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Physical Review B

Finite-temperature exact diagonalization cluster dynamical mean-field study of the two-dimensional Hubbard model: Pseudogap, non-Fermi-liquid behavior, and particle-hole asymmetry

A. Liebsch, Ning-Hua Tong

Published Oct 27, 2009• 92 citations
Publisher

Abstract

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 ${E}_{F}$, 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 ${E}_{F}$ 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 ${E}_{F}$. The critical doping, which marks the onset of non-Fermi-liquid behavior, is systematically smaller than for hole doping.

DOI
10.1103/physrevb.80.165126

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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.
Cluster dynamical mean-field calculations based on 2-, 4-, 8-, and 16-site clusters are used to analyze the doping-driven metal-insulator transition in the two-dimensional Hubbard model. Comparison of results obtained on different clusters enables a determination of those aspects of the physics that are common to all clusters and permits identification of artifacts associated with particular cluster geometries. A modest particle-hole asymmetry in the underlying band structure is shown to lead to qualitatively different behavior on the hole-doped side than on the electron-doped side. For particle-hole asymmetry of the sign and magnitude appropriate to high-Tc{T}_{c}Tc​ cuprates, the approach to the insulator from the hole-doping side is found to proceed in two stages from a high-doping region where the properties are those of a Fermi liquid with moderately renormalized parameters and very weak momentum dependence. As doping is reduced the system first enters an intermediate doping regime where the Fermi-liquid renormalizations are larger and the electron self-energy varies significantly around the Fermi surface and then passes to a small doping regime characterized by a gap on some parts of the Fermi surface but gapless behavior in other parts. On the electron-doped side the partially gapped regime does not occur, and the momentum dependence of the electron self-energy is less pronounced. Implications for the high-Tc{T}_{c}Tc​ cuprates and for the use of cluster dynamical mean-field methods in wider classes of problems are discussed.
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.
Cluster dynamical mean-field and maximum entropy analytical continuation methods are used to obtain theoretical estimates for the many-body density of states, electron self-energy, in-plane and ccc-axis optical conductivity and the B1g{B}_{1g}B1g​ and B2g{B}_{2g}B2g​ Raman-scattering spectra of the two-dimensional square lattice Hubbard model at intermediate interaction strengths and carrier concentrations near half filling. The calculations are based on an eight-site cluster approximation which gives access to both the zone-diagonal and zone-face portions of the Fermi surface. At low dopings, the zone-face regions exhibit a ``pseudogap,'' a suppression of the many-body density of states for energies near the Fermi surface. The pseudogap magnitude is largest near half filling and decreases smoothly with doping but as temperature is increased, the gap fills in rather than closes. The calculated response functions bear a strong qualitative resemblance to data taken in the pseudogap regime of high-Tc{T}_{c}Tc​ cuprates, strongly suggesting that the intermediate coupling Hubbard model accounts for much of the exotic behavior observed in high-Tc{T}_{c}Tc​ materials.
We present an exact diagonalization study of the self-energy of the two-dimensional Hubbard model. To increase the range of available cluster sizes we use a corrected ttt-JJJ model to compute approximate Green's functions for the Hubbard model. This allows to obtain spectra for clusters with 18 and 20 sites. The self-energy has several ``bands'' of poles with strong dispersion and extended incoherent continua with kkk-dependent intensity. We fit the self-energy by a minimal model and use this to extrapolate the cluster results to the infinite lattice. The resulting Fermi surface shows a transition from hole pockets in the underdoped regime to a large Fermi surface in the overdoped regime. We demonstrate that hole pockets can be completely consistent with the Luttinger theorem. Introduction of next-nearest-neighbor hopping changes the self-energy strongly and the spectral function with nonvanishing next-nearest-neighbor hopping in the underdoped region is in good agreement with angle-resolved photoelectron spectroscopy in the cuprates.
We study the normal-state, doping-driven phase diagram of the square-lattice Hubbard model using the dynamical cluster approximation combined with the numerical renormalization group as a cluster solver, which gives direct access to real-frequency dynamics at essentially zero temperature. In a parameter regime relevant for cuprates, U=7tU=7tU=7t and t′=−0.3tt'=-0.3tt′=−0.3t, we find a critical doping p∗p^{\ast}p∗ that marks a continuous quantum phase transition between a pseudogap metal and a normal Fermi liquid. The transition is identified by a continuous collapse, from both sides, of the Fermi-liquid scale extracted from charge, spin, and dx2−y2d_{x^2-y^2}dx2−y2​-wave pairing susceptibilities. This collapse produces a non-Fermi-liquid regime at intermediate energy scales, which appears to extend to arbitrarily low scales at p∗p^{\ast}p∗. As p∗p^{\ast}p∗ is crossed from the normal Fermi liquid at p>p∗p>p^{\ast}p>p∗ into the pseudogap metal at $p