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Effective singlet-triplet model for CuO2 plane in oxide superconductors: the change fluctuation regime
I ]lgA
Physica C 179 (1991) 159-166 North-Holland
Effective singlet-triplet model for C u O 2 plane in oxide superconductors: the charge fluctuation regime S.V. L o v t s o v
a
a n d V.Yu. Y u s h a n k h a i b
a Faculty of Physics, Irkutsk University, lrkutsk 664003, USSR b Laboratory o f Theoretical Physics, Joint Institute for Nuclear Research, Head Post office PO Box 79, Moscow, USSR
Received 4 June 1991
A reduction from the two-band Emery model to an effective Hamiltonian describing low-energyproperties of the CuO2 plane in oxide superconductors is presented. The approach is based on an expansion in the inter-site p--d hybridization amplitude 22~tdp (2ij < 0.14 ) between a Cu-orbital and a symmetric O-orbital in the Wannier representation. The resulting H hmiltonian is presented in terms of Hubbard operators for singlet and triplet local states. A further reduction to the single-band t-J model seems to be plausible
1. Introduction Since the discovery o f high-To superconductivity a great deal o f efforts have been made to obtain an effective Hamiltonian that describes the low-energy properties o f the CuO2 plane in oxide superconductors. As has been pointed out by Rice [ 1 ], the aim is to find the simplest Hamiltonian eliminating all terms which are not relevant to the determination o f the superconducting fixed point. Anderson was the first to propose [ 2 ] a Hamiltonian o f that sort which is now well-known as the t - J model. This model can be derived from the single-band H u b b a r d model in the large U-limit [ 3,4 ]. However, it is believed now that a generalized (multi-band) H u b b a r d model proposed by Emery [ 5 ] should be taken as a starting point o f this reduction. The first step in this way was made by Zhang and Rice [6] ( Z R hereafter) who suggested a perturbation scheme on the basis o f a simplified version o f the Emery model. They have found that the C u - O hybridization strongly binds a hole on each square o f O-ions with central Cu-ion to form a local singlet state with a binding energy (with respect to the triplet state and the nonbonding state) greater than the singlet bandwidth. After neglecting the triplet hole band ZR arrived at the t - J model. This has led to considerable discussion in the literature on the validity and limitations o f the reduction
from a multi-band description to a one-band model [ 7-13 ]. Particularly, it has been pointed out that the approach suggested by ZR is not controlled by a small parameter, i.e. the same physical quantities determine the binding energy o f the singlet, its band width, and the parameter of the singlet-triplet mixing as well. In this paper, starting with the same Hamiltonian as in ref. [ 6 ] we present a different perturbative approach to the reduction which allows us to overcome the objection mentioned above and to obtain the effective singlet-triplet Hamiltonian for the CuO2 plane in oxide superconductors. At the same time, our consideration based on the Wannier representation o f oxygen hole operators is similar to that by ZR. Thus, the Hamiltonian obtained, though somewhat more complicated, in main features resembles the t - J model. We believe that the latter is contained in our resulting Hamiltonian as a limiting case for a certain range of relevant model parameters and low temperatures.
2. Model Hamiltonian In the Emery model [ 5 ] the largest energy parameter is the on-site Coulomb interaction between copper d-electrons, Ud; and the largest hopping integral, t ap , corresponds to the o-bond between copper
0921-4534/91/$03.50 © 1991 Elsevier Science Publishers B.V. All rights reserved.
160
S. V. Lovtsov, V. Yu. Yushankhai / Effective singlet-triplet mode~for the CuO2plane
- and oxygen p(x, y)-orbitals. The simplified version of the Emery model we start with, is gained by keeping these parameters and ignoring the on-site Coulomb interaction on the O-ions, the intersite C u - O Coulomb interaction and direct O - O hopping terms. Two different regimes should be distinguished for hole dynamics. The first one, the socalled spin fluctuation regime, corresponds to a weak C u - O hybridization, tdP/A= t'l << 1 and tap~ ( Ud A ) = t [ << 1, where A=¢p--ed>0 is the energy distance between p- and d-atomic levels and (Ud--A) is the energy of the higher Hubbard sublevel at copper sites with respect to the p-level. Small t~ and t[ provide a commonly used perturbation expansion also used by Zr [ 6,10 ] and other authors [ 9,12,13 ] in deriving an effective one-band description. However, more representative values of the parameters in copper oxides do not give such weak hybridization. Actually, according to refs. [ 7,11,16 ] one has t~ ~ ½and t[ ~_ 7~and hence only the latter parameter may provide a good perturbation expansion. This is the charge fluctuation regime because the charge degrees of freedom at copper sites should be taken into account. Just this regime is treated below. After projecting out the upper Hubbard sublevel the two-band Emery model can be written to the second order in t[ as (in hole notation): d(x2-y2)
H=_AZ
X~+ io
where (N being the number of Cu sites) 2(i-j)=
1~
[ 1 - ½ ( c o s k x + c o s k y ) ] W 2 e ik(i-y) .
(4) One can check that 2 ( i - j ) rapidly decreases with the distance ( i - j ) . Particularly, several largest values of 2 ( i - j ) are presented in table 1. After introducing the notation V,j= 2tdP2(i--j), the Hamiltonian ( 1 ) can be represented in the following form: /40= ~ H~ i),
H=Ho+Hi,,,
i
Hi.~= E H t i ( ) ,
(5)
i~j
where U ~ ' ) = --A E X V + Vo E [X,~c,o+h.cl +JZ
+ [Xio~c i+o c i o - X i ~ c,,c~o],
(6)
+ ad + y[X,trOc,acjoX, c,~Cjo]+h.c.],
(7)
tY
H~g;J) = v o Y [ X ~ cj,, o
P~oP,o].
(1)
Here, Hubbard operators for a hole at Cu/-sites are introduced: XioO__ = d i+o ( 1 - n i _ o ) , X 7 ¢r -~ glia ( 1 - - n i _ o) ,
Clio,
where 8 = - G and t~m np = tdPS~m is the hybridization amplitude which is nonzero for nearest neighbours. The sign convention S , ~ - - + 1 is chosen in agreement with ZR [ 6 ]. The atomic energy of the p-level is chosen to be zero, ~p=0. According to ZR let us now define a symmetric combination of four O-hole states around a C U site p[s) _½ E SimPma' m
J
ima
, m , o ( U d - A ) [X, p ~ a p , ~ - X ,
=dio
(3)
p(d ) = ~ 2 ( i - j ) c j o ,
~. t~m[X~Pm~+h.c.] dp
i,dp ~,dp
Xi
which are not orthogonal. The Wannier representation provides the orthogonal symmetric O-states with corresponding operators c~o instead of p!g). The relation between them is familiar,
(2)
~_
and y = V o / ( U d - A ) , J = y Vo. Here H~ i) gives the onsite interaction with the C u - O hybridization term ~ V0=2tdP20 (where 2o=0.96), the hopping term Hi(~() with any distance between i a n d j sites is governed by the rather small parameters V~=2tPd2(i--j). Hence, the on-site Hamiltonian will be diagonalized Table 1 The values of the coefficients2(i-j) as functions of the intersite distance ( i - j ) = nxax+ nyay ny/n;
0 1 2
0
0.958 -0.140 0.014
1
-0.140 -0.023 -0.007
2
-0.014 -0.007 -0.003
S. V. Lovtsov, V.Yu. Yushankhai/ Effectivesinglet-tripletmodelfor the Cu02plane exactly and the hopping terms will be treated as a perturbation.
