Commensurate and incommensurate magnetic order in the doped two-dimensional Hubbard model: dynamical mean-field theory analysis

TL;DR

This paper develops a dynamical mean-field theory approach to study magnetic order in the doped two-dimensional Hubbard model, revealing the evolution from antiferromagnetic to incommensurate and paramagnetic phases, with results consistent with experimental observations.

Contribution

It introduces a DMFT method for spiral magnetic order and applies it to the doped Hubbard model, providing new insights into magnetic phase transitions and excitations.

Findings

Antiferromagnetic order transitions to incommensurate and then paramagnetic phases with doping.

Spectral weight at the Fermi level increases with doping, forming hole pockets.

Magnetic excitations are gapless transverse modes and gapped longitudinal modes, matching experimental spin-wave data.

Abstract

We develop a dynamical mean-field theory approach for the spiral magnetic order, changing to a local coordinate frame with preferable spin alignment along the $z$-axis, which can be considered with the impurity solvers treating the spin diagonal local Green's function. We furthermore solve the Bethe-Salpeter equations for nonuniform dynamic magnetic susceptibilities in the local coordinate frame. We apply this approach to describe the evolution of magnetic order with doping in the $t-t'$ Hubbard model with $t'=0.15$, which is appropriate for the description of the doped La$_2$CuO$_4$ high-temperature superconductor. We find that with doping the antiferromagnetic order changes to the $(Q,\pi)$ incommensurate one, and then to the paramagnetic phase. The spectral weight at the Fermi level is suppressed near half filling and continuously increases with doping. The dispersion of holes in the antiferromagnetic phase shows qualitative agreement with the results of the $t$-$J$ model consideration. In the incommensurate phase we find two branches of hole dispersions, one of which crosses the Fermi level. The resulting Fermi surface forms hole pockets. We also consider the dispersion of the magnetic excitations, obtained from the non-local dynamic magnetic susceptibilities. The transverse spin excitations are gapless, fulfilling the Goldstone theorem; in contrast to the mean-field approach the obtained magnetic state is found to be stable. The longitudinal excitations are characterized by a small gap, showing the rigidity of the spin excitations. For realistic hopping and interaction parameters we reproduce the experimentally measured spin-wave dispersion of La$_2$CuO$_4$.