Two-dimensional Heisenberg antiferromagnet: Analytic and numeric results

~

S o l i d S t a t e Communications, Vol. 70, No, 4. pp. 431-435, 1989. Printed in Great Britain.

0038-1098/8953.00+.00 Pergamon Press plc

TWO-DIMENSIONAL HEISENBERG ANTIFERROMA.GNET: ANALYTIC AND NUMERIC RESULTS M. Lagos and M. Kiwi Facultad de Ffsica, Unlversidad Cat61ica de Chile, Casilla 6177, Santiago 22, Chile E. R. Gagliano Centro At6mico Bariloche, Instituto Baiseiro, Comisidn Nacional de Energfa At6mica 8400 S. C. de Bariloche, R. N., Argentina and G. G. Cabrera Instituto de Fisica Gleb Wataghin, Universidade Estadual de Campinas (UNICAMP) 13081 Campinas, Brazil

(Received J ~ r y

24, 1989 by C. E. T. Goncalves da Silw )

An approximate analytic expression for the ground state of the anisotropic S = 1/2 Heisenberg model with antiferromagnetic exchange is studied. Its accuracy is determined for a wide range of the anisotropy parameter ,~. Restricting ourselves to the special case of the square lattice, we used the analytic ground state to calculate exactly the expectation values of several dynamical variables. The approximate analytic results thus obtained are compared with those given by spin wave theory and reliable numerical d a t a obtained by us and other authors. We conclude t h a t the analytic theory is asymptotically exact for a ~ 0 (Ising limit), is highly accurate for 0 < a < 0.5 (less than 1% uncertainty in all calculations), and is reasonably precise (less than 5% uncertainty) in the range 0.5 < a _< 1. ( a -- 1 : isotropic Heisenberg model). In particular, our asymptotic approach gives better results than spin wave theory for 0 < ,~ < 0.9.

The discovery of antiferromagnetic order in the layered perovskites has brought the study of Heisenberg antiferromagnets to the focus of interest. Although the ability of the Heisenberg model to provide a full understanding of the remarkable magnetic properties of the layered metal oxides is not well established, and is still a matter of controversy, 1,2 the field has become one of the most active ones in condensed matter physics. Important advances have been accomplished during the last two years, particularly in the knowledge of the ground state of the antiferromagnetic Heisenberg model in two dimensions, s - 8

and ii) The eventual connection of the two-dimensional antiferromagnetic correlations with the tetragonal-toorthorhombic structural transition in the layered perovskites, suggested by recent experiments. ~,1° Approaching them on the basis of an exact formalism seems at present hopeless or, at least, impractical. Thus, in order to avoid unnecessary complications, it is important to find out which of the many-body processes involved in t h e antiferromagnetic states are relevant. An analytic treatment of the anisotropic Heisenberg antiferromagnet, which satisfies to some extent the above stated requirements, has recently been published.11-14 It combines mathematical simplicity with numerical accuracy in the regime in which the system is antiferromagnetically ordered. The theory assumes a bipartite lattice whose dimensionality is kept as a parameter, 1' and yields a ground state with longranged antiferromagnetic order. In particular, the ground state energy of the isotropic square lattice case turns out to be quite close to the one obtained from nu-

However, parallel to the effort to obtain exact resuits, the development of a comprehensive analytical theory which provides reliable results and, in addition, be simple enough to handle problems in which the antiferromagnetic coupling is only one among several interactions, is a necessary goal. Two examples of problems in which the magnetic interactions are not alone are: i) The interaction between the antiferromagnetic order and the itinerant electrons or holes in doped samples; 431

