Renormalization group studies of phase diagrams of Ising and XY-model films of variable thickness

We compare the critical behavior of $\text{spin}\text{-}\text{1}\text{2}$ Ising model (ordinary phase transition in two dimensions) and classical XY-model (topological phase transition in two dimensions) films, with two flat surfaces and nearest neighbor couplings K_S between surface spins and K_B between all others, as a function of film thickness. We carry out a real space Migdal-style renormalization in two stages. In the bulk stage the film is first renormalized towards a double layer, with the renormalized parameters as inter- and intra-layer couplings. Then the double layer is renormalized with those couplings as initial parameters. From the RG-equations for the bulk stage we find a tricritical point, not only for the Ising model (in which case it is well known) but also for the XY-model. It signals the existence of distinct surface and bulk transitions for sufficiently large values of $\text{K}{}_{\mathrm{<mtext>S</mtext>}}\text{K}{}_{\mathrm{<mtext>B</mtext>}}$. For the Ising model the complete program can be carried out and the phase diagram for films of arbitrary thickness is constructed. For the double layer XY-model a sufficiently complex Migdal-style renormalization appears to be unfeasible, presumably due to the possible presence of strings between the layers. Therefore, in an alternative approach, two representations for the partition function of the double layer XY-model are given. The system can be described in terms of its topological excitations, i.e., vortices on each layer and strings, either closed or terminating in vortices, between them. The system is also written as three coupled Coulomb gases. Based on this representation a renormalization group is found, and used, together with a Griffiths inequality for the correlation functions, to obtain information on the phase diagram. If there is a single phase transition, there is one phase where the correlations exhibit power law behavior and another where they fall off exponentially. The transition temperature increases monotonously with interlayer coupling to twice its value in two dimensions, but the nature of the phase transition for any finite inter-layer coupling appears to be different from that at zero coupling. We suggest that this is associated with the behavior of the internal strings. These results should be relevant for the renormalization of the film with isotropic bulk couplings as well as for layered systems of finite thickness, with different inter- and intra-layer couplings.     

The ground-state rotational band in 159Tb

162 MeV ⁴⁰Ca ions have been used to Coulomb excite the ground-state band of ¹⁵⁹Tb up to spin $\text{25}\text{2}$. Lifetimes for levels up to spin $\text{25}\text{2}$ have been determined with DSA and recoil distance methods. Multipole mixing ratios for several cascade transitions were extracted from an analysis of γ-ray angular correlation data. Reduced transition probabilities thus deduced in a model-independent way were found to be in agreement with the rotational model with Q₀ = 7.41 ± 0.06 e · b and g_K?g_R = 1.377 ± 0.010.     

Approximate eigensolutions of Dirac equation for the superposition Hellmann potential under spin and pseudospin symmetries

The Hellmann potential is simply a superposition of an attractive Coulomb potential ?a/r plus a Yukawa potential be^{?δ r} /r. The generalized parametric Nikiforov–Uvarov (NU) method is used to examine the approximate analytical energy eigenvalues and two-component wave function of the Dirac equation with the Hellmann potential for arbitrary spin-orbit quantum number κ in the presence of exact spin and pseudospin (p-spin) symmetries. As a particular case, we obtain the energy eigenvalues of the pure Coulomb potential in the non-relativistic limit.     

Electromagnetic transitions in the ground-state rotational band of159Tb

The ground-state band of¹⁵⁹Tb has been Coulomb excited up to spin 23/2 by 151-MeV⁴⁰Ar ions. The lifetimes of the 9/2 to 23/2 levels have been determined by combining results obtained with the RD and DSA methods, theE2/M1 mixing ratios for ΔI=1 transitions have been measured using angular correlation techniques and the branching ratios for the levels up to spin 17/2 have been determined. The energies of the levels and the reducedM1 andE2 transition probabilities for their decay have been compared with the predictions of the rotational model, without and with ΔK=1 admixtures. No satisfactory agreement could be achieved for all available experimental data simultaneously.     

