the fluctuations of single charges by altering the original model to allow for non-neutral configurations. This leads to interesting physics and a new promising method for future studies of the KT transition.

13 Chapter 2 2D Coulomb Gas In this chapter, we give a brief overview of the 2D Coulomb gas, beginning with a derivation from the Ginzburg-Landau theory of superconductivity. We show the existence of vortices carrying quantized flux and the presence of phases containing these. The definition of the Coulomb gas and some immediate properties are then presented. Finally, we describe the discretization of the model and the two proposed modifications that allow a non-zero net vorticity. 2.1 Ginzburg-Landau Theory The Ginzburg-Landau theory of superconductivity [6] is described by the free energy functional [ F = F n + d d r α Ψ(r) 2 + β 2 Ψ(r) ] ( i 2eA)Ψ(r) 2 + B2, (2.1) 4m e 2µ 0 where F n is the normal phase free energy, α and β are phenomenological parameters, m e is the electron mass, e is the electron charge and B = A is the magnetic field. The Cooper pair density is given by Ψ 2, where Ψ is the order parameter of the system. By introducing a phase θ, we get the decomposition Ψ(r) = Ψ(r) e iθ(r). (2.2) Minimizing the free energy of Eq. (2.1) with respect to the order parameter leads to the famous Ginzburg-Landau equations αψ + β Ψ 2 Ψ + 1 ( i 2eA) 2 Ψ = 0 4m e (2.3) J = B = e Ψ 2 ( θ 2eA), µ 0 m e (2.4) 3

14 4 Chapter 2. 2D Coulomb Gas where J is the superconducting current density. We will ignore magnetic fields in the derivation that follows, meaning that B = A = J = 0, and reintroduce them in Ch Spin Waves In mean field theory, the superconducting order parameter is a constant and Eqs. (2.3) and (2.4) give the solutions α/β and 0. By extension from the Landau theory of phase transitions [7], it must be that α/β is only negative for temperatures T below the critical temperature T c. Hence, the mean field solution is { α/β if T < Tc Ψ 0 =. (2.5) 0 if T > T c Mean field theory neglects the importance of fluctuations in the system, which is not always a valid assumption. Indeed, the Ginzburg criterion [8] tells us that the mean field solution is accurate to great precision only near the critical point for a dimensionality d 4. The fluctuations of the order parameter become increasingly important for the physics of the phase transition as d decreases. To better account for these fluctuations, suppose that the order parameter has the full form Ψ(r) = Ψ 0 e iθ(r), (2.6) where Ψ 0 is the mean field solution. Up to a constant, the free energy of Eq. (2.1) is then F = const. + J 0 d d r( θ(r)) 2, (2.7) 2 where the (bare) superfluid stiffness is defined by J 0 = 2 Ψ 0 2 4m e. (2.8) Assuming small fluctuations of θ, the integration of phase field configurations in the partition function Z = Dθe F/(k BT ) can be extended to the real line. The system described by Eq. (2.7) then leads to a Gaussian integral for the expectation value of the order parameter. Using equipartition and assuming translational invariance, this leads to the correlation function Ψ(r)Ψ (0) r k B T 2πJ 0 (2.9) The correlation function has an algebraic dependence on the distance r, characterizing a quasi-long range order. The assumption of smoothly variating θ corresponds to so-called spin wave fluctuations and is typically valid for low temperatures. For high temperatures, ignoring the periodic nature of θ does not accurately reproduce the behavior of the system.

15 2.2. Coulomb Gas Picture Flux Quantization There are two length scales that characterize superconductors in Ginzburg-Landau theory. The first of these is obtained by noting that the quotient of α and /4m e has the dimension of a squared length according to Eq. (2.3). We can therefore define the superconducting coherence length as ξ = 2 4m e α. (2.10) The second length scale is defined from Eq. (2.4) in a similar manner and is called the London penetration depth, me λ = 2µ 0 e 2 Ψ 2. (2.11) The quotient κ = λ/ξ is known as the Ginzburg-Landau parameter. Superconductors for which κ < 1/2 are called type-i and those for which κ > 1/2 are called type-ii. Ginzburg and Landau showed that the energy of an interface between a normal and a superconductive region can only be negative for type-ii superconductors. Hence, such materials can display a mixed phase for which there are both normal and superconductive regions. These normal regions are known as vortices. Consider now a type-ii superconductor with a spatially fluctuating phase θ(r). According to Stokes law and Eq. (2.4), the magnetic flux through a vortex, where the current vanishes, is Φ = B ds = A dr = θ(r) dr. (2.12) 2e The order parameter is single valued, so its phase must change by a multiple of 2π along the contour of the region. For some q Z, it follows that where the flux quantum is defined as Φ = qφ 0, (2.13) Φ 0 = h 2e Tm 2. (2.14) This shows that vortices carry quantized magnetic flux. The index q is known as the winding number or vorticity of the vortex. 2.2 Coulomb Gas Picture Spin wave fluctuations are not always sufficient to describe a system like the 2D type-ii superconductor, because the presence of vortices is not accounted for. To

