Path-integral formulation of chiral invariant fermion models in two dimensions

We study the Thirring and chiral-invariant Gross-Neveu (CGN) models using the functional integral method. By introducing an auxiliary vector field we disclose a relation with two-dimensional gauge theories coupled to fermions and then extend a technique based on a chiral change in the functional variables to study purely fermionic models.We obtain the exact Klaiber solution for the massless Thirring model (for spin $\text{1}\text{2}$) in a very simple way and we then extend our technique to investigate the CGN model. We show the factorization of a free fermionic part at the level of Green functions on very general grounds. We then impose certain restrictions on the behavior of the fields — which render our treatment exact only in the zero winding number sector, but allow the computation of the U(1) part of the CGN Green functions exactly, showing, in particular, its complete decoupling from the color part and the almost long-range order behaviour in the infrared region.In our approach, the non-triviality of the jacobian arising from the chiral transformation — directly related to the topological density and the axial anomaly — appears to be crucial for the functional integral treatment of these models.     

Universal properties of the transition from quasi-periodicity to chaos in dissipative systems

An exact renormalization group transformation is developed for dissipative systems which describes how the transition to chaos may occur in a continuous and universal manner if the frequency ratio in the quasi-periodic regime is held at a fixed irrational value. Our approach is a natural extension of K.A.M. theory to strong coupling. Most of our analysis is for analytic circle maps. We have found a strong coupling fixed point where invertibility is lost, which describes the universal features of the transition to chaos. We find numerically that any two such critical maps with the same winding number are C¹ conjugate. It follows that the low frequency peaks in an experimental spectrum are universal and we determine how their envelope scales with frequency.When the winding number has a periodic continued fraction, our renormalization transform has a fixed point and spectra are self similar in addition. For a set of non-periodic winding numbers with full measure our renormalization transformation yields an ergodic trajectory in a sub-space of all critical maps. Physically one finds singular and universal spectra that do not scale.     

Long Cycles in a Perturbed Mean Field Model of a Boson Gas

In this paper we give a precise mathematical formulation of the relation between Bose condensation and long cycles and prove its validity for the perturbed mean field model of a Bose gas. We decompose the total density ρ=ρ_short+ρ_long into the number density of particles belonging to cycles of finite length (ρ_short) and to infinitely long cycles (ρ_long) in the thermodynamic limit. For this model we prove that when there is Bose condensation, ρ_long is different from zero and identical to the condensate density. This is achieved through an application of the theory of large deviations. We discuss the possible equivalence of ρ_long≠ 0 with off-diagonal long range order and winding paths that occur in the path integral representation of the Bose gas     

Winding Clusters in Percolation on the Torus and the Möbius Strip

Using a simulation technique introduced recently, we study winding clusters in percolation on the torus and the Möbius strip for different aspect ratios. The asynchronous parallelization of the simulation makes very large system and sample sizes possible. Our high accuracy results are fully consistent with predictions from conformal field theory. The numerical results for the Möbius strip and the number distribution of winding clusters on the torus await theoretical explanation. To our knowledge, this study is the first of its kind.     

The relation between properties of Gentile statistics and fractional statistics of anyon

In this paper, we discuss the relationship of two kinds of intermediate-statistics, the Gentile statistics and the fractional statistics of anyons. The anyon winding number representation is introduced. We construct the transformation between anyon winding number representation and the occupation number representation of particles of Gentile statistics. We study intermediate-statistics quantum bracket and coherent states for anyons in the winding number representation. We demonstrate that anyons can be simulated by Gentile statistics with a geometric phase.     

Universal scaling behaviour for iterated maps in the complex plane

According to the theory of Schröder and Siegel, certain complex analytic maps possess a family of closed invariant curves in the complex plane. We have made a numerical study of these curves by iterating the map, and have found that the largest curve is a fractal. When the winding number of the map is the golden mean, the fractal curve has universal scaling properties, and the scaling parameter differs from those found for other types of maps. Also, for this winding number, there are universal scaling functions which describe the behaviour asn→∞ of theQ _n ^th iterates of the map, whereQ _n is then ^th Fibonacci number.     

