Single-particle Kinetic Energy Research Articles

Entanglement entropy, single-particle occupation probabilities, and short-range correlations

For quantum many-body systems with short-range correlations (SRCs), the intimate relationship between their magnitude, the behavior of the single-particle occupation probabilities at momenta larger than the Fermi momentum, and the entanglement entropy is a new qualitative aspect not studied and exploited yet. A large body of recent condensed matter studies indicates that the time evolution of the entanglement entropy describes the nonequilibrium dynamics of isolated and strongly interacting many-body systems, in a manner similar to the Boltzmann entropy, which is strictly defined for dilute and weakly interacting many-body systems. Both theoretical and experimental studies in nuclei and cold atomic gases have shown that the fermion momentum distribution has a generic behavior $n(k)=C/{k}^{4}$ at momenta larger than the Fermi momentum, due to the presence of SRCs, with approximately 20% of the particles having momenta larger than the Fermi momentum. The presence of the long momentum tails in the presence of SRCs changes the textbook relation between the single-particle kinetic energy and occupation probabilities, ${n}_{\text{mf}}(k)=1/{1+exp\ensuremath{\beta}[\ensuremath{\epsilon}(k)\ensuremath{-}\ensuremath{\mu}]}$ for momenta very different form the Fermi momentum, particularly for dynamics processes. SRCs induced high-momentum tails of the single-particle occupation probabilities increase the entanglement entropy of fermionic systems, which in its turn affects the dynamics of many nuclear reactions, such as heavy-ion collisions and nuclear fission.

Hydrogen self-dynamics in diluted liquid mixtures with neon: An inelastic neutron scattering study.

We have measured the dynamic structure factor of liquid neon-hydrogen mixtures (T=30.1 K) at two different H_{2} concentration levels (namely, 3.4% and 10%) making use of inelastic neutron scattering. This system has been selected since the presence of heavy Ne atoms strongly influences the self-dynamics of the H_{2} centers of mass via the formation of short-lived cages, which act both on the vibrational and the diffusive parts of the single-particle motion. After operating a standard data reduction and the subtraction of the Ne signal, experimental neutron spectra were analyzed through a generalization of the Young and Koppel model, and the H_{2} center-of-mass self-dynamic structure factor was finally extracted for the two liquid samples. Important physical quantities (namely, single-particle mean kinetic energy and self-diffusion coefficient) were estimated from the experimental data and then compared with quantum dynamical calculations, which also provided simulations of the velocity autocorrelation functions for Ne atoms and H_{2} centers of mass. The latter estimates, in the framework of the well-known Gaussian approximation, were used to simulate the H_{2} center-of-mass self-dynamic structure factor in the same kinematic range and thermodynamic conditions of the neutron scattering one. The comparison between measured and calculated spectra turned out to be qualitatively good, but some discrepancies, especially in the low-frequency part, seem to reinforce the idea of the existence of relevant non-Gaussian effects as in the case of pure hydrogen and H_{2}-D_{2} mixtures.

Superfluidity in the absence of kinetics in spin-orbit-coupled optical lattices

At low temperatures bosons typically condense to minimize their single-particle kinetic energy while interactions stabilize superfluidity. Optical lattices with artificial spin-orbit coupling challenge this paradigm because here kinetic energy can be quenched in an extreme regime where the single-particle band flattens. To probe the fate of superfluidity in the absence of kinetics we construct and numerically solve interaction-only tight-binding models in flat bands. We find that novel superfluid states arise entirely from interactions operating in quenched kinetic energy bands, thus revealing a distinct and unexpected condensation mechanism. Our results have important implications for the identification of quantum condensed phases of ultracold bosons beyond conventional paradigms.

