Physica B 194-196 (1994) 175-176 North-Holland PHY$1CAI

Single particle excitations in itinerant antiferromagnets at small doping W. Brenig ~ *, A. P. Kampf b and K. W. BeckeV

~Institut fiir Theoretische Physik, Universit£t zu KSln, W-5000 KSln 41, Germany bIFF Forschungszentrum Jiilich, W-5170 Jiilich, Germany CMax-Planck-Institut fiir FestkSrperforschung, W-7000 Stuttgart 80, Germany

We present a selfconsistent strong coupling scheme to evaluate the single particle Green's function for the two dimensional Hubbard model in the doped spin density wave state. For small doping we analyze the quasiparticle properties including the dispersion and the quasipaxticle weight. Novel incoherent contributions to the spectral function resulting from multi spin wave processes axe discussed.

1. I N T R O D U C T I O N Experimental and theoretical evidence on the

cuprate perovskites has converged towards the two dimensional Hubbard model as a promising candidate to describe these systems which ex- hibit high temperature superconductivity [1] in the vicinity of a metal to antiferromagnetic (AF) insulator transition. Many investigations of the Hubbard model have focused on the interplay be- tween charge and spin degrees of freedom close to half filling starting from the localized limit. Ef- forts to study the elementary excitations in the itinerant regime for intermediate correlations are less exhaustive [2]. In this contribution we report results of a strong coupling theory for spin fluc- tuation induced single particle renormalization in the spin density wave (SDW) state of the Hub- bard model.

2. A F - P O L A R O N S IN T H E S D W STATE

The starting point is the SDW representation of the 2D Hubbard model

1 t? l HI~UB=~ y~ l. Eka k a k ~+HR~s (1)

k , a , t = ± l

Here atk(t)~ are conduction/valence band SDW particles for l==kl, respectively, and HRF, s is the residual Hubbard interaction Htr. The mean field kinetic energy E k is given by E k = (e~ + A2) 1/2 where e k is the 2D tight binding energy and A is the selfconsistent magnetic gap. Within the mag-

*Research performed within the program of the Son- derforschungsbereich 341 supported by the Deutsche Forschungsgemeinschaft.

netic Brillouin zone (MBZ)the operators 4 ( ~ a r e

given in terms of the original fermions c~ ~ by

azkt = V z k 4 -b la V l k + Q C~ + Q ~

Vzk = V-lk+Q -- [(1 +lek/Ek)/2]W2 (2)

Q is the square lattice nesting vector and 4 t is

extended to the 1st BriUouin zone (BZ) by ak? ~ =

l a 4 t _ Q . The resulting algebra is given by

l' t {a k ~,ak, ~,} = 6~,SW(Skk, + laSkk,+Q ). The bare fermions are expressed via c ~ =

~z=±l vz k 4 t a for all k E 1st BZ. To describe AF spin waves only the retarded

transverse spin susceptibility X +- (q, qr, t) = iO(t)([a+q(t),aq,]) is needed, where a~ =

~ k C~+q TCk 1" To evaluate this susceptibility

we employ the ttPA approximation. We have ex- tended the analysis of ref. [2] to include the possi- bility of Landau damping [3] at finite doping. We found the spin waves to be stable against relax- ation due to intra band scattering. Therefore we resort to a strong coupling expansion of the RPA susceptibility at zero doping. Extracting the spin wave poles in the large U-limit we find a 2x2 ma- trix representation in q space

X+-(q, q, z) = -2g(ek/(4t) + 1) Z 2 -- W~

z X + - ( q + Q,q ,z) = z2 _ C#~l (3)

where X+-(q ,q ,z) _-- x + - ( q + Q , q + Q,z) and X + - ( q + Q , q , z ) - X + - ( q , q + Q , z ) . J is

0921-4526/94/$07.00 © 1994 - Elsevier Science B.V. All rights reserved SSDI 0921-4526(93)E0666-5

176

= - . . 7+

I--'"" ~'"' `+~ II ,/, ~,.--., " +.-.~ 0.4

"-~ 0.2

I

-lO -5 0 w/t 5 10

Figure 1. Spectral function.

the AF exchange coupling given by 4t2/U where t is the bare hopping integral and OJq : 2 J [ 1 - e~/(16t2)] 1/2 is the spin wave dis- persion, z = w -t- i~ is a complex frequency.

