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A hole in a two-dimensional antiferromagnet
~
Solid State Communications, Vol. 76, No. 3, pp. 321-323, 1990. Printed in Great Britain.
0038-1098/9053.00+.00 Pergamon Press plc
A HOLE IN A T W O - D I M E N S I O N A L A N T I F E R R O M A G N E T A.V.Sherman Institute of Physics, E s t o n i a n A c a d e m y of Sciences, T a r t u 202400 USSR (Received
9 M~7 1990 by
A.A.Maradudin)
The t-J model of a hole in a spin-I/2 Heisenberg antiferromagnet on a square lattice is considered. In the spinwave a p p r o x i m a t i o n e i g e n e n e r g i e s and e i g e n f u n c t i o n s of the system are c a l c u l a t e d for a wide range of p a r a m e t e r s for points (0,0), (~/2,~/2), (~,0) and (~,~) of the B r i l l o u i n zone, where the band m a x i m u m and m i n i m u m are s u p p o s i n g l y located. The z c o m p o n e n t of the total spin of the states is d e t e r m i n e d r e v e a l i n g the formation of f e r r o m a g n e t i c regions around the hole.
Interest in strongly correlated e l e c t r o n i c s y s t e m s l h a s grown e s s e n t i a l ly after A n d e r s o n made a s u g g e s t i o n that the H u b b a r d model can explain the s u p e r c o n d u c t i n g t r a n s i t i o n in planarcuprate perovskites. One of essential problems here is the d e s c r i p t i o n of hole movement in an a n t i f e r r o m a g n e t i c plane m o d e l l i n g CuO z planes of these crystals. R e c e n t l y a number of papers devoted to this problem has appeared - . These papers, d e a l i n g with slightly d i f f e r e n t models (all being consequences of the Hubbard model at Umt, U is the Hubbard energy, t is the hopping matrix element), allow one to get a general i m p r e s s i o n about the character of the e n e r g y s p e c t r u m of the system at m o d e r a t e values of U/t. However, the mapping of a more r e a l i s t i c twoband Hubbard mqdel to the c o n s i d e r e d s i n g l e - b a n d one and available value~ of relevant parameters for La CuO 2 4 indicate that values of U/t may be la~get than those used in c a l c u l a t i o n s . Besides, in these papers the value of the total spin in the o b t a i n e d lowes~ states was not determined. It is known that at large values of U/t this quantity in the ground state may be larger than 1/2 (in a lattice with even n u m b e r of sites), which is c o n n e c t e d with a formation of a f e r r o m a g n e t i c a l l y ordered region around the hole. It is of interest to d e t e r m i n e the p a r a m e t e r regions where such f e r r o m a g n e t i c c l u s t e r s exist as their formation may have defin i t e c o n s e q u e n c e s for c a r r i e r t r a n s p o r t The aim of this paper is a calculation of energies of the system in h i g h - s y m m e t r i c points of the B r i l l o u i n zone and a b a n d w i d t h of the hole in the interval of the p a r a m e t e r U/t~200. The z component of the total spin, d e t e r m i ned in the course of calculations, is
used for i n d i c a t i n g the f e r r o m a g n e t i c cluster formation. In the large U limit, p r e s u m a b l y r e a l i z e d in cuprates, the Hubbard model may be t r a n s f o r m e d ~n~o s the t-J model with the H a m i l t o n i a n ' " H
=
P[-t
~ ' a l+ +aoalo
lao
+
2tz ~ Sl+ a.sllP, U I~
(i) where P p r o j e c t s out + states with d o u b l y o c c u p i e d sites, alo is the o p e r a t o r c r e a t i n g an e l e c t r o n in site 1 of a square l a t t i c e with spin p r o j e c t i o n a= ±I, the s u m m a t i o n over a p r o c e e d s over nearest neighbours, s z =~aa + a /2 I o I~ io ' Sl=
s~+ias[=a~ a-
i la I-~" The s p i n - w a v e approximation, being shown to be r e m a r k a b l y accurate in det e r m i n i n g the g r o u n d - s t a t e energy and s u b l a t t i c e m a g n e t i z a t i o n of HAF (the I
second term in r.h.s, of (1)) 1° , i~lint r o d u c e d by the f o l l o w i n g formulae+1I +PlY ~+--i s +I I =DIP 1 , sZ=ei,.l(nl/2_b~bl),~~
(2) where
b I is the s p i n - w a v e
(boson)
ope-
rator, n=(~,,) (the lattice peri+od is taken as a unit of length), n =~a a , 1 a la la P~=[l+a.exp(i,.l)]/2.
