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Physical review. B, Condensed matter

Anomalous paramagnetic phase of the Hubbard model

Takuya Okabe

Published Jan 15, 1999• 0 citations
Publisher

Abstract

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 ${T}_{\mathrm{N}}$ as a function of doping concentration x to find that it vanishes at $x\ensuremath{\simeq}0.05$ for $U/6t=4$ in a simple cubic lattice.

DOI
10.1103/physrevb.59.2564

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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=TNRPA\ensuremath∼U,{T=T}_{\mathrm{N}}^{\mathrm{RPA}}\ensuremath{\sim}U,T=TNRPA​\ensuremath∼U, we find a solution representing the paramagnetic phase of the Mott insulator, for which we obtain the N'eel temperature TN\ensuremath∼J{T}_{\mathrm{N}}\ensuremath{\sim}JTN​\ensuremath∼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≃0.05x\ensuremath{\simeq}0.05x\ensuremath≃0.05 for U/6t=4U/6t=4U/6t=4 in a simple cubic lattice.
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 $\ensuremath{\delta}>0$.
Antiferromagnetic and metal to insulator transitions in the ground state of two-dimensional half-filled and nearly half-filled Hubbard models with the nearest neighbor and the second nearest neighbor transfers (- t and t ′ , respectively) are studied by using the Green's function approach with the self-energy up to the second order in the coupling constant U . For a choice of the parameter value t ′ / t =0.2 the half-filled model undergoes the first order phase transition between paramagnetic metallic (PM) phase and antiferromagnetic insulator (AFI) phase at U / t =4.0 while the first order transition between PM and antiferromagnetic metallic (AFM) phases takes place for nearly half-filled band. The one-electron spectral density is calculated for the PM, AFM, and AFI phases near the phase boundaries. In the PM phase the quasi-particle dispersion indicates the band narrowing from the Hartree-Fock results and the flat band behavior as was indicated by previous stadies and was observed in high- T c cuprates. The quasi-particle dispersion in the AFM phase is seen to be close to the Hartree-Fock results.
We study the metal-insulator transition and magnetic ordering in the Hubbard model using the particle-hole mapping. The analysis simplifies near the ferromagnetic limit. We find that the two-dimensional Hubbard model has a charge excitation gap at half filling for any finite U in this region on both the bipartite square lattice and the nonbipartite triangular lattice. In some cases, the system goes through a first-order phase transition to become a paramagnetic metal as Sz{\mathit{S}}_{\mathit{z}}Sz​ is lowered. We also discuss the extension to the doped case. We find that in the large U limit, a single doped hole has a bandwidth of order of J rather than t at Sz{\mathit{S}}_{\mathit{z}}Sz​=0.
We present results of a systematic quantum Monte Carlo study for the single-band Hubbard model. Thereby we evaluated single-particle spectra (PES and IPES), two-particle spectra (spin and density correlation functions), and the dynamical correlation function of suitably defined diagnostic operators, all as a function of temperature and hole doping. The results allow us to identify different physical regimes. Near half-filling we find an anomalous ``Hubbard-I phase,'' where the band structure is, up to some minor modifications, consistent with the Hubbard-I predictions. At lower temperatures, where the spin response becomes sharp, additional dispersionless ``bands'' emerge due to the dressing of electrons/holes with spin excitations. We present a simple phenomenological fit that reproduces the band structure of the insulator quantitatively. The Fermi surface volume in the low-doping phase, as derived from the single-particle spectral function, is not consistent with the Luttinger theorem, but qualitatively in agreement with the predictions of the Hubbard-I approximation. The anomalous phase extends up to a hole concentration of $\ensuremath{\approx}15%,$ i.e., the underdoped region in the phase diagram of high-Tc{T}_{c}Tc​ superconductors. We also investigate the nature of the magnetic ordering transition in the single-particle spectra. We show that the transition to a spin-density wave-like band structure is not accomplished by the formation of any resolvable ``precursor bands,'' but rather by a (spectroscopically invisible) band of spin-3/2 quasiparticles. We discuss implications for the ``remnant Fermi surface'' in insulating cuprate compounds and the shadow bands in the doped materials.
We investigate the phase diagram of the three-dimensional Hubbard model at half filling using quantum Monte Carlo (QMC) simulations. The antiferromagnetic Neel temperature T_N is determined from the specific heat maximum in combination with finite-size scaling of the magnetic structure factor. Our results interpolate smoothly between the asymptotic solutions for weak and strong coupling, respectively, in contrast to previous QMC simulations. The location of the metal-insulator transition in the paramagnetic phase above T_N is determined using the electronic compressibility as criterion.