16 6 Chapter 2. 2D Coulomb Gas understand the role these flux quanta play, it is useful to rewrite the system in terms of a classical Coulomb gas. In doing so, we will assume from the start that the vortices are located at discrete points, although the essence of the argument also holds for a continuous density of vortices. By using Stokes law and setting the vortex normals along the z-axis, the defining flux relations of Eqs. (2.12) and (2.13) show that θ(r) = 2πẑ r q r δ(r r ), (2.15) where the system vortices are located at positions r and have winding numbers q r. Only the part of θ with a non-zero curl contributes to the flux quantization and it is therefore useful to decompose the vector as θ(r) = φ(r) + (ẑψ(r)), (2.16) where φ is curl free. The quantization condition of Eq. (2.15) then reads 2 ψ(r) = 2π r q r δ(r r ). (2.17) The system Hamiltonian corresponding to Eq. (2.7) can with this decomposition be written in two dimensions as H tot = J 0 d 2 r [ ( φ(r)) 2 + ( (ẑψ(r))) 2 2 φ(r) (ẑψ(r)) ]. (2.18) 2 The cross term of Eq. (2.18) cancels through partial integration because the phase factor φ (ẑψ(r)) vanishes along the system contour and (ẑψ(r)) is divergence free. Furthermore, the first term corresponds to the model mentioned earlier for spin waves. Since the term is Gaussian, its free energy is analytic for any temperature and it can therefore not be the cause of a phase transition. The spin waves completely decouple from the rest of the system, so it will be sufficient to only consider the vortex contribution to study the critical phenomena. By partial integration and the use of Eq. (2.17), the vortex contribution to Eq. (2.18) simplifies to H 0 = πj 0 r, r q r q r V (r r ), (2.19) where the sum runs over all pairs of vortices and the potential V (r) is the solution to the Poisson equation 2 V (r) = 2πδ(r). (2.20) Equations (2.19) and (2.20) define the Coulomb gas. The model is exactly that of a system of charged particles according to electrostatics.

17 2.3. Regularization 7 The creation of vortices involves thermal excitations that are not part of the electrostatic potential derived. The process can be included with a local contribution of a constant E c known as the core energy, which is added to the model with a chemical potential µ. We introduce this according to E c = µl 2 and make the full Hamiltonian H = H 0 µ N, (2.21) L2 where N is the total number of vortices. The choice to make µ scale with the size of the system does not follow the usual convention or understanding of a chemical potential, which can lead to some unexpected behavior. We will in the following almost exclusively focus on the case where µ = 0. We must also account for the spatial extension of the charges by some lower cutoff in length a and for the size of the system by an upper cutoff L. The potential of self-interaction according to Eq. (2.20) with these length scales in mind is V (0) = 2π d 2 k e ik 0 2π/a ( ) dk L (2π) 2 k 2 = 2π/L k = ln. (2.22) a The divergence for a 0 indicates that the model must be regularized, e.g. by being transferred to a lattice. The potential also diverges for N and this may be avoided by introducing additional constraints. One option is to require charge neutrality, which nullifies all self-interactions in Eq. (2.19). We will not force charge neutrality in what follows. This makes it possible for the system to have a non-zero net vorticity m = r q r. (2.23) 2.3 Regularization The model regularization adopted here is to use an L L square lattice with a cell width a. To approximate the behavior of a large system in a direct manner, we require the lattice to have periodic boundaries. On the periodic lattice, the potential of Eq. (2.20) can be written as a Fourier series according to V (r) = 2π L 2 V (k) cos(r k) (2.24) V (k) = k cos(k x ) 2 cos(k y ), (2.25)

