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arXiv:1004.1322 · Physical Review B

Spectral weight of doping-induced states in the two-dimensional Hubbard model

A. Liebsch

Published Jun 29, 2010• 21 citations
arXivHTMLPDF

Abstract

The spectral weight of states induced in the Mott gap via hole doping in the two-dimensional Hubbard model is studied within cluster dynamical mean-field theory combined with finite-temperature exact diagonalization. If the cutoff energy is chosen to lie just below the upper Hubbard band, the integrated weight per spin is shown to satisfy ${W}_{+}(\ensuremath{\delta})\ensuremath{\ge}\ensuremath{\delta}$ ($\ensuremath{\delta}$ denotes the total number of holes), in agreement with model predictions by Eskes et al. [Phys. Rev. Lett. 67, 1035 (1991)]. However, if the cutoff energy is chosen to lie in the range of the pseudogap, ${W}_{+}(\ensuremath{\delta})$ remains much smaller than $\ensuremath{\delta}$ and approximately saturates near $\ensuremath{\delta}\ensuremath{\approx}0.2,\dots{},0.3$. The analysis of recent x-ray absorption spectroscopy data therefore depends crucially on the appropriate definition of the integration window.

DOI
10.1103/physrevb.81.235133

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The spectral weight of states induced in the Mott gap via hole doping in the two-dimensional Hubbard model is studied within cluster dynamical mean-field theory combined with finite-temperature exact diagonalization. If the cutoff energy is chosen to lie just below the upper Hubbard band, the integrated weight per spin is shown to satisfy W+(\ensuremathδ)\ensuremath≥\ensuremathδ{W}_{+}(\ensuremath{\delta})\ensuremath{\ge}\ensuremath{\delta}W+​(\ensuremathδ)\ensuremath≥\ensuremathδ (\ensuremathδ\ensuremath{\delta}\ensuremathδ denotes the total number of holes), in agreement with model predictions by Eskes et al. [Phys. Rev. Lett. 67, 1035 (1991)]. However, if the cutoff energy is chosen to lie in the range of the pseudogap, W+(\ensuremathδ){W}_{+}(\ensuremath{\delta})W+​(\ensuremathδ) remains much smaller than \ensuremathδ\ensuremath{\delta}\ensuremathδ and approximately saturates near \ensuremathδ\ensuremath≈0.2,…,0.3\ensuremath{\delta}\ensuremath{\approx}0.2,\dots{},0.3\ensuremathδ\ensuremath≈0.2,…,0.3. The analysis of recent x-ray absorption spectroscopy data therefore depends crucially on the appropriate definition of the integration window.
The effect of doping in the two-dimensional Hubbard model is studied within finite-temperature exact diagonalization combined with cluster dynamical mean-field theory. By employing a mixed basis involving cluster sites and bath molecular orbitals for the projection of the lattice Green's function onto $2\ifmmode\times\else\texttimes\fi{}2$ clusters, a considerably more accurate description of the low-frequency properties of the self-energy is achieved than in a pure site picture. To evaluate the phase diagram, the transition from Fermi-liquid to non-Fermi-liquid behavior for decreasing hole doping is studied as a function of Coulomb energy, next-nearest-neighbor hopping, and temperature. The self-energy component ${\ensuremath{\Sigma}}_{X}$ associated with $X=(\ensuremath{\pi},0)$ is shown to develop a collective mode above EF{E}_{F}EF​, whose energy and strength exhibits a distinct dispersion with doping. This low-energy excitation gives rise to non-Fermi-liquid behavior as the hole doping decreases below a critical value ${\ensuremath{\delta}}_{c}$, and to an increasing particle-hole asymmetry, in agreement with recent photoemission data. This behavior is consistent with the removal of spectral weight from electron states above EF{E}_{F}EF​ and the opening of a pseudogap, which increases with decreasing doping. The phase diagram reveals that ${\ensuremath{\delta}}_{c}\ensuremath{\approx}0.15\dots{}0.20$ for various system parameters. For electron doping, the collective mode of ${\ensuremath{\Sigma}}_{X}(\ensuremath{\omega})$ and the concomitant pseudogap are located below the Fermi energy, which is consistent with the removal of spectral weight from the hole states just below EF{E}_{F}EF​. The critical doping, which marks the onset of non-Fermi-liquid behavior, is systematically smaller than for hole doping.
