- Email: [email protected]
Collective excitations in singlet-triplet paramagnet
Volume 62A, number 2
PHYSICS LETTERS
25 July 1977
C O L L E C T I V E E X C I T A T I O N S IN S I N G L E T - T R I P L E T P A R A M A G N E T E. SZCZEPANIAK Solid State Theory Division, Institute o f Physics, A. Mickiewicz University, Poznaff, Poland
Received 24 March 1977 The spectrum of the singlet-triplet model is investigated by the spectral density method. For high temperatures, the temperature dependence of the spectrum is obtained. The singlet-triplet model is highly interesting, because many rare-earth compounds exist having an A 1 singlet ground state, and a first excited T 1 triplet state (e.g. TmN [1], TbSb [2] and Pr3T1 [9]). This model has been considered by Hsieh and Blume [3] and by Hsieh [4] applying the RPA-MFA method. It appeared of interest to treat this model by the spectral density method (SDM), successfully applied to systems exhibiting phase transitions [5-7]. The Hamiltonian in the absence of an external magnetic field is of the form [3] t
tl
where S and T are - 1 / 2 operators. In order to investigate the collective excitations of this system, it is convenient to use the standard basis operators formalism [8]. In this formalism we have ^+_ 1 i . . . . . S i _~_I2(L14 + Z~ 4 _ LI31 + Z~2) ' T+ :N/~_~(_LI4 +L24i +z31i +L~2), ~i~z--~'U'121"ri+L~I +Lg3 _L~$4) '
i-~t-~x2-L21
-
3
" +Lg 3+L{14 '
where L~a, are standard basis operators, ct = 1 denotes the singlet state, ot = 2 - the triplet state I 1,0), and a = 3, 4 the other triplet states. The corresponding one-particle spectral density matrix for longitudinal excitations is defined as follows:
Aq(6O) = \ - < [ L q l ' L2q(1")] -)co
- ( [ L q l ' L12q(~')] ) w ) ,
(I)
and for transversal excitations as:
\ - ( [ L q l ' L31q(~')] -)to
- ( [ t l l ' / 1 4 q(z)] ,)to
where L q , are Fourier transforms of the standard-basis operators. This paper is restricted to the paramagnetic region (jz) = (aS z + bT z) = 0, meaning that (L33) = (/,44). On the approximation of the spectral density matrix Aq(cO) and ~kq(¢O)by one matrix Dirac 5-function each, and on calculation of the unknown functional parameters within the first two moments [5], we obtain the following equations for the energy Eq of longitudinal excitations and for the energy Uq of transversal excitations:
Eq =~/(Mq)2 _(Mq2)2iiL11)_(L22),
Uq = \/(Plq) 2 -(P~2)2/(L11>-(L33) ',
(3,4)
where Mlq, Mq, and P I ' P2q are diagonal and off-diagonal elements of the first moment matrix of the spectral den111
Volume 62A, number 2
PHYSICS LETTERS
25 July 1977
sity matrix Aq(CO) and, respectively the matrix ~q(CO). Neglecting transitions between triplet states, we obtain for the first moments the following expressions: M q ( ( L l l ) - (L22)) I - A +(~ =
+(a-b)2~j(k) 2N
(- (L22))J(q)
P~I :P'2
k
cth
x/(pk)2 _ (pk) 2
2((Lll )-(L22))
Mq = (_ (L22)) [ (a - b) 2 ((Zll>_ (Z22))j(q) 2
2N
( a - b ) 2x--~ Mk+M k 13x/~Mk)2 _ (Mk) 2 J(k) / , . ~ = ,_ ~ cth L,v k x/(M~) z _ (M~)a 2 ( ( L l l ) - (L22)
+~Z..I
V ~ 1 ) 2 _ (p~)2 cth
(5)
(a + b) 2"5"~_,. +_, P1k , ~x~Pkl )2 _(pk)2] ~-~ "L~J(q Ic)~/(Plk)2_(pk)2 ctn ~ - - ~ ],
..,, + m2 ..,, (a-_b) 2_ ~-~,.Igk'l '"--1_ 2N
~,/(g1~): _(M~): -~- - """ X/~lk):~ ~ ~ k ) 2 cth 2((LI1)-(L22> )
2(<~II > --)
pq1=(_)[_, (~(_)](q)+~3,1(k) ,+ (a - b) z 4N k
J" 1 '"2 cth x/(Mlk)2 _ (Mk) 2 2((Lll ) - (L22>)
(a +b)2 ~g(q M~ ~x/(Mkl)2 -(M~)2 ÷3(a_b)2~ j(k ) . Pkl - PI~ t3x/{Plg)2 - (P~2)2 4N +k)v/(Mk)2_(Mk2):icth ~ ( ~ 2 2 )) 4N k v/(pk)2_(pk)2Cth2((Lll)-(L22 ))
4N
k
x/(Pkl)2- (pk)2 c t h ~ 7 - - ( L ~ 2 >
) 3'
(7)
M~
~,/~M~I)~ - ( ~ ) ~ (8)
, (a - b) 2 ~
-r~
-
P 1k - P 2I,
. / 3 ~ / ( P 11,2 ) - ( P 2 k )2
. ~kJ(k)x/(Plk)2 _ (pk)2 ctla .2 (.( L.l l )._ . (L22>)
(f3 = 1/k B T, k B is Boltzmann's
(a + b ) 2
4N
v', .
