Order Parameter Correlation Function Research Articles

Effect of distribution shape on the melting transition, local ordering, and dynamics in a model size-polydisperse two-dimensional fluid

Abstract We study the effect of particle size polydispersity (δ) on the melting transition (T *), local ordering, solid–liquid coexistence phase and dynamics of two-dimensional Lennard–Jones fluids up to moderate polydispersity by means of computer simulations. The particle sizes are drawn at random from the Gaussian (G) and uniform (U) distribution functions. For these systems, we further consider two different kinds of particles, viz., particles having the same mass irrespective of size, and in the other case the mass of the particle scales with its size. It is observed that with increasing polydispersity, the value of T * initially increases due to improved packing efficiency (ϕ) followed by a decrease and terminates at δ ≈ 8% (U-system) and 14% (G-system) with no significant difference for both mass types. The interesting observation is that the particular value at which ϕ drops suddenly coincides with the peak of the heat capacity (CP ) curve, indicating a transition. The quantification of local particle ordering through the hexatic order parameter (Q 6), Voronoi construction and pair correlation function reveals that the ordering decreases with increasing δ and T. Furthermore, the solid–liquid coexistence region for the G-system is shown to be comparatively wider in the T–δ plane phase diagram than that for the U system. Finally, the study of dynamics reveals that polydisperse systems relax faster compared to monodisperse systems; however, no significant qualitative differences, depending on the distribution type and mass polydispersity, are observed.

Growth and aging in a few phase-separating active matter systems.

Via computer simulations we study evolution dynamics in systems of continuously moving active Brownian particles. The obtained results are discussed against those from the passive 2D Ising case. Following sudden quenches of random configurations to state points lying within the miscibility gaps and to the critical points, we investigate the far-from-steady-state dynamics by calculating quantities associated with structure and characteristic length scales. We also study aging for quenches into the miscibility gap and provide a quantitative picture for the scaling behavior of the two-time order-parameter correlation function. The overall structure and dynamics are consistent with expectations from the Ising model. This remains true for certain active lattice models as well, for which we present results for quenches to the critical points.

Shape and interfacial structure of droplets. Exact results and simulations

We consider the fluctuating interface of a droplet pinned on a flat wall. For such a system we compare results obtained within the exact field theory of phase separation in two dimensions and Monte Carlo (MC) simulations for the Ising model. The interface separating coexisting phases splits and hosts drops whose effect is to produce subleading corrections to the order parameter profile and correlation functions. In this paper we establish the first direct measurement of interface structure effects by means of high-performance MC simulations which successfully confirm the field-theoretical results. Simulations are found to corroborate the theoretical predictions for interface structure effects whose analytical expression has recently been obtained. It is thanks to these higher-order corrections that we are able to correctly resettle a longstanding discrepancy between simulations and theory for the order parameter profile. In addition, our results clearly establish the long-ranged decay of interfacial correlations in the direction parallel to the interface and their spatial confinement within the interfacial region also in the presence of a wall from which the interface is entropically repelled.

Boundary criticality of the O(N) model in d = 3 critically revisited

It is known that the classical O(N)O(N) model in dimension d > 3dgt;3 at its bulk critical point admits three boundary universality classes: the ordinary, the extra-ordinary and the special. For the ordinary transition the bulk and the boundary order simultaneously; the extra-ordinary fixed point corresponds to the bulk transition occurring in the presence of an ordered boundary, while the special fixed point corresponds to a boundary phase transition between the ordinary and the extra-ordinary classes. While the ordinary fixed point survives in d = 3d=3, it is less clear what happens to the extra-ordinary and special fixed points when d = 3d=3 and N \ge 2N≥2. Here we show that formally treating NN as a continuous parameter, there exists a critical value N_c > 2Ncgt;2 separating two distinct regimes. For 2 \leq N < N_c2≤N<Nc the extra-ordinary fixed point survives in d = 3d=3, albeit in a modified form: the long-range boundary order is lost, instead, the order parameter correlation function decays as a power of \log rlogr. For N > N_cNgt;Nc there is no fixed point with order parameter correlations decaying slower than power law. We discuss several scenarios for the evolution of the phase diagram past N = N_cN=Nc. Our findings appear to be consistent with recent Monte Carlo studies of classical models with N = 2N=2 and N = 3N=3. We also compare our results to numerical studies of boundary criticality in 2+1D quantum spin models.

