- Email: [email protected]
Magnetization curve and magnetic susceptibility of the d = ∞ Hubbard model on the Bethe lattice
Journal of Magnetism and Magnetic Materials 140-144 (1995) 1201-1202
~ H journalof magnetism ~ H and magnetic ~ H materials
ELSEVIER
Magnetization curve and magnetic susceptibility of the d = Hubbard model on the Bethe lattice Tetsuro Saso a,* Takumi Hayashi
b
a Departmentof Physics, Saitama University, Urawa338, Japan b Yamatake-HoneywellCo. Ltd, Fujisawa 251, Japan Abstract To investigate the magnetic properties of strongly correlated electron systems, the magnetization curve and the magnetic susceptibility of the infinite-dimensional Hubbard model are calculated using the quantum Monte Carlo method for solving the single-impurity part. For the Bethe lattice, it is found that the magnetization curve is similar to that obtained by the Gutzwiller approximation: the curve rises steeply towards the saturation value near the metal-insulator (MI) transition, in contrast with the result obtained by Georges and Krauth for the simple hypercubic lattice. The susceptibility is consistent with X ~ (Uc - U)-1 except near the MI transition, and rather close to the Gutzwiller result.
Strongly interacting electrons show a variety of interesting magnetic properties [1,2]. Recent progress in the study of the correlated lattice models in the limit of high spatial dimension [3] has opened a new possibility for understanding such systems. In the limit of infinite spatial dimension, d = at, of the Hubbard model,
H= -tEc+~c/~ + UEnir ijo"
hi+ ,
(1)
i
the self-energy becomes wavenumber-independent, so that the lattice problem is reduced to that of an impurity embedded in an effective medium, which must be determined self-consistently. In the case of the Bethe lattice, the self-consistent equations become rather simple [4,5]: the first one is G~(iEn)=[iEn-(wZ/4)G~(iE,)] -1, where (~,~(iEn) denotes the local Green functions (GF) for the effective medium and G,~(ie,) that of the whole system. The second equation is given by the quantum Monte Carlo (QMC) procedure [6] to solve the full Green function G,,(ie,) from (~tr as the unperturbed local GF: (~,(ie) G~,(e). We repeat these two steps until self-consistency is achieved. The density of states of the Bethe lattice is given by p(e) = (2/-rrW)[1 - ( e / W ) 2 ] 1/2, where W = 2~/-dt. We show the results for the case of half-filling in the following. Antiferromagnetic instability takes place for U > 0, but we assume that it is suppressed by certain modification of the hopping and that the system remains paramagnetic
[7]. We set W = 1, the temperature T = 1 / 3 2 and the Trotter size L = 64 in the QMC; hence, AT = 1 / L T = 0.5. We use 20000 sweeps for each single-impurity QMC calculation. At the present temperature, the MI transition takes place at U ~- 2.6 -= Uc. At lower temperatures, however, a sharp Kondo peak remains at the Fermi energy [5] and it is expected that Uc will become slightly larger (Uc = 3 has been reported in Refs. [8,9]). We first compared the values of (m 2) (m = n r - n ~ ) at H = 0 with the perturbational calculation up to o(U): ( m 2 ) = ~1 +
"^v~
' . . _ . ~
'
i 1
i 2
u
i 3
i 4
Fig. 1. Values of (m 2 ). The dotted line indicates perturbational result (m 2) = 1/2+0.167U.
0304-8853/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved
SSDI 0304-8853(94)00654-7
(2)
where A ( A ) = f0Wdep(E)e a* for d ~ ~. As shown in Fig. 1, our calculation agrees with this expression well for small U, while the calculation for the same quantity by Zhang et al. [9] gives a smaller value.
o..
* Corresponding author. Fax: +81-48-858-3698; email: saso @th.phy.saitama-u.ac.jp.
4U /o~d/~A4(/~) --{-o ( U 2 ) ,
T. Saso, T. Hayashi/Journal of Magnetism and Magnetic Materials 140-144 (1995) 1201-1202
1202
I'""
0., v
0
I
i
i
L
I 0.5
i
i
I
i
h
x0
Fig. 2. Magnetization curves for U = 0 (©), 1 (z~) and 2 (D). The solid lines indicates the spline interpolation of the data points. The dotted line near the U = 0 curve indicates the exact one.
