ELSEVIER Physica B 241-243 (1998) 573 575

Two-dimensional Heisenberg antiferromagnet in strong magnetic fields

M.E. Zhitomirsky*, T. Nikuni

Department of Physics, University oJ" Toronto, 60 St. George str., Toronto, ON, M5S IA 7, Canada

Abstract

We calculate magnetization and magnon dispersion for a spin-12 Heisenberg square lattice antiferromagnet at T = 0 in magnetic fields up to the saturation field in the framework of spin-wave theory. Magnetic field generally suppresses zero-point fluctuations, which leads to upward curvature of the magnetization curve at low fields and logarithmic singularity close to the saturation. New quantum effects appear due to noncollinearity of the spin structure. They consist of coupling between one- and two-magnon spectrum and affect strongly dispersion of excitations at finite fields. ~-) 1998 Elsevier Science B.V. All rights reserved.

Keywords: Quantum antiferromagnet; Spin waves

Recent progress in synthesis of molecular based magnets has provided a new family of spin-½, square lattice antiferromagnets (SAFMs): (5CAP)zCuBr(C1)4, (5MAP)zCuBr(C1)4, which have extremely small in- plane exchange constants in the range J ~ 1-8 K [1,2]. Moderate magnetic fields 4 - 2 0 T are suffi- cient to destroy AFM order in these materials and to saturate magnetic moments. So far, theoret- ical studies of the Heisenberg AFM on a square lattice have been focused on the zero-field proper- ties. The only finite field results for SAFM are the numerical simulation of the magnetization curve [3] and the hard core boson study of the region close to the saturation field Hc [4]. Our aim is to

*Corresponding author: Fax: (416) 978-2537; e-mail: mzh@physics.utoronto.ca.

advance in the theory of SAFM in strong magnetic fields at T -- 0 by using a technique of 1IS expansion.

We consider the Heisenberg Hamiltonian in a uniform field:

,2/t ~' = J ~ Si "Sj - H ~ ST. (1) (i.j) i

Since the crystal anisotropy is not included in Eq. (1), the only low-field phase at zero temper- ature is a state with AFM vector perpendicular to H and sublattices canted towards the field direction by the angle 0 : 0 = 0 at H = 0 , 0 = 9 0 ° at H = He = 8JS. Our spin-wave calculations are based on the standard procedure which transforms spin op- erators into bosons by the Hols te in-Primakoff transformation expanding square roots in powers of 1IS. Aiming to calculate the magnetization and the spectrum to the second and the first order in 1IS

0921-4526/98/$19.00 :.~ 1998 Elsevier Science B.V. All rights reserved PII S 0 9 2 1 - 4 5 2 6 ( 9 7 ) 0 0 6 4 7 - 9

574 M.E. Zhitomirsl~3'. T. Nikuni / Phvsica B 241 243 (1998) 573 575

respectively we use: S: = S - a + a, S + ~ x¢, 2/~(1 - a+a/4S)a. Applying this transformation to each spin in its local coordinate frame, we have

. . ~ = , ) ~ z ( 0 ) At - . ~ , ( 1 ) AV , ~ ( 2 1 A- , ~ ( 3 ) ~_ . ~ ( 4 1 + " ' ' ,

i2)

where ~~"~ is term of the order(I /S) "/2 consisting of products of n boson operators. Terms ~0~, .~2), ,;~4~ exist at H = 0, whereas ,;@'~ and ,~31 appear because of spin tilting in applied field.

