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arXiv:1712.07908 · Scientific Reports

Cooper Pairing in A Doped 2D Antiferromagnet with Spin-Orbit Coupling

Jingxiang Zhao, Qiang Gu

Published Jan 11, 2018• 4 citations
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Abstract

We study the two-dimensional Hubbard model with the Rashba type spin-orbit coupling within and beyond the mean-field theory. The antiferromagnetic ground state for the model at half-filling and the Cooper pairing induced by antiferromagnetic spin fluctuations near half-filling are examined based on the random-phase approximation. We show that the antiferromagnetic order is suppressed and the magnetic susceptibility turns out to be anisotropic in the presence of the spin-orbit coupling. Energy spectrums of transverse spin fluctuations are obtained and the effective interactions between holes mediated by antiferromagnetic spin fluctuations are deduced in the case of low hole doping. It seems that the spin-orbit coupling tends to form s+p-wave Cooper pairs, while the s+d-wave pairing is dominant when the spin-orbit coupling is absent.

DOI
10.1038/s41598-017-19038-7

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We study the two-dimensional Hubbard model with the Rashba type spin-orbit coupling within and beyond the mean-field theory. The antiferromagnetic ground state for the model at half-filling and the Cooper pairing induced by antiferromagnetic spin fluctuations near half-filling are examined based on the random-phase approximation. We show that the antiferromagnetic order is suppressed and the magnetic susceptibility turns out to be anisotropic in the presence of the spin-orbit coupling. Energy spectrums of transverse spin fluctuations are obtained and the effective interactions between holes mediated by antiferromagnetic spin fluctuations are deduced in the case of low hole doping. It seems that the spin-orbit coupling tends to form s+p-wave Cooper pairs, while the s+d-wave pairing is dominant when the spin-orbit coupling is absent.
The weak-coupling form of the effective interaction between two holes in the antiferromagnetic state of the two-dimensional Hubbard model near half-filling is derived. It is shown to predict a Cooper instability in the singlet ddd-wave channel, with binding temperature Tc>100{T}_{c}>100Tc​>100 K. Our results differ from those of the "spin-bag" theory of Schrieffer, Wen, and Zhang.
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 1/81/81/8 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 1/51/51/5 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 analyze the density dependence of the spin stiffnesses and the stability of magnetic order with respect to quantum fluctuations in the two-dimensional Hubbard model close to half-filling. The stiffnesses are computed from the spin susceptibility obtained from a random-phase approximation in a magnetically ordered state. For a sizable next-to-nearest-neighbor hopping amplitude and a moderate Hubbard interaction, the mean-field ground state is a Néel antiferromagnet in the electron-doped regime at and above half-filling and a planar circular spiral state in the hole-doped regime below half-filling. Upon electron doping, the Néel stiffness decreases smoothly and not very steeply. By contrast, the in-plane and out-of-plane stiffnesses in the spiral state drop abruptly at half-filling. The out-of-plane stiffness even drops to zero and then increases again very slowly upon increasing hole doping. At finite temperatures, the Néel-to-spiral transition is shifted into the hole-doped regime, the stiffnesses are continuous functions of the density, and they vanish at the transition. For small hole doping, the spin stiffnesses describe the quantum fluctuations only in a small momentum range, which shrinks to zero upon approaching half-filling. Using the above results, we show that the quantum ground state of the lightly electron-doped Hubbard model remains Néel ordered, while quantum fluctuations probably destroy the spiral long-range order in the hole-doped regime, giving rise to a quantum disordered state with a spin gap.
We study the effects of doping on the spin fluctuations of the two-dimensional Hubbard model. Within a harmonic approximation we calculate the linearized antiferromagnetic spin-wave spectra around a hole away from half-filling, which takes the form of a spin polaron in a mean-field approximation. The presence of the spin polaron breaks a symmetry of the problem, the product of time reversal and lattice translation. As a result, the double degeneracy of the magnon spectra of the N\'eel state is lifted. There also appear localized magnon modes depending on the ratio U/t of the Hubbard model. The quantum correction due to the spin fluctuations is found to increase the spin polaron binding energy for U/t13, but decreases it for U/t>13.
We study the interacting Fermi-Hubbard model in two spatial dimensions with synthetic gauge coupling of the spin-orbit Rashba type, at half-filling. Using real-space mean-field theory, we numerically determine the phase as a function of the interaction strength for different values of the gauge-field parameter. For a fixed value of the gauge field, we observe that when the strength of the repulsive interaction is increased, the system enters into an antiferromagnetic phase, then undergoes a first-order phase transition to a noncollinear magnetic phase. Depending on the gauge-field parameter, this phase further evolves to the one predicted from the effective Heisenberg model obtained in the limit of large interaction strength. We explain the presence of the antiferromagnetic phase at small interaction from the computation of the spin-spin susceptibility, which displays a divergence at low temperatures for the antiferromagnetic ordering. We discuss, how the divergence is related to the nature of the underlying Fermi surfaces. Finally, the fact that the first-order phase transitions for different gauge-field parameters occur at unrelated critical interaction strengths arises from a Hofstadter-like situation, i.e., for different magnetic phases, the mean-field Hamiltonians have different translational symmetries.
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.