161
1
-Iz2.>
3. Diagonalization of Ho and perturbation expansion in 2tdPA0/A The diagonalization of Ho is performed at each site independently. There are two different fermionic degrees of freedom at a site and we choose the auxiliary representation for them, following Long [ 14 ] in the main features. Namely, we define two kinds of creation (annihilation) operators: g+ (gi~) for an oxygen a n d f + (f~) for a copper hole, respectively. Then the complete set of basis vectors for one-site states can be subdivided into two sectors. The first one is given by (a site index is dropped):
10>-I~,>,
f f f 10>-=lf~> , f ~+g ~ + g~-4-
1
(11)
Analogously, the on-site Hamiltonian H~ ° is represented as H6 '> = ~ [dlg~>
+ [zll~, > <~a, I + x/~Vo( I~a,> < ~ I +h.c.) - 2 J l ~ > <~il ] +[dl0~><0~l-J~l~><~e~l-A].
(12)
~r
We diagonalize H6 i) by applying the canonical transformation
n~ i) =exp(Si)H~ i) e x p ( - S i )
2~rf~go 1 0 > =- I~u>,
(8)
with the generators Si
Si=O~ [ Ig~i> (f~, I - h . c ] +02[ I~ > <~u~I - h . c . ] , +
+
+
+
l
-4-
+
tan(201 ) = (2 Vo/A),
( f , gr , f , g, , - - ~ E f ~ ga )10> x/z a
tan(202) = x / ~ 2Vo/(Zt+2J).
-=(It, >, It_, >, Izo>) with one copper hole at the site; a = 4- ½. The vectors I~u> and Ir,~> in eq. (8) give the singlet and triplet states, respectively. The vector I~,> represents the only three-particle state at the site. The second sector is given by the following basis vectors [0>,g ff 1 0 > - Igo>,g+g+ 10>-= I~0>
(9)
with no copper hole operator. Now we present how the operators X ~ and c,+ act in the united state space defined above. Their representation is given by the following formulas: 2a X °°= Ifo> <01 + ~ I~u>
1
Izo > <¢1,
c~+ = Ig~> <01 +2trlq~>
2a
1
I o>
(10)
(13)
Finally, we obtain B~i)= ~ E(w)lwi,)
i)
(14) where w = ~, g, f, v= 0, q~, ~ and the energies of these localized states are defined as follows E(~) = - J ,
E(g,J)=½[A+_~/AZ+(2Vo)Z],
E(O) =A,
E(¢, ~) = ½[ (A-2J) _+~/(A+2J)E+2(2Vo) 2]. (15) We note that the three triplet states from eq. (8) possess the same energy which is zero in our notation, i.e. E ( z ) =0. The on-site energy spectrum given by eq. ( 15 ) is presented in fig. 1 for various values of the ratio tdP/A, (in numerical calculations throughout in this paper we choose Ud by putting tdP/ (Vd--A)= L One can clearly see that at any value of the on-site hybridization parameter Vo( = 2triP;to) the states from the sector (9) are separated from the f-sector (8) by
162
S. E
Lovtsov, E Yu. Yushankhai / Effective singlet-triplet model for the CuO2 plane
ent kinds of contributions: P,/~intPl and P2IT~.tP2 correspond to the lower and upper subspaces, respectively, while nmix=P,nimP2+P21Qi,tPI mixes these subspaces. Our aim is to obtain the effective Hamiltonian Herr describing the low-energy physics in the system. Requiring I Vo[/A << 1, it can be written formally to the second order in V~j/A as
3.00
2.00
1.00
<~
0.00
Ld
Heff=P,I~OPl + P, nintPl
- 1.00
1
-PtHintP21~o_ff
-2.00
-3.00
-4.00
11
.00
,11
1;
, i l l
~i
0.20
111
1 1 1 1 1 1 ,
i , i
,,
0.40
l [ ) l
Ir
111
0.60
,l
0.80
t~P/6 Fig. 1. Calculated energies of the localized states as functions of the ratio taP/A.Craves from ( 1) to (7) correspondto E(~u),E(f), E(~), E(z), E(0), E(g) and E(~), respectively.The singlet state energy E(~u) is the lowest. an energy distance not smaller than the charge transfer gap A. So in the following we shall call the former the upper subspace and the latter the lower subspace. Just the lower subspace determines the low-energy physics in the system. It is worth noting also that though the three-particle state (~-level) lies in the lower subspace, it can be neglected in the small doping limit we treat below. A doped hole goes to the singlet state, which provides the lowest energy in the system. This state is well-separated from the triplet states. Particularly, at tdP/zI ~--½ one has E ( z ) - E ( u / ) __ 1.5/I. As a next step, we apply the canonical transformation eq. (13) to the hopping Hamiltonian
aint=exp(~iSi)HinteXp(-~iSi)
,
XiOO__..~=eS~X~e-S,,
where/~(o) is the energy of a ground state referred to. Taking into account that the largest value of 12gjl is for the nearest (i, j)-sites and equals 1211 ___0.14 we have I Vijl/A<2td°2~/A. Hence, the expansion, eq. (18) is valid if
tdP/A<< 1/(212, I ) ~ 4 .
+ ~+ = e s,cioe + -s, cia~cia (17)
Let us now introduce the projection operator Pi onto the lower subspace and P2 onto the upper subspace. Then the hopping Hamiltonian/7i,, involves differ-
(19)
We emphasize that the requirement (19) on t dp is softer than the one commonly uses, i.e. tap~A<< 1, in the derivation of the effective second-order Hamiltonian in the spin fluctuation regime [3,4,6,8-13 ]. Moreover, the expansion procedure developed here is more suitable for copper oxides where tdP/A ~- ½.
