A QCA in 2 + 1 and 3 + 1 dimensions

Matrix product vectors form the appropriate framework to study and classify one-dimensional quantum systems. In this work, we develop the structure theory of matrix product unitary operators (MPUs) which appear e.g. in the description of time evolutions of one-dimensional systems. We prove that all MPUs have a strict causal cone, making them quantum cellular automata (QCAs), and derive a canonical form for MPUs which relates different MPU representations of the same unitary through a local gauge. We use this canonical form to prove an index theorem for MPUs which gives the precise conditions under which two MPUs are adiabatically connected, providing an alternative derivation to that of (Gross et al 2012 Commun. Math. Phys. 310 419) for QCAs. We also discuss the effect of symmetries on the MPU classification. In particular, we characterize the tensors corresponding to MPU that are invariant under conjugation, time reversal, or transposition. In the first case, we give a full characterization of all equivalence classes. Finally, we give several examples of MPU possessing different symmetries.

Much of the physics determined for hydrogen and hydrogen-like ions is relevant to atoms and ions with many electrons. In a multi-electron atom or ion, the quantum state energies and wavefunctions are dominated by the central potential arising from the nuclear charge as is the case for hydrogen and hydrogen-like ions. Consequently, quantum wavefunctions similar in form to those found for hydrogen or hydrogen-like ions are produced. There are some corrections needed to the exact electron energies associated with the nuclear electric field experienced by each electron due to shielding effects by other electrons – and some perturbing energy effects due to electron–electron Coulomb repulsion. The electron wavefunctions have similar orbital or angular shapes as for hydrogen and hydrogen-like ions (see Figures 7.2 and 7.3). The same electron-configuration system (i.e. 1s, 2s, 2p, …) can be used and the same degeneracies are associated with different values of principal quantum number n (degeneracy 2 n 2 ) and orbital angular momentum quantum number l (degeneracy 2×2 l(l +1 ) ) as found for hydrogen and hydrogen-like ions. As multi-electron atoms and ions have quantum states following the ‘rules’ established for hydrogen and hydrogen-like ions, the quantum state energy values are primarily determined by a principal quantum number n with values n =1, 2, 3, … . Angular quantum numbers l with l =0, 1, 2, … n − 1 are found to have a small effect on energy values due to shielding of the nuclear charge. As for hydrogen, the number of individual quantum states at a particular energy involves the different magnetic quantum numbers m with values m = − l ,− l + 1, … , 0, 1 … + l and the two possible spin orientations (relative to the orbital angular momentum). The states which the electrons occupy are called the electronic configuration for the atom or ion. An important feature of multi-electron atoms is established by the Pauli exclusion principle which requires that only one electron can occupy an individual quantum state. In the ground state of an atom or ion, the required number of electrons effectively fills the lowest energy quantum states designated by the n , l , m and s values according to the hydrogen degeneracies.

THE existence of the shell structure of nuclei has been derived from a considerable variety of empirical facts (stability and abundance of nuclei, spin, magnetic moment, quadrupole moment, isomerism and β-decay)1. The closed shells of neutrons as well as of protons correspond to the numbers 2, 8, 20, 28, 50, 82, 126. Several theories2 have been put forward, based on the model of a single nucleon moving in an average field of the rest. As it is not easy to conceive how a single particle embedded in matter of nuclear density could move in its own orbit without being disturbed by the others, we have searched for a more general explanation*. This is obtained with the help of a well-known relation first derived by Fermi, connecting the density ρ(r) of a degenerate system of spin-half particles to the numbers of particles with angular momentum quantum number l. If one assumes that the formation of a closed shell is associated with the filling up of states of definite angular momentum, it follows that the closed shells are characterized by .

