Order Parameter Field Research Articles

Criticality of O(N) symmetric models in the presence of discrete gauge symmetries.

We investigate the critical properties of the three-dimensional antiferromagnetic RP^{N-1} model, which is characterized by a global O(N) symmetry and a discrete Z_{2} gauge symmetry. We perform a field-theoretical analysis using the Landau-Ginzburg-Wilson (LGW) approach and a numerical Monte Carlo study. The LGW field-theoretical results are obtained by high-order perturbative analyses of the renormalization-group flow of the most general Φ^{4} theory with the same global symmetry as the model, assuming a gauge-invariant order-parameter field. For N=4 no stable fixed point is found, implying that any transition must necessarily be of first order. This is contradicted by the numerical results that provide strong evidence for a continuous transition. This suggests that gauge modes are not always irrelevant, as assumed by the LGW approach, but they may play an important role to determine the actual critical dynamics at the phase transition of O(N) symmetric models with a discrete Z_{2} gauge symmetry.

The Landau-de Gennes approach revisited: A minimal self-consistent microscopic theory for spatially inhomogeneous nematic liquid crystals.

We design a novel microscopic mean-field theory of inhomogeneous nematic liquid crystals formulated entirely in terms of the tensor order parameter field. It combines the virtues of the Landau-de Gennes approach in allowing both the direction and magnitude of the local order to vary, with a self-consistent treatment of the local free-energy valid beyond the small order parameter limit. As a proof of principle, we apply this theory to the well-studied problem of a colloid dispersed in a nematic liquid crystal by including a tunable wall coupling term. For the two-dimensional case, we investigate the organization of the liquid crystal and the position of the point defects as a function of the strength of the coupling constant.

Theories of binary fluid mixtures: from phase-separation kinetics to active emulsions

Binary fluid mixtures are examples of complex fluids whose microstructure and flow are strongly coupled. For pairs of simple fluids, the microstructure consists of droplets or bicontinuous demixed domains and the physics is controlled by the interfaces between these domains. At continuum level, the structure is defined by a composition field whose gradients – which are steep near interfaces – drive its diffusive current. These gradients also cause thermodynamic stresses which can drive fluid flow. Fluid flow in turn advects the composition field, while thermal noise creates additional random fluxes that allow the system to explore its configuration space and move towards the Boltzmann distribution. This article introduces continuum models of binary fluids, first covering some well-studied areas such as the thermodynamics and kinetics of phase separation, and emulsion stability. We then address cases where one of the fluid components has anisotropic structure at mesoscopic scales creating nematic (or polar) liquid-crystalline order; this can be described through an additional tensor (or vector) order parameter field. We conclude by outlining a thriving area of current research, namely active emulsions, in which one of the binary components consists of living or synthetic material that is continuously converting chemical energy into mechanical work. Such activity can be modelled with judicious additional terms in the equations of motion for simple or liquid-crystalline binary fluids. Throughout, the emphasis of the article is on presenting the theoretical tools needed to address a wide range of physical phenomena. Examples include the kinetics of fluid–fluid demixing from an initially uniform state; the result of imposing a steady macroscopic shear flow on this demixing process; and the diffusive coarsening, Brownian motion and coalescence of emulsion droplets. We discuss strategies to create long-lived emulsions by adding trapped species, solid particles, or surfactants; to address the latter, we outline the theory of bending energy for interfacial films. In emulsions where one of the components is liquid-crystalline, ‘anchoring’ terms can create preferential orientation tangential or normal to the fluid–fluid interface. These allow droplets of an isotropic fluid in a liquid crystal (orvice versa) to support a variety of topological defects, which we describe, altering their interactions and stability. Addition of active terms to the equations of motion for binary simple fluids creates a model of ‘motility-induced’ phase separation, where demixing stems from self-propulsion of particles rather than their interaction forces, altering the relation between interfacial structure and fluid stress. Coupling activity to binary liquid crystal dynamics creates models of active liquid-crystalline emulsion droplets. Such droplets show various modes of locomotion, some of which strikingly resemble the swimming or crawling motions of biological cells.