4. Effective Hamiltonian: zero- and first-order contributions
To obtain the effective Hamiltonian (18) explicitly, let us first note that the zero-order part PJ~oP, comes from eq. (14) after omitting the terms acting in the upper subspace and, of course, the three-particle contribution (~-term) due to a small doping. Further the first-order term, P l / ~ i n t P l is obtained from /tint by the following replacement
(16)
and H~,/ij) clearly is obtained from eq. (7) by replacing
(18)
~Xi, l , ~ + -+P, citr ~ + Pl -=- cia ~+'' City
(20)
After some simple algebra one obtains ~oO . ~ + 1 , (Xi,,, Cio
)=2aA~zclWi)(fa[
+ A ~ ; ¢ ( ~ [rio)(fa[+lri2a)(fo[), where
(21)
163
S. V. Lovtsov, V. Yu. Yushankhai / Effective singlet-triplet model for the CuO2 plane
fl A~c,x=t--~
- ~
COS 01 COS 0 2 + s i n 01 sin02;
,
)
sin 01cos 02 ,
A~;~ = ( - c o s 01; - s i n 01 ) •
(22)
To present the results in a more convenient form, we introduce a new type of H u b b a r d operators defined as follows: X?" =
Ifo>
I~,,>
(23)
X?"'--Iz.~>
PII4oP~ = E ( f ) ~ X ° ° + E ( ~ ) ~ X~, v, ia i
(24)
and the triplet sector in our notation is located at the zero energy E ( T ) = 0 . For the first-order contribution we finally obtain the form Pl/~intPl =
E
i~j,o
The results of a numerical analysis of the coefficients Ku, from eq. (26) are presented in fig. 2. However, more representative quantities are the hopping integrals between neighbouring i a n d j sites, vijguv = 2tdPk iKu,. Particularly, at ldP/A "~ ½ one has for the singlet-singlet hopping 2tdP21K~,~=0.16J (0.48 eV at d-~ 3 eV) and for triplet-triplet hopping 2tdP21K~-------0.12Zl. We note that these values are smaller by an order of magnitude than the relevant energy distance E ( r ) - E ( ~ , ) ~ - 1.5/1 (see fig. 1). Moreover, we obtain (at top/A ~- ½) that the singlettriplet mixing parameter is quite small: 2t°P21K~,~_ ~ --0.03A. These findings are consistent with the ZR result and hence make plausible a further reduction of the singlet-triplet description to the one-band t-J model [ 10 ]. In this case t = 2tdP2 IK~,~,. Recently, Hybertsen et al. [ 11 ] have used a cluster method to reduce a multi-band description to the one-band, the so-called t - t ' - J model. They added a t ' - t e r m describing the next nearest neighbour hopping of the singlet and obtained the following estimations: t--- 0.44 eV and t ' - - - 0 . 0 6 eV. In our consideration t' = 2taP,,].2K~ with '~2 ~ 0.02 (see table 1 ); hence t' (22/21)t ~- - 0 . 0 8 eV. It is remarkable that both the t and t' parameters obtained here are very close to that of ref. [ l 1 ]. Now we demonstrate more clearly that results ob-
Vij K~vXt X~
(
1.00
1I( r yOoyoO -1-")y2o, oytr,2tr "~2~xrzt~'ti ~tj i~..iti ~tj
+,/~(x°°s?2o +h.c.) ]
0.50
2 0.00 ::&
where K~,~ = 2A~A ~,
-0.50
K,~ = 2A ~A ~,
----A~A~+A~A~.
(26)
Here . ~ = A g - x/~7 c o s 01 COS 0 2 and all coefficients A~:~ are defined in eq. (22). Several contributions are involved in the Hamiltonian ofeq. (25): the first term ~ K ~ corresponds to singlet-singlet hoppings, the subsequent terms ~ K~ represent the hoppings from triplet to triplet states and the hybridization between them, and, finally, the terms ~ K~,~ represent the hybridization between singlet and triplet states.
-
1.00
-1.50
-2.00 0.00
0.20
0.40
tdP/A
0.60
0.80
Fig. 2. Calculated scale factors Ku, entering into the Hamiltonian ofeq. (25): (1) K~, (2) K~, (3) K~.
164
S.V. Lovtsov, V.Yu. Yushankhai /Effective singlet-triplet modelfor the Cu02 plane
tained by ZR [ 10] and others [ 12] are contained in ours as a limiting case. Actually, for small on-site hybridization 2 Vo= 4&P2o <
the following equation for the generator W of the canonical transformation
E ( r ) - E ( ¢ / ) =822(tl +t2)
n(2) elf
(27)
,
which is very close to the ZR result (here tl = (&p)2/ d and t2= (tdP)2/(Ud--d)). Analogously, for the nearest neighbours hopping and mixing parameters in the Hamiltonian of eq. (25) we obtain (i, j = n.n. )
VaK~,~,=-82o2,(½t, + t 2 ) ,
VaK~,=42o2, t,,
VaK~.,=4V/2 2o2, t2.
[/70, W] =/Tmix •
(29)
Then the second order contribution H~ 2) is formally given by ~
½P1 [ W,/Tmix]Pl
(30)
"
To present it explicitly, we decompose/7,,t, defined in eqs. (5), (7), and (16), into two parts: /Tint = Hi,~ + yHt,t, which is obvious. Then, particularly, the part PI/7~,,Pz can be written as Pl/7~nte2=
Z
g o [ ' ~ i ~ 1 2 ( e ) 2 d l - e ) al)
(28)
Taking into account that 82o2 t = - 1 we again come to the ZR result for the singlet-singlet hopping parameter. Moreover, all the parameters, eqs. (28), coincide with that of ref. [ 12 ] obtained in the spin fluctuation regime taP~d<< 1, tdP/(Ud--d) << 1. However, we emphasize that the estimations eqs. (27) and (28) follow from our consideration in a stronger limit, &p/d<< ~, while tdP/(Ud--A) << 1. Just the exact diagonalization of the on-site Hamiltonian, eq. (6), establishes this limit as a real weak hybridization (or spin fluctuation) regime. Strictly speaking, this limit is not realized in copper oxides, where tdP/~J~--½. The more general approach valid for rather strong hybridization taP~d<< 1/(2/211 )---4 is developed here, which gives the effective Hamiltonian ofeqs. (24) and (25).
