Free Energy Expansion Research Articles

Calculation of the T-X phase diagram for the first-order smectic–hexatic transitions in binary mixtures

This work gives the T-X phase diagram calculated from the Landau mean field model for the smectic–hexatic transitions in binary mixtures, particularly, in THI-13+THI-17. It is calculated near the SmC-SmF (Hex F)–Hex B triple point by considering transitions of SmC-SmF, SmF-HexB, and SmC-Hex B as the first order. The phase line equations are fitted to the experimental T-X phase diagram from the literature for this binary mixture by assuming the temperature and concentration dependence of the coefficients in the Landau free energy expansion. Calculations are given in some detail to describe the smectic–hexatic transitions in the THI-13+THI-17, which can also be considered for other binary mixtures under the Landau mean field model studied here.

Theory of the charge density wave in AV3Sb5 kagome metals

The family of metallic kagome compounds $A$V$_3$Sb$_5$ ($A$=K, Rb, Cs) was recently discovered to exhibit both superconductivity and charge order. The nature of the charge-density wave (CDW) phase is presently unsettled, which complicates the interpretation of the superconducting ground state. In this paper, we use group-theory and density-functional theory (DFT) to derive and solve a phenomenological Landau model for this CDW state. The DFT results reveal three unstable phonon modes with the same in-plane momentum but different out-of-plane momenta, whose frequencies depend strongly on the electronic temperature. This is indicative of an electronically-driven CDW, stabilized by features of the in-plane electronic dispersion. Motivated by the DFT analysis, we construct a Landau free-energy expansion for coupled CDW order parameters with wave-vectors at the $M$ and $L$ points of the hexagonal Brillouin zone. We find an unusual trilinear term coupling these different order parameters, which can promote the simultaneous condensation of both CDWs even if the two modes are not nearly-degenerate. We classify the different types of coupled multi-$\bf{Q}$ CDW orders, focusing on those that break the sixfold rotational symmetry and lead to a unit-cell doubling along all three crystallographic directions, as suggested by experiments. We determine a region in parameter space, characterized by large nonlinear Landau coefficients, where these phases - dubbed staggered tri-hexagonal and staggered Star-of-David - are the leading instabilities of the system. Finally, we discuss the implications of our results for the kagome metals.

Order parameter induced by an extremely short optical pulse in a medium with chiral carbon nanotubes

We study the effect of a three-dimensional extremely short optical pulse in a medium with chiral carbon nanotubes on the order parameter of this medium. The possibility of the appearance of a phase transition in such a medium is shown. We investigate the dependence of the pulse shape on the relaxation rate of the order parameter and on the parameter β in the Landau expansion of free energy.

Strong coupling expansion of free energy and BPS Wilson loop in mathcal{N} = 2 superconformal models with fundamental hypermultiplets

As a continuation of the study (in arXiv:2102.07696 and arXiv:2104.12625) of strong-coupling expansion of non-planar corrections in mathcal{N} = 2 4d superconformal models we consider two special theories with gauge groups SU(N) and Sp(2N). They contain N-independent numbers of hypermultiplets in rank 2 antisymmetric and fundamental representations and are planar-equivalent to the corresponding mathcal{N} = 4 SYM theories. These mathcal{N} = 2 theories can be realised on a system of N D3-branes with a finite number of D7-branes and O7-plane; the dual string theories should be particular orientifolds of AdS5 × S5 superstring. Starting with the localization matrix model representation for the mathcal{N} = 2 partition function on S4 we find exact differential relations between the 1/N terms in the corresponding free energy F and the frac{1}{2} -BPS Wilson loop expectation value leftlangle mathcal{W}rightrangle and also compute their large ’t Hooft coupling (λ » 1) expansions. The structure of these expansions is different from the previously studied models without fundamental hypermultiplets. In the more tractable Sp(2N) case we find an exact resummed expression for the leading strong coupling terms at each order in the 1/N expansion. We also determine the exponentially suppressed at large λ contributions to the non-planar corrections to F and leftlangle mathcal{W}rightrangle and comment on their resurgence properties. We discuss dual string theory interpretation of these strong coupling expansions.