PI
L.a J(q +k)V~/~l)2_(pk) k
. I x ~ i k ) 2 - ( / I ) 2 J1 cth . . . . 2 2 ( ( L l l ) - (L22))
constant) where the following decoupling was performed:
(L#~q >.
(9)
We now proceed to obtain the relation for the difference in population of the singlet state and of one of the triplet states (L 11 ) - (L22> or (L 11 > - (L33)" From the multiplication rule for LocL,r8 we get (Lll)
=~1 ~q(Lq12L21q) '
(L22)=_~q(LqlL;2q).
The correlations and can be calculated using the relation LSI u i ~ > - uf><~> = ( ([~,B(r)] ,) 1 - e~w 112
d___~_~ 2zr "
Volume 62A, number 2
PHYSICS LETTERS
25 July 1977
The average valu'es Of L l l and L22 take the form
2N q x/(Mq)2 _ (M~)2 cth ~
F
1
L 2
-
~
"J((Lll)
-
(L22))
1
M~I 3V(Mlq)2 - (Mq)2"l ' 2 2cth ~ _ < - ~ 2 2 ) ) J(
Assuming the conditions (L22) = (L33)= (L44)
(10)
or equivalently (Lll) +3(/,22 )= 1 we obtain 2
Mq
cth
2((L11)_(L22)) _j .
(11)
The sums,' occurring in eqs. (5)-(8) and (11) represent a new result specifically due to our use of the SDM approach. Our assumption of the condition (10)is justified by the circumstance that, in the paramagnetic region, all triplet states have the same energy. For T = 0 K we obtain, up to order A 2, the following expressions
Eq = Uq = A{1 -AT(q)- ~A 2 [@(q) -a/z] }
(Lll)- (L22)= 1 -A2/z,
E0 = (/-DT=0K = -- ~NA(3 +A2/z)
(12) (13, 14)
where z is the coordination number, and
A = (a - b)2J(O)/2A; T(q) =J(q)[J(O). It is worth noting that, on omission of the lattice sums and insertion of (Lll) = 1, (L22) = 0, our formulae (3) and (4) reduce to the weU-known RPA-MFA expressions [3], [4] for the excitation energies at T ='0 K E RPA'MFAq = URPA'MFA = AX/I -- 2A3'(q).
(15)
Also, as should be expected, our expressions (12)-(14) at z ~ oo go over into the RPA-MFA expressions up to order A 2 because two-site correlations neglected in the RPA-MFA give contributions proportional to z -1 in the SDM expressions up to order A 2. Thus, the difference between our SDM (3), (4) and the RPA-MFA (5) spectrum at T = 0 K, up to order A 2, amounts to E q - E qRPA-MFA =Uq __UqRPA-MFA=A2A2/z. In the high temperature region we obtain up to order A 2 A ( ( L l l ) - (L22)= 4 k B T - A 1
8A2kBT . ~ ) ( 4 k B T - A) 3
(16)
113
Volume 62A, number 2
Eq A-
Uq_1_A A
PHYSICS LETTERS
A 4kBT-A
7(q)
2AB kBT 3t(q)+SAz 2 . kBT z 4kBT-A 4kBT-A
25 July 1977
A2 2
A2
(17)
( 4 k B T _ A ) 2 "12(q)
where B = (a + b)2 J(O)/2A. Thus, in the high temperature region, our SDM spectrum (3), (4) and RPA-MFA spectrum, up to order A 2, differ by Eq ERPA'MFAq Uq uRPA'MFAq 8A 2 k BT 2AB k BT
A
A
- A
A
z
4kBT- A
z
4 k B T - A 7(q).
