Extended Hubbard model with renormalized Wannier wave functions in the correlated state: beyond the parametrized models

The method used earlier for analysis of correlated nanoscopic systems is extended to infinite (periodic) s-band-like systems described by the Hubbard model. The optimized single-particle Wannier wave functions contained in the parameters of the extended Hubbard model (in the nearest-neghbor hopping (-t), in the magnitude of the intraatomic interaction U, and in other parameters) are determined explicitly in the correlated state for the electronic systems of various symmetries and dimensions: Hubbard chain, square and triangular planar lattices, and the three cubic lattices (SC, BCC, FCC). In effect, the evolution of the electronic properties as a function of interatomic distance R is obtained. The model parameters in most cases do not scale linearly with the lattice spacing and hence, their solution as a function of microscopic parameters reflects only qualitatively the system evolution. Also, the atomic energy changes with R and therefore should be included in the model analysis. The solutions in one dimension (D = 1) can be analyzed both rigorously (by making use of the Lieb–Wu solution) and compared with the approximate Gutzwiller treatments. In higher dimensions (D = 2 and 3) only the latter approach is possible to implement within the scheme. The renormalized single particle wave functions are almost independent of the choice of the scheme selected to diagonalize the Hamiltonian in the Fock space in D = 1 case. For dimensions D > 1 the qualitative behavior is independent of the structure considered. The wave-function size increases above the Mott-Hubbard localization threshold and gradually reaches the atomic limit value. The method can be extended to other approximation schemes, as stressed at the end.     

Renormalized diffusion coefficients for electron-hole plasmas in crossed fields

The diffusion and drift of an excess plasma in a semiconductor is described with magnetohydrodynamic two-fluid equations including the fluctuating electric field produced by the equilibrium plasma in the sample. Using the weak coupling limit an equation of motion for the mean density of the excess plasma is established with renormalized drift and diffusion coefficients. With the aid of the fluctuation dissipation theorem these coefficients are expressed in terms of the dielectric function and discussed in detail for stable systems. The renormalized diffusion coefficient differs from the bare one by an additional term with thet ^–3/2-long time dependence. It is shown that this term in addition represents an anomalous diffusion rate proportionalB ^–1 which overweights the classical ambipolar diffusion for sufficiently strong fields, but decreases with increasing external electric field. The results are compared with experimental data.     

Spectral densities of response functions for the O(3) symmetric Anderson and two channel Kondo models

The O(3) symmetric Anderson model is an example of a system which has a stable low energy marginal Fermi liquid fixed point for a certain choice of parameters. It is also exactly equivalent, in the large U limit, to a localized model which describes the spin degrees of freedom of the linear dispersion two channel Kondo model. We first use an argument based on conformal field theory to establish this precise equivalence with the two channel model. We then use the numerical renormalization group (NRG) approach to calculate both one-electron and two-electron response functions for a range of values of the interaction strength U. We compare the behaviours about the marginal Fermi liquid and Fermi liquid fixed points and interpret the results in terms of a renormalized Majorana fermion picture of the elementary excitations. In the marginal Fermi liquid case the spectral densities of all the Majorana fermion modes display a dependence on the lowest energy scale, and in addition the zero Majorana mode has a delta function contribution. The weight of this delta function is studied as a function of the interaction U and is found to decrease exponentially with U for large U. Using the equivalence with the two channel Kondo model in the large U limit, we deduce the dynamical spin susceptibility of the two channel Kondo model over the full frequency range. We use renormalized perturbation theory to interpret the results and to calculate the coefficient of the ln divergence found in the low frequency behaviour of the T=0 dynamic susceptibility. Received 29 January 1999     

A statistical-mechanical theory of self-diffusion in colloidal suspensions — application to colloidal glass transitions

A statistical-mechanical theory of self-diffusion in colloidal suspensions is presented. A renormalized linear Langevin equation is derived from a nonlinear Langevin equation by employing the Tokuyama–Mori projection operator method. The friction constant is thus shown to be renormalized by the many-body correlation effects due to not only the direct interactions between particles, but also due to the hydrodynamic interactions between particles. The equations for the mean-square displacement and the non-Gaussian parameter are then derived. The present theory is applied to colloidal glass transitions to discuss the crossover phenomena in the dynamics of a single particle from a short-time self-diffusion process to a long-time self-diffusion process via a β (caging) stage. The effects of the renormalized friction coefficient on self-diffusion are thus explored with the aid of the analyses of the experimental data and the simulation results by the mean-field theory proposed by the present author. It is thus shown that the relaxation time of the renormalized memory function is given by the β-relaxation time. It is also shown that the non-Gaussian parameter is very small, even near the glass transition, because of the existence of the short-time self-diffusion coefficient caused by the hydrodynamic interactions.     