5. Second-order contributions:superexchange
interaction In this section our aim is to obtain the contributions of second order in Va/A to the effective Hamiltonian Herr, i.e. the last term in eq. (18). However, instead of the operator form, given by eq. (18), we have chosen an equivalent way of derivation based on the canonical Schriffer-Wolff transformation. This transformation is more convenient because it allows one to take into account all the necessary virtual processes on an equal footing. Requiring that the final effective Hamiltonian should not include the term linear i n /Trnix=Pl/Tintez+h.c., we obtain
~ + 1 2 [ ~oo ~0o +c,~ ~ j l 2 +X)11)
(31)
~ 0 0 7,12 "~- [~/Xillt, j~ .d_.~+ --t. ia I 1 ..1_/~01~ ) ]
and the operators entering into eq. (31 ) are defined by eq. (21 ) and the following formulas
( Ai12, ~oo . City -12~] = 2aFx;~ If*) ( ~oiI ~oO . ~+ 12 (X~12, c~o )
= L~;c []f~ ) ( 0 i 1 + ~
1
]rio ) (gia ] + ] ri2o ) (gio 1]
+ 2 t r M ~ [ ~u~) (gj~ [, ~:x= ~
(32)
cos O~sin 02 - s i n 01 cos 02;
1 ) -~sin01sin02
,
L~,~ = ( - sin 01 ; cos 01 ) , ~:~=
sin
01 COS 0 2 --COS
01
sin 02 ;
~ 2 COS01 cos 02) •
(33)
The part yP~/7~,,P2 can be represented in the same manner, which we omit for clarity. After some lengthy calculations one can check that the second order contributions can be finally written as
S. V. Lovtsov, V. Yu. Yushankhai / Effective singlet-triplet mode~for the CuO2p/ane
2V~j r,4a vg yuu q-r~
position. It is worth noting that the second order (in taP~(Ud--A) ) contribution from the upper Hubbard sublevel is taken into account in the starting Hamiltonian eq. ( 1 ). This contribution enters additively into the exchange constant as the last term in eq. (35). We have numerically analyzed the factors a, which is positive, and b as functions of td°/A. These factors allow one to estimate the superexchange antiferromagnetic integral Jdd( i, j) = 2a( 2tdPAd)2 / d , and the additional integral lad(i, j)=2b(2tdp2o)2/A, entering into Hamiltonian (34), for any distance between the i a n d j sites. For the neighbouring sites ( i , j = n . n . ) these quantities are presented in fig. 3. Particularly, a t tdP/A"~½ we obtain J d d ( i , j = n . n . ) = J d d = 0 . 0 5 A . Then if A = 3 eV, one has Jdd--~0.15 eV which is very close to the experimental value [ 15 ] and to the estimation given in ref. [ l 1 ]. We note that in the typical approach [ 6,16 ] one has as a rule a larger value of Jdd in the same range of model parameters. For the second neighbours (diagonal bond) the corresponding superexchange integral is
yr6 y ~ u]
i#j
+
~" H(2)(ijk).
(34)
iv~jv~k
Here S? (a=x, y, z) are the spin-½ operators defined through the Hubbard operators as
a,,,,,X,
,
N, = E X'y,
aa'
t7
where a '~ are the Pauli matrices. The operators X~ ~ (p, v, 7, ~=~/, 0, _+ l ) act in the space of singlet and triplet states. Summation over equal indices in eq. (34) is implied. We deliberately subdivide boson-like operators into two classes, {S '~} and {XU~}, to explicitly extract superexchange interaction between localized spins given by the first sum in eq. (34). Three-site interaction H t2) (ijk) is beyond our consideration. In this paper we also do not explicitly specify the c-number matrix factor du~ and c~,~ and restrict ourselves to an analysis of the magnetic interaction between localized spins. The magnetic term that is of the second order in V~jin our consideration, is analogous to the fourth-order (in/do) t e r m in the conventional expansion and, hence, to the J-term in the t-J model. The dimensionless factors a and b in eq. (34) are given by
Jdd(i,j=n.n.n.) --J'dd= J d d
(•2/•
1) 2"~ O.O01zJ •
I f , J - 3 eV, then J ~ i d - ~ 0 . 0 0 3 eV which is again close to that of ref. [ 11 ]. We note that in our approach 0.30
a= ; [2E(f) c°s220'
~--E(z)
0.25
(cos 02+ I/x/~ sin 20, sin 02)2 2E(f) - E ( 0 ) --E(~)
sin220,
0.20
.•
(sin 02 - I/x/~ sin 20, cos 02)2 2E(f) - E ( 0 ) -EUp)
0.15
"o
]
+ (U---U--~AA) J '
b=a-2
165
-
(35)
COS3201 .
The form of the denominators in eq. (35) shows which kinds of virtual processes contribute to the superexchange constant. Namely, as a first step, one of two particles should be excited from the f-level to the initially empty level (with the energy E ( 0 ) ) , while the other goes either to a triplet z-level (E(z) = 0 in our notation ), or a singlet ~-, or ~level starting from the same f-level. The second step should restore their
-
0.10
<:~ 0.05 "-D 0.00
]dd/A
-0.05
-0.10
, , , , , , , , , 1 , , ,
0.00
0.20
,
, , , , , l l l , , , s
0.40
tdP/A
,,
, i , , , , , , , , 1
0.60
0.80
Fig. 3. Calculated superexchange antiferromagnetic integral Jda and additional integral Idd taken for n.n. sites. The definitions are given in the text.
166
s.v. Lovtsov, V. Yu. Yushankhai /Effective singlet-triplet model for the CuOe plane
there is an obvious strict relation between the singlet-singlet h o p p i n g p a r a m e t e r s t, t' a n d the exchange integrals Jdd, J'dd, n a m e l y Jdd ~
Jdd T h e values for these p a r a m e t e r s as o b t a i n e d n u m e r ically by H y b e r t s e n et al. [ 1 1 ] on the basis o f a cluster m e t h o d a p p r o x i m a t e l y obey this relation as well.
triplet local states. We analyzed the resulting H a m iltonian in some limiting cases and e s t i m a t e d the relevant quantities for the most representative values o f the m o d e l parameters. In this way we o b t a i n e d a good agreement with other considerations [ 6, l 0 - 1 2 ]. So a further reduction o f the effective H a m i l t o n i a n to the o n e - b a n d t - J , or t - t ' - J m o d e l [6,11 ] seems to be very plausible. However, to justify this reduction, some subtle questions r e m a i n to be e x a m i n e d carefully.
6. Conclusion References In this p a p e r we d e v e l o p e d an a p p r o a c h to the reduction o f the two-band p - d - m o d e l o f the CuO2 plane to an effective H a m i l t o n i a n giving the low-energy b e h a v i o u r o f the system. Following the m a i n idea o f Z R [6 ] we e l i m i n a t e the operators associated with the oxygen c o o r d i n a t e s a n d describe the system in terms o f local singlet a n d triplet states. However, our c o n s i d e r a t i o n is valid in a wider range o f the p - d hyb r i d i z a t i o n p a r a m e t e r t dp, including the charge-fluct u a t i o n regime, in c o m p a r i s o n with Z R who treated the limit o f relatively weak hybridization, i.e. the spin-fluctuation regime. Two successive steps are used in our approach: first, the on-site interactions involving the h y b r i d i z a t i o n t e r m Vo = 2tdP2o are kept exact and, second, intersite interactions ~ v- o = 2tdP2, ~ << Vo are treated perturbatively. This a p p r o a c h is d e v e l o p e d to the second o r d e r in V J A a n d the effective H a m i l t o n i a n is presented in terms o f H u b b a r d operators associated with singlet a n d
[ 1] T.M. Rice, Phys. Scripta T29 (1989) 72. [2] P.W. Anderson, Science 235 (1987) 1196. [3] C, Gros, R. Joynt and T.M. Rice, Phys. Rev. B 36 (1987) 381. [4] K. Huang and E. Manousakis, Phys. Rev. B 36 (1987) 8302. [5l V.J. Emery, Phys. Rev. Lett. 58 (1987) 2794. [6] F.C. Zhang and T.M. Rice, Phys. Rev. B 37 (1988) 3759. [7] F. Mila, Phys. Rev. B 38 (1988) 11358. [8] H. Eskes and G.A. Sawatzky, Phys. Rev. Lett. 61 (1988) 1415. [9] V.J. Emery and G. Reiter, Phys. Rev. B 38 (1988) 11938; ibid.,B 41 (1990) 7247. [ 10] F.C. Zhang and T.M. Rice, Phys. Rev. B 41 (1990) 7243. [ll]M.S. Hybertsen, E.B. Stechel, M. Schluter and D.R. Jennison, Phys. Rev. B 41 (1990) 11068. [ 12] J.L. Shen and C.S. Ting, Phys. Rev. B 41 (1990) 1969. [13] H.B. Pang, T. Xiang, Z.B. Su and L. Yu, Phys. Rev. B 41 (1988) 7209. [14] M.W. Long, Z. Phys. B 71 (1988) 23. [ 15 ] R.P. Singh et al., Phys. Rev. Len. 62 ( 1989 ) 2736. [ 16] T.M. Rice, F. Mila and F.C. Zhang, Preprint ETH-TH/9027.
