Ursell operators in statistical physics of dense systems: the role of high order operators and of exchange cycles

The purpose of this article is to discuss cluster expansions in dense quantum systems as well as their interconnection with exchange cycles. We show in general how the Ursell operators of order 3 or more contribute to an exponential which corresponds to a mean-field energy involving the second operator U2, instead of the potential itself as usual. In a first part, we consider classical statistical mechanics and recall the relation between the reducible part of the classical cluster integrals and the mean-field; we introduce an alternative method to obtain the linear density contribution to the mean-field, which is based on the notion of tree-diagrams and provides a preview of the subsequent quantum calculations. We then proceed to study quantum particles with Boltzmann statistics (distinguishable particles) and show that each Ursell operator Un with n greater or equal to 3 contains a ``tree-reducible part'', which groups naturally with U2 through a linear chain of binary interactions; this part contributes to the associated mean-field experienced by particles in the fluid. The irreducible part, on the other hand, corresponds to the effects associated with three (or more) particles interacting all together at the same time. We then show that the same algebra holds in the case of Fermi or Bose particles, and discuss physically the role of the exchange cycles, combined with interactions. Bose condensed systems are not considered at this stage. The similarities and differences between Boltzmann and quantum statistics are illustrated by this approach, in contrast with field theoretical or Green's functions methods, which do not allow a separate study of the role of quantum statistics and dynamics.Comment: 31 pages, 7 figure

Large amplitude spin waves in ultra-cold gases

We discuss the theory of spin waves in non-degenerate ultra-cold gases, and compare various methods which can be used to obtain appropriate kinetic equations. We then study non-hydrodynamic situations, where the amplitude of spin waves is sufficiently large to bring the system far from local equilibrium. In the first part of the article, we compare two general methods which can be used to derive a kinetic equation for a dilute gas of atoms (bosons or fermions) with two internal states (treated as a pseudo-spin 1/2). The collisional methods are in the spirit of Boltzmann's original derivation of his kinetic equation where, at each point of space, the effects of all sorts of possible binary collisions are added. We discuss two different versions of collisional methods, the Yvon-Snider approach and the S matrix approach. The second method uses the notion of mean field, which modifies the drift term of the kinetic equation, in the line of the Landau theory of transport in quantum liquids. For a dilute cold gas, it turns out that all these derivations lead to the same drift terms in the transport equation, but differ in the precise expression of the collision integral and in higher order gradient terms. In the second part of the article, the kinetic equation is applied to spin waves in trapped ultra-cold gases. Numerical simulations are used to illustrate the strongly non-hydrodynamic character of the spin waves recently observed with trapped Rb87 atoms. The decay of the phenomenon, which takes place when the system relaxes back towards equilibrium, is also discussed, with a short comment on decoherence.Comment: To appear in Eur. Phys. J.

Flow curves of colloidal dispersions close to the glass transition: Asymptotic scaling laws in a schematic model of mode coupling theory

The flow curves, viz. the curves of stationary stress under steady shearing, are obtained close to the glass transition in dense colloidal dispersions using asymptotic expansions in a schematic model of mode coupling theory. The shear thinning of the viscosity in fluid states and the yielding of glassy states is discussed. At the transition between fluid and shear-molten glass, simple and generalized Herschel-Bulkley laws are derived with power law exponents that can be computed for different particle interactions from the equilibrium structure factor.Comment: 14 pages, 14 figures, 4 tables, Eur. Phys. J. E (submitted

Ab-initio description of heterostructural alloys: Thermodynamic and structural properties of Mg_x Zn_{1-x} O and Cd_x Zn_{1-x} O

Pseudobinary heterostructural alloys of ZnO with MgO or CdO are studied by composing the system locally of clusters with varying ratio of cations. We investigate fourfold (wurtzite structure) and sixfold (rocksalt structure) coordination of the atoms. By means of density functional theory we study a total number of 256 16-atom clusters divided into 22 classes for the wurtzite structure and 16 classes for the rocksalt structure for each of the alloy systems. The fraction with which each cluster contributes to the alloy is determined for a given temperature T and composition x within (i) the generalized quasi-chemical approximation, (ii) the model of a strict-regular solution, and (iii) the model of microscopic decomposition. From the cluster fractions we derive conclusions about the miscibility and the critical compositions at which the average crystal structure changes. Thermodynamic properties such as the mixing free energy and the mixing entropy are investigated for the three different statistics. We discuss the consequences of the two different local lattice structures for characteristic atomic distances, cohesive energies, and the alloys' elasticities. The differences in the properties of Mg_x Zn_{1-x} O and Cd_x Zn_{1-x} O are explained and discussed.Comment: 15 pages, 13 figure

