Functional Integral Theory Research Articles

Integrable and absolutely continuous vector-valued functions

The theory of integration for functions with values in a topological vector space is a bit of a messy subject. There are many different choices for the integral, most distinct from one another in general, and with no compelling reason to always adopt one definition or another as the “right” one. No attempt is made here to resolve this question of which integral is the “right” integral, but we overview a few of the common notions and prove a couple of useful properties for the notion of integrability by seminorm. For this notion of integrability, the completeness of the space of integrable functions with values in a complete locally convex space is established. This space is then easily seen to be isomorphic to the completion of the projective tensor product of the usual L1 space of scalar functions with the vector space. Absolute continuity for this notion of integrability is also considered. Here, it is shown that the familiar properties of differentiation for absolutely continuous scalar functions holds in the vector-valued case.

Toward an optimal theory of integration for functions taking values in quasi-Banach spaces

We present a new approach to define a suitable integral for functions with values in quasi-Banach spaces. The integrals of Bochner and Riemann have deficiencies in the non-locally convex setting. The study of an integral for p-Banach spaces initiated by Vogt is neither totally satisfactory, since there are quasi-Banach spaces which are p-convex for all 0<p<1, so it is not always possible to choose an optimal p to develop the integration. Our method puts the emphasis on the galb of the space, which permits a precise definition of its convexity. The integration works for all spaces of galbs known in the literature. We finish with a fundamental theorem of calculus for our integral.

Thermal fluctuations in superconducting phases with chiral d + id and s symmetry on a triangular lattice

The behavior of thermal fluctuations of a superconducting order parameter with extended s and chiral d + id symmetry is investigated. The study is carried out on a triangular lattice within the framework of the quasi-two-dimensional single-band model with attraction between electrons at neighboring sites. The method of consistent consideration of the order parameter fluctuations and the charge carrier scattering by fluctuations of coupled electron pairs, based on the theory of functional integration is used. The distribution functions of the phase fluctuation probabilities depending on temperature and charge carrier concentration are obtained. The temperature dependences of the amplitudes of the averaged superconducting order parameter are calculated. A phase diagram of superconducting states is constructed for the entire range of variation in the charge carrier concentration 0 < n < 2. Near the boundaries of this range, topologically trivial superconducting states with extended s symmetry are realized, while a superconducting state with topologically nontrivial chiral d + id symmetry is realized between them. The calculated anomalous self-energies are compared with the experimental ones obtained using machine learning techniques

Self-Consistent Consideration of Fluctuations in Singlet Superconducting Phases with s and d Symmetry

We analyze the behavior of thermal fluctuations of the superconducting order parameter with extended s-wave and $${{d}_{{{{x}^{2}} - {{y}^{2}}}}}$$-wave symmetry. For this purpose, we develop a method of self-consistent consideration of the order parameter fluctuations and charge carrier scatterers by fluctuations of coupled electron pairs using the theory of functional integration. The study is performed based on the quasi-two-dimensional one-band model with attraction between electrons located at neighboring sites. We obtain the distribution functions of the phase fluctuation probabilities depending on temperature, charge carrier concentrations, and model parameters. It is shown that the phase of the order parameter in the superconducting region is coherent, and the density of states has a dip at the Fermi level. In approaching the incoherent region of the phase diagram, the dip in the density of states disappears simultaneously with the loss of phase coherence. At the same time, the order parameter amplitude averaged over fluctuations remains finite at any temperature and concentration of charge carriers. Our results show that the pseudogap state cannot be explained in the frames of this scenario.

Effect of atomic disorder on the magnetic phase separation

The effect of disorder on the magnetic phase separation between the antiferromagnetic and incommensurate helical and phases is investigated. The study is based on the quasi-two-dimensional single-band Hubbard model in the presence of atomic disorder (the Anderson–Hubbard model). A model of binary alloy disorder is considered, in which the disorder is determined by the difference in energy between the host and impurity atomic levels at a fixed impurity concentration. The problem is solved within the theory of functional integration in static approximation. Magnetic phase diagrams are obtained as functions of the temperature, the number of electrons and impurity concentration with allowance for phase separation. It is shown that for the model parameters chosen, the disorder caused by impurities whose atomic-level energy is greater than that of the host atomic levels, leads to qualitative changes in the phase diagram of the impurity-free system. In the opposite case, only quantitative changes occur. The peculiarities of the effect of disorder on the phase separation regions of the quasi-two-dimensional Hubbard model are discussed.