The
classical
Neel state IN>, the basis state of the s p i n - w a v e expansion, is d e t e r m i n e d by the c o n d i t i o n bllN>=0. In this approximation
the H a m i l t o n i a n
HAF
can be dia-
g o n a l i z e d by the u n i t a r y t r a n s f o r m a t i o n U = e x p { ~ u (b b -b + b + )I k k k-k k-k'
1 l+~k ak=~in i-~ k (3)
VOl. 76, NO. 3
A TWO-DIMENSIONAL ANTIFERROMAGNET
322 where
Wk-~.exp(ik.a)14, [U,P]=0. Transa formation (3) is equivalent to the Bog o l i u b o v t r a n s f o r m a t i o n but more convenient because the t ~ a n s f o r m e d Hamilionian HAp=U H l w U = ~ e k b k b k, ek=4~(l-lk), has
very
simple
ground
states
alo[NT.
i
To obtain the result of unitary transformation (3) applied to the first term in r.h.s, of (i) note that the action of this term on the c o n s i d e r e d states with one hole and w i t h o u t d o u b l y occupied sites is equiv~alent t q the actio~ -, where T~ of an o p e r a t o r t~ T](b] a + b]) la - --acts onlYaOn the index of the hole operator: T l a L a = ~ i L a L _ a _G. As a result, H=U+HU = - t
y/N
I
~ _a, ik. (l-a) [bkUk+b+kVk] + _ A Tlte
'
fficient constants are omitted• H a m i l t o n i a n (4) is c o n v e n i e n t for c a r r y i n g out c a l c u l a t i o n s 1 ~ i t h the help of the Lanczos a l g o r i t h m . As we are i n t e r e s t e d only in the lowest states the a l g o r i t h m is following. Let 10> be some initial n o r m a l i z e d state with proper symmetry. An o r t h o g o n a l c o m p l e m e n t to it I17 is built in accord with the prescription
Eo=, VIIT=(H-Eo)I07,
C 0 = { I + [ V / I E ~ - E I I ] 2 I -I/2
<1117=1.
(5) c1=coV/(E~-E~),
10'>=c~I17+CoI0>, #
EI=
\
%\
lak
2t z eik'l[b_+kUk+bkVk ] ] + =~..o b+b. , (4) U k k K K w h e r e U k = C h ( 2 a k ) , V k = - S h ( 2 ~ k) a n d u n s u -
State
\\
Fig.l.
The
*
20 energy
40 E0/t
of
tVt
| ~l i
60" 200
a hole
in
a
2d a n t i f e r r o m a g n e t as a function of U/t for points (0,0) (solid line), (~/2, x/2) (dashed line), (x,0) (dash-dotted line) of the B r i l l o u i n zone. There is a break i n the U/t axes at U/t>60.
I
L
t
c o r r e s p o n d s to the energy
_.+.0([..0].v.).,.+
°
E°
2
lower than the state E 0. Hence
2 energy of the initial 10'> is a better appro-
x i m a t i o n to the lowest (at a given symmetry, in p a r t i c u l a r at a given wave vector) state than ]07. By using i0'7 in (5) and (6) instead of 107 one can obtain the next approximation. In an idealized s i t u a t i o n this process should be p r o l o n g e d until a relative error of two subsequent values of E 0 becomes lower than a given value. However a number of spin c o n f i g u r a t i o n s n e c e s s a r y for d e s c r i b i n g states i0> and I17 grows from step to step and to avoid o v e r f l o wing the c o m p u t e r storage and g r o w i n g the c o m p u t a t i o n time the program used on each step selects j largest components and c o n t i n u e s c a l c u l a t i o n s (5), (6) as if only these c o m p o n e n t s are nonzero. By c h a n g i n g j an a c c u r a c y of o b t a i n e d e n e r g i e s may be estimated. At
o'
zb
4b
U/t
8b:
Fig.2. The hole b a n d w i d t h B/t as a function of U/t.