Proton vibrational dynamics in lithium imide investigated through incoherent inelastic and Compton neutron scattering

In the present study we report neutron spectroscopic measurements on polycrystalline lithium imide, namely, incoherent inelastic neutron scattering at 20 K, and neutron Compton scattering from 10 K up to room temperature. From the former technique the H-projected density of phonon states up to 100 meV is derived, while the latter works out the spherically averaged single-particle (i.e., H, Li, and N) momentum distributions and, from this, the mean kinetic energies. Only for H at the lowest investigated temperature, non-gaussian components of its momentum distribution are detected. However, these components do not seem directly connected to the system anharmonicity, being fully compatible with the simple N-H bond anisotropy. Neutron data are also complemented by ab initio lattice dynamics simulations, both harmonic and, at room temperature, carried out in the framework of the so-called "quantum colored noise thermostat" method. The single-particle mean kinetic energies in lithium imide as a function of temperature show a quite peculiar behavior at the moment not reproduced by ab initio lattice dynamics methods, at least as far as H and Li are concerned. As matter of fact, neither their low temperature values nor their temperature trends can be precisely explained in terms of standard phonon calculations.

Towards an exact orbital-free single-particle kinetic energy density for the inhomogeneous electron liquid in the Be atom

Holas and March [Phys. Rev. A51, 2040 (1995)] wrote the gradient of the one-body potential V(r) in terms of low-order derivatives of the idempotent Dirac density matrix built from a single Slater determinant of Kohn–Sham orbitals. Here, this is first combined with the study of Dawson and March [J. Chem. Phys. 81, 5850 (1984)] to express the single-particle kinetic energy density of the Be atom ground-state in terms of both the electron density n(r) and potential V(r). While this is the more compact formulation, we then, by removing V(r), demonstrate that the ratio t(r)/n(r) depends, though non-locally, only on the single variable n′(r)/n(r), no high-order gradients entering for the spherical Be atom.

Extraction of single-particle mean kinetic energy from macroscopic thermodynamic data

We have devised a simple and practical method for the extraction of single-particle mean kinetic energy from the Helmholtz free energy, when this quantity is known for a pair of isotopes of the element under investigation. This approach is originated from a property of the mass derivative of the Helmholtz free energy, but makes use of a mass incremental ratio evaluated for the aforementioned isotopes. The transformation from mass incremental ratio to mass derivative is done using a self-consistent method based on two approximations, the latter of which is quite surprising and has been thoroughly tested. Practical calculations on solid and liquid H 2 and D 2, where the center-of-mass mean kinetic energy is known from reliable quantum simulation techniques, are accomplished, obtaining a remarkable agreement, in spite of a slight overestimate for the calculated data set over the simulated one. The last part of the study is devoted to mixed systems, like water and ice, where thermodynamic data are abundant and proton mean kinetic energy has been measured via deep inelastic neutron scattering. Here results are less accurate than earlier, with an overestimate of about 9–10%, and, in addition, no substantial variation going from ice to liquid water. Further investigations on the H 2O/D 2O system are suggested.

A form of the single-particle kinetic energy density of an inhomogeneous electron liquid from a combination of one-body potential and ground-state electron density

Gál and March have recently proposed a form of the single-particle kinetic energy density in density functional theory in terms of the one-body potential V(r) and the ground-state electron density n(r) generated thereby. Here, with a minor modification of the GM form, examples are given for (a) harmonic trapping and (b) a bare Coulomb potential. The case of the He atom is also considered, via the Chandrasekhar variational wave function. Finally, the use of the semiempirical fine-tuned Hartree–Fock n(r) for spherical atoms due to Cordero et al. is briefly referred to.

The role of the von Weizsäcker kinetic energy gradient term in independent harmonically confined fermions for arbitrary two-dimensional closed-shell occupancy

The search for the single-particle kinetic energy functional TS[n] continues to be of major interest for density functional theory. Since it is expected to be generally applicable, exactly solvable models are of obvious interest. Here we focus on one, which is also of interest experimentally in magnetic trapping of ultracold fermion vapours. This is the model of independent harmonically trapped fermions in two dimensions. Here, the role of the von Weizsäcker inhomogeneity kinetic energy is a focal point, prompted also by the work of Delle Site (2005 J. Phys. A: Math. Gen. 38 7893).

