The main focus of this thesis lies on the critical properties of twodimensional (2D) superconductors in zero magnetic field. Simulations based on variants of the 2D XY model are shown to give characteristic features close to the phase transition which agree qualitatively with experimental data. Thus, it is concluded that these common characteristic features are caused by two-dimensional vortices. The thesis consists of an introductory part and five separate publications. In the introductory part of the thesis the basic results of the Ginzburg-Landau model, which gives a phenomenological description of superconductors, are described. In 2D systems, the superconductive phase transition in the absence of a magnetic field is governed by the unbinding of thermally created vortices and is called the Kosterlitz-Thouless (KT) phase transition. An introduction to this kind of transition is given. The important features of the current-voltage (IV) characteristics and the nonlinear conductivity, which can be used to study the KT transition, are discussed. The scaling analysis procedure, a powerful tool for the analysis of the properties of a system in the vicinity of phase transition, is reviewed. A scaling form for the nonlinear dc conductivity, which takes into account finite-size e ects, is discussed. The static 2D XY model, which is usually used to describe superfluids, superconducting films as well as the high-Tc superconductors with high anisotropy, is introduced. Three different types of dynamic models, namely resistively shunted junction, relaxational, and Monte Carlo dynamics are superimposed on the 2D XY model for the evaluation of the dynamic properties. TheVillain model and a modifiedXY model using a p-type interaction potential exhibit different densities of the thermally created vortices. Since the dominant characteristic physical features close to the KT transition are associated with vortex pair fluctuations these two models are investigated. The introductory part closes with a short introduction to each of the five published articles.

The present thesis reports computer simulation studies of several phase transition related phenomena in a range of soft-condensed matter systems. A coherent unifying theme of the thesis is the understanding of dynamics of phase transitions through free energy calculations using recently developed efficient non-Boltzmann sampling methods. Based on the system/phenomena of interest, the thesis has been classified into four major parts: I. Isotropic-nematic (IN) phase transition in liquid crystals. II. Nucleation phenomena in gas-liquid transition with particular emphasis on the systems close to the spinodal curve. III. Collapse transition in linear hydrocarbon (n-alkane) chains for a varying range of length, solvent and temperature. IV. Crystallization of unbranched polymer chains in dilute solution, with particular emphasis on the temperature dependent crossover between the rod-like crystalline state and spherical molten globule state. The thesis has been further divided into ten chapters running through the four parts mentioned before. In the following we provide a brief chapter-wise outline of the thesis. Part I deals with the power law relaxation and glassy dynamics in thermotropic liquid crystals close to the IN transition and consists of two chapters. To start with, Chapter I.1 provides an introduction to thermotropic liquid crystals. Here we briefly introduce various liquid crystalline phases, the order parameter used to characterize the IN transition, a few well established theoretical models, and we conclude with describing the recent experimental and computer simulation studies that have motivated the work described in the next chapter. In Chapter I.2, we present our molecular dynamics simulation studies on single particle and collective orientational dynamics across the IN transition for Lebwohl Lasher model, which is a well-known lattice model for thermotropic liquid crystals. Even this simplified model without any translational degrees of freedom successfully captures the short-tointermediate time power law decay recently observed in optical heterodyne detected optical Kerr effect (OHDOKE) measurements near the IN transition. The angular velocity time correlation function also exhibits a rather pronounced power law decay near the IN boundary. In the mean squared angular displacement at comparable time scales, we observe the emergence of a sub-diffusive regime which is followed by a super-diffusive regime before the onset of the longtime diffusive behavior. We observe signature of dynamical heterogeneity through pronounced non-Gaussian behavior in the orientational motion particularly at lower temperatures. Interestingly, this behavior closely resembles what is usually observed in supercooled liquids. We obtain the free energy as a function of orientational order parameter by the use of recently developed transition matrix Monte Carlo (TMMC) method. The free energy surface is flat for the system considered here and the barrier between isotropic and nematic phases is vanishingly small for this weakly first-order transition, hence allowing for large scale, collective, and