Soluble fermionic quantum critical point in two dimensions

We study a model for a quantum critical point in two spatial dimensions between a semimetallic phase, characterized by a stable quadratic Fermi node, and an ordered phase, in which the spectrum develops a band gap. The quantum critical behavior can be computed exactly, and we explicitly derive the scaling laws of various observables. While the order-parameter correlation function at criticality satisfies the usual power law with anomalous exponent $\eta_\phi = 2$, the correlation length and the expectation value of the order parameter exhibit essential singularities upon approaching the quantum critical point from the insulating side, akin to the Berezinskii-Kosterlitz-Thouless transition. The susceptibility, on the other hand, has a power-law divergence with non-mean-field exponent $\gamma = 2$. On the semimetallic side, the correlation length remains infinite, leading to an emergent scale invariance throughout this phase.

Spatio-Temporal Spreading of Correlations in the Bose–Hubbard Model

We have developed a formalism that allows the study of correlations in space and time in the Bose–Hubbard model, both in equilibrium and out of equilibrium. We obtain a two-particle irreducible effective action in the contour-time formalism that allows us to obtain equations of motion for the superfluid order parameter and two-point correlation functions. We use these equations to calculate equilibrium phase diagrams that are significant improvements on mean field theory and apply them to study the spreading of correlations after a quench in the Mott insulator phase. We study the single-particle density matrix and find velocities for the spreading of correlations in one, two and three dimensions. We discuss the prospects of generalizing this approach to study the disordered Bose–Hubbard model.

Geometry of bounded critical phenomena

The quest for a satisfactory understanding of systems at criticality in dimensions d > 2 is a major field of research. We devise here a geometric description of bounded systems at criticality in any dimension d. This is achieved by altering the flat metric with a space dependent scale factor γ(x), x belonging to a bounded domain Ω. γ(x) is chosen in order to have a scalar curvature to be constant and matching the one of the hyperbolic space, the proper notion of curvature being—as called in the mathematics literature—the fractional Q-curvature. The equation for γ(x) is found to be the fractional Yamabe equation (to be solved in Ω) that, in absence of anomalous dimension, reduces to the usual Yamabe equation in the same domain. From the scale factor γ(x) we obtain novel predictions for the scaling form of one-point order parameter correlation functions. A (necessary) virtue of the proposed approach is that it encodes and allows to naturally retrieve the purely geometric content of two-dimensional boundary conformal field theory. From the critical magnetization profile in presence of boundaries one can extract the scaling dimension of the order parameter, Δ ϕ . For the 3D Ising model we find Δ ϕ = 0.518 142(8) which favorably compares (at the fifth decimal place) with the state-of-the-art estimate. A nontrivial prediction is the structure of two-point spin-spin correlators at criticality. They should depend on the fractional Q-hyperbolic distance calculated from the metric, in turn depending only on the shape of the bounded domain and on Δ ϕ . Numerical simulations of the 3D Ising model on a slab geometry are found to be in agreement with such predictions.

Fluctuation-dominated phase ordering at a mixed order transition

Mixed order transitions are those which show a discontinuity of the order parameter as well as a divergent correlation length at the transition (critical) point. We show that the behavior of the order parameter correlation function along the transition line of mixed order transitions can change from normal critical behavior with power law decay, to fluctuation-dominated phase ordering as a parameter is varied. The defining features of fluctuation-dominated order are anomalous fluctuations which remain large in the thermodynamic limit, and correlation functions which approach a finite value through a cusp singularity as the separation scaled by the system size approaches zero. We demonstrate that fluctuation-dominated order sets in along a portion of the transition line of an Ising model with truncated long-range interactions which was earlier shown to exhibit mixed order transitions. We also argue that this connection may hold more generally.