Relation between single-particle kinetic energy and exchange energy in DFT for the inhomogeneous electron liquid in the Be atom

In a recent study, the authors have used the semi empirical fine-tuned Hartree–Fock ground-state electron density n(r) of Cordero et al. [Phys. Rev. A 75, 052502 (2007)] for the Be atom to calculate the phase θ(r) from a non-linear pendulum-like equation. Since the density amplitude n(r)1/2 plus θ(r) determine, in turn, the idempotent Dirac density matrix γ(r, r′), we use n(r) and θ(r) first of all to calculate the exchange energy density e X (r) of the density functional theory (DFT). This enables us to obtain the Slater (Sl) approximation to the exchange-only potential. A comparison can then be made, by integrating the earlier predicted exchange-correlation force −∂V XC (r)/∂r, of V XC (r) with . Relationship to the Becke semiempirical density gradient approximation for exchange is also established. Some brief discussion of the Perdew–Burke–Ernzerhof density functional is added.

Pauli potential in terms of kinetic energy density and electron density in the leading Coulombic term of the nonrelativistic1/Zexpansion of spherical atomic ions

The Pauli potential ${V}_{P}$ in density functional theory is known to be the difference between the functional derivative of the single-particle kinetic energy ${T}_{s}[n]$ with respect to the electron density $n$ and its von Weizs\"acker counterpart. For the leading Coulombic term in the $1/Z$ expansion for spherical atomic ions, ${V}_{P}[n]$ is written in terms of the kinetic energy density plus $n(r)$ and its low-order derivatives. For comparison, the example of an arbitrary number of closed shells with purely harmonic confinement is also treated.

Exchange energy for two closed shells generated by a bare Coulomb potential energy −Ze 2/r in the limit of large Z, in two dimensions

Two-dimensional (2D) inhomogeneous electron assemblies are becoming increasingly important in Condensed Matter and associated technologies. Here, therefore, we contribute to the Density Functional Theory of such 2D electronic systems by calculating, analytically, (i) the idempotent Dirac density matrix γ(r, r′) generated by two closed shells for the bare Coulomb potential −Ze2/r and (ii) the exchange energy density \({\varepsilon_x({\bf r})}\) . Some progress is also possible concerning the exchange potential Vx(r), one non-local approximation being the Slater potential \({2\varepsilon_x(r)/n(r)}\) , with n(r) the ground state electron density. However, to complete the theory of Vx(r), the functional derivative of the single-particle kinetic energy per unit area δt(s)/δn(r) is still required.

Differential equation for the electron density in large molecules

Developments in density functional theory involve, almost inevitably, the knowledge of the single-particle kinetic energy Ts[p] as a functional of the electron density p(r). Here, the possibilities of approximating this quantity through a one-body potential inserted in a Schrodinger equation for the density amplitude {p(r)}1/2 are explored in both finite and extended inhomogeneous electron gases. Nonlocal theory relating the contribution VPauli(r) of the one-body potential arising from the Fermi statistics obeyed by electrons is contemplated and some examples given.

Use of the differential virial theorem to estimate the spatial variation of the exchange-correlation force−∂VXC(r)∕∂rin the ground states of the spherical atoms He and Be

We use the differential virial theorem (DVT) directly to display the approximate spatial dependence of the exchange-correlation (XC) force in He and Be, applying an exact integral constraint on the XC force, recently established by March and Nagy. In He, an analytic ground-state density $n(r)$, combined with the DVT plus the von Weizs\"acker single-particle kinetic energy, suffices to determine an approximate XC force. For Be, the XC force is calculated for the semiempirical fine-tuned Hartree-Fock density, as proposed by Cordero et al. [Phys. Rev. A 75, 052502 (2007)]. However, for the single-particle kinetic energy, following Dawson and March, a phase $\ensuremath{\theta}(r)$ must be obtained by solving numerically a nonlinear pendulumlike equation.

