Diagram analysis of the Hubbard model: Stationarity property of the thermodynamic potential

The diagram approach proposed many years ago for the strongly correlated Hubbard model is developed with the aim to analyze the thermodynamic potential properties. A new exact relation between renormalized quantities such as the thermodynamic potential, the one-particle propagator, and the correlation function is established. This relation contains an additional integration of the one-particle propagator with respect to an auxiliary constant. The vacuum skeleton diagrams constructed from the irreducible Green’s functions and tunneling propagator lines are determined and a special functional is introduced. The properties of this functional are investigated and its relation to the thermodynamic potential is established. The stationarity property of this functional with respect to first-order variations of the correlation function is demonstrated; as a consequence, the stationarity property of the thermodynamic potential is proved.     

The concept of effective density and the closed integrofunctional equation for a radial function of a simple liquid

The thermodynamically consistent concept of effective density, similar to the idea of an acting field in statistical thermodynamics, is introduced. Assuming a linear functional relation between the effective and true densities (the packet approximation), a generating functional is constructed and from it, in the usual way, the closed integrofunctional equation for a radial function is obtained. Some properties of the obtained equation are investigated, including its relation to known integral equations, and a possible scheme for its solution, by the method of iteration, is presented.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 96–100, September, 1975.     

Contact value relations and density functional theory for the electrical double layer

The recently developed contact-corrected density functional theory is applied to an electric double layer containing a valence and size symmetric electrolyte. The restricted primitive model is used to describe the electrolyte, and a non-polarisable, smooth, planar, uniformly charged hard surface is used to model the electrode. Comparison with Monte Carlo simulation results for this system shows that the theory satisfies the first contact value relation in line with the premise of the approach. Additionally, the theory is found to satisfy the semi-empirical second contact value relation quite closely. This is a welcome result, since no reference is made to this second relation nor is the relation used in the theoretical development. The second contact relation is closely related to the anomalous positive temperature derivative of the double layer capacitance at low temperature, low ion concentration, and low electrode charge, and indeed, the contact-corrected density functional theory predicts this effect.     

Row transfer matrix functional relations for Baxter's eight-vertex and six-vertex models with open boundaries via more general reflection matrices

The functional relations of the transfer matrices of fusion hierarchies for six- and eight-vertex models with open boundary conditions have been presented in this paper. We have shown the su(2) fusion rule for the models with more general reflection boundary conditions, which are represented by off-diagonal reflection matrices. Also we have discussed some physics properties, which are determined by the functional relations. Finally, the intertwining relation between the reflection K-matrices for the vertex and SOS models is discussed.     

Off-diagonal Bethe ansatz solution of the XXX spin chain with arbitrary boundary conditions

Employing the off-diagonal Bethe ansatz method proposed recently by the present authors, we exactly diagonalize the XXX spin chain with arbitrary boundary fields. By constructing a functional relation between the eigenvalues of the transfer matrix and the quantum determinant, the associated T–Q$T\text{–}Q$ relation and the Bethe ansatz equations are derived.     

Integrable boundary Boltzmann weights and surface free energy inversion relations of the chiral Potts model

The integrability of the chiral Potts model with boundaries is considered in this paper. The boundary star-triangle relation determining the boundary Boltzmann weights for the chiral Potts model is presented. By solving the boundary star-triangle relation the boundary Boltzmann weights are obtained. The fusion procedure is then applied to derive the functional relations of the transfer matrices of the model with boundaries. From these functional relations the inversion relations of the surface free energies are extracted when the system size is big enough. Surprisingly, the inversion relation of the local surface free energy is as simple as those of other non-chiral models, but it has still to be solved.     

Nuclear energy density functionals constrained by low-energy QCD

A microscopic framework of nuclear energy density functionals is reviewed, which establishes a direct relation between low-energy QCD and nuclear structure, synthesizing effective field theory methods and principles of density functional theory. Guided by two closely related features of QCD in the low-energy limit: a) in-medium changes of vacuum condensates, and b) spontaneous breaking of chiral symmetry; a relativistic energy density functional is developed and applied in studies of ground-state properties of spherical and deformed nuclei.     

Supersymmetry in Euclidean Quantum Field Theory

Supersymmetry is formulated within a functional approach of euclidean quantum field theory. The rǒle of the euclidean Lie superalgebra and its relation to the Poincaré Lie superalgebra are investigated in detail. As example we study the Wess-Zumino model in two dimensions.     

Free energy of the solvable chiral Potts model

Very recently, it has been shown that there are chiralN-state Potts models in statistical mechanics that satisfy the star-triangle relation. Here it is shown that the relation implies that the free energy (and its derivatives) satisfies certain functional relations. These can be used to obtain the free energy: in particular, we expand about the critical case and find that the exponent is 1–2/N.     

Classical stability and semiclassical ground state of a monmoving,rigid glue loop

Polyakov's action for rigid strings provides a semiclassical approach to a phenomenological glue loop. The static energy functional is discussed via its relation to knot theory. The lowest order mass correction is calculated by expanding around the classical worldsheet.     

