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arXiv:2012.07382 · AIP Advances

Antiferromagnetic fluctuations in the one-dimensional Hubbard model

V. Janiš, Antonín Klíč, Jiawei Yan

Published Dec 1, 2020• 1 citation
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Abstract

We study the low-temperature critical behavior of the one-dimensional Hubbard model near half filling caused by enhanced antiferromagnetic fluctuations. We use a mean-field-type approximation with a two-particle self-consistency renormalizing the bare interaction. It allows us to control a transition from high to low temperatures as well as from weak to strong-coupling. We show that there is a crossover temperature T0 = t exp{−1/Uρ(0)} for arbitrary interaction U > 0 and the bare density of states at the Fermi energy ρ(0) > 0. The solution at lower temperatures goes over to strong coupling and approaches a quantum critical point with the diverging staggered susceptibility and a gap in the excitation spectrum at zero temperature.

DOI
10.1063/9.0000019

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We study the low-temperature critical behavior of the one-dimensional Hubbard model near half filling caused by enhanced antiferromagnetic fluctuations. We use a mean-field-type approximation with a two-particle self-consistency renormalizing the bare interaction. It allows us to control a transition from high to low temperatures as well as from weak to strong-coupling. We show that there is a crossover temperature T0 = t exp{−1/Uρ(0)} for arbitrary interaction U > 0 and the bare density of states at the Fermi energy ρ(0) > 0. The solution at lower temperatures goes over to strong coupling and approaches a quantum critical point with the diverging staggered susceptibility and a gap in the excitation spectrum at zero temperature.
We present fully self-consistent calculations in the fluctuation exchange approximation for the half-filled Hubbard model in two dimensions. A non-Fermi-liquid state evolves with decreasing temperature in this self-consistent model of coupled spin fluctuations and quasiparticles. The mean-field phase transition to long-range antiferromagnetic order is suppressed and we find no evidence of a phase transition to long-range magnetic order. We show that the real part of the self-energy at zero energy shows a positive slope and the imaginary part of the self-energy shows a local minimum. The scale of this structure is set by the zero-temperature gap in mean-field theory. The growth of spin fluctuations is reflected in the evolution of sharply peaked structure in the spin fluctuation propagator around zero energy and Q = (π, π). We present calculations for the Hubbard model in one dimension in a two-particle self-consistent parquet approximation. A second-moment sum rule suggests that vertex corrections to the self-energy are responsible for the greater accuracy of the parquet approximation compared with other self-consistent perturbation theories.
We present single-particle and thermodynamic properties of the half-filled single-band Hubbard model in 2D calculated in the self-consistent fluctuation exchange approximation. The low-energy excitations at moderate temperatures and small UUU are quasiparticles with a short lifetime. As the temperature is lowered, coupling to evolving spin fluctuations leads to the extinction of these quasiparticles, signaled by a weak pseudogap in the density of states and by a positive slope in $\mathrm{Re}\ensuremath{\Sigma}({\mathbf{k}}_{F},\ensuremath{\varepsilon})$ and a local maximum in $|\mathrm{Im}\ensuremath{\Sigma}({\mathbf{k}}_{F},\ensuremath{\varepsilon})|$ at $\ensuremath{\varepsilon}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0$. We explain these results using a simple spin-fluctuation model.
An effect of antiferromagnetic fluctuations on single-particle excitations is examined on the basis of an extended dynamical mean-field theory that takes account of long-range spin fluctuations onto the local self energy. We demonstrate that the single-particle spectral intensity in the two-dimensional Hubbard model at half-filling exhibits the pseudo-gap behavior near the Fermi energy due to the critical antiferromagnetic fluctuations. We also show that for the symmetric case in the Anderson lattice model, the hybridized-band gap is enlarged by the antiferromagnetic fluctuations. In both models the antiferromagnetic fluctuations suppress the double occupancy of the correlated electrons.
We study the temperature and doping dependence of the two-dimensional (2D) Hubbard model in the intermediate coupling regime (U=4).(U=4).(U=4). As temperature is decreased for a half-filled band, 2D critical spin fluctuations give rise to a strong local maximum in $|\mathrm{Im}\ensuremath{\Sigma}({k}_{F},\ensuremath{\omega})|$ at $\ensuremath{\omega}=0,$ leading to a split peak in the spectral function. These strong fluctuations dominating near half filling are completely destroyed by 12% doping concentration. Below this concentration the imaginary part of the self-energy vanishes nearly quadratically in frequency at the Fermi energy, a Fermi-liquid-like feature. With varying the temperature at this concentration, we present numerical evidence for the band Kondo effect in the doped 2D Hubbard model by showing at low temperatures a growing peak in the density of states, nearly quadratically vanishing (in frequency) scattering rates at the Fermi energy and a linearly vanishing (in temperature) effective magnetic moment.
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 present an analytic solution of the self consistent random phase approximation equations for the half-filled Hubbard model, in the strong coupling limit. In the half-filled limit the Hubbard model shows antiferromagnetic ordering. The self energy is calculated by considering processes whereby the electrons emit and absorb low frequency spin waves, which are the Goldstone modes of the broken symmetry phase. We examine the electronic spectrum, which is dominated by the broad incoherent portions of the spectrum associated with the upper and lower Hubbard bands. The quasi-particle spectrum has a divergent wave function renormalization, indicating a divergent renormalization of the quasi-particle mass and zero quasi-particle weight, as in a non-Fermi liquid. The self consistency equation determined from the Goldstone modes in the susceptibility shows that the antiferromagnetism is prohibited in one and two dimensions but not in three. The magnetic ordering temperature is determined by dimensionality and the magnitude of the Anderson kinetic exchange of the effective Heisenberg Hamiltonian.