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Electrical resistivity and Hall coefficient of the Hubbard model in high dimension
ELSEVIER
Physica B 199&200 (1994) 219- 221
Electrical resistivity and Hall coefficient of the Hubbard model in high dimension Q. Qin*, G. Czycholl institut fiir Theoretische Physik, Universitiit Bremen. 28334 Bremen. Germany
Abstract We discuss the conductivity tensor for the Hubbard model in the limit of large spatial dimension D. Using the standard perturbation treatment with respect to the Coulomb correlation U, we calculate the self-energy. As vertex corrections vanish for large D, the transport quantities can be determined immediately; the resistance as a functien efthe ~.e~perature T s h o w s a T z dependence only for very low T a n d a linear T dependence for intermediate and large T. A maximum is found in the T dependence of the Hall coefficient Rn.
I. Introduction The linear temperature (T) dependence of the resistivity is interpreted as one of the most striking normal state anomalies of high-To superconductors and as an indication of non-Fermi liquid behaviour in these systems [1], which are usually modelled by (Hubbard-like) correlated electron models in two dimension. However, in this paper we will show that a linear T dependence of the resistivity over a wide T regime is not as unusual as one might expect and may be obtained even for high dimensions within a treatment, which fulfils all Fermi liquid properties. We study the Hubbard model in the limit of high spatial dimensions D [2] within the self-consistent second-¢.:aer U-perturbation treatment, which certainly c • ills all Luttinger sum rules [3] and thus the Fermi liq properties. Starting from the Kubo formula, we den ~, expressions for the static electrical conductivity tensor of a correlated electron system in the presence of a weak * Corresponding author.
magnetic field under the assumption that vertex corrections can be neglected, which is valid in the O ---, ~ limit. This allows us to derive a formula for the Hall coefficient Rn, which depends only on two on-site and off-site Green's function matrix elements.
2. The conductivity tensor Being interested in the Hall coefficient, we study only a diamagnetic coupling of the magnetic field. Then the model is given by
H = Ho + I." + H',
(1)
where Ho = ~ , e k Ck+~Cko, V = U ~ R n e T n R , , and H' describes the (diamagnetic) coupling to tl~e n, agnetic field B. i.e. in lowest order in B: H ' = - { l / c ) J A , and J = e / m ( p - (e/c)A) the total current including the diamagnetic part.
0921-4526/94/$07.00 ~:C] 1994 Elsevier Science B.V. All rights reserved SSDI 0921-452o~93)k0346-I
Q. Qin, G. C:ycholl/ PhysicaB 199&200(1994) 219-221
220
According to the Kubo linear response theory, the conductivity tensor is given by
with
iI
1 iOGnj,s,(c° + i0),
1
(2)
where f~ = Na ° is the volume of the (D-dimensional simple-cubic) system and
Gj, g,iz)
=
--"
e i'; ([J,,(t), J~,(0)])dt
(3)
is the current-current response function. As vertex corrections vanish in the limit D ~ or, the current-current response function is essentially given by products of one-particle Green's functions [4]. We choose in the following calculation a magnetic field in the z-direction B = (0, 0, B) and work in the Landau gauge, i.e. with the vector potential A = (0, Bx, 0). One can show that, to lowest order in the magnetic field, the contribution of the diamagnetic current to the conductivity vanishes. After some manipulations the matrix elements of the conductivity tensor can explicitly be written [5] as
~
W(z) = -~ _Do
;-r2
-- t dr"
Energies and temperatures are measured in units of the effective width of the unperturbed conduction band, i.e. setting 2t2D = 1. In the following we will only present the numerical results for the resistance and the Hall coefficient. For the details on the calculation of the self-energy, we refer to Ref. [6]. For a given total number of electrons per lattice site ntot, the chemical potential # has been determined self-consistently; therefore, a change of nlot leads to a shift of #. The parameters in Figs. 1 and 2 are U = 2, and the occupation for the three curves in each figure is (1) ntot = 1.2, (2) nloT = 1.4, i3) n,ot= 1.6. In Fig. 1 the resistance p(T) is plotted for different n,ot. The conductivity unit is eZ/(a ° - Zh.O), which vanishes in the limit D - , ~ . However, we are interested in the transport quantities in leading non-vanishing order in 1/D. For D = 3 and the lattice constant a of a few
0.3
4t2 e z g
,
v
-
i
-
..s s'°°
¢Txx= -- 7zal)_ 2h Id~f'(~)(001ilm G(~ + i0))2100), (4) 16t2e2~ ~d
0.2
]
0.~ (5)
-
s'~""
.~:6.e:N"<:'"
0.0
where
~=lelaZB/hc, fie) is the Fermi function, G(z) = (z - Ho - Z i z ) ) - t the one-particle Green's function, Z(z) the self-energy (for D--, ~ local). Thus, the
0.1
0.2
Fig. 1. The resistance
diagonal and off-diagonal elements of the conductivity tensor are given by simple-site matrix elements of the one-particle Green's function, if the vertex corrections vanish and a linearization in the magnetic field is justified.
