We show that the renormalization group decimation of modern nucleon potential models to low momenta results in a unique nucleon interaction V_{low k}. This interaction is free of short-ranged singularities and can be used directly in many-body calculations. The RG scaling properties follow directly from the invariance of the scattering phase shifts. We discuss the RG treatment of Fermi liquids. The RG equation for the scattering amplitude in the two particle-hole channels is given at zero temperature. The flow equations are simplified by retaining only the leading term in an expansion in small momentum transfers. The RG flow is illustrated by first studying a system of spin-polarized fermions in a simple model. Finally, results for neutron matter are presented by employing the unique low momentum interaction V_{low k} as initial condition of the flow. The RG approach yields the amplitude for non-forward scattering, which is of great interest for calculations of transport properties and superfluid gaps in neutron star interiors. The methods used can also be applied to condensed matter systems in the absence of long-ranged interactions.
We show that the renormalization group decimation of modern nucleon potential models to low momenta results in a unique nucleon interaction V_{low k}. This interaction is free of short-ranged singularities and can be used directly in many-body calculations. The RG scaling properties follow directly from the invariance of the scattering phase shifts. We discuss the RG treatment of Fermi liquids. The RG equation for the scattering amplitude in the two particle-hole channels is given at zero temperature. The flow equations are simplified by retaining only the leading term in an expansion in small momentum transfers. The RG flow is illustrated by first studying a system of spin-polarized fermions in a simple model. Finally, results for neutron matter are presented by employing the unique low momentum interaction V_{low k} as initial condition of the flow. The RG approach yields the amplitude for non-forward scattering, which is of great interest for calculations of transport properties and superfluid gaps in neutron star interiors. The methods used can also be applied to condensed matter systems in the absence of long-ranged interactions.
Renormalization group methods can be applied to the nuclear many-body problem using the approach proposed by Shankar. We start with the two-body low momentum interaction V_{low k} and use the RG flow from the particle-hole channels to calculate the full scattering amplitude in the vicinity of the Fermi surface. This is a new straightforward approach to the many-body problem which is applicable also to condensed matter systems without long-ranged interactions, such as liquid 3He. We derive the one-loop renormalization group equations for the quasiparticle interaction and scattering amplitude at zero temperature. The RG presents an elegant method to maintain all momentum scales and preserve the antisymmetry of the scattering amplitude. As a first application we solve the RG equations for neutron matter. The resulting quasiparticle interaction includes effects due to the polarization of the medium, the so called induced interaction of Babu and Brown. We present results for the Fermi liquid parameters, the equation of state of neutron matter and the 1S0 superfluid pairing gap.
In the present thesis superfluid many-fermion systems are investigated in the framework of the Renormalization Group (RG). Starting from an experimentally determined two-body interaction this scheme provides a microscopic approach to strongly correlated many-body systems at low temperatures. The fundamental objects under investigation are the two-point and the four-point vertex functions. We show that explicit results for simple separable interactions on BCS-level can be reproduced in the RG framework to high accuracy. Furthermore the RG approach can immediately be applied to general realistic interaction models. In particular, we show how the complexity of the many-body problem can be reduced systematically by combining different RG schemes. Apart from technical convenience the RG framework has conceptual advantage that correlations beyond the BCS level can be incorporated in the flow equations in a systematic way. In this case however the flow equations are no more explicit equations like at BCS level but instead a coupled set of implicit equations. We show on the basis of explicit calculations for the single-channel case the efficacy of an iterative approach to this system. The generalization of this strategy provides a promising strategy for a non-perturbative treatment of the coupled channel problem. By the coupling of the flow equations of the two-point and four-point vertex self-consistency on the one-body level is guaranteed at every cutoff scale.
