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Ground state magnetic properties of the symmetric periodic Anderson model with finite U
PHYSICA ELSEVIER
Physica B 199&200 (1994) 222-223
,
Ground state magnetic properties of the symmetric periodic Anderson model with finite U T.M. Hong*, S.J. Sun, M.F. Yang Department of Physics. National Tsing Hua University. Hsinchu 30043. Taiwan
Abstract For the symmetric periodic Anderson model at half-filling, we emplloy the multiple-slave-boson technique, suitable for finite local Coulomb repulsion, at the mean-field level and build in the antiferromagnetism by dividing the lattice into two sublattices. For the one-dimensional case, antiferromagnetism appears for large U. In infinite dimension~, the Vdependence of the N6el temperature is tongue-like, and antiferromagnetism is stable only when Vis below a critical value. The magnetism appears to be local-moment-like for small V, but itinerant close to the critical I,:
One main weakness of the slave-boson mean-field theory [1] is its incapability to predict the magnetic properties. In addition to calculating perturbations up to the second order, there is another way to obtain magnetism at the mean-field level, i.e. firstly, assume the ground state to be magnetic, knowing that the internal field is related to the ordered moment, and then determine the size of the moment self-consistently via the mean-field equations. The latter approach has been applied to the Kondo lattice [2] model and the infinite..U periodic Anderson model [3] (PAM). In the second example, antiferromagnetism is found to be stable since total energy is lowered by a new energy gap from lifting of the degeneracy at the reduced Brillouin-zone boundary. Heavy-fermion compounds, such as the high-temperature superconductors, belong to the category of strongly correlated many-electron systems. The d- or f-orbital electrons involved experience a large or, site Coulomb repulsion U when doubly occupied. In contrast to the usual slave-boson approach [1] for infinite U, there are speculations [4] that a [inite, although large, U might be crucial to obtain the antiferromagnetism.
We will employ here a special kind of slave-boson mean-field approximation (SBMFJ for finite U, which allows for the symmetry-broken states appTcp~iate for a bipartitie lattice. For convenience, we concentrate., on the symmetric PAM (i.e., the bare f-electron one-body energy equal to - U/2) at half-filling to make possible connections with the newly discovered Kondo insulators. Following the slave-boson formalism introduced by Kotliar and Ruckenstein [5], the partition function for the PAM can be formulated in a functional-integral form with four types of bosons on each site. These bosons act as projectors on empty, singly occupied by one f-electron of either spin, and doubly occupied gtate~; . . . . . .
| Tnder
th¢~
. . . . . . . . . . . . .
,:t.~t~r . ~ , * ~
,~ ~, 1., r ~I Jr. r ,V ~ Hv;,.r,.,*;,-,,~ H~tt'UZt,
,111
t..lU
ku ~v oo ~v
a
I
fields are treated as numbers. In order to discuss antiferromagnetism, we divide the lattice into two sublattices [3]. For the symmetric PAM at half-filling, the effective hybridization is shown [6] to ren.orma!ize by a factor -: . _ v/6(l - rnf - 26)+ v'6(l + m f - 26)
* Corresponding author. 0921-4526/94/$07.00 'c) 1994 Elsevier Science B.V. All rights reserved SSDI 0921-4526(93}EO262-F
(l)
T.M. Hong et al. / Physica B 199&200 ( 1 9 ~ ) 222-223
dominates, in contrast to being itinerant for V close to |,', when K o n d o screening finally sets in to kill the magnetism at V~ (this is the picture by Doniach [1 I]). As in the Kondo impurity problem, the dotted line in Fig. i marks the.h~lse phase boundary for a nonzero 6 (on the right side). Again similar to the impurity problem, the effective hybridization equals zero when 6 = 0 because of Eq. I1), and therefore the conduction and f-electrons become decoupled. It is worth noting that this dotted line seems to overlap with the magnetic phase boundary in region I.
I
0.08
I
s /--
U=2.0
0.08
/
/
,
°.,
°.,
o.,
223
i
0.6
O.e
|
O.7
V Fig. 1. Solid line shows the V-dependence of the N~I temperature. Dotted line is the boundary for nonzero effective hybridization {on lhe right).
