A path-functional field theory of lattice gauge models and the large-N limit

We transform lattice gauge models to a theory of functional fields defined on a set of closed paths. Some relevant properties of the formalism are discussed in detail, with emphasis on symmetry and topological structure. We then investigate the large-N limit of the U(N) lattice gauge model in arbitrary dimensions using this formalism. Assuming the existence of the limit, we show, to arbitrary order of the strong coupling expansion parameter (g²N)^?, which is kept fixed, that for the leading contribution in the limit: (i) the flow of indices in color space can be represented by planar diagrams; (ii) when the diagrams are immersed in space-time they are random surfaces without handles; (iii) there are interactions of the surfaces which can be depicted as the formation of multisheet bubblesw in the surfaces. This formalism also makes it possible to set up a gauge-invariant mean-field approximation.     

The weak-coupling phase of lattice spin and gauge models

We analyze the lattice weak-coupling (w.c.) expansion of O(N), CP^N?1 and chiral spin models, and of large-N reduced chiral and gauge models.We find that the w.c. expansion always agrees with mean field results, whenever comparable, for arbitrary space-time dimensions, and that the expansion of the reduced models agrees with that of the original ones. However, w.c. results disagree with one-dimensional large-N and (old and new) exact results. We explain this phenomenon as a failure of the analytic continuation from higher dimensions that defines lattice w.c. perturbation theory for massless models (even if infrared singularities always cancel).We use an improved version of the mean field (m.f.) technique suitable for reduced models. We compute the m.f. approximation of chiral models and use this result to determine the large-d (m.f.) behaviour of reduced gauge models, finding agreement with standard Wilson theory results.We give a new characterization of large-N chiral models in terms of the single-link integral for the adjoint representation of SU(N).     

Generalized Boltzmann equation for lattice gas automata

In this paper a theory is formulated that predicts velocity and spatial correlations between occupation numbers that occur in lattice gas automata violating semi-detailed balance. Starting from a coupled BBGKY hierarchy for then-particle distribution functions, cluster expansion techniques are used to derive approximate kinetic equations. In zeroth approximation the standard nonlnear Boltzmann equation is obtained; the next approximation yields the ring kinetic equation, similar to that for hard-sphere systems, describing the time evolution of pair correlations. The ring equation is solved to determine the (nonvanishing) pair correlation functions in equilibrium for two models that violate semidetailed balance. One is a model of interacting random walkers on a line, the other one is a two-dimensional fluid-type model on a triangular lattice. The numerical predictions agree very well with computer simulations.     

Variational principle for regular and random Ising models on the cactus tree or on the usual lattice in the “cactus approximation”

The distribution functions and the free energy are expressed in terms of the effective fields for the regular and random Ising models of an arbitrary spin S on the generalized cactus tree. The same expressions apply to systems on the usual lattice in the “cactus approximation” in the cluster variation method. For an ensemble of random Ising models of an arbitrary spin S on the generalized cactus tree, the equation for the probability distribution function of the effective fields is set up and the averaged free energy is expressed in terms of the probability distribution. The same expressions apply to the system on the usual lattice in the “cactus approximation”. We discuss the quantities on the usual lattice when the system or the ensemble of random systems has the translational symmetry. Variational properties of the free energy for a system and of the averaged free energy for an ensemble of random systems are noted. The “cactus approximations” are applicable to the Heisenberg model as well as to the Ising model of an arbitrary spin, and to ensembles of random systems of these models.     

How to do mean field theory in Feynman gauge and doing it for U(1) with corrections to fourth order

It is demonstrated how mean field theory with corrections from fluctuations may be applied to lattice gauge theories in covariant gauges. By fixing the gauge at tree level the importance of fluctuations is decreased. This is understood as inclusion of terms of next-to-leading-order in d in the definition is the mean field tree approximation, d being the dimension of the lattice. The gauge group U(1) and Wilson's action are used as testing ground. Tree and one-loop results comparable to those previously obtained in axial gauge are obtained for d = 4. The next three correction terms to the free and plaquette energies are evaluated in Feynman gauge. The truncated asymptotic series thus obtained is compared to that of the ordinary weak coupling expansion. The mean field series gives, to those orders studied, a much better approximation. The location of phase transitions in 4d and 5d are predicted with 1% error bars.     

Chiral symmetry breakdown,anomaly, and the PCAC relation on a lattice

Lattice fermion formulation is investigated using a solvable model which resembles quantum chromodynamics. CP₂^N?1 models with quarks are formulated on a lattice. For dynamical quarks, a generalized formulation of the Wilson and the Osterwalder-Seiler lattice fermion is used. In the $\text{1}\text{N}$ expansion, the spontaneous breakdown of chiral symmetry (which is softly broken by the quark mass) apparently occurs in this model, and the “pion” mass is calculated. From the above results, it is shown that the above lattice fermion formulations have the desired continuum limit. The axial-vector current is investigated and it is proved that the usual anomaly appears in the continuum limit and the PCAC relation is satisfied.     

