A renormalization procedure for the correlation function theory of finite Fermi systems

Avishay Efrat, Moshe Schwartz

We describe how to evaluate approximately various physical interesting quantities in random Ising systems by direct renormalization of a finite system. The renormalization procedure is used to reduce the number of degrees of freedom to a number that is small enough, enabling direct summing over the surviving spins. This procedure can be used to obtain averages of functions of the surviving spins. We show how to evaluate averages that involve spins that do not survive the renormalization procedure. We show, for the random field Ising model, how to obtain $\overline{\ensuremath{\Gamma}(\stackrel{\ensuremath{\rightarrow}}{r})}=\overline{〈\ensuremath{\sigma}(0)\ensuremath{\sigma}(\stackrel{\ensuremath{\rightarrow}}{r})〉\ensuremath{-}〈\ensuremath{\sigma}(0)〉〈\ensuremath{\sigma}(\stackrel{\ensuremath{\rightarrow}}{r})〉},$ the ``connected'' correlation function, and $\overline{S(\stackrel{\ensuremath{\rightarrow}}{r})}=\overline{〈\ensuremath{\sigma}(0)\ensuremath{\sigma}(\stackrel{\ensuremath{\rightarrow}}{r})〉},$ the ``disconnected'' correlation function. Consequently, we show how to obtain the average susceptibility and the average energy. For an Ising system with random bonds and random fields, we show how to obtain the average specific heat. We conclude by presenting our numerical results for the average susceptibility and the function $\overline{\ensuremath{\Gamma}(r)}$ along one of the principal axes. (In this work, the full three-dimensional (3D) correlation is calculated and not just parameters such $\ensuremath{\nu}$ or $\ensuremath{\eta}).$ The results for the average susceptibility are used to extract the critical temperature and critical exponents of the 3D random field Ising system.

Walter Metzner

Emmanuel Pereira, Aldo Procacci

A. Liam Fitzpatrick, Gonzalo Torroba, Huajia Wang

We study the renormalization of the Fermi surface coupled to a massless boson near three spatial dimensions. For this, we set up a Wilsonian RG with independent decimation procedures for bosons and fermions, where the four-fermion interaction ``Landau parameters'' run already at tree level. Our explicit one-loop analysis resolves previously found obstacles in the renormalization of finite-density field theory, including logarithmic divergences in nonlocal interactions and the appearance of multilogarithms. The key aspects of the RG are the above tree-level running, and a UV-IR mixing between virtual bosons and fermions at the quantum level, which is responsible for the renormalization of the Fermi velocity. We apply this approach to the renormalization of $2{k}_{F}$ singularities, and to Fermi surface instabilities in a companion paper, showing how multilogarithms are properly renormalized. We end with some comments on the renormalization of finite-density field theory with the inclusion of Landau damping of the boson.

R. M. Noack, Steven R. White, Douglas J. Scalapino

Klaus Morawetz, Vinod Ashokan, Kare Narain Pathak +2 more

Expanding the two-particle Green's functions determines the self-energy and the polarization as well as the response function on the same footing. The correlation energy is calculated with the help of the extended quasiparticle picture, which accounts for off-shell effects. The corresponding response function leads to the same correlation energy as the self-energy in agreement with perturbation theory, provided one works in the extended quasiparticle picture. A one-dimensional quantum wire of fermions is considered and ground-state properties are calculated in the high-density regime within the extended quasiparticle picture and Born approximation. While the on-shell selfenergies are strictly zero due to Pauli-blocking of elastic scattering, the off-shell behavior shows a rich structure of a gap in the damping of excitation, which is closed when the momentum approaches the Fermi one. The consistent spectral function is presented, completing the first two energy-weighted sum rules. The excitation spectrum shows a splitting due to holons and antiholons as non-Fermi liquid behavior. A renormalization procedure is proposed by subtracting an energy constant to render the Fock exchange energy finite. The effective mass derived from meanfield approximation shows a dip analogous to the onset of Peierls instability. The reduced density matrix or momentum distribution is calculated with the help of a Pad\'e regularization repairing deficiencies of the perturbation theory. A seemingly finite step at the Fermi energy indicating Fermi-liquid behavior is repaired in this way.