correlated orientational density fluctuations. We attribute this large scale fluctuations as the reason for the observed power law decay of the orientational time correlation functions. Part II consists of three chapters, where we focus on the age old problem of nucleation and growth, both from the perspective of thermodynamics and kinetics. We account for the rich history of the problem in the introductory Chapter II.1. In this chapter we describe various types and examples of the nucleation phenomena, and a brief account of the major theoretical approaches used so far. We begin with the most successful Classical Nucleation Theory (CNT), and then move on to more recent applications of Density Functional Theory (DFT) and other mean-field types of models. We conclude with a comparison between the experiments, theories and computational studies. In the next chapter (Chapter II.2) we attempt to elucidate the mechanism of nucleation near the gas-liquid spinodal from a microscopic point of view. Here we construct a multidimensional free energy surface of nucleation of the liquid phase from the parent supercooled and supersaturated vapor phase near the gas-liquid spinodal. In particular, we remove the Becker-Doring constraint of having only one growing cluster in the system. The free energy, as a function of the size of the largest cluster, develops a pronounced minimum at a subcritical cluster size close to the spinodal. This signifies a two step nature of the process of nucleation, where the rapid formation of subcritical nuclei is followed by further growth by slower density fluctuations on an uphill free energy surface. An alternative free energy pathway involving the participation of many subcritical clusters is envisaged near the spinodal where the growth of the nucleus is found to be promoted by a coalescence mechanism in contrast to the single particle addition assumption within CNT. The growth of the stable phase becomes progressively collective and spatially diffuse, and the significance of a “critical nucleus” is lost for deeper quenches. In this chapter we present our studies both in 3dimensional Lennard-Jones (LJ) system and Ising model (both 2and 3dimensions). Our general findings seem to be independent of the model chosen. While the previous chapter focuses on relatively well-studied 3-dimensional (3D) LJ system, in Chapter II.3 we present our studies on the characteristics of the nucleation phenomena in 2dimensional (2D) Lennard-Jones fluid. To the best of our knowledge this is the first extensive computer simulation study to check the accuracy of CNT in 2D. Using various Monte Carlo methods, we calculate the free energy barrier for nucleation, line tension, and bulk densities of equilibrium liquid and vapor phases, and also investigate the size and shape of the critical nucleus. The study is carried out at an intermediate level of supersaturation (away from the spinoidal limit). In 2D, a surprisingly large cutoff (rc ≥ 7.0σ where σ is the diameter of LJ particles) in the truncation of the LJ potential is required to obtain converged results. A lower cutoff leads to a substantial error in the values of the line tension, nucleation barrier, and characteristics of the critical cluster. Note that typically 2.5σ is sufficient for 3D LJ fluids. We observe that in 2D system CNT fails to provide a reliable estimate of the free energy barrier. While it is known to slightly overestimate the nucleation barrier in 3D, it underestimates the barrier by as much as 50% at the saturation ratio S = 1.1(defined as S = P/Pc, where Pc is the coexistence pressure) and at the reduced temperature T* = 0.427(defined as T* = KBT/ ε, where ε is the depth of the potential well). The reason for the marked inadequacy of the CNT in 2D can be attributed to the non-circular nature of the critical clusters. Although the shape becomes increasingly circular and the clusters become more compact with increase in cutoff radius, an appreciable non-circular nature remains even without any cutoff to make the simple CNT inaccurate. Part III again consists of three chapters and focuses on the conformational equilibria. Collapse transition and self-organized structures of n-alkanes in solution. In Chapter III.1 we carry out a brief survey of the existing theories of polymer in solution, with particular emphasis on the collapse process in poor solvents. We also introduce the concept of “hydrophobicity” and “hydrophobic collapse”, which is now a subject enormous interest, partly because it my help in understanding the initial processes involved in protein folding. We briefly discuss the subject of formation of beautiful self-organized structures by block copolymers, and also simple homopolymers which is essentially the focus of the work embodied in the next two chapters. In Chapter III.2 we demonstrated a chain length dependent crossover in the structural properties of linear hydrocarbon (n-alkane) chains using detailed atomistic simulations in explicit water. We identify a number of exotic structures o the polymer chain through energy minimization of representative snapshots collected from molecular dynamics trajectory. While the collapsed state