Similarly, eqs. (16), (17) go over into RPA-MFA at z -+ oo. The investigated spectrum is expressed up to the order A 2 and we have obtained corrections proportional to z - 1 in contradistinction to the RPA-MFA method at T = 0 K. In the high temperature region, our correction depends on temperature, although this dependence is very weak. The difference between the population of the singlet state and that of the triplet state (see eq. (13)) is now endowed with a correction proportional to z -1 and is no longer equal to 1 even at T-- 0 K. This reflects the fact that the crystal ground state is not the real ground state of the singlet-triplet Hamiltonian [3]. Generally, SDM offers improvements in comparison with RPA-MFA, but leads to a highly complicated set o f self-consistent equations. Particularly, we expect important differences in comparison with RPA-MFA in the behaviour of the elementary excitations generated by transitions between the triplet states. Thanks are due to Prof. dr L. Kowalewski, A. Popielewicz, R. Micnas and G. Kamieniarz for their very helpful discussions.
References [1] [2] [3] [4] [5] [6] [7] [8] [9]
114
B.R. Cooper, Phys. Rev. 163 (1967) 444. B.R. Cooper and O. Vogt, Phys. Rev. B 1 (1970) 1218. Y.Y. Hsieh and M. Blume, Phys. Rev. B 6 (1972) 2684. Y.Y. Hsieh, Can. Jour. Phys. 51 (1973) 50. O. Kalashnikov and E.S. Fradkin, Phys. Stat. Sol. b 59 (1973) 9. G. Kamieniarz, A. Popielewicz, R. Micnas and E. Szezepaniak, Phys. Lett. A 55 (1976) 329. A. Popielewicz and E. Szczepaniak, Phys. Stat. Sol. b 74 (1976) K81. S.B. Haley and P. ErdSs, Phys. Rev. B 5 (1972) 1106. E. Bucher, I.P. Malta and A.8. Cooper, Phys. Rev. B 6 (1972) 2710.
PHYSICS LETTERS
25 July 1977
C O L L E C T I V E E X C I T A T I O N S IN S I N G L E T - T R I P L E T P A R A M A G N E T E. SZCZEPANIAK Solid State Theory Division, Institute o f Physics, A. Mickiewicz University, Poznaff, Poland
Received 24 March 1977 The spectrum of the singlet-triplet model is investigated by the spectral density method. For high temperatures, the temperature dependence of the spectrum is obtained. The singlet-triplet model is highly interesting, because many rare-earth compounds exist having an A 1 singlet ground state, and a first excited T 1 triplet state (e.g. TmN [1], TbSb [2] and Pr3T1 [9]). This model has been considered by Hsieh and Blume [3] and by Hsieh [4] applying the RPA-MFA method. It appeared of interest to treat this model by the spectral density method (SDM), successfully applied to systems exhibiting phase transitions [5-7]. The Hamiltonian in the absence of an external magnetic field is of the form [3] t
tl
where S and T are - 1 / 2 operators. In order to investigate the collective excitations of this system, it is convenient to use the standard basis operators formalism [8]. In this formalism we have ^+_ 1 i . . . . . S i _~_I2(L14 + Z~ 4 _ LI31 + Z~2) ' T+ :N/~_~(_LI4 +L24i +z31i +L~2), ~i~z--~'U'121"ri+L~I +Lg3 _L~$4) '
i-~t-~x2-L21
-
3
" +Lg 3+L{14 '
