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arXiv:2310.17302 · SciPost Physics

Mott transition and pseudogap of the square-lattice Hubbard model: Results from center-focused cellular dynamical mean-field theory

Michael Meixner, Henri Menke, Marcel Klett, Sarah Heinzelmann, Sabine Andergassen, P. Hansmann, Thomas Schäfer

Published Feb 27, 2024• 14 citations
arXivHTMLPDF

Abstract

The recently proposed center-focused post-processing procedure [Phys. Rev. Res. 2, 033476 (2020)] of cellular dynamical mean-field theory suggests that central sites of large impurity clusters are closer to the exact solution of the Hubbard model than the edge sites. In this paper, we systematically investigate results in the spirit of this center-focused scheme for several cluster sizes up to 8×8 in and out of particle-hole symmetry. First we analyze the metal-insulator crossovers and transitions of the half-filled Hubbard model on a simple square lattice. We find that the critical interaction of the crossover is reduced with increasing cluster sizes and the critical temperature abruptly drops for the 4×4 cluster. Second, for this cluster size, we apply the center-focused scheme to a system with more realistic tight-binding parameters, investigating its pseudogap regime as a function of temperature and doping, where we find doping dependent metal-insulator crossovers, Lifshitz transitions and a strongly renormalized Fermi-liquid regime. Additionally to diagnosing the real space origin of the suppressed antinodal spectral weight in the pseudogap regime, we can infer hints towards underlying charge ordering tendencies.

DOI
10.21468/scipostphys.16.2.059

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The recently proposed center-focused post-processing procedure [Phys. Rev. Res. 2, 033476 (2020)] of cellular dynamical mean-field theory suggests that central sites of large impurity clusters are closer to the exact solution of the Hubbard model than the edge sites. In this paper, we systematically investigate results in the spirit of this center-focused scheme for several cluster sizes up to 8×8 in and out of particle-hole symmetry. First we analyze the metal-insulator crossovers and transitions of the half-filled Hubbard model on a simple square lattice. We find that the critical interaction of the crossover is reduced with increasing cluster sizes and the critical temperature abruptly drops for the 4×4 cluster. Second, for this cluster size, we apply the center-focused scheme to a system with more realistic tight-binding parameters, investigating its pseudogap regime as a function of temperature and doping, where we find doping dependent metal-insulator crossovers, Lifshitz transitions and a strongly renormalized Fermi-liquid regime. Additionally to diagnosing the real space origin of the suppressed antinodal spectral weight in the pseudogap regime, we can infer hints towards underlying charge ordering tendencies.
We revisit the cellular dynamical mean-field theory (CDMFT) for the single-band Hubbard model on the square lattice at half filling, reaching real-space cluster sizes of up to 9 × 9 sites.Using benchmarks against direct lattice diagrammatic Monte Carlo at high temperature, we show that the self-energy obtained from a cluster center-focused extrapolation converges faster with the cluster size than the periodization schemes previously introduced in the literature.The same benchmark also shows that the cluster spin susceptibility can be extrapolated to the exact result at large cluster size, even though its spatial extension is larger than the cluster size.
The four-site "DCA" method of including intersite correlations in the dynamical mean-field theory is used to investigate the metal-insulator transition in the Hubbard model. At half-filling a gap-opening transition is found to occur as the interaction strength is increased beyond a critical value. The gapped behavior found in the 4-site DCA approximation is shown to be associated with the onset of strong antiferromagnetic and singlet correlations and the transition is found to be potential energy driven. It is thus more accurately described as a Slater phenomenon (induced by strong short-range order) than as a Mott phenomenon. Doping the gapped phase leads to a non–Fermi-liquid state with a Fermi surface only in the nodal regions and a pseudogap in the antinodal regions at lower dopings x ≲0.15 and to a Fermi-liquid phase at higher dopings.
The transition between a metallic and a pseudogap phase in high-Tc{T}_{c}Tc​ cuprate superconductors, which is the subject of recent experimental investigations, is studied theoretically within the Hubbard model. We apply the cluster dynamical impurity approximation to the anisotropic Hubbard model on the square lattice at zero temperature and finite doping. This approach can detect a first-order transition between two metallic states: a pseudogap state at low doping with a depleted density of states at the Fermi level and a correlated metal at higher doping. This transition was seen first in the cluster dynamical mean-field theory at finite temperatures by Sordi et al. [Phys. Rev. Lett. 108, 216401 (2012)] on the isotropic Hubbard model. Here we investigate this transition at zero temperature and as a function of on-site interaction UUU, anisotropy ty/tx{t}_{y}/{t}_{x}ty​/tx​, and doping. We find a first-order transition line which ends at a critical point, which occurs around ${t}_{y}/{t}_{x}\ensuremath{\sim}0.5$.
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.
In this study, the metal-insulator transition in the square-lattice Hubbard model at half-filling is revisited in relation to the DOS and spectral functions by means of the ladder dual fermion approximation (LDFA). For this purpose, a new expression of the two-body Green's function in the form of resolvents is proposed, which provides tractable and efficient means to calculate the local vertex function with the Lanczos exact diagonalization (ED) method. This makes it possible to use the Lanczos ED method as a solver of the effective impurity Anderson model for LDFA, opening up the way to access low temperatures for these perturbative extensions of the dynamical mean-field theory and to obtain accurate DOS and spectral functions on the real frequency axis by a new variant of the maximum entropy method. It is found that for $U\ensuremath{\le}3.5t$, as temperature decreases, the pseudogap formation due to antiferromagnetic correlations in the quasiparticle peak of the spectral function occurs at the XXX point $[\mathbit{k}=(\ensuremath{\pi},0)]$, spreads through the Fermi surface and ends at the M2{M}_{2}M2​ point $[\mathbit{k}=(\ensuremath{\pi}/2,\ensuremath{\pi}/2)]$. The almost simultaneous creation of the pseudogap and the loss of the Fermi liquid feature is consistent with that expected in the Slater regime. Although the pseudogap still appears in the quasi-particle-like single peak for $U\ensuremath{\ge}4.0t$, the Fermi-liquid feature is partially lost on the Fermi surface already at higher temperatures as expected in the Mott-Heisenberg regime, in which local spins are preformed at high temperatures. A sharp crossover from a pseudogap phase to a Mott insulator at finite ${U}^{*}\ensuremath{\approx}4.7t$ is found to occur below the temperature of the pseudogap formation similar to a previous study with the nonlinear $\ensuremath{\sigma}$ model approach.
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.