V. A. POPOV and A. A. LOGINOV: Generalization of the Heisenberg Hamiltonian 83

phys. stat. sol. (b) 84, 83 (1977)

Subject classification: 13.5.1 and 13.5.4

Physico- Technical Institute of Low Temperatures, Ukrainian Academy of Sciences, Kharkov

Generalization of the Heisenberg Hamiltonian €or the Case of an Atomic System with a Variable Atomic Spin Value

BY V. A. POPOV and A. A. LOGINOV

The representation invariant relative to spin rotations of the Hamiltonian of electrostatic elec- tron interactions is derived for a dielectric. The temperature dependence of the exciton spectrum and that of the value of the site spin moment, (S,’), due to exciton-magnon interactions are considered.

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1. Introduction The Dirac vector operator [l] for an exchange-coupled electron pair is widely used

as the Heisenberg Hamiltonian to describe the properties of magnetically ordered substances. The generalization of the Heisenberg Hainiltonian for non-zero orbital inonienta of states of the electron pair involved in the exchange has been given in

Bogolyubov and Tyablikov [4, 51 indicated that taking into account the non-ortho- gonality of atomic wave functions of exchange electrons does not affect the form of the Heisenberg Hamiltonian. Many attempts have been undertaken [6 to 81 to use the Dirac operator for describing the interaction of a system of many-electron atoms. With a minimum number of assumptions on the value of electron interactions, Irkhin [9] generalized the Dirac operator for a system of many-electron atoms with only sniall variations in the operator.

It is typical that the Dirac operator and the Irkhin operator coninlute with the operator Sj of electron or atom spin, respectively, and therefore, they describe only such changes in the spin system for which S; remains constant. The constancy of Sj in the case of many-electron atoms imposes restrictions on the theory which should he removed.

The purpose of our paper is to derive (using a more successive account of electro- static interactions) a new Haiiiiltonian which, on the one hand, is explicitly invariant relative to spin rotations and, on the other hand, imposes no rigid restrictions on the Sf’ value of a many-electron atom. With the Hamiltonian derived, we calculate the temperature dependence of both the statistical average (Sj) and the energy of a differ- ent type of excitons in a ferrodielectric.

[ 2 ) a].

2. The System Hamiltonian in Terms of Transition Operators

We consider a dielectric with n, magnetic electrons on every lattice site (d, f elec- trons). The system of electrons is described in ternis of second quantization using a set fi*

84 V. A. POPOV and A. A. LOGINOV

of orthonormalized states, { O f a } , which can be taken as orthogonalized atoniic orbitals [5] (a stands for states of site f ) . Such a set is invariant for each site with respect to the the operator of the electron spin s . I n the fixed particle number approximation, nf = nf,, the Hamiltonian of the system can be given in the following form:

a

(1)

where k..., s... have the same indices as the corresponding products of Ferriii opera- tors a+, a and are equal to

s... = ( O f ; , ; ( l ) Of;a;(% U(1, 2 ) @f,*,(2) Ofi&) . Here V,(r ) is the energy of the interaction between an electron and the f th lattice ion, U(1, 2 ) = e2(/rl - v21)-l = U(lrl - r21).

In the Haniiltonian (1) we can take a new representation using the identity

where Pfil is the projection operator on states with n electrons in site f . When suh- stituting (3) into (1) for each site it is easy to see that the determinant of the nf par- ticle states, @ f A , originating from { O f % } in the f-site, appears in the integrals in ( 2 ) , A =I (al, ... , anf) and at being arranged in a certain order. The Hamiltonian ( I ) then will have the form

where A)A = aftx, *.. ahn 3 A,, = (A; , )+ ,

and K... and F... are written as follows:

is the permutation operator for l i and 2i' electrons, 1 = (rllsll, ... , ~ 1 ~ p l , ~ , ) ,

2 = (r2rs21, ... , ~ 2 ~ ~ ~ 2 ~ ~ ) . The operators A + , A are many-particle operators of secondary quantization due to Irkhin "31. They have complicate coniniutation relations and therefore are inconvenient for use. More suitable are the operators B{j,= A,+,,Af,, using them the operator (4) takes the form

Heisenberg Hamiltonian for the Case of an Atomic System a i th Variable Spin 85

They are the operator of a A + A‘ transition on the f-site in the basis of states of the whole electron system defined by the projectors 17 B2,Ar The coefficients W define

the exchange Hamiltonian JE,,. The difference JE - JEe, = JEc is the Coulomb part of (1). The specific features of {Ofa} relative to spin rotations permit one to consider A = ( y , S , p) , where AS is the total spin of nf electrons, p its projection, and y are the other quantum numbers.