is ring-like(circular) for small chains(CnH2n+2; n ≤ 20) and spherical for very long ones( n = 100), we find the emergence of ordered helical structures at intermediate lengths (n ~ 40). We find different types of disordered helices and toroid-like structures at n = 60. We also report a sharp transition in the stability of the collapsed state as a function of the chain length through relevant free energy calculations. While the collapsed state is only marginally metastable for C20H42, a clear bistable free energy surface emerges only when the chain is about 30 monomers long. For n = 30, the polymer exhibits an intermittent oscillation(characterized by well-developed 1/f noise, where f is the frequency ) between the collapsed and the coil structures, characteristic of two stable states separated by a small barrier. This appears to support a weakly first order phase transition between the extended and the collapsed states. Chapter III.3 extends the study of previous chapter to much longer chains (n ≥ 100), which irreversibly collapse in water into globular forms. Even though the collapsed form has a nearly spherical shape, close inspection shows a propensity towards local ordering in the alignment of the polymer segments. This tendency to maintain alignment in order to maximize the number of contacts leads to a core-shell like structure, where the shell is often characterized by a bent rod-like shape consisting of two adjacent segments running in parallel. A key event associated with the initial stage of collapse seems to be the formation of a skewed ring (or loop) that serves as a “nucleation center” for rest of the chain to collapse into. Time evolution of the radial distribution function of water surrounding the polymer, shows that the density of neighboring water decreases by only about 15-20% from that of bulk water. Even though interior of the ting-like structures is fully devoid of water, solvent accessible surface representation shows that these regions are geometrically/spatially inaccessible to water molecules. We suggest that the role of water is to stabilize such ring-like structures once formed by natural conformational fluctuations of the polymer chain. This view is confirmed by observation of spontaneous formation and melting away of such ring-like entities in a polar aprotic solvent(DMSO). We also comment on the role of the flexibility of polymer chains in determining the collapse kinetics. The last part(Part IV) of the thesis consists of two chapters that deal with the crystallization of linear polymer chains from dilute solution. The way long chain polymers crystallize is drastically different from their small molecule counterparts due to their topological connectivity. Linear polymers often crystallize from dilute solution in the form of thin lamellae with well-defined crystallographic features. In Chapter IV.1 we briefly survey the current theoretical understanding and confusions associated with the highly debated field of polymer crystallization. While the last few decades have seen the development of many successful phenomenological theories, the molecular mechanism of formation of such self-organized lamellae is extremely complex and very poorly understood. There are clearly two distinct steps in polymer crystallization. Firstly, the individual linear polymers must self-organize into bundles of somewhat regular structures. These structures then further aggregate to lamellar form and crystallize into a lattice. In this respect , it has marked similarity to the problem of protein crystallization. In chapter IV.2 we present Brownian dynamics simulation studies of a single polythelene chain of length 500. Such systems can reasonably mimic the process of crystallization from dilute solutions. Our simulations could successfully reproduce some of the interesting phenomena observed in experiments and very recent computer simulation studies, including multi-center nucleation of rod-like structures within a single polymer chain, an inverse relation between lamellar thickness and temperature etc. But our primary focus has been to understand the nature of the phase transition as one traverses along the melting temperature and the underlying free energy surface. Near the melting temperature we observe a very intriguing fluctuation between the disordered molten globule state and the ordered rod-like crystalline, where these two forms have highly different shape and structure. These fluctuations have strong signature of 1/f noise or intermittency. This clearly indicates the existence of a weakly first order transition, where two widely different states with large difference in values of order parameter are separated by a rather small free energy barrier. This can be related to the experimentally observed density fluctuations that resemble spinodal decomposition. It is important to note that very similar fluctuations have been observed in our previous studies on liquid crystals (Chapter 1.2) and intermediate sized alkalines in water(Chapter III.2) that signifies a universal underlying energy landscape for these systems. We have discussed the scope of future work at the end of each chapter whenever appropriate.

