Motion of a single hole in a disordered magnetic background

30January1995 PHYSICS LETTERS A

Physics Letters A 197 (1995) 353-360

ELSEVIER

Motion of a single hole in a disordered magnetic background A. Belkasri

a,b, J.L.

Richard a

a Centre de Physique Thborique, CNRS Luminy, case 907, F-13288 Marseille Cedex 9, France b Universit~d'Aix-Marseille II, Marseille, France

Received 19 July 1994;accepted for publication 22 November 1994 Communicated by J. Flouquet

Abstract

The spectrum of a single hole is calculated within the spin-hole model using a variational method. This calculation is done for any rotational invariant magnetic background. We have found that when the magnetic background changes from a disordered to a locally ordered state, the spectrum changes qualitatively. We have also found that the spin pattern around the hole is polarized. This problem is related to the study of copper oxide planes CuO2 doped with a small number of holes.

1. Introduction

The study of the motion o f one hole in strongly correlated systems is very important because, in principle, it may contain some features of the properties of the C u O : planes doped with a small number of holes. The one hole motion in a quantum antiferromagnetic ( A F M ) background has been studied by several authors [ 1-4 ] in the framework of the t - J model. These studies were carried out within an AFM ordered state. The main result of this analysis is the existence of a well-defined quasiparticle description for a coherent hole propagation with the band minima at ( + n/2, + n / 2 ) and the bandwidth W o f t h e order of J a n d not t. These results are in good agreement with numerical exact diagonalization for small systems [5,6 ]. All these investigations concern the magnetic ordered case, and they were carried out in the framework of the t - J model which has the difficulty o f the constraint o f no double occupancy at the same site. In this paper we return to the Hamiltonian H~_2~ derived in, e.g., Refs. [ 7 - 1 0 ] . This Hamiltonian is obtained from the Emery model by means o f a perturbative expansion up to the second order in tpd (the hopping parameter between Cu and O sites). We point out that the t term of the t - J Hamiltonian is obtained from np_~"t2)by projection on local singlet states [12]. We will not consider here the fourth order term which gives the super-exchange energy between spins located on the Cu sites. This term will not be relevant in our calculation since we want to study essentially the motion of a hole in a disordered magnetic background. In the framework o f the H~2d) Hamiltonian we will compute the dispersion relation E ( k ) by applying a trial wave function. The results agree with previous calculations for the t - J model when we consider the limit o f an ordered magnetic background. Our starting point is the HptZ_d ) Hamiltonian (for shortness we use H instead ofHp~2_~) which can be written as H = Z To f t . , f J , , + g t,J,a

Z

wowo'f]~,.~,'fJ'~"S~,

i,j,jt, a,a'

0375-9601/95/$09.50 9 1995 Elsevier Science B.V. All rights reserved SSD10375-9601 (94)00958-9

(1)

A. Belkasri.J.L. Richard/ PhysicsLettersA 197(1995)353-360

354

+ where (f~o,fo) is a fermion field with the spin index a. St is the spin operator on the Cu site i, lr= (rx, Zr, r,) are the Pauli matrices and

T,, = T 1 ~ ,e e x p [ i k ( / - j ) ] ,

(2)

1

wij = ~ ~ Wkexp [ i k ( i - j ) ]

(3)

where ~k= 4 ( 1 -- 7k), COg= ~ and 7k= (COSkx + cos ky) / 2. The parameters T and g are related to the parameters d = % - ~d, Ud and tpd by the following relations,

T=_89

1

ualfl),

(4,

Hamiltonian ( 1 ) describes a system of itinerant holes interacting with a magnetic background. We notice that if we restrict the g term to a local spin-spin interaction (i.e. take w~j= Woe,j) we get the phenomenological Hamiltonian considered by Monthoux and Pines [ 13 ].

2. Construction of the trial state for one hole motion

The most general state for a single hole is

~u= ~f~,,,A(ia)q~,

(6)

i,o"

where q~ is some magnetic background and A (ia) is some combination of spin operators. The meaning of Eq. ( 6 ) is clear: the motion of the hole will perturb the magnetic background ~, generating all sorts of excited magnetic states provided by the operator A (ia). In what follows, we will suppose that the magnetic background 9 is rotationally invariant, i.e. S, ot r

0.