This representation of the Hainiltonian (6) agress with that of Bogolyubov and Tyablikov 141 for a system with one “valence” electron per atom and differs from that of Hubbard [lo] in the explicit expressions for the coefficients for the case of several

Representations (4) and (6) do not take into account the explicit invariance of electrostatic interactions under spin rotations and are not suitable in many respects. Therefore, we consider now the Hamiltonian representation (6) which takes the above explicit invariance into account.

f

valence” electrons per atom. “

3. Invariant Representation of the Hamiltonian According to the form of the basis

{Ofa } = { Of?.(Y) x u ( 4 } the space @fin) is invariant with respect to the spin operator s for the systeiii of nf electrons. This permits to consider A = ( y , S, p ) = (T, p) , where X is the total spin of nf electrons, p its projection, and y are the other quantum numbers which deter- mine the @fA state. Therefore, using the identity (3), the total spin operator Sn for 2 nf electrons can be written as follows: f

(8) can be treated as the expansion of Sn in a sum of commuting site spin operators because sf x sf, = iSff,Sf,

Taking account of the general features of the irreducible representations of basis vec- tors, @fA can be given in the form

S;BL = S(S + 1) BL , S;BL = pB5A , A = ( y , 8, PI (9)

s, s, ’ s are the Clebsch-Gordan coefficients. (6) can be reduced to the explicit

where (PI PLZ i,) invariant form. Firstly, we introduce the operators

%,p;r,p == ( - l ) s - p & , I L ; r , - p (11) which for fixed r, r’ are the components of the tensor operator of type D(s’) @ D(s) relative t o spin rotations [ I l l . Using (ll), one can write the first-order irreducible tensor operators

86 V. A. Porov and A. A. LOGINOV

where " ") are the Wigner coefficients. The factors f(S', 8 ) are taken in the form Pl Pz P

x 6(s;s;' f ) 6(Szs; $) a(s;s; f ) , (17)

where 6(x, y, z ) = 1 when x, y, z satisfy the integer perimeter triangle condition, 6(x, y, z ) = 0 for other cases,

e2 gf,&J.n f*% = / OXdr1) 65PZ) jli - r 2 ( 6f1Az(r2) 6fzA,(r1) dr1 drz

1 ( i=l nf i+j

(17b)

and Cf,?,, are taken from the expansion (10). nf n, ns

Since the operators C 2 U( lrli - rziti) and 2 h(i) + 2 U ( i , j ) commute with

S,, the Coulomb part of the Hamiltonian (6) can be expressed in terms of the operators i=l i'=l

B5.r:

Heisenberg Hamiltonian for the Case of an Atomic System with Variable Spin 87

The operators B5.r coniinute with the operators Sn, Sf for S J = X and are therefore scalar relative to spin rotations. They satisfy the equalities (7) for the transition oper- ators.

Expressions (14) and (18) solve the problem of the explicit representation of the Haniiltonian (6) in a form which is invariant relative to spin rotations. I n the exchange part, X,, (14), the first term as well as 3, is scalar relative to Sf. The second term con- tains a part which is associated with the products

for all f , and is a Hamiltonian of Heisenberg type commuting with Sf rather than with Sj and thus retains the value of the atomic spin squared, Sf”. The other terms in (14) contain, only for one site, the operators ZIf,,, with S’ =# S which commute neither with Sf nor withsf . The part of X,, containing such terms does not retain S: and therefore, the system states for definite values of Sj for each site f are in general not eigenstates of the total Hamiltonian. However, it should be noted that the system states with maximum values Sf = Xf” and projections pf = Xof (as well as p, = -Sf”) for all f form two subspaces invariant relative to the Hamiltonian (6). Hence, among the superpositions of states from these subspaces there are eigenstates of the Hamiltonian (6)

