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Ground state of ferromagnet with antiferromagnetic impurity
Solid State Commumcatlons, Vol 17, pp 867-870,
Pergamon Press,
1975.
Prmted m Great Bntam
GROUND STATE OF FERROMAGNET WITH ANTIFERROMAGNETIC IMPURITY V.1 Peresada, V N Afanasyev and V.S Borotiov Physical-Techmcal Institute of Low Temperatures of the Ukramlan Academy of Sciences, Kharkov, USSR (Recened
28 May 1975 by E.A Kaner)
The Helsenberg ferromagnet with an antlferromagnetlc nnpunty and arbitrary spm IS considered The method 1s suggested for constructing the ground state of such a system, the method usmg the Jacob1 matnx techmque As an example, there has been mvestlgated the ferromagnet wth a sunple cubic lattice and the matnx spm S = l/2, and the nnpunty spm S’ = 1 The energy and wave function of the ground state are found as dependent on the system parameters
THE PROBLEM of fmdmg the local states of a ferromagnetic crystal with the antiferromagnetic nnpurtty IS on of the basic problems m the theory of magnetlcally ordered systems, as is also the problem of constructmg the ground state of such a system The mvestlgatlon 1s all the more lfficult because of the non-one-particle nature of the problem when the nnpunty spm S > l/2 These questions have been constdered m a number of papers, but for the problem to be solved it was assumed either to involve a weakly coupled nnpurlty1-4 or to be one-dnnenslonal 6
the host lattice atoms and unpunty atoms, respectively, E. IS the energy of the total lattice maxnnum spm
state It 1s clear at once that the ground state of the ferromagnet urlth an antlferromagnetlc nnpunty should be close to the so-called “N6el” state m which the Z-proJections of the host lattice spms are ldentlcal and equal to S and the Z-proJection of the unpurity atom spm 1sequal to (- S’) At the same tune the wavefunctlon descnbmg the Nbel state is not the elgenfunctlon of the operator (1) The genume ground state 1s characterized by the same value of the total spm as the Nobelstate, I e S,, - 2.S’,‘*’ where S,, 1s the maxnnum spm of the lattice Consequently, the ground state ~nll be situated m the subspace of (Zs’) spm devlatlons the basis of which may be represented as a multitude of vectors
Tim paper suggests a new and powerful method of findmg the energy and the wave function of the ground state of a Helsenberg ferromagnet wtth an ant]ferromagnetic unpunty, tis method holdmg for arbitrary values of the matnx spm S and the nnpurlty spm s’
h,
The Ham&Oman of the Helsenberg ferromagnet Hrlth an antlferromagnetlc lmpunty located at the ongm of coordinates may be wntten as
m=
CiJ. ,TtlSX
s, IO>.
(2)
IO>denotes the state 111wluch all the spms have the mamum proJectIon on the quantlzatlon axis Z, s;=s; -by I’ c u, m are normahzmg factors, I, 1, m are indices numbermg the magnetic lattice pomts , these m&ces may comclde with each other
Here the summation 1sperformed over the magnetic lattice pomts, the origm of coordmates excepted, T,,:> 0 (T,, = 0), T& < 0 - the exchange mteractlon constants, S,(r f 0) and Sh are the spin operators of
On any of the vectors (2) the subspace can be built that 1s mvanant Hrlth respect to the operator (1) and 1sspanned by the vectors 867
868
FERROMAGNET WITH ANTIFERROMAGNETIC IMPURITY
h,* ,mffh,,
,m-*h,,,
.m
(3)
We wrll refer to tlus subspace as generated by the vector h,,, ,,, The smallest ergenvalue of the operator H 111such a subspace wrll correspond to the ground state rf the vectors (3) are not orthogonal to thts ground state It can be shown that the subspace generated by any of the vectors (2) contams the ground state and that thrs state ISseparated by a gap from the contmuous spectrum m such a subspace The vector descnbmg the “Neel” state ho = ce(S;)ss’ 10)
with mrtral condrtrons P-l(E)
PO(E) = 1
= 0,
