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arXiv:cond-mat/0102387 · Physical review. B, Condensed matter

Antiferromagnetic phase in the Hubbard model by means of the composite operator method

Adolfo Avella, Ferdinando Mancini, Roland Münzner

Published Jun 7, 2001• 24 citations
arXivHTMLPDF

Abstract

We have investigated the antiferromagnetic phase of two-dimensional (2D), three-dimensional (3D), and extended Hubbard models on a bipartite cubic lattice by means of the composite operator method within a two-pole approximation. This approach yields a fully self-consistent treatment of the antiferromagnetic state that respects the symmetry properties of both the model and the algebra. The complete phase diagram, as regards the antiferromagnetic and paramagnetic phases, has been drawn. We first reported, within a pole approximation, three kinds of transitions at half-filling: Mott-Hubbard, Mott-Heisenberg, and Heisenberg transitions. We have also found a metal-insulator transition, driven by doping, within the antiferromagnetic phase. This latter is restricted to a very small region near half-filling, and has, in contrast to what has been found by similar approaches, a finite critical Coulomb interaction as a lower bound at half-filling. Finally, it is worth noting that our antiferromagnetic gap has two independent components: one due to the antiferromagnetic correlations, and another coming from the Mott-Hubbard mechanism.

DOI
10.1103/physrevb.63.245117

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We have investigated the antiferromagnetic phase of two-dimensional (2D), three-dimensional (3D), and extended Hubbard models on a bipartite cubic lattice by means of the composite operator method within a two-pole approximation. This approach yields a fully self-consistent treatment of the antiferromagnetic state that respects the symmetry properties of both the model and the algebra. The complete phase diagram, as regards the antiferromagnetic and paramagnetic phases, has been drawn. We first reported, within a pole approximation, three kinds of transitions at half-filling: Mott-Hubbard, Mott-Heisenberg, and Heisenberg transitions. We have also found a metal-insulator transition, driven by doping, within the antiferromagnetic phase. This latter is restricted to a very small region near half-filling, and has, in contrast to what has been found by similar approaches, a finite critical Coulomb interaction as a lower bound at half-filling. Finally, it is worth noting that our antiferromagnetic gap has two independent components: one due to the antiferromagnetic correlations, and another coming from the Mott-Hubbard mechanism.
The cluster slave-spin method is employed to investigate systematically the ground state properties of the Hubbard model on a square lattice with doping $\ensuremath{\delta}$ and coupling strength UUU being its parameters. At half-filling, a relation between the staggered magnetization MMM and the antiferromagnetic (AFM) gap ${\mathrm{\ensuremath{\Delta}}}_{\text{AFM}}$ is established in the small UUU limit to compare with that from the Hartree-Fock theory, and a first-order metal-insulator Mott transition in the paramagnetic state is substantiated, which is characterized by discontinuities and hystereses at UMott=10t{U}_{\text{Mott}}=10tUMott​=10t. The interaction Uc{U}_{c}Uc​ for the crossover in the AFM state, separating the weak- and strong-coupling regimes, is found to remain almost unchanged with large dopings, and smaller than UMott{U}_{\text{Mott}}UMott​ at half-filling because of long range AFM correlations. Finally, an overall phase diagram in the $U\ensuremath{-}\ensuremath{\delta}$ plane is presented, which is composed of four regimes: the AFM insulator at half-filling, the AFM metal with the compressibility $\ensuremath{\kappa}>0$ or $\ensuremath{\kappa}<0$, and the paramagnetic metal, as well as three phase transitions: (i) From the AFM metal to the paramagnetic metal, (ii) between the AFM metal phases with positive and negative $\ensuremath{\kappa}$, and (iii) separating the AFM insulating phase at $\ensuremath{\delta}=0$ from the AFM metal phase for $\ensuremath{\delta}>0$.