4. Structure of the Exciton Energy Spectrum and Its Temperature Dependence

In order to find out the structure of the energy spectruni of single-particle excita- tions described by the Hamiltonian (6), we firstly define exactly an atomic (site) basis used for estimating the coefficients of the Hamiltonian (6). The state Co = I... @fn ...) is taken as a trial ground state of a crystal so that the average energy (Co, 3,Co) x x (C,, C0)-l is minimum. Atomic states corresponding to Co can be designated by the index fyoSo G fro. The excited states are obtained by separating the single-site part from X,, using the relation .Z B$r = 1

r

and by reducing it to a diagonal form 2 A$B$r. Such values of the site state energy,

A$, are in agreement with the crystal field approximation. These energy levels are degenerate relative to spin rotations. Then X , can be given as follows:

r(w0o)

(20 a)

That part of X e , (14) which is scalar relative to Sf can be included, if necessary, in (19) and (20).

From (20) it follows that there are two types of excited states which are due to changes in the number y and belong to a class of Frenkel excitons. The first type of excited states occurs when the ground-state multiplicity So is fitted by several values of y. Each excitation with y + yo is translationally degenerate. It is transferred from site to site through the operators B~so~yS,B~So;y,So of (20). This results in splitting the l e v e l & , into a band, the width of which is large compared to the exchange parameters

88 V. A. POPOV and A. A. LOOIKOV

and is determined by the Coulomb integrals ~ f z ~ f l j r 8 ~ ~ : ~ ~ ~ . According to [ 2 ) . such

excitations of different y , which belong to the ground-state multiplicity So, will be called orbitons.

Another type of excited states occurs a t transitions from (yoso) to ( y S ) for X + So. These single-particle excitations which are due to a change both in y and in S, appear in X , as localized within a zero-width band (translational degeneracy). Unlike orbitons, such excitations can be called mixitons. The characteristic property of mixitons is that their translational degeneracy is lifted only by non-Heisenberg terms of X e x (14). The width of a band to which the level 2:s of a mixiton is split is considerably smaller than that of an orbiton and decreases proportionally to Jn/6@-1) as IS - Sol = n increases. J ' s are the exchange parameters of a non-Heisenberg type,

Besides orbitons and mixitons there are pure spin excitations (niagnons) which are due to a deviation of the atomic spin from the quantization axis for fixed y,So. The lowest part of the energy spectrum of a crystal is due to magnons (along with phonons). The interaction between excitons (or orbitons or mixitons) and thermal niagnons results in a temperature dependence of the exciton energy spectrum

The temperature dependence of the exciton spectrum can be found in first approx- iiiiation by averaging X',, over magnon excitations with their equilibrium density matrix. To first order in J/&,s, 8' =/= S (AS'S is a value of the order of 2r,Ljf - account should be taken only of terms of the type (ZI$;rlZI$;r2) (6s:,s,-i6s:-1,,s2 + + 6s;s16s;s2). The operators for niagnon and exciton excitations are introduced by the expressions

B&r = bf+pbp for T', T + To; BGoT0 = 1 - bf'rbfr. r(iroo)

The Hamiltonian is expressed in terms of the above operators by the equalities ( { A 2 , A , , A _ } EE A , A , = A, f illu):

Ilf,,, = {B&r 1/2S - kf - 1 a f , B$,ra;, -Bf,,r j (2S - hf - 1) (28 - GI)} , S ' = S - l ;

SjBf,,r = { (S - G j ) B$r, i 2 X - kj ajBfrfr, a; 1/28 - kf Bf,,,} , S' = S . (23) Expressions (21), ( 2 2 ) for magnon and exciton operators indicate that for sufficiently large S a class of states can be separated, above which rnagnons and excitons are found to be almost kinematically independent. I n this case the commutator [a f , a;] becomes

Heisenberg Hamiltoninn for the Case of an Atomic System with Variable Spin 89

very neaily a Bose one and [ b f r , bf;.,] diffeis not so much from the Bogolyahov-Tyabll- kov commutator of excitation operators [4]. The excitation operators related to dif- ferent sites are all commutative.