David Rees, FRS 1918–2013

David Rees, FRS 1918–2013

Improvement of airfoil design using smooth curvature technique

The newly developed generalized function of airfoil profiles of wind turbine based on Trajkovski conformal transform theory can be used to fit the existing airfoil profiles and create the new ones by adjusting the coefficients of the generalized function. In this approach, the geometrical scale factor a， which was taken as a constant 0.25, has a significant impact on the curvature smooth continuity which will affect the aerodynamic performances of the airfoil. In this paper, the functional integral theory of wind turbine airfoils is studied. Furthermore, the advantage and the importance of curvature issue for airfoil surface are discussed in detail. It is found that, when different existing airfoils were analyzed using the generalized function, the geometrical scale factor a reaches an unexpected lower value. Based on curvature smooth continuity theory, a new method is presented to correct the geometrical scale factor a. As a result, the curvature smooth continuity of the fitting profile has been greatly improved, compared with that of the original profile. As an application of this new method, the DU93-W-210 airfoil is improved with the corrected geometrical scale factor a, and optimized using genetic algorithm (GA) method by controlling the coefficients of the shape function, leading to a new airfoil. Comparatively, the aerodynamic performances of the new airfoil such as maximum lift coefficient, maximum lift-drag ratio, roughness insensitivity and so forth are better than the DU93-W-210 airfoil performances. The achieved results show that this novel method is feasible to optimize airfoils of wind turbine.

Temperature dependence of the magnetization of transition-metal thin films: Fe- monolayers and bilayers

The functional integral theory of bulk itinerant magnetism is extended to be applied to low-dimensional transition-metal systems. Using a real-space recursive expansion of the local Green’s function we calculate the magnetization curves and the Curie temperature TC for the bulk, monolayer, and bilayer of Fe. Disorder is treated by using the virtual crystal approximation. The local-environment dependence of TC is analyzed by comparing the results for bulk and the (001) and (110) oriented films of Fe. Finally, the applicability of the theory is discussed.

Microscopic functional integral theory of quantum fluctuations in double-layer quantum Hall ferromagnets

We present a microscopic theory of zero-temperature order parameter and pseudospin stiffness reduction due to quantum fluctuations in the ground state of double-layer quantum Hall ferromagnets. Collective excitations in this systems are properly described only when interactions in both direct and exchange particle-hole channels are included. We employ a functional integral approach which is able to account for both, and comment on its relation to diagrammatic perturbation theory. We also discuss its relation to Gaussian fluctuation approximations based on Hubbard-Stratonovich-transformation representations of interactions in ferromagnets and superconductors. We derive remarkably simple analytical expressions for the correlation energy, renormalized order parameter and renormalized pseudospin stiffness.

On the functional integral theory of systems with kinematical interaction

We propose a systematic way to investigate the low-temperature thermodynamic properties of quantum spin systems subject to the restriction that only a finite number of bosons may occupy a single lattice site. Such a kinematical interaction results in the appearance of a temperature dependent chemical potential. Its low-temperature asymptotics is calculated self-consistently using the functional integration technique.

Operator calculations in the theory of path integrals

A method is developed in the theory of functional integration that makes it possible to use the group-theoretical formalism to calculate Feynman integrals. The method is used to consider some problems that have important applications in many branches of quantum mechanics. It particular, the formalism makes it possible to solve problems with quadratic Lagrangians with arbitrary dependence of the parameters on the time.

From the Hubbard model to classical spin-fluctuation theory.