Physica C 179 (1991) 159-166 North-Holland
Effective singlet-triplet model for C u O 2 plane in oxide superconductors: the charge fluctuation regime S.V. L o v t s o v
a
a n d V.Yu. Y u s h a n k h a i b
a Faculty of Physics, Irkutsk University, lrkutsk 664003, USSR b Laboratory o f Theoretical Physics, Joint Institute for Nuclear Research, Head Post office PO Box 79, Moscow, USSR
Received 4 June 1991
A reduction from the two-band Emery model to an effective Hamiltonian describing low-energyproperties of the CuO2 plane in oxide superconductors is presented. The approach is based on an expansion in the inter-site p--d hybridization amplitude 22~tdp (2ij < 0.14 ) between a Cu-orbital and a symmetric O-orbital in the Wannier representation. The resulting H hmiltonian is presented in terms of Hubbard operators for singlet and triplet local states. A further reduction to the single-band t-J model seems to be plausible
1. Introduction Since the discovery o f high-To superconductivity a great deal o f efforts have been made to obtain an effective Hamiltonian that describes the low-energy properties o f the CuO2 plane in oxide superconductors. As has been pointed out by Rice [ 1 ], the aim is to find the simplest Hamiltonian eliminating all terms which are not relevant to the determination o f the superconducting fixed point. Anderson was the first to propose [ 2 ] a Hamiltonian o f that sort which is now well-known as the t - J model. This model can be derived from the single-band H u b b a r d model in the large U-limit [ 3,4 ]. However, it is believed now that a generalized (multi-band) H u b b a r d model proposed by Emery [ 5 ] should be taken as a starting point o f this reduction. The first step in this way was made by Zhang and Rice [6] ( Z R hereafter) who suggested a perturbation scheme on the basis o f a simplified version o f the Emery model. They have found that the C u - O hybridization strongly binds a hole on each square o f O-ions with central Cu-ion to form a local singlet state with a binding energy (with respect to the triplet state and the nonbonding state) greater than the singlet bandwidth. After neglecting the triplet hole band ZR arrived at the t - J model. This has led to considerable discussion in the literature on the validity and limitations o f the reduction
from a multi-band description to a one-band model [ 7-13 ]. Particularly, it has been pointed out that the approach suggested by ZR is not controlled by a small parameter, i.e. the same physical quantities determine the binding energy o f the singlet, its band width, and the parameter of the singlet-triplet mixing as well. In this paper, starting with the same Hamiltonian as in ref. [ 6 ] we present a different perturbative approach to the reduction which allows us to overcome the objection mentioned above and to obtain the effective singlet-triplet Hamiltonian for the CuO2 plane in oxide superconductors. At the same time, our consideration based on the Wannier representation o f oxygen hole operators is similar to that by ZR. Thus, the Hamiltonian obtained, though somewhat more complicated, in main features resembles the t - J model. We believe that the latter is contained in our resulting Hamiltonian as a limiting case for a certain range of relevant model parameters and low temperatures.
2. Model Hamiltonian In the Emery model [ 5 ] the largest energy parameter is the on-site Coulomb interaction between copper d-electrons, Ud; and the largest hopping integral, t ap , corresponds to the o-bond between copper
0921-4534/91/$03.50 © 1991 Elsevier Science Publishers B.V. All rights reserved.
160
S. V. Lovtsov, V. Yu. Yushankhai / Effective singlet-triplet mode~for the CuO2plane
- and oxygen p(x, y)-orbitals. The simplified version of the Emery model we start with, is gained by keeping these parameters and ignoring the on-site Coulomb interaction on the O-ions, the intersite C u - O Coulomb interaction and direct O - O hopping terms. Two different regimes should be distinguished for hole dynamics. The first one, the socalled spin fluctuation regime, corresponds to a weak C u - O hybridization, tdP/A= t'l << 1 and tap~ ( Ud A ) = t [ << 1, where A=¢p--ed>0 is the energy distance between p- and d-atomic levels and (Ud--A) is the energy of the higher Hubbard sublevel at copper sites with respect to the p-level. Small t~ and t[ provide a commonly used perturbation expansion also used by Zr [ 6,10 ] and other authors [ 9,12,13 ] in deriving an effective one-band description. However, more representative values of the parameters in copper oxides do not give such weak hybridization. Actually, according to refs. [ 7,11,16 ] one has t~ ~ ½and t[ ~_ 7~and hence only the latter parameter may provide a good perturbation expansion. This is the charge fluctuation regime because the charge degrees of freedom at copper sites should be taken into account. Just this regime is treated below. After projecting out the upper Hubbard sublevel the two-band Emery model can be written to the second order in t[ as (in hole notation): d(x2-y2)
H=_AZ
X~+ io
where (N being the number of Cu sites) 2(i-j)=
1~
[ 1 - ½ ( c o s k x + c o s k y ) ] W 2 e ik(i-y) .
(4) One can check that 2 ( i - j ) rapidly decreases with the distance ( i - j ) . Particularly, several largest values of 2 ( i - j ) are presented in table 1. After introducing the notation V,j= 2tdP2(i--j), the Hamiltonian ( 1 ) can be represented in the following form: /40= ~ H~ i),
H=Ho+Hi,,,
i
Hi.~= E H t i ( ) ,
(5)
i~j
where U ~ ' ) = --A E X V + Vo E [X,~c,o+h.cl +JZ
+ [Xio~c i+o c i o - X i ~ c,,c~o],
(6)
+ ad + y[X,trOc,acjoX, c,~Cjo]+h.c.],
(7)
tY
H~g;J) = v o Y [ X ~ cj,, o
P~oP,o].