2.0i
3. Numerical results for the Hubbard model
t
Of
T
Using the standard self-consistent second-order U-perturbation technique [3,4], we calculate the self-energy Z(z), which enters the Green's function for large D according to
..°
r'.." s" • s'.," ,j,.,t"o" • s.." s£."
ef(~:)
x Re ?G(e&+ i0) I 1 i),
.°"
s"s~'s" °." "" s'°" °~".." s°.'
J
rtaO_Zh
•
'
f-
•
I
|
0.3
T
0.5
pITt for different
,~
i, ~
,
'
O.G
n,o.
,
~-~.
c~ ..o,
,%
) Goo(:)=
--i
~
rc
2.0
_ w(Z--_ _Z(z)~
v'5 /'
Fig. 2. The Hall coet]icienl Rid T) for different n...
Q. Qin, G. C-vchoH / Physica B 199&200 (1994) 219-221 angstroms the resistivity unit is of the magnitude of a few mf~ cm. We find for very small T a T 2 dependence, as expected from the Fermi liquid behaviour of the selfenergy. However, there is a large T regime, in which the resistance shows a linear T dependence. The Hall coetiieient in a weak magnetic field may be written as Rn = trxx/Baxxaxy, which remains a welldefined quantity in the limit D ~ oo, because trx~ --, lID and a~r "" 1/D2. In Fig. 2 we present the results for Rn(T). The unit of RH is aa/ec, which is about 1.0 × 10- 3 cm s C - J, if we again choose O = 3 and a of a few angstroms. The Hall coefficient shows a maximum at relative high temperature. F o r large tltot, Rlt is sensitive to the temperature and shows clearly a maximum. For small n,ot, it changes very slowly with increasing T which
221
is consistent with the Fermi liquid behaviour of the Hall coefficient.
References [1] C.M. Varma, P.B. Littlewood, S. Schmitt-Rink, E. Abrahams and A.E. Ruekenstein, Phys. Rev. Lett. 63 (1989) 1996. [2] W. Metzner and D. VoUhardt, Phys. Rev. Lett. 62 (1989) 324. [3] J.M. Luttinger and J.C. Ward, Phys. Rev. 118 (1960) 1417: J.M. Luttinger, Phys. Rev. 119 (1960) 1!53; 121 (1960) 942. [4] H. Schweitzer and G. Czycho!l, Phys, Rev. Let! f,7 (1991) 3724. [5] Q. Qin and G. Czycholl, to be published. [6] H. Schweitzer at~d G. Czycholl, Z. Phys. B 79 (1990) 377.
Physica B 199&200 (1994) 219- 221
Electrical resistivity and Hall coefficient of the Hubbard model in high dimension Q. Qin*, G. Czycholl institut fiir Theoretische Physik, Universitiit Bremen. 28334 Bremen. Germany
Abstract We discuss the conductivity tensor for the Hubbard model in the limit of large spatial dimension D. Using the standard perturbation treatment with respect to the Coulomb correlation U, we calculate the self-energy. As vertex corrections vanish for large D, the transport quantities can be determined immediately; the resistance as a functien efthe ~.e~perature T s h o w s a T z dependence only for very low T a n d a linear T dependence for intermediate and large T. A maximum is found in the T dependence of the Hall coefficient Rn.
I. Introduction The linear temperature (T) dependence of the resistivity is interpreted as one of the most striking normal state anomalies of high-To superconductors and as an indication of non-Fermi liquid behaviour in these systems [1], which are usually modelled by (Hubbard-like) correlated electron models in two dimension. However, in this paper we will show that a linear T dependence of the resistivity over a wide T regime is not as unusual as one might expect and may be obtained even for high dimensions within a treatment, which fulfils all Fermi liquid properties. We study the Hubbard model in the limit of high spatial dimensions D [2] within the self-consistent second-¢.:aer U-perturbation treatment, which certainly c • ills all Luttinger sum rules [3] and thus the Fermi liq properties. Starting from the Kubo formula, we den ~, expressions for the static electrical conductivity tensor of a correlated electron system in the presence of a weak * Corresponding author.
magnetic field under the assumption that vertex corrections can be neglected, which is valid in the O ---, ~ limit. This allows us to derive a formula for the Hall coefficient Rn, which depends only on two on-site and off-site Green's function matrix elements.
2. The conductivity tensor Being interested in the Hall coefficient, we study only a diamagnetic coupling of the magnetic field. Then the model is given by
H = Ho + I." + H',
(1)
where Ho = ~ , e k Ck+~Cko, V = U ~ R n e T n R , , and H' describes the (diamagnetic) coupling to tl~e n, agnetic field B. i.e. in lowest order in B: H ' = - { l / c ) J A , and J = e / m ( p - (e/c)A) the total current including the diamagnetic part.
