PHYSICAL REVIEW 8 VOLUME 25, NUMBER 1 1 JANUARY 19S2

Antiferromagnetism, projected , and the Bogoliubov transformation for bosons

* Miguel Kiwi, Tsung-han Lin, and L. M. Falicov Department of , University of California, Berkeley, California 94720 (Received 23 June 1981)

We investigate the spectral density of an antiferromagnet described by a Heisenberg Hamiltonian and find that the procedure commonly followed to evaluate the localized density of states leads to inconsistent results. These inconsistencies are illustrated with a detailed discussion of the one-dimensional antiferromagnetic chain.

I. INTRODUCTION by the Holstein-Primakoff transformation:

S+=&2$ (1 l2—' (2a) In recent years, the development of sophisticated aj at $) aj, detection techniques gave strong impetus to experi- St =&2$at (1 at aj l—2$)'~ (2b) mental and theoretical studies of surface properties. — In particular, the magnetic excitation spectrum in St+ &2$—bt (1 b, bt /2$ —)' (2c) the vicinity of the surface, which seems to be an =&2$(1 bt btl2—$)'~ bt, (2d) important factor in some catalytic properties of some substances, has received a good deal of atten- SJz=—S —aj.aj. , (2e) tion. ' SI'——S+bI b In this contribution we report some surprising diAiculties which are encountered in the study of The introduction of running waves by means of an antiferromagnetic surface, when standard Green's-functions techniques are used. In fact, we (3a) show below how the procedure usually followed J the can lead to unacceptable negative weights of (3b) spectral density of magnon states.

yields for the Hamiltonian of Eq. (1) the form

H = —2XzJS —4EpgHg S+Hp+H (4) II. ANALYTIC FORMULATION i, where The starting point of our calculation is the usual Ho ey(akak +bkbk ) Heisenberg Hamiltonian given by H =JQS;.St~s 2pttHg +Sf +—2pttH„+St', +QTk«kbk+bkak» k where J ~ 0 is the nearest-neighbor exchange in- e=—2(JzS+pttH„), tegral; S; is the angular momentum operator "' at site i, pz is the Bohr magneton, and Hz g 0 is a Tk =2JS+e' 5 fictitious , which simulates the crys- tal anisotropy. The anisotropy field tends to align From now on we restrict our interest to the one- spins in the predominantly up-sublattice E "up") magnon Hamiltonian which is (j Hp, neglecting H & in the +z direction and spins in the down-sub- of fourth order in the t ak, bk I operators. lattice (l E "down") in the —z direction. The vec- The usual procedures for handling the Hamil- tor 5 connects each with its nearest neighbors tonian Ho are as follows: (a) to perform a Bogo- on the magnetic lattice. liubov transformation or (b) to use Green's func- The inconvenient commutation relations of the tions. The former method is best suited to treat spin operators S; are converted into Bose relations the translationally invariant case, while the latter,

432 1982 The American Physical Society ANTIFERROMAGNETISIM, PROJECTED DENSITY OF STATES, . . . more general and more cumbersome, has to be in- One readily obtains(&krak voked to solve nonuniform problems. HO =QEk +pk pk ) +const, (10) III. THE BOGOLIUBOV TRANSFORMATION where Ek =(e —Tk)' y0; the parameters can The Hamiltonian. Ho can be diagonalized exactly be determined to be in terms of the operators Qk =(E+Ek)/2Ek, (1la) &k =—ukak —vk&k (Sa) ukvk Tk~~sk ~ (1lb) pk =~kbk uk&k (gb) uk =(e Ek)—~'-Ek, (1lc) where uk and vk are real parameters which satisfy where one is forced to choose the product 2 2 uk vk k&0

IV. GREEN'S FUNCTIONS TREATMENT

The rdevant Green's function for the above case is

& & = & & ukbk(r) — « ~k(t) I ~k(O) ~kqk(r) I ~kak(0) ukbk(0) & & (12)

uk ~k & & & &bk ~k & & & & & & & bk & & . I ~k 1 « ~kuk I ~kuk ~k I bk &+uk bk The equations of motion for the four Green's functions on the right-hand side of Eq. (12) are readily solved and yield

~k &k i(~' (13a « l ».=(~+e) Ek»— — «be~ k&.&= Tk~(~ Ek) &&~k lb &k& (13b)

bk bk & = (~ F)l(~'— —) . — (13c) « ~ &~ Ek Having derived these expressions we turn to exhibit the results they lead to.

