Gaussian Fluctuation Theory Research Articles

BCS–BEC crossover in a quasi-two-dimensional Fermi superfluid

We study the crossover from the Bardeen–Cooper–Shrieffer regime to the Bose–Einstein-condensation regime in an anisotropic trapped quantum gas of ultracold fermionic atoms. Using an effective two-dimensional Hamiltonian with renormalized interactions between atoms and dressed molecules, we extend the Gaussian pair fluctuation theory to quasi-two-dimensional systems. The equation of state, pair size and sound velocity as well as the convexity parameters of the Goldstone mode are investigated to show how Fermi superfluidity is affected by reduced dimensionality and Gaussian fluctuations at zero temperature in a wide range of crossover. We compare our results with the recent experiment of atomic gases and find good agreements.

Reversal of the Temperature Dependence of Hydrophobic Hydration in Supercooled Water.

Using simulations and theory, we examine the enthalpy and entropy of hydrophobic hydration which exhibit minima in supercooled water, contrasting with the monotonically increasing temperature dependence traditionally ascribed to these properties. The enthalpy/entropy minima are marked by a negative to positive sign change in the heat capacity at a size-dependent reversal temperature. A Gaussian fluctuation theory accurately captures the reversal temperature, tracing it to water’s distinct thermal expansivity and compressibility influenced by its metastable liquid–liquid critical point.

Bridging Gaussian Density Fluctuations from Microscopic to Macroscopic Volumes: Applications to Non-Polar Solute Hydration Thermodynamics.

The hydration of hydrophobic solutes is intimately related to the spontaneous formation of cavities in water through ambient density fluctuations. Information theory-based modeling and simulations have shown that water density fluctuations in small volumes are approximately Gaussian. For limiting cases of microscopic and macroscopic volumes, water density fluctuations are known exactly and are rigorously related to the density and isothermal compressibility of water. Here, we develop a theory—interpolated gaussian fluctuation theory (IGFT)—that builds an analytical bridge to describe water density fluctuations from microscopic to molecular scales. This theory requires no detailed information about the water structure beyond the effective size of a water molecule and quantities that are readily obtained from water’s equation-of-state—namely, the density and compressibility. Using simulations, we show that IGFT provides a good description of density fluctuations near the mean, that is, it characterizes the variance of occupancy fluctuations over all solute sizes. Moreover, when combined with the information theory, IGFT reproduces the well-known signatures of hydrophobic hydration, such as entropy convergence and solubility minima, for atomic-scale solutes smaller than the crossover length scale beyond which the Gaussian assumption breaks down. We further show that near hydrophobic and hydrophilic self-assembled monolayer surfaces in contact with water, the normalized solvent density fluctuations within observation volumes depend similarly on size as observed in the bulk, suggesting the feasibility of a modified version of IGFT for interfacial systems. Our work highlights the utility of a density fluctuation-based approach toward understanding and quantifying the solvation of non-polar solutes in water and the forces that drive them toward surfaces with different hydrophobicities.

Polariton-polariton interaction beyond the Born approximation: A toy model study

We theoretically investigate the polariton-polariton interaction in microcavities beyond the commonly used Born approximation (i.e., mean-field), by adopting a toy model with a contact interaction to approximately describe the attraction between electrons and holes in quantum well and by using a Gaussian pair fluctuation theory beyond mean-field. We obtain a density or chemical potential independent polariton-polariton interaction strength even in two-dimensions, which result from coupling to the photon field. We show that quantum fluctuations lead to about a factor of two reduction in the polariton-polariton interaction strength within our toy model. Together with corrections to the 1s exciton approximation at very strong light-matter coupling, we find the polariton-polariton interaction strength under typical experimental conditions is overestimated by a factor three in the widely used theories, if our toy model can qualitatively simulate the polariton interaction in GaAs quantum wells. We compare our prediction with the most recent measurement and argue that the beyond-Born-approximation effect to the polariton-polariton interaction strength is crucial for a quantitative understanding of the experimental data by E. Estrecho \textit{et al.}, Phys. Rev. B \textbf{100}, 035306 (2019).