18 8 Chapter 2. 2D Coulomb Gas where the positions r = (r x, r y ) are all pairs of integers between 0 and L 1 and k = (k x, k y ) are the corresponding phase space combinations, k x, y = 2πr x, y /L. To facilitate numeric treatment, we also introduce the local function U(r) = r q r V (r r ) (2.26) The system Hamiltonian of Eqs. (2.19) and (2.21) then takes the form H = πj 0 r q r U(r) µ N. (2.27) L2 2.4 Magnetic Field Fluctuations Since we are considering a non-neutral system, Eq. (2.13) tells us that there will be a total flux mφ 0 that is potentially non-zero. The flux corresponds to a magnetic field B = A that we neglected in deriving the Hamiltonian of Eq. (2.19). To reintroduce the magnetic field, we add the last term of Eq. (2.1) to Eq. (2.21) and modify the remaining expression to be gauge invariant. The only factor that needs to be changed is the charge q, since θ 2eA is an invariant and q θ. Hence, the Hamiltonian with a magnetic field is H B = πj 0 (q r B(r))(q r B(r ))V (r r ) µ L 2 N + λ L B(r) 2. (2.28) 2 r, r r The constant λ L is the magnetic susceptibility (for a system of size L) and measures the degree of magnetization in response to the magnetic field. Since we do not consider any external magnetic fields, we will allow all variations of B and can remove their explicit mention in the Hamiltonian by integrating out their contribution to the partition function as DBe βh B. This is most easily done using the phase space expression H B = πj 0 2L 2 V (k) q k B(k) 2 λ2 L 2L 2 B(k) 2. (2.29) k The result is exactly the Hamiltonian of Eq. (2.27), but with the phase space potential of Eq. (2.25) slightly modified depending on the local properties of the field. k

19 2.4. Magnetic Field Fluctuations Model A In model A, we assume the magnetic field B is constant over the entire system. The resulting modified phase space potential is λ 2 L if k = 0 V (k) = 1 if k 0. (2.30) 4 2 cos(k x ) 2 cos(k y ) The modification does the bare minimum of lifting the self-interaction divergence in Eq. (2.22) and makes the magnetic fluctuations uniform across the system in a sense Model B Model B allows B to have any combination of values at the lattice sites, which leads to 1 V (k) = 4 2 cos(k x ) 2 cos(k y ) + λ 2. (2.31) L This also corresponds to using a modified Poisson equation ( 2 λ 2 L )V (r) = 2πδ(r), (2.32) which means that λ L fits the definition of a screening length.

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21 Chapter 3 Kosterlitz-Thouless Transition This chapter introduces the basics of the Kosterlitz-Thouless transition. We begin with a simple argument that shows why the transition should occur and its general classification. We then present some thermodynamic properties that will be of use in the data analysis as well as the major results of renormalization group theory. The last part of the chapter is dedicated to introducing a finite-size scaling analysis of the magnetic permeability that will be exploited to interpret the results. 3.1 Nature of the Transition In the early theoretical study of phase transitions, their only known cause was spontaneous symmetry breaking. To this day, the vast majority of known critical behaviors can be explained by such a mechanism. A system with dimensionality d 2 and short-range interactions, however, can not exhibit a continuous phase transition within this framework. This is because the continuous symmetries that they have simply cannot be spontaneously broken, as rigorously stated by the Mermin-Wagner theorem [9]. The 2D Coulomb gas is an example of such a system. The net vorticity m averages to 0 even in the algebraically ordered phase and there is thus no broken symmetry. A different kind of phase transition can still be found when including the contribution of vortex excitations. The transition takes the system from the low temperature quasi-ordered phase to a high temperature disordered phase and is known as the Kosterlitz-Thouless (KT) transition. Instead of relying on symmetry breaking, the KT transition is a direct cause of the topological nature of its available vortex configurations. 11

22 12 Chapter 3. Kosterlitz-Thouless Transition A simple thermodynamic argument allows us to understand the essence of the KT transition. The energy of a single vortex according to Eqs. (2.19) and (2.22) is E 1 = πj 0 ln(l/a). (3.1) The number of places to insert such a vortex for a cutoff length scale a in a system of size L is (L/a) 2, so the entropy of a single vortex is S 1 = k B ln(l/a) 2. (3.2) The free energy E 1 T S 1 is therefore positive only for temperatures below T KT = πj 0 2k B. (3.3) Above this critical temperature, it is favorable to have single vortices. On the other hand, vortex-antivortex pairs separated by a distance r have an energy cost 2πJ 0 ln(r/a). These pairs are favorable at any temperature if r is sufficiently small. The transition that occurs is therefore one that unbinds pairs of vortices as the temperature exceeds a critical value. 3.2 Thermodynamics There are a number of thermodynamic properties that showcase the critical behavior of the model, but we will here focus on a few that are of use to us in checking the agreement between simulation and theory. The basic quantities extracted from simulations are the system energy E = H and the net vorticity m = r q r as well as their powers and we therefore only consider derivatives of these. The heat capacity per lattice point is defined by C V = 1 L 2 T 2 ( E2 E 2 ). (3.4) This generally has an infinite peak at the critical point in an unbounded system. For a bounded system, the peak has a finite value that is shifted away from the temperature at which the transition occurs. A key quantity to our investigation is the magnetic permeability. This effectively measures the degree of magnetization the system can obtain in response to an applied magnetic field. For the model at hand, it is given by µ V = L2 T ( B2 B 2 ). (3.5) The magnetic field B is connected to m by the vortex quantization of Eq. (2.13). Since m = 0 always holds, we are left with µ V = Φ2 0 L 2 T m2. (3.6)

