Final Report Summary - ERG (Exact Renormalisation Group and Phases of Many Fermion Systems) The outgoing phase of the project was devoted to both developments of the exact renormalisation group (ERG) formalism and applications to some concrete physical systems. Namely, the approach was used to study the energy density at zero temperature of a one-component system of fermions in the unitary regime. It was shown that applying the ERG formalism to energy-density calculations requires some modifications to remove some theoretical uncertain- ties. The system of modified evolution equations has been derived and analysed. It was shown that modified ERG approach provides the sensible values of the universal coefficients consistent with both the experiment and existing theoretical calculation. The fundings have been summarised in the J. Phys.A 42, 465002, (2009). In this paper, possible further developments of the approach were also suggested. One example is ultra-cold gases of fermionic atoms with attractive fermion-fermion interactions. These interactions lead to composite bosons which carry fermionic quantum numbers. The crucial quantity to look at is the boson-boson scattering length aB. We have calculated the ratio of fermionic and bosonic scattering lengths of a B/aF and found that it decreases when the three-body term is included. We have also studied the sensitivity of the results to some scale parameters and shown that the three-body term significantly reduces the sensitivity of the results to this parameter. Our results were reported in the web e-prints arXiv:0911.4608 [cond-mat.quant-gas] and arXiv:0912.3665 [nucl-th] and published in Phys.Rev.A 81, 043628, (2009) and in EPJ Web Conf.3:02003,2010. In a major extension of our work, we have included all local (non-derivative) four-body interactions in the effective action. We find that these lead to results for the ratio of fermionic and bosonic scattering lengths aB/aF very close to those obtained by groups solving numerically the Schrodinger equation for four particles. Given the relative simplicity of our approach this result is indeed very encouraging. Note that our ERG calculations can relatively easily be extended to include finite-density effects. We have analyzed the sensitivity of the results to the scale parameter entering the regulator and found that the inclusion of both the three-body and four-body terms makes the results practically independent of the scale. Our findings have been reported in e-print: arXiv:1011.5852 and have been published Phys.Rev.A 83,2011. During the return phase we generalised our ERG approach for the case of the nuclear few-body systems. We have derived the corresponding flow equations under assumption of the exact SU(4) symmetry taking into account the three-body interactions and applied the developed for- malism to the deuteron-deuteron system. We have calculated the deuteron deuteron scattering length in different spin-isospin channels. We found that in the channel with zero total spin (singlet channel) the dynamics is driven by the Efimov physics and that one needs to fix one universal three-body parameter in order to extract the four-body observables. On the contrary, there is no need to fix the three body parameters in the quintet channel. Our conclusions agree with those, reached in the effective field theory treatment. The preliminary results will shortly be published in 'Progress of Particle and Nuclear Physics' and the more complete version is now submitted for publication Another extension of the application of the ERG method to few-body systems is to take into account the energy dependencies of the couplings. In order to do that we have introduced the effective 'trimer field' and derived the evolution equations. These calculations are currently in progress. We have also derived the flow equations describing the few-nucleon system beyond the SU(4) limit. These equations are now being analysed. We have also continued applying the ERG approach to studies of other many-body systems. In particular, the approach has been used to study the phenomenon of kaon condensation which may occur in the system of interacting kaons at finite isospin chemical potential. The corresponding flow equations have been derived and analysed. We have obtained the dispersion relations including the modes with spontaneously broken symmetry in agreement with the Goldstone theorem for the systems at finite density It has been found that the phenomena of kaon condensation develops at some critical value of density. The corresponding paper is now being prepared for submission. The other system we have studied was asymmetric many-fermion system with mismatched Fermi surfaces and different masses. It is known that such system can have a gapless phase (Sarma phase) which competes with the standard BCS pairing pattern. We have shown that the ERG approach is able to reproduce the mean-field results and estimated the beyond-mean-field corrections. We found that the effect of quantum fluctuations leaves no room for the Sarma phase. The corresponding paper is now being prepared for submission. During the reported period, Dr B. Krippa has also taken part in several scientific meetings where the presentations of the results obtained have been given. These meetings included: 1. 'Frontiers of many-body physics',international workshop, 2009, Seattle, United States of America (USA). The talk 'Exact RG and many-body problem has been given'. 2. International Conference on cold atoms, 2009, Trieste, Italy. The oral presentation 'Cold Fermi gases in unitary regime in the framework of exact RG' has been given. 3. 19th International IUPAP Conference on Few-Body Problems in Physics (FB 19), 2009, Bonn, Germany. The invited talk 'Nonperturbative renormalisation group: Applications to the few and many-body systems' has been given. 4. 'Many-body physics and cold atoms', international Workshop, 2010, Seattle, USA. The talk 'Few-body physics and Exact RG' was delivered. 5. 'Exact Renormalisation Group', 2010, Corfu, Greece. The talk 'Exact renormalisation group for few-body systems: Application to dimer-dimer scattering' was given. 6. 'Exact renormalisation group and few-body physics', talk given on 19th Particles and Nuclei International Conference, 2011, MIT, USA. 7. 'Nonperturbative renormalisation group and few-nucleon problem', reported on Erice School on Nuclear Physics, 2011, Erice, Italy.