Near-Critical Phase Behavior in Polyelectrolyte Solutions: Effect of Charge Fluctuations.

The critical behaviors in polyelectrolyte (PE) solutions are studied by a renormalized Gaussian fluctuation theory. PEs are fully charged linear wormlike chains. The electrostatic interactions are considered in a continuum solvent while ignoring other interactions. The effects of temperature on the criticality are explored in a salt-free polyanion solution, and the effects of salt are explored in a symmetric mixed solution of polycation and polyanion. An unphysical phase coexistence (UPPC), in which a metastable dense phase coexists with an unstable dilute phase, always exists in the phase diagram. In the near-critical region, the UPPC could interfere with the real phase coexistence. The classical critical point is replaced by a "critical line", giving a flat top for the phase diagram. These behaviors are due to the effect of charge fluctuation in the near-critical region. Our results explain why the mean field approach overestimates the critical salt density by about 10% to experimental data in the study of coacervation between oppositely charged PEs.

Fluctuation diamagnetic susceptibility in type-II superconductors under magnetic field

Strong fluctuation effects were found in both low- and high-field regimes by recent measurements of magnetization on [Formula: see text] (LCCO) single crystals. The low-field fluctuation diamagnetic susceptibility data could not be fitted by simple Gaussian fluctuation theory using the lowest Landau level (LLL) approximation because of the slightly nonlinear behavior around the mean-field transition temperature [Formula: see text]. Self-consistent calculation of fluctuation diamagnetic susceptibility in high-temperature superconductors, based on the Ginzburg–Landau (GL) two-dimensional model and including all Landau levels, is presented. Our results are valid for arbitrary values of the magnetic field not too close to [Formula: see text]. The results agree well with the experimental data in a wide region around [Formula: see text], including both below and above [Formula: see text].

Resonantly interacting p -wave Fermi superfluid in two dimensions: Tan's contact and the breathing mode

Inspired by the renewed experimental activities on $p$-wave resonantly interacting atomic Fermi gases, we theoretically investigate some experimental observables of such systems at zero temperature in two dimensions, using both mean-field theory and Gaussian pair fluctuation theory. These observables include the two $p$-wave contact parameters and the breathing mode frequency, which can be readily measured in current cold-atom setups with $^{40}$K and $^{6}$Li atoms. We find that the many-body component of the two contact parameters exhibits a pronounced peak slightly above the resonance and consequently leads to a dip in the breathing mode frequency. In the resonance limit, we discuss the dependence of the equation of state and the breathing mode frequency on the dimensionless effective range of the interaction, $k_{F}R_{p}\ll1$, where $k_{F}$ is the Fermi wavevector and $R_{p}$ is the effective range. The breathing mode frequency $\omega_{B}$ deviates from the scale-invariant prediction of $\omega_{c}=2\omega_{0}$, where $\omega_{0}$ is the trapping frequency of the harmonic potential. This frequency shift is caused by the necessary existence of the effective range. In the small range limit, we predict that the mode frequency deviation at the leading order is given by, $\delta\omega_{B}\simeq-(\omega_{0}/4)\ln^{-1}(k_{F}R_{p})$.

Quantum fluctuations of a resonantly interacting p -wave Fermi superfluid in two dimensions

Using the Gaussian pair fluctuation theory, we investigate quantum fluctuations of a strongly interacting two-dimensional chiral \textit{p}-wave Fermi superfluid at the transition from a Bose-Einstein condensate (BEC) to a topologically non-trivial Bardeen-Cooper-Schrieffer (BCS) superfluid. Near the topological phase transition at zero chemical potential, $\mu=0$, we observe that quantum fluctuations strongly renormalize the zero-temperature equations of state, sound velocity, pair-breaking velocity, and Berezinskii-Kosterlitz-Thouless (BKT) critical temperature of the Fermi superfluid, all of which can be non-analytic functions of the interaction strength. The indication of non-analyticity is particularly evident in the BKT critical temperature, which also exhibits a pronounced peak near the topological phase transition. Across the transition and towards the BEC limit we find that the system quickly becomes a trivial interacting Bose liquid, whose properties are less dependent on the interparticle interaction. The qualitative behavior of composite bosons in the BEC limit remains to be understood.