This thesis presents a theoretical study of materials under high pressure using ab initio lattice dynamics based on density functional theory and density functional perturbation theory using both super-cell and linear response approach. Ab initio lattice dynamics using super-cell approach is applied to compare our theoretical predictions with experimental findings. Phonon dispersion curves of fcc α-γ cerium are calculated and compared with inelastic X-ray scattering data. Pressure dependency of phonon density of states in two cubic phases TiO2 allows us to assign the observed cubic phase in experiments to be of fluorite rather than pyrite structure. Dynamical stability of cotunnite TiO2 phase at low pressure can explain the observed quenching phenomena in experiments. Our calculated O2 vibron mode in both ε-ζ phases of solid oxygen supports the hypothesis that both phases are iso-structural. Hydrogen-dense materials attract great attention not only because they open a path to study phenomena related to metallization (superconductivity) of solid hydrogen but also because they are closely related to important industrial applications (hydrogen storage). Using linear response method, we find that metallic fcc-AlH3 is dynamically stabilized in the range of 72-106 GPa and can persist at ambient pressure if finite temperature effects are considered. For SiH4, we test dynamical stability, Raman spectra, zero point energy, and utilize GW calculations for self energy correction. We find that a metallic tetragonal phase of SiH4 can be assigned to the experimentally observed one. Our ab initio lattice dynamics calculations based on density functional perturbation theory predict that fcc-YH3 is a pressure-induced superconductor with a high transition temperature of 40 K at 17.7 GPa. With increasing pressure this material undergoes a superconductor-metal-superconductor transition and the underlying mechanism of this transition can simultaneously explains also the observed metal-insulator transition at 25 GPa in YH3-δ.

HTML clipboard Statistical physics seeks to explain macroscopic properties of matter in terms of microscopic interactions. Of particular interest is the phenomenon of phase transition: the sudden changes in macroscopic properties as external conditions are varied. Two models in particular are of great interest to mathematicians, namely the Ising model of a magnet and the percolation model of a porous solid. These models in turn are part of the unifying framework of the random-cluster representation, a model for random graphs which was first studied by Fortuin and Kasteleyn in the 1970’s. The random-cluster representation has proved extremely useful in proving important facts about the Ising model and similar models. In this work we study the corresponding graphical framework for two related models. The first model is the transverse field quantum Ising model, an extension of the original Ising model which was introduced by Lieb, Schultz and Mattis in the 1960’s. The second model is the space–time percolation process, which is closely related to the contact model for the spread of disease. In Chapter 2 we define the appropriate space–time random-cluster model and explore a range of useful probabilistic techniques for studying it. The space– time Potts model emerges as a natural generalization of the quantum Ising model. The basic properties of the phase transitions in these models are treated in this chapter, such as the fact that there is at most one unbounded fk-cluster, and the resulting lower bound on the critical value in <img src="http://upload.wikimedia.org/math/a/b/8/ab820da891078a8245d7f4f3252aee4f.png" />. In Chapter 3 we develop an alternative graphical representation of the quantum Ising model, called the random-parity representation. This representation is based on the random-current representation of the classical Ising model, and allows us to study in much greater detail the phase transition and critical behaviour. A major aim of this chapter is to prove sharpness of the phase transition in the quantum Ising model—a central issue in the theory— and to establish bounds on some critical exponents. We address these issues by using the random-parity representation to establish certain differential inequalities, integration of which gives the results. In Chapter 4 we explore some consequences and possible extensions of the results established in Chapters 2 and 3. For example, we determine the critical point for the quantum Ising model in <img src="http://upload.wikimedia.org/math/a/b/8/ab820da891078a8245d7f4f3252aee4f.png" /> and in ‘star-like’ geometries. / HTML clipboard Statistisk fysik syftar till