’t Hooft expansion of multi-boundary correlators in 2D topological gravity

AbstractWe study multi-boundary correlators of Witten–Kontsevich topological gravity in two dimensions. We present a method of computing an open string like expansion, which we call the ’t Hooft expansion, of the $n$-boundary correlator for any $n$ up to any order by directly solving the Korteweg–De Vries equation. We first explain how to compute the ’t Hooft expansion of the one-boundary correlator. The algorithm is very similar to that for the genus expansion of the open free energy. We next show that the ’t Hooft expansion of correlators with more than one boundary can be computed algebraically from the correlators with a lower number of boundaries. We explicitly compute the ’t Hooft expansion of the $n$-boundary correlators for $n=1, 2, 3$. Our results reproduce previously obtained results for Jackiw–Teitelboim gravity and also the ’t Hooft expansion of the exact result of the three-boundary correlator which we calculate independently in the Airy case.

Metal-insulator transition and antiferromagnetism in the generalized Hubbard model: Treatment of correlation effects

The ground state for the half-filled $t-t'$ Hubbard model is treated within the Hartree-Fock approximation and the slave boson approach including correlations. The criterium for the metal-insulator transition in the Slater scenario is formulated using an analytical free-energy expansion in the next-nearest-neighbor transfer integral $t'$ and in direct antiferromagnetic gap $\Delta$. The correlation effects are generally demonstrated to favor the first-order transition. For a square lattice with a strong van Hove singularity, accidental close degeneracy of antiferromagnetic and paramagnetic phases is analytically found in a wide parameter region. As a result, there exists an interval of $t'$ values for which the metal-insulator transition is of the first order due to the existence of the van Hove singularity. This interval is very sensitive to model parameters (direct exchange integral) or external parameters. For the simple and body-centered cubic lattices, the transition from the insulator antiferromagnetic state with increasing $t'$ occurs to the phase of an antiferromagnetic metal and is a second-order transition which is followed by a transition to a paramagnetic metal. These results are quantitatively modified when taking into account the intersite Heisenberg interaction, which can induce first-order transitions. A comparison with the Monte Carlo results is performed.

The Low-Temperature Expansion of the Casimir-Polder Free Energy of an Atom with Graphene

We consider the low-temperature expansion of the Casimir-Polder free energy for an atom and graphene by using the Poisson representation of the free energy. We extend our previous analysis on the different relations between chemical potential μ and mass gap parameter m. The key role plays the dependence of graphene conductivities on the μ and m. For simplicity, we made the manifest calculations for zero values of the Fermi velocity. For μ>m, the thermal correction ∼T2, and for μ<m, we confirm the recent result of Klimchitskaya and Mostepanenko, that the thermal correction ∼T5. In the case of exact equality μ=m, the correction ∼T. This point is unstable, and the system falls to the regime with μ>m or μ<m. The analytical calculations are illustrated by numerical evaluations for the Hydrogen atom/graphene system.

Genus expansion of open free energy in 2d topological gravity

We study open topological gravity in two dimensions, or, the intersection theory on the moduli space of open Riemann surfaces initiated by Pandharipande, Solomon and Tessler. The open free energy, the generating function for the open intersection numbers, obeys the open KdV equations and Buryak’s differential equation and is related by a formal Fourier transformation to the Baker-Akhiezer wave function of the KdV hierarchy. Using these properties we study the genus expansion of the free energy in detail. We construct explicitly the genus zero part of the free energy. We then formulate a method of computing higher genus corrections by solving Buryak’s equation and obtain them up to high order. This method is much more efficient than our previous approach based on the saddle point calculation. Along the way we show that the higher genus corrections are polynomials in variables that are expressed in terms of genus zero quantities only, generalizing the constitutive relation of closed topological gravity.

Helmholtz free-energy high-temperature perturbation expansion for square-well and square-shoulder potentials

ABSTRACT The Helmholtz free-energy high-temperature perturbation terms of the square-shoulder and the square-well potentials of variable range were calculated using the Microcanonical Ensemble simulation method up to the eight-order. The numerical evaluation of the coefficients can be useful to build improved equations of state for these systems. To our knowledge, this is the first work that reports values for the seventh and eighth expansion coefficients for these discrete potentials. The square-shoulder high temperature expansion coefficients are numerically compared with those of the square-well potential. We confirmed that there exists a simple relation among them: even terms are equal for both potentials and odd terms differ by a minus sign, for all the densities and ranges considered. This relation was first proposed in the work of Benavides and Gil-Villegas (Molecular Physics 97, 1225 (1999)) in the context of Discrete Perturbation theory but it was only analytically tested for the first and second-order perturbation terms, and, for the second-order term, it has only been analysed within the Barker and Henderson Local and Macroscopic Compressibility approximations.