0921-4526/94/$07.00 ~:C] 1994 Elsevier Science B.V. All rights reserved SSDI 0921-452o~93)k0346-I
Q. Qin, G. C:ycholl/ PhysicaB 199&200(1994) 219-221
220
According to the Kubo linear response theory, the conductivity tensor is given by
with
iI
1 iOGnj,s,(c° + i0),
1
(2)
where f~ = Na ° is the volume of the (D-dimensional simple-cubic) system and
Gj, g,iz)
=
--"
e i'; ([J,,(t), J~,(0)])dt
(3)
is the current-current response function. As vertex corrections vanish in the limit D ~ or, the current-current response function is essentially given by products of one-particle Green's functions [4]. We choose in the following calculation a magnetic field in the z-direction B = (0, 0, B) and work in the Landau gauge, i.e. with the vector potential A = (0, Bx, 0). One can show that, to lowest order in the magnetic field, the contribution of the diamagnetic current to the conductivity vanishes. After some manipulations the matrix elements of the conductivity tensor can explicitly be written [5] as
~
W(z) = -~ _Do
;-r2
-- t dr"
Energies and temperatures are measured in units of the effective width of the unperturbed conduction band, i.e. setting 2t2D = 1. In the following we will only present the numerical results for the resistance and the Hall coefficient. For the details on the calculation of the self-energy, we refer to Ref. [6]. For a given total number of electrons per lattice site ntot, the chemical potential # has been determined self-consistently; therefore, a change of nlot leads to a shift of #. The parameters in Figs. 1 and 2 are U = 2, and the occupation for the three curves in each figure is (1) ntot = 1.2, (2) nloT = 1.4, i3) n,ot= 1.6. In Fig. 1 the resistance p(T) is plotted for different n,ot. The conductivity unit is eZ/(a ° - Zh.O), which vanishes in the limit D - , ~ . However, we are interested in the transport quantities in leading non-vanishing order in 1/D. For D = 3 and the lattice constant a of a few
0.3
4t2 e z g
,
v
-
i
-
..s s'°°
¢Txx= -- 7zal)_ 2h Id~f'(~)(001ilm G(~ + i0))2100), (4) 16t2e2~ ~d
0.2
]
0.~ (5)
-
s'~""
.~:6.e:N"<:'"
0.0
where
~=lelaZB/hc, fie) is the Fermi function, G(z) = (z - Ho - Z i z ) ) - t the one-particle Green's function, Z(z) the self-energy (for D--, ~ local). Thus, the
0.1
0.2
Fig. 1. The resistance
diagonal and off-diagonal elements of the conductivity tensor are given by simple-site matrix elements of the one-particle Green's function, if the vertex corrections vanish and a linearization in the magnetic field is justified.
2.0i
3. Numerical results for the Hubbard model
t
Of
T
Using the standard self-consistent second-order U-perturbation technique [3,4], we calculate the self-energy Z(z), which enters the Green's function for large D according to
..°
r'.." s" • s'.," ,j,.,t"o" • s.." s£."
ef(~:)
x Re ?G(e&+ i0) I 1 i),
.°"
s"s~'s" °." "" s'°" °~".." s°.'
J
rtaO_Zh
•
'
f-
•
I
|
0.3
T
0.5
pITt for different
,~
i, ~
,
'
O.G
n,o.
,
~-~.
c~ ..o,
,%
) Goo(:)=
--i
~
rc
2.0
_ w(Z--_ _Z(z)~
v'5 /'
Fig. 2. The Hall coet]icienl Rid T) for different n...
Q. Qin, G. C-vchoH / Physica B 199&200 (1994) 219-221 angstroms the resistivity unit is of the magnitude of a few mf~ cm. We find for very small T a T 2 dependence, as expected from the Fermi liquid behaviour of the selfenergy. However, there is a large T regime, in which the resistance shows a linear T dependence. The Hall coetiieient in a weak magnetic field may be written as Rn = trxx/Baxxaxy, which remains a welldefined quantity in the limit D ~ oo, because trx~ --, lID and a~r "" 1/D2. In Fig. 2 we present the results for Rn(T). The unit of RH is aa/ec, which is about 1.0 × 10- 3 cm s C - J, if we again choose O = 3 and a of a few angstroms. The Hall coefficient shows a maximum at relative high temperature. F o r large tltot, Rlt is sensitive to the temperature and shows clearly a maximum. For small n,ot, it changes very slowly with increasing T which
221
is consistent with the Fermi liquid behaviour of the Hall coefficient.
References [1] C.M. Varma, P.B. Littlewood, S. Schmitt-Rink, E. Abrahams and A.E. Ruekenstein, Phys. Rev. Lett. 63 (1989) 1996. [2] W. Metzner and D. VoUhardt, Phys. Rev. Lett. 62 (1989) 324. [3] J.M. Luttinger and J.C. Ward, Phys. Rev. 118 (1960) 1417: J.M. Luttinger, Phys. Rev. 119 (1960) 1!53; 121 (1960) 942. [4] H. Schweitzer and G. Czycho!l, Phys, Rev. Let! f,7 (1991) 3724. [5] Q. Qin and G. Czycholl, to be published. [6] H. Schweitzer at~d G. Czycholl, Z. Phys. B 79 (1990) 377.