where L~a, are standard basis operators, ct = 1 denotes the singlet state, ot = 2 - the triplet state I 1,0), and a = 3, 4 the other triplet states. The corresponding one-particle spectral density matrix for longitudinal excitations is defined as follows:
Aq(6O) = \ - < [ L q l ' L2q(1")] -)co
- ( [ L q l ' L12q(~')] ) w ) ,
(I)
and for transversal excitations as:
\ - ( [ L q l ' L31q(~')] -)to
- ( [ t l l ' / 1 4 q(z)] ,)to
where L q , are Fourier transforms of the standard-basis operators. This paper is restricted to the paramagnetic region (jz) = (aS z + bT z) = 0, meaning that (L33) = (/,44). On the approximation of the spectral density matrix Aq(cO) and ~kq(¢O)by one matrix Dirac 5-function each, and on calculation of the unknown functional parameters within the first two moments [5], we obtain the following equations for the energy Eq of longitudinal excitations and for the energy Uq of transversal excitations:
Eq =~/(Mq)2 _(Mq2)2iiL11)_(L22),
Uq = \/(Plq) 2 -(P~2)2/(L11>-(L33) ',
(3,4)
where Mlq, Mq, and P I ' P2q are diagonal and off-diagonal elements of the first moment matrix of the spectral den111
Volume 62A, number 2
PHYSICS LETTERS
25 July 1977
sity matrix Aq(CO) and, respectively the matrix ~q(CO). Neglecting transitions between triplet states, we obtain for the first moments the following expressions: M q ( ( L l l ) - (L22)) I - A +(~ =
+(a-b)2~j(k) 2N
(
P~I :P'2
k
cth
x/(pk)2 _ (pk) 2
2((Lll )-(L22))
Mq = (
2N
( a - b ) 2x--~ Mk+M k 13x/~Mk)2 _ (Mk) 2 J(k) / , . ~ = ,_ ~ cth L,v k x/(M~) z _ (M~)a 2 ( ( L l l ) - (L22)
+~Z..I
V ~ 1 ) 2 _ (p~)2 cth
(5)
(a + b) 2"5"~_,. +_, P1k , ~x~Pkl )2 _(pk)2] ~-~ "L~J(q Ic)~/(Plk)2_(pk)2 ctn ~ - - ~ ],
..,, + m2 ..,, (a-_b) 2_ ~-~,.Igk'l '"--1_ 2N
~,/(g1~): _(M~): -~- - """ X/~lk):~ ~ ~ k ) 2 cth 2((LI1)-(L22> )
2(<~II > --
pq1=(
J" 1 '"2 cth x/(Mlk)2 _ (Mk) 2 2((Lll ) - (L22>)
(a +b)2 ~g(q M~ ~x/(Mkl)2 -(M~)2 ÷3(a_b)2~ j(k ) . Pkl - PI~ t3x/{Plg)2 - (P~2)2 4N +k)v/(Mk)2_(Mk2):icth ~ ( ~ 2 2 )) 4N k v/(pk)2_(pk)2Cth2((Lll)-(L22 ))
4N
k
x/(Pkl)2- (pk)2 c t h ~ 7 - - ( L ~ 2 >
) 3'
(7)
M~
~,/~M~I)~ - ( ~ ) ~ (8)
, (a - b) 2 ~
-r~
-
P 1k - P 2I,
. / 3 ~ / ( P 11,2 ) - ( P 2 k )2
. ~kJ(k)x/(Plk)2 _ (pk)2 ctla .2 (.( L.l l )._ . (L22>)
(f3 = 1/k B T, k B is Boltzmann's
(a + b ) 2
4N
v', .
PI
L.a J(q +k)V~/~l)2_(pk) k
. I x ~ i k ) 2 - ( / I ) 2 J1 cth . . . . 2 2 ( ( L l l ) - (L22))
constant) where the following decoupling was performed:
(9)
We now proceed to obtain the relation for the difference in population of the singlet state and of one of the triplet states (L 11 ) - (L22> or (L 11 > - (L33)" From the multiplication rule for LocL,r8 we get (Lll)
=~1 ~q(Lq12L21q) '
(L22)=_~q(LqlL;2q).
The correlations
d___~_~ 2zr "
Volume 62A, number 2
PHYSICS LETTERS
25 July 1977
The average valu'es Of L l l and L22 take the form
2N q x/(Mq)2 _ (M~)2 cth ~
F
1
L 2
-
~
"J((Lll)
-
(L22))
1
M~I 3V(Mlq)2 - (Mq)2"l ' 2 2cth ~ _ < - ~ 2 2 ) ) J(
Assuming the conditions (L22) = (L33)= (L44)
(10)
or equivalently (Lll) +3(/,22 )= 1 we obtain 2
Mq
cth
2((L11)_(L22)) _j .