23 3.3. Renormalization Group 13 For completeness, it is worth also mentioning the dielectric response function ɛ, which can be defined in phase space as ɛ 1 (k) = 1 V (k) L 2 T q kq k, (3.7) Because of the correlation function involved, ɛ cannot be calculated as efficiently as any quantity related only to m and E, but it has the advantage of a clearly defined discontinuity at the critical point. The so-called universal jump of the KT transition is given by ɛ 1 (0) T = { 4 if T = T c 0 if T = T c+. (3.8) Beyond the general idea used in this thesis, the universal jump is the main method to detect the KT transition by measurements. Through a finite-size version of Eq. (3.8) with a logarithmic correction, Monte Carlo simulations estimate the critical temperature to T c = πJ 0 /k B, which is just below that of Eq. (3.3). 3.3 Renormalization Group One of the main results in the theoretical study of the KT transition is the renormalization group (RG) analysis of Kosterlitz [10]. Although we will not go into detail here, the main idea is to integrate a renormalized form of superfluid stiffness by gradually increasing the lower cut-off a with a scale factor b and hence excluding short-range fluctuations. The main variables involved are the superfluid stiffness J = 2 ρ R 2m e, (3.9) where ρ R is the fully renormalized superfluid areal density, and the vortex fugacity ζ = e ER c /(k BT ), (3.10) where E R c is the renormalized core energy. The final results of the analysis are the RG flow equations. In terms of the reduced variables x = 1 πj β/2 and y = 2πζ, the equations are to lowest order dx dl = 2y2 (3.11) dy = 2xy, dl (3.12)

24 14 Chapter 3. Kosterlitz-Thouless Transition where l = ln b. For a constant C depending on the initial conditions, the equations obey the simple relation x 2 y 2 = C 2. (3.13) The RG flow is plotted from Eq. (3.13) in Fig. 3.1 with the flow direction indicated by the arrows. The curves are mirrored around x = 0 and we will focus on x < 0, which is fulfilled for the systems explored by our simulations. Figure 3.1. Flow diagram of the Kosterlitz RG equations. For negative x, we have the presence of two distinct regions divided by a separatrix y = x flowing to the critical point x = y = 0. The separatrix corresponds to C = 0 and serves as the critical line of the model, where T = T c. Below the line, C 2 > 0 and the RG flow ends at x = C on the line of fixed points x < 0, y = 0. Since the flow tends to zero fugacity, this region corresponds to the low temperature phase with T < T c. The region above the critical line, where C 2 < 0, must therefore correspond to T > T c. Its flow continues through x = 0 on to diverge in positive x and y. The critical point emerges from the RG treatment by taking x = y to get the temperature TKT R πj = 2k B (1 + y 0 ), (3.14) where y 0 corresponds to the bare fugacity. By neglecting the small y 0, the temperature agrees with that of Eq. (3.3), but with J instead of its bare counterpart J 0.

25 3.4. Finite-Size Scaling 15 The solutions to Eqs. (3.11) and (3.12) below and on the critical line are 2C(b/b 0 ) 2C y(b) = 1 (b/b 0 ) 4C if T < T c 1, (3.15) if T = T c 2 ln(b/b 0 ) where the scaling constant b 0 depends on the initial values. Furthermore, the constant C can be related for T < T c to the fully renormalized superfluid stiffness J R according to 3.4 Finite-Size Scaling C = πj R 2k B T 1. (3.16) Since the net vorticity m averages to zero for all temperatures, it is not a meaningful property to be used in interpreting the simulation data. However, its second power m 2 can easily be calculated and contains information about the system fluctuations. From Eq. (3.6), we also know that m 2 is proportional to the magnetic permeability µ V, which can be measured on a macroscopic scale and is therefore of interest in order to get a connection to experiments. This leads us to the question of what form µ V takes near the critical point for systems of finite size. Let b be a scaling factor and suppose µ V depends on x, y, λ L and L. We know that λ L and L scale as lengths and we have the scaling of x and y from RG theory. According to Eq. (3.5), µ V itself scales as an area. When we rescale the system by b, we therefore end up with ( µ V (x, y, λ, L) = b 2 µ V x(b), y(b), λ L b, L ). (3.17) b We now choose L = b, which sets the effective system size to L = 1. We also take λ L = λl, (3.18) by introducing a new dimensionless quantity λ. This eliminates two of the variables involved in Eq. (3.17) and the resulting relation is µ V (x, y, λ, L) = L 2 µ V (x(l), y(l), λ, 1). (3.19) We also have an explicit solution of y(l) in terms of x(l), so we are left with µ V scaling only in relation to y. However, the fugacity tends to zero with the vortex flow for large scales when T T c. This is not what we expect of µ V m 2 /L 2 for any temperatures and y is therefore dangerously irrelevant.

