The quantum theory of nucleon correlation in finite nuclei. II. Higher order correlation effects and additive pair approximation

In the first part of this article it was shown that the variational solution of the Schroedinger equation of a finite Fermion system can be written as a finite sum of A terms (for A particles) the first of which is the Hartree-Fock energy, while the rest represent the correlation effects. In the first part explicit formulas for the 2-particle correlation energy were given. In this paper explicit formulas are given for the higher order correlation energies. It is shown that two different models can be developed depending on the orthogonality condition used. Beginning with the 4th order effects the “linked” and “unlinked” correlation terms are separated. An exact formula is given for the case in which only the 2-particle effects, linked and unlinked are taken into account. The “additive pair approximation” in which the correlation energy is given as the sum of 2-particle energies is investigated and it is shown to be related to the exact formula by a clearly defined set of approximations. Various possible applications of the model are discussed.     

Robust thermal quantum correlation and quantum phase transition of spin system on fractal lattices

We investigate the quantum correlation measured by quantum discord (QD) for thermalized ferromagnetic Heisenberg spin systems in one-dimensional chains and on fractal lattices using the decimation renormalization group approach. It is found that the QD between two non-nearest-neighbor end spins exhibits some interesting behaviors which depend on the anisotropic parameter Δ, the temperature T, and the size of system L. With increasing Δ continuously, the QD possesses a cuspate change at Δ = 0 which is a critical point of quantum phase transition (QPT). There presents the “regrowth” tendency of QD with increasing T at Δ < 0, in contrast to the “growth” of QD at Δ > 0. As the size of the system L becomes large, there still exists considerable thermal QD between long-distance end sites in spin chains and on the fractal lattices even at unentangled states, and the long-distance QD can spotlight the presence of QPT. The robustness of QD on the diamond-type hierarchical lattices is stronger than that in spin chains and Koch curves, which indicates that the fractal can affect the behaviors of quantum correlation.     

A topological expansion for high-energy hadronic collisions: (I). General properties and connection with the reggeon calculus

In this and in a companion paper a “topological expansion” for high-energy hadronic processes is proposed and discussed. In this first paper the general properties of the expansion and its connection with Gribov's reggeon calculus are presented. The topological expansion is first defined mathematically for a large class of theories and is shown to be equivalent to a “large N expansion” in some theories which include planar dual models and non-Abelian gauge theories. Next, the definition of the bare parameters is given in terms of graphs on a sphere. The bare pomeron pole and its couplings are thus introduced. The (inclusive) form of s-channel unitarity and its consequences fo the above couplings are recalled. It is then shown how the expansion in the number of “handles” of the graph can be related to Gribov's reggeon calculus and how, with the aid of discontinuity equations in the J-plane, scaling solutions can be obtained and critical indices can be computed to yield known results. Our main new results, including the study of s-channel discontinuities and of positivity constraints as well as the definition of a new fireball expansion, and the discussion of the relevance of this theory at finite (present) energies are presented in the second paper.     

Continuum Nonsimple Loops and 2D Critical Percolation

Substantial progress has been made in recent years on the 2D critical percolation scaling limit and its conformal invariance properties. In particular, chordal SLE ₆(the Stochastic Loewner Evolution with parameter κ=6) was, in the work of Schramm and of Smirnov, identified as the scaling limit of the critical percolation “exploration process.” In this paper we use that and other results to construct what we argue is the fullscaling limit of the collection of allclosed contours surrounding the critical percolation clusters on the 2D triangular lattice. This random process or gas of continuum nonsimple loops in Bbb R²is constructed inductively by repeated use of chordal SLE ₆. These loops do not cross but do touch each other—indeed, any two loops are connected by a finite “path” of touching loops.     