We now formulate a Dyson equation for the mt Green's function G~m(k , r )= - (T~aZk a(T)a k ~)

of the SDW particles. Strong coupling to the spin degrees of freedom is approximated by including multiple spin wave scattering in the

E~ (k,T) as sketched in the in- self energy z m set of Fig. 1. In the large U limit we find G ~ - I (k , r ) : G~ ~ l(k, T) -- 0. At small doping we focus on the Green's function of a single hole. In that case particle hole symmetry leaves only the retarded valence band self energy to be de- termined

~ - t ( k , z ) = V 2 ~ {

q¢0

(1-1- 2_~J) f~dwA~_.-~(k-_q,w) wq Jo Wq+WTZ

2 . / _ r ° A -~ - 1 ( k ~ (,,.,~ • a ,, - - N [ , J ~ -F(1 - - - ) l d w . . . . z I ( 4 ) a~q j _ ~ ~ q - ~ -

Here A~ z - l (k ,w) : -Im[G~ 1 - l ( k , w + i~)]/~r is the spectral function of the retarded propaga- tor. The primed q-summation is restricted to the MBZ.

3 . R E S U L T S A N D D I S C U S S I O N We have solved Eqn.(4) by iteration on finite

lattices. In Fig. 1 a characteristic spectrum of

0 . 6

0 . 5

0 , 4

- 3

- 4

" i. ' t t ' i ' I ' I ' I ~ . . . . h - - - - ' ~

i ~ ~ x ? . - i , i .... i u = 6 I- ' ". i ~' i " ". i : " i 1 6 x 1 6

I ; , / ' i i. r ~ ! l • I : . -

i ".., ! / t Fi "., l ~ ! I "x i / t ,, SC

. " , ' x ~ I , , r - $ ; , , i ; , , I "

I ~ . . . . ! ~ - . ~ . . . . . . : . . . . . I

-3 l-J. '<. ! ,,.'"., ..... ~. .................. +-I

! ".. "". i ~" ~" !

- 4 '~ i ~' ! I ! ... ! / . s c i -I

" ~ , ." ~ b a r e S D W !

- 5 ~ - , I , ",a-., I , I , I ;, i , , ,,-1 ( ~ / 2 , ~ / 2 ) (o ,o ) ( ~ , o ) ( ~ / 2 , ~ / 2 )

F i g u r e 2 . Q u a s i p a r t i c l e p r o p e r t i e s

the valence band propagator is shown on the MBZ boundary for U / t = 6 on a 16x16 lattice. Besides a renormalized quasiparticle peak the spectral den- sity displays considerable incoherent weight due to spin wave shake off, both, above the magnetic gap and below the quasiparticle pole. Earlier in- vestigations [4] have missed the latter effect due to a lack of selfconsistency. The valence band in- coherent weight is reminiscent of similar findings on t - J type models. Fig. 2 depicts quasiparti- cle properties along the irreducible wedge of the MBZ for parameters identical to those of Fig. 1. The Z-factor is smallest at the valence band bot- tom where the spin wave shake off is strongest. A pronounced band narrowing of Eqp(k) is evident. Moreover we find a non linear dependence of the effective gap on U. For all values of U investigated we observed Eqp(k) to have pockets at ( r /2 , 7r/2) at the l-loop level, however, they are shifted to (~, 0) for an infinite number of loops.

R E F E R E N C E S

1. J.G. Bednorz and K.A. Miiller, Z. Phys. B 64 (1986) 189.

2. J.R. Schrieffer, X.G. Wen and S.C. Zhang, Phys. Rev. B 39 (1989) 11663; A. Singh and Z. Te~anoviE, Phys. Rev. B 41 (1990) 614.

3. W. Brenig, A.P. Kampf, and K. Becker to be published

4. G. Vignale and M.R. Hedayati, Phys. Rev. B 42 (1990) 786.