(1)
Here, Hubbard operators for a hole at Cu/-sites are introduced: XioO__ = d i+o ( 1 - n i _ o ) , X 7 ¢r -~ glia ( 1 - - n i _ o) ,
Clio,
where 8 = - G and t~m np = tdPS~m is the hybridization amplitude which is nonzero for nearest neighbours. The sign convention S , ~ - - + 1 is chosen in agreement with ZR [ 6 ]. The atomic energy of the p-level is chosen to be zero, ~p=0. According to ZR let us now define a symmetric combination of four O-hole states around a C U site p[s) _½ E SimPma' m
J
ima
, m , o ( U d - A ) [X, p ~ a p , ~ - X ,
=dio
(3)
p(d ) = ~ 2 ( i - j ) c j o ,
~. t~m[X~Pm~+h.c.] dp
i,dp ~,dp
Xi
which are not orthogonal. The Wannier representation provides the orthogonal symmetric O-states with corresponding operators c~o instead of p!g). The relation between them is familiar,
(2)
~_
and y = V o / ( U d - A ) , J = y Vo. Here H~ i) gives the onsite interaction with the C u - O hybridization term ~ V0=2tdP20 (where 2o=0.96), the hopping term Hi(~() with any distance between i a n d j sites is governed by the rather small parameters V~=2tPd2(i--j). Hence, the on-site Hamiltonian will be diagonalized Table 1 The values of the coefficients2(i-j) as functions of the intersite distance ( i - j ) = nxax+ nyay ny/n;
0 1 2
0
0.958 -0.140 0.014
1
-0.140 -0.023 -0.007
2
-0.014 -0.007 -0.003
S. V. Lovtsov, V.Yu. Yushankhai/ Effectivesinglet-tripletmodelfor the Cu02plane exactly and the hopping terms will be treated as a perturbation.
161
1
-Iz2.>
3. Diagonalization of Ho and perturbation expansion in 2tdPA0/A The diagonalization of Ho is performed at each site independently. There are two different fermionic degrees of freedom at a site and we choose the auxiliary representation for them, following Long [ 14 ] in the main features. Namely, we define two kinds of creation (annihilation) operators: g+ (gi~) for an oxygen a n d f + (f~) for a copper hole, respectively. Then the complete set of basis vectors for one-site states can be subdivided into two sectors. The first one is given by (a site index is dropped):
10>-I~,>,
f f f 10>-=lf~> , f ~+g ~ + g~-4-
1
(11)
Analogously, the on-site Hamiltonian H~ ° is represented as H6 '> = ~ [dlg~>
+ [zll~, > <~a, I + x/~Vo( I~a,> < ~ I +h.c.) - 2 J l ~ > <~il ] +[dl0~><0~l-J~l~><~e~l-A].
(12)
~r
We diagonalize H6 i) by applying the canonical transformation
n~ i) =exp(Si)H~ i) e x p ( - S i )
2~rf~go 1 0 > =- I~u>,
(8)
with the generators Si
Si=O~ [ Ig~i> (f~, I - h . c ] +02[ I~ > <~u~I - h . c . ] , +
+
+
+
l
-4-
+
tan(201 ) = (2 Vo/A),
( f , gr , f , g, , - - ~ E f ~ ga )10> x/z a
tan(202) = x / ~ 2Vo/(Zt+2J).
-=(It, >, It_, >, Izo>) with one copper hole at the site; a = 4- ½. The vectors I~u> and Ir,~> in eq. (8) give the singlet and triplet states, respectively. The vector I~,> represents the only three-particle state at the site. The second sector is given by the following basis vectors [0>,g ff 1 0 > - Igo>,g+g+ 10>-= I~0>
(9)
with no copper hole operator. Now we present how the operators X ~ and c,+ act in the united state space defined above. Their representation is given by the following formulas: 2a X °°= Ifo> <01 + ~ I~u>
1
Izo >
c~+ = Ig~> <01 +2trlq~>
2a
1
I o>
(10)
(13)
Finally, we obtain B~i)= ~ E(w)lwi,)
i)
(14) where w = ~, g, f, v= 0, q~, ~ and the energies of these localized states are defined as follows E(~) = - J ,
E(g,J)=½[A+_~/AZ+(2Vo)Z],
E(O) =A,
E(¢, ~) = ½[ (A-2J) _+~/(A+2J)E+2(2Vo) 2]. (15) We note that the three triplet states from eq. (8) possess the same energy which is zero in our notation, i.e. E ( z ) =0. The on-site energy spectrum given by eq. ( 15 ) is presented in fig. 1 for various values of the ratio tdP/A, (in numerical calculations throughout in this paper we choose Ud by putting tdP/ (Vd--A)= L One can clearly see that at any value of the on-site hybridization parameter Vo( = 2triP;to) the states from the sector (9) are separated from the f-sector (8) by
162
S. E
Lovtsov, E Yu. Yushankhai / Effective singlet-triplet model for the CuO2 plane
ent kinds of contributions: P,/~intPl and P2IT~.tP2 correspond to the lower and upper subspaces, respectively, while nmix=P,nimP2+P21Qi,tPI mixes these subspaces. Our aim is to obtain the effective Hamiltonian Herr describing the low-energy physics in the system. Requiring I Vo[/A << 1, it can be written formally to the second order in V~j/A as
3.00
2.00
1.00
<~
0.00
Ld
Heff=P,I~OPl + P, nintPl
- 1.00
1
-PtHintP21~o_ff
-2.00
-3.00
-4.00
11
.00
,11
1;
, i l l
~i
0.20
111
1 1 1 1 1 1 ,
i , i
,,
0.40
l [ ) l
Ir
111
0.60
,l
0.80
t~P/6 Fig. 1. Calculated energies of the localized states as functions of the ratio taP/A.Craves from ( 1) to (7) correspondto E(~u),E(f), E(~), E(z), E(0), E(g) and E(~), respectively.The singlet state energy E(~u) is the lowest. an energy distance not smaller than the charge transfer gap A. So in the following we shall call the former the upper subspace and the latter the lower subspace. Just the lower subspace determines the low-energy physics in the system. It is worth noting also that though the three-particle state (~-level) lies in the lower subspace, it can be neglected in the small doping limit we treat below. A doped hole goes to the singlet state, which provides the lowest energy in the system. This state is well-separated from the triplet states. Particularly, at tdP/zI ~--½ one has E ( z ) - E ( u / ) __ 1.5/I. As a next step, we apply the canonical transformation eq. (13) to the hopping Hamiltonian
aint=exp(~iSi)HinteXp(-~iSi)
,
XiOO__..~=eS~X~e-S,,
where/~(o) is the energy of a ground state referred to. Taking into account that the largest value of 12gjl is for the nearest (i, j)-sites and equals 1211 ___0.14 we have I Vijl/A<2td°2~/A. Hence, the expansion, eq. (18) is valid if
tdP/A<< 1/(212, I ) ~ 4 .
+ ~+ = e s,cioe + -s, cia~cia (17)
Let us now introduce the projection operator Pi onto the lower subspace and P2 onto the upper subspace. Then the hopping Hamiltonian/7i,, involves differ-
(19)
We emphasize that the requirement (19) on t dp is softer than the one commonly uses, i.e. tap~A<< 1, in the derivation of the effective second-order Hamiltonian in the spin fluctuation regime [3,4,6,8-13 ]. Moreover, the expansion procedure developed here is more suitable for copper oxides where tdP/A ~- ½.
