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arXiv:2108.01315 · Physical review. B./Physical review. B

Multipolar ordering in the three-orbital Hubbard model

Naoya Chikano, Shintaro Hoshino, Hiroshi Shinaoka

Published Dec 14, 2021• 9 citations

Abstract

The ground-state phase diagrams of the three-orbital ${t}_{2\mathrm{g}}$ Hubbard model are studied using a Hartree-Fock approximation. First, a complete set of multipolar order parameters for ${t}_{2\mathrm{g}}$ models defined in terms of the effective total angular momentum ${j}_{\mathrm{eff}}$ is theoretically derived. These order parameters can classify off-diagonal orders between ${j}_{\mathrm{eff}}=1/2$ and ${j}_{\mathrm{eff}}=3/2$ manifolds. Second, through extensive Hartree-Fock calculations, the ground-state phase diagrams in the space of (1) the on-site Coulomb repulsion $U$, (2) the spin-orbit coupling $\ensuremath{\lambda}$, and (3) the number of electrons $n$ are mapped out. A variety of nontrivial quantum phases with ${j}_{\mathrm{eff}}$-diagonal and ${j}_{\mathrm{eff}}$--off-diagonal multipole orders are found. Finally, future studies using more numerically expensive methods, such as dynamical mean-field theory, are discussed.

DOI: 10.1103/physrevb.104.235125

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The ground-state phase diagrams of the three-orbital

{t}_{2\mathrm{g}}

Hubbard model are studied using a Hartree-Fock approximation. First, a complete set of multipolar order parameters for

{t}_{2\mathrm{g}}

models defined in terms of the effective total angular momentum

{j}_{\mathrm{eff}}

is theoretically derived. These order parameters can classify off-diagonal orders between

{j}_{\mathrm{eff}}=1/2

and

{j}_{\mathrm{eff}}=3/2

manifolds. Second, through extensive Hartree-Fock calculations, the ground-state phase diagrams in the space of (1) the on-site Coulomb repulsion

U

, (2) the spin-orbit coupling

\ensuremath{\lambda}

, and (3) the number of electrons

n

are mapped out. A variety of nontrivial quantum phases with

{j}_{\mathrm{eff}}

-diagonal and

{j}_{\mathrm{eff}}

--off-diagonal multipole orders are found. Finally, future studies using more numerically expensive methods, such as dynamical mean-field theory, are discussed.

We study ordered phases with broken translational symmetry in the half-filled three-orbital Hubbard model with antiferromagnetic Hund coupling by means of dynamical mean-field theory (DMFT) and continuous-time quantum Monte Carlo simulations. The stability regions of the antiferro-orbital (AFO), antiferromagnetic (AFM), and charge density wave (CDW) states are determined by measuring the corresponding order parameters. We introduce two symmetrically distinct AFO order parameters and show that these are the primary order parameters in the phase diagram. The CDW and AFM states appear simultaneously with these two types of AFO orders in the weak and strong coupling region, respectively. The DMFT phase diagram is consistent with the results obtained by the Hartree approximation and strong-coupling perturbation theory. In the weak coupling regime, a nontrivial exponent $\ensuremath{\beta}=3/2$ is found for the CDW order parameter, which is related to the coupling between the CDW and AFO orders in the Landau theory characteristic for the three-orbital model. We also demonstrate the existence of a metallic AFO state without any charge disproportions and magnetic orders, which appears only at finite temperatures.

We consider an extended Hubbard model of interacting fermions on a lattice. The fermion kinetic energy corresponds to a tight-binding Hamiltonian with nearest-neighbor (-t) and next-nearest-neighbor (

{\mathrm{t}}^{|\mathrm{I}\mathrm{H}}

) hopping matrix elements. In addition to the on-site Hubbard interaction (U) we also consider a nearest-neighbor repulsion (V). We obtain the zero-temperature phase diagram of our model within the Hartree-Fock approximation. We consider ground states having charge- and spin-density-wave ordering as well as states with orbital antiferromagnetism or spin nematic order. The latter two states correspond to particle-hole binding with ${\mathrm{d}}_{{\mathrm{x}}^{2}\mathrm{\ensuremath{-}}{\mathrm{y}}^{2}}$ symmetry in the charge and spin channels, respectively. For

{\mathrm{t}}^{|\mathrm{I}\mathrm{H}}

=0, only the charge-density-wave and spin-density-wave states are energetically stable. For nonzero

{\mathrm{t}}^{|\mathrm{I}\mathrm{H}}

, we find that orbital antiferromagnetism (or spin nematic) order is stable over a finite portion of the phase diagram at weak coupling. This region of stability is seen to grow with increasing values of

{\mathrm{t}}^{|\mathrm{I}\mathrm{H}}

.

Using the density matrix renormalization group technique we study the effect of spin-orbit coupling on a three-orbital Hubbard model in the

{({t}_{2g})}^{4}

sector and in one dimension. Fixing the Hund coupling to a robust value compatible with some multiorbital materials, we present the phase diagram varying the Hubbard

U

and spin-orbit coupling $\ensuremath{\lambda}$, at zero temperature. Our results are shown to be qualitatively similar to those recently reported using the dynamical mean-field theory in higher dimensions, providing a robust basis to approximate many-body techniques. Among many results, we observe an interesting transition from an orbital-selective Mott phase to an excitonic insulator with increasing $\ensuremath{\lambda}$ at intermediate

U

. In the strong

U

coupling limit, we find a nonmagnetic insulator with an effective angular momentum $\ensuremath{\langle}{({\mathbf{J}}^{\mathrm{eff}})}^{2}\ensuremath{\rangle}\ensuremath{\ne}0$ near the excitonic phase, smoothly connected to the $\ensuremath{\langle}{({\mathbf{J}}^{\mathrm{eff}})}^{2}\ensuremath{\rangle}=0$ regime. We also provide a list of quasi-one-dimensional materials where the physics discussed in this paper could be realized.