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TWO-DIMENSIONAL HEISENBERG ANTIFEREOMAGNET

merical computations, z' An important feature of the formulation, which is neither variational nor perturbative, is that the calculation of matrix elements between the stationary states can be accomplished analyticaily,is In a previous publication the ground state of the one-dimensionai Heisenberg model with anisotropic antiferromagnetic exchange, its structure, energy and correlation coefficients were numerically calculated and compared with the corresponding results of the appreximate analytic approach. ~s It was concluded there that our one-dimensionai ground state is practically exact for anisotropies within the range 0 < ,~ < 0.5, where a = 0 corresponds to the Ising limit and ,~ = 1 to the isotropic Heisenberg model. When ,, > 0.5 the errors start to grow. Specializing the general theory to the square lattice case, and comparing the ground state energy with reliable numerical data, 16 a similar behavior is apparent, but the discrepancies with exacts results start to grow z' beyond a = 1, instead of ~, = 0.5. Therefore, the analytic approach applied to the square lattice seems to be highly accurate over the whole range 0 _< a _< 1, and to be valid even for the isotropie Heisenberg antiferromagnet." Unfortunately, reliable numerical results in two dimensions are only available for the ground state energy. In this communication we perform an extensive test of the closed-form analytic ground state in twodimensions. Our aim is to use it to calculate several physical quantities and compare the results with the corresponding ones obtained numerically for a finite lattice. The numerical computations were carried out using a modified Lanczos method, ze'l~ which allows to obtain exact results for a 4 × 4 periodic lattice within reasonable computing time. The numerical procedure used by us cannot handle with ease sample periodicities larger than 4 × 4. However, it has the advantage of yielding the ground state, from which mean values of dynamical variables can readily be obtained. As is shown below this sample size is large enough to accurately describe the infinite system for 0 < a < 0.5. In the vicinity of a -- 1 the system goes through a parametric crossover from a quasi-localized spin phase to a regime with extended spin wave functions. Whether the transitio~ is abrupt, as recent results seem to indicate, z5 or smooth, is not yet clear. However, in both cases, one expects that the finite size of the sample will widen the transition and the results for the anisotropy parameter in the neighborhood of a = 1 may differ significatively from the infinite system ones. Thus, if it is not :feasible to extrapolate the results to N --* co, the determination of the domain in which the finite size calculations are reliable becomes an important issue. The interest of the numerical computations presen~.ed here is twofold: On the one hand, our results for energies and st~ggered magnetizations of the 4 × 4 periodic lattice are in excellent agreement (i.e. within

0.1%) with those obtained by Barnes et al.15 for anisotroples in the range 0.4 _< a _< 1.0, and for the same sample size. This provides an independent confirmation of the results obtained by these authors. Such confirmation is particularly interesting because Barnes et al. , using a different procedure, also provide results for 6 × 6 and 8 × 8 systems, which allow to extrapolate ~s to N = co. On the other hand, our numerical results prove not to be affected by sample size when a is in the range 0 < a < 0.5, and can be used to compare with those obtained from the analytic formulation mentioned previously. The ground state energy eigenvalues for N -- co, obtained by Barnes et al. on the basis of extrapolating Montecarlo results for the square lattice with periodicity L × L, (L = 4, 6, 8), allow to determine the range of a in which the L = 4 case can safely be used to represent the infinite lattice. Within the error bounds of the Montecarlo calculation (less than 0.1%) the ground state energy data for 0 < a < 0.9 is described by the quadratic relation",z5

Eg

1(_~)

Nj=-~

1+

(1)

.

For a = 1 Eq. (1) gives a slightly larger error ",16 of 0.3%. The broken line in Figure 1 represents Eq. (1), the open circles are the extrapolated data of Barnes et al., and the crosses the results we obtained for the 4 × 4 periodic lattice. Expression (1) turns out to be a special case of the general equation1' E,

z

( '=') 1+

(2)

valid for any lattice with coordination number z and values of a which are small enough. In any event Eq. (1) can be considered as an empirical result. Applying the Hellmann-Feynmantheorem to the Heisenberg Ham;Itonian

+

+

+ ,,-,-

+

(31

where the vector i~ runs over one of the two sublattices determined by the antiferromagnetic order, and {6-'} is a set of z vectors connecting each lattice site with its nearest neighbors, one obtains that the ground state energy satisfies

8Egaa= zNJ2('s+ (R + ~8_ (/Z)) .