For Hamiltonian ( 1 ), if we set g = 0 , the following state for one hole with spin up,

E ~o(i)f~,~qb,

(7)

i

is an eigenstate for the T term of Hamiltonian ( 1 ), if ~0 (i) = e ik~ for a given k. On the other hand, the g term can be rewritten in the following form,

- ]g E w,jwtj'(dr

-at, f),) ()~j,,d,,-fj,,a,,)+ 89 Z w,jw,j,tdr

i,j,j'

+ 1 (d~rfj, +dr, f j,) @,,d,, +fj.rd,~) ] .

+d~,,f~J,fJ,,d**

i,j,j'

(8)

In this form we have split the Hamiltonian into two parts, one which contains singlet operators and another which contains triplet operators. These singlet and triplet operators are more general than those considered by Zhang and Rice [ 121, because here they are nonlocal. From Eq. (8) we see also that the singlet states and the triplet states are separated by an energy 2g. If we want to look for low energy excitations and we neglect mixing between the two bands, we can write the following state for one hole with spin up,

A. Belkasri,J.L. Richard/ PhysicsLettersA 197 (1995)353-360

t,J

~, ( i, j) ( dt,,f ~, -dt,,f ]t )4, ~ .

355 (9)

Then a natural trial state for the full Hamiltonian ( 1 ) can be written as i

Vo(i)ft,~+ ~ ~l(i,j)(dt,,fJ,-dt,,f~,)d~, ~ . t,y

(10)

To simplify the calculation we take ~u~(i,j) = ~ 0") wj~.Then the state (10) can be rewritten in a more transparent way,

(

~= ~i r176

)

2 o,(J)wo(ft, tS]+f~* S+) qb, J

(11)

where q~o(i) = ~Uo(i)- 89 (J)wo. Examples for the magnetic state qb are the singlet resonating-valence-bond states considered by Anderson, Dou~ot and Liang [ 14 ] which may be long range or short range ordered.

3. Dispersion relation for a single hole

The dispersion relation E(k) will be calculated by using the variational method. We will minimize (~, H~) keeping the norm (u g) fixed. Straight-forward calculation, using the property El wilwo= To, leads to

- g Y~ f't~,[~(l)qh(l')($rSt,) +h.c.]+g ~ ft~,~t~,~(l')qh(l)(SrS~,) z,t'

+ig

U'

E

h,12,13(h~12~13)

~hh'~hl2~l(13)~gl(lE)(Sl3"(ShXSh)),

with 7~j= To~ T and the average (

(12)

) defined as

( S a ... S b) - (~, S a ... S b ~ ) .

(13)

Now if we suppose that the magnetic background 9 (in Eq. ( 11 ) ) is invariant under the time reversal operation K, we will have ( K ~ , S~ S~2S~3g~ ) = ( ~, S lXiS~2S lZ3~) ) = - ( ~, SlX S~2S lZ3~) )

(14)

for any It,/2, 13 (all different), since KSK-1 = - S . And finally we have (Sl3"(Sh XSl2) > =O .

(15)

Consequently the last term of Eq. (12) vanishes. The energy will be calculated by minimizing the energy functional ~r

2) = (~u, n~u) - 2 ( ~ , ~u).

(16)

Using Eq. (12) and (r v/)_-- ~ 1(#o(i)12+ ~, Tjj,~j(J')~I(]")(Sy'Sj,), i

j,j~

we get the following equation for the energy E,

(17)

356

A. Belkasri, J.L. Richard/Physics Letters A 197 (1995) 353-360

)(coo(k))__0

gwka

gWkak

(18)

TCk + g d k - - E a k J \CO.(k)J

where C0o(k) and COl(k) are the Fourier components of coo(i) and COt(i) respectively. On the other hand, ak=3--4Zol?k,

Ck=15--32Zol~k+4Zo2(4y~--l)+472k(Zll--Zo2)

,

dk=12+Zol--8ak,

(19)

with •01

=

--

(So "Sel ),

Zll = (So'Se,+e2),

Zoz = -- ( S o ' S 2 e , ) ,

(20)

e~ and e2 denoting the two unit vectors of the square lattice. F r o m Eq. (18), the lowest energy band is given by E k / g = 89[q(ek +Ck/ak) +dk/ak --X/[,(Ek --Ck/ak) --dk/ak]2 +4ekak ] ,

(21)

with , = T / g and for a given k we have the following solutions for (ao(i) and (o~(i), coo(i; k) = W k a k U ( k ) e igi ,

COl(i; k) = - ( q e k - - E k / g ) U ( k ) e ik~ ,

(22)

where the function U ( k ) is determined by writing (~v, ~,) = 1 and we get 1

U(k)2= eka 2 + ~ ( , e k - - E k / g ) 2"

(23)

The dispersion relation Ek depends only on the spin-spin correlation function for nearest and next nearest neighbors. This is due to our expression for the trial state. Introducing more and more spin excitations would essentially produce new spin correlation functions at larger distance. However, since we are interested essentially in a background with short correlation functions, this would merely not change our result at least qualitatively. Knowing Zol. Z~ and ~o2 we evaluate then the energy E k. If we s e t , = O. we can have a qualitative idea about the position of the m i n i m u m Of Ek. Ek/ g = -- I ( Idk/ak I + d l dk/ ak 12+ 4Ekak ) .