Hidden fluctuations close to a quantum bicritical point

In this paper we describe physical properties arising in the vicinity of two coupled quantum phase transitions. We consider a phenomenological model based on two scalar order parameter fields locally coupled biquadratically and having a common quantum critical point as a function of a quantum tuning parameter such as pressure or magnetic field. A self-consistent treatment suggests that the uniform static susceptibilities of the two order parameter fields may have the same qualitative form at low temperature even where the forms differ sharply in the absence of the biquadratic coupling. The possible limitations of the self-consistent analysis leading to this prediction are considered.

Universality of domain growth in antiferromagnets with spin-exchange kinetics.

We study phase ordering kinetics in symmetric and asymmetric binary mixtures, undergoing an order-disorder transition below the critical temperature. Microscopically, we model the kinetics via the antiferromagnetic Ising model with Kawasaki spin-exchange kinetics. This conserves the composition while the order parameter (staggered magnetization) is not conserved. The order-parameter correlation function and structure factor show dynamical scaling, and the scaling functions are independent of the mixture composition. The average domain size shows a power-law growth: [Formula: see text]. The asymptotic growth regime has [Formula: see text], though there can be prolonged transients with [Formula: see text] for asymmetric mixtures. Our unambiguous observation of the asymptotic universal regime is facilitated by using an accelerated Monte Carlo technique. We also obtain the coarse-grained free energy from the Hamiltonian, as a function of two order parameters. The evolution of these order parameters is modeled by using Model C kinetics. As for the microscopic dynamics, the average domain size of the nonconserved order-parameter (staggered magnetization) field exhibits a power-law growth: [Formula: see text] at later times, irrespective of the mean value of the conserved order-parameter (composition) field.

Phase-field study on geometry-dependent migration behavior of voids under temperature gradient in UO2 crystal matrix

In this work, a phase-field model is established to capture the void migration behavior under a temperature gradient within a crystal matrix, with an appropriate consideration of the surface diffusion mechanism and the vapor transport mechanism. The interfacial energy and the coupling between the vacancy concentration field and the crystal order parameter field are carefully modeled. Simulations are performed on UO2. The result shows that for small voids (with an area ≤ πμm2), the well-known characteristics of void migration, in consistence with the analytical model, can be recovered. The migration is manifested by a constant velocity and a minor change of the void shape. In contrast, for large voids (with an area of ∼10 μm2) initially in circular shapes, significant deformation of the void from a circular to cashew-like shape is observed. After long-time migration, the deformed void would split into smaller voids. The size-dependent behavior of void migration is due to the combined effect of the interfacial energy (which tends to keep the void in circular shape) and the surface diffusion flow (which tends to deform the void due to the nonuniform diffusion along the surface). Moreover, the initial shape of the void modifies the migration velocity and the time point when splitting occurs (for large voids) at the beginning of migration due to the shape relaxation of the void. However, it has a minor effect on the long-time migration. Our work reveals novel void migration behaviors in conditions where the surface-diffusion mechanism is dominant over the vapor transport mechanism; meanwhile, the size of the void lies at a mediate size range.

Finite-deformation phase-field chemomechanics for multiphase, multicomponent solids

The purpose of this work is the development of a framework for the formulation of geometrically non-linear inelastic chemomechanical models for a mixture of multiple chemical components diffusing among multiple transforming solid phases. The focus here is on general model formulation. No specific model or application is pursued in this work. To this end, basic balance and constitutive relations from non-equilibrium thermodynamics and continuum mixture theory are combined with a phase-field-based description of multicomponent solid phases and their interfaces. Solid phase modeling is based in particular on a chemomechanical free energy and stress relaxation via the evolution of phase-specific concentration fields, order-parameter fields (e.g., related to chemical ordering, structural ordering, or defects), and local internal variables. At the mixture level, differences or contrasts in phase composition and phase local deformation in phase interface regions are treated as mixture internal variables. In this context, various phase interface models are considered. In the equilibrium limit, phase contrasts in composition and local deformation in the phase interface region are determined via bulk energy minimization. On the chemical side, the equilibrium limit of the current model formulation reduces to a multicomponent, multiphase, generalization of existing two-phase binary alloy interface equilibrium conditions (e.g., KKS). On the mechanical side, the equilibrium limit of one interface model considered represents a multiphase generalization of Reuss-Sachs conditions from mechanical homogenization theory. Analogously, other interface models considered represent generalizations of interface equilibrium conditions consistent with laminate and sharp-interface theory. In the last part of the work, selected existing models are formulated within the current framework as special cases and discussed in detail.

