Weak-coupling treatment of correlated electron systems in finite dimensions

Physica B 165&166 North-Holland

(1990)

WEAK-COUPLING

Heinz

TREATMENT

SCHWEITZER

Institut

415-416

OF CORRELATED

ELECTRON

IN FINITE DIMENSIONS

SYSTEMS

and Gerd CZYCHOLL

fur Physik,

Universitiit

Dortmund,

Postfach

500500,

D-4600

Dortmund

SO, Germany

namely Hubbard model (HMl and periodic The standard models for correlated electron systems, Anderson model (PAM) are investigated within the second order U-perturbation treatment around For a d-dimensional system the full k-dependent the non-magnetic Hartree-Fock solution. starting from the local selfenergy can efficiently be calculated by means of a l/d-expansion approximation, which is correct in the limit d+a. Models of correlated electron systems as the Hubbard model (HM) and the periodic Anderson model (PAM1 are presently of great interest, in particular for a theoretical description of high temperature superconductors and of heavy fermion systems. The HM is given by H=~E,&~

+ I.I,.&+,,ci~c&~ i

ko

and the PAM is given by H= skc;acka+C(Ef fi’of,,+V(~;f,,+ h.c.)+~f;,fiOf;,fi_,) id F0 For the conduction band we assume a simple cubic tight binding band structure for dimension d, i.e

(site-diagonal) Z(z) has been used as an additional ad hoc approximation without having a criterion for the validity of the LA. But recently it has been pointed out (3,4) that the LA becomes correct only in the limit d+a 61. So the corrections to the LA for finite d can be classified as l/d contributions. manner, i.e. We found that in this by a i/d-expansion, which converges even for d=l, Z,(z) can much more efficiently be calculated than by the direct p,q double summation. Fourier start with a specific, we To be transformation of the selfenergy Zk(zl=ZR,O(z) The site given by

Due to the prefactor l/d energies are measured in units of the half conduction band width. - The simplest approximation for these models beyond the (trivial) Hartree-Fock treatment is provided by the second order perturbation theory (SOPT) with respect to the Coulomb correlation II. Within the SOPT relative to the non-magnetic Hartree-Fock solution the selfenergy is given by Zk(iw_)=-(~~CGk+q(lol)Gp_q(iwn+iw2-iwl)Gp(iw2) YY P9

Here N denotes the number of lattice sites, T the Matsubara frequencies, and temperature, wi are Gk(zl is the Hartree-Fock Green function, which for the HM is given by Gk(z)=

and for

1 z - u

the PAM by

G,(z)

z-Er-I.I

- -

vz z - Ek

calculation of the the explicit But usually k-dependent SOPT selfenergy E.,(z) is relatively complicated because of the double momentum (p,q-Isummation. Therefore, in most of the existing SOPT selfenergy computations (1,21 the “local k-independent approximation” (LA) of a

0921-4526/90/$03.50

matrix

-I?T’

1

ZRVz) the

GR( iwn+io2-iw,)G&,l

selfenergy

are

G_R(iw,)

Qo> - _

where

Jd<$I$ GR(zl = k 1 exp(ikRl G,(z)= k denotes the site matrix elements of the Hartree Green function, p&J= - Im G&+iO)/rc its Fock spectral function. the Matsubara Replacing function summations Fermi frequency by integrations and using a Laplace transformation as in (4) we finally obtain 2,&z)= U2 (-8 JdX eixz with A&)=

Jd[

o&J

( Ai((XlB_&-XI

+ B&((XIA_R(-ll

I

f(c) emixc

BR(X) = s dc pR(c) (l-f@) - sk

1

??

Z&J=

+ 1 exp(ikRl R40 elements of

e-ixT

Thus the calculation of each site matrix element of the selfenergy Z~(zl can as efficiently be performed as the calculation of the LA or the d=a selfenergy (4,S), and provided that the R-sum can be terminated after a finite number of neighbor shells, i.e. provided that the l/d-expansion converges, we the full have a quick algorithm to determine k-dependent selfenergy Z,(z). The convergence of the l/d-expansion depends on the model under consideration, i.e. on the particular spectral function pR(zl, and it depends, of course, on the dimension d. Empirically we found out that

@ 1990 - Elsevier Science Publishers B.V. (North-Holland)

416

H. Schweitzer,

for the HM one has to extend the R-sum up to the 3rd neighbor shell for d=3, to the 6th neighbor shell for d=Z, and even up to the SOth neighbor shell for d=l. For the PAM the convergence is somewhat quicker: one has to go up to the 2nd neighbor shell for d=3, to the 5th neighbor shell for d=2 and to the 20th neighbor shell for d=l. So even for d=l the i/d-expansion converges, but the is weak. convergence But for d=2 and 3 the convergence is quick and the present R-summation method is certainly superior to the direct p,q double summation. For an explicit example we present a few results obtained for the PAM in two dimensions (d=21. We choose the following parameters: E,= -0.5, U=l., V=O.2 and a total number of 2 electrons per lattice site (“symmetric PAM”). Then the chemical potential must be at zero and the total f-electron occupation must be nf= fi+> + =l. , i.e. the effective Hartree-Fock f-level is at zero, too: Ef+Ufib> = 0. Figures 1 a.and b. show the results Par the

G. Czycholl

momentum (k-1 and frequency (z-) dependence of the SOPT selfenergy real and imaginary part for T=O. In the frequency dependence we observe, in particular, the vanishing of Im&(z+iOl at the Fermi level, z=O., in accordance with the Luttinger theorem. The real part has a corresponding structure resulting in a strong negative slope at the Fermi level, which automatically provides for a strong mass enhancement. Furthermore we observe a weak (cosine like) momentum (k-ldependence. The resulting f-electron spectral function is shown in observe, strong Fig.2. We in particular, quasi-particle resonance peaks near to the chemical potential and a hybridisation gap around zero. The width of this hybridisation gap and the exact the resonance peaks position of is strongly influenced by the explicit k-dependence of the selfenergy. 4.0

-

3.0 -

2.0 -

1.0 -

-1.0 FIGURE

-0.5

0.0

I ,

0.5

, E

2: Corresponding frequency dependence the SOPT f-electron spectral function the symmetric PAM in dimension d=2

4 1. 0

of of

In summary we have demonstrated that within selfenergy can the SOPT the full k-dependent efficiently be determined by means of a Fourier neighbor shell transformation , i.e. a nearest be interpreted as a which can summation, l/d-expansion. Of course this is not only valid for the SOPT, but it remains true also for higher order summations (e.g. T-matrix approximation and RPA). presented will be results and More details elsewhere

(6).

REFERENCES

FiGURE

1 a and b: Momentum (k-1 and frequency (z-1 dependence of the real and imaginary part of the SOPT selfenergy &(z+iOl of the symmetric PAM in dimension d=2 for E,=-0.5, U=l., V=O.2

(11 G.Treglia, F. Ducastelle, D.Spanjaard, Phys. Rev. B 21. 3729 (1980); B 22, 6472 (1980); Journ. de Physique 41, 281 (1980) (2) R.Taranko, E.Taranko, J.Malek, J.Phys. F 18, L87 (1988); J. Phys.:Condensed Matter 1, 2935 (1989) (31 E. Miiller-Hartmann, Z. Phys. B 3, 211 (1989) (41 H. Schweitzer, G.CzycholI, Solid State Commun. 69, 171 (19891 (Sl W. Metzner, D. Vollhardt, Phys. Rev. Letters t& 324 (1989) (6) H.Schweitzer, G.Czycholl, Solid State Commun. (19901, in print, and to be published