26 16 Chapter 3. Kosterlitz-Thouless Transition To conclude the scaling argument we must proceed with systems slightly larger than just one vortex. When we allow two vortices, the partition function is Z = 1+ ζ + ζ and to first order we get µ V B 2 = ζ ζ ( 1) ζ 2 2 ζ y(l). (3.20) Hence, it holds for large systems that µ V L 2 y when T T c. By applying Eq. (3.6), we finally obtain the simple scaling relation m 2 y Scaling Formulas With the scaling of m 2 established, we are now in the position to predict how the models at hand act near the critical point. Equation (3.15) give us the expressions 1 m 2 = A 1 (L/L 0 ) 4C l (L/L 0 ) 2C if T < T c (3.21) 1 m 2 = A c ln(l/l 0 ) if T = T c, (3.22) where A l and A c are unknown proportionality constants and L 0 is a scaling constant. The formulas obtained are well suited for comparison to the values of m 2 sampled from simulations. We could in practice also relate the simulations to µ V, which is a more physical property, but this requires an additional factor of L 2 to be added. Equation (3.21) can be used to see the general agreement of simulations with the expected small temperature behavior of the system and supplies the constant C connected to J R by Eq. (3.16). Since m 2 has been inversed, the formula has the nice feature of roughly being a straight line in logarithmic scale of L near T c (where C is small). The distinct logarithmic form of Eq. (3.22) is also of great interest. Since it should only hold to good accuracy at precisely the critical point, it allows simulations of multiple system sizes to pinpoint the location of T c. The main aim of this project is to explore how well these two formulas fit simulations of the modified Coulomb gas models. If there is a good agreement, we also want to see if it can be exploited to efficiently obtain values of T c, C and L 0.

27 Chapter 4 Simulation Method We dedicate this chapter to the main aspects of our approach to the simulation of a non-neutral 2D Coulomb gas. The first section introduces the basics of the Monte Carlo method and the Metropolis-Hastings algorithm as well as the specifics for our simulation program. We then detail the processing performed before and after the simulations and conclude with some insight into the implementation of the programs. Natural units are used in this chapter and throughout the chapters that follow, which means that = k B = e = 1. The bare superfluid density is also assigned the value J 0 = 1/(2π). 4.1 Monte Carlo Method The naive approach to determine the properties of a physical system is to calculate the contributions of all possible configurations. For a system with a large amount of coupled degrees of freedom, this is not feasible. A commonly used alternative is that of Monte Carlo methods, which utilize random sampling to reduce the need for computations. The class of techniques we focus on here are based on importance sampling, which means that the samples are generated in a way that reflects how strongly they contribute to the calculation of physical quantities. For a system such as ours, the states Γ are connected to a distribution ρ(γ) such that the expectation value of a quantity A(Γ) is A = dγρ(γ)a(γ). (4.1) Instead of performing this integration, we can use ρ as a probability distribution to generate a number of samples Γ (i), where i = 1,..., n. These samples produce an 17

28 18 Chapter 4. Simulation Method estimate A n = 1 n which by the law of large numbers obeys lim n A n = A. n A(Γ (i) ), (4.2) i=1 By utilizing estimates as in Eq. (4.2), an efficient simulation method can be found if we can create samples distributed according to ρ with a small number of computations. One way of doing so is to set the samples in a Markov chain, defined by having the probability of generating a new sample only dependent on the previously generated sample. We denote the transition probability from the state Γ s to the state Γ t as π st and the probability of the state Γ s after m steps as ρ (m) s given initial values of ρ (1) s ). (with some The rows and columns defined by the indices in π st and ρ (m) s give us a stochastic matrix π with t π st = 1 and a set of probability distributions ρ (m) with s ρ(m) s = 1. According to the Markov chain recipe of transitions, the probability distribution changes after one step according to ρ (m+1) = ρ (m) π. To get the correct distribution after a long chain of samples, we must therefore require that ρ = lim m ρ(1) π m. (4.3) In particular, the limiting distribution ρ must fulfill the eigenvalue equation ρπ = ρ, which in index form reads ρ s π st = ρ s. (4.4) A sufficient condition for Eq. (4.4) to hold is that of detailed balance, t ρ s π st = ρ t π ts. (4.5) This has a number of solutions, but we will here focus on the most common one Metropolis-Hastings Algorithm We can decompose the transitional probability as π st = g st α st, (4.6) where g st is the probability of proposing Γ t to be the new state given that Γ s is the current state and α st is the corresponding probability of accepting the new state. The Metropolis choice of acceptance distribution [11] is α st = min ( 1, ρ t ρ s g st g ts ), (4.7) which clearly obeys the detailed balance condition of Eq. (4.5) for any choice of g and any given ρ.

