On the limit cycle for the 1/r potential in momentum space

The renormalization of the attractive 1/r² potential has recently been studied using a variety of regulators. In particular, it was shown that renormalization with a square well in position space allows multiple solutions for the depth of the square well, including, but not requiring a renormalization group limit cycle. Here, we consider the renormalization of the 1/r² potential in momentum space. We regulate the problem with a momentum cutoff and absorb the cutoff dependence using a momentum-independent counterterm potential. The strength of this counterterm is uniquely determined and runs on a limit cycle. We also calculate the bound state spectrum and scattering observables, emphasizing the manifestation of the limit cycle in these observables.     

Critical properties of the biaxial Ising model with both the longitudinal crystal field and the transverse dilution crystal field

Within the effective field theory (EFT), the critical properties of the biaxial Ising model with both longitudinal crystal field and transverse dilution crystal field are investigated for a simple cubic lattice. The tricritical point (TCP) and its trajectory are discussed in T-D_x and T-D_z space. A new phenomenon of two TCPs is found in T-D_x space. There exists a second-order line between two first-order lines, separated by two TCPs. The change of dilution concentration leads to a complex relation of the trajectory of the TCP. The degenerate patterns at the ground state appear by changing the longitudinal crystal field. The range of the ordered phase for transition lines labelled as a positive or (negative) value of D_x/J becomes larger or (smaller) with the decrease of t_x in T-D_z space. Some results have not been revealed in previous works.     

Dynamical role of the degree of intraspecific cooperation: A simple model for prebiotic replicators and ecosystems

We present a simple mean field model to analyze the dynamics of competition between two populations of replicators in terms of the degree of intraspecific cooperation (i.e., autocatalysis) in one of these populations. The first population can only replicate with Malthusian kinetics while the second one can reproduce with Malthusian or autocatalytic replication or with a combination of both reproducing strategies. The model consists of two coupled, nonlinear, autonomous ordinary differential equations. We investigate analytically and numerically the phase plane dynamics and the bifurcation scenarios of this ecologically coupled system, focusing on the outcome of competition for several degrees of intraspecific cooperation, σ, in the second population of replicators. We demonstrate that the dynamics of both populations can not be governed by a limit cycle, and also that once cooperation is considered, the topology of phase space does not allow for coexistence. Even for low values of the degree of intraspecific cooperation, for large enough autocatalytic replication rates, the second population of replicators is able to outcompete the first one, having a wide basin of attraction in state space. We characterize the same power law dependence between the outcompetition extinction times, τ, and the degree of intraspecific cooperation for both populations, given by τ∼c_iσ⁻¹. Our results suggest that, under some kinetic conditions, the appearance of autocatalysis might be favorable in a population of replicators growing with Malthusian kinetics competing with another population also reproducing exponentially.     

ABCD rule for Gaussian beam propagation in the context of quantum optics derived by the IWOP technique

The development of technique of integration within an ordered product (IWOP) of operators extends the Newton-Leibniz integration rule, originally applying to permutable functions, to the non-commutative quantum mechanical operators composed of Dirac’s ket-bra, which enables us to obtain the images of directly mapping symplectic transformation in classical phase space parameterized by [A, B; C, D] into quantum mechanical operator through the coherent state representation, we call them the generalized Fresnel operators (GFO) since they correspond to Fresnel transforms in Fourier optics. Based on GFO we find the ABCD rule for Gaussian beam propagation in the context of quantum optics (both in one-mode and two-mode cases) whose classical correspondence is just the ABCD rule in matrix optics. The entangled state representation is used in discussing the two-mode case.     

Localization at the energy threshold in non-commutative space

We study non-commutative quantum mechanics and exploit the non-commutative parameter as a scale for a scale symmetric system. The Hamiltonian in non-commutative space allows an unusual bound state at the threshold of the energy, E=0. The so(2,1) algebra for the system is also studied in non-commutative space.     

Fermion to boson mapping

The enveloping algebra, D_n, of fermions is extended on the lattice to include the discrete space invariance. This extended algebra, denoted X, has the space symmetry as a factor: X/D_n = space group.     

A relativistic wave equation for the Skyrmion

We propose a relativistically invariant wave equation for the Skyrme soliton. It is a differential equation on the space R^1,3×S³ which is invariant under the Lorentz group and isospin. The internal variable valued in SU(2)≡S³ describes the orientation of the soliton. The mass of a particle of spin and isospin both equal to is predicted to be which agrees with the known spectrum for low angular momentum. The iso-scalar magnetic moment is predicted to be , where Σ is the spin.     

