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Physical review. B, Condensed matter

Linearized antiferromagnetic spin-wave spectra around a hole in the two-dimensional Hubbard model

Han-Yong Choi

Published Aug 1, 1991• 7 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.44.2609

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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 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.
Spin and hole excitation spectra and spectral weights are calculated for the half-filled Hubbard model, as functions of t∕Ut∕Ut∕U. We find that the high energy spin spectra are sensitive to charge fluctuations. The energy difference $\ensuremath{\Delta}(\ensuremath{\pi},0)\ensuremath{-}\ensuremath{\Delta}(\ensuremath{\pi}∕2,\ensuremath{\pi}∕2)$ in the single-magnon dispersion relation, which is negative for the Heisenberg model, changes sign at a fairly small $t∕U\ensuremath{\approx}0.053(5)$. The hole bandwidth is proportional to JJJ, and considerably larger than in the $t\text{\ensuremath{-}}J$ models. It has a minimum at $(\ensuremath{\pi}∕2,\ensuremath{\pi}∕2)$ and a very weak dispersion along the antiferromagnetic zone boundary. A good fit to the measured spin spectra in La2CuO4{\mathrm{La}}_{2}\mathrm{Cu}{\mathrm{O}}_{4}La2​CuO4​ at T=10KT=10\phantom{\rule{0.3em}{0ex}}\mathrm{K}T=10K is obtained with the parameter values U=3.1eVU=3.1\phantom{\rule{0.3em}{0ex}}\mathrm{eV}U=3.1eV, t=0.35eVt=0.35\phantom{\rule{0.3em}{0ex}}\mathrm{eV}t=0.35eV.
We calculate the excitation spectrum and the response functions of the textures induced by doping in the half-filled Hubbard model. Localized states appear within the antiferromagnetic gap. The spin wave spectrum is shifted to lower frequencies, although the antiferromagnetic peak at zero energy is depleted. Charge excitations appear at energies comparable to the spin modes. These excitations mediate effective interactions which can be attractive for next-nearest-neighbour sites. The overall features of the results are inconsistent with normal metal behaviour, as described by the same scheme.
A scheme of symmetry breaking is proposed for the two-dimensional Hubbard model at less than half filling. The self-induced mean field is a random spin-dependent potential. The energy of the spin-disordered state is shown to be lower than that of commensurate or incommensurate antiferromagnetic states. Long-wavelength charge and spin fluctuations obey the diffusion equation, leading to deviations from Fermi-liquid behavior. At wave vector q=(\ensuremath{\pi},\ensuremath{\pi}) the transverse spin-excitation spectrum has a gap.
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.
The interplay between charge and spin dynamics is at the heart of strongly correlated materials. Inspired by recent quantum simulation experiments, we develop a conserving diagrammatic method to describe the Fermi-Hubbard model for strong repulsion and small hole doping away from the half-filled anti-ferromagnetic ground state. We show that doping leads to four hole pockets in the Brillouin zone formed by magnetic polarons, which become increasingly damped with hole concentration. Likewise, the magnon spectrum of the anti-ferromagnet softens and dampens with doping due to hole-induced magnetic frustration. This gives rise to a suppression of the anti-ferromagnetic correlations in agreement with recent experiments. We then calculate the response of the system to a lattice modulation and recover the qualitative difference between in-phase and out-of-phase modulations seen in experiments, which was interpreted as signs of pseudogap physics. Our results indicate that the complex competition between spin and charge degrees of freedom and the emergence of the pseudogap phase may be usefully analyzed for small dopings, where systematic theories can be developed.