29 4.1. Monte Carlo Method 19 If we take the acceptance of Eq. (4.7), one step in the algorithm can be summarized as 1. Generate a state Γ t at random according to g st, where Γ s is the current state. 2. Generate a number r at random from a uniform distribution of numbers between 0 and Accept Γ t as the current state if α st r. 4. Keep Γ s as the current state if α st < r. Before the process begins, the system must be initialized with some state. The choice does not affect the limit probability distribution, but can influence the rate of convergence. Furthermore, the states generated by the early steps of the Markov chain will not be distributed as wanted. We therefore need to discard the first n warm states as a warmup before generating the final n samp states to be sampled Main Program The simulation program made for the purposes of this thesis is based on the Metropolis-Hastings algorithm. The choices left free from the general recipe of the algorithm have been tailored to obtain a decent convergence to the limit probability distribution and a low overall complexity. The algorithm is based on a common choice for 2D Coulomb gases [12]. The simulation starts from a cold state, where the vorticity is null at all sites, and proceeds with n warm + n samp steps in total. For each step, two updates to the system are proposed: 1. A single lattice site x is selected at random (from a uniform distribution of all possible lattice sites). The corresponding vorticity is changed at random by q = ±1. 2. A lattice site x and one of its four nearest neighbors y are selected at random. The vorticity at x is changed by +1 and the vorticity at y is changed by -1. The transition probability g of the proposed state does not depend on the current state and is uniform in terms of both the lattice sites and the vorticity changes. The factor g st /g ts of Eq. (4.7) therefore cancels and we are left with a quotient ρ t /ρ s of the state probability distribution ρ to determine the acceptance probability α. For our system of particles, we have the Boltzmann factors ρ(γ) = e βh(γ), (4.8) where β = 1/T is the inverse temperature and H(Γ) is the system Hamiltonian. Suppose E is the energy change in the system as a result of the two proposed

30 20 Chapter 4. Simulation Method updates. Equation (4.7) then tells us that the update is accepted if either E 0 or e β E is larger than a uniformly generated random number between 0 and 1. To calculate E, we must add the contributions of both updates. We leave out the details of the calculations, but these values are found from Eq. (2.27). The energy change corresponding to update 1 is For update 2, we have E 1 = q(u(x) + V (0) q/2) µ ( q(x) + q q(x) ). (4.9) L2 E 2 =U(x) U(y) + V (0) V (x y)+ + µ (4.10) ( q(x) + 1 q(x) + q(y) 1 q(y) ). L2 It should here be noted that the actual program used in the simulation included a minor error. The implementation of Eq. (4.9) contained an extra term q m+ q/2 L, 2 related to a magnetic field as in Eq. (2.28). This roughly corresponds to a value λ L = 1 that has not been integrated to be part of the potential. In all but the very smallest systems considered, the value of the the term is extremely negligible and should not have any impact on the physics. Once an update is accepted, the algorithm updates the properties of the system. The process includes changing all values of the function U of Eq. (2.26), which requires a loop through all L 2 sites of the lattice. This corresponds to the bulk of the work from the entire algorithm and its contribution to the complexity of the program is O((n warm + n samp )L 2 ). The full complexity depends on how often the updates are performed, which has a complicated dependence on all input parameters. The program saves the observables required to estimate the thermodynamics, which is the system energy E and its power E 2 as well as the net vorticity m and its powers m 2 and m 4. These are sampled even if both update attempts fail, but only if the warmup phase has concluded. There is also an option to save the full system information in terms of the vorticities of all sites after all steps of the algorithm, which allows us to make snapshots and animations of the entire configuration. 4.2 Pre- and Post-Processing Beyond the main simulation algorithm described, the program consists of two additional parts. These can be run independently and take care of pre-processing the input parameters and post-processing the output data.

31 4.3. Implementation 21 Calculating all values of the potential V given by Eqs. (2.24), (2.30) and (2.31) has a complexity of O(L 4 ), which is relatively large for the system sizes considered. To save time on multiple calls for simulations with the same set of potential parameters, V is calculated and saved ahead of the main simulation program. The values are loaded when a simulation is initialized and used to update U whenever a proposed lattice update is accepted. The statistics are calculated by specifying a number of bins n bins for error analysis and dividing the gathered samples from many separate simulation runs accordingly. The error bars of the thermodynamic properties are estimated according to a scheme known as Jackknife resampling [13]. If we suppose a quantity A has a set of values A i with a mean A, its Jackknife means are defined as A i = 1 n bins 1 A j. (4.11) The Jackknife deviance of A is then given by nbins 1 A = ( A i A ) 2. (4.12) n bins The advantage of using Jackknife resampling is that it assures that all thermodynamic properties derived from the simulation data are calculated correctly. It is not the most efficient method available, but the complexity of the calculations are still very low compared to the other parts of the program. Some additional processing is required to fit the sampled inverse square net vorticity 1/m 2 to the scaling formulas of Eqs. (3.21) and (3.22). To begin with, we linearly interpolate the values of 1/m 2 to get a better resolution in β than that gathered from the simulations. At any of the values of β obtained, we then calculate the sum of squared errors scaled by the error estimates as i j i χ 2 (β) = L ( ( ) ( 1 1 m 2 β, L m 2 ) scaling β, L ) 2 ( / 1 ) 2 m 2, (4.13) β, L where the sum runs over all system sizes L for which data is available. The minima of these χ 2 values indicate the critical point β = β c. The fitting process also supplies values of the constants A l, A c and C. 4.3 Implementation The main program used for simulations, pre-processing and most of the postprocessing was written in C11 and compiled with GCC All calculations