Time-series analysis of a collective variable in high-dimensional cellular automata

We perform a maximum-entropy frequency analysis of the occupation-density time series for a recently proposed totalistic cellular automaton rule in five dimensions. This new information complements partial knowledge coming from winding number measurements. We discuss the possible phenomenology of the model in terms of our findings.     

The Transition to a Giant Vortex Phase in a Fast Rotating Bose-Einstein Condensate

We study the Gross-Pitaevskii (GP) energy functional for a fast rotating Bose-Einstein condensate on the unit disc in two dimensions. Writing the coupling parameter as 1/ε ² we consider the asymptotic regime ε → 0 with the angular velocity Ω proportional to (ε ²|log ε|)⁻¹. We prove that if Ω = Ω₀(ε ²|log ε|)⁻¹ and Ω₀ > 2(3π)⁻¹ then a minimizer of the GP energy functional has no zeros in an annulus at the boundary of the disc that contains the bulk of the mass. The vorticity resides in a complementary ‘hole’ around the center where the density is vanishingly small. Moreover, we prove a lower bound to the ground state energy that matches, up to small errors, the upper bound obtained from an optimal giant vortex trial function, and also that the winding number of a GP minimizer around the disc is in accord with the phase of this trial function.     

Left-right correlation energy

We first attempt to determine a local exchange functional E_x[p] which accurately reproduces the Hartree-Fock (HF) energies of the 18 first and second row atoms. E_x[p is determined from p and |δp|, and we find that we can improve significantly upon Becke's original generalized gradient approximation functional (commonly called B88X) by allowing the coefficient of the Dirac exchange term to be optimized (it is argued that molecules do not behave like the uniform electron gas). We call this new two parameter exchange functional OPTX. We find that neither δ p or t = Σ δ _i |² improve the fit to these atomic energies. These exchange functionals include not only exchange, but also left-right correlation. It is therefore proposed that this functional provides a definition for exchange energy plus left-right correlation energy when used in Kohn-Sham (KS) calculations. We call this energy the Kohn-Sham exchange (or KSX) energy. It is shown that for nearly all molecules studied these KSX energies are lower than the corresponding HF energies, thus giving values for the non-dynamic correlation energy. At stretched geometries, the KSX energies are always lower than the HF energies, and often substantially so. Furthermore all bond lengths from the KSX calculations are longer than HF bond lengths and experimental bond lengths, which again demonstrates the inclusion of left-right correlation effects in the functional. For these reasons we prefer to split the correlation energy into two parts: left-right correlation energy and dynamic correlation energy, arguing that the usage of the words ‘non-dynamic’ or ‘static’ or ‘near-degeneracy’ is less meaningful. We recognize that this definition of KSX is not precise, because the definition of a local E_x[p] can never be precise. We also recognize that these ideas are not new, but we think that their importance has been insufficiently recognized in functional determination. When we include third row atoms in our analysis, we are unable to find a local exchange functional which is a substantial improvement over B88X for the reproduction of HF energies. This must arise from the effects of the core orbitals, and therefore we do not consider that this detracts from the improved accuracy of OPTX. We report some MCSCF calculations constructed from bonding-antibonding configurations, from which we attempt to calculate ab initio left-right correlation. There is only moderate agreement between the two approaches. Finally we combine the OPTX functional with established correlation functionals (LYP, P86, P91) to form OLYP, OP86 and OP91; OLYP is a great improvement on BLYP for both energy and structure, and OP86, OP91 are an improvement over BP86, BP91 for structure. The importance of the exchange functional for molecular structure is therefore underlined.     

Padé approximations of quantized-vortex solutions of the Gross–Pitaevskii equation

Quantized vortices are important topological excitations in Bose–Einstein condensates. The Gross–Pitaevskii equation is a widely accepted theoretical tool. High accuracy quantized-vortex solutions are desirable in many numerical and analytical studies. We successfully derive the Padéapproximate solutions for quantized vortices with winding numbers ω = 1, 2, 3, 4, 5, 6 in the context of the Gross–Pitaevskii equation for a uniform condensate. Compared with the numerical solutions, we find that(1) they approximate the entire solutions quite well from the core to infinity;(2) higher-order Padé approximate solutions have higher accuracy;(3) Padé approximate solutions for larger winding numbers have lower accuracy. The healing lengths of the quantized vortices are calculated and found to increase almost linearly with the winding number. Based on experiments performed with ~(87)Rb cold atoms, the healing lengths of quantized vortices and the number of particles within the healing lengths are calculated, and they may be checked by experiment. Our results show that the Gross–Pitaevskii equation is capable of describing the structure of quantized vortices and physics at length scales smaller than the healing length.     