c h a n g i n g j twice (the number of config u r a t i o n s used in d e t e r m i n i n g E O varies a p p r o x i m a t e l y from 1000 to 2000) the a c c u r a c y appears to be of 2% for U/t= 200, 1% for U/t=70 and 0.3% for U/t=20. This gives an a c c u r a c y in d e t e r m i n i n g the e x c i t a t i o n b a n d w i d t h for these values of U/t of a p p r o x i m a t e l y 100%, 30% and 2% c o r r e s p o n d i n g l y . A more d e t a i l e d d e s c r i p t i o n of an analogous p r o c e d u r e applied ~ an analogous system may be found in In c a l c u l a t i n g E 0 in accord with + Eqs. (5), (6) the operators b I are con-
A TWO-DIMENSIONAL ANTIFERROMAGNET
VOI. 76, NO. 3
sidered as being determined ~ y.T the first formula in (2), so that blbl=0. Only
the
(±I,±I),
components (±2,0),
~m
with
(0,+2)
and
m =(0,0} , Um_a+Vm
with m=(±l,0), (0,±I), being much larger than other c o m p o n e n t s of the Fourier transform w_=N- ~.w, .exp(ik.m) (w= m
k
~
~, u or v) , have been c o n s i d e r e d as nonzero. Since [H, Sz]=0, where S z=~Slz , the e i g e n s t a t e s of H, besides a wave v e c t o r are c h a r a c t e r i z e d by the value of the z c o m p o n e n t of the total spin (note that [U,Sz]=O) . Because this value in the lowest state is unknown, a m i x t u r e of s t a t e s with d i f f e r e n t S should be taz ken as a s t a r t i n g state in (5) . The p r o c e d u r e selects itself essential components. The mixture of c o m p o n e n t s with S equal to 0.5 to 4.5 has been used in z our calculations. Results of c a l c u l a t i o n s are shown in Figs. 1 and 2 where the energies E 0 in the points (0,0) , (N/2,N/2) and (~,0) of the B r i l l o u i n zone and the width of the band (the largest of these energies minus the lowest one) are d e p i c t e d in d e p e n d e n c e on U/t. In Fig.l the zero of energy is the energy of the system with one hole and t=0. Values of E and S in the point (~,~) equal to 0 z those in the point (0,0). 5 In the region U/t~16 c o n s i d e r e d in our results are close to those obtained in that paper. However the energies o b t a i n e d by us for H a m i l t o n i a n (4) are systematical~ higher than the energles found in , and the d e v i a t i o n is different for d i f f e r e n t points: for U/t=4 it equals 5% in the point (0,0) and 16% in the points (N/2,N/2) and (N, 0). As a consequence, the values of the b a n d w i d t h are s l i g h t l y lower than those in . The origin of this d i f f e r e n c e is
323
s u p p o s e d to be due to the fact that the spin-wave a p p r o x i m a t i o n used does not take into account a static spin d i s t o r tion c a u s e d by the hole i n j e c t i o n and its c o n t r i b u t i o n to the energy. The e x c i t a t i o n band m i n i m u m is positioned in the point (~/2,~/2) for U/t ~70 and in the point (0,0) for larger U/t. It means that the hole m o v e m e n t with the most probable jumps to the next nearest sites for U/t>70 is replaced by a more complex e x c i t a t i o n transport, i n d i c a t i n g increased spin distortions c o n n e c t e d with the f e r r o m a g n e t i c cluster formation. As the lowest eigenstates of H at a given value of the wave vector these clusters (states with Sz>0.5) appear near U/t~10, at the beginning, in the points (0,0), (~,~), at the top of the band (Sz=l.5). Near the band bottom and in the point (~/2,~/2) the clusters appear when U/t~70. With the growth of U/t S z grows, acquiring h a l f - i n t e g e r values and at U/t=200 in the points (0,0) and (K,0) it equals to 3.5, whereas for (~/2,~/2) S =1.5. The z S z growth indicates the enlargement of the cluster size. It can be supposed that It~e t r a n s i t i o n to the N a g a o k a ' s limit occurs when the cluster size becomes c o m p a r a b l e with the sample size (cf w~%h the results for small systems ). By a c c e p t i n g 2 t 2 / U = J / 2 ~ 0 . 0 5 e V (see (I)) and (taking into accoun~ the mapping 6) t~0.3÷leV for LazCuO4-, one comes to a c o n c l u s i o n that the r e p r e s e n tative point of this crystal is in the region 12&U/t~40 in Figs. I, 2. In this region B~0.2eV and the states near the bottom and the top of the band are char a c t e r i z e d by Sz=0.5 and 1.5, respectively.