(4)

Since for 0 < a < 0.9 E~ is quadratic in a, the mean value of Eq.(4) turns out to be linear in the anisotropy parameter. Combining Eqs. (I) and (4) one obtains for the square lattice (z = 4)

Vol. 70, No.

433

TWO-DIMENSIONAL HEISENBERG ANTIFERROMAGNET

4

-0.50

0.45 ,4"

-0.56

4-

0.30

~-0.62

'~->

~" 0.15

+ O,'..

-0.68

0.00

-0.74 0.00

, 0.25

, 0.50

, 0.75

1.00

-0.15 0.00

,

J

i

0.25

0.50

0.75

O~

Q~

Figure 1 Ground state energy per spin as a function of the anisotropy parameter ,~. The broken line is the energy eigenvalue of the asymptotic (,y --* 0) theory of Ref. 14, which in two dimensions reduce to Eq. (1). The open circles represent Montecarlo r e s u l t s of Barnes eta/. for the N = eo system. Results for the 4 × 4 periodic lattice, obtained by the Lanezos method, are indicated by crosses. The dotted line depicts the ground state energy given by spin wave theory. The solid line represents the exact mean values (15) of H with respect to the ground state vector (11) of the asymptotic (a - , 0) theory.

-~.

1.00

(5)

Eq. (5) is highly accurate over the whole range 0 _< a < 0.9. Small deviations are expected close to ,, = 1. This way, departures from linearity of the transverse correlation (gla+ (/~ + ~ 8 _ (/~)[g) as a function of the anisotropy parameter a are an indication of the presence of size effects. The crosses in Fig.2 represent (a/6 + (g[a+ C~ + as given by our ground state of the 4 x 4 system. The onset of size effects at a = 0.5 is apparent from the plot. However, it is also apparent that the regime 0 _< cz _< 0.5 is reasonably free from disturbances originating in the small size of the sample. The solid lines in Figs. 1, 3, 4 and 5 represent the mean values of the energy

Figure 2 Values of Q = (~/6-t-(gl8+

(K'+~,8_ (R)lg))/(a/6)

obtained by numerical diagonalization of H in the 4 × 4 lattice (crosses), and those derived exactly from the state (11) (solid line). Departures from Q : 0 indicate the presence of size effects in the numerical computations. The plot shows that numerical results for the 4 x 4 lattice are reliable in the interval 0 < ~ < 0.5.

s~ = (gl~+ (-~ + ~),- (~)I~},

(s)

and the correlation functions

G1 = {gls.(/~+ g~)8.(/~)lg},

(9)

G2 = (gI,%(R +~z +'~)'.(.R)Ig),

(1o)

between nearest and next-nearest neighbors, respectively, with respect to the analytical expression

~,- (~)lg))/(~/6),

(H) = (giHlg),

(6)

Ig) = e x p [ - ~ c ~ - ~N( ~ b g

~t - ~b~)][//)

for the ground state obtained in Ref. 14. Vector [HI represents the N4e] state which assigns spin up to the sublattice {R). In the limit ~, ~ 0 the operators

the staggered magnetization

~. = 2 (gls, (-~)lg), the transverse correlation

(~)

(11)

2

• R

a

N

434

TWO-DIMENSIONAL HEISENBERG ANTIFERROMAGNET

1.00

Vol.

70,

No.

4

0.28 ' " : ' " ' ".......... """..i..........,..... ~

" '""-...

o.1

0.89 o

+

~" 0.78

4-

.~

..

0.00

eJ

-

8-0. 4

0.67

0.56 0.00

Q) r,.,

, 0.25

, 0.50

. 0.75

! .00

Figure 3 Staggered magnetization as a function of the anisotropy parameter a. The solid line depicts Eq. (16). Crosses represent numerical d a t a for the 4 ×4 periodic lattice. For a > 0.5 the data indicated by crosses incorporate size effects. Open circles are the Montecarlo results of Barnes e t a / . extrapolated to N --* co. The dotted line is obtained from spin wave theory.