Geometric diagnostics of complex patterns: spiral defect chaos

Motivated by the observation of spiral patterns in a wide range of physical, chemical, and biological systems, we present an automated approach that aims at characterizing quantitatively spiral-like elements in complex stripe-like patterns. The approach provides the location of the spiral tip and the size of the spiral arms in terms of their arc length and their winding number. In addition, it yields the number of pattern components (Betti number of order 1), as well as their size and certain aspects of their shape. We apply the method to spiral defect chaos in thermally driven Rayleigh-Benard convection and find that the arc length of spirals decreases monotonically with decreasing Prandtl number of the fluid and increasing heating. By contrast, the winding number of the spirals is nonmonotonic in the heating. The distribution function for the number of spirals is significantly narrower than a Poisson distribution. The distribution function for the winding number shows approximately an exponential decay. It depends only weakly on the heating, but strongly on the Prandtl number. Large spirals arise only for larger Prandtl numbers (Pr approximately > 1). In this regime the joint distribution for the spiral length and the winding number exhibits a three-peak structure, indicating the dominance of Archimedean spirals of opposite sign and relatively straight sections. For small Prandtl numbers the distribution function reveals a large number of small compact pattern components.     

“磁通浓缩器”无浓缩磁通作用

本文通过解麦克斯韦方程给出了把“浓缩体”插入初级线圈前后的磁场分布。方程的解表明,如果在“浓缩体”插入前后,保持初级线圈中的电流不变,则二者在中空区产生的磁场不变;如果供应M者的能量不变(即C,V固定),则插入“浓缩体”前后场值之比B/B₀∝(R₀)^1/2,但这完全是由于磁场的再分布引起的。这时中空区的总磁通减小,磁通并不守恒,不存在使磁通浓缩的问题,因而用它产生高强脉冲磁场存在着原理性错误。本文指出了把“浓缩器”看作脉冲变压器的错误所在。理论计算还表明用端面为截头圆锥形装置能产生较高场是结构效应所致。 关键词：     

Definition and statistical distributions of a topological number in the lattice O(3) σ-model

We define a topological number Q for spin fields on a two-dimensional lattice. Q assumes integer values only and reduces to the well-known winding number in the classical continuum limit. A Monte Carlo measurement of the topological susceptibility χ_t = 〈Q²〉 / volume on a 100 × 100 lattice reveals that it decreases exponentially with increasing β( = inverse bare coupling constant). The corresponding prediction of the perturbative renormalization group is not matched, however.     

Winding up of the wave-function phase by an insulator-to-superfluid transition in a ring of coupled Bose-Einstein condensates

We study phase transition from the Mott insulator to superfluid in a periodic optical lattice. Kibble-Zurek mechanism predicts buildup of winding number through random walk of BEC phases, with the step size scaling as a third root of transition rate. We confirm this and demonstrate that this scaling accounts for the net winding number after the transition.     

Field-induced winding of chiral polymers

We propose a microscopic model of a chiral polymer chain with permanent transverse dipoles interacting with an external electric field. Its behaviour has been investigated by computer simulation in the limit of weak chirality. Large-scale (tertiary) helical winding induced along the field direction has been found above a threshold field E_c, and the helix parameters have been calculated as functions of the field strength. Below E_c there is no coherent helical structure of the chain conformation. We find a characteristic scaling of the threshold and the winding radius a with the chain bending modulus , and . Received: 15 November 1997 / Accepted: 16 February 1998     

Incubation in the UV irradiation of condensed CHCl<Subscript>3</Subscript> solids