From the theory of lmear operators the rmportant property of roots of the polynomrals P,(E) is known.1° The energres of locahzed states comcrde with the roots of the equation P,(e) = 0 when n + = It allows to approxrmately fmd the energes of locahzed states using the roots of the polynomrals P,(E) with finite n It appears that the drfference between thrs found energy value and the genume one vamshes exponentially with n growmg (when n + - )
(4)
1smost surtable to choose as a generatmg vector. Other locahzed states5’6 may also exrst m thrs system which are orthogonal to the subspace generated by the vector ho The energres of these states he considerably hrgher than the energy of the genume ground state 6 The suggested method allows to find these states as well Accordmg to the general theory’-’ the separated subspaces help to reduce the problem of fmdmg the ground state of the system under consrderatron to the calculatron of the smallest ergenvalue of a certam Jacobi matnx and the correspondmg ergenfuctron In the basrs obtamed by a consequential orthogonalrzatron of the vectors (3) the operator H matnx has the Jacob! form
To illustrate the method, let us find the energy and the wave function of the ground state of a ferromagnet wrth a snnple cubic magnetic lattice and the nearest nerghbour mteractron The host lattice spm 1s S = l/2 At the orrgm of coordmates the antrferromagnetic rmpunty 1slocated wrth S’ = 1 The constants of the exchange mteractron are equal to I and I’, respectively As shown by the calculations, due to the good convergence of the result for thrs system, rt 1ssufficrent to consrder the polynomral P,(e) with n = 3 It 1s not drffcult to calculate the first three basic vectors (not normahzed) m the subspace generated by the vector of the type (4) lo =
(W2 IO),
11 = Ho -Hoolo
(5:1
&, =
Vol 17,No 7
l2 = Hli -Hllll
= F s;&Io), -Hello
=
The ergenfunctron of the operator H correspondmg to the ergenvalue E may be cast m the form8 G(e) = ‘4 j.
P,(e)&,
(6)
where 1,~ the orthonormahzed basis m whrch the operator H matrrx has the form (5), A 1sthe normalrzmg factor The coefficrents P,(E) has polynomrals expressed by means of the matrrx elements of the Jacobi matnx (5) accordmg to the recurrent relation
Here the summation over 6, S 1,6 2 IS performed over the first coordmation sphere of the lmpunty spm, the mdrces A,, A2 run over the second and thrrd coordmatron spheres, respectively From these expressrons the matrrc elements (I, and b, of the matnx (5) are obtamed a0
L,Pn(e)
= (e -a,-,
= Ho0
E
(109~0)
-
Ill0
)F,-, (e) - LPn-2(e) (7)
bo
=
Ho,
II2
=
-q.fo,
FERROMAGNET WITH ANTIFERROMAGNETIC IMPURITY
Vol 17, No. 7
0
FIG 1 Ground state energy of Hetsenberg ferromagnet urlth the antiferromagnetic unpurtty S’ = 1
111 =
Hll = $5
bl
=
H12
a2
=
Hz2 =
27(1-
-677).eg
r)) + 20~7~ (5 - 27)). Eg 3 54 + 120772
( \ N
,-
1
24684
FIG 2 The coeffclents A , = AP, of the ground state wave function development (6) of the ferromagnet ~rlth the antlferromagnetlc unpunty S’ = 1
,
where 9 = I T’/TI, e. = 6T, llli II1s the norm of the vector 1,. The energy IScounted from the level
Eo = -
869
61S2 - 6I’SS’.
N is the number of pomts m the spm system With the help of these matnx elements and the recurrent relation (7) the polynomuds P,(e) are constructed. The smallest root of the equation Pa(e) = 0 1s approxnnately equal to the energy of the ground state. In Fig 1 the dependence 1s plotted of the system ground state energy on the parameter q. Accountmg for the expression (6) and the obtamed energy values the wave function of the ground state 1s found Figure 2 presents the results of these calculations The magmtudes PO, P, , P2 approach certam positive values when II’/11 + 0~ Also, the weights of the states Pi with I > 3 approach zero It can be shown that for the unpunty spm S’(2S’ + 1) of the coefficients Pi m the wave function of the ground state (6) approach the non-zero hnuts.