Considering the temperature dependence of exciton excitations in the range T < T,. where T , is the spin ordering temperature, we can restrict ourselves to ternis quadratic in a+, a of the Hamiltonian exchange part. The condition IS - Sol < So a t 7' < T , provides a sufficient kinematic independence of exciton and niagnon excitations as well as their Bose hehaviour.

Taking these remarks into account for a ferrodielectric, the part of the Haniiltonian (14) related to operator IZf,,, can be given as follows:

The part of Xk, which involves both the a and B operators, describes the exciton- magnon interaction and vanishes a t kl --f 0 , k, + 0. Such a hehaviour of the exciton- niagnon interaction amplitude is due to the Goldstone character of the magnons con- sidered and expresses the invariance of the Haniiltonian relative to spin rotations.

Averaging the interaction over the equilibrium density matrix of inagnons for a cubic lattice in the nearest-neighbour approximation, we obtain the following effec- tive Haniiltonian of excitons:

At T < T , the temperature corrections for all spectrum paraiiieters are generally proportional to (T/T,)512. For a ( y ,So - 1)-type exciton corresponding to a non- degenerate crystalline term the following spectrum can be obtained in the Heitler- London approximation :

where J > = J$$$o - J k 2 2 and J r ( k ) is the Fourier transform of J ~ ~ ~ o ~ o T ~ ~ ~ L s ~ . It should be mentioned that the effective mass and its temperature correction are deter- niined by the same parameter.

4 k ) = 1, - GJ>q(T) - J d k ) [So + T(T)I, (26)

5. Statistical Average (S;) at Low Temperatures As mentioned above, the appearance of non-Heisenberg terms in the exchange part

of the Hanliltonian disturbs the law of Sj conservation, leading to a difference between the statistical average (S,)?and Xo(So + 1). Let us calculate (S,) for a ferrodielectric. To do this, the operator Sf can be given as follows:

S; = So(So + 1) - (28 , + 1) mf + mj , where

nzf = 2 (8, - S) B$r. (27) r

90 V. A. POPOV and A. A. LOGINOV

Suppose that the true ground state of X c has niaxiniuin spin So on all sites and is degen- erate only with respect to the site spin projection and separated by a gap A, 3 J from the rest of the spectrum for Xc . To second order in J / A 0 , for T < A , the calcula- tion of (Sf") reduces to a statistical averaging of the operator

where P is the projection operator on the subspace of Co-states, related to the ground level EA of the HaniiltonianXE (the prime denotes that the part of X,, which is scalar relative to Sf is included in X ; , but eliminated froni Xk,), m = 2 mr.

After reduction of (28) the calculation of (Sj) reduces to the statistical averaging of the operator

m = PX&l - P ) (XE - Eh)-1 m(XL - EA)-1 (1 - P) Xk,P, (28)

f

(29) + 2 gAf1 -f!A [St - 2s:+ (SfPfY - 2fio(So - 1) ( S f P f J I

(J(fi - fi) = Jf2r,r0)

f,=+f, with the Heisenberg Haniiltonian

f l r o r o Xa = + 2 J(f1 - 1 2 ) SfLSfz fx=+fz

operating in C,. Using for a ferrodielectric the Holstein-Prirnakoff representation [ 121 of spin opera-

tors and the approximation So > 1, it is easy to see that only the first term of (29) containing the factors q, contributes to the part of f i which is quadratic in magnon operators u+, a. After the Fourier transformation we obtain

'h 2 2 (glk, -k - 2glk,0 + gl0,O) SiaLak = 2Si 2 p k a h k >

(30) k k

glkl,k, = 2 ql(h,, 11,) ei(hlkl-khzkJ . hiha

For atoms with half-filled shells in the ground state when only one terni conforms to the maxiniuni atom multiplicity, we have

gi = Jf:r~rdf:r,r,~s,s,-i~ (31) 2 f r r frr,

r Since in our case all states

Therefore, for a cubic lattice with parameter a , p , = p(ak)4 in the nearest-neighbour approximation. When the shell occupation is different, g, has only the property

and therefore P k = ~ ( a k ) ~ . As seen from (31)

at every site f are of the same parity, - fl> f3 - fl) = m(f1 - fi, f3 - fl) = m(f1 - fz, fl - f 3 ) *

gAf2 - fl, f 3 - fl) = m(f1 - fi> fl - f3)

P = Z AF ( Jg rOrJz If-91 = a .

