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Magnetic properties of the three-band Hubbard model
Physica B 259—261 (1999) 747—748
Magnetic properties of the three-band Hubbard model Th. Maier*, M.B. Zoelfl, Th. Pruschke, J. Keller Institut fu( r Theoretische Physik, Universita( t Regensburg, 93040 Regensburg, Germany
Abstract The magnetic properties of the three-band Hubbard model are calculated within the dynamical mean-field theory. The static spin-susceptibilities are discussed for different dopings d. The d—¹-phase-diagram exhibits an enhanced stability of the antiferromagnetic state for electron-doping in comparison with hole-doping. Deep in the antiferromagnetic phase multipeak-structures in the spectral functions are observed which can be related to bound states for one hole in the Nee´l-background. 1999 Elsevier Science B.V. All rights reserved. Keywords: Three-band Hubbard model; Dynamical mean-field theory; Magnetic ordering
The strong asymmetry in the magnetic phase diagram of high-¹ materials concerning the enhanced stability of the antiferromagnetic state for electron-doping in comparison with hole-doping [1] is a direct consequence of the oxygen degrees of freedom in the CuO -planes [2]. The simplest model which includes the oxygen sites and the effect of strong correlations at the Cu-sites is the three-band Hubbard model,
ing a resolvent technique [5]. To allow for solutions with broken spin-symmetry we extended this approach to lattices with AB-like structure. Fig. 1a shows the d-part of the spectrum for majorityand minority-spin. The part at u(0 consists of the lower Hubbard band which has mainly d-character. Right above the gap is the Zhang—Rice band which is generated by the singlet combination of the d- and
H" e dR d # e pR p B GN GN N HN HN G N H N # t (dR p #h.c.)# º nB nB , (1) GH GN HN B Gj Gi 6GH7 N G where the fermion operators d and p refer to Cu and N N O hole states. In this paper this model is analyzed within the dynamical mean-field theory (DMFT) [3], where the effective impurity is chosen as a cluster consisting of one Cu d-orbital and the orthonormalized combination of the four surrounding O p-orbitals, in order to treat local spin and charge correlations better than on a mean-field level [4]. The local states of this cluster which are coupled to a surrounding medium are calculated selfconsistently us-
* Corresponding author. Tel.: #49-941-943-2034; fax: #49941-943-4382; e-mail: [email protected].
Fig. 1. (a) d-Spectral function for the majority (solid lines) and minority (dashed lines) spin for º "2(e !e )"7t, d"!0.05, b"22/t. (b) Comparison of the low energy part with exact results (vertical lines).
0921-4526/99/$ — see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 8 ) 0 1 0 2 1 - 7
748
Th. Maier et al. / Physica B 259—261 (1999) 747—748
Fig. 2. Magnetic phase diagram of the three-band Hubbard model for º "2(e !e )"7t.
p-states on one plaquette. The nature of the low-energy bands has already been pointed out in Refs. [4,6]. The transfer of spectral weight between the different spin bands yields a finite sublattice magnetization. In addition a pronounced multipeak structure can be observed. These regular resonances can be related to bound states of a single hole in the Nee´l background. This special problem can be solved exactly for d"R and ¹"0 within the t—J-model [7]. Fig. 1b shows a comparison of the low-energy part of the spectrum with these exact results (vertical lines). We find quite a good agreement with the distance of the peak positions. The broadening results from finite temperature, sublattice magnetization and doping effects.
Furthermore, we calculated the static homogeneous (q"0) and staggered (q"(n, n, 2)) magnetic susceptibility for the d-spins. For the chosen parameters we find a finite and slowly varying ferromagnetic susceptibility, which does not show any tendency towards an instability, whereas the antiferromagnetic susceptibility diverges at a finite temperature ¹"¹ . The d—¹-phase diagram , in Fig. 2 is obtained by calculating the inverse staggered susceptibility for different dopings d. Note that the metal—insulator transition does not occur precisely at d"0 because of the missing particle hole symmetry at half filling and the used values for the parameters in the three-band Hubbard model. The stronger sensistivity of the antiferromagnetic phase in the case of hole-doping is qualitatively in agreement with experiments. To conclude, we have shown that the three-band Hubbard model exhibits the same low energy physics as the t—J model in the antiferromagnetic state, and the inclusion of the oxygen degrees of freedom leads to the asymmetry of the magnetic d—¹-phase diagram. References [1] C. Almasan, M.B. Maple, in: C.M. Rao (Ed.), Chemistry of High-Temperature Superconductors, World Scientific, Singapore, 1991. [2] A. Aharony et al., Phys. Rev. Lett. 60 (1988) 1330. [3] E. Mu¨ller-Hartmann, Z. Phys. 74 (1989) 507. [4] P. Lombardo, M. Avignon, J. Schmalian, K.H. Bennemann, Phys. Rev. B 54 (1996) 5317. [5] H. Keiter, J.C. Kimball, Phys. Rev. Lett. 25 (1970) 672. [6] L.F. Feiner, Phys. Rev. B 48 (1993) 16 857. [7] R. Strack, D. Vollhardt, Phys. Rev. B 46 (1992) 13 852.