On the evaluation of a certain class of Feynman diagrams in x-space: Sunrise-type topologies at any loop order

We review recently developed new powerful techniques to compute a class of Feynman diagrams at any loop order, known as sunrise-type diagrams. These sunrise-type topologies have many important applications in many different fields of physics and we believe it to be timely to discuss their evaluation from a unified point of view. The method is based on the analysis of the diagrams directly in configuration space which, in the case of the sunrise-type diagrams and diagrams related to them, leads to enormous simplifications as compared to the traditional evaluation of loops in momentum space. We present explicit formulae for their analytical evaluation for arbitrary mass configurations and arbitrary dimensions at any loop order. We discuss several limiting cases in their kinematical regimes which are e.g. relevant for applications in HQET and NRQCD. We completely solve the problem of renormalization using simple formulae for the counterterms within dimensional regularization. An important application is the computation of the multi-particle phase space in D-dimensional space-time which we discuss. We present some examples of their numerical evaluation in the general case of D-dimensional space-time as well as in integer dimensions D = D₀ for different values of dimensions including the most important practical cases D₀ = 2, 3, 4. Substantial simplifications occur for odd integer space-time dimensions where the final results can be expressed in closed form through elementary functions. We discuss the use of recurrence relations naturally emerging in configuration space for the calculation of special series of integrals of the sunrise topology. We finally report on results for the computation of an extension of the basic sunrise topology, namely the spectacle topology and the topology where an irreducible loop is added.     

Quantum anticentrifugal force for wormhole geometry

We show the existence of an anticentrifugal force in a wormhole geometry in R³. This counterintuitive force was shown to exist in a flat R² space. The role the geometry plays in the appearance of this force is discussed.     

The phase diagrams and compensation behaviors of mixed spin Blume-Capel model in a trimodal magnetic field

The phase diagrams and compensation behaviors of mixed spin-1/2 and spin-1 Blume-Capel model in a trimodal magnetic field are investigated in the framework of the effective field theory on simple cubic lattice. The change of negative crystal field and trimodal concentration can affect the TCP, the second-order phase and the magnetic field degeneration at ground state in T-H space. In T-D space, the trajectory of the TCP takes on the acre curve and there exist the two TCPs under certain condition. In addition to giving one or two compensation temperature points in M-T space, the mixed spin Blume-Capel model also provides one or two novel compensation magnetic field points in M-H space. Some results are not revealed in previous works.     

An optimum Hamiltonian for non-Hermitian quantum evolution and the complex Bloch sphere

For a quantum system governed by a non-Hermitian Hamiltonian, we studied the problem of obtaining an optimum Hamiltonian that generates nonunitary transformations of a given initial state into a certain final state in the smallest time τ. The analysis is based on the relationship between the states of the two-dimensional subspace of the Hilbert space spanned by the initial and final states and the points of the two-dimensional complex Bloch sphere.     

Effects of the substitution of gadolinium for neodymium on the crystal and magnetic properties of the Nd1−xGdxCo4B compounds

The crystal and magnetic properties of the Nd_1−xGd_xCo₄B compounds for 0?x?1 have been studied by X-ray powder diffraction, magnetization and differential scanning calorimetry (DSC) measurements. These compounds crystallize in a hexagonal CeCo₄B-type structure with the P6/mmm space group. The substitution of Gd for Nd leads to a decrease of the unit-cell parameter a and the unit-cell volume V, while the unit-cell parameter c remains almost constant. Magnetic measurements indicate that all samples are ordered magnetically below room temperature. The Curie temperatures determined by the DSC technique increase linearly as Nd is substituted by Gd. The saturation magnetization at 5 K decreases upon the Gd substitution up to x=0.6, and then increases again.     