A renormalization group analysis of the Kosterlitz-Thouless phase

We consider a classical Coulomb gas with a short distance cutoff in two dimensions; equivalently a Sine-Gordon field theory. For low temperature β^-1 and low activityz the gas is in a multipole phase, the Kosterlitz-Thouless phase. For β>8π andz sufficiently small we give a complete renormalization group analysis for this phase and show that the flow of the effective measures is toward a free field (infrared asymptotic freedom). This should lead to control over the long distance behavior of the theory. Research supported by the Natural Sciences and Engineering Research Council     

Spatial correlations in the electron gas: Path integral Monte Carlo simulation

Thermally excited states of the three-dimensional electron gas in a neutralizing background are computed by path integral Monte Carlo simulation for values of the Wigner-Seitz radius within the interval 5 < r _s < 15. Coulomb and exchange interactions, permutation symmetry, and spin state are treated explicitly. Variation of electron correlation functions with density and temperature is analyzed. Quantum effects suppress and enhance spatial correlation at low and high densities, respectively. Transition between the electron-gas states characterized by these opposite trends corresponds to a density of approximately 2.5 × 10²¹ cm^?3. A transition line between liquid-like and gaslike phases is determined in the temperature-density diagram. Weak anisotropy of many-body correlations in the liquid-like state stimulates excitation of spherically symmetric collective rotational modes. The effective short-range pseudopotential exhibits strong temperature dependence due to exchange effects. For strongly correlated systems, the characteristic screening length deviates from that predicted by the Thomas-Fermi screening model ( $ \sim \sqrt {r_s } $ ), approaching a linear function of r _s. The effective short-range interaction substantially differs from the Yukawa potential in mean field theory. Coulomb interaction shifts the Fermi level up by an order of magnitude or higher, and this effect becomes stronger with decreasing density.     

Periodic-orbit theory of level correlations

We present a semiclassical explanation of the so-called Bohigas-Giannoni-Schmit conjecture which asserts universality of spectral fluctuations in chaotic dynamics. We work with a generating function whose semiclassical limit is determined by quadruplets of sets of periodic orbits. The asymptotic expansions of both the nonoscillatory and the oscillatory part of the universal spectral correlator are obtained. Borel summation of the series reproduces the exact correlator of random-matrix theory.     

Classical Coulomb fluids in a confined geometry

It has already been argued that a classical (three-dimensional) Coulomb fluid confined between two parallel walls exhibits ideal gas features when the distance between the walls becomes small; this is confirmed in the present paper. Two-dimensional models of Coulomb fluids (with a logarithmic interaction), confined in a strip, are also studied. These models do not become ideal gases in the narrow strip limit. The correlation functions are also studied. There is a special temperature at which exact results are obtained. At that temperature, the two-dimensional, two-component plasma (two-dimensional Coulomb gas), which is a conductor when unconfined, becomes a dielectric as soon as it is confined in a strip of noninfinite width. This can be understood as a displacement of the Kosterlitz-Thouless transition by the confinement.     

Upper bounds on the transition temperature of classical n.n. ferromagnetic systems for general n and d

A differential equation is obtained for the spin-spin correlation function of a system of n-component classical spins interacting via a nearest neighbour ferromagnetic interaction. The differential equation makes it possible to obtain upper bounds on the correlation length and the transition temperature. It also makes it possible to relate the correlation length with the energy per spin and to obtain a necessary condition for the existence of the phase transition in terms of the energy per spin.     

A many-electron perturbation theory study of the hexagonal boron nitride bilayer system*

In this article we explore methods to reduce the computational cost in many-electron wave function expansions including explicit correlation and compact one-electron basis sets for the virtual orbitals. These methods are applied to the calculation of the interlayer binding energy of the h-BN bilayer system. We summarize the optimized interlayer distances as well as their binding energies for various stacking faults on different levels of theory including second-order Møller-Plesset perturbation theory and the random phase approximation. Furthermore, we investigate the asymptotic behavior of the binding energy at large interlayer separation and find that it decays as D^-4 in agreement with theoretical predictions, where D is the interlayer distance.     

The Quantum and Klein-Gordon Oscillators in a Non-commutative Complex Space and the Thermodynamic Functions

In this work we study the quantum and Klein-Gordon oscillators in a non-commutative complex space. We show that a particle described by such oscillators behaves similarly as an electron with spin in a commutative space in an external uniform magnetic field. Therefore the wave-function $\psi (z,\bar{z} )$ takes values in C ⁴, spin up, spin down, particle, antiparticle, a result which is obtained by the Dirac theory. We obtain the energy levels by exact solutions. We also derive the thermodynamic functions associated to the partition function, and show that the non-commutativity effects are manifested in energy at the high temperature limit.     

