The Luttinger sum rule is usually considered for Landau Fermi liquids in which the single particle Green’s function G(k, 0) changes sign at the the Fermi surface by passing through infinity. However the general proof allows also for a sign change at which G has a zero. A recent analysis by Konik and coworkers considers a model of 2‐leg Hubbard ladders weakly coupled by a small long range interladder tunneling. At half‐filling a semimetallic state with small Fermi pockets is induced beyond a threshold tunneling strength. The sign changes in G(k, 0) relevent for the Luttinger sum rule now take place at surfaces with both zeros and infinities. The zero surfaces differ from the minimum gap surfaces. The latter are often used in ARPES experiments on underdoped cuprates to obtain an underlying Fermi surface but this procedure leads to problems with the Luttinger sum rule. Some speculations on how the Luttinger sum rule should be applied to the pseudogap phase of the underdoped cuprates are included.
We study the effect of doping away from half-filling in weakly (but finitely) interacting N-leg Hubbard ladders using renormalization-group and bosonization techniques. For a small on-site repulsion U, the N-leg Hubbard ladders are equivalent to an N-band model, where at half-filling the Fermi velocities are ${v}_{1}{=v}_{N}<{v}_{2}{=v}_{N\ensuremath{-}1}<\ensuremath{\cdot}\ensuremath{\cdot}\ensuremath{\cdot}.$ We then obtain a hierarchy of energy scales, where the band pairs $(j,N+1\ensuremath{-}j)$ are successively frozen out. The low-energy Hamiltonian is then the sum of
N/2 [or $(N\ensuremath{-}1)/2$ for N odd] two-leg ladder Hamiltonians without gapless excitations (plus a single chain for N odd with one gapless spin mode)---similar to the N-leg Heisenberg spin ladders. The energy scales lead to a hierarchy of gaps. Upon doping away from half-filling, the holes first enter the band(s) with the smallest gap: For odd N, the holes enter first the nonbonding band
(N+1)/2 and the phase is a Luttinger liquid, while for even N, the holes enter first the band pair
(N/2,N/2+1) and the phase is a Luther-Emery liquid, similar to numerical treatments of the $t\ensuremath{-}J$ model, i.e., at and close to half-filling, the phases of the Hubbard ladders for small and large U are the same. For increasing doping, hole pairs subsequently enter at critical dopings the other band pairs $(j,N+1\ensuremath{-}j)$ (accompanied by a diverging compressibility): The Fermi surface is successively opened by doping, starting near the wave vector $(\ensuremath{\pi}/2,\ensuremath{\pi}/2).$ Explicit calculations are given for the cases
N=3 and 4.
One of the fundamental questions about high-temperature cuprate superconductors is the size of the Fermi surface underlying the superconducting state. By analyzing the single-particle spectral function for the Fermi-Hubbard model as a function of repulsion
U and chemical potential $\ensuremath{\mu}$, we find that the Fermi surface in the normal state undergoes a transition from a large Fermi surface matching the Luttinger volume as expected in a Fermi liquid, to a Fermi surface that encloses fewer electrons that we dub the ``Luttinger breaking'' phase, as the Mott insulator is approached. This transition into a non-Fermi-liquid phase that violates the Luttinger count occurs at a critical density in the absence of any other broken symmetry. We obtain the Fermi-surface contour from the spectral weight ${A}_{\mathbf{k}}(\ensuremath{\omega}=0)$ and from an analysis of the singularities of the Green's function
ReGk(E=0), calculated using determinantal quantum Monte Carlo and analytic continuation methods. We discuss our numerical results in connection with experiments on Hall measurements, scanning tunneling spectroscopy, and angle-resolved photoemission spectroscopy.
We use higher-dimensional bosonization to study the normal state of electrons in weakly coupled metallic chains interacting with long-range Coulomb forces. Particular attention is paid to the crossover between Luttinger and Fermi-liquid behavior as the interchain hopping ${t}_{\ensuremath{\perp}}$ is varied. Although in the physically interesting case of finite but small ${t}_{\ensuremath{\perp}}$ the quasiparticle residue does not vanish, the single-particle Green's function exhibits the signature of Luttinger-liquid behavior (i.e., anomalous scaling and spin-charge separation) in a large intermediate parameter regime. Using realistic parameters, we find that the scaling behavior in this regime is characterized by an anomalous dimension of the order of unity, as suggested by recent experiments on quasi-one-dimensional conductors. Our calculation also gives insights into the approximations inherent in higher-dimensional bosonization; in particular, we show that the replacement of a curved Fermi surface by a finite number
M of flat patches can give rise to unphysical nesting singularities in the single-particle Green's function, which disappear only in the limit $M\ensuremath{\rightarrow}\ensuremath{\infty}$ or if curvature effects are included. We also compare our approach with other methods. This work extends our recent letter [Phys. Rev. Lett. 74, 2997 (1995)].
The two-dimensional Hubbard model with a single spin-up electron interacting with a finite density of spin-down electrons is studied by use of the quantum Monte Carlo technique, the standard second-order perturbation theory and a very good variational wave function introduced recently by Edwards. We also present a new conjugate gradient scheme for the evaluation of this variational state. We performed simulations up to 242 sites at U/t=4 reaching the zero-temperature properties with no ``fermion-sign problem'' and found a surprisingly good accuracy of the Edwards state at low density or low doping. The conjugate gradient method was then applied to the system up to 3698 sites and infinite U. Fermi-liquid theory seems to remain stable in two dimensions for all cases studied with the exception of the half-filling case where a Luttinger-like behavior survives in the Hubbard model, yielding a vanishing quasiparticle weight in the thermodynamic limit.