Structural, thermal and morphological studies of magnesium substituted-lithium manganese oxide spinels

Coexistence of two phases having space groups of Fd3m and P4₃32 in the Mg-doped LiMn₂O₄ spinel is being reported for the first time in this article. Mg-doped LiMn₂O₄ powders have been synthesized by sol-gel method using citric acid as a chelating agent. X-ray powder diffraction (XRD) studies show that the crystal structure of LiMg_xMn_2−xO₄ for x<0.25 is a single-phase cubic spinel, which has space group of Fd3m. The cubic spinel structures having space group of Fd3m and P4₃32 are found to coexist in the compound for x=0.25. The structure becomes single-phase cubic spinel with space group P4₃32 for x>0.25. Field emission scanning electron microscopy (FESEM) shows that particle size of various synthesized powders ranges from 100 to 350 nm. Particle size decreases with increase in Mg content. Differential thermal analysis (DTA) and thermogravimetry (TG) studies show an exponential decay relationship between Mg-doping content and the decomposition temperature to form nonstoichiometry (LiMg_xMn_2−xO_4−δ) in air atmosphere. Fourier transform infrared spectroscopy (FTIR) analysis shows increase in the number of vibrational bands with increase in Mg content, which indicates ordering of the ions in the case of ordered spinel structure, and consequent reduction of the space group symmetry from O_h⁷ to O⁷.     

Phase transition properties of the spin-1 Blume-Emery-Griffiths model with random transverse crystal field

Phase transition properties of a spin-1 Blume-Emery-Griffiths model (BEGM) with random transverse crystal field is studied by the effective field theory for a simple cubic lattice. In T−D_x space, we obtain the phase diagrams with the ratio α between the biquadratic interaction and the exchange interaction as well as a tunable parameter l of the transverse crystal field. The tricritical point (TCP) appears at α<0, which undergoes a crossover from positive to negative direction of the transverse crystal field when l<0. The TCP cannot be observed for α>0. The maximum critical temperature increases with the increase of α. The position of the peak value tends to the drift of negative or positive direction for a different magnitude or an imperfect (±) transverse crystal field distribution. In T−α space, the range of ordered phase is magnified when the ratio is changed from α<0 to α>0. The random transverse crystal field obviously affects the TCP.     

Influence of Ho substitution for Nd on magnetic properties of the Nd1−xHoxMn2Ge2 germanides

The structure and magnetic properties of Nd_1−xHo_xMn₂Ge₂ (0.0≤x≤1.0) germanides were studied by X-ray diffraction (XRD), differential scanning calorimetry (DSC) techniques and AC magnetic susceptibility measurements. All compounds crystallize in the ThCr₂Si₂-type structure with the space group I4/mmm. Substitution of Ho for Nd leads to a linear decrease in the lattice constants and the unit cell volume, and the magnetic interactions in the Mn sublattice cross over from a ferromagnetic character to an antiferromagnetic one. A typical SmMn₂Ge₂-like behavior is observed for x=0.6 and 0.8. The results are collected in a phase diagram.     

On the partial trace over collective spin degrees of freedom

We derive analytical properties for the degeneracy ν(N,j) occurring in the decomposition of the state space C2⊗N. We also investigate the dynamics of two qubits coupled via Ising interactions to separate spin baths, and we study the thermodynamic limit.     

The inverse problem in the case of bound states

We investigate the inverse problem for bound states in the D = 3 dimensional space. The potential is assumed to be local and spherically symmetric. The present method is based on relationships connecting the moments of the ground state density to the lowest energy of each state of angular momentum ?. The reconstruction of the density ρ(r) from its moments is achieved by means of the series expansion of its Fourier transform F(q). The large q-behavior is described by Padé approximants. The accuracy of the solution depends on the number of known moments. The uniqueness is achieved if this number is infinite. In practice, however, an accuracy better than 1% is obtained with a set of about 15 levels.The method is tested on a simple example, and applied to three different spectra.     

Magnetic properties of the Pr1−xGdxCo4B compounds

In this work, we have investigated the effect of the substitution of Gd for Pr on the crystal structure and magnetic properties of the Pr_1−xGd_xCo₄B compounds for 0?x?1 using X-ray powder diffraction, magnetic measurements, and differential scanning calorimetry (DSC). These compounds have hexagonal CeCo₄B-type structure with the space group P6/mmm. The substitution of Gd for Pr leads to a decrease of the unit-cell parameters a and the unit-cell volume V, while the unit-cell parameter c increases slightly. Magnetic measurements indicate that all samples are ordered magnetically below room temperature. The Curie temperatures determined by DSC technique increase as Pr is substituted by Gd. The saturation magnetization at 5 K decreases upon Gd substitution up to x=0.6, and then increases again.