Acknowledgement - I am indebted to V.V. H i z h n y a k o v for s t i m u l a t i n g discussions.
REFERENCES
i. 2. 3.
4. 5. 6. 7.
8.
P.W.Anderson, Science 235, 1196 (1987). S.A.Trugman, Phys. Rev. B37, 1597 (1988). S.Schmitt-Rink, C . M . V a r m a & A.E Ruckenstein, Phys. Rev. Lett. 60, 2793 (1988). C.L.Kane, P.A.Lee & N.Read, Phys. Rev. B39, 6880 (1989). S.Sachdev, Phys. Rev. B39, 12232 (1989). F . C . Z h a n g & T.M.Rice, Phys. Rev. B37, 3759 (1988). L.F.Mattheiss, Phys. Rev. Lett. 58, 1028 (1987); J.Q.Liu, S.I.Tian,Y.F. Yah, Z.X.Liu, D . W . C h a n g & Z.X.Zhao, Modern Phys. Lett. B3, 257 (1989). E.L.Nagaev, Soy. Phys. JETP 27, 122
9. I0. II. 12.
13. 14. 15.
(1968) [Zh. Eksp. Teor. Fiz. 54, 228 (1968)]; V.Hizhnyakov & E.Sigmund, Physica C156, 655 (1988). J.E.Hirsch, Phys. Rev. Left. 54, 1317 (1985). D.A.Ruse, Phys. Rev. B37, 2380 (1988). E.L.Nagaev, Physics of Magnetic Semiconductors, Nauka, M o s c o w (1979). B.N.Parlett, The Symmetric Eigenvalue Problem, Prentice-Hall, New York (1980). A.V.Sberman, Phys. Status Solidi b135, 697 (1986); b145, 319 (1988). Y.Nagaoka, Phys. Rev. 147, 392 (1966). J.Bon~a, P . P r e l o v ~ e k & I.Sega, Phys. Rev. B39, 7074 (1989).
Solid State Communications, Vol. 76, No. 3, pp. 321-323, 1990. Printed in Great Britain.
0038-1098/9053.00+.00 Pergamon Press plc
A HOLE IN A T W O - D I M E N S I O N A L A N T I F E R R O M A G N E T A.V.Sherman Institute of Physics, E s t o n i a n A c a d e m y of Sciences, T a r t u 202400 USSR (Received
9 M~7 1990 by
A.A.Maradudin)
The t-J model of a hole in a spin-I/2 Heisenberg antiferromagnet on a square lattice is considered. In the spinwave a p p r o x i m a t i o n e i g e n e n e r g i e s and e i g e n f u n c t i o n s of the system are c a l c u l a t e d for a wide range of p a r a m e t e r s for points (0,0), (~/2,~/2), (~,0) and (~,~) of the B r i l l o u i n zone, where the band m a x i m u m and m i n i m u m are s u p p o s i n g l y located. The z c o m p o n e n t of the total spin of the states is d e t e r m i n e d r e v e a l i n g the formation of f e r r o m a g n e t i c regions around the hole.