(24)

It is easy to see that Idk/akl is maximal for k-- (0, 0) and 4~kak is maximal for k = (~, n). Then since to lower Ek both of the quantities should be maximal, there will be a competition between them. For Z01 = 0, I d k / a k ] = 4 and consequently the m i n i m u m will be at k = (~z, zc). For Zot # 0 the position of the m i n i m u m will be in between (0, 0) and (n, 7t). 3.1. Paramagnetic case For a completely disordered magnetic system, we have ( S i ' S j ) = ]6,j.

(25)

Therefore Zoo, Z~ and Zo2 vanish and we obtain for the energy E k / g = , ( 9 --2~k) -- 2--X/12 ( 1 --?k) + [2 - - , ( 89+ 2~'k) ]2.

(26)

The dispersion relation (26) is plotted in Fig. 1 for the physical value ,=-~. The band m i n i m u m is reached at points ( _+n, + ~). The center of the Brillouin zone F(0, 0) is a m a x i m u m . The bandwidth is given by W = E ( 0 , 0) - E ( n , zt) = 2 g [ x / 6 + (1 + 3,)2 _ 1 - 30] .

(27)

For the physical value ,=-~, we have W=3.14g. From Ref. [9] we can get the relation between g and the t parameter of the t - J model, and we have t = g (3 - 20)/4. We end then with a bandwidth W = 4.71 t. We obtain a bandwidth of the order of t. We notice that even in this disordered case the bandwidth was reduced from 8t

A. Belkasri, J.L. Richard / Physics Letters A 197 (1995) 353-360 -3

.

.

.

.

I

.

.

.

.

!

.

357 .

.

.

n=I/a -4

~t-5 -8

M

X

r

M

Fig. 1. (for the t - J model with J = 0: the free motion) to 4.71 t, which expresses the strong correlation in the system. We expect that in the ordered state the bandwidth will be reduced further. Now we want to examine the spin pattern around the hole. We started from a state without order and we want to calculate the local magnetization at site j, if the hole is at site i for the trial wave function q/(k) with energy Ek. Algebraic calculation leads to Wkak ( tlEk -- E k / g ) eka~+3(qEk--Ek/g) 2

(q,,(k), niS]~,(k)) (~u(k), ~,(k)) = - w o c ~

(28)

where we have used Eqs. (22), (23). Since the minimum of energy is at Q = (n, n), the hole will have the lowest energy E e and then we get the following correlation function between the hole and surrounding spins, (.is;>

=

(~(k),~(k))

L{

fdg-8,1

-- V / ~ \ 6 + 8 8

]

o cos[Q(i-j)].

(29)

For r/= ~ we obtain ( niS] ) ~- - ~w 0 cos[ Q( i - j ) ] .

(30)

Eq. (30) means that the hole polarizes the spins in its vicinity. The spin pattern around the hole is antiferromagnetic and the correlation decreases like wo when the distance between the hole and neighbor spin increases (see Fig. 2 ). The trial state ~ corresponds to a state with a total spin 89and Slot = + 89 The state corresponding to Slot = - 89 is just given by S~ot~. The correlation function for a hole with spin down is related to the case of spin up by ( n , S j ) " ) = (S~o~,, n,SjS?o, ~,) = - ( n i S j ) ( ' .

(31)

This implies that a hole with spin down will polarize spins in its vicinity with opposite direction with respect to a hole with spin up. Naively we may say that when two holes with opposite spins are introduced in the system it will be preferable to have them close to each other in order to cancel their spin polarizations and obtain again a system without magnetic order which favours the motion of the holes. This scenario might be similar to the Cooper pairing.

A. Belkasri, J.L. Richard/Physics LettersA 197 (1995) 353-360

358

+0.007

+0.007

+0.011

-0.070

+0.011

-0.070

S z = 1/2

-0.070

j.

+0.007

j.

+0.011

-0.070

4-0.011

+0.007 Fig. 2.