The results of the magnetization curve are shown in Fig. 2 The dotted line indicates the exact curve for U = 0. The initial magnetization curve becomes steeper when U increases from 0, and the curve at U = 2 shows slight upturn, indicating a first-order transition. These behaviors agree qualitatively with the variational calculation within the Gutzwiller approximation [10], but differ from those in Ref. [11] for the hypercubic lattice, where the magnetization curve increases only smoothly.
i,,,
~
,,,,
,,,,i,fll
o.5
0 0
ll~llllIJlll'll"~d' t 2
u
If there is no coupling between the local magnetic moments, the susceptibility g would diverge at the MI transition point, since the width T K of the Kondo peak at E F becomes infinitely small [5,9], and X ~ TK 1. But even in the limit d ~ w, the magnetic coupling remains finite since the exchange J ~ ~ j ( t / ~ / d ) 2 / U ~ t 2 / U , so that x ~ ( T K + J ) -1 does not diverge [7]. Except near the transition point, however, we find that X-~, calculated from the initial slopes of the magnetization curves in Fig. 2, is rather close to the Gutzwiller variational calculation [10]:
3
Fig. 3. Inverse magnetic susceptibility (@) compared with the Gutzwiller calculation (dotted line).
= [1 - ~ u / u c ) 2]
XGw
- p ~ 0 ) u {1 + (U/2Uc) } {1 + (U/Uc)} ~ (3)
(Uc = 3 2 / 3 w = 3.40) quantitatively (Fig. 3). Acknowledgements: The present work was supported by the Grants-in-Aid for Scientific Research on Priority Areas, 'Computational Physics as a New Frontier in Condensed Matter Research' and 'Science of High-Temperature Superconductivity' from the Ministry of Education, Science and Culture of Japan.
References [1] Y. Tokura, Y. Taguchi, Y. Okada, Y. Fujimori, T. Arima, K. Kumagai and Y. lye, Phys. Rev. Lett. 70 (1993) 2126. [2] J.M. Mignot, J. Flouquet, P. Haen, F. Lapierre, L. Puech and J. Voiron, J. Mag. Magn. Mater. 76 & 77 (1988) 97. [3] W. Metzner and D. Vollhardt, Phys. Rev. Lett. 62 (1989) 324. [4] A. Georges and W. Krauth, Phys. Rev. Lett. 69 (1992) 1240. [5] M. Jarrell, Phys. Rev. Lett. 69 (1992) 168. [6] J.E. Hirsch and R.M. Fye, Phys. Rev. Lett. 56 (1986) 2521. [7] M.J. Rozenberg, G. Kotliar and X.Y. Zhang, preprint (1993). [8] M.J. Rozenberg, X.Y. Zhang and G. Kotliar, Phys. Rev. Lett. 69 (1992) 1236. [9] X.Y. Zhang, MJ. Rozenberg and G. Kotliar, Phys. Rev. Lett. 70 (1993) 1666. [10] D. Vollhardt, Rev. Mod. Phys. 56 (1984) 99.
~ H journalof magnetism ~ H and magnetic ~ H materials
ELSEVIER
Magnetization curve and magnetic susceptibility of the d = Hubbard model on the Bethe lattice Tetsuro Saso a,* Takumi Hayashi
b
a Departmentof Physics, Saitama University, Urawa338, Japan b Yamatake-HoneywellCo. Ltd, Fujisawa 251, Japan Abstract To investigate the magnetic properties of strongly correlated electron systems, the magnetization curve and the magnetic susceptibility of the infinite-dimensional Hubbard model are calculated using the quantum Monte Carlo method for solving the single-impurity part. For the Bethe lattice, it is found that the magnetization curve is similar to that obtained by the Gutzwiller approximation: the curve rises steeply towards the saturation value near the metal-insulator (MI) transition, in contrast with the result obtained by Georges and Krauth for the simple hypercubic lattice. The susceptibility is consistent with X ~ (Uc - U)-1 except near the MI transition, and rather close to the Gutzwiller result.
Strongly interacting electrons show a variety of interesting magnetic properties [1,2]. Recent progress in the study of the correlated lattice models in the limit of high spatial dimension [3] has opened a new possibility for understanding such systems. In the limit of infinite spatial dimension, d = at, of the Hubbard model,
H= -tEc+~c/~ + UEnir ijo"
hi+ ,
(1)
i
the self-energy becomes wavenumber-independent, so that the lattice problem is reduced to that of an impurity embedded in an effective medium, which must be determined self-consistently. In the case of the Bethe lattice, the self-consistent equations become rather simple [4,5]: the first one is G~(iEn)=[iEn-(wZ/4)G~(iE,)] -1, where (~,~(iEn) denotes the local Green functions (GF) for the effective medium and G,~(ie,) that of the whole system. The second equation is given by the quantum Monte Carlo (QMC) procedure [6] to solve the full Green function G,,(ie,) from (~tr as the unperturbed local GF: (~,(ie) G~,(e). We repeat these two steps until self-consistency is achieved. The density of states of the Bethe lattice is given by p(e) = (2/-rrW)[1 - ( e / W ) 2 ] 1/2, where W = 2~/-dt. We show the results for the case of half-filling in the following. Antiferromagnetic instability takes place for U > 0, but we assume that it is suppressed by certain modification of the hopping and that the system remains paramagnetic
[7]. We set W = 1, the temperature T = 1 / 3 2 and the Trotter size L = 64 in the QMC; hence, AT = 1 / L T = 0.5. We use 20000 sweeps for each single-impurity QMC calculation. At the present temperature, the MI transition takes place at U ~- 2.6 -= Uc. At lower temperatures, however, a sharp Kondo peak remains at the Fermi energy [5] and it is expected that Uc will become slightly larger (Uc = 3 has been reported in Refs. [8,9]). We first compared the values of (m 2) (m = n r - n ~ ) at H = 0 with the perturbational calculation up to o(U): ( m 2 ) = ~1 +
"^v~
' . . _ . ~
'
i 1
i 2
u
i 3
i 4
Fig. 1. Values of (m 2 ). The dotted line indicates perturbational result (m 2) = 1/2+0.167U.