In this paper we develop a microscopic foundation for the Murata-Doniach model of spin fluctuations which has been widely used in connection with band-structure calculations. The main result of the paper is the formulation of the partition function of an itinerant system as a functional integral over magnetization modes, and an explicit formula for the energy functional appearing in the exponent of the Boltzmann factor. Such a derivation is made since former theoretical investigations focus on the interaction part of the parition function Z/${\mathit{Z}}_{0}$, whereas the Murata-Doniach model is formulated in terms of a functional integral for the complete partition function Z. We start with an approximate description of magnetic excitations of noninteracting fermions with collective modes, and derive a bosonlike partition function for these magnetization modes. This is combined with the well-known result for the interaction part of the partition function in the Hubbard model obtained by functional-integral theory. The leading term of the energy functional appearing in the exponent of the parition function agrees with that of the Ginzburg-Landau expansion for the energy of a classical magnetization field. In the course of the transformation to a bosonlike system we predict that the cutoff wave vector ${\mathit{q}}_{\mathit{c}}$ which must be introduced in the classical model is temperature dependent with ${\mathit{q}}_{\mathit{c}}$\ensuremath{\sim}${\mathit{T}}^{1/3}$. It is shown that the frequencies of the collective modes are reduced by the Stoner enhancement factor compared with the one-particle excitation energies of Stoner theory.

Electron correlations in transition metals

The influence of electron correlations on the ground-state properties of transition metals is discussed. A number of corresponding effects are considered. Among them are a suppression of charge fluctuations, build-up of Hund's rule correlations, modification of the Stoner-Wohlfarth criterium for the onset of magnetic order, modification of magnetic volume effects, etc. Furthermore we discuss briefly the treatment of correlation effects at finite temperatures. Thereby it is important to go beyond the static approximation to the functional integral theory in order to property include the ground state correlations.

Critical Study of the Static Functional Integral Method in the Hubbard Model

AbstractThe effect of the static approximation in the functional integral representation of the paramagnetic Hubbard model is critically investigated. The exactly solvable two‐site model is treated within the one‐ and two‐field static functional integral method and some thermodynamic properties (local magnetic moment, specific heat cv) are compared with the exact solution for various interaction strengths and electron densities. It is shown that the static functional integral theory yields unphysical results for cv in the strong interacting system at small densities. The two‐field and one‐field methods are preferable in the low‐ and high‐temperature regions, respectively.

On the validity of the static approximation in the spin-fluctuation theory for itinerant electrons

The static approximation in the functional integral theory of itinerant magnetism is critically investigated. For the two-site Hubbard model the local magnetic moment and the specific heat are calculated as functions of temperature and interaction strength. By comparison with the exact results the error in the static approximation is found to be large at low temperatures and for small interaction strengths. It is shown that the subdivision of the ground state into a phase without a local moment (small interaction) and a phase with local moment (large interaction) is induced by the static approximation. The specific heat is underestimated and the zero-point entropy is finite. The implications for the infinite Hubbard model are briefly discussed.

Mesures coniques et intégrale de Daniell

AbstractLet X be a weakly complete proper cone contained in a weak space E and h(E) the Riesz space generated by the continuous linear forms on E. A positive conical measure μ on X is a positive linear form on h(E)|x. G. Choquet has proved μ is a Daniell integral on E when E is weakly complete, but μ is not generally a Daniell integral on X. However we give an integration theory for functions on X and compare this theory with the classical Daniell theory. The case where μ is maximal in the sense of G. Choquet is remarkable.

Integration in a convex linear topological space

This paper is concerned with a theory of integration for functions with values in a convex linear topological space. We consider an integral which is essentially an extension to this general space of the integral studied by Garrett Birkhoff [1] in a Banach space. By imposing different convex neighborhood topologies on a Banach space, we obtain as instances of our integral those defined by Birkhoff [1], Dunford [2], Gelfand [3], and Pettis [4]. Let f(s) be a function on an abstract set S to the real numbers, and let a(o-) be a nonnegative measure function on an additive family of subsets , of S. A necessary and sufficient condition for the Lebesgue integral to exist is that for each e >0 there exist a partition A0 of S into a denumerable set of sets (oi) such that for any two orderings of these sets, (ai) and (U-2),