Static and thermodynamic properties of low-density supercritical 4He—breakdown of the Feynman–Hibbs approximation

We study the applicability of the semiclassical Feynman and Hibbs (FH) (second-order or fourth-order) effective potentials to the description of the thermodynamic properties of quantum fluids at finite temperatures. First, we use path integral Monte Carlo (PIMC) simulations to estimate the thermodynamic/static properties of our model quantum fluid, i.e. low-density 4He at 10 K. With PIMC we obtain the experimental equation of state, the single-particle mean kinetic energy, the single-particle density matrix and the single-particle momentum distribution of this system at low densities. We show that our PIMC results are in full agreement with experimental data obtained with deep inelastic neutron scattering at high momentum transfers (D. Colognesi, C. Andreani, R. Senesi, Europhys. Lett., 2000, 50, 202). As expected, in this region of the 4He phase diagram, quantum effects modify the width of the single-particle momentum distribution but do not alter its Gaussian shape. Knowing the exact values of density, pressure and single-particle mean kinetic energy for our model quantum fluid, we investigate the limitations of the semiclassical FH effective potentials. We show that commonly used 'short-time' approximations to the high-temperature density matrix due to Feynman and Hibbs can only be applied in a very limited range of the 4He phase diagram. We found that FH effective potentials reproduce the experimental densities of 4He at 10 K for Lambda/a < 0.45 (Lambda = 2.73 A denotes the thermal de Broglie wavelength, a = rho(-1/3) is the mean nearest-neighbor distance in the fluid and rho denotes fluid density). Moreover, semiclassical FH effective potentials are able to correctly predict the single-particle mean kinetic energy of 4He at 10 K in a very limited range of fluid densities, i.e.Lambda/a < 0.17. We show that the ad hoc application of the semiclassical FH effective potentials for the calculation of the thermodynamic properties of dense liquid-like para-hydrogen (para-H2) adsorbed in nanoporous materials below 72 K is questionable.

The Pauli potential in relation to the differential virial theorem with application to experiments on ultracold atomic gases of fermions

In early work by the writer introducing the Pauli potential V P ( r ) into density functional theory, the relation of V P ( r ) to the, as yet unknown, single-particle kinetic energy density functional was emphasized. Here, because of ongoing experiments on ultracold atomic gases of fermions, an explicit expression for the first derivative of V P ( r ) for an arbitrary number of closed shells generated by harmonic confinement is derived in terms of the spherically symmetric particle density n ( r ) and the confining potential.

He4adsorbed in cylindrical silica nanopores: Effect of size on the single-atom mean kinetic energy

This paper reports a study of the short-time dynamics of helium confined in silica nanopores (xerogel powder), with average pore diameters of 24 and $160\phantom{\rule{0.3em}{0ex}}\mathrm{\AA{}}$. The longitudinal momentum distribution of helium adsorbed in xerogels has been determined via deep inelastic neutron scattering (DINS) measurements performed on the VESUVIO spectrometer at the ISIS spallation source. DINS measurements, in the attosecond time scale, (i.e., ${10}^{\ensuremath{-}16}--{10}^{\ensuremath{-}15}\phantom{\rule{0.3em}{0ex}}\mathrm{s}$), were performed at a temperature of $T=2.5\phantom{\rule{0.3em}{0ex}}\mathrm{K}$ and saturated vapor pressure conditions, with 95% pore volume filling. The average wave-vector transfer $q$ was about $130\phantom{\rule{0.3em}{0ex}}{\mathrm{\AA{}}}^{\ensuremath{-}1}$. For confined helium, significant changes in the values of the single-particle mean kinetic energies $⟨{E}_{K}⟩$ are found in the bulk phase. These are $32.6\ifmmode\pm\else\textpm\fi{}8.7\phantom{\rule{0.3em}{0ex}}\mathrm{K}$ for the $24\phantom{\rule{0.3em}{0ex}}\mathrm{\AA{}}$ and $24.4\ifmmode\pm\else\textpm\fi{}5.3\phantom{\rule{0.3em}{0ex}}\mathrm{K}$ for the $160\phantom{\rule{0.3em}{0ex}}\mathrm{\AA{}}$ pore diameters, remarkably higher than $⟨{E}_{K}⟩=16.2\ifmmode\pm\else\textpm\fi{}0.4\phantom{\rule{0.3em}{0ex}}\mathrm{K}$, the value of normal liquid $^{4}\mathrm{He}$ at $T=2.5\phantom{\rule{0.3em}{0ex}}\mathrm{K}$ and saturated vapor pressure conditions. The results are interpreted in terms of a model where $^{4}\mathrm{He}$ atoms are arranged in concentric annuli along the cylindrical pore axis, with $⟨{E}_{K}⟩$ mainly dependent on the ratio between the atomic ``effective'' diameter and the pore diameter. The number of solid layers close to pore surface is found to be strongly pore-size dependent with one single solid layer for $24\phantom{\rule{0.3em}{0ex}}\mathrm{\AA{}}$ diameter pore and three solid layers for $160\phantom{\rule{0.3em}{0ex}}\mathrm{\AA{}}$ diameter pore.