Statistical field theory with constraints: Application to critical Casimir forces in the canonical ensemble.

The effect of imposing a constraint on a fluctuating scalar order parameter field in a system of finite volume is studied within statistical field theory. The canonical ensemble, corresponding to a fixed total integrated order parameter (e.g., the total number of particles), is obtained as a special case of the theory. A perturbative expansion is developed which allows one to systematically determine the constraint-induced finite-volume corrections to the free energy and to correlation functions. In particular, we focus on the Landau-Ginzburg model in a film geometry (i.e., in a rectangular parallelepiped with a small aspect ratio) with periodic, Dirichlet, or Neumann boundary conditions in the transverse direction and periodic boundary conditions in the remaining, lateral directions. Within the expansion in terms of ε=4-d, where d is the spatial dimension of the bulk, the finite-size contribution to the free energy of the confined system and the associated critical Casimir force are calculated to leading order in ε and are compared to the corresponding expressions for an unconstrained (grand canonical) system. The constraint restricts the fluctuations within the system and it accordingly modifies the residual finite-size free energy. The resulting critical Casimir force is shown to depend on whether it is defined by assuming a fixed transverse area or a fixed total volume. In the former case, the constraint is typically found to significantly enhance the attractive character of the force as compared to the grand canonical case. In contrast to the grand canonical Casimir force, which, for supercritical temperatures, vanishes in the limit of thick films, in the canonical case with fixed transverse area the critical Casimir force attains for thick films a negative value for all boundary conditions studied here. Typically, the dependence of the critical Casimir force both on the temperaturelike and on the fieldlike scaling variables is different in the two ensembles.

Effective field theory of an anomalous Hall metal from interband quantum fluctuations

We construct an effective field theory, a two-dimensional two-component metallic system described by a model with two Fermi surfaces ("pockets"). This model describes a translationally invariant metallic system with two types of fermions, each with its own Fermi surface, with forward scattering interactions. This model, in addition to the $O(2)$ rotational invariance, has a $U(1) \times U(1)$ symmetry of separate charge conservation for each Fermi surface. For sufficiently attractive interactions in the $d$-wave (quadrupolar) channel this model has an interesting phase diagram that includes a spontaneously generated anomalous Hall metal phase. We derive the Landau-Ginzburg effective action of quadrupolar order parameter fields which enjoys an $O(2)\times U(1)$ global symmetry associated to spatial isotropy and the internal $U(1)$ relative phase symmetries respectively. We show that the order parameter theory is dynamically local with a dynamical scaling of $z=2$ and perform a one-loop renormalization group analysis of the Landau-Ginzburg theory. The electronic liquid crystal phases that result from spontaneous symmetry breaking are studied and we show the presence of Landau damped Nambu-Goldstone modes at low momenta that is a signature of non-Fermi liquid behavior. Electromagnetic linear response is also analyzed in both the normal and symmetry broken phases from the point of view of the order parameter theory. The nature of the coupling of electromagnetism to the order parameter fields in the normal phase is non-minimal and decidedly contains a precursor to the anomalous Hall response in the form of a order-parameter-dependent Chern-Simons term in the effective action.