Phase stability and Landau phenomenological model of relaxor ferroelectric single crystals 0.78Pb(Mg1/3Nb2/3)O3-0.22PbTiO3

Relaxor-based ferroelectric single crystal with lower PbTiO3 content has complex phase transitions and mixed ferroelectric and relaxor characteristics, making it very challenging for the construction of Landau free energy expansion crucially used for theoretical investigation of this system. In this work, relaxor behavior in 0.78Pb(Mg1/3Nb2/3)O3-0.22PbTiO3 (PMN-0.22PT) single crystals was well suppressed through careful experimental design, leaving preserved single domain state for the Landau parameter extraction in this rhombohedral-phase composition. Using these extracted Landau free energy expansion parameters, energy profile and phase coexistence were theoretically studied and compared with those of PMN-0.30PT and PMN-0.36PT single crystals. Results show that rhombohedral phase dominates in PMN-0.22PT, and tetragonal phase dominates in PMN-0.36PT around room temperature. While, rhombohedral phase and tetragonal phase coexist in MPB composition PMN-0.30PT with small discrepancy of both free energies and polarization vectors.

Contributions to polarization and polarization switching in antiphase boundaries of SrTiO3 and PbZrO3

We use a recently developed method—based on layer group analysis combined with the Landau theory—to investigate the polar properties of antiphase boundaries (APBs) in SrTiO3 and PbZrO3. For SrTiO3, we find that, in addition to the biquadratic, Houchmandazeh-Laizerowicz-Salje (HLS) coupling bijklPiPjϕkϕl in the Landau-Ginzburg free energy expansion, additional rotopolar terms of the form WijklPiϕk∂ϕl∂xj contribute considerably to the polarization of antiphase boundaries in these materials. The rotopolar terms can be split into a symmetric flexoelectric part and an antisymmetric one. The antisymmetric Lifshitz term leads to a macroscopic polarization of APBs, which can be switched by application of an external electric field. For PbZrO3, the observed polarization profiles [Wei et al., Mater. Res. Bull. 62, 101 (2015)] are fully compatible with the symmetries of the corresponding layer groups. Unlike in SrTiO3, there exists no Lifshitz invariant WijklPiηk∂ηl∂xj for the order parameter ηi(i=1,…,12) describing the displacements of lead atoms. However, a detailed group theoretical treatment indicates that the polarity of APBs in PbZrO3 is driven by higher order interactions between polarization Pi, order parameter ηk, and order parameter gradients ∂ηl∂xj.

Polarization of domain boundaries in SrTiO3 studied by layer group and order-parameter symmetry

Based on a recently developed combination of layer group analysis with order-parameter symmetry, we study the polarity of antiphase domain boundaries (APBs) and ferroelastic twin boundaries (TBs) in $\mathrm{Sr}\mathrm{Ti}{\mathrm{O}}_{3}$. In addition to the celebrated layer group analysis of domain twins, the present method allows us to investigate tensor properties of domain walls also for the case where order-parameter variables other than the spontaneous ones are active. We find that antiphase boundaries in $\mathrm{Sr}\mathrm{Ti}{\mathrm{O}}_{3}$ can carry a polarization if in addition to the spontaneous order parameter $(0,0,{\ensuremath{\phi}}_{s})$ a second component, i.e., $({\ensuremath{\phi}}_{1},0,{\ensuremath{\phi}}_{s})$, develops within the domain wall. This result, which is solely based on symmetry arguments, strongly suggests that polarization in APBs is possible if a phase transition from an Ising-type wall to a N\'eel- or Bloch-like wall occurs. This is in very good agreement with previous calculations based on Landau-Ginzburg free energy expansions including biquadratic ($\ensuremath{\propto}{P}_{i}{P}_{j}{\ensuremath{\phi}}_{k}{\ensuremath{\phi}}_{l}$) [A. K. Tagantsev et al. Phys. Rev. B 64, 224107 (2001)] and flexoelectric ($\ensuremath{\propto}{P}_{k}\frac{\ensuremath{\partial}{u}_{ij}}{\ensuremath{\partial}\ensuremath{\xi}}$) coupling terms [A. N. Morozovska et al. Phys. Rev. B. 85, 094107 (2012)]. The present results also unveil a close connection between the recently discovered macroscopic polarization in antiferrodistortive cycloids of ferroelastic domain walls of $\mathrm{Sr}\mathrm{Ti}{\mathrm{O}}_{3}$ and a mechanism for explaining polarization of APBs.

