ELSEVIER Physica B 199&200 (1994) 275-277

Single particle excitations in itinerant antiferromagnets at small doping t

Wolfram Brenig Arno P. Kampf a, Klaus W. Becker b

Institut fiir Theoretische Physik, Universitiit zu K61n. 50937 K61n, Germany b Max-Planck-lnstitut fiir Festk6rperforschung, 70569 Stuttgart, Germany

Abstract

We present a self-consistent strong coupling scheme to evaluate the single particle Green's function for the two- dimensional Hubbard model in the spin density wave state. We analyze the single quasi-hole properties including its dispersion and the spectral weight. Novel incoherent contributions to the spectral function resulting from multi-spin wave processes are discussed.

1. Introduction 2. AF-polarons in the SDW state

The vicinity of antiferromagnetism and superconduc- tivity in the phase diagrams of the cuprate perovskites has stimulated research to understand the properties of carriers doped into an antiferromagnetic (AF) insulating state in two dimensions. The single band Hubbard model and in particular the t - J Hamiltonian have been focused on in many investigations. In the strong coupling limit the competition between AF order- ing of localized spin degrees of freedom and the delocal- ization of doped holes is well understood [1]. Efforts to study the elementary excitations in the itinerant regime for intermediate correlations are less exhaustive [2]. In this contribution we report results of a theory for spin fluctuation induced single pamcle renormalization in the spin density wave (SDW) state of the Hubbard model.

* Corresponding author. ' Research performed within the program of the Sonderfor- schungsbereich 341 supported by the Deutsche Forschung> gemeinschaft.

The starting point is the SDW representation of the two-dimensional (2D) Hubbard model [2, 3]:

1 HHUB 2 E 1" + HRE s. (2.1)

k.a.I = +_ I

Here a~'~ ' are conduction or valence band SDW particles for l = 1 or t = - 1, reapectively. They

t are linear transforms of the original fermions c~, via a ~ = v~ c*k. + lov~ + ec~ + O~ with the nesting vector Q = (rt, n) [2, 3]. HRES is the residual interaction. The mean field kinetic energy E~ is given by E, =(e~ + d21 ~2 where e., is the 2D tight binding energy and A is the self-consistent magnetic gap.

Beyond the mean field description the SDW particles are expected to couple strongly to AF spin wave excitations. Therefore, we have iavestigated a Dyscn equation for the SDW propagator-matrix summing all noncrossiny dia'.4ra,ts

for scattering from transverse spin fluctuations (Fig. t}:

¢t - Za,k l, k2, eu)

= - T U 2 ~ [G'--,,(kl - q, k2 - q, c,, - ¢o, IZ °°~q, q, ,~, t q, t,>,

+G~_~(~ l_q , k 2 _ q . c , _ ~ o , l - / - ° " ( ~ , q , eg,}]. ~2.2~

0921-4526..'94/$07.00 ~, 1994 Elsevmr ~cmnce B.~. All rights reserved SSDI 0921-4526193}E0207-W

276 IV. Brenig et a/. I Physica B 199&200 (1994) 275-277

16 = 16, U,=6 O d i i ..... i ! i i i I ' (~ )=

. (~/il,~.nl)

0.2

' ' 1 ' ' ' ' 1 . . . .

[ = , . . ÷ : ~ - * ' : _

" 0 . I ' ' ' ' ' I ' ' ' ' I ' ' ' '

~ o . ~ ( ~ , ~ ) = ( o . o ) -

O , , i , , ,

- 1 0 - 5 0 5

in the SDW state. In that case particle hole symmetry leaves only the retarded valence band self-energy to be determined:

Z ; " i - ~ (k, z)

- f( Z3Io ' " ' "°" = U z Y" 1 + dta' A ; - q ~ o k \ ( O q / ~ ¢ + ( t ) ' + Z

+ ( I _ 2 £ ) f 0 d,o, A21-1(k~__q, to,)~" (2.4) k ¢o U J _ ~ c o q - c o - z )

Here AZ, l - t ( k , ( o ) = - I m [ G 2 l - l ( k , to + it/)]/n is the spectral function of the retarded propagator. The primed q-summation is restricted to the magnetic Brillouin zone (MBZ).