06: 0 2
4
6
8
10
?wd)
FIG 3 The mean value of Z-projection of the unpunty atom spm S’ = 1 When the ground state wave function 1sknown, tt IS not hfficult to find any charactenstlc of this state, e.g. 111Fig. 3 shows the magnitude ($ ), 1.e the mean value of Z-prqection of the unpunty atom spin as a function of 9 For the nearest nelghbours the mean value of spm Z-projection differs from the maxunum one by not more than 4% For the next nelghbours the devlatlon does not exceed several ten’s parts of one per cent The method suggested m th.u paper may be successfully applied to the mvestlgatlon of the localized states appearmg m the magnetically ordered systems m the presence of vanolls defects (pomt defects of substltutlon or penetration, hnear defects, the presence of the surface etc ). It is unportant that the method can be apphed to the problem wth many-particle excitations
FERROMAGNET WITH ANTIFERROMAGNETIC IMPURITY
870
Vol 17,No
7
REFERENCES ISHII H , KANAMORI J & NAKAMURA T ,Proq
Theor. Phys 33,795 (1965)
2.
IZYUMOV Yu.A. & MEDVEDEV M V., Zhum. Exptl. Teor Fzz. 51,5 17 (m Russian) (1966)
3
WANG Y L. & CALLEN H , Phrs Rev. 160,358 (1966)
4
BALAGUROV B Ya., VAKS V G , Zhurn. Expfl
5
ON0 I & END0 Y ,J Phys Sot. Japan 35,661
6
IZYUMOV Yu.A. & MEDVEDEV M V , Teorza Magnztoupoqadochennykh Knstallov s Pnmeslamz (Theory of Magnetzcally Ordered C’?ystalswzth Impuntzes) p 153 “Nauka”, Moscow (in Russian) (1970) PERESADA V.1 , Zhum. Exptl. Teor Fzz 53,605 PERESADA
V I & AFANASJEV
Teor. Fzz 66, 1135 (in Russian) (1974) (1973)
(m Russian) (1967)
V N., Zhum Exptl Teor Fzz 57, 135 (m Russian) (1970)
PERESADA V I , AFANASYEV V N & BOROVIKOV V S , Fzzzka nzzkzkh temperatur (Physzcs of Low Temperatures) 1,443 (m Russian) (1975) 10
PERESADA V I , Trudy FTINT AN USSR (Proceedzng of the Physzcotechnzcai Instztute of Low Temperatures, Academy of Sciences of the Ukramtan SSR), II, 172 (m Russian) (1968)
Pergamon Press,
1975.
Prmted m Great Bntam
GROUND STATE OF FERROMAGNET WITH ANTIFERROMAGNETIC IMPURITY V.1 Peresada, V N Afanasyev and V.S Borotiov Physical-Techmcal Institute of Low Temperatures of the Ukramlan Academy of Sciences, Kharkov, USSR (Recened
28 May 1975 by E.A Kaner)
The Helsenberg ferromagnet with an antlferromagnetlc nnpunty and arbitrary spm IS considered The method 1s suggested for constructing the ground state of such a system, the method usmg the Jacob1 matnx techmque As an example, there has been mvestlgated the ferromagnet wth a sunple cubic lattice and the matnx spm S = l/2, and the nnpunty spm S’ = 1 The energy and wave function of the ground state are found as dependent on the system parameters
THE PROBLEM of fmdmg the local states of a ferromagnetic crystal with the antiferromagnetic nnpurtty IS on of the basic problems m the theory of magnetlcally ordered systems, as is also the problem of constructmg the ground state of such a system The mvestlgatlon 1s all the more lfficult because of the non-one-particle nature of the problem when the nnpunty spm S > l/2 These questions have been constdered m a number of papers, but for the problem to be solved it was assumed either to involve a weakly coupled nnpurlty1-4 or to be one-dnnenslonal 6
the host lattice atoms and unpunty atoms, respectively, E. IS the energy of the total lattice maxnnum spm
state It 1s clear at once that the ground state of the ferromagnet urlth an antlferromagnetlc nnpunty should be close to the so-called “N6el” state m which the Z-proJections of the host lattice spms are ldentlcal and equal to S and the Z-proJection of the unpurity atom spm 1sequal to (- S’) At the same tune the wavefunctlon descnbmg the Nbel state is not the elgenfunctlon of the operator (1) The genume ground state 1s characterized by the same value of the total spm as the Nobelstate, I e S,, - 2.S’,‘*’ where S,, 1s the maxnnum spm of the lattice Consequently, the ground state ~nll be situated m the subspace of (Zs’) spm devlatlons the basis of which may be represented as a multitude of vectors
Tim paper suggests a new and powerful method of findmg the energy and the wave function of the ground state of a Helsenberg ferromagnet wtth an ant]ferromagnetic unpunty, tis method holdmg for arbitrary values of the matnx spm S and the nnpurlty spm s’
h,
The Ham&Oman of the Helsenberg ferromagnet Hrlth an antlferromagnetlc lmpunty located at the ongm of coordinates may be wntten as
m=
CiJ. ,TtlSX
s, IO>.