V. RESULTS AND CRITIQUE

Spectral weights are obtained through the imaginary part of the corresponding Green's functions. The density of antiferromagnetic magnon states in our case is related to

1 2 2 2 2 2 Im« & & = uk5(~+— — [5(~—Ek) —5(~+Ek)1 « I ~k +;0+=~k [~k5(~ Ek) Ek)l »kuk

+uk [uk5(co Ek) uk5(co+—Ek) ]—=5(co Ek)— (14)

Since the form of Ho as written in Eq. (10) implies ed to each of the Green's functions of Eqs. (13) that have had to be incorporated, which in itself is a — cause of concern. D (co) =2+5(co Ek ), But moreover, when we related our results with what can be found in the literature, we do find ad- where the factor of 2 takes into account both the ditional difficulties. In fact, the local density of 0& and 0& states, we find that the end states has been obtained ak ~ pk ~ magnon using result of Eq. (14) is indeed correct. However, in 1 D;(co) = ——Im [G~+ (co)+G~+ (co) (16) obtaining this result negative spectral weights relat- ], MIGUEL KIWI, TSUNG-HAN LIN, AND L. M. FALICOV 25

where D;(ro)= ——TrkIm {((ak ~ak)+ ((bk ~bp&&)

(18) If, for the sake of clarity and to keep the algebra as In the uniform (translationally invariant) case, simple as possible we restrict ourselves to one relation (16) is equivalent to dimension, then Eq. (18) leads to

1 dE 2V yE 2V — 2V — 2V+E8( ) b( E) E~( E) Eb( E) )

(19)

I where we have defined V=—2JS; the same expres- lems in the nonuniform antiferromagnet. In fact sion is derived using Eq. (16) with the localized when the spectral density of a nonuniform one- Green's functions obtained through Fourier dimensional antiferromagnetic chain is evaluated transformation of (13a) and (13c). on the basis of Eq. (16},with the incorrect choice Carrying out the trivial integration in Eq. (19) of signs which yield Eq. (21) in the uniform case, yields the expression unphysical negative weights of states localized at co=0 are obtained. In addition, if one carries out 22V 1 D;(co) =- an analogous calculation for a Bethe lattice with 1r co V2 (4 coordination z & 2 {the one-dimensional chain can which is not the correct density of states. Howev- be pictured as a Bethe lattice with z =2), then not er, if the of the square root for the two last only localized, but also extended states with nega- terms of the integrand in (19) are chosen to be neg- tive spectral weight are obtained. This fact violates ative then the correct density of states the fundamental rule that D;(ro) &0, obtained from hermiticity considerations. 2 1 D(r0) (21) These results [in particular D;(t0)+D(co)] consti- =- 2)1/2 (4V2 tute a clear indication that relation (16), which ex- is obtained. While this is the choice followed by actly or with slight variations has often been used previous authors it does not constitute a formally in the literature, does not provide the correct way correct procedure to obtain the density of magnon to project magnon states onto a localized basis set. states. On the other hand, there seems to be no trivial It is important to emphasize that the correct way to generalize the expression of Eq. (18}to al- choice of signs is that of Eq. (19), which leads to low the treatment of nonuniform antiferromagnetic Eq. (20), i.e., to the conclusion that D;(c0)@D(co). systems. In other words, D;(ru) is not a projected density of states. ACKNOWLEDGMENT The incorrect choice of signs, which produces Eq. (21) and appears formally correct, is in fact This work was supported in part by National completely wrong and leads to very serious prob- Science Foundation Grant No. DMR8106494.

'Now returned from sabbatical leave to Universidad 28. Laks and C. E. T. Gonyalves da Silva, Surf. Sci. 71, Simon Bolivar, Caracas, Venezuela. 563 (1978). Now returned from leave to Beijing University, Beijing, 3J. L. Moran-Lopez and L. M. Falicov, Phys. Rev. B 20, China. 3900 (1979). ~J. B. Salzberg, C. E. T. Gonqalves da Silva, and L. M. 4J. L. Moran-Lopez and L. M. Falicov, J. Magn. Magn. Falicov, Phys. Rev. 8 14, 1314 (1976). Mater. 15-18, 1077 (1980). 25 ANTIFERROMAGNETISIM, PROJECTED DENSITY OF STATES, . . . 435

5See for example G. A. Gonzalez de la Cruz, and C. E. from [nk Hol =sk+k. T. Gonqalves da Silva, Rev. Bras. Fis. 9, 193 (1979). ~We use double-time-retarded Green's functions and fol- The sign of the product ukvk is unambiguously deter- low the notation of D. N. Zubarev, Usp. Fiz. Nauk mined by the set of equations for uk and vk derived 71, 71 (1960) [Sov. Phys. —Usp. 3, 320 (1970).