A strong-coupling regime of the Hubbard model around half filling is investigated in the Hartree-Fock random-phase approximation. At half filling, besides a normal solution that gives rise to antiferromagnetic instability at ${T=T}_{\mathrm{N}}^{\mathrm{RPA}}\ensuremath{\sim}U,$ we find a solution representing the paramagnetic phase of the Mott insulator, for which we obtain the N\'eel temperature ${T}_{\mathrm{N}}\ensuremath{\sim}J$ in three-dimensional lattices. The solution is interpreted as the ferromagnetic band insulator depolarized by spin wave, the bound state of a particle-hole pair. Similarly, an anomalous metallic phase is derived from the Nagaoka ferromagnetic state. We evaluate TN{T}_{\mathrm{N}}TN​ as a function of doping concentration x to find that it vanishes at $x\ensuremath{\simeq}0.05$ for U/6t=4U/6t=4U/6t=4 in a simple cubic lattice.
We investigate the metal-insulator transition in the half-filled Hubbard model on the two-dimensional triangular lattice using both the Kotliar-Ruckenstein slave-boson technique and an exact numerical diagonalization of finite clusters. Contrary to the case of a square lattice, where a perfect nesting of the Fermi surface leads to a metal-insulator transition at arbitrarily small values of U, always accompanied by antiferromagnetic ordering, on a triangular lattice, due to the lack of perfect nesting, the transition takes place at a finite value of U, and frustration induces a nontrivial competition among different magnetic phases. Indeed, within the mean-field approximation in the slave-boson approach, as the interaction grows the paramagnetic metal turns into a metallic phase with incommensurate spiral ordering. Increasing the interaction further, a linear spin-density wave is stabilized, and finally for strong coupling the latter phase undergoes a first-order transition toward an antiferromagnetic insulator. No trace of the intermediate phases is seen in the exact diagonalization results, indicating a transition between a paramagnetic metal and an antiferromagnetic insulator.
The antiferromagnetic phase of two-dimensional (2D) and three-dimensional (3D) Hubbard model with nearest neighbor hopping is studied on a bipartite cubic lattice by means of the quantum $\mathrm{SU}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{U}(1)$ rotor approach that yields a fully self-consistent treatment of the antiferromagnetic state that respects the symmetry properties of the model and satisfy the Mermin-Wagner theorem. The collective variables for charge and spin are isolated in the form of the space-time fluctuating U(1) phase field and rotating spin-quantization axis governed by the SU(2) symmetry, respectively. As a result interacting electrons appear as composite objects consisting of bare fermions with attached U(1) and SU(2) gauge fields. An effective action consisting of a spin-charge rotor and a fermionic field is derived as a function of the Coulomb repulsion UUU and hopping parameter ttt. At zero temperature, our theory describes the evolution from a Slater (U⪡t)(U⪡t)(U⪡t) to a Mott-Heisenberg (U⪢t)(U⪢t)(U⪢t) antiferromagnet. The results for zero-temperature sublatice magnetization (2D) and finite temperature (3D) phase diagram of the antiferromagnetic Hubbard model as a function of the crossover parameter U∕tU∕tU∕t are presented and the role of the spin Berry phase in the interaction driven crossover is analyzed.
We study the antiferromagnetic phase of three-dimensional Hubbard model with nearest neighbors hopping on a bipartite cubic lattice.We use the quantum SU(2)×U(1) rotor approach that yields a fully self-consistent treatment of the antiferromagnetic state that respects the symmetry properties of the model and satisfies the Mermin-Wagner theorem.As our theory describes the evolution from a Slater (U t) to a Mott-Heisenberg (U t) antiferromagnet, we present the phase diagram of the antiferromagnetic Hubbard model as a function of the crossover parameter U/t.
We study antiferromagnetism and single-particle properties in the two-dimensional half-filled Hubbard model at low temperature. Collective spin fluctuations are governed by a non-linear sigma model that we derive from the Hubbard model for any value of the Coulomb repulsion. As the Coulomb repulsion increases, the ground state progressively evolves from a Slater to a Mott-Heisenberg antiferromagnet. At finite temperature, we find a metal-insulator transition between a pseudogap phase at weak coupling and a Mott-Hubbard insulator at strong coupling.