(2)
IO>denotes the state 111wluch all the spms have the mamum proJectIon on the quantlzatlon axis Z, s;=s; -by I’ c u, m are normahzmg factors, I, 1, m are indices numbermg the magnetic lattice pomts , these m&ces may comclde with each other
Here the summation 1sperformed over the magnetic lattice pomts, the origm of coordmates excepted, T,,:> 0 (T,, = 0), T& < 0 - the exchange mteractlon constants, S,(r f 0) and Sh are the spin operators of
On any of the vectors (2) the subspace can be built that 1s mvanant Hrlth respect to the operator (1) and 1sspanned by the vectors 867
868
FERROMAGNET WITH ANTIFERROMAGNETIC IMPURITY
h,* ,mffh,,
,m-*h,,,
.m
(3)
We wrll refer to tlus subspace as generated by the vector h,,, ,,, The smallest ergenvalue of the operator H 111such a subspace wrll correspond to the ground state rf the vectors (3) are not orthogonal to thts ground state It can be shown that the subspace generated by any of the vectors (2) contams the ground state and that thrs state ISseparated by a gap from the contmuous spectrum m such a subspace The vector descnbmg the “Neel” state ho = ce(S;)ss’ 10)
with mrtral condrtrons P-l(E)
PO(E) = 1
= 0,
From the theory of lmear operators the rmportant property of roots of the polynomrals P,(E) is known.1° The energres of locahzed states comcrde with the roots of the equation P,(e) = 0 when n + = It allows to approxrmately fmd the energes of locahzed states using the roots of the polynomrals P,(E) with finite n It appears that the drfference between thrs found energy value and the genume one vamshes exponentially with n growmg (when n + - )
(4)
1smost surtable to choose as a generatmg vector. Other locahzed states5’6 may also exrst m thrs system which are orthogonal to the subspace generated by the vector ho The energres of these states he considerably hrgher than the energy of the genume ground state 6 The suggested method allows to find these states as well Accordmg to the general theory’-’ the separated subspaces help to reduce the problem of fmdmg the ground state of the system under consrderatron to the calculatron of the smallest ergenvalue of a certam Jacobi matnx and the correspondmg ergenfuctron In the basrs obtamed by a consequential orthogonalrzatron of the vectors (3) the operator H matnx has the Jacob! form
To illustrate the method, let us find the energy and the wave function of the ground state of a ferromagnet wrth a snnple cubic magnetic lattice and the nearest nerghbour mteractron The host lattice spm 1s S = l/2 At the orrgm of coordmates the antrferromagnetic rmpunty 1slocated wrth S’ = 1 The constants of the exchange mteractron are equal to I and I’, respectively As shown by the calculations, due to the good convergence of the result for thrs system, rt 1ssufficrent to consrder the polynomral P,(e) with n = 3 It 1s not drffcult to calculate the first three basic vectors (not normahzed) m the subspace generated by the vector of the type (4) lo =
(W2 IO),
11 = Ho -Hoolo
(5:1
&, =
Vol 17,No 7
l2 = Hli -Hllll
= F s;&Io), -Hello
=
The ergenfunctron of the operator H correspondmg to the ergenvalue E may be cast m the form8 G(e) = ‘4 j.