Field-driven dynamics and time-resolved measurement of Dzyaloshinskii-Moriya torque in canted antiferromagnet YFeO3

Electrical spin switching in an antiferromagnet is one of the key issues for both academic interest and industrial demand in new-type spin devices because an antiferromagnetic system has a negligible stray field due to an alternating sign between sub-lattices, in contrast to a ferromagnetic system. Naturally, questions arise regarding how fast and, simultaneously, how robustly the magnetization can be switched by external stimuli, e.g., magnetic field and spin current. First, the exploitation of ultrafast precessional motion of magnetization in antiferromagnetic oxide has been studied intensively. Regarding robustness, the so-called inertia-driven switching scenario has been generally accepted as the switching mechanism in antiferromagnet system. However, in order to understand the switching dynamics in a canted antiferromagnet, excited by magnetic field, accurate equation of motion and corresponding interpretation are necessary. Here, we re-investigate the inertia-driven switching process, triggered by the strict phase matching between effective driving field, dh/dt, and antiferromagnetic order parameters, l. Such theoretical approaches make it possible to observe the static parameters of an antiferromagnet, hosting Dzyaloshinskii–Moriya (DM) interaction. Indeed, we estimate successfully static parameters, such as DM, exchange, and anisotropy energies, from dynamical behaviour in YFeO3, studied using terahertz time-domain spectroscopy.

Dynamic zero modes of Dirac fermions and competing singlet phases of antiferromagnetic order

In quantum spin systems, singlet phases often develop in the vicinity of an antiferromagnetic order. Typical settings for such problems arise when itinerant fermions are also present. In this work, we develop a theoretical framework for addressing such competing orders in an itinerant system, described by Dirac fermions strongly coupled to an O(3) nonlinear sigma model. We focus on two spatial dimensions, where upon disordering the antiferromagnetic order by quantum fluctuations the singular tunneling events also known as (anti)hedgehogs can nucleate competing singlet orders in the paramagnetic phase. In the presence of an isolated hedgehog configuration of the nonlinear sigma model field, we show that the fermion determinant vanishes as the dynamic Euclidean Dirac operator supports fermion zero modes of definite chirality. This provides a topological mechanism for suppressing the tunneling events. Using the methodology of quantum chromodynamics, we evaluate the fermion determinant in the close proximity of magnetic quantum phase transition, when the antiferromagnetic order parameter field can be described by a dilute gas of hedgehogs and antihedgehogs. We show how the precise nature of emergent singlet order is determined by the overlap between dynamic fermion zero modes of opposite chirality, localized on the hedgehogs and antihedgehogs. For a Kondo-Heisenberg model on the honeycomb lattice, we demonstrate the competition between spin Peierls order and Kondo singlet formation, thereby elucidating its global phase diagram. We also discuss other physical problems that can be addressed within this general framework.

Dynamical evolution of critical fluctuations and its observation in heavy ion collisions

We study time evolution of critical fluctuations of conserved charges near the QCD critical point in the context of relativistic heavy ion collisions. A stochastic diffusion equation is employed in order to describe the diffusion property of the critical fluctuation arising from the coupling of the order parameter field to conserved charges. We show that the diffusion property gives rise to a possibility of probing the early time fluctuations through the rapidity window dependence of the second-order cumulant and correlation function of conserved charges. It is pointed out that their non-monotonic behaviors as functions of the rapidity interval are robust experimental signals for the existence of the critical enhancement around the QCD critical point.

Calculation of the magnetotransport for a spin-density-wave quantum critical theory in the presence of weak disorder

We compute the Hall angle and the magnetoresistance of the spin-fermion model, which is a successful phenomenological theory to describe the physics of the cuprates and iron-based superconductors within a wide range of doping regimes. We investigate both the role of the spin-fermion interaction that couples the large-momentum antiferromagnetic fluctuations to the so-called “hot-spots” at the Fermi surface and also of an effective higher-order composite operator in the theory. The latter operator provides a scattering mechanism such that the momentum transfer for the fermions close to the Fermi surface can be small. We also include weak disorder that couples to both the bosonic order-parameter field and the fermionic degrees of freedom. Since the quasiparticle excitations were shown in recent works to be destroyed at the “hot-spots” in the low-energy limit of the model, we employ the Mori-Zwanzig memory matrix approach that permits the evaluation of all transport coefficients without assuming well-defined Landau quasiparticles in the system. We then apply this transport theory to discuss universal metallic-state properties as a function of temperature and magnetic field of the cuprates from the perspective of their fermiology, which turn out to be in qualitative agreement with key experiments in those materials.