The structure and dipolar properties of CO2 adsorbed in a porous glassy chalcogel: Insights from first-principles molecular dynamics

A thorough understanding of the interactions of CO2 with the hosting porous network is crucial for the design of new adsorbent materials with enhanced gas adsorption capacity and selectivity. In this paper, first-principles molecular dynamics simulations are performed to assess the interactions of CO2 adsorbed in a nanoporous glassy chalcogenide (i.e. chalcogel) under relevant laboratory conditions. The structure and local organization of the confined CO2 molecules are analyzed in terms of atomic density, orientational order parameter and pair correlation functions. Maximally localized Wannier functions are used to unravel the electronic structure, the local molecular dipole and the nature of the chemical bonding at the interface between CO2 and the glassy surface. Our results provide a useful insight on the complex interplay between the chemical interactions competing at the buried interface made by the CO2 molecules adsorbed in g-GeS2 chalcogel.

Contour-time approach to the Bose–Hubbard model in the strong coupling regime: Studying two-point spatio-temporal correlations at the Hartree–Fock–Bogoliubov level

We develop a formalism that allows the study of correlations in space and time in both the superfluid and Mott insulating phases of the Bose–Hubbard Model. Specifically, we obtain a two particle irreducible effective action within the contour-time formalism that allows for both equilibrium and out of equilibrium phenomena. We derive equations of motion for both the superfluid order parameter and two-point correlation functions. To assess the accuracy of this formalism, we study the equilibrium solution of the equations of motion and compare our results to existing strong coupling methods as well as exact methods where possible. We discuss applications of this formalism to out of equilibrium situations.

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.

Different growth rates for spin and superfluid order in a quenched spinor condensate

In this paper we study the coarsening dynamics of a spinor condensate quenched into an easy-axis ferromagnetic phase by a sudden change in the quadratic Zeeman energy. We show that applying a spin rotation prior to changing the Zeeman energy accelerates the development of local order and reduces heating. We examine the longitudinal spin ordering and the superfluid ordering of the system and show that the respective order parameter correlation functions exhibit dynamic scaling in the late time dynamics. Our results also demonstrate that these two types of order grow at different rates, i.e.~with different dynamic critical exponents. The spin domain area distribution is calculated and is shown to have power law scaling behavior expected from percolation theory.

Kinetics of Order-Disorder Transitions in Binary Mixtures: A Monte Carlo Study

We study ordering kinetics in symmetric and asymmetric binary mixtures, undergoing an order-disorder transition below the critical temperature. We model the microscopic dynamics via an anti-ferromagnetic Ising model with Kawasaki spin-exchange kinetics. This conserves the composition but the order parameter, (staggered magnetization) is not a conserved quantity. The order-parameter correlation function shows dynamical scaling, and the scaling function is independent of mixture composition. The average domain size exhibits power law growth: L(t) ∼ tα. The asymptotic growth regime has α = ½, though there can be prolonged transients with α < ½ for asymmetric mixtures. Our unambiguous observation of the asymptotic universal regime is facilitated by use of an accelerated Monte Carlo technique.

Aging in ferromagnetic ordering: full decay and finite-size scaling of autocorrelation

Nonequilibrium dynamics in Ising and Ginzburg–Landau models were studied for a nonconserved order parameter that mimics ordering in ferromagnets. The focus was on the understanding of the decay of the two time (t, tw; t > tw) order-parameter correlation function. For this quantity, a full form has been obtained empirically which, for t ≫ tw, provides a power-law ∼ (ℓ/ℓw)−λ, ℓ and ℓw being the characteristic lengths at t and tw, respectively. This empirical form was used for a finite-size scaling analysis to obtain the exponent λ in space dimensions d = 2 and 3. Our estimates of λ and understanding of the finite-size effects, for the models considered, provide useful information on the relevance of thermal noise. The values of λ obtained are in good agreement with the predictions of a theory based on Gaussian auxiliary field ansatz.

Specific heat in two-dimensional melting.

We report the specific heat cN around the melting transition(s) of micrometer-sized superparamagnetic particles confined in two dimensions, calculated from fluctuations of positions and internal energy, and corresponding Monte Carlo simulations. Since colloidal systems provide single particle resolution, they offer the unique possibility to compare the experimental temperatures of the peak position of cN(T) and symmetry breaking, respectively. While order parameter correlation functions confirm the Kosterlitz-Thouless-Halperin-Nelson-Young melting scenario where translational and orientational order symmetries are broken at different temperatures with an intermediate so called hexatic phase, we observe a single peak of the specific heat within the hexatic phase, with excellent agreement between experiment and simulation. Thus, the peak is not associated with broken symmetries but can be explained with the total defect density, which correlates with the maximum increase of isolated dislocations. The absence of a latent heat strongly supports the continuous character of both transitions.

