The relationship between the pseudogap and underlying ground-state phases has not yet been rigorously established. We investigated the doped two-dimensional Hubbard model at finite temperature using controlled diagrammatic Monte Carlo calculations, allowing for the computation of spectral properties in the infinite-size limit and with arbitrary momentum resolution. We found three distinct regimes as a function of doping and interaction strength: a weakly correlated metal, a correlated metal with strong interaction effects, and a pseudogap regime at low doping. We show that the pseudogap forms both at weak coupling, when the magnetic correlation length is large, and at strong coupling, when it is shorter. As the temperature goes to zero, the pseudogap regime extrapolates precisely to the ordered stripe phase found by ground-state methods.
We investigate the doped two-dimensional Hubbard model at finite temperature using controlled diagrammatic Monte Carlo calculations allowing for the computation of spectral properties in the infinite-size limit and, crucially, with arbitrary momentum resolution. We show that three distinct regimes are found as a function of doping and interaction strength, corresponding to a weakly correlated metal with properties close to those of the non-interacting system, a correlated metal with strong interaction effects including a reshaping of the Fermi surface, and a pseudogap regime at low doping in which quasiparticle excitations are selectively destroyed near the antinodal regions of momentum space. We study the physical mechanism leading to the pseudogap and show that it forms both at weak coupling when the magnetic correlation length is large and at strong coupling when it is shorter. In both cases, we show that spin-fluctuation theory can be modified in order to account for the behavior of the non-local component of the self-energy. We discuss the fate of the pseudogap as temperature goes to zero and show that, remarkably, this regime extrapolates precisely to the ordered stripe phase found by ground-state methods. This handshake between finite temperature and ground-state results significantly advances the elaboration of a comprehensive picture of the physics of the doped Hubbard model.
Long-wavelength spin fluctuations prohibit antiferromagnetic long-range order at finite temperature in two dimensions. Nevertheless, the correlation length starts to grow rapidly at a crossover temperature, leading to critical slowing down and to a renormalized-classical regime over a wide range of temperature, between a fraction of the mean-field transition temperature and the zero-temperature ordered state. This leads to a single-particle pseudogap of the kind observed in electron-doped cuprates. Very few theoretical methods can claim an accurate description of this regime. The challenge is that in this regime Fermi-liquid quasiparticles are already destroyed and new quasiparticles of the ordered state are not fully formed yet. Here, we study this problem for the two-dimensional Hubbard model by first generalizing the two-particle self-consistent approach. Using a special algorithm, spin fluctuations are treated in the thermodynamic limit even for large correlation lengths. The effects of Kanamori-Br\"uckner screening, of classical and of quantum fluctuations are taken into account. Results are presented at half filling for the one-band Hubbard model with nearest-neighbor hopping. They agree well with available benchmark diagrammatic quantum Monte Carlo at high temperature where the pseudogap opens up. In addition to temperature-dependent spectral properties, we find quantum corrections to the zero-temperature renormalized mean-field antiferromagnetic gap. Finally, analytic continuation of the Matsubara results for spectral functions show that the pseudogap opens up at significantly higher temperature than was previously identified based on the Matsubara data only.
We determine the spin and charge orders in the ground state of the doped two-dimensional (2D) Hubbard model in its simplest form, namely with only nearest-neighbor hopping and on-site repulsion. At half-filling, the ground state is known to be an antiferromagnetic Mott insulator. Doping Mott insulators is believed to be relevant to the superconductivity observed in cuprates. A variety of candidates have been proposed for the ground state of the doped 2D Hubbard model. A recent work employing a combination of several state-of-the-art numerical many-body methods established the stripe order as the ground state near
1/8 doping at strong interactions. In this paper, we apply one of these methods, the cutting-edge constrained-path auxiliary field quantum Monte Carlo (AFQMC) method with self-consistently optimized gauge constraints, to systematically study the model as a function of doping and interaction strength. With careful finite size scaling based on large-scale computations, we map out the ground state phase diagram in terms of its spin and charge order. We find that modulated antiferromagnetic order persists from near half-filling to about
1/5 doping. At lower interaction strengths or larger doping, these ordered states are best described as spin-density waves, with essentially delocalized holes and modest oscillations in charge correlations. When the charge correlations are stronger (large interaction or small doping), they are best described as stripe states, with the holes more localized near the node in the antiferromagnetic spin order. In both cases, we find that the wavelength in the charge correlations is consistent with so-called filled stripes.
We investigate the momentum-resolved spin and charge susceptibilities, as well as the chemical potential and double occupancy in the two-dimensional Hubbard model as functions of doping, temperature, and interaction strength. Through these quantities, we identify a weak-coupling regime, a strong-coupling regime with short-range correlations and an intermediate-coupling regime with long magnetic correlation lengths. In the spin channel, we observe an additional crossover from commensurate to incommensurate correlations. In contrast, we find charge correlations to be only short ranged for all studied temperatures, which suggests that the spin and charge responses are decoupled. These findings were obtained by a connected determinant diagrammatic Monte Carlo algorithm for the computation of double expansions, which we introduce in this paper. This permits us to obtain numerically exact results at temperatures as low as
T=0.067 and interactions up to
U=7, while working on arbitrarily large lattices. Our method also allows us to gain physical insights from investigating the analytic structure of perturbative series. We connect to previous work by studying smaller lattice geometries and report substantial finite-size effects.
Exact diagrammatic quantum Monte Carlo (DiagMC) results for the nearest-neighbor Hubbard model motivate a closer study of the pseudogap. Using the improved two-particle self-consistent approach (TPSC+), we analyze Energy (EDC) and Momentum Distribution Curves (MDC) simultaneously. We show that Fermi-liquid terminology breaks down in the pseudogap regime, requiring a distinction between true and false Fermi surfaces and zero-energy quasiparticle (ZEQ) lines. A false Fermi surface has a momentum-space spectral maximum but a frequency-space depression at zero energy, while a false ZEQ line violates the standard quasiparticle condition
∂Σ′(k,ω)/∂ω∣ω=0<0.The pseudogap is driven by critical thermal spin fluctuations, which occur in two dimensions because of the Mermin-Wagner theorem. Commensurate fluctuations first open an antinodal pseudogap, leaving true Fermi arcs. With decreasing temperature, the Fermi surface evolves into hole- and electron-like false Fermi surfaces. Incommensurate fluctuations generate hot spots near the diagonal, where the pseudogap persists to the quantum critical point (QCP), whereas at
kAN it disappears before the QCP doping. There, the spectrum has two precursor antiferromagnetic (AFM) bands, both in the unoccupied (
ω>0) region. We benchmark TPSC+ against DiagMC results for the Matsubara spectral proxy
−Im[G(k,iπT)]/π. TPSC+ underestimates pseudogap suppression in the strong-interaction regime, reproducing DiagMC behavior at lower temperatures or doping. This proxy is equivalent to the spectral function with thermal broadening
η=πT, which obscures features when
πT is not the smallest energy scale. Using
A(k,0), we find that hole-like Fermi surfaces emerge at any interaction strength at low temperature, even at weak coupling.
The transition between a metallic and a pseudogap phase in high-
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
U, anisotropy
ty/tx, and doping. We find a first-order transition line which ends at a critical point, which occurs around ${t}_{y}/{t}_{x}\ensuremath{\sim}0.5$.