A new lattice approximation forthe Høegh-Krohn quantum field model

The lattice approximation for the exponential interaction model (Høbegh-Krohn quantum field model) with different lattice cutoffs a, a′ in the free and interacting parts is discussed. It is shown that the continuum limit exists under certain conditions on the dependence a′(a).     

Pressure–temperature dependence of thermodynamic properties of rutile (TiO2): A first-principles study

Ab-initio calculations of thermal properties of rutile (TiO₂) have been performed by using the projector augmented-wave (PAW) method within the generalized gradient approximation (GGA). Both pressure- and temperature-dependent thermodynamic properties such as the bulk modulus, thermal expansion, thermal expansion coefficient, heat capacity at constant volume and constant pressure were calculated using two different models based on the quasiharmonic approximation (QHA): the Debye–Slater and Debye–Grüneisen model with Dugdale–MacDonald (DM) approximation. Also, the empirical energy corrections were applied to the results to correct the systematic errors introduced by the functional. It is found that the Debye–Grüneisen model provides more accurate estimates than the Debye-Slater models, especially after empirical energy correction.     

Gitterkonstanten- und Dichteänderung durchF-Zentren in KBr

F centers were created in KBr by additive coloration. Change of lattice parameter was determined by the X-ray diffraction method and change of mass density by a hydrostatic flotation method. Combination of these results led to a lattice expansion of 0.6 atomic volumes perF center. Using elastic approximation the microscopic dilatation around anF center is obtained.     

Progress in lattice gauge theory

Two topics of lattice gauge theory are reviewed. They include string tension and β-function calculations by strong coupling Hamiltonian methods for SU(3) gauge fields in 3 + 1 dimensions, and a 1/N-expansion for discrete gauge and spin systems in all dimensions. The SU(3) calculations give solid evidence for the coexistence of quark confinement and asymptotic freedom in the renormalized continuum limit of the lattice theory. The crossover between weak and strong coupling behavior in the theory is seen to be a weak coupling but non-perturbative effect. Quantitative relationships between perturbative and non-perturbative renormalization schemes are obtained for the O(N) nonlinear sigma models in 1 + 1 dimensions as well as the range theory in 3 + 1 dimensions. Analysis of the strong coupling expansion of the β-function for gauge fields suggests that it has cuts in the complex 1/g²-plane. A toy model of such a cut structure which naturally explains the abruptness of the theory's crossover from weak to strong coupling is presented. The relation of these cuts to other approaches to gauge field dynamics is discussed briefly.The dynamics underlying first order phase transitions in a wide class of lattice gauge theories is exposed by considering a class of models-P(N) gauge theories - which are soluble in the N → ∞ limit and have non-trivial phase diagrams. The first order character of the phase transitions in Potts spin systems for N #62; 4 in 1 + 1 dimensions is explained in simple terms which generalizes to P(N) gauge systems in higher dimensions. The phase diagram of Ising lattice gauge theory coupled to matter fields is obtained in a $\text{1}\text{N}$ expansion. A one-plaquette model (1 time-0 space dimensions) with a first-order phase transitions in the N → ∞ limit is discussed.     

Perturbative and nonperturbative correspondences between compact and noncompact sigma-models

Compact (ferro- and antiferromagnetic) sigma-models and noncompact (hyperbolic) sigma-models are compared in a lattice formulation in dimensions d?2$d?2$. While the ferro- and antiferromagnetic models are essentially equivalent, the qualitative difference to the noncompact models is highlighted. The perturbative and the large N expansions are studied in both types of models and are argued to be asymptotic expansions on a finite lattice. An exact correspondence between the expansion coefficients of the compact and the noncompact models is established, for both expansions, valid to all orders on a finite lattice. The perturbative one involves flipping the sign of the coupling and remains valid in the termwise infinite volume limit. The large N correspondence concerns the functional dependence on the free propagator and holds directly only in finite volume.     

Renormalization techniques for quantum spin systems. Ground-state energies

Projection of the Hamiltonian of an antiferromagnetic lattice of spins $\text{1}\text{2}$, without external fields, onto a subspace of the total spinor space gives an approximation for the lowest eigenvalue of this Hamiltonian. Repeated projection results in a series expansion for this approximation. In each projection the form of the Hamiltonian is conserved. The formal structure of this projection technique shows a strong analogy with the Wilson theory or renormalization-group theory of phase transitions. Numerical results are given for linear chains and triangular lattice.Analogous techniques apply to Ising and isotropic XY models in transverse fields.     

