5 research outputs found
Reduction formula for fermion loops and density correlations of the 1D Fermi gas
Fermion N-loops with an arbitrary number of density vertices N > d+1 in d
spatial dimensions can be expressed as a linear combination of (d+1)-loops with
coefficients that are rational functions of external momentum and energy
variables. A theorem on symmetrized products then implies that divergencies of
single loops for low energy and small momenta cancel each other when loops with
permuted external variables are summed. We apply these results to the
one-dimensional Fermi gas, where an explicit formula for arbitrary N-loops can
be derived. The symmetrized N-loop, which describes the dynamical N-point
density correlations of the 1D Fermi gas, does not diverge for low energies and
small momenta. We derive the precise scaling behavior of the symmetrized N-loop
in various important infrared limits.Comment: 14 pages, to be published in Journal of Statistical Physic
Fermion loops, loop cancellation and density correlations in two dimensional Fermi systems
We derive explicit results for fermion loops with an arbitrary number of
density vertices in two dimensions at zero temperature. The 3-loop is an
elementary function of the three external momenta and frequencies, and the
N-loop can be expressed as a linear combination of 3-loops with coefficients
that are rational functions of momenta and frequencies. We show that the
divergencies of single loops for low energy and small momenta cancel each other
when loops with permuted external variables are summed. The symmetrized N-loop,
i.e. the connected N-point density correlation function of the Fermi gas, does
not diverge for low energies and small momenta. In the dynamical limit, where
momenta scale to zero at fixed finite energy variables, the symmetrized N-loop
vanishes as the (2N-2)-th power of the scale parameter.Comment: 24 pages (including 3 EPS figures), LaTeX2e; submitted to Phys. Rev.
Renormalized perturbation theory for Fermi systems: Fermi surface deformation and superconductivity in the two-dimensional Hubbard model
Divergencies appearing in perturbation expansions of interacting many-body
systems can often be removed by expanding around a suitably chosen renormalized
(instead of the non-interacting) Hamiltonian. We describe such a renormalized
perturbation expansion for interacting Fermi systems, which treats Fermi
surface shifts and superconductivity with an arbitrary gap function via
additive counterterms. The expansion is formulated explicitly for the Hubbard
model to second order in the interaction. Numerical soutions of the
self-consistency condition determining the Fermi surface and the gap function
are calculated for the two-dimensional case. For the repulsive Hubbard model
close to half-filling we find a superconducting state with d-wave symmetry, as
expected. For Fermi levels close to the van Hove singularity a Pomeranchuk
instability leads to Fermi surfaces with broken square lattice symmetry, whose
topology can be closed or open. For the attractive Hubbard model the second
order calculation yeilds s-wave superconductivity with a weakly momentum
dependent gap, whose size is reduced compared to the mean-field result.Comment: 18 pages incl. 6 figure
Instabilitäten und spontane Symmetriebrechung in zweidimensionalen Fermisystemen
Since the discovery of the superconducting cuprates and their hitherto unknown electronic properties there has been a continuously growing interest in a theoretical treatment of suitable models of strongly correlated fermions which at least qualitatively reflect the essential aspects of these physical systems. The Hubbard model in two dimensions plays an important role in this context. The present thesis examines instabilities and possible symmetry broken phases of the two-dimensional repulsive Hubbard model with the help of numeric and semi analytic methods. While in the first part it is pointed out how such instabilities can be treated by a mean field theory, the second part of the work supplies the theoretical bases of the numeric procedures developed in part III. For the case of a pure nearest neighbour hopping it is shown that the second order contributions of perturbation theory generate an effective attraction between the electrons. The energy scale of the superconducting instabilities is also examined; it is shown that the vanHove singularity drastically enhances this critical scale at half filling. The investigation of a possible coexistence of two different kinds of spontaneous symmetry breaking forms the emphasis of the thesis: Lately accomplished renormalization group calculations showed (for suitably chosen parameters) the occurrence of a Pomeranchuk instability of the two-dimensional Hubbard model with nonvanishing next nearest neighbour hopping, which should give rise to a ground state that spontaneously breaks the model's lattice symmetry as well as its U(1) symmetry, i.e. the ground state could be characterized by two co-existing order parameters. For treatment of the problem of co-existing instabilities the ansatz of a renormalized perturbation theory is used, which had been suggested in order to avoid the occurrence of unphysical singularities arising within the framework of naive perturbation theory; these singularities are related to interaction-induced shifts of the Fermi surface, which cannot be captured by simply shifting the chemical potential. The attractive Hubbard model is also treated within the framework of renormalized perturbation theory; the well-known reduction of the superconducting order parameter with respect to its BCS value reported in earlier investigations can be confirmed. A spontaneously broken lattice symmetrie does not arise here, in contrast to the repulsive case