We prove regularity properties of the self-energy, to all orders in perturbation theory, for systems with singular Fermi surfaces which contain Van Hove points where the gradient of the dispersion relation vanishes. In this paper, we show for spatial dimensions d ≥ 3 that despite the Van Hove singularity, the overlapping loop bounds we proved together with E. Trubowitz for regular non-nested Fermi surfaces [J. Stat. Phys.84 (1996) 1209–1336] still hold, provided that the Fermi surface satisfies a no-nesting condition. This implies that for a fixed interacting Fermi surface, the self-energy is a continuously differentiable function of frequency and momentum, so that the quasiparticle weight and the Fermi velocity remain close to their values in the noninteracting system to all orders in perturbation theory. In a companion paper, we treat the more singular two-dimensional case.
I established the criteria and obtained analytical results for the pseudogap at the Van Hove (antinodal) point on the Fermi surface in two dimensions. The original criterion $\ensuremath{\xi}\ensuremath{\gg}{\ensuremath{\xi}}_{th_db}={v}_{F}/\ensuremath{\pi}T$ is not applicable in this case since Fermi velocity
vF=0. It turns out that the characteristic length for the pseudogap crossover at the Van Hove point ${\ensuremath{\xi}}_{th_vh}\ensuremath{\propto}1/{T}^{1/2}$ is significantly shorter than the one at the regular Fermi surface points ${\ensuremath{\xi}}_{th_db}\ensuremath{\propto}1/T$. In particular, ${\ensuremath{\xi}}_{th_vh}$ is between one and two lattice spacings in the intermediate interaction regime of the Hubbard model. I have also identified the regime where there is still a single maximum in the spectral function, but single particle properties are abnormal. Specifically, the imaginary part of the self-energy has a minimum instead of maximum at the Fermi level and the slope of the real part of the self-energy is positive instead of negative. The important advantages of an analytical approach is that it provides results in both the Matsubara frequency representation and in the real frequencies representation. I compare Matsubara frequency results with the exact numerical results of the Monte Carlo methods in the Hubbard model and then show what they correspond to in the real frequency self-energy and spectral function.
We analyze in detail the fermionic self-energy $\ensuremath{\Sigma}(\ensuremath{\omega},T)$ in a Fermi liquid (FL) at finite temperature
T and frequency $\ensuremath{\omega}$. We consider both canonical FLs, i.e., systems in spatial dimension
D>2, where the leading term in the fermionic self-energy is analytic [the retarded $\mathrm{Im}{\ensuremath{\Sigma}}^{R}(\ensuremath{\omega},T)=C({\ensuremath{\omega}}^{2}+{\ensuremath{\pi}}^{2}{T}^{2})]$, and noncanonical FLs in
1<D<2, where the leading term in $\mathrm{Im}{\ensuremath{\Sigma}}^{R}(\ensuremath{\omega},T)$ scales as
TD or ${\ensuremath{\omega}}^{D}$. We relate the ${\ensuremath{\omega}}^{2}+{\ensuremath{\pi}}^{2}{T}^{2}$ form to a special property of the self-energy, ``the first-Matsubara-frequency rule,'' which stipulates that ${\ensuremath{\Sigma}}^{R}(i\ensuremath{\pi}T,T)$ in a canonical FL contains an
O(T) but no
T2 term. We show that in any
D>1, the next term after
O(T) in ${\ensuremath{\Sigma}}^{R}(i\ensuremath{\pi}T,T)$ is of order
TD (
T3lnT in
D=3). This
TD term comes from only forward and backward scattering, and is expressed in terms of fully renormalized amplitudes for these processes. The overall prefactor of the
TD term vanishes in the ``local approximation,'' when the interaction can be approximated by its value for the initial and final fermionic states right on the Fermi surface. The local approximation is justified near a Pomeranchuk instability, even if the vertex corrections are non-negligible. We show that the strength of the first-Matsubara-frequency rule is amplified in the local approximation, where it states that not only the
TD term vanishes, but also that ${\ensuremath{\Sigma}}^{R}(i\ensuremath{\pi}T,T)$ does not contain any terms beyond
O(T). This rule imposes two constraints on the scaling form of the self-energy: upon replacing $\ensuremath{\omega}$ by $i\ensuremath{\pi}T$, $\mathrm{Im}{\ensuremath{\Sigma}}^{R}(\ensuremath{\omega},T)$ must vanish and $\mathrm{Re}{\ensuremath{\Sigma}}^{R}(\ensuremath{\omega},T)$ must reduce to
O(T). These two constraints should be taken into consideration in extracting scaling forms of ${\ensuremath{\Sigma}}^{R}(\ensuremath{\omega},T)$ from experimental and numerical data.
The use of effective field theories to attack new and seemingly disparate problems has proliferated in the past several decades. In this thesis, we develop effective field theories for systems of fermionic quasiparticles possessing Fermi surfaces, with a particular focus on Fermi surfaces proximal to Van Hove singularities. Such systems are a fruitful source of complex and novel behavior in condensed matter physics. We begin with an overview of the renormalization group procedure at the heart of effective field theory by analyzing a simple example. We emphasize the concept that the RG relates the observables of one theory to those of another theory with precisely the same form but different numerical parameters. We also note the generality and extensibility of these concepts. We then apply this perspective to the study of quasiparticles with a round Fermi surface, employing the technique of binning the quasiparticle fields in momentum space to translate previous treatments into a more modern form. We next develop an effective field theory describing the excitations of modes around a Fermi surface with a Van Hove singularity. We resolve lingering questions about the presence of nonlocal interactions in similar models. We find a rich and complicated theory capable of describing deviations from typical Fermi liquid behavior that nonetheless displays some universal dependence on the interactions involving modes in the vicinity of the Van Hove point. We close with an analysis of the instabilities of this Van Hove effective field theory.
Divergencies appearing in perturbation expansions of interacting many-body systems can often be removed by expanding around a suitably chosen renormalized (instead of the noninteracting) 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 solutions 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 a broken square lattice symmetry, whose topology can be closed or open. For the attractive Hubbard model the second-order calculation yields an s-wave superconductivity with a weakly momentum dependent gap, whose size is reduced compared to the mean-field result.
We consider a system of interacting fermions in two dimensions beyond the second-order perturbation theory in the interaction. It is shown that the mass-shell singularities in the self-energy, arising already at the second order of the perturbation theory, manifest a nonperturbative effect: an interaction with the zero-sound mode. Resumming the perturbation theory for a weak, short-range interaction and accounting for a finite curvature of the fermion spectrum, we eliminate the singularities and obtain the results for the quasiparticle self-energy and the spectral function to all orders in the interaction with the zero-sound mode. A threshold for emission of zero-sound waves leads a nonmonotonic variation of the self-energy with energy (or momentum) near the mass shell. Consequently, the spectral function has a kinklike feature. We also study in detail a nonanalytic temperature dependence of the specific heat $C(T)\ensuremath{\propto}{T}^{2}$. It turns out that although the interaction with the collective mode results in an enhancement of the fermion self-energy, this interaction does not affect the nonanalytic term in
C(T) due to a subtle cancellation between the contributions from the real and imaginary parts of the self-energy. For a short-range and weak interaction, this implies that the second-order perturbation theory suffices to determine the nonanalytic part of
C(T). We also obtain a general form of the nonanalytic term in
C(T), valid for the case of a generic Fermi liquid, i.e., beyond the perturbation theory.