Salt Partitioning in Complex Coacervation of Symmetric Polyelectrolytes

We perform a general thermodynamic analysis for the salt partitioning behavior in the coexisting phases for symmetric mixtures of polycation and polyanion solutions. We find that salt partitioning is determined by the competition between two factors involving the ratio of the polyelectrolyte concentration in the coacervate phase to that in the supernatant phase and the difference in the exchange excess chemical potential Δμex—the excess chemical potential difference between PE segments and small ions—between the coexisting phases. The enrichment of salt ions in the coacervate phase predicted by the Voorn–Overbeek theory is shown to arise from its neglect of chain connectivity in the excess free energy which results in Δμex = 0 under all conditions. We argue that chain connectivity in general leads to a finite value of Δμex, which decreases with increasing PE concentration. Explicit calculations using theories that include the chain connectivity correlations—a simple liquid-state theory and a renormalized Gaussian fluctuation theory—show nonmonotonic behavior of the salt-partitioning coefficient (the ratio of salt ion concentration in the coacervate phase to that in the supernatant phase): it is larger than 1 at very low salt concentrations, reaches a minimum at some intermediate salt concentration, and approaches 1 at the critical point. This behavior is consistent with recent computer simulation and experimental results.

Breathing-mode frequency of a strongly interacting Fermi gas across the two- to three-dimensional crossover

We address the interplay between dimension and quantum anomaly on the breathing mode frequency of a strongly interacting Fermi gas harmonically trapped at zero temperature. Using a beyond-mean-field Gaussian pair fluctuation theory, we employ periodic boundary conditions to simulate the dimensionality of the system and impose a local-density approximation, with two different schemes, to model different trapping potentials in the tightly confined axial direction. By using a sum-rule approach, we compute the breathing mode frequency associated with a small variation of the trapping frequency along the weakly confined transverse direction and describe its behavior as a function of the dimensionality, from two to three dimensions, and of the interaction strength. We compare our predictions with previous calculations on the two-dimensional breathing mode anomaly and discuss their possible observation in ultracold Fermi gases of $^{6}\mathrm{Li}$ and $^{40}\mathrm{K}$ atoms.

Superfluid density and critical velocity near the Berezinskii-Kosterlitz-Thouless transition in a two-dimensional strongly interacting Fermi gas

We theoretically investigate superfluidity in a strongly interacting Fermi gas confined to two dimensions at finite temperature. Using a Gaussian pair fluctuation theory in the superfluid phase, we calculate the superfluid density and determine the critical temperature and chemical potential at the Berezinskii-Kosterlitz-Thouless transition. We propose that the transition can be unambiguously demonstrated in cold-atom experiments by stirring the superfluid Fermi gas using a red detuned laser beam, to identify the characteristic jump in the local Landau critical velocity at the superfluid-normal interface, as the laser beam moves across the cloud.

Non-Arrhenius form of the Henry adsorption on inhomogeneous substrates: the effect of frozen disorder

Usually, when considering sub-monolayer adsorption at different temperatures T, the logarithm of the Henry constant is represented as $$\ln {K_H}\sim {U_0}/T$$ with some constant “heat of adsorption” U 0 > 0. However, such Arrhenius-type form has a clear base only for an ideal translational-invariant crystal substrate. In more real situation, e.g. for a structurally disordered substrate, the “heat of adsorption” will be some random function of two-dimensional coordinates characterized by some distribution function of its different values. Starting from general principles of the theory of Gaussian fluctuations, we show that the adsorption at the substrate with bulk inhomogeneous structure leads to expression $$\ln {K_H}\sim {U_0}/T+{\Delta ^2}/{T^2}$$ with some fluctuation quantity Δ. A possibility to observe such “non-Arrhenius” additive seems most favorable at rather low temperatures for substrates modeled by substitutional solid solutions. The theoretical predictions are illustrated by comparison with the experimental data obtained for some disordered adsorbents.

