Abstract:
The application of the exact renormalisation group to a many-fermion system with a short-range attractive force is studied. We assume a simple ansatz for the effective action with effective bosons, describing pairing effects and derive a set of approximate flow equations for the effective coupling including boson and fermionic fluctuations. The phase transition to a phase with broken symmetry is found at a critical value of the running scale. The mean-field results are recovered if boson-loop effects are omitted. The calculations with two different forms of the regulator was shown to lead to similar results.

Abstract:
The effect of mixing of the vector and axial vector correlation functions in the nuclear medium arising from the interaction of nuclear pions with corresponding interpolating currents is considered. It is shown that the mass difference between $\rho$ and $a_1$ meson gets smaller with the increase of the nuclear density reflecting the phenomena of partial restoration of chiral symmetry whereas the absolute values of meson masses may both decrease and increase in nuclear medium depending on the model used for the phenomenological spectral density.

Abstract:
The constraints imposed by chiral symmetry on hadron correlation functions in nuclear medium are discussed. It is shown that these constraints implies the certain structure of the in-medium hadron correlators and lead to the cancellation of the order $\rho m_\pi$ term in the in-medium nucleon correlator. We also consider the effect of mixing of the chiral partners correlaton functions arising from the interaction of nuclear pions with corresponding interpolating currents. It reflects the phenomena of partial restoration of chiral symmetry. The different scenarios of such restoration are briefly discussed.

Abstract:
The application of the exact renormalisation group to symmetric as well as asymmetric many-fermion systems with a short-range attractive force is studied. Assuming an ansatz for the effective action with effective bosons, describing pairing effects a set of approximate flow equations for the effective coupling including boson and fermionic fluctuations has been derived. The phase transition to a phase with broken symmetry is found at a critical value of the running scale. The mean-field results are recovered if boson-loop effects are omitted. The calculations with two different forms of the regulator are shown to lead to a similar results. We find that, being quite small in the case of the symmetric many-fermion system the corrections to mean field approximation becomes more important with increasing mass asymmetry.

Abstract:
We study the exact renormalisation group flow for ultracold Fermi-gases in unitary regime. We introduce a pairing field to describe the formation of the Cooper pairs, and take a simple ansatz for the effective action. Set of approximate flow equations for the effective couplings including boson and fermionic fluctuations is derived. At some value of the running scale, the system undergoes a phase transition to a gapped phase. The values of the energy density, chemical potential, pairing gap and the corresponding proportionality constants relating the interacting and non-interacting Fermi gases are calculated. Standard mean field results are recovered if we omit the boson loops.

Abstract:
Functional renormalisation group approach is applied to a imbalanced many- fermion system with a short-range attractive force. Composite boson field is introduced to describe the pairing between different flavour fermions. A set of approximate flow equations for the effective couplings is derived and solved. We identify the critical values of mass and particle number density mismatch when the system undergoes a phase transition to a normal state and determine the phase diagram both at unitary regime and nearby.

Abstract:
The application of the nonperturbative renormalisation group approach to a system with two fermion species is studied. Assuming a simple ansatz for the effective action with effective bosons, describing pairing effects we derive a set of approximate flow equations for the effective coupling including boson and fermionic fluctuations. The case of two fermions with different masses but coinciding Fermi surfaces is considered. The phase transition to a phase with broken symmetry is found at a critical value of the running scale. The large mass difference is found to disfavour the formation of pairs. The mean-field results are recovered if the effects of boson loops are omitted. While the boson fluctuation effects were found to be negligible for large values of $p_{F} a$ they become increasingly important with decreasing $p_{F} a$ thus making the mean field description less accurate.

Abstract:
We study a dynamics of ultracold Fermi-gases near the unitary limit in the framework of Effective Field Theory. It is shown that, while one can obtain a reasonable description of the universal proportionality constant both in the narrow and the broad Feshbach resonance limits, the reguirement of the reparametrisation invariance leads to appearance of the three body forces needed to cancel the otherwise arising off-shell uncertainties. The size of the unsertainties is estimated.

Abstract:
The Functional Renormalisation Group approach is applied the imbalanced many-fermion systems. The system is found to exhibit the first order phase transition from the superfluid to normal phase when the density (chemical potential) mismatch becomes larger then some critical values. The perspectives of using fermionic cold atoms to study nuclear/quark matter is briefly discussed.

Abstract:
The quark condensate in nuclear matter contains a term of order $\rho m_\pi$, arising from the contribution of low-momentum virtual pions to the $\pi N$ sigma commutator. Standard treatments of QCD sum rules for a nucleon in matter generate a similar term in the nucleon effective mass, although this is inconsistent with chiral perturbation theory. We show how an improved treatment of pionic contributions on the phenomenological side of the sum rules can cancel out this unwanted piece. Our results also show that factorisation ansatz for the four-quark condensate cannot be valid in matter.