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arXiv:1812.04805 · Physical review. B./Physical review. B

Metal-insulator transition in the two-dimensional Hubbard model: Dual fermion approach with Lanczos exact diagonalization

Arata Tanaka

Published May 20, 2019• 26 citations
arXivHTMLPDF

Abstract

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 $X$ point $[\mathbit{k}=(\ensuremath{\pi},0)]$, spreads through the Fermi surface and ends at the ${M}_{2}$ 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.

DOI
10.1103/physrevb.99.205133

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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≤3.5tU\ensuremath{\le}3.5tU\ensuremath≤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 [\mathbitk=(\ensuremathπ,0)][\mathbit{k}=(\ensuremath{\pi},0)][\mathbitk=(\ensuremathπ,0)], spreads through the Fermi surface and ends at the M2{M}_{2}M2​ point [\mathbitk=(\ensuremathπ/2,\ensuremathπ/2)][\mathbit{k}=(\ensuremath{\pi}/2,\ensuremath{\pi}/2)][\mathbitk=(\ensuremathπ/2,\ensuremathπ/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≥4.0tU\ensuremath{\ge}4.0tU\ensuremath≥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≈4.7t{U}^{*}\ensuremath{\approx}4.7tU∗\ensuremath≈4.7t is found to occur below the temperature of the pseudogap formation similar to a previous study with the nonlinear \ensuremathσ\ensuremath{\sigma}\ensuremathσ model approach.
In the half-filled one-orbital Hubbard model on a square lattice, we find pseudogap-like features in the form of two-peak structures associated with the momentum-resolved spectral function. These features exist within the temperature window ${T}_{N}\ensuremath{\lesssim}T\ensuremath{\lesssim}{T}^{*}$, where TN{T}_{N}TN​ is the N\'eel temperature and T∗{T}^{*}T∗ is the temperature below which there exists a well-formed dip in the density of state. Inside the window ${T}_{N}\ensuremath{\lesssim}T\ensuremath{\lesssim}{T}^{*}$, the peak-to-peak separation in the two-peak structure of the momentum-resolved spectral function grows on moving away from the point ($\ensuremath{\pi}/2,\ensuremath{\pi}/2$) along the normal state Fermi surface toward $(\ensuremath{\pi},0)$, a behavior with remarkable similarities to what is observed in the ddd-wave state and pseudogap phase of high-Tc{T}_{c}Tc​ cuprates. We unveil these features by using a parallelized cluster-based Monte Carlo method for simulating the magnetic order parameter fields on a superlattice. The method enables us to access the momentum-resolved single-particle spectral function corresponding to a lattice size of $\ensuremath{\approx}240 \ifmmode\times\else\texttimes\fi{}$ 240 with an almost negligible finite-size effect.
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.
Though most fermionic Mott insulators order at low temperatures, ordering is ancillary to their insulating behavior. Our emphasis here is on disentangling ordering from the intrinsic strongly correlated physics of a doped half-filled band. To this end, we focus on the two-dimensional Hubbard model. Because the charge gap arises from on-site correlations, we have been refining the nonperturbative approach of Matsumoto and Mancini [Phys. Rev. B 55, 2095 (1997)] which incorporates local physics. Crucial to this method is a self-consistent two-site dynamical cluster expansion which builds in the nearest-neighbor energy scale J. At half-filling, we find that the spectral function possesses a gap of order U and is devoid of any coherent quasiparticle peaks although ordering or charge fractionalization are absent. At low temperatures, local antiferromagnetic correlations emerge. In the doped case, we find that the Fermi surface exceeds the Luttinger volume. The breakdown of Luttinger's theorem in the underdoped regime is traced both to the dynamically generated Mott gap as well to a nonvanishing of the imaginary part of the self-energy at the Fermi level. Spectral weight transfer across the Mott gap also emerges as a ubiquitous feature of a doped Mott insulator and suggests that high- and low-energy scales are inseparable. Additionally in the underdoped regime, we find that a pseudogap exists in the single-particle density of states as well as in the heat capacity. The pseudogap (which