IN two previous notes1, Prof. Max Born and I have shown that one can obtain a theory of superconductivity by taking account of the fact that the interaction of the electrons with the ionic lattice is appreciable only near the boundaries of Brillouin zones, and particularly strong near the corners of these. This leads to the criterion that the metal should be superconductive if a set of corners of a Brillouin zone is lying very near the Fermi surface, considered as a sphere, which limits the region in the momentum space completely filled with electrons.

On the theory of superconductivity

The two point integrals contributing to the self energy of a particle in a three dimensional quantum field theory are calculated to two loop order in perturbation theory as well as the vacuum ones contributing to the effective potential to three loop order. For almost every integral an expression in terms of elementary and dilogarithm functions is obtained. For two integrals, the master integral and the Mercedes integral, a one dimensional integral representation is obtained with an integrand consisting only of elementary functions. The results are applied to a scalar $\lambda\phi^4$ theory.

Feynman diagrams to three loops in three dimensional field theory

We study properties of magnetic field-induced Bose-Einstein condensate of triplons as a function of temperature and the field within the Hartree-Fock-Bogoliubov approach including the anomalous density. We show that the magnetization is continuous across the transition, in agreement with the experiment. In sufficiently strong fields the condensate becomes unstable due to triplon-triplon repulsion. As a result, the system is characterized by two critical magnetic fields: one producing the condensate and the other destroying it. We show that nonparabolic triplon dispersion arising due to the gapped bare spectrum and the crystal structure has a strong influence on the phase diagram.

High-field instability of a field-induced triplon Bose-Einstein condensate

Conditions are studied under which there can exist a quasiequilibrium mixture of itinerant and localized bosonic atoms in an optical lattice, even at zero temperature and at integer filling factor, when such a coexistence is impossible for an equilibrium lattice. The consideration is based on a model having the structure of a two-band, or two-component, boson Hubbard Hamiltonian. The minimal value for the ratio of on-site repulsion to tunneling parameter, necessary for the occurrence of such a mixture, is found.

Quasiequilibrium Mixture of Itinerant and Localized Bose Atoms in Optical Lattice

Conditions are studied under which there can exist a quasi-equilibrium mixture of itinerant and localized bosonic atoms in an optical lattice, even at zero temperature and at integer filling factor, when such a coexistence is impossible for an equilibrium lattice. The consideration is based on a model having the structure of a two-band, or two-component, boson Hubbard Hamiltonian. The minimal value for the ratio of onsite repulsion to tunneling parameter, necessary for the occurrence of such a mixture, is found.

Quasi-equilibrium mixture of itinerant and localized bose atoms in optical lattice

The properties of the uniform Bose gas are studied within the optimized variational perturbation theory (Gaussian approximation) in a self-consistent way. It is shown that the atomic Bose-Einstein condensate with a repulsive interaction becomes unstable when the gas parameter $\ensuremath{\gamma}=\ensuremath{\rho}{a}^{3}$ exceeds a critical value ${\ensuremath{\gamma}}_{crit}\ensuremath{\approx}0.01$. The quantum corrections beyond the Bogoliubov-Popov approximation to the energy density, chemical potential, and pressure in powers of $\sqrt{\ensuremath{\gamma}}$ expansions are presented.

Stability of the homogeneous Bose-Einstein condensate at large gas parameter

The post Gaussian effective potential in D = 3 dimensions and the Gaussian effective potential in $D = 2 + \varepsilon$
 are evaluated for the Ginzburg-Landau theory of superconductivity. It is shown that, the next order correction to the Gaussian approximation of the Ginzburg-Landau parameter $\kappa$
 is significant, whereas contribution from the two dimensionality is rather small. This strongly indicates that strong correlation plays a more dominant role than the two dimensionality does in high T
 c
 superconductivity.

Post Gaussian effective potential in the Ginzburg Landau theory of superconductivity

The post-Gaussian effective potential in D=2+2{epsilon} dimensions is evaluated for the Ginzburg-Landau theory of superconductivity. Two- and three-loop integrals for the post-Gaussian correction terms in D=2+2{epsilon} dimensions are calculated and {epsilon} expansions for these integrals are constructed. In D=2+2{epsilon} fractal dimensions the Ginzburg-Landau parameter turned out to be sensitive to {epsilon} and the contribution of the post-Gaussian term is larger than that for D=3. Adjusting {epsilon} to the recent experimental data on {kappa}(T) for the high-T{sub c} cuprate superconductor Tl{sub 2}Ca{sub 2}Ba{sub 2}Cu{sub 3}O{sub 10}, we found that {epsilon}=0.21 is the best choice for this material. The result clearly shows that, in order to understand high-T{sub c} superconductivity, it is necessary to include the fluctuation contribution as well as the contribution from the dimensionality of the sample. The method gives a theoretical tool to estimate the effective dimensionality of samples.

Ginzburg-Landau theory of superconductivity at fractal dimensions

We have extended the variational perturbative theory based on the back ground field method to include the optimized expansion of Okopinska and the post Gaussian effective potential of Stansu and Stevenson. This new method provides much simpler way to compute the correction terms to the Gausssian effective action (or potential). We have also renormalized the effective potential in 3+1 dimensions by introducing appropriate counter terms in the lagrangian

Optimized post Gaussian approximation in the background field method

We have extended the variational perturbative theory based on the background field method to include the optimized expansion of Okopinska and the post Gaussian effective potential of Stansu and Stevenson. This new method provides a much simpler way to compute the correction terms to the Gaussian effective action (or potential). We have also renormalized the effective potential in 3+1 dimensions by introducing appropriate counterterms in the Lagrangian.

Abdulla Rakhimov

Papers

Stability of the homogeneous Bose-Einstein condensate at large gas parameter

Post Gaussian effective potential in the Ginzburg Landau theory of superconductivity

Ginzburg-Landau theory of superconductivity at fractal dimensions

Optimized post Gaussian approximation in the background field method

Optimized post gaussian approximation in the background field method