Quantization of intrinsic localized modes: Effective linear lattice method

A working method is proposed to quantize intrinsic localized modes (ILMs) in lattices with hard quartic anharmonicity in the framework of the rotating-wave approximation. This is done by reducing a nonlinear eigenvalue problem to a linear one by averaging slowly varying reduced effective force constants over frequencies, enabling us to quantize the ILM in a straightforward manner. Such an effective linear lattice (ELL) method is first applied to an analytically tractable d-dimensional cubic lattice to show that the concept of the ELL holds exactly in the strong localization limit. Next, general lattice models are investigated to achieve quantization of the ILM in an approximate manner. The obtained analytical results are tested by solving numerically a model one-dimensional lattice to show that phase-space trajectory of an ILM-bearing atom is of elliptic type with finite but small width. The numerical result confirms the validity of the ELL leading to its semi-classical quantization. On the other hand, orbits of all the remaining atoms exhibits complex non-periodic trajectory to which a direct application of the semi-classical quantization rule appears impossible.     

A method for estimating glueball-meson mixing in lattice QCD

A systematic expansion of lattice QCD amplitudes based on the replica trick is discussed, the leading term of which is the quenched approximation. A parameter is defined that estimates the mixing between glueball and q$\text{q}$ meson states and provides a test for the reliability of the quenched approximation. The procedure is illustrated by an explicit Monte Carlo calculation for a model system on a one-dimensional lattice.     

First order transition in three-dimensional systems with fully broken O(3) symmetry

Systems with fully broken O(3) symmetry are studied by Monte Carlo simulations. Two models of classical Heisenberg magnets in three dimensions are considered: the stacked three-exchange model on a simple cubic lattice and the model on a stacked-triangular lattice with two competing interlayer exchanges. The first order transition is found in both models. Pseudouniversal behavior of the models is found for small size lattices.     

Nonphysical noises and instabilities in plasma simulation due to a spatial grid

A series of plasma numerical simulation has been performed in order to understand the enhancement of nonphysical noises and instabilities due to the use of a spatial grid. Several different superparticle models including the Nearest Grid Point (NGP) model, Cloud-in-Cell (CIC) or Particle-in-Cell (PIC) models, Lewis energy conserving code, and the multipole expansion code have been examined for a Maxwellian plasma and a one beam plasma using a one-dimensional, one-specie (electron) plasma. An instability was observed for all of the models when the Debye length was too small compared with the grid size. When the Debye length is comparable to the grid size, no instabilities were observed. However, the enhancement of noises at high frequencies (ω > 3ω_pe may not always be negligible- even for long wavelength modes for the NGP model. For the NGP and CIC, PIC models, the experimental results are in good agreement with Langdon's theory. It is observed that the dipole expansion model, which is the first-order approximation to the multipole expansion scheme, is similar to CIC, PIC models in many respects and appears to be the same order of approximation.     

Strongly coupled massive QCD3 and frustrated Heisenberg antiferromagnets: fractional statistics and chiral-spin order

Strongly coupled massive SU(N_C) and U(N_C) QCD₃ on a lattice is studied using the 1/N_C expansion. The quark mass terms have a definite sign in the present model, and therefore the system explicitly breaks the parity symmetry. The continuum counterpart generates the Maxwell + Chern-Simons theory by integrating out the quark field. In the present paper, we shall integrate out the gauge fields using the strong-coupling expansion and obtain a frustrated quantum Heisenberg model as an effective model. The ground state of the above effective quantum spin model is studied using the large-N_C approximation. There are two phases; one is a Neel-ordered state and the other is a state with a chiral-spin order. It is explicitly shown that the chiral-spin ordered state corresponds to a state with spontaneous generation of color magnetic flux in the original theory and fractional statistics appears in that phase. This result strongly suggests that there are (at least) two phases in the massive QCD₃ and Maxwell-CS theory. One is the confinement phase and the other is the perturbative deconfinement phase with fractional-statistics excitations.     

CPN−1 coupled to gauge fields: Mean field theory and monte carlo results

We define a two parameter lattice field theory which interpolates between the O (2N) Heisenberg model, pure U(1) gauge theory, and a lattice version of the CP^N?1 model. The phase diagram in space-time dimension d=4 is obtained by Monte Carlo simulation on a 4⁴ lattice, and the nature of the phases is discussed in mean field approximation.     

Symmetry Breaking in Finite Volume (Ⅱ)

Using a Hamiltonian approach and the adiabatic approximation, the low-lying spectrum of the lattice O (N) model in the broken phase is analyzed to the next to leading order. It is seen that these corrections appear as a systematic expansion in the inverse power of the lattice size L in the large volume limit.     

Electronic structure, optical and thermodynamic properties of orthorhombic UCoGe under pressure

The electronic structures, optical and thermodynamic properties of orthorhombic UCoGe are investigated using the generalized gradient approximation (GGA) formalism in the framework of the density functional theory (DFT). The obtained lattice parameters, bulk modulus B and its pressure derivative B′ of UCoGe are in agreement with the available experimental data. From the analysis of band structure and density of states coming out from our calculations, we can see that UCoGe in the ground state belongs to a typical metallic alloy. Various optical properties, including the dielectric function and absorption coefficient as functions of the photon energy are calculated. The thermodynamic properties of UCoGe are predicted using the quasi-harmonic Debye model for the first time. The Debye temperature, the Grüneisen parameter, the heat capacity and the thermal expansion coefficient are obtained at high pressures and temperatures.