Temperature dependence of the pair coherence and healing lengths for a fermionic superfluid throughout the BCS-BEC crossover

We calculate the pair correlation function and the order parameter correlation function, which probe, respectively, the intrapair and interpair correlations of a Fermi gas with attractive interparticle interaction, in terms of a diagrammatic approach as a function of coupling throughout the BCS-Bose-Einstein condensation (BEC) crossover and of temperature, both in the superfluid and normal phase across the critical temperature ${T}_{c}$. Several physical quantities are obtained from this calculation, including the pair coherence and healing lengths, the Tan's contact, the crossover temperature ${T}^{*}$ below which interpair correlations begin to build up in the normal phase, and the signature for the disappearance of the underlying Fermi surface which tends to survive in spite of pairing correlations. A connection is also made with recent experimental data on the temperature dependence of the normal coherence length as extracted from the proximity effect measured in high-temperature (cuprate) superconductors.

Dynamical properties of random-field Ising model

Extensive Monte Carlo simulations are performed on a two-dimensional random field Ising model. The purpose of the present work is to study the disorder-induced changes in the properties of disordered spin systems. The time evolution of the domain growth, the order parameter, and the spin-spin correlation functions are studied in the nonequilibrium regime. The dynamical evolution of the order parameter and the domain growth shows a power law scaling with disorder-dependent exponents. It is observed that for weak random fields, the two-dimensional random field Ising model possesses long-range order. Except for weak disorder, exchange interaction never wins over pinning interaction to establish long-range order in the system.

Projection operator approach to the Bose-Hubbard model

We develop a projection operator formalism for studying both the zero temperature equilibrium phase diagram and the non-equilibrium dynamics of the Bose-Hubbard model. Our work, which constitutes an extension of Phys. Rev. Lett. {\bf 106}, 095702 (2011), shows that the method provides an accurate description of the equilibrium zero temperature phase diagram of the Bose-Hubbard model for several lattices in two- and three-dimensions (2D and 3D). We show that the accuracy of this method increases with the coordination number $z_0$ of the lattice and reaches to within 0.5% of quantum Monte Carlo data for lattices with $z_0=6$. We compute the excitation spectra of the bosons using this method in the Mott and the superfluid phases and compare our results with mean-field theory. We also show that the same method may be used to analyze the non-equilibrium dynamics of the model both in the Mott phase and near the superfluid-insulator quantum critical point where the hopping amplitude $J$ and the on-site interaction $U$ satisfy $z_0J/U \ll 1$. In particular, we study the non-equilibrium dynamics of the model both subsequent to a sudden quench of the hopping amplitude $J$ and during a ramp from $J_i$ to $J_f$ characterized by a ramp time $\tau$ and exponent $\alpha$: $J(t)=J_i +(J_f-J_i) (t/\tau)^{\alpha}$. We compute the wavefunction overlap $F$, the residual energy $Q$, the superfluid order parameter $\Delta(t)$, the equal-time order parameter correlation function $C(t)$, and the defect formation probability $P$ for the above-mentioned protocols and provide a comparison of our results to their mean-field counterparts. We find that $Q$, $F$, and $P$ do not exhibit the expected universal scaling. We explain this absence of universality and show that our results for linear ramps compare well with the recent experimental observations.

Quantum quench in the transverse field Ising chain: I. Time evolution of order parameter correlators

We consider the time evolution of order parameter correlation functions after a sudden quantum quench of the magnetic field in the transverse field Ising chain. Using two novel methods based on determinants and form factor sums respectively, we derive analytic expressions for the asymptotic behaviour of one- and two-point correlators. We discuss quenches within the ordered and disordered phases as well as quenches between the phases and to the quantum critical point. We give detailed accounts of both methods.

Domain growth and ordering kinetics in dense quark matter

The kinetics of chiral transitions in quark matter is studied in a two flavor Nambu-Jona-Lasinio model. We focus on the phase ordering dynamics subsequent to a temperature quench from the massless quark phase to the massive quark phase. We study the dynamics by considering a phenomenological model (Ginzburg-Landau free-energy functional). The morphology of the ordering system is characterized by the scaling of the order-parameter correlation function.