Interest in strongly correlated e l e c t r o n i c s y s t e m s l h a s grown e s s e n t i a l ly after A n d e r s o n made a s u g g e s t i o n that the H u b b a r d model can explain the s u p e r c o n d u c t i n g t r a n s i t i o n in planarcuprate perovskites. One of essential problems here is the d e s c r i p t i o n of hole movement in an a n t i f e r r o m a g n e t i c plane m o d e l l i n g CuO z planes of these crystals. R e c e n t l y a number of papers devoted to this problem has appeared - . These papers, d e a l i n g with slightly d i f f e r e n t models (all being consequences of the Hubbard model at Umt, U is the Hubbard energy, t is the hopping matrix element), allow one to get a general i m p r e s s i o n about the character of the e n e r g y s p e c t r u m of the system at m o d e r a t e values of U/t. However, the mapping of a more r e a l i s t i c twoband Hubbard mqdel to the c o n s i d e r e d s i n g l e - b a n d one and available value~ of relevant parameters for La CuO 2 4 indicate that values of U/t may be la~get than those used in c a l c u l a t i o n s . Besides, in these papers the value of the total spin in the o b t a i n e d lowes~ states was not determined. It is known that at large values of U/t this quantity in the ground state may be larger than 1/2 (in a lattice with even n u m b e r of sites), which is c o n n e c t e d with a formation of a f e r r o m a g n e t i c a l l y ordered region around the hole. It is of interest to d e t e r m i n e the p a r a m e t e r regions where such f e r r o m a g n e t i c c l u s t e r s exist as their formation may have defin i t e c o n s e q u e n c e s for c a r r i e r t r a n s p o r t The aim of this paper is a calculation of energies of the system in h i g h - s y m m e t r i c points of the B r i l l o u i n zone and a b a n d w i d t h of the hole in the interval of the p a r a m e t e r U/t~200. The z component of the total spin, d e t e r m i ned in the course of calculations, is
used for i n d i c a t i n g the f e r r o m a g n e t i c cluster formation. In the large U limit, p r e s u m a b l y r e a l i z e d in cuprates, the Hubbard model may be t r a n s f o r m e d ~n~o s the t-J model with the H a m i l t o n i a n ' " H
=
P[-t
~ ' a l+ +aoalo
lao
+
2tz ~ Sl+ a.sllP, U I~
(i) where P p r o j e c t s out + states with d o u b l y o c c u p i e d sites, alo is the o p e r a t o r c r e a t i n g an e l e c t r o n in site 1 of a square l a t t i c e with spin p r o j e c t i o n a= ±I, the s u m m a t i o n over a p r o c e e d s over nearest neighbours, s z =~aa + a /2 I o I~ io ' Sl=
s~+ias[=a~ a-
i la I-~" The s p i n - w a v e approximation, being shown to be r e m a r k a b l y accurate in det e r m i n i n g the g r o u n d - s t a t e energy and s u b l a t t i c e m a g n e t i z a t i o n of HAF (the I
second term in r.h.s, of (1)) 1° , i~lint r o d u c e d by the f o l l o w i n g formulae+1I +PlY ~+--i s +I I =DIP 1 , sZ=ei,.l(nl/2_b~bl),~~
(2) where
b I is the s p i n - w a v e
(boson)
ope-
rator, n=(~,,) (the lattice peri+od is taken as a unit of length), n =~a a , 1 a la la P~=[l+a.exp(i,.l)]/2.