Influence of Pr doping in non-stoichiometric NiFe2O4 on magnetic response and electronic properties: Magnetic Compton scattering and ab-initio studies

X-ray magnetic circular dichroism measurements with the photon helicity modulation technique at Ir ${L}_{2,3}$ absorption edges and sum rule analysis show that Ir develops a composition-dependent induced spin and orbital magnetic moments in hcp ${\mathrm{Co}}_{100\ensuremath{-}x}{\mathrm{Ir}}_{x}$ $(x=5,17,25,32)$ alloys. The total moment per Ir is found to be in the range of $0.39{\ensuremath{\mu}}_{B}--0.1{\ensuremath{\mu}}_{B}$---i.e., decreasing with the increase of $x$. The spin and orbital moments of Ir in the alloys are found to be aligned antiparallel, showing the violation of the third Hund's rule at these compositions. Electronic structure calculations of Co and Ir magnetic moments in ${\mathrm{Co}}_{100\ensuremath{-}x}{\mathrm{Ir}}_{x}$ at $x=25$ within the local spin density approximation illustrate that significant $3d\text{\ensuremath{-}}5d$ hybridization results in the formation of an induced moments at the Ir sites, with the spin moments on the order of $0.2{\ensuremath{\mu}}_{B}--0.3{\ensuremath{\mu}}_{B}$ that are antiparallel to the Ir orbital moment concomitant with a reduced moment on the neighboring Co sites.

The magnetic properties (field range H = 0–50 kOe, temperature range T = 3–300 K), and structural features of a sodium-doped carbon composite material based on fullerene C60 and thermally exfoliated graphite (TEG) are studied. The material is obtained with different ratios of the components by sintering at a pressure of 7 GPa and T = 600°C, at which it is found that significant amorphization of the crystal lattice of the initial C60 occurs. The dia-, para-, and ferromagnetic components (MD, MPM, and MFM) were separated from the total magnetic moment of the samples under study. It is found that a sodium dopant has no effect on the magnetic properties of the composite. Analysis of the MPM(H) field dependences by using the Brillouin function for the fullerene-containing sample (i.e., without TEG) makes it possible to determine the quantum number of the total angular momentum of paramagnetic (PM) centers. Its value is found to be J = 1, which corresponds to elementary magnetic moment μPM = 2μB of a PM center. The concentration of PM centers is estimated at the level of NPM ≈ (2–5) × 1018 g–1 for most samples, including the material without TEG. The introduction of TEG into the initial composition and an increase in its proportion in the composite leads to a strong increase in the magnetic moment, which is explained by an increase in both the J value and the concentration of PM centers.

A self-consistent-field theory is given for the electronic collective modes of a chain containing a finite number, N, of Coulomb-coupled spherical two-dimensional electron gases arranged with their centers along a straight line, for simulating electromagnetic response of a narrow-ribbon of metallic shells. The separation between nearest-neighbor shells is arbitrary and because of the quantization of the electron energy levels due to their confinement to the spherical surface, all angular momenta L of the Coulomb excitations, as well as their projections M on the quantization axis, are coupled. However, for incoming light with a given polarization, only one angular momentum quantum number is usually required. Therefore, the electromagnetic response of the narrow-ribbon of metallic shells is expected to be controlled externally by selecting different polarizations for incident light. We show that, when N = 3, the next-nearest-neighbor Coulomb coupling is larger than its value if they are located at opposite ends of a right-angle triangle forming the triad. Additionally, the frequencies of the plasma excitations are found to depend on the orientation of the line joining them with respect to the axis of quantization since the magnetic field generated from the induced oscillating electric dipole moment on one sphere can couple to the induced magnetic dipole moment on another. Although the transverse inter-shell electromagnetic coupling can be modeled by an effective dynamic medium, the longitudinal inter-shell Coulomb coupling, on the other hand, can still significantly modify the electromagnetic property of this effective medium between shells.

Ab initio density functional calculations including spin-orbit coupling (SOC) have been performed for Pt(n), n = 2-6 clusters. The strong SOC tends to stabilize planar structures for n = 2-5, whereas for clusters consisting of six atoms, three-dimensional structures remain preferred. SOC leads to the formation of large orbital magnetic moments and to a mixing of different spin states. Due to the spin-mixing the total magnetic moment may be larger or smaller than the spin moment in the absence of SOC. Both spin and orbital moments are found to be anisotropic. Because of the strong SOC the energy differences between coexisting magnetic isomers can be comparable to or even smaller than their magnetic anisotropy energies. In this case the lowest barrier for magnetization reversal can be determined by a magnetic isomer which is different from the ground state configuration.