Entropy of quantum field in toroidal black hole without brick wall

In this paper the entropy of a toroidal black hole due to a scalar field is investigated by using the DLM scheme. The entropy is renormalized to the standard Bekenstein-Hawking formula with a one-loop correction arising from the higher curvature terms of the gravitational action. For the scalar field, the renormalized Newton constant and two renormalized coupling constants in the toroidal black hole are the same as those in the Reissner-Nordstrom black hole except for other one.     

Renormalized vertex modification of Hall conductivity for dilute magnetic alloys

By means of the renormalized vertex procedure for the motion of Green's function developed by the authors, the vertex function of magnetic alloys, based on thes-d exchange interaction, is solved exactly and the corresponding Hall conductivity tensors are obtained. It is found that the value of the renormalized Hall conductivity is (1+h ²)^–1 times less than that before the renormalization (hereh is a reduced magnetic field). It is shown that the renormalized modification of the conductivity is very important in the cases with not too weak external magnetic field and slow relaxation time.     

A unified grand canonical description of the nonextensive thermostatistics of the quantum gases: Fractal and fractional approach

In this paper, the particles of quantum gases, that is, bosons and fermions are regarded as g-ons which obey fractional exclusion statistics. With this point of departure the thermostatistical relations concerning the Bose and Fermi systems are unified under the g-on formulation where a fractal approach is adopted. The fractal inspired entropy, the partition function, distribution function, the thermodynamics potential and the total number of g-ons have been found for a grand canonical g-on system. It is shown that from the g-on formulation; by a suitable choice of the parameters of the nonextensivity q, the parameter of the fractional exclusion statistics g, nonextensive Tsallis as well as extensive (q=1) standard thermostatistical relations of the Bose and Fermi systems are recovered. Received 17 September 1999     

Lagrangian for the t–J Model Constructed from the Generators of the Supersymmetric Hubbard Algebra

In order to describe the dynamics of the t–J model, two different families of first-order Lagrangians in terms of the generators of the Hubbard algebra are found. Such families correspond to different dynamical second-class constrained systems. The quantization is carried out by using the path-integral formalism. In this context the introduction of proper ghost fields is needed to render the model renormalizable. In each case the standard Feynman diagrammatics is obtained and the renormalized physical quantities are computed and analyzed. In the first case a nonperturbative large-N expansion is considered with the purpose of studying the generalized Hubbard model describing N-fold-degenerate correlated bands. In this case the 1/N correction to the renormalized boson propagator is computed. In the second case the perturbative Lagrangian formalism is developed and it is shown how propagators and vertices can be renormalized to each order. In particular, the renormalized ferromagnetic magnon propagator coming from our formalism is studied in details. As an example the thermal softening of the magnon frequency is computed. The antiferromagnetic case is also analyzed, and the results are confronted with previous one obtained by means of the spin-polaron theories.     

Scale transformations for renormalized field operators: Discussion of the soluble model

We discuss the operator formulation of the Zachariasen-Thirring model, describing the chain approximation to the propagator (the sum of three-particle massless bubbles) in massless λ⁴ theory. Such a model is formally scale-invariant and explicitly soluble. All intermediate steps of conventional renormalization procedure, regularization, introduction of appropriate counterterms, and cut-off free limit, are explicitly performed. In every step the scaling properties are discussed and respective dilatation currents are written down. After the proper choice of scale transformations for the renormalized field operator, we obtain the nonlocal dilatation current, defining the renormalized dilatation generator D_Λ^R(t). In the cut-off free limit Λ → ∞ the ET commutator of D_Λ^R(t) with renormalized field operators reproduces the Callan-Symanzik modification of “naive” canonical scale transformations. The renormalized scale transformations coincide in the cut-off free limit with renormalized dimensional transformations and define the exact symmetry of the renormalized theory.     

On the Connections between the Non-linearity of Equations of Motion in Quantum Theory,Asymptotic Behavior and Renormalization

It is shown that the ordinary perturbation expressions used in quantum mechanics lead to the wrong asymptotic behavior of the Heisenberg observables as function of time. This difficulty is traced to the non-linearity of the Heisenberg equations of motion and is studied in the context of a one-dimensional non-linear oscillator problem. It is found that the correct asymptotic behavior can be obtained by a process of renormalization analogous to renormalization theory in quantum field theory. It turns out that the renormalized parameters analogous to mass and wave-function renormalization are not c-numbers but are instead q-numbers. It is suggested that the renormalization parameters of quantum field theory are also q-numbers.     