P,(e)&,
(6)
where 1,~ the orthonormahzed basis m whrch the operator H matrrx has the form (5), A 1sthe normalrzmg factor The coefficrents P,(E) has polynomrals expressed by means of the matrrx elements of the Jacobi matnx (5) accordmg to the recurrent relation
Here the summation over 6, S 1,6 2 IS performed over the first coordmation sphere of the lmpunty spm, the mdrces A,, A2 run over the second and thrrd coordmatron spheres, respectively From these expressrons the matrrc elements (I, and b, of the matnx (5) are obtamed a0
L,Pn(e)
= (e -a,-,
= Ho0
E
(109~0)
-
Ill0
)F,-, (e) - LPn-2(e) (7)
bo
=
Ho,
II2
=
-q.fo,
FERROMAGNET WITH ANTIFERROMAGNETIC IMPURITY
Vol 17, No. 7
0
FIG 1 Ground state energy of Hetsenberg ferromagnet urlth the antiferromagnetic unpurtty S’ = 1
111 =
Hll = $5
bl
=
H12
a2
=
Hz2 =
27(1-
-677).eg
r)) + 20~7~ (5 - 27)). Eg 3 54 + 120772
( \ N
,-
1
24684
FIG 2 The coeffclents A , = AP, of the ground state wave function development (6) of the ferromagnet ~rlth the antlferromagnetlc unpunty S’ = 1
,
where 9 = I T’/TI, e. = 6T, llli II1s the norm of the vector 1,. The energy IScounted from the level
Eo = -
869
61S2 - 6I’SS’.
N is the number of pomts m the spm system With the help of these matnx elements and the recurrent relation (7) the polynomuds P,(e) are constructed. The smallest root of the equation Pa(e) = 0 1s approxnnately equal to the energy of the ground state. In Fig 1 the dependence 1s plotted of the system ground state energy on the parameter q. Accountmg for the expression (6) and the obtamed energy values the wave function of the ground state 1s found Figure 2 presents the results of these calculations The magmtudes PO, P, , P2 approach certam positive values when II’/11 + 0~ Also, the weights of the states Pi with I > 3 approach zero It can be shown that for the unpunty spm S’(2S’ + 1) of the coefficients Pi m the wave function of the ground state (6) approach the non-zero hnuts.
06: 0 2
4
6
8
10
?wd)
FIG 3 The mean value of Z-projection of the unpunty atom spm S’ = 1 When the ground state wave function 1sknown, tt IS not hfficult to find any charactenstlc of this state, e.g. 111Fig. 3 shows the magnitude ($ ), 1.e the mean value of Z-prqection of the unpunty atom spin as a function of 9 For the nearest nelghbours the mean value of spm Z-projection differs from the maxunum one by not more than 4% For the next nelghbours the devlatlon does not exceed several ten’s parts of one per cent The method suggested m th.u paper may be successfully applied to the mvestlgatlon of the localized states appearmg m the magnetically ordered systems m the presence of vanolls defects (pomt defects of substltutlon or penetration, hnear defects, the presence of the surface etc ). It is unportant that the method can be apphed to the problem wth many-particle excitations
FERROMAGNET WITH ANTIFERROMAGNETIC IMPURITY
870
Vol 17,No
7
REFERENCES ISHII H , KANAMORI J & NAKAMURA T ,Proq
Theor. Phys 33,795 (1965)
2.
IZYUMOV Yu.A. & MEDVEDEV M V., Zhum. Exptl. Teor Fzz. 51,5 17 (m Russian) (1966)
3
WANG Y L. & CALLEN H , Phrs Rev. 160,358 (1966)
4
BALAGUROV B Ya., VAKS V G , Zhurn. Expfl
5
ON0 I & END0 Y ,J Phys Sot. Japan 35,661
6
IZYUMOV Yu.A. & MEDVEDEV M V , Teorza Magnztoupoqadochennykh Knstallov s Pnmeslamz (Theory of Magnetzcally Ordered C’?ystalswzth Impuntzes) p 153 “Nauka”, Moscow (in Russian) (1970) PERESADA V.1 , Zhum. Exptl. Teor Fzz 53,605 PERESADA
V I & AFANASJEV
Teor. Fzz 66, 1135 (in Russian) (1974) (1973)
(m Russian) (1967)
V N., Zhum Exptl Teor Fzz 57, 135 (m Russian) (1970)
PERESADA V I , AFANASYEV V N & BOROVIKOV V S , Fzzzka nzzkzkh temperatur (Physzcs of Low Temperatures) 1,443 (m Russian) (1975) 10
PERESADA V I , Trudy FTINT AN USSR (Proceedzng of the Physzcotechnzcai Instztute of Low Temperatures, Academy of Sciences of the Ukramtan SSR), II, 172 (m Russian) (1968)