4-

+

Gt

-0.28 0.00

i

|

|

0.25

0.50

0.75

1.00

Figure 5 Comparison of Eqs. (18) and (19) for the nearest and next nearest neighbor correlation functions (solid lines) with numerical data (crosses). satisfy

[~bZ, ~b~, ] = ~,~,~'i

(13)

~ig) =0.

(14)

and

0.24

The mean values (6)-(10) were calculated exactly, on the basis of Eqs.(11) and (12), without resorting to the approximate relations (13) and (14), by the method of the unitar3, transformation detailed in Refs. 12 and 13. The generalization to more than one-dimension is quite straightforward, in spite of the labor it requires. The analytic expressions illustrated by the solid lines of Figs. 1, 3 - 5 are the following:

0.18

r,~ 0.12

(H) =-=~--(J~ + j;j~2

0.06

0.00

0.00

i

t

=

0.25

0.50

0.75

2aJoJ1),

(15)

Mo -- Jo2 ,

(16)

$I = - I Jo ./i

(17)

1.00

Figure 4 Transverse correlation in terms of the a.nisotropy parameter ~,. The solid line represents Eq. (17). Crosses are numericaliy obtained values.

=

I

4

+J;j~),

(18)

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TWO-DIMENSIONAL HEISENBERG ANTIFERROMAGNET

Barnes et a/. This departure ranges from less than 1 for a = 0.5 to about 5 ~ for a = 1.

and 1

4

G, = ~(J~ + 2J~o + J~ - 2,/0).

(19)

In Eqs. (15) - (19) Jo and ./1 denote the Beseel functious

Jo ~---Jo (2a),

435

J1 -~-J1 (2/I~) •

(20)

The crosses in Figs. 1 - 5 represent numerically calculated values for the 4 x 4 periodic lattice. The agreement of the results (15) - (19) of the asymptotic analytic theory with the computer calculations is excellent in the range 0 < a < 0.5 of the anisotropy parameter. For ~, > 0.5 finite size effects become appreciable and the numerical results for the 4 x 4 periodic lattice lose reliability. However, this is not the only source for the discrepancies between the formulae (15) - (19) with the numerical calculations for a > 0.5. The solid line in Fig. 2 represents (a/6 + 0.5. This indicates that expression {11) for the ground state loses accuracy for a > 0.5. Figure 1 allows to compare the highly reliable values of Ref. 15 for the ground state energy (circles) with
It is quite surprising that the approximate theory yielding Eq. (11) for the ground state 14 gives energy eigenvahes which agree within 0.5 % with the results of Barnes eta/. 15 over the whole range 0 < a < 1. Instead, the mean value (15) of the energy, which follows from solving exactly the bracket (g]HJg), where [g) is the ket (11), yields much larger errors for a in the range 0.5 < a < 1. The same applies to the transverse correlation $1. The mean value $1 can be calculated directly from the approximate formalism without resorting to the Hellmann-Feynman theorem, which applies only to the exact eigenenergies. From the definition (12) of the ~b-operators and the asymptotic (a --* 0) property (14) one can readily show that Sx = - ~ / 6 . This is the same result obtained from applying the Hellmann-Feynman theorem to the energy eigenvalue (1). Therefore, the approximate formalism, valid in principle in the limit a --* 0, is consistent with both the Hellmann-Feynman theorem and the results of Barnes eta/. over the whole range 0 _< ~ _< 1 with high accuracy. This suggests that our bosonization of the Heisenberg Hamiltonian is essentially correct for 0 ~ a _< 1, but the explicit form (12) for the Bose operators, which was used in the derivation of Eqs. ( 1 5 ) - (19), is still too simple, and has to be refined in order to achieve better accuracy in the range 0.5 < a < 1.

Acknowledgements Two of the authors (M. L. and M. K.) acknowledge financial s u p p o r t from F O N D E C Y T (Chile). One of them (E. R. G.) received support from the Consejo Naeional de Investigaciones Cientfficas y T&nicas, Argentina, and the other (G. G. C.) is recognized to the brazilian agencies C N P q and F I N E P . They axe also grateful to Mr. J. Rogan for the artwork.

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