Bound states in gauge theories as the Poincaré group representations

The bound-state generating functional is constructed in gauge theories. This construction is based on the Dirac Hamiltonian approach to gauge theories, the Poincaré group classification of fields and their nonlocal bound states, and the Markov-Yukawa constraint of irreducibility. The generating functional contains additional anomalous creations of pseudoscalar bound states: para-positronium in QED and mesons inQCDin the two-gamma processes of the type of γ + γ → π ₀ +para-positronium. The functional allows us to establish physically clear and transparent relations between the perturbativeQCD to its nonperturbative low-energy model by means of normal ordering and the quark and gluon condensates. In the limit of small current quark masses, the Gell-Mann-Oakes-Renner relation is derived from the Schwinger-Dyson and Bethe-Salpeter equations. The constituent quark masses can be calculated from a self-consistent nonlinear equation.     

Extension of Variable Separable Solutions for Nonlinear Evolution Equations

We give the generalized definitions of variable separable solutions to nonlinear evolution equations, and characterize the relation between the functional separable solution and the derivative-dependent functional separable solution. The new definitions can unify various kinds of variable separable solutions appearing in references. As application, we classify the generalized nonlinear diffusion equations that admit special functional separable solutions and obtain some exact solutions to the resulting equations.     

Analytical properties of the anisotropic cubic Ising model

We combine an exact functional relation, the inversion relation, with conventional high-temperature expansions to explore the analytic properties of the anisotropic Ising model on both the square and simple cubic lattice. In particular, we investigate the nature of the singularities that occur in partially resummed expansions of the partition function and of the susceptibility.     

Padé approximant in radiative transfer

A functional relation is obtained between radiative transfer in an inhomogeneous medium with internal sources and diffuse reflection. The intensity of the emerging radiation for a linear source is obtained by using the Padé approximation. The single scattering albedo is assumed to decrease exponentially with optical depth. Numerical results are given.     

Aspects of the functional renormalisation group

We discuss structural aspects of the functional renormalisation group. Flows for a general class of correlation functions are derived, and it is shown how symmetry relations of the underlying theory are lifted to the regularised theory. A simple equation for the flow of these relations is provided. The setting includes general flows in the presence of composite operators and their relation to standard flows, an important example being NPI quantities. We discuss optimisation and derive a functional optimisation criterion. Applications deal with the interrelation between functional flows and the quantum equations of motion, general Dyson-Schwinger equations. We discuss the combined use of these functional equations as well as outlining the construction of practical renormalisation schemes, also valid in the presence of composite operators. Furthermore, the formalism is used to derive various representations of modified symmetry relations in gauge theories, as well as to discuss gauge-invariant flows. We close with the construction and analysis of truncation schemes in view of practical optimisation.     

Vibrational and thermodynamic properties of orthorhombic CaSnO3 from DFT and DFPT calculations

Density functional theory (DFT) and density functional perturbation theory (DFPT) calculations were used to investigate the vibrational and thermodynamic properties of orthorhombic stannate CaSnO₃ compound. Our approach was based on the generalized gradient approximation with dispersion correction (GGA+D), considering the norm-conserved pseudopotentials. The phonon dispersion relation as well as theoretical peaks of the infrared (IR) and Raman spectrum in the frequency range of 100–800 cm⁻¹ was analyzed and assigned. The thermodynamic potentials and the specific heat at constant volume of the CaSnO₃ compound are also calculated, whose dependence with the temperature are discussed.     

Quantum work relations and response theory

A universal quantum work relation is proved for isolated time-dependent Hamiltonian systems in a magnetic field as the consequence of microreversibility. This relation involves a functional of an arbitrary observable. The quantum Jarzynski equality is recovered in the case this observable vanishes. The Green-Kubo formula and the Casimir-Onsager reciprocity relations are deduced thereof in the linear response regime.     

Chaos control and reduced-order generalized synchronization for the Chen——Liao system

This paper deals with the problem of chaos control and synchronization of the Chen--Liao system. From rigorous mathematic justification, the chaotic trajectories of the Chen--Liao system are led to a type of points whose four-dimensional coordinates have a particular functional relation among them. Meanwhile, a new synchronization manner, reduced-order generalized synchronization (RGS), is proposed which has the characteristic of having a functional relation between the slave and the partial master systems. It is shown that this new synchronization phenomenon can be realized by a novel technique. Numerical simulations have verified the effectiveness of the proposed scheme.     

Nonlinear Evolution Related to Invariant Functionals and the SUSY Transformations

A supersymmetrized formalism of the nonlinear evolution equations and the relation to the invariant functional are discussed.     

Riemann surfaces,Clifford algebras and infinite dimensional groups

We consider functionals on one dimensional subshifts which have prescribed Randon-Nikodym derivative under transportation by conjugating homeomorphisms, and investigate their relation to Ruelle's transfer operator. In particular we show that two-sided functionals essentially are products of a functional which are supported on stable and unstable leaves. We also prove the meromorphicity of the Fourier transform of correlation functions for AxiomA follows in a more general setting.