Phase Transition for the Maki–Thompson Rumour Model on a Small-World Network

We consider the Maki–Thompson model for the stochastic propagation of a rumour within a population. In this model the population is made up of “spreaders”, “ignorants” and “stiflers”; any spreader attempts to pass the rumour to the other individuals via pair-wise interactions and in case the other individual is an ignorant, it becomes a spreader, while in the other two cases the initiating spreader turns into a stifler. In a finite population the process will eventually reach an equilibrium situation where individuals are either stiflers or ignorants. We extend the original hypothesis of homogenously mixed population by allowing for a small-world network embedding the model, in such a way that interactions occur only between nearest-neighbours. This structure is realized starting from a k-regular ring and by inserting, in the average, c additional links in such a way that k and c are tuneable parameters for the population architecture. We prove that this system exhibits a transition between regimes of localization (where the final number of stiflers is at most logarithmic in the population size) and propagation (where the final number of stiflers grows algebraically with the population size) at a finite value of the network parameter c. A quantitative estimate for the critical value of c is obtained via extensive numerical simulations.     

Self-trapping of electron pairs in materials with tunneling states. Electrons in “perfect” covalent glasses

It is shown that in perfect covalent glasses (and some local structures) the self-trapping of negative-U electron (on hole) pairs is realized on selected “flexible” atomic configurations. The basic concepts of the theory of the phenomenon are presented; the criteria and some associated effects are analyzed. The selected configuration contains an appropriate atom (atomic group) in “critical local potential” with unusually small spring constant in respective well. The phenomenon (self-trapping of electron or hole pair) can generate both Anderson negative-U centres in “perfect” glass (i.e. not associated with defects in glasses) and, possibly, a class of deep-level impurity centres in some crystals.     

Quasiparticle picture of quarks near chiral phase transition

We study how the quasiparticle picture of the quark can be modified near but above the critical temperature (T _c) of the chiral phase transition; we incorporate into the quark self-energy the effects of the precursory soft modes of the phase transition, i.e. ‘para-σ(π) meson’. It is found that the quark spectrum has a three-peak structure near T _c: We show that the additional new spectra originate from the mixing between a quark (antiquark) and an antiquark-hole (quark-hole) caused by a “resonant scattering” of the quasi-fermions with the thermally-excited soft modes.     

Structural phase transitions in ionic molecular solids

An overview is presented of our studies on the nature of structural instabilities in relatively complex ionic solids. These are based on parameter-free interionic potentials based on the Gordon-Kim modified electron gas formalism extended to molecular ions.

We describe the manner in which there emerge from these studies quite general concepts of “size” and “shape” as structural determinants. In particular, we discuss how these, and the approximate symmetries that they can produce, can provide a relatively simple structure-based explanation of the origins of incommensurate phases in these systems. However, we also emphasize that the existence of such symmetries does not guarantee an incommensurate phase. This can only be realized if long-range correlations are sufficiently strong to overcome random local disordering. Thus, either the molecular units are partially linked and/or there exist long-range Coulomb interactions between individual units.     

Investigation of the 4f-electron-phonon-coupling in the ferromagnet LiTbF4 by inelastic scattering of light

The polarized Raman spectra of LiTbF₄ in an external magnetic field (B≦8 T) have been recorded in the wavenumber interval 0≦500 cm^?1 and at temperatures of 1.8 K and 4.2 K. We have studied effects of the 4f electron-phonon (magnetoelastic) coupling manifesting itself in the splitting of a twofold degenerate phonon mode and in anticrossing effects between phonons and electronic transitions. In the spectra probing the scattering tensor elements (xz) and (zy) this anticrossing shows an asymmetry with respect to the frequencies and scattering intensities of the quasi-degenerate components. These effects are discussed in detail and can be interpreted by the theory of magnetoelastic interaction and by taking into consideration the finite widths of the electronic and phonon components (“critical coupling”). Brillouin scattering has been used to determine the sound velocities in thea-direction. No effect of the magneto-elastic interaction could be detected in this case.     

Finite-dimensional quantum systems: Complementarity,phase space,and all that

A complete set of d + 1 mutually unbiased bases exists in a Hilbert space of dimension d whenever d is power of a prime. We discuss a simple construction of d + 1 disjoint classes (each one having d ? 1 commuting operators) such that the corresponding eigenstates form sets of unbiased bases. One of these classes is diagonal and can be mapped to “ladder” operators by means of the finite Fourier transform. Using this idea, we naturally introduce the notion of quantum phase as complementary to the inversion. Relevant examples involving qubits and qutrits are discussed.     