Correlation functions in the multiple Ising model coupled to gravity

The model of p Ising spins coupled to 2d gravity, in the form of a sum over planar φ³ graphs, is studied and in particular the two-point and spin-spin correlation functions are considered. We first solve a toy model in which only a partial summation over spin configurations is performed and, using a modified geodesic distance, various correlation functions are determined. The two-point function has a diverging length scale associated with it. The critical exponents are calculated and it is shown that all the standard scaling relations apply. Next the full model is studied, in which all spin configurations are included. Many of the considerations for the toy model apply for the full model, which also has a diverging geometric correlation length associated with the transition to a branched polymer phase. Using a transfer function we show that the two-point and spin-spin correlation functions decay exponentially with distance. Finally, by assuming various scaling relations, we make a prediction for the critical exponents at the transition between the magnetized and branched polymer phases in the full model.     

Critical exponents predicted by grouping of Feynman diagrams in φ4 model

Different perturbation theory treatments of the Ginzburg‐Landau phase transition model are discussed. This includes a criticism of the perturbative renormalization group (RG) approach and a proposal of a novel method providing critical exponents consistent with the known exact solutions in two dimensions. The usual perturbation theory is reorganized by appropriate grouping of Feynman diagrams of φ⁴ model with O(n) symmetry. As a result, equations for calculation of the two‐point correlation function are obtained which allow to predict possible exact values of critical exponents in two and three dimensions by proving relevant scaling properties of the asymptotic solution at (and near) the criticality. The new values of critical exponents are discussed and compared to the results of numerical simulations and experiments.     

The Berezinskii-Kosterlitz-Thouless transition and correlations in the XY kagomé antiferromagnet

The problem of the Berezinskii-Kosterlitz-Thouless transition in the highly frustrated XY kagomé antiferromagnet is solved. The transition temperature is found. It is shown that the spin correlation function exponentially decays with distance even in the low-temperature phase, in contrast to the order parameter correlation function, which decays algebraically with distance.     

Low energy behavior of the overscreened N-channel Kondo model

With eigen-functional method, we study the low energy behavior of the over screened N-channel Kondo model, and under linearized approximation, we give a general asymptotic expression of the local spin–spin correlation function, in which the correlation exponent is determined by phase shifts produced by scattering potentials. Moreover, we demonstrate that the temperature dependence of the local spin susceptibility and specific heat shows non-Fermi liquid behavior for large channel number N2, which is consistent with the exact results obtained by the Bethe ansatz and conformal field theory.     

Exact multiple scattering of waves from random rough surfaces: Speckle contrast

An exact formula for the mean multiple scattered intensity is presented by using normal statistics, as well as a gaussian correlation function, for the random rough surface in the limit of large correlation distance, T → ∞. Comparisons with the Kirchoff-Beckman approximation are also done. Further, we obtain an exact expression for the speckle contrast which explains experimental data and draws conclusions on the physical properties of the surface.     

Spin dynamics in a one-dimensional ferromagnetic bose gas

We investigate the propagation of spin excitations in a one-dimensional ferromagnetic Bose gas. While the spectrum of longitudinal spin waves in this system is soundlike, the dispersion of transverse spin excitations is quadratic, making a direct application of the Luttinger liquid theory impossible. By using a combination of different analytic methods we derive the large time asymptotic behavior of the spin-spin dynamical correlation function for strong interparticle repulsion. The result has an unusual structure associated with a crossover from the regime of trapped spin wave to an open regime and does not have analogues in known low-energy universality classes of quantum 1D systems.     

The Kosterlitz-Thouless transition in two-dimensional Abelian spin systems and the Coulomb gas

We rigorously establish the existence of a Kosterlitz-Thouless transition in the rotator, the Villain, the solid-on-solid, and the _n models, forn large enough, and in the Coulomb lattice gas, in two dimensions. Our proof is based on an inductive expansion of the Coulomb gas in the sine-Gordon representation, extending over all possible distance scales, which expresses that gas as a convex superposition of dilute gases of neutral molecules whose activities are small if is sufficiently large. Such gases are known not to exhibit screening. Abelian spin systems are related to a Coulomb gas by means of a duality transformation.Work supported in part by Grant NSF-DMR 79-04355 and 81-00417     

Analytic theory of correlation energy and spin polarization in the 2D electron gas

We present an analytic theory of the pair distribution function and the ground-state energy in a two-dimensional (2D) electron gas with an arbitrary degree of spin polarization. Our approach involves the solution of a zero-energy scattering Schrödinger equation with an effective potential which includes a Fermi term from exchange and kinetic energy and a Bose-like term from Jastrow-Feenberg correlations. The form of the latter is assessed from an analysis of data on a 2D gas of charged bosons. We obtain excellent agreement with data from quantum Monte Carlo studies of the 2D electron gas. In particular, our results for the correlation energy show a quantum phase transition occurring at coupling strength r_s≈24 from the paramagnetic to the fully spin-polarized fluid.