It is instructive to look first at the results of the harmonic (linear spin-wave) approximation, which amounts to keeping only the first three terms in the expansion of ,~. Minimization of the classical en- ergy ~ o l yields sin 0 = H/H~, which is equivalent to ,;~1~ _ 0. Quadratic form of boson operators .;@~2~ can be easily diagonalized giving classical spin-wave spectrum

~o°(k) = 2JSx/(l + ;',)(1 - cos 207k) (3)

and first order correction to the magnetization:

M = ~ 1 - ~ 7k l - c o s 2 0 ? , , ' (4)

where 7, = ½ (cos kx + cos kx). The upward curva- ture of M(H) for S = ½ seen in Fig. 1 is a result of gradual suppression of zero-point oscillations in external field. Expanding Eq. (4) in powers of H/Hc we obtain

2 H 2 M(H) ~ z H + - - - ~ . (5)

n H~

This form of M(H) implies that quantum fluctu- ations in SAFM yield nonanalytic dependence of the ground state energy on applied field. In the opposite limit, for fields close to the saturation field Eq. (4) gives

M ( H ) = S 1 - ~ l o g ~ , (6)

where 6 = ( H ¢ - H)/H~. The magnetization curve has an infinite slope at H = H~. Analogous result for singularity in M(H) close to H~ was previously obtained in Ref. [4], though with much smaller logarithmic prefactor.

The next order 1IS corrections to the ground state energy come from ~ 4 ~ (first-order contribu-

0 . 5 i i i i i i i i i i i i i i i i

0.4

e- ._o *" 0.3 .s

. ~ 0 . 2

0.1

0.0 0 1 2 3 4

f ield

Fig. 1. Magnet iza t ion curve for spin-½ SAFM. Circles are nu- merical da ta of Ref. [3]. Dashed and solid lines are the results of first and second order app rox ima t ion in 1/S, respectively. Field is measured in units of J.

tion) and from ,~t3) (second-order contribution). The latter can be written as .~?t~13) + .~3~ + ~31, where ,~.J~3j is linear in boson operators and re- normalizes canting angle, ~23) ~ ~ ( k , q)(b [ ~ + e bqbk +h.c.) represents coupling between one-mag- non and two-magnon excitations, and ,~331~

+ + + 3@'~3) ~ ~.c~ (33) ~F(k , qXbq , qbq bk + h.c.). O n l y . and yield nonzero corrections to the ground state en- ergy. After estimating corresponding four-dimen- sional integrals, the derivative of the energy with respect to field is taken numerically and the result for S = ½ is presented in Fig. 1. The second spin- wave correction to the magnetization is small in the whole field range 0 < H < Hc and improves an agreement with numerical data obtained on finite clusters [3].

In order to find renormalization of spectrum, we calculated first non-zero correction to the magnon self-energy and used in the first order 6o~(k)= Z1 ~(k, ~o°(k)). At H = 0 the only nonzero correction is from ,~,@l~, which shifts up spin-wave energies with respect to the classical value (3). With increas- ing field the coupling to the two-magnon con- tinuum ~ ~23~ ~ sin 20 grows and gives an opposite effect, i.e., pushes down magnon energies. The latter

M.E. Zhitomirslcv, 7". Nikuni / Physica B 241 243 (1998) 573 575 575

3.0

2.5

2.0

g 1.5

0.5

0.() 0.0

, i , i , i , i , , • , , , • , , , ,

" ~ .. ... H=O. 7H c

-.,.,%\ \NX

, /H=O. 1Hc

0 . | 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 . 0

w a v e v e c t o r

Fig. 2. Linear (dashed lines) and renormalized (solid lines) mag- non energies along the diagonal (0,0t (r~, re). Renormalized and classical energies have the same gap at k = 0. Field is measured in units of He, energy in units of J.

effect becomes dominant at intermediate fields H ~ 0.5He and leads to unusual wave-vector dis- persions shown in Fig. 2.

In conclusion, we have demonstrated that the physics of spin-½ SAFM can be understood in the framework of spin-wave theory. Noncollinearity of spin structure at finite fields leads to new quantum effects which can be seen in unusual magnon dis- persion.

References

[1] P.R. Hammer et al., J. Appl. Phys. 81 (1997) 4615. [2] A.S. Albrecht et al., preprint (1997). [3] M.S. Yang, K. Miitter, preprint cond-mat/9610092 (1996). [4] S. Gluzman, Z. Phys. B 90(1993) 313.