Ordering and phase transitions in ising systems with competing short range and dipolar interactions

As a simple model of order-disorder ferroelectrics or dipolar magnets we consider a simple cubic Ising-system with nearest neighbor exchangeJ and dipolar interaction of strengthµ ²/a ³. ForJa ³/µ ²Ja ³/µ ²<0.1270 the ground state consists of ferromagnetic rows (in spin direction) arranged antiferromagnetically in the plane perpendicular to it. AtJa ³/µ ²=0.1279 the structure changes to a layered antiferromagnetic structure with a twocomponent order parameter, while forJa ³/µ ²>0.16429 the ferromagnetic phase becomes stable (with domain arrangements depending on the shape of the sample). For all critical values ofJa ³/µ ² where the bulk energies of two phases become equal also the interface energy between these phases is found to be zero. The ordering at nonzero temperature is studied by means of mean-field approximations (MFA) and Monte Carlo (MC) calculations. It turns out that forJa ³/µ ² of order unity the MFA overestimates ordering temperatures by about a factor of two, and predicts multicritical points (between the disordered and two ordered phases) at nonzero temperature, including two biaxial Lifshitz points which the MC work suggests to occur atT=0. In contrast to MFA the layered antiferromagnetic structure is found to be stable only at extremely lowT, because a metastable spin-glass phase (with random arrangement of ferromagnetic rows in the spin direction) has only slightly higher energy. The MFA also yields two regimes of helical phases which are “locked in” to the antiferromagnetic phases at uniaxial Lifshitz points occurring at the Brillouin zone boundary. In the MC-work various methods of treating the long-range interaction are investigated. While all kinds of truncations as well as compensating field methods are rather unsatisfactory in our case, Ewald summation techniques yield satisfactory results. Nevertheless strong fluctuations as well as strong finite size effects prevent us from making accurate exponent estimates, but arguments are given that there is no regime of broad visibility of Landaulike critical behavior. Finally the extension of our results to other lattices as well as experimental applications are briefly discussed.     

A Quantum Computational Semantics for Epistemic Logical Operators. Part I: Epistemic Structures

Some critical open problems of epistemic logics can be investigated in the framework of a quantum computational approach. The basic idea is to interpret sentences like “Alice knows that Bob does not understand that π is irrational” as pieces of quantum information (generally represented by density operators of convenient Hilbert spaces). Logical epistemic operators (to understand, to know…) are dealt with as (generally irreversible) quantum operations, which are, in a sense, similar to measurement-procedures. This approach permits us to model some characteristic epistemic processes, that concern both human and artificial intelligence. For instance, the operation of “memorizing and retrieving information” can be formally represented, in this framework, by using a quantum teleportation phenomenon.     

How Big Is Too Big? Critical Shocks for Systemic Failure Cascades

External or internal shocks may lead to the collapse of a system consisting of many agents. If the shock hits only one agent initially and causes it to fail, this can induce a cascade of failures among neighboring agents. Several critical constellations determine whether this cascade affects the system in part or as a whole which, in the second case, leads to systemic risk. We investigate the critical parameters for such cascades in a simple model, where agents are characterized by an individual threshold θ _i determining their capacity to handle a load αθ _i with 1?α being their safety margin. If agents fail, they redistribute their load equally to K neighboring agents in a regular network. For three different threshold distributions P(θ), we derive analytical results for the size of the cascade, X(t), which is regarded as a measure of systemic risk, and the time when it stops. We focus on two different regimes, (i) EEE, an external extreme event where the size of the shock is of the order of the total capacity of the network, and (ii) RIE, a random internal event where the size of the shock is of the order of the capacity of an agent. We find that even for large extreme events that exceed the capacity of the network finite cascades are still possible, if a power-law threshold distribution is assumed. On the other hand, even small random fluctuations may lead to full cascades if critical conditions are met. Most importantly, we demonstrate that the size of the “big” shock is not the problem, as the systemic risk only varies slightly for changes in the number of initially failing agents, the safety margin and the threshold distribution, which further gives hints on how to reduce systemic risk.     