where 6 denotes the probability of the double occupancy and m r is the magnetic moment of the f-electron. Final form of the free energy and the four new energy bands depends on three parameters 6, m r, and the internal field h, which are determined by minimizing the free energy. For the one-dimensiom "ase [6], our results agree with those by the Quantm., Monte Carlo [7] (QMC) simulation and the Gutzwiller-correlated spin-densitywave function [8]. Anti-ferromagnetic states appear for large U/t (2t being the bare conduction bandwidth), which is expected to become short ranged by including the fluctuations of the boson fields. In infinite dimensions [9], antiferromagnetism is found to be stable for V < V~, and the value of I.~, falls within the range set by Q M C [10]. As a function of increasing V, m f decreases monotonically from ",he full moment, while m ~, of a different sign, increases from zero initially (region I) and then decreases to null at V~ (region II). Further analysis on the tongue-like !," dependence of the N6el temperature, as shown in Fig. i, leads us to believe that there are two different mechanisms for the magnetism: local-moment like for V ~. 0 where the RKKY effect
T,M.H. acknowledges the support by NSC-82-0208M007-164 and 149. SJ,S. and M.F.Y. are supported by NSC 82-0511-M-007-140.
References
[i] P. Coleman, Phys. Rev. B 29 (1984) 3035; J. Rasul and H.U. Desgranges, J. Phys. C 19 119861 L671. [2] V. Yu. lrkhin and MI. Katsnelson, J. Phys. C 2 {1990j 8715. [3] T.M. Hong and G.A. Gehring, J. Magn. Magn. Mater. 108 11992) 93. [4] Ted Hsu, private communication. This approach often has trouble with tile antiferromagnetic order, e.g. in: I.K. Affleck and J.B. Marstom Phys. Rev. B 37 (198813774 for the Hubbard model. These authors found a staggered flux phase but could not treat antiferromagnetism in their formalism. [5] G. Kotliar and A.E Ruckenstein, Phys. Re~. LctL 57 (1986l 1362. [6] M.F. Yang, S.J. Sun and T.M Hong. to be published in Phys. Re~. B. [7] R. Blankerbecler. J.R. Fulco, W. Gill anti D,J. Scalapino, Phys. Rev. [.ell. 58 (1987)411. [8] Zs. Gulacsi, R. Strack and O. Voilhardt. Phys. Roy. B 47 (1993) 8594. [9] S.J. Sun, M.F. Yang and T.M. Hong, to be published in Phys. Rev. B. [10] M. Jarrell, H. Akhlaghgour and Th. Pruschke, Phys. Re';. Lett. 70 (19931 1670. [1 !] S. Doniach, Physica P. 91 (19771 231: in: Valence Instabilities and Related NaNow Band Phenomena, ed. R.D. Park (Plenum, New York. 1977) p. 16o.
Physica B 199&200 (1994) 222-223
,
Ground state magnetic properties of the symmetric periodic Anderson model with finite U T.M. Hong*, S.J. Sun, M.F. Yang Department of Physics. National Tsing Hua University. Hsinchu 30043. Taiwan
Abstract For the symmetric periodic Anderson model at half-filling, we emplloy the multiple-slave-boson technique, suitable for finite local Coulomb repulsion, at the mean-field level and build in the antiferromagnetism by dividing the lattice into two sublattices. For the one-dimensional case, antiferromagnetism appears for large U. In infinite dimension~, the Vdependence of the N6el temperature is tongue-like, and antiferromagnetism is stable only when Vis below a critical value. The magnetism appears to be local-moment-like for small V, but itinerant close to the critical I,:
One main weakness of the slave-boson mean-field theory [1] is its incapability to predict the magnetic properties. In addition to calculating perturbations up to the second order, there is another way to obtain magnetism at the mean-field level, i.e. firstly, assume the ground state to be magnetic, knowing that the internal field is related to the ordered moment, and then determine the size of the moment self-consistently via the mean-field equations. The latter approach has been applied to the Kondo lattice [2] model and the infinite..U periodic Anderson model [3] (PAM). In the second example, antiferromagnetism is found to be stable since total energy is lowered by a new energy gap from lifting of the degeneracy at the reduced Brillouin-zone boundary. Heavy-fermion compounds, such as the high-temperature superconductors, belong to the category of strongly correlated many-electron systems. The d- or f-orbital electrons involved experience a large or, site Coulomb repulsion U when doubly occupied. In contrast to the usual slave-boson approach [1] for infinite U, there are speculations [4] that a [inite, although large, U might be crucial to obtain the antiferromagnetism.
We will employ here a special kind of slave-boson mean-field approximation (SBMFJ for finite U, which allows for the symmetry-broken states appTcp~iate for a bipartitie lattice. For convenience, we concentrate., on the symmetric PAM (i.e., the bare f-electron one-body energy equal to - U/2) at half-filling to make possible connections with the newly discovered Kondo insulators. Following the slave-boson formalism introduced by Kotliar and Ruckenstein [5], the partition function for the PAM can be formulated in a functional-integral form with four types of bosons on each site. These bosons act as projectors on empty, singly occupied by one f-electron of either spin, and doubly occupied gtate~; . . . . . .
| Tnder
th¢~
. . . . . . . . . . . . .