3. Resu l t s and discussion

Fig. I. Spectral functions.

Here Z - " " and Go_. are the dynamical transverse spin susceptibility and the dressed c-propagator. These ar~ 2 x 2 matrices in q space by virtue of Umklapp from the SDW exchange potential with k t . 2 e { k , k + Q } and /5 = p + Q. Standard finite-temperature notation has been used in Eq. (2.2).

To evaluate the retarded transverse spin suscepti- bility Z~-"(q,q' , t )=iOlt l([S"_qlt) .Sq."]) we have employed the RPA approximation [2]. Focussing on the collective dynamics and extracting the spin wave poles in the large U-limit one finds a 2 x 2 matrix representation in q space:

Z'-" (q, q, : } = - 2J(%/(4t) + I) _2 _ f, Oq

G 2 ;~:o ~tq -t- Q, q, : ) = _ _z _ ~,~,~" {2.3}

where /° "(q + Q,q + Q,:) and "/f-"(q,q + Q,:) are obtained by q--, q + Q. J is the AF" exchange cnupling given by 4tX/u where t is the bare hopping integral and ~,~q = 2 J [ 1 - ~:2/(16ta)]t'2 is the spin wave dispersion. : = ~,~ + it/is a complex frequency.

To simplify the four integral eqs. (2.2) we first transform to the a-particle Green's functions G~'(k, c,) approximating the coefficient c[ by its large L'

wdue 1/ \ 2. This leads to G~ -~ (k. ;;,,} = 6~, ~ ~k, ;:,,) - 0. Secondly, at low doping, we focus on a sinqle hob,

We have solved Eq. (2.4) (SC equation) on finite lattices. In Fig. I characteristic spectra of the valence band propagator are shown for U/t = 6 on a 16 x 16 lattice. Besides a renormalized quasi-hole peak the spectral density displays considerable incoherent weight due to spin wave shake off, both, above the magnetic gap and below the quasi-hole pole. Earlier investigations [4] have missed the latter effect due to a lack of self-consistency. The valence-band incoherent weight is reminiscent of similar findings on t J type models. Figure 2 depicts the quasi-hole

U = 4 , 2 4 x 24 -2.0. Lq

-2.5" '

,,,~ - 3 . 0 - ,

- 3 . 5 - o',

- 4 . 0 - o o., o

0 " 9 1 , ~

0.8 "~

0.7

0 6 "~,

(n/2.n/2)

A

N

~,e88- d

, d e~

,43

$o z

JSr ° 0

0

¢ ¢

o' /

j 8

:-8-.8.8-8-.8:,1

• SC o SDW

~ 0 _ 0 . . . 0 - D "0" "q

(0,0) (n.O) (n/2,n/2)

Fig. 2. Quasi-hole properties.

w. Brenig et al./ Physica B 199&200 (1994) 275-277 277

properties along the irreducible wedge of the MBZ for the largest system we have studied and for U/t = 4. The Z-factor is clearly k-dependent and smallest at the zone center where the spin wave shake off is strongest. A pronounced band narrowing is evident. We find a nonlinear dependence of the effective gap on U. For all U and all systems sizes the degeneracy of the SDW bands along the MBZ boundary is barely lifted by the SC equations with the maximum occurring at k=(n ,0) . Analyzing data from system sizes N z = 4 z,8 z . . . . . 242 for U / t = 4 [3], we have extrapolated a thermodynamic limit for the bandwidth

Wsc/WsDw ~- 0.65 and the z-factors z(lt/2, r~/2) = 0.80 and z(O, 0) -- 0.51.

References

[1] G. Martir~ez and P. Horsch, Phys. Rev. B 44 (1991) 317. [2] J.R. Schri~,ffer, X.G. Wen and S.C. Zhang, Phys. Rev. B 39

(1989) 11653; A. Singh and Z. Te~anovi~, Phys. Rev. B 41 (1990) 614.

[3] W. Brcnig and A.P. Kampf, Europhys. Lett., in press. [4] G. Vignale and M.R. Hedayati, Phys. Rev. B 42 (1990) 786.