Dimensional crossover in a strongly interacting ultracold atomic Fermi gas

We theoretically explore the crossover from three dimensions (3D) to two (2D) in a strongly interacting atomic Fermi superfluid through confining the transverse spatial dimension. Using the gaussian pair fluctuation theory, we determine the zero-temperature equation of state and Landau critical velocity as functions of the spatial extent of the transverse dimension and interaction strength. In the presence of strong interactions, we map out a dimensional crossover diagram from the location of maximum critical velocity, which exhibits distinct dependence on the transverse dimension from 2D to quasi-2D, and to 3D. We calculate the dynamic structure factor to characterize the low-energy excitations of the system and propose that the intermediate quasi-2D regime can be experimentally probed using Bragg spectroscopy.

General Linearized Theory of Quantum Fluctuations around Arbitrary Limit Cycles.

The theory of Gaussian quantum fluctuations around classical steady states in nonlinear quantum-optical systems (also known as standard linearization) is a cornerstone for the analysis of such systems. Its simplicity, together with its accuracy far from critical points or situations where the nonlinearity reaches the strong coupling regime, has turned it into a widespread technique, being the first method of choice in most works on the subject. However, such a technique finds strong practical and conceptual complications when one tries to apply it to situations in which the classical long-time solution is time dependent, a most prominent example being spontaneous limit-cycle formation. Here, we introduce a linearization scheme adapted to such situations, using the driven Van der Pol oscillator as a test bed for the method, which allows us to compare it with full numerical simulations. On a conceptual level, the scheme relies on the connection between the emergence of limit cycles and the spontaneous breaking of the symmetry under temporal translations. On the practical side, the method keeps the simplicity and linear scaling with the size of the problem (number of modes) characteristic of standard linearization, making it applicable to large (many-body) systems.

Superconducting fluctuations in a thin NbN film probed by the Hall effect

We present a comprehensive study of how superconducting fluctuations in the normal state contribute to the conductivity tensor in a thin (119 $\AA$) film of NbN. It is shown how these fluctuations drive a sign change in the Hall coefficient $R_\mathrm{H}$ for low magnetic fields near the superconducting transition. The scaling behaviours as a function of distance to the transition $\epsilon=\ln(T/T_\mathrm{c})$ of the longitudinal ($\sigma_\mathrm{xx}$) and transverse ($\sigma_\mathrm{xy}$) conductivity is found to be consistent with Gaussian fluctuation theory. Moreover, excellent quantitative agreement between theory and experiment is obtained without any adjustable parameters. Our experimental results thus provide a case study of the conductivity tensor originating from short-lived Cooper pairs.

Electrostatic correlations and the polyelectrolyte self energy.

We address the effects of chain connectivity on electrostaticfluctuations in polyelectrolyte solutions using a field-theoretic, renormalizedGaussian fluctuation (RGF) theory. As in simple electrolyte solutions [Z.-G. Wang,Phys. Rev. E 81, 021501 (2010)], the RGF provides a unified theory forelectrostatic fluctuations, accounting for both dielectric and charge correlationeffects in terms of the self-energy. Unlike simple ions, the polyelectrolyte self energydepends intimately on the chain conformation, and our theory naturally provides aself-consistent determination of the response of intramolecular chain structure topolyelectrolyte and salt concentrations. The effects of the chain-conformation on theself-energy and thermodynamics are especially pronounced for flexiblepolyelectrolytes at low polymer and salt concentrations, where application of thewrong chain structure can lead to a drastic misestimation of the electrostaticcorrelations. By capturing the expected scaling behavior of chain size from dilute tosemi-dilute regimes, our theory provides improved estimates of the self energy at lowpolymer concentrations and correctly predicts the eventual N-independenceof the critical temperature and concentration of salt-free solutions of flexiblepolyelectrolytes. We show that the self energy can be interpreted in terms of aninfinite-dilution energy μm,0el and a finite concentrationcorrelation correction μcorr which tends to cancel out the formerwith increasing concentration.