4. Effective Hamiltonian: zero- and first-order contributions
To obtain the effective Hamiltonian (18) explicitly, let us first note that the zero-order part PJ~oP, comes from eq. (14) after omitting the terms acting in the upper subspace and, of course, the three-particle contribution (~-term) due to a small doping. Further the first-order term, P l / ~ i n t P l is obtained from /tint by the following replacement
(16)
and H~,/ij) clearly is obtained from eq. (7) by replacing
(18)
~Xi, l , ~ + -+P, citr ~ + Pl -=- cia ~+'' City
(20)
After some simple algebra one obtains ~oO . ~ + 1 , (Xi,,, Cio
)=2aA~zclWi)(fa[
+ A ~ ; ¢ ( ~ [rio)(fa[+lri2a)(fo[), where
(21)
163
S. V. Lovtsov, V. Yu. Yushankhai / Effective singlet-triplet model for the CuO2 plane
fl A~c,x=t--~
- ~
COS 01 COS 0 2 + s i n 01 sin02;
,
)
sin 01cos 02 ,
A~;~ = ( - c o s 01; - s i n 01 ) •
(22)
To present the results in a more convenient form, we introduce a new type of H u b b a r d operators defined as follows: X?" =
Ifo>
I~,,>
(23)
X?"'--Iz.~>
PII4oP~ = E ( f ) ~ X ° ° + E ( ~ ) ~ X~, v, ia i
(24)
and the triplet sector in our notation is located at the zero energy E ( T ) = 0 . For the first-order contribution we finally obtain the form Pl/~intPl =
E
i~j,o
The results of a numerical analysis of the coefficients Ku, from eq. (26) are presented in fig. 2. However, more representative quantities are the hopping integrals between neighbouring i a n d j sites, vijguv = 2tdPk iKu,. Particularly, at ldP/A "~ ½ one has for the singlet-singlet hopping 2tdP21K~,~=0.16J (0.48 eV at d-~ 3 eV) and for triplet-triplet hopping 2tdP21K~-------0.12Zl. We note that these values are smaller by an order of magnitude than the relevant energy distance E ( r ) - E ( ~ , ) ~ - 1.5/1 (see fig. 1). Moreover, we obtain (at top/A ~- ½) that the singlettriplet mixing parameter is quite small: 2t°P21K~,~_ ~ --0.03A. These findings are consistent with the ZR result and hence make plausible a further reduction of the singlet-triplet description to the one-band t-J model [ 10 ]. In this case t = 2tdP2 IK~,~,. Recently, Hybertsen et al. [ 11 ] have used a cluster method to reduce a multi-band description to the one-band, the so-called t - t ' - J model. They added a t ' - t e r m describing the next nearest neighbour hopping of the singlet and obtained the following estimations: t--- 0.44 eV and t ' - - - 0 . 0 6 eV. In our consideration t' = 2taP,,].2K~ with '~2 ~ 0.02 (see table 1 ); hence t' (22/21)t ~- - 0 . 0 8 eV. It is remarkable that both the t and t' parameters obtained here are very close to that of ref. [ l 1 ]. Now we demonstrate more clearly that results ob-
Vij K~vXt X~
(
1.00
1I( r yOoyoO -1-")y2o, oytr,2tr "~2~xrzt~'ti ~tj i~..iti ~tj
+,/~(x°°s?2o +h.c.) ]
0.50
2 0.00 ::&
where K~,~ = 2A~A ~,
-0.50
K,~ = 2A ~A ~,
----A~A~+A~A~.
(26)
Here . ~ = A g - x/~7 c o s 01 COS 0 2 and all coefficients A~:~ are defined in eq. (22). Several contributions are involved in the Hamiltonian ofeq. (25): the first term ~ K ~ corresponds to singlet-singlet hoppings, the subsequent terms ~ K~ represent the hoppings from triplet to triplet states and the hybridization between them, and, finally, the terms ~ K~,~ represent the hybridization between singlet and triplet states.
-
1.00
-1.50
-2.00 0.00
0.20
0.40
tdP/A
0.60
0.80
Fig. 2. Calculated scale factors Ku, entering into the Hamiltonian ofeq. (25): (1) K~, (2) K~, (3) K~.
164
S.V. Lovtsov, V.Yu. Yushankhai /Effective singlet-triplet modelfor the Cu02 plane
tained by ZR [ 10] and others [ 12] are contained in ours as a limiting case. Actually, for small on-site hybridization 2 Vo= 4&P2o <
the following equation for the generator W of the canonical transformation
E ( r ) - E ( ¢ / ) =822(tl +t2)
n(2) elf
(27)
,
which is very close to the ZR result (here tl = (&p)2/ d and t2= (tdP)2/(Ud--d)). Analogously, for the nearest neighbours hopping and mixing parameters in the Hamiltonian of eq. (25) we obtain (i, j = n.n. )
VaK~,~,=-82o2,(½t, + t 2 ) ,
VaK~,=42o2, t,,
VaK~.,=4V/2 2o2, t2.
[/70, W] =/Tmix •
(29)
Then the second order contribution H~ 2) is formally given by ~
½P1 [ W,/Tmix]Pl
(30)
"
To present it explicitly, we decompose/7,,t, defined in eqs. (5), (7), and (16), into two parts: /Tint = Hi,~ + yHt,t, which is obvious. Then, particularly, the part PI/7~,,Pz can be written as Pl/7~nte2=
Z
g o [ ' ~ i ~ 1 2 ( e ) 2 d l - e ) al)
(28)
Taking into account that 82o2 t = - 1 we again come to the ZR result for the singlet-singlet hopping parameter. Moreover, all the parameters, eqs. (28), coincide with that of ref. [ 12 ] obtained in the spin fluctuation regime taP~d<< 1, tdP/(Ud--d) << 1. However, we emphasize that the estimations eqs. (27) and (28) follow from our consideration in a stronger limit, &p/d<< ~, while tdP/(Ud--A) << 1. Just the exact diagonalization of the on-site Hamiltonian, eq. (6), establishes this limit as a real weak hybridization (or spin fluctuation) regime. Strictly speaking, this limit is not realized in copper oxides, where tdP/~J~--½. The more general approach valid for rather strong hybridization taP~d<< 1/(2/211 )---4 is developed here, which gives the effective Hamiltonian ofeqs. (24) and (25).