3.2. Locally ordered case Since experiments on copper oxides show that the spin susceptibility presents a sharp peak near the wave vector Q [ 11 ] in the metallic phase, we shall take a Lorentzian shape for the static susceptibility to describe the case of a locally ordered state. We write then

c(~) •q= 1 +~2[ 1 + 1 (COSqx +COS qe) ] ,

(32)

which may be correct only for short correlation length ~. C ( ( ) is determined by writing (Si'S~) = ] which gives 3(1~ C(~)= ~ ~

1

-i-+r189

)-1

]

.

(33)

The inverse Fourier transform of Eq. (32) gives

z(i-j)= 1~

C(~)exp [ i q ( i - j ) ]

l+r189176

_

4C(r

r

zo(i-J) exp[-iQ(i-j)]

(34)

and z e ( i - J ) satisfies the equation 4

~t ( - A )i,ze(l-j) + ~XQ(i--j) =Jij, with A the Laplacian on the square lattice. The asymptotic solution of this equation (i.e. for l i - j l - , ~ ) known [ 15 ] and we have

ze( i " j) ~exp( - l i-jl/~L) where ~L is given by

(35) is (36)

A. Belkasri, J.L. Richard/PhysicsLettersA 197(1995) 353-360

1

~LL = c h - I ( l +

359

1/~2) "

(37)

For ~ sufficiently large we have ~L-- ~/V/~. NOW having the phenomenological form o f the static susceptibility (Eq. ( 3 2 ) ) , we can compute XOl, Xll and ;(02 for different correlation lengths9 Table 1 contains some values o f these quantities. We notice that by increasing ~ we can have an increasingly ordered state. For example we almost recover the values of Z01, Zl~ and Zo2 predicted by the linear spin wave approximation for ~= 5. Inserting the values o f Zo0, Z~l and Zo2 in Eq. (21) we get the dispersion plotted in Fig. 3 for different values o f ~. We see from Fig. 3 that the spectrum changes drastically when the magnetic background changes from a disordered state to an ordered one. The m i n i m u m of Ek shifts from ( + n, + n) in the case o f a disordered background (~= 0) to ( + n~ 2, + n~ 2 ) for ~= 3. We see clearly that the bandwidth has been strongly reduced in comparison with the disordered case. It was reduced from I F ~- 3.14g (for ~= 0 ) to W = 1.3g (for ~ = 3 ). In terms o f t we get a bandwidth W = 1.8t. But it is still of the order o f t and not J. This is a discrepancy with previous calculations done in the framework o f the t - J model for an AFM background [ 1-4 ] which may be explained by the fact that no magnetization is present in our calculation since we considered our background as a singlet state. Another reason might be that with our variational approach we do not include the incoherent motion o f the quasi-particle which is the consequence of retarded effects. Table 1 Correlation functions Xo~,XHand Zo2calculated from the phenomenological form of the static susceptibility Eq. (32), for different

1 2 3 4 5

I , I*

XOl

XI 1

Xo2

-0.105 -0.199 -0.259 -0.299 -0.328

0.0278 0.094 0.150 0.192 0.223

0.015 0.058 0.101 0.138 0.167

-2

.....

77 = 1 / 6

~=0

.......... ~ = 1 -3

__~=3 I 9

-4

ii "',

I e 9 ~,

I ." s t .;

I 9149 " " . % 9 9149

.....

9

o'

~=5

,-" I

9

-5 ........

.' 9

. 9

II

-6

o ~

-7 M

X

r Fig. 3.

M

360

A. Belkasri, J.L. Richard/Physics Letters A 197 (i995) 353-360

4. Conclusion In this section we have c o m p u t e d the dispersion relation E ( k ) for a single hole in the framework o f the s p i n hole model using a variational method. The calculation was done for any rotational invariant magnetic background. A n d we have shown how the hole spectrum changes when the magnetic e n v i r o n m e n t goes from a diso r d e r e d background to an ordered one. Especially we have found that the m i n i m u m energy shifts from ( + rr, + n) for the completely disordered case to ( + n/2, + n / 2 ) in the case o f ordered magnetic background. The b a n d w i d t h is strongly reduced by a factor l, when the magnetic background changes from a disordered to an o r d e r e d state. An other interesting result that we have found is the spin polarization a r o u n d the hole. N a m e l y when a hole moves in a completely disordered background it will polarize spins in its vicinity. This polarization is decreasing with the distance as a power law a n d has an A F M ordering. We expect that two holes with opposite spins will tend to p a i r in o r d e r to r e m o v e their respective spin polarizations. This question deserves further studies and is now u n d e r investigation.

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