Finite-size correction to the scaling of free energy in the dimer model on a hexagonal domain

We consider dimer model on a hexagonal lattice. This model can be seen as a "pile of cubes in the box". The energy of configuration is given by the volume of the pile and the partition function is computed by the classical MacMahon formula or, more formally, by the determinant of Kasteleyn matrix. We use the expression for the partition function to derive the scaling behavior of free energy in the limit of lattice mesh tending to zero and temperature tending to infinity. We consider the cases of finite hexagonal domain, of infinite height box and coordinate-dependent Boltzmann weights. We obtain asymptotic expansion of free energy and discuss the universality and physical meaning of the expansion coefficients.

Approximation of correlation functions in phase‐field crystal model by machine learning approach

Two different phase‐field crystal (PFC) free excess energy expansions have been analyzed with respect to the order of a truncation term and to the accuracy of a pair correlation functions fitting. The coefficients of the correlation function in PFC model was found for the aluminum crystal based on the molecular dynamics data. A machine learning (ML) approach has been utilized to construct a neural network (NN) for the PFC approximation of correlation functions. The effective iteratomic potentials of the embedded atom model were used as an input data for NN training. Obtained NN predicts the coefficients of the correlation functions with the potentials data as input. Predictions were studied and verified with respect to the properties of NN and parameters of ML.

On the Singularity of the Liquid-Gas Coexistence Curve Diameter

A simplified Anisimov–Wang variant of the complete scaling approach makes it possible to determine the amplitudes of singularities for the diameter of the phase coexistence curve (CXC) on the basis of the coefficients in the power series expansion of the mean-field free energy in the reduced temperature and pressure near the critical point. This method is applied to obtain the amplitudes for the leading critical singularities of the CXC diameter in the case of a fluid described in the framework of the mesoscopic mean-field model. The results obtained demonstrate that the amplitudes of leading singularities of the CXC diameter are determined by the mesoscopic asymmetry parameters of the heterophase fluid.

Energy Stable Numerical Schemes for Ternary Cahn-Hilliard System

We present and analyze a uniquely solvable and unconditionally energy stable numerical scheme for the ternary Cahn-Hilliard system, with a polynomial pattern nonlinear free energy expansion. One key difficulty is associated with presence of the three mass components, though a total mass constraint reduces this to two components. Another numerical challenge is to ensure the energy stability for the nonlinear energy functional in the mixed product form, which turns out to be non-convex, non-concave in the three-phase space. To overcome this subtle difficulty, we add a few auxiliary terms to make the combined energy functional convex in the three-phase space, and this, in turn, yields a convex-concave decomposition of the physical energy in the ternary system. Consequently, both the unique solvability and the unconditional energy stability of the proposed numerical scheme are established at a theoretical level. In addition, an optimal rate convergence analysis in the $$\ell ^\infty (0,T; H_N^{-1}) \cap \ell ^2 (0,T; H_N^1)$$ norm is provided, with Fourier pseudo-spectral discretization in space, which is the first such result in this field. To deal with the nonlinear implicit equations at each time step, we apply an efficient preconditioned steepest descent (PSD) algorithm. A second order accurate, modified BDF scheme is also discussed. A few numerical results are presented, which confirm the stability and accuracy of the proposed numerical scheme.

Interparticle correlations in the simple cubic lattice of ferroparticles: Theory and computer simulations

Anisotropic interparticle correlations in the simple cubic lattice of single-domain ferroparticles (SCLF) are studied using both theory and computer simulation. The theory is based on the Helmholtz free energy expansion like classical virial series up to the second virial coefficient. The analytical formula for the Helmholtz free energy is incorporated in a logarithmic form to minimize the effects of series truncation. The new theoretical approach, including discrete summation over lattice nodes coordinates, is compared critically against the classical virial expansion of the Helmholtz free energy for the dipolar hard sphere fluid; the main differences between the Helmholtz free energy of SCLF and dipolar hard sphere fluid are discussed. The theoretical results for the Helmholtz free energy, the magnetization, and the initial magnetic susceptibility of the SCLF are compared against Molecular Dynamic simulation data. In all cases, theoretical predictions using logarithmic form of the Helmholtz free energy are seen to be superior, but they only have an applicability range of the effective dipolar coupling constant λe<1.5. For highest values of λe, the structural transition of the magnetic dipoles in SCLF is observed in Molecular Dynamic simulation. It has been shown that for λe≳2, an antiferromagnetic order appears in the system.