32 22 Chapter 4. Simulation Method made with this were run on the computational cluster Octopus at the department of Theoretical Physics at KTH using Intel Xeon ES-2620 processors. The final parts of the post-processing and all plotting was done with Python and run locally on a laptop. The curve fitting was handled by the scipy package of Python. The C part of the code is written with a largely modular approach and has been thoroughly unit tested. The results of each function were compared to those of calculations made by hand for a number of different scenarios. The program used to collect the data presented in this thesis has been accepted by all such tests. A rudimentary comparison was also made between the major thermodynamic properties obtained by this program and those of a program independently developed by Mats Wallin, which resulted in a satisfactory agreement. The random number generator used is PCG [14] and seeding is done with the system time. The PCG generator was developed relatively recently and is therefore not as widely tested as alternatives like the Mersenne Twister, but it does have advantages like a very low complexity, an arbitrary period and simple implementation.

33 Chapter 5 Results The results of the Coulomb gas simulations are presented here. A brief explanation is given for the parameters chosen and the rest of the chapter is devoted to comparing the data obtained to the expected behavior of the system. The aim is to verify that models A and B of Eqs. (2.30) and (2.31) display a KT transition and to determine which model is most effective in deriving its properties. The plots shown indicate the model used by its corresponding letter in the legend. If nothing is explicitly stated, the chemical potential µ is set to Choice of Parameters The parameters involved in the simulation are the inverse temperature β, the system size L, the number of warmup samples n warm, the number of gathered samples n samp, the number of bins for error analysis n bins, the potential parameter λ and the chemical potential µ. For the purposes of finding interesting physics and making efficient simulations, certain combinations of parameters can be excluded. We here discuss the choices that have been made and the reasons behind the decisions. One aspect of choosing appropriate parameters is tuning the acceptance rate α sim of the simulations, i.e. the fraction of proposed lattice updates that are accepted. If updates are more easily accepted, the variance of the samples is greater. On the other hand, the computation involved in generating vortex changes is wasted to a greater degree if a large amount of updates are rejected. For our system, the absolute majority of the work is updating the state is changed and we should therefore expect a low acceptance rate to be more efficient. Figure 5.1 shows how α sim varies as a function of the inverse temperatures for a few fixed parameters and a system size of L = 128. The corresponding plot for a 23

34 24 Chapter 5. Results function of the system size with β = 4.7 fixed is given in Fig The rates are generally higher for higher temperatures, which should indeed be the case since a system that allows more unbound vortices will accept a greater amount of states. Smaller systems have higher acceptance rates, unless a negative chemical potential is included. In terms of the parameter λ, smaller values generally lead to more acceptance. Figure 5.1. Acceptance rate for n warm = 10 8, n samp = 10 9 and L = 128. As mentioned in Ch. 3.2, simulations using the universal jump put the critical point around β c = and we mainly want to investigate β in the vicinity of this value. To see how the system acts for some displacement in the temperature, we have chosen a range of β = 4 to β = 5.5 as the main field of investigation. It is beneficial to use as many and as high L as possible, but we are limited by the algorithm complexity of L 2. To get a logarithmic scale suitable to the scaling expression, L was chosen in powers of 2 from L = 4 up to L = 128. A system size of 256 was used for one case of parameters and doing so required several days of computation to get acceptable error estimations. A sufficient number of warmup samples from a cold start was determined from trial and error. For all extreme cases of parameters, multiple simulations with different numbers of warmup runs were made and compared to check at which point no systematic error could be observed. This was found to be about at about 10 million

