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 investigate the response of an electron system which exhibits ideal nesting features. Using the standard Matsubara formalism we derive analytic expressions for the imaginary and real parts of the bare particle–hole susceptibility. The imaginary part has sharp peaks whose maxima at the nesting momenta approximately scale with (ω/T). The peak lineshapes resemble neutron scattering data on chromium and some copper oxide superconductors. The real part of the bare susceptibility at the nesting vectors diverges logarithmically at low temperatures. Analytic formulas for the first vertex correction to the susceptibility are derived for a Hubbard interaction and its momentum and temperature variations are calculated numerically. This term detracts substantially from the ordinary RPA terms for intermediate values of the Coulomb repulsion. Exact cancellation of a certain class of diagrams at half filling is shown to result from particle–hole symmetry. We discuss the consequences of these results for spin fluctuation theories of high temperature superconductors and spin density wave instabilities.
We present the expression for the quasiparticle vertex function ${\ensuremath{\Gamma}}^{\ensuremath{\omega}}({K}_{F},{P}_{F})$ (proportional to the Landau interaction function) in a 2D Fermi liquid (FL) near an instability towards antiferromagnetism. This function is relevant in many ways in the context of metallic quantum criticality. Previous studies have found that near a quantum critical point, the system enters into a regime in which the fermionic self-energy is large near hot spots on the Fermi surface [points on the Fermi surface connected by the antiferromagnetic ordering vector ${q}_{\ensuremath{\pi}}=(\ensuremath{\pi},\ensuremath{\pi})$] and has much stronger dependence on frequency than on momentum. We show that in this regime, which we termed a critical FL, the conventional random-phase-approximation- (RPA) type approach breaks down, and to properly calculate the vertex function one has to sum up an infinite series of terms which were explicitly excluded in the conventional treatment. Besides, we show that, to properly describe the spin component of ${\ensuremath{\Gamma}}^{\ensuremath{\omega}}({K}_{F},{P}_{F})$ even in an ordinary FL, one has to add Aslamazov-Larkin (AL) terms to the RPA vertex. We show that the total ${\ensuremath{\Gamma}}^{\ensuremath{\omega}}({K}_{F},{P}_{F})$ is larger in a critical FL than in an ordinary FL, roughly by an extra power of magnetic correlation length $\ensuremath{\xi}$, which diverges at the quantum critical point. However, the enhancement of ${\ensuremath{\Gamma}}^{\ensuremath{\omega}}({K}_{F},{P}_{F})$ is highly nonuniform: It holds only when, for one of the two momentum variables, the distance from a hot spot along the Fermi surface is much larger than for the other one. This fact renders our case different from quantum criticality at small momentum, where the enhancement of ${\ensuremath{\Gamma}}^{\ensuremath{\omega}}({K}_{F},{P}_{F})$ was found to be homogeneous. We show that the charge and spin components of the total vertex function satisfy the universal relations following from the Ward identities related to the conservation of the particle number and the total spin. We show that in a critical FL, the Ward identity involves ${\ensuremath{\Gamma}}^{\ensuremath{\omega}}({K}_{F},{P}_{F})$ taken between particles on the FS. We find that the charge and spin components of ${\ensuremath{\Gamma}}^{\ensuremath{\omega}}({K}_{F},{P}_{F})$ are identical to leading order in the magnetic correlation length. We use our results for ${\ensuremath{\Gamma}}^{\ensuremath{\omega}}({K}_{F},{P}_{F})$ and for the quasiparticle residue to derive the Landau parameters
Fcl=0=Fsl=0, the density of states, and the uniform (
q=0) charge and spin susceptibilities ${\ensuremath{\chi}}_{c}^{l=0}={\ensuremath{\chi}}_{s}^{l=0}$. We show that the density of states
NF diverges as $log\ensuremath{\xi}$; however,
Fc,sl=0 also diverge as $log\ensuremath{\xi}$, such that the total ${\ensuremath{\chi}}_{c,s}^{(l=0)}\ensuremath{\propto}{N}_{F}/(1+{F}_{c}^{l=0})$ remain finite at $\ensuremath{\xi}=\ensuremath{\infty}$. We show that at weak coupling these susceptibilities are parametrically smaller than for free fermions.
The pairing instability induced by either a given attractive interaction or particle-hole excitations is examined by using the perturbational diagram approach. A graphical functional derivative method based on Ward's identity is developed to represent the contributions to the interparticle interactions from different channels. It is shown that pairing instability for the electrons excited from the Fermi sea may happen due to the presence of holes even when the original interaction is repulsive. Thus, in contrast to the Cooper instability caused by a given attractive interaction, the pairing instability can be induced by the particle-hole excitations. The scattering diagrams from this model allows one to evaluate the two particle vertex function and Tc by using the pole condition of the vertex function. It has been found that Tc can reach quite high values and gives rise to the bell shape. The application of the results to a layered 2D Fermi gas is considered.
The application of Brueckner's perturbation approach to interacting Fermion systems is prevented by the poles of the t-matrix which can be present for finite as well as infinite systems if the interaction is attractive. It is discussed that the singularity is removed if we take account of the contribution to the energy from diagrams involving two hole labels a number of times. These diagrams are not necessarily of ladder type and correspond to the possibility that a scattered pair of particles can go to the levels which they have left in the unperturbed state. Though the removal of poles makes one entertain a hope for the perturbation approach to finite systems, it is discussed that the same is not the case with infinite systems. It is pointed out that the difficulty in dealing with infinite systems is not caused by the pairs corresponding to the poles as is usually considered to be, but is due to pairs lying more closely to the Fermi surface.