att förklara ett materials makroskopiska egenskaper i termer av dess mikroskopiska struktur. En särskilt intressant egenskap är är fenomenet fasövergång, det vill säga en plötslig förändring i de makroskopiska egenskaperna när externa förutsättningar varieras. Två modeller är särskilt intressanta för en matematiker, nämligen Ising-modellen av en magnet och perkolationsmodellen av ett poröst material. Dessa två modeller sammanförs av den så-kallade fk-modellen, en slumpgrafsmodell som först studerades av Fortuin och Kasteleyn på 1970-talet. fk-modellen har sedermera visat sig vara extremt användbar för att bevisa viktiga resultat om Ising-modellen och liknande modeller. I den här avhandlingen studeras den motsvarande grafiska strukturen hos två näraliggande modeller. Den första av dessa är den kvantteoretiska Isingmodellen med transverst fält, vilken är en utveckling av den klassiska Isingmodellen och först studerades av Lieb, Schultz och Mattis på 1960-talet. Den andra modellen är rumtid-perkolation, som är nära besläktad med kontaktmodellen av infektionsspridning. I Kapitel 2 definieras rumtid-fk-modellen, och flera probabilistiska verktyg utforskas för att studera dess grundläggande egenskaper. Vi möter rumtid-Potts-modellen, som uppenbarar sig som en naturlig generalisering av den kvantteoretiska Ising-modellen. De viktigaste egenskaperna hos fasövergången i dessa modeller behandlas i detta kapitel, exempelvis det faktum att det i fk-modellen finns högst en obegränsad komponent, samt den undre gräns för det kritiska värdet som detta innebär. I Kapitel 3 utvecklas en alternativ grafisk framställning av den kvantteoretiska Ising-modellen, den så-kallade slumpparitetsframställningen. Denna är baserad på slumpflödesframställningen av den klassiska Ising-modellen, och är ett verktyg som låter oss studera fasövergången och gränsbeteendet mycket närmare. Huvudsyftet med detta kapitel är att bevisa att fasövergången är skarp—en central egenskap—samt att fastslå olikheter för vissa kritiska exponenter. Metoden består i att använda slumpparitetsframställningen för att härleda vissa differentialolikheter, vilka sedan kan integreras för att lägga fast att gränsen är skarp. I Kapitel 4 utforskas några konsekvenser, samt möjliga vidareutvecklingar, av resultaten i de tidigare kapitlen. Exempelvis bestäms det kritiska värdet hos den kvantteoretiska Ising-modellen på <img src="http://upload.wikimedia.org/math/a/b/8/ab820da891078a8245d7f4f3252aee4f.png" /> , samt i ‘stjärnliknankde’ geometrier. / QC 20100705

We study Gaussian sample covariance matrices with population covariance a bounded-rank perturbation of the identity, as well as Wigner matrices with bounded-rank additive perturbations. The top eigenvalues are known to exhibit a phase transition in the large size limit: with weak perturbations they follow Tracy-Widom statistics as in the unperturbed case, while above a threshold there are outliers with independent Gaussian fluctuations. Baik, Ben Arous and Péché (2005) described the transition in the complex case and conjectured a similar picture in the real case, the latter of most relevance to high-dimensional data analysis. Resolving the conjecture, we prove that in all cases the top eigenvalues have a limit near the phase transition. Our starting point is the work of Rámirez, Rider and Virág (2006) on the general beta random matrix soft edge. For rank one perturbations, a modified tridiagonal form converges to the known random Schrödinger operator on the half-line but with a boundary condition that depends on the perturbation. For general finite-rank perturbations we develop a new band form; it converges to a limiting operator with matrix-valued potential. The low-lying eigenvalues describe the limit, jointly as the perturbation varies in a fixed subspace. Their laws are also characterized in terms of a diffusion related to Dyson's Brownian motion and in terms of a linear parabolic PDE. We offer a related heuristic for the supercritical behaviour and rigorously treat the supercritical asymptotics of the ground state of the limiting operator. In a further development, we use the PDE to make the first explicit connection between a general beta characterization and the celebrated Painlevé representations of Tracy and Widom (1993, 1996). In particular, for beta = 2,4 we give novel proofs of the latter. Finally, we report briefly on evidence suggesting that the PDE provides a stable, even efficient method for numerical evaluation of the Tracy-Widom distributions, their general beta analogues and the deformations discussed and introduced here. This thesis is based in part on work to be published jointly with Bálint Virág.