Electronic properties of small metallic particles

Metallic particles with linear dimensions d small compared with other characteristic lengths (like the wavelength of electromagnetic radiation, the de Broglie wavelength of the conduction electrons, the coherence length or the penetration depth in the superconducting state, etc.) show interesting effects which are usually unobservable in bulk metals. The electronic properties of these particles with diameters of a few nm can be analysed by considering the microcrystals not as “giant molecules” but as “small solids”, i.e. by using the familiar methods of solid state physics with some properly defined boundary conditions. Due to the smallness of the particles, the customary quasi-continuous electronic excitation spectrum splits up into discrete energy levels with an average energy splitting δ of a few meV. If then the relevant energies (like the thermal energy kT, the Zeeman energy μ₀μ_BH, the electrostatic energy edE, the photon energy $\text{h}\text{?}\text{ω}$, the condensation energy for the superconducting state Δ, etc.) are comparable with δ, novel effects are to be expected, called “quantum size effects” (QSE). In an ensemble of small particles, it is expected that the discrete energy levels are statistically distributed; therefore, methods of level statistics can be employed to calculate the different electronic properties of small particles.In this report, the more phenomenological aspects of the physics of small particles are discussed, where e.g. the interaction of the electromagnetic radiation with the particle is described by a dielectric constant, also characteristic for the bulk metal. The more microscopic quantum size effects in small particles are then analysed theoretically, mainly from the point of view of the statistics of discrete energy levels, and the existing experimental results are discussed. Superconductivity in small metallic particles is reviewed with emphasis on the critical fields in small particles, the magnetic field dependence of their microscopic properties (e.g. density of states), the problem of a lower size limit of a superconductor, and fluctuations in small superconductors. Finally, the most commonly used experimental methods to produce small particles are described.     

Phase diagrams of weakly anisotropic Heisenberg antiferromagnets: III. Nonlinear excitations and random fields in quasi 2-dimensional systems

The concept of effective field-dependent anisotropy is applied to the “spinflop” transition in the quasi 2-d Heisenberg antiferromagnet with weak orthorhombic anisotropy. From the correspondence between the “spinflop” problem and the commensurate-incommensurate transtion it follows that the “spinflop” is not first order and that random fields may cause domain-wall formation. This would explain the observed broadening of the “spinflop” in K₂MnF₄. In 3-d antiferromagnets such anomalous broadening is not observed, which would agree with the critical dimensionality d_c = 2 for the random-field problem.     

Vertex models on Feynman diagrams

We discuss the statistical mechanics of vertex models on both generic (“thin”) and planar (“fat”) random graphs. Such models can be formulated as the N → 1 and N → ∞ limits of N × N complex matrix models, respectively. From the graph theoretic perspective one is using matrix model and field theory inspired methods to count various classes of directed graphs. For the thin random graphs we use saddle point methods to solve the models in the thermodynamic, large number of vertices limit and note that, as in the case of the eight-vertex model on the square lattice, various other models such as the Ising model appear as particular limits. The generic solution of the fat graph model is rather more elusive, but we show that for several choices of the couplings the models can be reduced to eigenvalue integrals and their critical behaviour deduced.     

Random walks on sparsely periodic and random lattices: I. Random walk properties from lattice bond enumeration

We have developed a new technique for calculating certain asymptotic random walk properties on sparsely periodic and related random lattices in two and three dimensions. This technique is based on an ansatz which relates the number of lattice bonds in “irreducible lattice fragments” to the number of steps along these bonds. We show that certain random walk properties can be calculated very simply on the basis of this ansatz and that they depend only on the density of bonds and not on the arrangement of the bonds within the lattice. The random walk properties calculated here (mean square displacements, number of distinct sites visited, probability of return to the origin) are in complete agreement with results obtained earlier via generating function techniques. A subsequent paper contains generating function calculations which verify a number of new results presented here, such as mean occupation frequency of lattice sites, and a proof of our basic assumption on the relation between the number of lattice bonds and random walk steps.