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In this work, a new functional is introduced to treat pairing correlations in finite many-body systems. Guided by the projected BCS framework, the energy is written as a functional of occupation numbers. It is shown to generalize the BCS approach and to provide an alternative to Variation After Projection framework. Illustrations of the new approach are given for the pairing Hamiltonian for various particle numbers and coupling strengths. In all case, a very good agreement with the exact solution is found.

We propose a local energy density functional for global description of pairing correlations by focusing on the neutron excess dependence. We show the clear correlation between pairing gaps and effective mass parameters as a function of neutron excess. This effect can be taken into account to the density functional by the isovector density dependence in the particle-particle channel.

The exact renormalization group method is applied to many-fermion systems with short-range attractive forces. The strength of the attractive fermion-fermion interaction is determined from the vacuum scattering length. A set of approximate flow equations is derived including fermionic bosonic fluctations. The numerical solutions show a phase transition to a gapped phase. The inclusion of bosonic fluctuations is found to be significant only in the small-gap regime.

The temperature and field dependence of the penetration depth was determined from muon spin rotation (μ⁺SR) measurements on a single crystal of YBa₂Cu₃O₇ having a superconducting transition at T_c ≈ 91.3 K. Data were acquired at applied magnetic fields of 0.05, 1.0, 3.0, and 6.0 Tesla, yielding results inconsistent with any pairing state requiring nodes, including d-wave pairing. These data are, however, completely consistent with s-wave (or extended s-wave) superconductivity, with clear evidence of field-dependent, temperature-activated vortex pinning. Our results confirm the s-wave character originally observed in 1989, and show that the features of μ⁺SR (and microwave) data used by other authors as evidence for d-wave superconductivity are instead due to temperature- and field-dependent vortex pinning/reordering, resulting in significant distortion of the flux lattice.

Experiments reveal the existence of metallic bands at surfaces of metals and insulators. The bands can be doped externally. We review properties of surface superconductivity that may set up in such bands at low temperatures and various means of superconductivity defection. The fundamental difference as compared to the ordinary superconductivity in metals, besides its two-dimensionality lies in the absence of the center of space inversion. This results in mixing between the singlet and triplet channels of the Cooper pairing.

The idea of spontaneous symmetry breaking in many-body physics from personal perspective (Bose-gas, nuclear structure and a new approach of Generalized Density Matrix).

Treating competing fluctuations, e.g., density, spin, current, need a tractable, self-consistent approach. One method that treats particle-particle and particle-hole correlations self-consistently is the diagrammatic "crossing-symmetric equations" method. In a general calculation for pairing, non-local interaction plays an important role in enhancing certain quantum fluctuations and thereby determining the pairing symmetry.

We introduce a new class of exactly solvable boson pairing models using the technique of Richardson and Gaudin. Analytical expressions for all energy eigenvalues and the first few energy eigenstates are given. In addition, another solution to Gaudin's equation is also mentioned. A relation with the Calogero–Sutherland model is suggested.

The physical properties of the nuclear shape have been investigated through the charge square radius (<r²>) and the quadrupole (Q₂) and hexadecapole (Q₄) moments of the even–even neutron-rich rare-earth nuclei. The single-particle energies used are those of a deformed Woods–Saxon mean-field. The pairing effects have been included by means of an exact projection method. The model has been tested for the "ordinary" nuclei near the shell closure N = 82 and has correctly reproduced the experimental data and particularly the "Kink" effect. The study has then been extended to the neutron-rich nuclei and has shown a stability of the <r²> and Q₂ results for N≃100 which may be attributed to the existence of a new magic number. On the other hand, a saturation of the prolate shape appears around N = 108 for the elements Nd, Sm and Gd and near N = 102 for the Dy, Er and Yb. These observations could not be confirmed by the investigation of the hexadecapole moment.

The variation of the two-neutron separation energy (S_2N), as a function of the neutron number N, is studied using a microscopic model that includes the pairing effects rigorously within the Fixed-Sharp-BCS method. The model was first tested on "ordinary" nuclei and allowed one to suitably reproduce the experimental data and to confirm the results of previous studies. The model was then applied to the even–even neutron-rich isotopes in the rare-earth region and showed, on the one hand, a relatively important variation of S_2N, when N = 100, that could lead to the assumption of the existence of a new magic number in this region, and on the other hand, a weak variation of S_2N when N > 100. These findings corroborate the previously obtained results for the charge mean square radius and the quadrupole and hexadecapole moments within the same model.