-

2 f r o r r

The coefficient 13 can be approximately given in the form - P Z: (ar + jLr,)-2 , If-gl = a ,

s, s1(*ro) when only the single-site part is taken in XE. It should be noted that in any case

q l ( f 2 -fl,f3 -fl) = ql(f3 - f l , f 2 -fl) *

With the above remarks in mind, we can obtain from (30) 15

p N 8n"iz [ ( 7 / 2 ) (T/BC)7l2

- 3 p f l m 5 ( 5 / 2 ) ( T/0,)5/2

in the absence of I' + I', a t S = So;

in the presence of I' + I', at 8 = So . (fib =

Heisenberg Hamiltonian for the Case of an Atomic System with Variable Spin 91

Here 0, = J ( a ) 8,. Because in the approxiniation considered with respect to JIA (mf> x (rn?) , we get

(33) Hence, we can conclude that in a ferrodielectric the average (Sj) decreases in different ways as the temperature increases, and this depends on the atom shell occupation. At T = 0 (Sf) = S,(S,, + 1) due to the fact that the crystal state of maxiinuni X,, and S; = So at all lattice sites is an eigenstate of the Hamiltonian X .

(S?) = So(So +1) - 2s0 h'-l(?%)G .

6. Conclusions The main results of the theoretical study considered are : 1. In the approximation of a fixed number of "valence" electrons per atom in

the system the Hamiltonian of electrostatic interactions between the atoms of the system is derived in a form explicitly invariant relative to spin rotations. The exchange part of the Hamiltonian involves both Heisenberg and non-Heisenberg ternis, the latter breaking the law of conservation of the site spin squared, S;.

2 . Using the Hainiltonian derived, the energy of exciton-magnon interaction and the temperature dependence of the exciton energy spectrum are calculated for a ferro- dielectric. The exciton-magnon interaction is found to cause a renormalization of the parameters of the exciton Haniiltonian due to corrections proportional to J r ( T/1',)5'2 at T < T , ( J r is the exchange parameter which is dependent on the type of I?-exci- tons, T , is the Curie temperature).

3 . The non-Heisenberg part of the exchange Hamiltonian is shown to cause a de- crease in the statistical average (Sj) of the atomic spin as the temperature rises. This is due to the atom shell occupation by electrons. When the shell is half-filled, the decrease is proportional to ( 5!'/5!'c)7/2, in other cases it is proprotional to ( 5!'/5!'c)5/2. Since the proportionality factors contain sinall values of the order of (J /A)z the con- dition S! = const, which is used in the spin wave theory based on the Heisenberg exchange Hamiltonian, is met rather exactly in a wide temperature range, T < T,, provided that the crystal field does not give rise to mixitons with energy of the order of the exchange energy.

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[4] N. N. BOGOLYUBOV and S. V. TYABLIKOV, Zh. exper. teor. Fiz. 19, 256 (1949). [5] N. N. BOGOLYUBOV, Izbrannye trudy, Vol. 2, Naukova Dumka, Kiev 1970 (p. 400). [6] Yu. A. IZYU~~OV and E. N. YAKOVLEV, Fiz. Metallov i Metallovedenie 9, 667 (1960); 8, 3

[7] J. H. VAN VLECK, Matem. fisica teorica 1411, 189 (1962). [8] R. K. NESBET, Ann. Phys. (U.S.A.) 4, 87 (1958). [9] Yu. P. IRKHIN, Zh. exper. teor. Fiz. 50, 379 (1966).

Vol. 4, VINITI, Moskva 1967.

(1959).

[lo] J. HUBBARD, Proc. Roy. SOC. A285, 542 (1965). [ll] B. R. JUDD, SecondQuantization and Atomic Spectroscopy, The John Hopkins Press, Bal-

[12] T. HOLSTEIN and H. PRIMAKOBB, Phys. Rev. 68, 1098 (1940). timore, Maryland, 1967.

(Received M a y 9, 1977)