The
classical
Neel state IN>, the basis state of the s p i n - w a v e expansion, is d e t e r m i n e d by the c o n d i t i o n bllN>=0. In this approximation
the H a m i l t o n i a n
HAF
can be dia-
g o n a l i z e d by the u n i t a r y t r a n s f o r m a t i o n U = e x p { ~ u (b b -b + b + )I k k k-k k-k'
1 l+~k ak=~in i-~ k (3)
VOl. 76, NO. 3
A TWO-DIMENSIONAL ANTIFERROMAGNET
322 where
Wk-~.exp(ik.a)14, [U,P]=0. Transa formation (3) is equivalent to the Bog o l i u b o v t r a n s f o r m a t i o n but more convenient because the t ~ a n s f o r m e d Hamilionian HAp=U H l w U = ~ e k b k b k, ek=4~(l-lk), has
very
simple
ground
states
alo[NT.
i
To obtain the result of unitary transformation (3) applied to the first term in r.h.s, of (i) note that the action of this term on the c o n s i d e r e d states with one hole and w i t h o u t d o u b l y occupied sites is equiv~alent t q the actio~ -, where T~ of an o p e r a t o r t~ T](b] a + b]) la - --acts onlYaOn the index of the hole operator: T l a L a = ~ i L a L _ a _G. As a result, H=U+HU = - t
y/N
I
~ _a, ik. (l-a) [bkUk+b+kVk] + _ A Tlte
'
fficient constants are omitted• H a m i l t o n i a n (4) is c o n v e n i e n t for c a r r y i n g out c a l c u l a t i o n s 1 ~ i t h the help of the Lanczos a l g o r i t h m . As we are i n t e r e s t e d only in the lowest states the a l g o r i t h m is following. Let 10> be some initial n o r m a l i z e d state with proper symmetry. An o r t h o g o n a l c o m p l e m e n t to it I17 is built in accord with the prescription
Eo=
C 0 = { I + [ V / I E ~ - E I I ] 2 I -I/2
<1117=1.
(5) c1=coV/(E~-E~),
10'>=c~I17+CoI0>, #
EI=
\
%\
lak
2t z eik'l[b_+kUk+bkVk ] ] + =~..o b+b. , (4) U k k K K w h e r e U k = C h ( 2 a k ) , V k = - S h ( 2 ~ k) a n d u n s u -
State
\\
Fig.l.
The
*
20 energy
40 E0/t
of
tVt
| ~l i
60" 200
a hole
in
a
2d a n t i f e r r o m a g n e t as a function of U/t for points (0,0) (solid line), (~/2, x/2) (dashed line), (x,0) (dash-dotted line) of the B r i l l o u i n zone. There is a break i n the U/t axes at U/t>60.
I
L
t
c o r r e s p o n d s to the energy
_.+.0([..0].v.).,.+
°
E°
2
lower than the state E 0. Hence
2 energy of the initial 10'> is a better appro-
x i m a t i o n to the lowest (at a given symmetry, in p a r t i c u l a r at a given wave vector) state than ]07. By using i0'7 in (5) and (6) instead of 107 one can obtain the next approximation. In an idealized s i t u a t i o n this process should be p r o l o n g e d until a relative error of two subsequent values of E 0 becomes lower than a given value. However a number of spin c o n f i g u r a t i o n s n e c e s s a r y for d e s c r i b i n g states i0> and I17 grows from step to step and to avoid o v e r f l o wing the c o m p u t e r storage and g r o w i n g the c o m p u t a t i o n time the program used on each step selects j largest components and c o n t i n u e s c a l c u l a t i o n s (5), (6) as if only these c o m p o n e n t s are nonzero. By c h a n g i n g j an a c c u r a c y of o b t a i n e d e n e r g i e s may be estimated. At
o'
zb
4b
U/t
8b:
Fig.2. The hole b a n d w i d t h B/t as a function of U/t.