Stereo atomscope and diffraction spectroscopy—Atomic site specific property analysis

The electric field control of $d$-electron magnetism in multiferroic transition metal oxides is attracting widespread interest for the underlying fundamental physics and for next generation spintronic devices. Here, we report an extensive study of the $3d$ magnetism in magnetoelectric ${\mathrm{Ga}}_{0.6}{\mathrm{Fe}}_{1.4}{\mathrm{O}}_{3}$ (GFO) epitaxial films by polarization-dependent x-ray absorption spectroscopy. We found a nonzero integral of the x-ray magnetic circular dichroism (XMCD), with the sign depending upon the relative orientation between the external magnetic field and the crystallographic axes. This finding translates into a sign reversal between the average Fe magnetic orbital and spin moments. Large Fe displacements, among inequivalent octahedral sites, lower the symmetry of the system producing anisotropic paths for the Fe-O bondings giving rise to a large orbital-lattice interaction akin to a preferential crystallographic direction for the uncompensated, or averaged among different sites, magnetic orbital moment. The latter may lead to a partial reorientation of the magnetic orbital moment under an external magnetic field that, combined with the ferrimagnetic nature of the GFO, can qualitatively explain the observed sign reversal of the XMCD integral. The results suggest that control over the local symmetry of the oxygen octahedra in transition metal oxides can offer a suitable leverage over the manipulation of the effective orbital and spin moments in magnetoelectric systems.

Fractal (or transfractal) features are common in real-life networks and are known to influence the dynamic processes taking place in the network itself. Here, we consider a class of scale-free deterministic networks, called (u, v)-flowers, whose topological properties can be controlled by tuning the parameters u and v; in particular, for u > 1, they are fractals endowed with a fractal dimension df, while for u = 1, they are transfractal endowed with a transfractal dimension d̃f. In this work, we investigate dynamic processes (i.e., random walks) and topological properties (i.e., the Laplacian spectrum) and we show that, under proper conditions, the same scalings (ruled by the related dimensions) emerge for both fractal and transfractal dimensions.

We use x-ray magnetic circular dichroism to evidence the effect of correlations on the local impurity magnetic moment in an archetypal Kondo system, namely, a dilute Cu:Fe alloy. Applying the sum rules on the Fe ${L}_{2,3}$ absorption edges, the evolution of the spin and orbital moments across the Kondo temperature are determined separately. The spin moment presents a crossover from a nearly temperature-independent regime below the Kondo temperature to a paramagneticlike regime above. Conversely, the weak orbital moment shows a temperature-independent behavior in the whole temperature range, suggesting different Kondo screening temperature scales for the spin and orbital moments.

International audience

This work presents electronic and magnetic properties of selected members in the cubic Laves phase series ${\mathrm{Nd}}_{1\ensuremath{-}x}{\mathrm{Pr}}_{x}\mathrm{CoNi}$ (0 $\ensuremath{\le} x \ensuremath{\le}$ 1) and ${\mathrm{Ce}}_{0.25}{\mathrm{Pr}}_{0.75}\mathrm{CoNi}$, together with the corresponding binary compositions (${\mathrm{NdCo}}_{2}, {\mathrm{NdNi}}_{2}, {\mathrm{PrCo}}_{2}, {\mathrm{PrNi}}_{2}, {\mathrm{CeCo}}_{2}, {\mathrm{CeNi}}_{2}$), using soft x-ray absorption spectroscopy, x-ray magnetic circular dichroism (XMCD), density-functional theory, and crystal field multiplet calculations. All transition-metal moments saturate below 1 T, while the rare-earth moments do not saturate even at 5 T, consistent with van Vleck paramagnetic contributions and crystal field suppression. While the sum rules are widely used to extract element-specific magnetic moments from XMCD, we show that for $3d$ transition metals, their application requires accurate estimates of the number of unoccupied $3d$ states. We observe a finite magnetic moment on Ni, challenging the common assumption of its nonmagnetic character in Laves phases. The orbital magnetic moments were determined using the spin rules, while the spin moments were estimated from single-ion values from multiplet calculations, due to the invalidity of the spin sum rule for light rare-earth elements. The magnetic moments of Nd and Pr are found to be suppressed relative to their free-ion values, with multiplet theory indicating that this is due to crystal field effects. Our results confirm that Nd and Pr maintain localized $4{f}^{3}$ and $4{f}^{2}$ configurations, respectively, and that their element-specific magnetic moments are robust to rare-earth substitution. Ce, on the other hand, exhibits a tuneable mixed-valent ground state with both magnetic $4{f}^{1}$ and nonmagnetic $4{f}^{0}$ components. The relative fraction of these states varies with the electronegativity of the surrounding $3d$ transition metals, revealing a pathway to tune Ce magnetism via composition. This work establishes a framework for accurately interpreting XMCD in light rare-earth-based intermetallics and provides insight for designing light rare-earth-based magnetocaloric materials.