The renormalization-group (RG) method is applied to study interacting fermions at finite temperature. A model based on the [psi][superscript]4-Grassmann effective action with SU(N )-invariant short-range interaction and a rotationally invariant Fermi surface in spatial dimensions d = 2, 3 is studied. We show how the key results of the Landau Fermi liquid theory can be recovered by this finite-temperature RG technique. Applying the RG to response functions, we find the compressibility and the spin susceptibility as solutions of the RG flow equations. We discuss subtleties associated with the symmetry properties of the four-point vertex (the implications of the Pauli principle). We point out distinctions between three quantities: the bare interaction of the low-energy effective action, the Landau function and the forward scattering vertex.The bare interaction of the effective action is not a RG fixed point, but a common starting point of the flow trajectories of two limiting forms of the four-point vertex. We have derived RG equations for the Landau channel that take into account both contributions of the direct (ZS) and the exchange (ZS' ) particle-hole graphs at one-loop level.The basic quantities of Fermi Liquid theory, the Landau interaction function and the forward scattering vertex, are calculated as fixed points of these flows in terms of the effective action's interaction function. The classic derivations of Fermi Liquid theory applying the Bethe-Salpeter equation and other analogous approaches, tantamount to some sort of RPA-type (decoupled) approximation, neglect the zero-angle singularity in the ZS' graph. As a consequence, the antisymmetry of the forward scattering vertex is not guaranteed in the final result, and the RPA sum rule must be imposed by hand on the components of the Landau function to satisfy the Pauli principle. This sum rule, not indispensable in the original phenomenological formulation of the Landau FLT, is equivalent, from the RG point of view, to a fine tuning of the effective interaction. Our results show that the strong interference of the direct and exchange processes of the particle-hole scattering near zero angle invalidates the RPA (decoupled) approximation in this region, resulting in temperature-dependent narrow-angle anomalies in the Landau function and scattering vertex, revealed by the RG analysis. In the present RG approach the Pauli principle is automatically satisfied. As follows from the RG solution, the amplitude sum rule, being an artefact of the RPA approximation, is not needed to respect statistics and, moreover, is not valid.
The renormalization-group (RG) approach proposed earlier by Shankar for interacting spinless fermions at T=0 is extended to the case of nonzero temperature and spin. We study a model with SU(N)-invariant short-range effective interaction and rotationally invariant Fermi surface in two and three dimensions. We show that the Landau interaction function of the Fermi liquid, constructed from the bare parameters of the low-energy effective action, is RG invariant. On the other hand, the physical forward-scattering vertex is found as a stable fixed point of the RG flow. We demonstrate that in d=2 and 3, the RG approach to this model is equivalent to Landau's mean-field treatment of the Fermi liquid. We discuss subtleties associated with the symmetry properties of the scattering amplitude, the Landau function, and the low-energy effective action. Applying the RG to response functions, we find the compressibility and the spin susceptibility as fixed points.
We apply the finite-temperature renormalization group (RG) to a model based on an effective action with a short-range repulsive interaction and a rotation-invariant Fermi surface. The basic quantities of Fermi-liquid theory, the Landau function, and the scattering vertex are calculated as fixed points of the RG flow in terms of the effective action's interaction function. The classic derivations of Fermi-liquid theory, which apply the Bethe-Salpeter equation and amount to summing direct particle-hole ladder diagrams, neglect the zero-angle singularity in the exchange particle-hole loop. As a consequence, the antisymmetry of the forward scattering vertex is not guaranteed and the amplitude sum rule must be imposed by hand on the components of the Landau function. We show that the strong interference of the direct and exchange processes of particle-hole scattering near zero angle invalidates the ladder approximation in this region, resulting in temperature-dependent narrow-angle anomalies in the Landau function and scattering vertex. In this RG approach the Pauli principle is automatically satisfied. The consequences of the RG corrections on Fermi-liquid theory are discussed. In particular, we show that the amplitude sum rule is not valid.
We present two novel relations between the quasiparticle interaction in nuclear matter and the unique low momentum nucleon-nucleon interaction in vacuum. These relations provide two independent constraints on the Fermi liquid parameters of nuclear matter. Moreover, the new constraints define two combinations of Fermi liquid parameters, which are invariant under the renormalisation group flow in the particle-hole channels. Using empirical values for the spin-independent Fermi liquid parameters, we are able to compute the major spin-dependent ones by imposing the new constraints as well as the Pauli principle sum rules.