D -wave superconductivity and Bogoliubov-Fermi surfaces in Rarita-Schwinger-Weyl semimetals

We uncover the properties of complex tensor (d-wave) superconducting order in three-dimensional Rarita-Schwinger-Weyl semimetals that host pseudospin-3/2 fermions at a fourfold linear band crossing point. Although the general theory of d-wave order was originally developed for materials displaying quadratic band touching, it directly applies to the case of semimetals with linear dispersion, several candidate compounds of which have been discovered experimentally very recently. The spin-3/2 nature of the fermions allows for the formation of spin-2 Cooper pairs which may be described by a complex second-rank tensor order parameter. In the case of linear dispersion, for the chemical potential at the Fermi point and at strong coupling, the energetically preferred superconducting state is the uniaxial nematic state, which preserves time-reversal symmetry and provides a full (anisotropic) gap for quasiparticle excitations. In contrast, at a finite chemical potential, we find that the usual weak-coupling instability is towards the "cyclic state", well known from the studies of multicomponent Bose-Einstein condensates, which breaks time reversal symmetry maximally, has vanishing average value of angular momentum, and features 16 small Bogoliubov--Fermi surfaces. The Rarita-Schwinger-Weyl semimetals provide therefore the first example of weakly coupled, three-dimensional, isotropic d-wave superconductors where the ground state is uniquely determined by the quartic expansion of the mean-field free energy, and not afflicted by the accidental degeneracy first noticed by Mermin over 40 years ago. We discuss the appearance and stability of the Bogoliubov-Fermi surfaces in absence of inversion symmetry in the electronic Hamiltonian, as in the case at hand.

Surfactant/biosurfactant mixing: Adsorption of saponin/nonionic surfactant mixtures at the air-water interface

Saponins are naturally occurring biosurfactants present in a wide range of plant species. They are highly surface active glycosides, and are used to stabilise foams and emulsions in foods, beverages and cosmetics. They have great potential for an even wider range of applications, especially when mixed with different synthetic surfactants. Understanding those mixing properties are key to the exploitation of saponins in that wider range of potential applications.The surface adsorption properties of the saponin, escin, with two conventional nonionic surfactants, polyethylene glycol surfactants, have been studied at the air-water interface using neutron reflectivity, NR, and surface tension, ST.Although the saponin and polyethylene glycol, CnEOm, surfactants are both nonionic the disparity in the relative surface activities and packing constraints result in non-ideal mixing. Comparison with the predictions of the pseudo phase approximation requires the inclusion of the quadratic, cubic and quartic terms in the expansion of the excess free energy of mixing to explain the variations in the surface composition. For escin/pentaethylene glycol monododecyl ether, C12EO5, the interaction is attractive and close to ideal. For escin/octaethylene glycol monododecyl ether, C12EO8, it is repulsive and close to the criteria for demixing. The differences in mixing behaviour are attributed to greater packing constraints imposed by the larger ethylene oxide headgroup of the C12EO8 compared to C12EO5.

Spin-orbit coupling and spin-triplet pairing symmetry in Sr2RuO4

Spin-orbit coupling (SOC) plays a crucial role in determining the spin structure of an odd parity psedospin-triplet Cooper pairing state. Here, we present a thorough study of how SOC lifts the degeneracy among different p-wave pseudospin-triplet pairing states in a widely used microscopic model for $\mathrm{Sr_2 Ru O_4}$, combining a Ginzburg-Landau (GL) free energy expansion, a symmetry analysis of the model, and numerical weak-coupling renormalization group (RG) and random phase approximation (RPA) calculations. These analyses are then used to critically re-examine previous numerical results on the stability of chiral p-wave pairing. The symmetry analysis can serve as a guide for future studies, especially numerical calculations, on the pairing instability in $\mathrm{Sr_2 Ru O_4}$ and can be useful for studying other multi-band spin-triplet superconductors where SOC plays an important role.