Density-potential relation via the single-particle kinetic energy density for one-dimensional two-level fermion systems with application to Be-like atomic ions in the(1s)2(2s)2configuration

Using the Dawson-March transformation of two-level wave functions in terms of density amplitude and phase, a differential equation is constructed relating the single-particle kinetic energy to the fermion density. Employing then the differential form of the virial theorem derived by March and Young [Nucl. Phys. 12, 237 (1959)], a nonlinear electron-density--one-body potential relation emerges. Application to the Be-like atomic ions is briefly considered.

Can density functional theory make use of experimentally determined ground-state electron densities via the Dirac density matrix for molecules of biological interest?

Abstract More than four decades ago, March and Murray gave a perturbation theory of the single-particle(s) Dirac density matrix γ s ( r , r ′ ) to all orders in a given one-body potential energy V ( r ) . However, for density functional theory in orbital-free form, one requires the functional γ s [ ρ ] where ρ ( r ) is the ground-state electron density. Therefore, in the present study, a first-order non-linear differential equation is proposed for γ s in terms of ρ ( r ) and ∇ ρ ( r ) , plus the single-particle kinetic energy. Since this latter quantity is itself known to be a functional of ρ, the existence of such an equation for γ s would be a significant step along the road to determining the desired functional γ s [ ρ ] . As yet, we have succeeded in giving a rigorous proof of the proposed differential equation for γ s ( r , r ′ ) only for one- and two-level molecules. If it is subsequently proved for an arbitrary number of levels, which we believe should be possible, it would then allow γ s to be calculated for molecules of biological interest, from experimentally measured ground-state densities ρ ( r ) , as the approach is entirely orbital-free.

Single-particle kinetic energy in density-functional theory: Harmonic confinement in two and three dimensions

Using the known perturbation series for the idempotent Dirac density matrix in powers of a given one-body potential $V(\mathbf{r})$, Stoddart and March (SM) generated a corresponding series for the kinetic energy density. While the general term of the SM series is known, a summation has not been achieved to date. A contribution to solve this problem is made here by exhibiting an explicit form of the above kinetic energy density for Fermions filling an arbitrary number of closed shells, when the confinement is harmonic. This example is of considerable current interest because of ongoing experiments on ultracold atomic gases of Fermions.

Dirac density matrix and the Legendre transform of the kinetic energy generated by one-dimensional model potentials

In density functional theory, the single-particle kinetic energy is still not known in an orbital-free form. The natural tool to calculate such kinetic energy for a given one-body potential V is the Dirac density matrix y. Here, by first taking solvable one-dimensional examples, and in particular the sech 2 (x) potential, forms for the Dirac density matrix are exhibited in terms of the potential.This in turn generates the Legendre transform of the single-particle kinetic energy. Finally, one-dimensional perturbation theory for the kinetic energy density, summed to all orders in V, is presented in the appendix.