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arXiv:cond-mat/0609280 · Physical Review B

Luttinger sum rule for finite systems of correlated electrons

J. Kokalj, Peter Prelovšek

Published Jan 10, 2007• 20 citations
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Abstract

The validity of the Luttinger sum rule is considered for finite systems of interacting electrons, where the Fermi volume is determined by location of zeros of Green's function. It is shown that the sum rule in the paramagnetic state is evidently violated within the planar $t\text{\ensuremath{-}}J$ model at low doping while for the related Hubbard model, even in the presence of next-nearest-neighbor hopping, no clear-cut exception is found.

DOI
10.1103/physrevb.75.045111

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The validity of the Luttinger sum rule is considered for finite systems of interacting electrons, where the Fermi volume is determined by location of zeros of Green's function. It is shown that the sum rule in the paramagnetic state is evidently violated within the planar t\ensuremath-Jt\text{\ensuremath{-}}Jt\ensuremath-J model at low doping while for the related Hubbard model, even in the presence of next-nearest-neighbor hopping, no clear-cut exception is found.
The validity of the Luttinger sum rule is investigated within the prototype tight-binding model of interacting fermions in one dimension, i.e., the $t\text{\ensuremath{-}}V$ model including the next-nearest-neighbor hopping ${t}^{\ensuremath{'}}$, in order to break the particle-hole symmetry. Scaling analysis of finite-system results at half filling reveals evident breakdown of the sum rule in the large gap regime at V⪢tV⪢tV⪢t while the sum rule appears to recover together with vanishing of the Mott-Hubbard gap.
We show exactly with an SU(N) interacting model that even if the ambiguity associated with the placement of the chemical potential, μ, for a T=0 gapped system is removed by using the unique value μ(T→0), Luttinger's sum rule is violated even if the ground-state degeneracy is lifted by an infinitesimal hopping. The failure stems from the nonexistence of the Luttinger-Ward functional for a system in which the self-energy diverges. Since it is the existence of the Luttinger-Ward functional that is the basis for Luttinger's theorem which relates the charge density to sign changes of the single-particle Green function, no such theorem exists. Experimental data on the cuprates are presented which show a systematic deviation from the Luttinger count, implying a breakdown of the electron quasiparticle picture in strongly correlated electron matter.
We analyze a model of two-leg Hubbard ladders weakly coupled by interladder tunneling. At half filling a semimetallic state with small Fermi pockets is induced beyond a threshold tunneling strength. The sign changes in the single electron Green's function relevant for the Luttinger sum rule now take place at surfaces with both zeros and infinities with important consequences for the interpretation of angle-resolved photoemission spectroscopy experiments. Residual interactions between electron and holelike quasiparticles cause a transition to long range order at low temperatures. The theory can be extended to small doping leading to superconducting order.
By on the analyzing the general structure of the Green function of a strongly correlated electron system, it is shown that, for the regime of strong correlations, Luttinger’s theorem should be generalized in the following way: the volume of the Fermi surface of the system of noninteracting particles is equal to that of the quasiparticles in the strongly correlated system with due regard for the spectral weight of the quasiparticles. An investigation of the t-J model and of the Hubbard model, as applied to the paramagnetic nonsuperconducting phase, shows that the generalized Luttinger theorem is valid for these models.
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 obtain a controlled description of a strongly correlated regime of electronic behavior. We begin by arguing that there are two ways to characterize the electronic degree of freedom, either by the canonical fermion algebra or the graded Lie algebra su(2∣2)\mathfrak{su}(2|2)su(2∣2). The first underlies the Fermi liquid description of correlated matter, and we identify a regime governed by the latter. We exploit an exceptional central extension of su(2∣2)\mathfrak{su}(2|2)su(2∣2) to employ a perturbative scheme recently developed by Shastry and obtain a series of successive approximations for the electronic Green's function. We then focus on the leading approximation, which reveals a splitting in two of the electronic dispersion. The Luttinger sum rule is violated, and a Mott metal-insulator transition is exhibited. We offer a perspective.