Tunable orbital susceptibility in α\alpha-T3{\cal T}_3 tight-binding models

We study the importance of interband effects on the orbital susceptibility of three bands $\alpha$-${\cal T}_3$ tight-binding models. The particularity of these models is that the coupling between the three energy bands (which is encoded in the wavefunctions properties) can be tuned (by a parameter $\alpha$) without any modification of the energy spectrum. Using the gauge-invariant perturbative formalism that we have recently developped, we obtain a generic formula of the orbital susceptibility of $\alpha$-${\cal T}_3$ tight-binding models. Considering then three characteristic examples that exhibit either Dirac, semi-Dirac or quadratic band touching, we show that by varying the parameter $\alpha$ and thus the wavefunctions interband couplings, it is possible to drive a transition from a diamagnetic to a paramagnetic peak of the orbital susceptibility at the band touching. In the presence of a gap separating the dispersive bands, we show that the susceptibility inside the gap exhibits a similar dia to paramagnetic transition.Comment: 15 pages,5 figs. Proceedings of the International Workshop on Dirac Electrons in Solids 2015Proceedings of the International Workshop on Dirac Electrons in Solids 201

Structural relaxation of polydisperse hard spheres: comparison of the mode-coupling theory to a Langevin dynamics simulation

We analyze the slow, glassy structural relaxation as measured through collective and tagged-particle density correlation functions obtained from Brownian dynamics simulations for a polydisperse system of quasi-hard spheres in the framework of the mode-coupling theory of the glass transition (MCT). Asymptotic analyses show good agreement for the collective dynamics when polydispersity effects are taken into account in a multi-component calculation, but qualitative disagreement at small $q$ when the system is treated as effectively monodisperse. The origin of the different small-$q$ behaviour is attributed to the interplay between interdiffusion processes and structural relaxation. Numerical solutions of the MCT equations are obtained taking properly binned partial static structure factors from the simulations as input. Accounting for a shift in the critical density, the collective density correlation functions are well described by the theory at all densities investigated in the simulations, with quantitative agreement best around the maxima of the static structure factor, and worst around its minima. A parameter-free comparison of the tagged-particle dynamics however reveals large quantiative errors for small wave numbers that are connected to the well-known decoupling of self-diffusion from structural relaxation and to dynamical heterogeneities. While deviations from MCT behaviour are clearly seen in the tagged-particle quantities for densities close to and on the liquid side of the MCT glass transition, no such deviations are seen in the collective dynamics.Comment: 23 pages, 26 figure

Quasiparticle band structure based on a generalized Kohn-Sham scheme

We present a comparative full-potential study of generalized Kohn-Sham schemes (gKS) with explicit focus on their suitability as starting point for the solution of the quasiparticle equation. We compare $G_0W_0$ quasiparticle band structures calculated upon LDA, sX, HSE03, PBE0, and HF functionals for exchange and correlation (XC) for Si, InN and ZnO. Furthermore, the HSE03 functional is studied and compared to the GGA for 15 non-metallic materials for its use as a starting point in the calculation of quasiparticle excitation energies. For this case, also the effects of selfconsistency in the $GW$ self-energy are analysed. It is shown that the use of a gKS scheme as a starting point for a perturbative QP correction can improve upon the deficiencies found for LDA or GGA staring points for compounds with shallow $d$ bands. For these solids, the order of the valence and conduction bands is often inverted using local or semi-local approximations for XC, which makes perturbative $G_0W_0$ calculations unreliable. The use of a gKS starting point allows for the calculation of fairly accurate band gaps even in these difficult cases, and generally single-shot $G_0W_0$ calculations following calculations using the HSE03 functional are very close to experiment