5. Second-order contributions:superexchange
interaction In this section our aim is to obtain the contributions of second order in Va/A to the effective Hamiltonian Herr, i.e. the last term in eq. (18). However, instead of the operator form, given by eq. (18), we have chosen an equivalent way of derivation based on the canonical Schriffer-Wolff transformation. This transformation is more convenient because it allows one to take into account all the necessary virtual processes on an equal footing. Requiring that the final effective Hamiltonian should not include the term linear i n /Trnix=Pl/Tintez+h.c., we obtain
~ + 1 2 [ ~oo ~0o +c,~ ~ j l 2 +X)11)
(31)
~ 0 0 7,12 "~- [~/Xillt, j~ .d_.~+ --t. ia I 1 ..1_/~01~ ) ]
and the operators entering into eq. (31 ) are defined by eq. (21 ) and the following formulas
( Ai12, ~oo . City -12~] = 2aFx;~ If*) ( ~oiI ~oO . ~+ 12 (X~12, c~o )
= L~;c []f~ ) ( 0 i 1 + ~
1
]rio ) (gia ] + ] ri2o ) (gio 1]
+ 2 t r M ~ [ ~u~) (gj~ [, ~:x= ~
(32)
cos O~sin 02 - s i n 01 cos 02;
1 ) -~sin01sin02
,
L~,~ = ( - sin 01 ; cos 01 ) , ~:~=
sin
01 COS 0 2 --COS
01
sin 02 ;
~ 2 COS01 cos 02) •
(33)
The part yP~/7~,,P2 can be represented in the same manner, which we omit for clarity. After some lengthy calculations one can check that the second order contributions can be finally written as
S. V. Lovtsov, V. Yu. Yushankhai / Effective singlet-triplet mode~for the CuO2p/ane
2V~j r,4a vg yuu q-r~
position. It is worth noting that the second order (in taP~(Ud--A) ) contribution from the upper Hubbard sublevel is taken into account in the starting Hamiltonian eq. ( 1 ). This contribution enters additively into the exchange constant as the last term in eq. (35). We have numerically analyzed the factors a, which is positive, and b as functions of td°/A. These factors allow one to estimate the superexchange antiferromagnetic integral Jdd( i, j) = 2a( 2tdPAd)2 / d , and the additional integral lad(i, j)=2b(2tdp2o)2/A, entering into Hamiltonian (34), for any distance between the i a n d j sites. For the neighbouring sites ( i , j = n . n . ) these quantities are presented in fig. 3. Particularly, a t tdP/A"~½ we obtain J d d ( i , j = n . n . ) = J d d = 0 . 0 5 A . Then if A = 3 eV, one has Jdd--~0.15 eV which is very close to the experimental value [ 15 ] and to the estimation given in ref. [ l 1 ]. We note that in the typical approach [ 6,16 ] one has as a rule a larger value of Jdd in the same range of model parameters. For the second neighbours (diagonal bond) the corresponding superexchange integral is
yr6 y ~ u]
i#j
+
~" H(2)(ijk).
(34)
iv~jv~k
Here S? (a=x, y, z) are the spin-½ operators defined through the Hubbard operators as
a,,,,,X,
,
N, = E X'y,
aa'
t7
where a '~ are the Pauli matrices. The operators X~ ~ (p, v, 7, ~=~/, 0, _+ l ) act in the space of singlet and triplet states. Summation over equal indices in eq. (34) is implied. We deliberately subdivide boson-like operators into two classes, {S '~} and {XU~}, to explicitly extract superexchange interaction between localized spins given by the first sum in eq. (34). Three-site interaction H t2) (ijk) is beyond our consideration. In this paper we also do not explicitly specify the c-number matrix factor du~ and c~,~ and restrict ourselves to an analysis of the magnetic interaction between localized spins. The magnetic term that is of the second order in V~jin our consideration, is analogous to the fourth-order (in/do) t e r m in the conventional expansion and, hence, to the J-term in the t-J model. The dimensionless factors a and b in eq. (34) are given by
Jdd(i,j=n.n.n.) --J'dd= J d d
(•2/•
1) 2"~ O.O01zJ •
I f , J - 3 eV, then J ~ i d - ~ 0 . 0 0 3 eV which is again close to that of ref. [ 11 ]. We note that in our approach 0.30
a= ; [2E(f) c°s220'
~--E(z)
0.25
(cos 02+ I/x/~ sin 20, sin 02)2 2E(f) - E ( 0 ) --E(~)
sin220,
0.20
.•
(sin 02 - I/x/~ sin 20, cos 02)2 2E(f) - E ( 0 ) -EUp)
0.15
"o
]
+ (U---U--~AA) J '
b=a-2
165
-
(35)
COS3201 .
The form of the denominators in eq. (35) shows which kinds of virtual processes contribute to the superexchange constant. Namely, as a first step, one of two particles should be excited from the f-level to the initially empty level (with the energy E ( 0 ) ) , while the other goes either to a triplet z-level (E(z) = 0 in our notation ), or a singlet ~-, or ~level starting from the same f-level. The second step should restore their
-
0.10
<:~ 0.05 "-D 0.00
]dd/A
-0.05
-0.10
, , , , , , , , , 1 , , ,
0.00
0.20
,
, , , , , l l l , , , s
0.40
tdP/A
,,
, i , , , , , , , , 1
0.60
0.80
Fig. 3. Calculated superexchange antiferromagnetic integral Jda and additional integral Idd taken for n.n. sites. The definitions are given in the text.
166
s.v. Lovtsov, V. Yu. Yushankhai /Effective singlet-triplet model for the CuOe plane
there is an obvious strict relation between the singlet-singlet h o p p i n g p a r a m e t e r s t, t' a n d the exchange integrals Jdd, J'dd, n a m e l y Jdd ~
Jdd T h e values for these p a r a m e t e r s as o b t a i n e d n u m e r ically by H y b e r t s e n et al. [ 1 1 ] on the basis o f a cluster m e t h o d a p p r o x i m a t e l y obey this relation as well.
triplet local states. We analyzed the resulting H a m iltonian in some limiting cases and e s t i m a t e d the relevant quantities for the most representative values o f the m o d e l parameters. In this way we o b t a i n e d a good agreement with other considerations [ 6, l 0 - 1 2 ]. So a further reduction o f the effective H a m i l t o n i a n to the o n e - b a n d t - J , or t - t ' - J m o d e l [6,11 ] seems to be very plausible. However, to justify this reduction, some subtle questions r e m a i n to be e x a m i n e d carefully.
6. Conclusion References In this p a p e r we d e v e l o p e d an a p p r o a c h to the reduction o f the two-band p - d - m o d e l o f the CuO2 plane to an effective H a m i l t o n i a n giving the low-energy b e h a v i o u r o f the system. Following the m a i n idea o f Z R [6 ] we e l i m i n a t e the operators associated with the oxygen c o o r d i n a t e s a n d describe the system in terms o f local singlet a n d triplet states. However, our c o n s i d e r a t i o n is valid in a wider range o f the p - d hyb r i d i z a t i o n p a r a m e t e r t dp, including the charge-fluct u a t i o n regime, in c o m p a r i s o n with Z R who treated the limit o f relatively weak hybridization, i.e. the spin-fluctuation regime. Two successive steps are used in our approach: first, the on-site interactions involving the h y b r i d i z a t i o n t e r m Vo = 2tdP2o are kept exact and, second, intersite interactions ~ v- o = 2tdP2, ~ << Vo are treated perturbatively. This a p p r o a c h is d e v e l o p e d to the second o r d e r in V J A a n d the effective H a m i l t o n i a n is presented in terms o f H u b b a r d operators associated with singlet a n d
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