0304-8853/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved
SSDI 0304-8853(94)00654-7
(2)
where A ( A ) = f0Wdep(E)e a* for d ~ ~. As shown in Fig. 1, our calculation agrees with this expression well for small U, while the calculation for the same quantity by Zhang et al. [9] gives a smaller value.
o..
* Corresponding author. Fax: +81-48-858-3698; email: saso @th.phy.saitama-u.ac.jp.
4U /o~d/~A4(/~) --{-o ( U 2 ) ,
T. Saso, T. Hayashi/Journal of Magnetism and Magnetic Materials 140-144 (1995) 1201-1202
1202
I'""
0., v
0
I
i
i
L
I 0.5
i
i
I
i
h
x0
Fig. 2. Magnetization curves for U = 0 (©), 1 (z~) and 2 (D). The solid lines indicates the spline interpolation of the data points. The dotted line near the U = 0 curve indicates the exact one.
The results of the magnetization curve are shown in Fig. 2 The dotted line indicates the exact curve for U = 0. The initial magnetization curve becomes steeper when U increases from 0, and the curve at U = 2 shows slight upturn, indicating a first-order transition. These behaviors agree qualitatively with the variational calculation within the Gutzwiller approximation [10], but differ from those in Ref. [11] for the hypercubic lattice, where the magnetization curve increases only smoothly.
i,,,
~
,,,,
,,,,i,fll
o.5
0 0
ll~llllIJlll'll"~d' t 2
u
If there is no coupling between the local magnetic moments, the susceptibility g would diverge at the MI transition point, since the width T K of the Kondo peak at E F becomes infinitely small [5,9], and X ~ TK 1. But even in the limit d ~ w, the magnetic coupling remains finite since the exchange J ~ ~ j ( t / ~ / d ) 2 / U ~ t 2 / U , so that x ~ ( T K + J ) -1 does not diverge [7]. Except near the transition point, however, we find that X-~, calculated from the initial slopes of the magnetization curves in Fig. 2, is rather close to the Gutzwiller variational calculation [10]:
3
Fig. 3. Inverse magnetic susceptibility (@) compared with the Gutzwiller calculation (dotted line).
= [1 - ~ u / u c ) 2]
XGw
- p ~ 0 ) u {1 + (U/2Uc) } {1 + (U/Uc)} ~ (3)
(Uc = 3 2 / 3 w = 3.40) quantitatively (Fig. 3). Acknowledgements: The present work was supported by the Grants-in-Aid for Scientific Research on Priority Areas, 'Computational Physics as a New Frontier in Condensed Matter Research' and 'Science of High-Temperature Superconductivity' from the Ministry of Education, Science and Culture of Japan.
References [1] Y. Tokura, Y. Taguchi, Y. Okada, Y. Fujimori, T. Arima, K. Kumagai and Y. lye, Phys. Rev. Lett. 70 (1993) 2126. [2] J.M. Mignot, J. Flouquet, P. Haen, F. Lapierre, L. Puech and J. Voiron, J. Mag. Magn. Mater. 76 & 77 (1988) 97. [3] W. Metzner and D. Vollhardt, Phys. Rev. Lett. 62 (1989) 324. [4] A. Georges and W. Krauth, Phys. Rev. Lett. 69 (1992) 1240. [5] M. Jarrell, Phys. Rev. Lett. 69 (1992) 168. [6] J.E. Hirsch and R.M. Fye, Phys. Rev. Lett. 56 (1986) 2521. [7] M.J. Rozenberg, G. Kotliar and X.Y. Zhang, preprint (1993). [8] M.J. Rozenberg, X.Y. Zhang and G. Kotliar, Phys. Rev. Lett. 69 (1992) 1236. [9] X.Y. Zhang, MJ. Rozenberg and G. Kotliar, Phys. Rev. Lett. 70 (1993) 1666. [10] D. Vollhardt, Rev. Mod. Phys. 56 (1984) 99.