We study Gaussian sample covariance matrices with population covariance a bounded-rank perturbation of the identity, as well as Wigner matrices with bounded-rank additive perturbations. The top eigenvalues are known to exhibit a phase transition in the large size limit: with weak perturbations they follow Tracy-Widom statistics as in the unperturbed case, while above a threshold there are outliers with independent Gaussian fluctuations. Baik, Ben Arous and Péché (2005) described the transition in the complex case and conjectured a similar picture in the real case, the latter of most relevance to high-dimensional data analysis. Resolving the conjecture, we prove that in all cases the top eigenvalues have a limit near the phase transition. Our starting point is the work of Rámirez, Rider and Virág (2006) on the general beta random matrix soft edge. For rank one perturbations, a modified tridiagonal form converges to the known random Schrödinger operator on the half-line but with a boundary condition that depends on the perturbation. For general finite-rank perturbations we develop a new band form; it converges to a limiting operator with matrix-valued potential. The low-lying eigenvalues describe the limit, jointly as the perturbation varies in a fixed subspace. Their laws are also characterized in terms of a diffusion related to Dyson's Brownian motion and in terms of a linear parabolic PDE. We offer a related heuristic for the supercritical behaviour and rigorously treat the supercritical asymptotics of the ground state of the limiting operator. In a further development, we use the PDE to make the first explicit connection between a general beta characterization and the celebrated Painlevé representations of Tracy and Widom (1993, 1996). In particular, for beta = 2,4 we give novel proofs of the latter. Finally, we report briefly on evidence suggesting that the PDE provides a stable, even efficient method for numerical evaluation of the Tracy-Widom distributions, their general beta analogues and the deformations discussed and introduced here. This thesis is based in part on work to be published jointly with Bálint Virág.

Recently a periodic surface model was developed to assist geometric construction in computer-aided nano-design. This implicit surface model helps create super-porous nano structures parametrically and support crystal packing. In this thesis, a new approach for pathway search in phase transition simulation of crystal structures is proposed. The approach relies on the interpolation of periodic loci surface models. Respective periodic plane models are reconstructed from the positions of individual atoms at the initial and final states, and surface correspondence is found using a Simulated Annealing-like algorithm. With geometric constraints imposed based on physical and chemical properties of crystals, two surface interpolation methods are used to approximate the intermediate atom positions on the transition pathway in the full search of the minimum energy path. This hybrid approach integrates geometry information in configuration space and physics information to allow for efficient transition pathway search. The methods are demonstrated by examples of FeTi, VO2, and FePt. Additionally, two new particle swarm optimization (PSO) algorithms are developed and applied to crystal structure relaxation of the initial and final states. The PSO algorithms are integrated into the Quantum-Espresso open-source software package and tested against the default Broyden-Fletcher-Goldfarb-Shanno relaxation method.

Phase transitions and other intrinsic properties (shape, size, architecture) of molecularly structured aerosol particles are important for understanding their role in planetary atmospheres and for technical applications. By combining bath gas cooling with time resolved mid-infrared spectroscopy and modeling, information is obtained on dynamic processes and intrinsic properties of fluoroform and ethane aerosol particles. The distinct infrared spectral features of fluoroform aerosol particles make it a particularly suitable model system. Homogeneous crystallization rates of the sub-micron sized aerosol particles are determined (JV = 10⁸ - 10¹⁰ cm-³s-¹ or JS = 10³ – 10⁵ cm-²s-¹ at a temperature of T = 78 K), and the controversial question regarding volume versus surface nucleation in freezing aerosols is addressed. It is demonstrated that current state of the art measurements of droplet ensembles cannot distinguish between the two mechanisms due to inherent experimental uncertainties. The evolution of particle shape from spherical supercooled droplets to cube-like crystalline particles and eventually to elongated crystalline particles is recorded and analyzed in detail with the help of vibrational exciton model calculations. Phase behaviour of pure ethane aerosols and ethane aerosols formed in the presence of other ice nuclei under conditions mimicking Titan’s atmosphere provide evidence for the formation of supercooled liquid ethane aerosol droplets, which subsequently crystallize. The observed homogeneous freezing rates (JV = 10⁷ – 10⁹ cm-³s-¹) imply that supercooled ethane could play a similar role in ethane rich regions of Titan’s atmosphere as supercooled water does in the Earth’s atmosphere.