Magnetic properties of the three-band Hubbard model Th. Maier*, M.B. Zoelfl, Th. Pruschke, J. Keller Institut fu( r Theoretische Physik, Universita( t Regensburg, 93040 Regensburg, Germany
Abstract The magnetic properties of the three-band Hubbard model are calculated within the dynamical mean-field theory. The static spin-susceptibilities are discussed for different dopings d. The d—¹-phase-diagram exhibits an enhanced stability of the antiferromagnetic state for electron-doping in comparison with hole-doping. Deep in the antiferromagnetic phase multipeak-structures in the spectral functions are observed which can be related to bound states for one hole in the Nee´l-background. 1999 Elsevier Science B.V. All rights reserved. Keywords: Three-band Hubbard model; Dynamical mean-field theory; Magnetic ordering
The strong asymmetry in the magnetic phase diagram of high-¹ materials concerning the enhanced stability of the antiferromagnetic state for electron-doping in comparison with hole-doping [1] is a direct consequence of the oxygen degrees of freedom in the CuO -planes [2]. The simplest model which includes the oxygen sites and the effect of strong correlations at the Cu-sites is the three-band Hubbard model,
ing a resolvent technique [5]. To allow for solutions with broken spin-symmetry we extended this approach to lattices with AB-like structure. Fig. 1a shows the d-part of the spectrum for majorityand minority-spin. The part at u(0 consists of the lower Hubbard band which has mainly d-character. Right above the gap is the Zhang—Rice band which is generated by the singlet combination of the d- and
H" e dR d # e pR p B GN GN N HN HN G N H N # t (dR p #h.c.)# º nB nB , (1) GH GN HN B Gj Gi 6GH7 N G where the fermion operators d and p refer to Cu and N N O hole states. In this paper this model is analyzed within the dynamical mean-field theory (DMFT) [3], where the effective impurity is chosen as a cluster consisting of one Cu d-orbital and the orthonormalized combination of the four surrounding O p-orbitals, in order to treat local spin and charge correlations better than on a mean-field level [4]. The local states of this cluster which are coupled to a surrounding medium are calculated selfconsistently us-
* Corresponding author. Tel.: #49-941-943-2034; fax: #49941-943-4382; e-mail: [email protected].
Fig. 1. (a) d-Spectral function for the majority (solid lines) and minority (dashed lines) spin for º "2(e !e )"7t, d"!0.05, b"22/t. (b) Comparison of the low energy part with exact results (vertical lines).
0921-4526/99/$ — see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 8 ) 0 1 0 2 1 - 7
748
Th. Maier et al. / Physica B 259—261 (1999) 747—748
Fig. 2. Magnetic phase diagram of the three-band Hubbard model for º "2(e !e )"7t.
p-states on one plaquette. The nature of the low-energy bands has already been pointed out in Refs. [4,6]. The transfer of spectral weight between the different spin bands yields a finite sublattice magnetization. In addition a pronounced multipeak structure can be observed. These regular resonances can be related to bound states of a single hole in the Nee´l background. This special problem can be solved exactly for d"R and ¹"0 within the t—J-model [7]. Fig. 1b shows a comparison of the low-energy part of the spectrum with these exact results (vertical lines). We find quite a good agreement with the distance of the peak positions. The broadening results from finite temperature, sublattice magnetization and doping effects.
Furthermore, we calculated the static homogeneous (q"0) and staggered (q"(n, n, 2)) magnetic susceptibility for the d-spins. For the chosen parameters we find a finite and slowly varying ferromagnetic susceptibility, which does not show any tendency towards an instability, whereas the antiferromagnetic susceptibility diverges at a finite temperature ¹"¹ . The d—¹-phase diagram , in Fig. 2 is obtained by calculating the inverse staggered susceptibility for different dopings d. Note that the metal—insulator transition does not occur precisely at d"0 because of the missing particle hole symmetry at half filling and the used values for the parameters in the three-band Hubbard model. The stronger sensistivity of the antiferromagnetic phase in the case of hole-doping is qualitatively in agreement with experiments. To conclude, we have shown that the three-band Hubbard model exhibits the same low energy physics as the t—J model in the antiferromagnetic state, and the inclusion of the oxygen degrees of freedom leads to the asymmetry of the magnetic d—¹-phase diagram. References [1] C. Almasan, M.B. Maple, in: C.M. Rao (Ed.), Chemistry of High-Temperature Superconductors, World Scientific, Singapore, 1991. [2] A. Aharony et al., Phys. Rev. Lett. 60 (1988) 1330. [3] E. Mu¨ller-Hartmann, Z. Phys. 74 (1989) 507. [4] P. Lombardo, M. Avignon, J. Schmalian, K.H. Bennemann, Phys. Rev. B 54 (1996) 5317. [5] H. Keiter, J.C. Kimball, Phys. Rev. Lett. 25 (1970) 672. [6] L.F. Feiner, Phys. Rev. B 48 (1993) 16 857. [7] R. Strack, D. Vollhardt, Phys. Rev. B 46 (1992) 13 852.