Topological Fermi-liquid theory for interacting Weyl metals with time reversal symmetry breaking

Introducing both Berry curvature and chiral anomaly into Landau's Fermi-liquid theory, we construct a topological Fermi-liquid theory, applicable to interacting Weyl metals in the absence of time reversal symmetry. Following the Landau's Fermi-liquid theory, we obtain an effective free-energy functional in terms of the density field of chiral fermions. The density field of chiral fermions is determined by a self-consistent equation, minimizing the effective free-energy functional with respect to the order-parameter field. Beyond these thermodynamic properties, we construct Boltzmann transport theory to encode both the Berry curvature and the chiral anomaly in the presence of forward scattering of a Fermi-liquid state, essential for understanding dynamic correlations in interacting Weyl metals. This generalizes the Boltzmann transport theory for the Landau's Fermi-liquid state in the respect of incorporating the topological structure and extends that for noninteracting Weyl metals in the sense of introducing the forward scattering. Finally, we justify this topological Fermi-liquid theory, generalizing the first-quantization description for noninteracting Weyl metals into the second-quantization representation for interacting Weyl metals. First, we derive a topological Fermi-gas theory, integrating over high-energy electronic degrees of freedom deep inside a pair of chiral Fermi surfaces. As a result, we reproduce a topological Drude model with both the Berry curvature and the chiral anomaly. Second, we take into account interactions between such low-energy chiral fermions on the pair of chiral Fermi surfaces. We perform the renormalization group analysis, and find that only forward scattering turns out to be marginal above possible superconducting transition temperatures, justifying the topological Fermi-liquid theory of interacting Weyl metals with time reversal symmetry breaking.

Modeling and simulation of non-isothermal rate-dependent damage processes in inhomogeneous materials using the phase-field approach

We present a continuum model that incorporates rate-dependent damage and fracture, a material order-parameter field and temperature within a phase-field approach. The models covers partial damage as well as the formation of macro-cracks. For the material order parameter we assume a Cahn Larche-type dynamics, which makes the model in particular applicable to binary alloys. We give thermodynamically consistent evolution equations resulting from a unified variational approach. Diverse coupling mechanisms can be covered within the model, such as heat dissipation, thermal-expansion-induced failure and crack deflection due to inhomogeneities. With help of an adaptive finite element code we conduct numerical experiments of different complexity in order to study the possibilities and limitations of the presented model. We furthermore include anisotropic linear elasticity in our model and investigate the effect on the crack pattern.

Longitudinal fluctuations in the Berezinskii-Kosterlitz-Thouless phase

We analyze the interplay of longitudinal and transverse thermal fluctuations in a $\text{U}(1)$ symmetric two-dimensional ${\ensuremath{\phi}}^{4}$ theory. Toward this end, we derive coupled renormalization-group equations for both types of fluctuations obtained from a linear (Cartesian) decomposition of the order-parameter field. Discarding the longitudinal fluctuations, the expected Berezinskii-Kosterlitz-Thouless phase characterized by a finite stiffness and an algebraic decay of order-parameter correlations is recovered. Renormalized by transverse fluctuations, the longitudinal mass scales to zero so that longitudinal fluctuations become increasingly important for small momenta. Within our expansion of the effective action, they generate a logarithmic decrease of the stiffness, in agreement with previous functional renormalization-group calculations. The logarithmic terms imply a deviation from the vanishing $\ensuremath{\beta}$ function for the stiffness in the nonlinear $\ensuremath{\sigma}$ model describing the phase fluctuations at three-loop order. To gain further insight, we also compute the flow of the parameters characterizing longitudinal and transverse fluctuations from a density-phase representation of the order-parameter field, with a cutoff on phase fluctuations. The power-law flow of the longitudinal mass and other quantities is thereby confirmed, but the stiffness remains finite in this approach. We conclude that the marginal flow of the stiffness obtained in the Cartesian representation is an artifact of the truncated expansion of momentum dependences.