(11)
The sums,' occurring in eqs. (5)-(8) and (11) represent a new result specifically due to our use of the SDM approach. Our assumption of the condition (10)is justified by the circumstance that, in the paramagnetic region, all triplet states have the same energy. For T = 0 K we obtain, up to order A 2, the following expressions
Eq = Uq = A{1 -AT(q)- ~A 2 [@(q) -a/z] }
(Lll)- (L22)= 1 -A2/z,
E0 = (/-DT=0K = -- ~NA(3 +A2/z)
(12) (13, 14)
where z is the coordination number, and
A = (a - b)2J(O)/2A; T(q) =J(q)[J(O). It is worth noting that, on omission of the lattice sums and insertion of (Lll) = 1, (L22) = 0, our formulae (3) and (4) reduce to the weU-known RPA-MFA expressions [3], [4] for the excitation energies at T ='0 K E RPA'MFAq = URPA'MFA = AX/I -- 2A3'(q).
(15)
Also, as should be expected, our expressions (12)-(14) at z ~ oo go over into the RPA-MFA expressions up to order A 2 because two-site correlations neglected in the RPA-MFA give contributions proportional to z -1 in the SDM expressions up to order A 2. Thus, the difference between our SDM (3), (4) and the RPA-MFA (5) spectrum at T = 0 K, up to order A 2, amounts to E q - E qRPA-MFA =Uq __UqRPA-MFA=A2A2/z. In the high temperature region we obtain up to order A 2 A ( ( L l l ) - (L22)= 4 k B T - A 1
8A2kBT . ~ ) ( 4 k B T - A) 3
(16)
113
Volume 62A, number 2
Eq A-
Uq_1_A A
PHYSICS LETTERS
A 4kBT-A
7(q)
2AB kBT 3t(q)+SAz 2 . kBT z 4kBT-A 4kBT-A
25 July 1977
A2 2
A2
(17)
( 4 k B T _ A ) 2 "12(q)
where B = (a + b)2 J(O)/2A. Thus, in the high temperature region, our SDM spectrum (3), (4) and RPA-MFA spectrum, up to order A 2, differ by Eq ERPA'MFAq Uq uRPA'MFAq 8A 2 k BT 2AB k BT
A
A
- A
A
z
4kBT- A
z
4 k B T - A 7(q).
Similarly, eqs. (16), (17) go over into RPA-MFA at z -+ oo. The investigated spectrum is expressed up to the order A 2 and we have obtained corrections proportional to z - 1 in contradistinction to the RPA-MFA method at T = 0 K. In the high temperature region, our correction depends on temperature, although this dependence is very weak. The difference between the population of the singlet state and that of the triplet state (see eq. (13)) is now endowed with a correction proportional to z -1 and is no longer equal to 1 even at T-- 0 K. This reflects the fact that the crystal ground state is not the real ground state of the singlet-triplet Hamiltonian [3]. Generally, SDM offers improvements in comparison with RPA-MFA, but leads to a highly complicated set o f self-consistent equations. Particularly, we expect important differences in comparison with RPA-MFA in the behaviour of the elementary excitations generated by transitions between the triplet states. Thanks are due to Prof. dr L. Kowalewski, A. Popielewicz, R. Micnas and G. Kamieniarz for their very helpful discussions.
References [1] [2] [3] [4] [5] [6] [7] [8] [9]
114
B.R. Cooper, Phys. Rev. 163 (1967) 444. B.R. Cooper and O. Vogt, Phys. Rev. B 1 (1970) 1218. Y.Y. Hsieh and M. Blume, Phys. Rev. B 6 (1972) 2684. Y.Y. Hsieh, Can. Jour. Phys. 51 (1973) 50. O. Kalashnikov and E.S. Fradkin, Phys. Stat. Sol. b 59 (1973) 9. G. Kamieniarz, A. Popielewicz, R. Micnas and E. Szezepaniak, Phys. Lett. A 55 (1976) 329. A. Popielewicz and E. Szczepaniak, Phys. Stat. Sol. b 74 (1976) K81. S.B. Haley and P. ErdSs, Phys. Rev. B 5 (1972) 1106. E. Bucher, I.P. Malta and A.8. Cooper, Phys. Rev. B 6 (1972) 2710.