is set by the energy scale t2/U){t}^{2}/U)t2/U) is argued to be a ubiquitous feature of a lightly doped Mott state and simply represents the fact that hole transport involves double occupancy. In analogy with the Mott gap and antiferromagnetism, we propose that ordering may also accompany the formation of a pseudogap. We suggest a current pattern within a one-band model that preserves translational but breaks time-reversal symmetry along the canonical x and y axes but not along $x=\ifmmode\pm\else\textpm\fi{}y$ that is consistent with the experimental observations. Finally, we show that the Hall coefficient in a doped Mott insulator must change sign at a doping level x<1/3.x<1/3.x<1/3. The sign change is tied to a termination of strong correlation physics in the doped Mott state.
We study the two-dimensional square-lattice Hubbard model for small to moderate interaction strengths $1\ensuremath{\le}U/t\ensuremath{\le}4$ by means of the ladder dual fermion approach. The nonlocal correlations beyond dynamical mean-field theory lower the potential energy, lead to a maximum in the uniform susceptibility, and induce a pseudogap in the density of states. While the self-energy exhibits precursors of a possible insulating phase linked to the appearance of long-range fluctuations, the metallic phase persists within the accessible temperature range. Finite-size effects affect results qualitatively. Upper bounds on the crossover temperature are found to be significantly lower than previously reported dynamical vertex approximation results at U/t=1U/t=1U/t=1.
To clarify the origin of the pseudogap and strange metal states as well as their mutual relationship in cuprate superconductors, a comprehensive study on the spectral function, Fermi surface, resistivity and dynamical spin susceptivity of the Hubbard model on the square lattice has been conducted by means of the ladder dual-fermion approximation with an electron self-energy correction. It is found that the appearance of these two states requires that the characteristic hole concentration below which the Mott-Heisenberg and Slater mechanisms of electron localization occurs pMSp_{\rm MS}pMS​ nearly coincides with the hole concentration where the Van Hove singularity (VHS) point is placed at the vicinity of the Fermi level. When this condition is met the VHS point is pinned at which the nesting condition of the antiferromagnetic (AFM) fluctuation is fulfilled almost everywhere on the Fermi surface in wide range of the hole concentration in a metallic state, i.e., the strange metal state. The spin fluctuation of the strange metal state is nearly quantum critical and the dynamical spin susceptivity is well described by overdamped spin wave having the ω/Tω/Tω/T scaling with the relaxation rate at the Planckian limit. Because of these distinctive features of the strange metal state, the kkk dependence of scattering rate of electrons is small and electrons behave as the marginal Fermi liquid, resulting in TTT-linear resistivity. In contrast, the pseudogap state is magnetically in the renormalized classical regime and the pseudogap is formed near the X point where the nesting condition of the short-range AFM order is fulfilled.
In the half-filled one-orbital Hubbard model on a square lattice, we find pseduogap features in the form of two-peak structures associated with the momentum-resolved spectral function, which exists within the temperature window TN≲T≲T∗T_N \lesssim T \lesssim T^*TN​≲T≲T∗. T∗T^*T∗ is the temperature below which there exists a well-formed dip in the density of state. Inside the window TN≲T≲T∗T_N \lesssim T \lesssim T^*TN​≲T≲T∗, the peak-to-peak separation in the two-peak structure of the momentum-resolved spectral function rises on moving away from the point (π/2,π/2π/2, π/2π/2,π/2) along the normal state Fermi surface towards (π,0)(π, 0)(π,0), a behavior remarkably similar to what is observed in the pseudogap phase. We unveil these features by using a parallelized cluster-based Monte-Carlo method for simulating the magnetic order parameter fields on a superlattice, which enables us to access the momentum-resolved single-particle spectral function corresponding to a lattice size of ∼\sim∼ 240 ×\times× 240 with almost negligible finite-size effects.