A phenomenological order approach to the volume phase transition in microgel particles.

We proposed a model to describe the volume phase transition of microgel particles, based on constitutive relations that consider the mechano-chemical coupling of deformation and solvency processes, as well as the phenomenological cooperative effect of molecular orientation of solvatable segments in the particles. We regard a nematic-isotropic like transition and define a tensor order parameter allowing analysis of the collective behaviour of water solvation mechanisms based on the Landau-De Gennes theory for liquid crystal elastomers. A complementary approach to the energy balance density is obtained with the order parameter field considering the deformation by a continuum mechanics approximation; our approach aims to extend the comprehensive understanding of the elastic behaviour and water sorption related phenomena in microgel particles. With the proposed model, we reproduce the behaviour of swelling curves through a volume phase transition of thermosensitive microgel particles by assessing characteristic parameters belonging to particular physical properties. The parameters defined within the model such as elastic and solvency related constants and characteristic temperatures were obtained by fitting of a data set mimicking the experimentally observed swelling behaviour. The obtained values of the parameters were consistent with the discussed phenomenology which shows the importance of molecular ordering interpretation as a source to describe the volume phase transition in microgel particles.

Disorder trapping by rapidly moving phase interface in an undercooled liquid

Non-equilibrium phenomena such as the disappearance of solute drag, the origin of solute trapping and evolution of disorder trapping occur during fast transformations with originating metastable phases [D.M. Herlach, P.K. Galenko, D. Holland-Moritz, Metastable solids from undrercooled melts (Elsevier, Amsterdam, 2007)]. In the present work, a theoretical investigation of disorder trapping by a rapidly moving phase interface is presented. Using a model of fast phase transformations, a system of governing equations for the diffusion of atoms, and the evolution of both long-range order parameter and phase field variable is formulated. First numerical solutions are carried out for a congruently melting binary alloy system.

Phase-field modeling of chemical control of polarization stability and switching dynamics in ferroelectric thin films

Phase-field simulation (PFS) have revolutionized the understanding of domain structure and switching behavior in ferroelectric thin films and ceramics. Generally, PFS is based on solution of a (set) of Ginzburg-Landau equations for a defined order parameter field(s) under physical boundary conditions (BCs) of fixed potential or charge. While well-matched to the interfaces in bulk materials and devices, these BCs are generally not applicable to free ferroelectric surfaces. Here, we developed a self-consistent phase-field model with boundary conditions based on electrochemical equilibria. We chose Pb(Zr0.2Ti0.8)O3 ultrathin film consisting of (001) oriented single tetragonal domain (Pz) as a model system, and systematically studied the effects of oxygen partial pressure, temperature and surface ion on the ferroelectric state, and compared it with the case of complete screening. We have further explored the polarization switching induced by the oxygen partial pressure and observed pronounced size effect induced by chemical screening. Our work thus helps to understand the emergent phenomena in ferroelectric thin films brought about by the electrochemical ionic surface compensations.

Incorporating Noise Quantitatively in the Phase Field Crystal Model via Capillary Fluctuation Theory.

A tacit assumption underlying most phase field models of nonequilibrium phase transformations is that of scale separation. Stochastic order parameter field theories utilize noise to separate atomic-scale fluctuations from the slowly varying fields that describe microstructure patterns. The mesoscale distribution of such stochastic variables is generally assumed to follow Gaussian statistics, with their magnitude following fluctuation-dissipation relations. However, there is still much debate about how atomic-scale fluctuations map onto the mesoscale upon coarse graining of microscopic theories. This Letter studies interface fluctuations in the phase field crystal (PFC) model and proposes a self-consistent method for relating how the effective noise strength and spectral filtering of the noise in the PFC model, and similar types of microscopic models, should be defined so as to attain the spectrum of mesoscale capillary fluctuations quantitatively.