Gaussian fluctuations of spatially inhomogeneous polymers.

Inhomogeneous polymers, such as partially cofilin-bound actin filaments, play an important role in various natural and biotechnological systems. At finite temperatures, inhomogeneous polymers exhibit non-trivial thermal fluctuations. More broadly, these are relatively simple examples of fluctuations in spatially inhomogeneous systems, which are less understood compared to their homogeneous counterparts. Here we develop a statistical theory of torsional, extensional and bending Gaussian fluctuations of inhomogeneous polymers (chains), where the inhomogeneity is an inclusion of variable size and stiffness, using both continuum and discrete approaches. First, we analytically calculate the complete eigenvalue and eigenmode spectra within a continuum field theory. In particular, we show that the wavenumber inside and outside of the inclusion is nearly linear in the eigenvalue index, with a nontrivial coefficient. Second, we solve the corresponding discrete problem and highlight fundamental differences between the continuum and discrete spectra. In particular, we demonstrate that above a certain wavenumber the discrete spectrum changes qualitatively and discrete evanescent eigenmodes, which do not have continuum counterparts, emerge. The implications of these differences are explored by calculating fluctuation-induced forces associated with free-energy variations with either the inclusion properties (e.g. inhomogeneity formed by adsorbing molecules) or with an external geometric constraint. The former, which is the fluctuation-induced contribution to the adsorbing molecule binding force, is shown to be affected by short wavelengths and thus cannot be calculated using the continuum approach. The latter, on the other hand, is shown to be dominated by long wavelength shape fluctuations and hence is properly described by the continuum theory.

Scattering and Gaussian Fluctuation Theory for Semiflexible Polymers.

The worm-like chain is one of the best theoretical models of the semiflexible polymer. The structure factor, which can be obtained by scattering experiment, characterizes the density correlation in different length scales. In the present review, the numerical method to compute the static structure factor of the worm-like chain model and its general properties are demonstrated. Especially, the chain length and persistence length involved multi-scale nature of the worm-like chain model are well discussed. Using the numerical structure factor, Gaussian fluctuation theory of the worm-like chain model can be developed, which is a powerful tool to analyze the structure stability and to predict the spinodal line of the system. The microphase separation of the worm-like diblock copolymer is considered as an example to demonstrate the usage of Gaussian fluctuation theory.

Beyond Gaussian pair fluctuation theory for strongly interacting Fermi gases

Interacting Fermi systems in the strongly correlated regime play a fundamental role in many areas of physics and are of particular interest to the condensed matter community. Though weakly inter- acting fermions are understood, strongly correlated fermions are difficult to describe theoretically as there is no small interaction parameter to expand about. Existing strong coupling theories rely heavily on the so-called many-body T-matrix approximation that sums ladder-type Feynman diagrams. Here, by acknowledging the fact that the effective interparticle interaction (i.e., the vertex function) becomes smaller above three dimensions, we propose an alternative way to reorganize Feynman diagrams and develop a theoretical framework for interacting Fermi gases beyond the ladder approximation. As an application, we solve the equation of state for three- and two-dimensional strongly interacting fermions and find excellent agreement with experimental [Science 335, 563 (2012)] and other theoretical results above the temperature 0.5TF .

Pseudogap phenomena in a two-dimensional ultracold Fermi gas near the Berezinskii-Kosterlitz-Thouless transition

We investigate single-particle excitations and strong-coupling effects in a twodimensional Fermi gas. Including pairing fluctuations within a Gaussian fluctuation theory, we calculate the density of states ρ(ω) near the Berezinskii-Kosterlitz-Thouless (BKT) transition temperature TBKT. Near TBKT, we show that superfluid fluctuations induce a pseudogap in ρ(ω). The pseudogap structure is very similar to the BCS superfluid density of states, although the superfluid order parameter is absent in the present two-dimensional case. Since a twodimensional 40K Fermi gas has recently been realized, our results would contribute to the understanding of single-particle properties near the BKT instability.