c h a n g i n g j twice (the number of config u r a t i o n s used in d e t e r m i n i n g E O varies a p p r o x i m a t e l y from 1000 to 2000) the a c c u r a c y appears to be of 2% for U/t= 200, 1% for U/t=70 and 0.3% for U/t=20. This gives an a c c u r a c y in d e t e r m i n i n g the e x c i t a t i o n b a n d w i d t h for these values of U/t of a p p r o x i m a t e l y 100%, 30% and 2% c o r r e s p o n d i n g l y . A more d e t a i l e d d e s c r i p t i o n of an analogous p r o c e d u r e applied ~ an analogous system may be found in In c a l c u l a t i n g E 0 in accord with + Eqs. (5), (6) the operators b I are con-
A TWO-DIMENSIONAL ANTIFERROMAGNET
VOI. 76, NO. 3
sidered as being determined ~ y.T the first formula in (2), so that blbl=0. Only
the
(±I,±I),
components (±2,0),
~m
with
(0,+2)
and
m =(0,0} , Um_a+Vm
with m=(±l,0), (0,±I), being much larger than other c o m p o n e n t s of the Fourier transform w_=N- ~.w, .exp(ik.m) (w= m
k
~
~, u or v) , have been c o n s i d e r e d as nonzero. Since [H, Sz]=0, where S z=~Slz , the e i g e n s t a t e s of H, besides a wave v e c t o r are c h a r a c t e r i z e d by the value of the z c o m p o n e n t of the total spin (note that [U,Sz]=O) . Because this value in the lowest state is unknown, a m i x t u r e of s t a t e s with d i f f e r e n t S should be taz ken as a s t a r t i n g state in (5) . The p r o c e d u r e selects itself essential components. The mixture of c o m p o n e n t s with S equal to 0.5 to 4.5 has been used in z our calculations. Results of c a l c u l a t i o n s are shown in Figs. 1 and 2 where the energies E 0 in the points (0,0) , (N/2,N/2) and (~,0) of the B r i l l o u i n zone and the width of the band (the largest of these energies minus the lowest one) are d e p i c t e d in d e p e n d e n c e on U/t. In Fig.l the zero of energy is the energy of the system with one hole and t=0. Values of E and S in the point (~,~) equal to 0 z those in the point (0,0). 5 In the region U/t~16 c o n s i d e r e d in our results are close to those obtained in that paper. However the energies o b t a i n e d by us for H a m i l t o n i a n (4) are systematical~ higher than the energles found in , and the d e v i a t i o n is different for d i f f e r e n t points: for U/t=4 it equals 5% in the point (0,0) and 16% in the points (N/2,N/2) and (N, 0). As a consequence, the values of the b a n d w i d t h are s l i g h t l y lower than those in . The origin of this d i f f e r e n c e is
323
s u p p o s e d to be due to the fact that the spin-wave a p p r o x i m a t i o n used does not take into account a static spin d i s t o r tion c a u s e d by the hole i n j e c t i o n and its c o n t r i b u t i o n to the energy. The e x c i t a t i o n band m i n i m u m is positioned in the point (~/2,~/2) for U/t ~70 and in the point (0,0) for larger U/t. It means that the hole m o v e m e n t with the most probable jumps to the next nearest sites for U/t>70 is replaced by a more complex e x c i t a t i o n transport, i n d i c a t i n g increased spin distortions c o n n e c t e d with the f e r r o m a g n e t i c cluster formation. As the lowest eigenstates of H at a given value of the wave vector these clusters (states with Sz>0.5) appear near U/t~10, at the beginning, in the points (0,0), (~,~), at the top of the band (Sz=l.5). Near the band bottom and in the point (~/2,~/2) the clusters appear when U/t~70. With the growth of U/t S z grows, acquiring h a l f - i n t e g e r values and at U/t=200 in the points (0,0) and (K,0) it equals to 3.5, whereas for (~/2,~/2) S =1.5. The z S z growth indicates the enlargement of the cluster size. It can be supposed that It~e t r a n s i t i o n to the N a g a o k a ' s limit occurs when the cluster size becomes c o m p a r a b l e with the sample size (cf w~%h the results for small systems ). By a c c e p t i n g 2 t 2 / U = J / 2 ~ 0 . 0 5 e V (see (I)) and (taking into accoun~ the mapping 6) t~0.3÷leV for LazCuO4-, one comes to a c o n c l u s i o n that the r e p r e s e n tative point of this crystal is in the region 12&U/t~40 in Figs. I, 2. In this region B~0.2eV and the states near the bottom and the top of the band are char a c t e r i z e d by Sz=0.5 and 1.5, respectively.
Acknowledgement - I am indebted to V.V. H i z h n y a k o v for s t i m u l a t i n g discussions.
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