Matrix-Assisted-Pulsed-Laser-Evaporation (MAPLE) has emerged as a very promising technique for the deposition of polymers and biopolymers in intact and functional form. However, our understanding of the mechanism of the procedure is still limited. Here, we examine laser-induced (248 nm) desorption from condensed CHCl₃ solid, which has been employed as a potential matrix in MAPLE. We find that the absorption of the condensed halocarbon increases significantly with successive laser pulses, as a result of the formation and accumulation of strongly absorbing products. This results in a significant increase of the ejection efficiency in the irradiation with successive laser pulses. Thus, in studies employing multi-pulse irradiation protocols, the attained laser-induced temperatures are considerably higher than what is estimated on the basis of the absorption coefficient of CHCl₃. Thus, contrary to previous suggestions, ablation of CHCl₃ frozen solid at 248 nm may be due to explosive boiling. A number of additional implications are also discussed.     

SLE on Doubly-Connected Domains and the Winding of Loop-Erased Random Walks

Two-dimensional loop-erased random walks (LERWs) are random planar curves whose scaling limit is known to be a Schramm-Loewner evolution SLE _κ with parameter κ=2. In this note, some properties of an SLE _κ trace on doubly-connected domains are studied and a connection to passive scalar diffusion in a Burgers flow is emphasised. In particular, the endpoint probability distribution and winding probabilities for SLE₂ on a cylinder, starting from one boundary component and stopped when hitting the other, are found. A relation of the result to conditioned one-dimensional Brownian motion is pointed out. Moreover, this result permits to study the statistics of the winding number for SLE₂ with fixed endpoints. A solution for the endpoint distribution of SLE₄ on the cylinder is obtained and a relation to reflected Brownian motion pointed out.     

Statistical Properties of the 2D Attached Rouse Chain

We study various dynamical properties (winding angles, areas) of a set of harmonically bound Brownian particles (monomers), one endpoint of this chain being kept fixed at the origin 0. In particular, we show that, for long times t, the areas {A _i} enclosed by the monomers scale like t ^1/2, with correlated gaussian distributions. This has to be compared to the winding angles { _i} around fixed points that scale like t and are distributed according to independent Cauchy laws.     

Weak Power-Counting Theorem for the Renormalization of the Nonlinear Sigma Model in Four Dimensions

The formulation of the non linear σ-model in terms of flat connection allows the construction of a perturbative solution of a local functional equation by means of cohomological techniques which are implemented in gauge theories. In this paper we discuss some properties of the solution at the one-loop level in D = 4. We prove the validity of a weak power-counting theorem in the following form: although the number of divergent amplitudes is infinite only a finite number of counterterms parameters have to be introduced in the effective action in order to make the theory finite at one loop, while respecting the functional equation (fully symmetric subtraction in the cohomological sense). The proof uses the linearized functional equation of which we provide the general solution in terms of local functionals. The counterterms are expressed in terms of linear combinations of these invariants and the coefficients are fixed by a finite number of divergent amplitudes. These latter amplitudes contain only insertions of the composite operators φ₀ (the constraint of the non linear σ-model) and F _μ (the flat connection). The structure of the functional equation suggests a hierarchy of the Green functions. In particular once the amplitudes for the composite operators φ₀ and F _μ are given all the others can be derived by functional derivatives. In this paper we show that at one loop the renormalization of the theory is achieved by the subtraction of divergences of the amplitudes at the top of the hierarchy. As an example we derive the counterterms for the four-point amplitudes. PACS numbers: 11.10.Gh, 11.30.Rd     

Projective Module Description of the q-Monopole

The Dirac q-monopole connection is used to compute projector matrices of quantum Hopf line bundles for arbitrary winding number. The Chern–Connes pairing of cyclic cohomology and K-theory is computed for the winding number −1. The non-triviality of this pairing is used to conclude that the quantum principal Hopf fibration is non-cleft. Among general results, we provide a left-right symmetric characterization of the canonical strong connections on quantum principal homogeneous spaces with an injective antipode. We also provide for arbitrary strong connections on algebraic quantum principal bundles (Hopf–Galois extensions) their associated covariant derivatives on projective modules. Received: Received: 4 September 1998 / Accepted: 16 October 1998