Refined holographic entanglement entropy for the AdS solitons and AdS black holes

We consider the refinement of the holographic entanglement entropy for the holographic dual theories to the AdS solitons and AdS black holes, including the corrected ones by the Gauss–Bonnet term. The refinement is obtained by extracting the UV-independent piece of the holographic entanglement entropy, the so-called renormalized entanglement entropy which is independent of the choices of UV cutoff. Our main results are: (i) the renormalized entanglement entropies of the _AdSd+1${\mathrm{AdS}}_{d+1}$ soliton for d=4,5$d=4,5$ are neither monotonically decreasing along the RG flow nor positive-definite, especially around the deconfinement/confinement phase transition; (ii) there is no topological entanglement entropy for AdS₅ soliton even with Gauss–Bonnet correction; (iii) for the AdS black holes, the renormalized entanglement entropy obeys an expected volume law at IR regime, and the transition between UV and IR regimes is a smooth crossover even with Gauss–Bonnet correction; (iv) based on AdS/MERA conjecture, we postulate that the IR fixed-point state for the non-extremal AdS soliton is a trivial product state.     

Superconducting State in the Lagrangian Formalism of the Generalized Hubbard Model

The nonperturbative large-N expansion applied to the generalized Hubbard model describing N-fold-degenerate correlated bands is considered. Our previous results, obtained in the framework of the Lagrangian formalism for the normal-state case, are extended to the superconducting state. The standard Feynman diagrammatics is obtained and the renormalized physical quantities are computed and analyzed. Our purpose is to obtain the 1/N corrections to the renormalized boson and fermion propagators when a state with Cooper-pair condensation (i.e., the superconducting state) is considered.     

Entropy maximum in a nonlinear system with the 1/<Emphasis Type="Italic">f</Emphasis> fluctuation spectrum

Analysis of the control and subordination is carried out for the system of nonlinear stochastic equations describing fluctuations with the 1/f spectrum and with the interaction of nonequilibrium phase transitions. It is shown that the control equation of the system has a distribution function that decreases upon an increase in the argument in the same way as the Gaussian distribution function. Therefore, this function can be used for determining the Gibbs-Shannon informational entropy. The local maximum of this entropy is determined, which corresponds to tuning of the stochastic equations to criticality and indicates the stability of fluctuations with the 1/f spectrum. The values of parameter q appearing in the definition of these entropies are determined from the condition that the coordinates of the Gibbs-Shannon entropy maximum coincide with the coordinates of the Tsallis entropy maximum and the Renyi entropy maximum for distribution functions with a power dependence.     

On the Hamiltonian of the Cubic Boson Selfinteraction in Four Dimensional Space Time

In four dimensional space time a boson field with cubic selfinteraction is considered. It is shown, that if a space cutoff function is included in the interaction term, the renormalized Hamiltonian H_ren exists as a densely defined symmetric operator.     

Effect of Coulomb forces on the position of the pole in the scattering amplitude and on its residue

The vertex constant for the virtual decay of a nucleus to two charged particles having an arbitrary orbital angular momentum l was explicitly obtained for the first time in the case of a standard expansion of the effective-range function K _l (k ²), as well as in the case where the function K ₀(k ²) has a pole. The bound state of the 3He nucleus and the resonance states of the ²He and ³He nuclei in the S wave, as well as the resonance states of the ⁵He and ⁵Li nuclei in the P wave, are studied as physical examples. For the Np and N d systems, the trajectories of the pole in the complex planes of the momentum and the corresponding renormalized vertex constant are constructed for the transition from a resonance state to a virtual state as the Coulomb interaction is switched off gradually.     

Field Equations in Quantum Electrodynamics

A formulation of quantum electrodynamics is presented, based on finite local field equations. These Dirac and Maxwell equations have the usual form except that the current operators f(x) and j^μ (x) are explicitly expressed as local limits of sums of non-local field products and suitable subtraction terms. These limits are shown to exist and to yield finite operators in the sense that the iterative solutions to the field equations are equivalent to conventional renormalized perturbation theory. The various invariance properties of the theory, including Lorentz invariance, gauge invariance, charge conjugation invariance, and renormalization invariance, are discussed and related directly to the field equations and current definitions. Initially only the general forms of the currents, based on dimensional arguments, are given. The electric current, for example, contains the (suitably defined) term :A³(x) :.The corresponding field equations are used to derive renormalized Dyson-Schwinger-type integral equations for the renormalized proper part functions ∑, II^μν, Λ^μ, and X^αβγδ (the four-photon vertex function), etc. Application of the boundary conditions ∑(p̀ = m) = ∑′(p̀ = m) = II(O) = II′(O) = II″(O) = Λ(p̀ = m, o) = X(O, O, O, O) = O completely specifies the current operators. Consistency is established by deriving the same equations from rigorous renormalization theory so that their iterative solutions are proved to reproduce the correct renormalized perturbation expansion. The electric current operator is exhibited in a manifestly gauge invariant form and in a form which is manifestly negative under charge conjugation. It is shown, in fact, that much of j^μ (x) can be determined directly from the requirements of gauge invariance and charge conjugation covariance, without recourse to the integral equations. It is suggested that equal time commutation relations can serve to similarly specify the rest of the current.