35 5.1. Choice of Parameters 25 Figure 5.2. Acceptance rate for n warm = 10 8, n samp = 10 9 and β = 4.7. updates and n warm was set to 10 8 to be certain the limit distribution would always be attained. As a base case, n samp at does well enough for any parameters considered in order assure a hardly noticeable error margin. The size of the error should scale with the system size, but we found this relation to be at most linear in L. We therefore did not make the number of gathered samples scale with full sweeps of the system, which has the advantage of an increased accuracy for small system samples. The low acceptance rate for large β makes collecting enough samples more troublesome and n samp was therefore increased to for β above 5. The data binning requires each bin of samples to be large enough for there to be next to no correlation between the series and small enough for noise to be canceled. Accurately calculating correlation times is a difficult task for this system, but some basic trials were run for a small number of updates. The results indicate that n bins = 100 is sufficiently small to remove correlation effects and to provide acceptable estimations for the errors bars. With the other parameters set, some appropriate values for λ were determined by initial guesses and experimentation. Small values of λ lead to cumbersomely large acceptance rates and large values seem to shift the critical point from its expected value, which leads to λ between about 0.1 and 0.8 to be most appropriate. The

36 26 Chapter 5. Results influence of a chemical potential µ could be relevant, but has not been considered in detail. The value chosen is µ = 0 in all cases, except for one that instead uses a relatively large value µ = Potential The potential depends on the system size L and the scaled magnetic susceptibility λ according to Eqs. (2.30) and (2.31). As a function of of one dimension x, the potential V (x, 0) is plotted in Fig. 5.3 for L = 128 at a few different λ. Other values of L produce identical plots because of the periodicity. Figure 5.3. System potential V (x, 0) in one dimension x. It can be noted that the λ of model A merely shifts the potential by a constant value in all positions and relatively quickly converges to its form at λ = 0. For model B, the dominating impact of λ is the zero point of the potential in phase space, where the potential has the same form as in model A. The two models are therefore almost indistinguishable for λ above Configuration Examples Figures 5.4, 5.5 and 5.6 show four sample configurations taken at n samp = 1000 updates after the warmup phase for model B with λ = 0.4 and µ = 0. Since the

37 5.3. Configuration Examples 27 configurations are chosen ergodically, each plot is representative of the parameter setup used to produce it. It can be noted that the relative frequency of bound pairs increases as the temperature does and it seems that a phase transition occurs somewhere between β = 4 and β = 5. In all of the cases, large vorticities are heavily suppressed and almost all of the sites have vorticity 0 or ±1. Furthermore, the number of vortices N changes as expected with temperature and the net vorticity m is near 0 in all cases, despite this not being required by the model. Figure 5.4. Sample configuration from model B after n samp = 1000 updates with λ = 0.4, µ = 0, L = 64 and β = 3. Colors indicate the site vorticity.

38 28 Chapter 5. Results Figure 5.5. Sample configuration from model B after n samp = 1000 updates with λ = 0.4, µ = 0, L = 64 and β = 4. Colors indicate the site vorticity. Figure 5.6. Sample configuration from model B after n samp = 1000 updates with λ = 0.4, µ = 0, L = 64 and β = 5. Colors indicate the site vorticity.

39 5.4. Basic Quantities Basic Quantities In this section we show the main results of a simulation with λ = 0.4 and µ = 0 using model B. There are notable differences with other parameters and with model A, but these are most easily seen in the analysis of Ch Figure 5.7 shows the system energy per lattice site, E/L 2. Fewer configurations are accepted at lower temperatures, which can be seen from the decreasing energy. There is also a convergence to a fixed curve for large systems, indicating the behavior for an infinite lattice. Figure 5.7. Energy per lattice site of model B for λ = 0.4 and µ = 0. The heat capacity of Eq. (3.4) is displayed in Fig Peaks can be seen for very high temperatures, but they are shifted from the critical one due to the finite system size and should not be seen as an actual indicator of the critical point. Although not directly essential to our analysis, we show the magnetic permeability of Eq. (3.6) in Fig In contrast to the squared net vorticity m 2, to which µ V is proportional, we note that the quantity is heavily shifted for different system sizes.

40 30 Chapter 5. Results Figure 5.8. Heat capacity of model B for λ = 0.4 and µ = 0. Figure 5.9. Magnetic permeability of model B for λ = 0.4 and µ = 0.

41 5.4. Basic Quantities 31 The inverse squared net vorticity 1/m 2 is shown in Fig as a function of β. It can be seen that the curves nearly cross each other at some temperature slightly higher than the critical. Such a crossing would have been seen if y had not been dangerously irrelevant for the scaling argument of Ch. 3.4, because the result would have been a value of m 2 independent of L at the critical temperature. Figure Inverse squared net vorticity of model B for λ = 0.4 and µ = 0. To make use of the scaling formulas, we also want to look at 1/m 2 as a function of L. Figure 5.11 displays the sampled values of 1/m 2 in full lines along with fits obtained from the low temperature formula of Eq. (3.21) in dashed lines. The corresponding plot for fits to the critical temperature formula of Eq. (3.22) is shown in Fig In both cases, the data fits well in the regimes for which the formulas are meant and the critical point seems to be at about β = 4.7.