The LiHo_xY_{1-x}F_4 magnetic material in a transverse magnetic field B_x perpendicular to the Ising spin direction has long been used to study tunable quantum phase transitions in pure and random disordered systems. We first present analytical and numerical evidences for the validity of an effective spin-1/2 approach to the description of a general dipolar spin glass model with strong uniaxial Ising anisotropy and subject to weak B_x. We relate this toy model to the LiHo_xY_{1-x}F_4 transverse field Ising material. We show that an effective spin-1/2 model is able to capture both the qualitative and quantitative aspects of the physics at small B_x. After confirming the validity of the effective spin-1/2 approach, we show that the field-induced magnetization along the x direction, combined with the local random dilution-induced destruction of crystalline mirror symmetries generates, via the predominant dipolar interactions between Ho^{3+} ions, random fields along the Ising z direction. This identifies LiHo_xY_{1-x}F_4 in B_x as a new random field Ising system. We show that the random fields explain the smearing of the nonlinear susceptibility at the spin glass transition with increasing B_x. In this thesis, we also investigate the phase diagram of non-diluted LiHoF_4 in the presence of B_x, by performing Monte-Carlo simulations. A previous quantum Monte Carlo (QMC) simulation found that even for small B_x where quantum fluctuations are small, close to the classical critical point, there is a discrepancy between experiment and the QMC results. We revisit this problem, focusing on weak B_x close to the classical T_c, using an alternative approach. For small B_x, by applying a so-called cumulant expansion, the quantum fluctuations around the classical T_c are taken into account perturbatively. We derived an effective perturbative classical Hamiltonian, on which MC simulations are performed. With this method we investigate different proposed sources of uncertainty which can affect the numerical results. We fully reproduce the previous QMC results at small B_x. Unfortunately, we find that none of the modifications to the microscopic Hamiltonian that we explore are able to provide a B_x-T phase diagram compatible with the experiments in the small semi-classical B_x regime.

The importance of soft matter research, as a major class of materials including liquid crystals, polymers, colloids, emulsions, and forms, is attributed to the behavior resemblances in each branch of soft matter responding to the external perturbations. Hence, one of the most required inquiries in soft matter physics is understanding how the structures with characteristic length scales evolve in response to external perturbations, and concomitant phase transitions. We have focused on adopting polarized Raman spectroscopy to probe phase transitions in soft materials consisting of anisometric components and the evolution of molecular orientational ordering as a complementary tool to other methodologies, but distinct in some respects. The primary task is quantifying the degree of molecular orientation, i.e., obtaining orientational order parameters, in liquid crystal (LCs) system. Thermal evolution of orientation degree in a hitherto elusive biaxial nematic (Nb) phase as well as a commonly known uniaxial nematic (Nu) phase were interrogated from the measurements of anisotropy in polarized Raman intensities. We demonstrated reliable and applicable method to quantify the orientation degree for systems possessing anisotropic ordering. We also addressed a strong potential of Raman spectroscopy that the changes of vibrational energy reflect the variations of intermolecular interactions and structural changes on the molecular level induced by phase transitions. As a subfield of soft matter, we characterized phase transitions and anisotropic ordering observed in an evaporating conjugated polymer solution and elucidated the mechanism of the entities undergoing phase transitions using mainly polarized Raman spectroscopy. In addition, we have shown that tracking Raman spectral changes can provide valuable information for understanding structure-property relations when the measurements of the evolution in physical properties are carried out simultaneously.