Determination of the initial saturation of a hydrocarbon reservoir requires resistivity log data and saturation exponent. A number of experimental investigations has shown that reservoir wettability affect the exponent value. A remarkable divergence conclusions, however, still appear in the literature. The objective of the present study was therefore to investigate the effect of rock wettability on saturation exponent. Fractal concepts have been applied to the image of thin-sections of a core sample and used to derive the resistivity equation. The advantage of this approach is that it is independent of rock-fluids equilibrium problems that commonly influence laboratory measurements. A general equation of electrical resistivity has been developed in this study. Electrical tortuosity, clay content, and rock wettability are incorporated in the equation. The present work employed twenty thin-sections of limestone and sandstones. It was found that the lowest exponent of 1.61 was obtained for strongly water-wet shaly sandstone and the highest value of 4.99 is for strongly oil-wet. The exponent consistently increases as the wetting condition is shifted from strongly water-wet toward oil-wet. It is close to 2.0 for clean sandstones at strongly water-wet, supporting the empirical formula of Archie. Introduction Estimation of a hydrocarbon reserve is strongly dependent of electric log data and the value of saturation exponent used. This exponent is usually either assumed to be 2.0 regardless of the reservoir wettability or is derived from laboratory measurements of electrical properties of the cores. Saturation exponent has been the subject to many laboratory investigations. The researchers arrived at considerably different quantitative results, particularly for wetting conditions at oil-wet side. Detailed review of this matter has been made by Anderson and Sondenaa et al. A lack of analytical investigation has been conducted to study the effect of wettability on electrical properties of rocks containing fluids. Fractal concepts have been applied to a wide range of research area related to porous media at various scales, from microscale (pore level) to megascale (field scale). A detailed review of fractal concepts applications has been presented by Sahimi and Yortsos. Briefly, a fractal object is an object that has a geometric dimension greater than its topological (Euclidian) dimension. Fractal geometry may describe the irregurality and fragmentation of natural objects and patterns. To characterize a fractal object or set, the fractal dimension D must be determined. A simple method is by "box counting". This is simply done by drawing a mesh of squares of side dimension that covers the set. The fractal dimension is determined from the slope of the relation between count and square side dimension. which is the fractal dimension that can yield any real value. To handle the complexity of the problems, the present study was directed to applying concepts of fractal analysis to determination of the electrical properties and thus saturation exponent. Pore structures and geometries can be quite complex in nature. The complexity stems from the irregularity of pore wall surfaces. The presence of clays may increase the complexity of pore structures. As a matter of fact fluids distribution within the pores is influenced by the wetting condition. This in turn would affect the fluids connectivity. Therefore, electric tortuosity depends not only on pores and fluids connectivity but also on the irregurality of main grain surfaces and volume and distribution of clays within the pores. As pores of a porous medium can be magnified by employing microscope, identification of fine details of the pores can be performed through the use of thin sections made. Ultimately, mathematical analysis coupled with fractal analysis is thus possible to study electrical properties of porous media. P. 143