,:t.~t~r . ~ , * ~
,~ ~, 1., r ~I Jr. r ,V ~ Hv;,.r,.,*;,-,,~ H~tt'UZt,
,111
t..lU
ku ~v oo ~v
a
I
fields are treated as numbers. In order to discuss antiferromagnetism, we divide the lattice into two sublattices [3]. For the symmetric PAM at half-filling, the effective hybridization is shown [6] to ren.orma!ize by a factor -: . _ v/6(l - rnf - 26)+ v'6(l + m f - 26)
* Corresponding author. 0921-4526/94/$07.00 'c) 1994 Elsevier Science B.V. All rights reserved SSDI 0921-4526(93}EO262-F
(l)
T.M. Hong et al. / Physica B 199&200 ( 1 9 ~ ) 222-223
dominates, in contrast to being itinerant for V close to |,', when K o n d o screening finally sets in to kill the magnetism at V~ (this is the picture by Doniach [1 I]). As in the Kondo impurity problem, the dotted line in Fig. i marks the.h~lse phase boundary for a nonzero 6 (on the right side). Again similar to the impurity problem, the effective hybridization equals zero when 6 = 0 because of Eq. I1), and therefore the conduction and f-electrons become decoupled. It is worth noting that this dotted line seems to overlap with the magnetic phase boundary in region I.
I
0.08
I
s /--
U=2.0
0.08
/
/
,
°.,
°.,
o.,
223
i
0.6
O.e
|
O.7
V Fig. 1. Solid line shows the V-dependence of the N~I temperature. Dotted line is the boundary for nonzero effective hybridization {on lhe right).
where 6 denotes the probability of the double occupancy and m r is the magnetic moment of the f-electron. Final form of the free energy and the four new energy bands depends on three parameters 6, m r, and the internal field h, which are determined by minimizing the free energy. For the one-dimensiom "ase [6], our results agree with those by the Quantm., Monte Carlo [7] (QMC) simulation and the Gutzwiller-correlated spin-densitywave function [8]. Anti-ferromagnetic states appear for large U/t (2t being the bare conduction bandwidth), which is expected to become short ranged by including the fluctuations of the boson fields. In infinite dimensions [9], antiferromagnetism is found to be stable for V < V~, and the value of I.~, falls within the range set by Q M C [10]. As a function of increasing V, m f decreases monotonically from ",he full moment, while m ~, of a different sign, increases from zero initially (region I) and then decreases to null at V~ (region II). Further analysis on the tongue-like !," dependence of the N6el temperature, as shown in Fig. i, leads us to believe that there are two different mechanisms for the magnetism: local-moment like for V ~. 0 where the RKKY effect
T,M.H. acknowledges the support by NSC-82-0208M007-164 and 149. SJ,S. and M.F.Y. are supported by NSC 82-0511-M-007-140.
References
[i] P. Coleman, Phys. Rev. B 29 (1984) 3035; J. Rasul and H.U. Desgranges, J. Phys. C 19 119861 L671. [2] V. Yu. lrkhin and MI. Katsnelson, J. Phys. C 2 {1990j 8715. [3] T.M. Hong and G.A. Gehring, J. Magn. Magn. Mater. 108 11992) 93. [4] Ted Hsu, private communication. This approach often has trouble with tile antiferromagnetic order, e.g. in: I.K. Affleck and J.B. Marstom Phys. Rev. B 37 (198813774 for the Hubbard model. These authors found a staggered flux phase but could not treat antiferromagnetism in their formalism. [5] G. Kotliar and A.E Ruckenstein, Phys. Re~. LctL 57 (1986l 1362. [6] M.F. Yang, S.J. Sun and T.M Hong. to be published in Phys. Re~. B. [7] R. Blankerbecler. J.R. Fulco, W. Gill anti D,J. Scalapino, Phys. Rev. [.ell. 58 (1987)411. [8] Zs. Gulacsi, R. Strack and O. Voilhardt. Phys. Roy. B 47 (1993) 8594. [9] S.J. Sun, M.F. Yang and T.M. Hong, to be published in Phys. Rev. B. [10] M. Jarrell, H. Akhlaghgour and Th. Pruschke, Phys. Re';. Lett. 70 (19931 1670. [1 !] S. Doniach, Physica P. 91 (19771 231: in: Valence Instabilities and Related NaNow Band Phenomena, ed. R.D. Park (Plenum, New York. 1977) p. 16o.