The effect of doping in the two-dimensional Hubbard model is studied within finite temperature exact diagonalization combined with cluster dynamical mean field theory. By employing a mixed basis involving cluster sites and bath molecular orbitals for the projection of the lattice Green's function onto 2x2 clusters, a considerably more accurate description of the low frequency properties of the self-energy is achieved than in a pure site picture. The transition from Fermi-liquid to non-Fermi-liquid behavior for decreasing hole doping is studied as a function of Coulomb energy, next-nearest neighbor hopping, and temperature. In particular, the self-energy component Sigma_X associated with X=(pi,0) is shown to exhibit an onset of non-Fermi-liquid behavior as the hole doping decreases below a critical value delta_c. The imaginary part of Sigma_X(omega) then develops a collective mode above E_F, which exhibits a distinct dispersion with doping. Accordingly, the real part of Sigma_X(omega) has a positive slope above E_F, giving rise to an increasing particle-hole asymmetry as the system approaches the Mott transition. This behavior is consistent with the removal of spectral weight from electron states close to E_F and the opening of a pseudogap which increases with decreasing doping. The phase diagram reveals that delta_c = 0.15 ... 0.20 for various system parameters. For electron doping, the collective mode of Sigma_X(omega) and the concomitant pseudogap are located below the Fermi energy which is consistent the removal of spectral weight from hole states just below E_F. The critical doping which marks the onset of non-Fermi-liquid behavior, is systematically smaller than for hole doping.
The Drude weight of the Hubbard model on the two-dimensional square lattice is studied by the exact diagonalizations applied to clusters up to 20 sites. We carefully examine finite-size effects by consideration of the appropriate shapes of clusters and the appropriate boundary condition beyond the imitation of employing only the simple periodic boundary condition. We successfully capture the behavior of the Drude weight that is proportional to the squared hole doping concentration. Our present result gives a consistent understanding of the transition between the Mott insulator and doped metals. We also find, in the frequency dependence of the optical conductivity, that the mid-gap incoherent part emerges more quickly than the coherent part and rather insensitive to the doping concentration in accordance with the scaling of the Drude weight.
Since the discovery of half-filled cuprate to be a Mott insulator, the excitation spectra above the chemical potential for the unoccupied states has attracted much research attention. There were many theoretical works using different numerical techniques to study this problem, but many have reached different conclusions. One of the reasons is the lack of very detailed high-resolution experimental results for the theories to be compared with. Recently, the scanning tunneling spectroscopy [P. Cai et al., Nat. Phys. 12, 1047 (2016); C. Ye et al., Nat. Commun. 4, 1365 (2013)] on lightly doped Mott insulator with an antiferromagnetic order found the presence of in-gap states with energy of order half an eV above the chemical potential. The measured spectral properties with doping are not quite consistent with earlier theoretical works. Although the experiment has disorder and localization effect, but for the energy scale we will study here, a model without disorder is sufficed to illustrate the underlying physics. We perform a diagonalization method on top of the variational Monte Carlo calculation to study the evolution of antiferromagnetic Mott state with doped hole concentration in the Hubbard model. Our results found in-gap states that behave similarly with ones reported by STS. These in-gap states acquire a substantial amount of dynamical spectral weight transferred from the upper Hubbard band. The in-gap states move toward chemical potential with increasing spectral weight as doping increases. Our result also provides information about the energy scale of these in-gap states in relation with the Coulomb coupling strength UUU.
We have generalized the dynamical mean-field theory to study the doping dependence of the crossover from antiferromagnetic to short-range order modeled by an incommensurate spin density wave in the Hubbard model. The local self-energy which includes spin fluctuations gives quasiparticle weights and spectral properties in good agreement with quantum Monte Carlo and exact diagonalization data in two dimensions. The spectra at finite doping are characterized by a Mott-Hubbard ``gap'' accompanied by a pseudogap induced by the local spin order.
The optical conductivity $\ensuremath{\sigma}(\ensuremath{\omega})$ in the two-dimensional Hubbard model is examined by applying the exact diagonalization technique to small square clusters with periodic boundary conditions up to $\sqrt{20}\ifmmode\times\else\texttimes\fi{}\sqrt{20}$ sites. Spectral-weight distributions at half-filling and their doping dependence in the 20-site cluster are found to be similar to those in a $\sqrt{18}\ifmmode\times\else\texttimes\fi{}\sqrt{18}$ cluster, but different from $4\ifmmode\times\else\texttimes\fi{}4$ results. The results for the 20-site cluster enable us to perform a systematic study of the doping dependence of the spectral-weight transfer from the region of the Mott-gap excitation to lower-energy regions. We discuss the dependence of the Drude weight and the effective carrier number on the electron density at a large on-site Coulomb interaction.