Using HF + BCS method we study light nuclei with nuclear charge in the range 2 ≤ Z ≤ 8 and lying near the neutron drip line. The HF method uses effective Skyrme forces and allows for axial deformations. We find that the neutron drip line forms stability peninsulas at ¹⁸He and ⁴⁰C. These isotopes are found to be stable against one neutron emission and possess the highest known neutron to proton ratio in stable nuclei.

Nuclear level densities of ${}^{207}$Pb and ${}^{89}$Y are calculated using the Lipkin–Nogami (LN) method and Bradeen–Cooper–Schrieffer (BCS) model. It is revealed that the calculated nuclear level densities are highly matched with the experimental data of Oslo group. The excitation energy and entropy are calculated for mentioned nuclei. In the case of two studied nuclei the characteristic of being magic for the number of neutrons or protons causes the decrease of the excitation energy and entropy contribution of magic system at low temperatures.

Gap parameter of Lipkin–Nogami (LN) model is replaced by order parameter of the exact Ginzburg–landau (EGL) theory. Thermodynamic quantities such as energy, entropy and heat capacity for ${}^{96,97}$Mo nuclei are calculated using this modified form of the LN model (MLN). In the LN model, the gap parameter suddenly decreases to zero at critical temperature. This causes singular points in the graph of heat capacity. However, in the MLN method, the order parameter does not become zero at critical temperature and gradually decreases along with the temperature. This causes the singular points, which are predicted in the heat capacity of LN model to be eliminated. Therefore, the heat capacity as a function of temperature becomes continuous and S-shaped, which is qualitatively in agreement with the experimental data.

Treating competing fluctuations, e.g., density, spin, current, need a tractable, self-consistent approach. One method that treats particle-particle and particle-hole correlations self-consistently is the diagrammatic "crossing-symmetric equations" method. In a general calculation for pairing, non-local interaction plays an important role in enhancing certain quantum fluctuations and thereby determining the pairing symmetry.

Experiments reveal the existence of metallic bands at surfaces of metals and insulators. The bands can be doped externally. We review properties of surface superconductivity that may set up in such bands at low temperatures and various means of superconductivity defection. The fundamental difference as compared to the ordinary superconductivity in metals, besides its two-dimensionality lies in the absence of the center of space inversion. This results in mixing between the singlet and triplet channels of the Cooper pairing.

The idea of spontaneous symmetry breaking in many-body physics from personal perspective (Bose-gas, nuclear structure and a new approach of Generalized Density Matrix).

The quark shell model has been successful in describing properties of hadrons. Because of color the quark shell model with 3n valence quarks has many more states which are singlet in color than the nuclear shell model with n valence nucleons. However, the quark interaction has been shown to favor two quarks coupled to spin zero and isospin zero and color , called diquarks. We show that the color singlet states in the quark shell model which have the maximal number of diquarks consistent with the Pauli symmetry are in one to one correspondence with the states of the nuclear shell model. We also investigate the implications of the quark interactions on the nuclear shell model interaction.

The solutions of the Wigner-transformed time-dependent Hartree–Fock–Bogoliubov equations are studied in the constant-Δ approximation, in spite of the fact that this approximation is known to violate both local and global particle-number conservation. As a consequence of this symmetry breaking, the longitudinal response function given by this approximation contains spurious contributions. A simple prescription for restoring both broken symmetries and removing the spurious strength is proposed. It is found that the semiclassical analogue of the quantum single-particle spectrum, has an excitation gap of 2Δ, in agreement with the quantum result. The effects of pairing correlations on the density response functions of three one-dimensional systems of different size are shown.

We study the application of the exact renormalisation group to a many-body system consisting of fermions interacting through a short-range attractive force. This is modelled through an effective range expansion using an effective field theory inspired approach. We investigate a systematic description of many-body effects in such systems.

Results of the semi-microscopic self-consistent approach to describe the ground state properties of the inner crust of a neutron star developed recently within the Wigner-Seitz method with pairing effects taken into account are briefly reviewed.