Magnetization curve of a square-lattice Heisenberg antiferromagnet

PHYSICAL REVIEW B

VOLUME 57, NUMBER 9

1 MARCH 1998-I

Magnetization curve of a square-lattice Heisenberg antiferromagnet M. E. Zhitomirsky* and T. Nikuni Department of Physics, University of Toronto, Toronto M5S 1A7, Canada ~Received 25 August 1997! We calculate the uniform magnetization of a spin-1/2 Heisenberg square lattice antiferromagnet at T50 in magnetic fields up to the saturation field in the framework of spin-wave theory including quantum corrections of the second order in 1/S. External field generally suppresses zero-point fluctuations, which leads to an upward curvature of the magnetization curve at low fields and logarithmic singularity close to the saturation. @S0163-1829~98!03610-8#

The magnetic properties of the copper oxide materials in their insulating phase are believed to be described by a square lattice spin-1/2 Heisenberg antiferromagnet ~AFM! with a rather large exchange constant J;1500 K. This fact has renewed interest in the low-dimensional quantum AFMs.1 On the other hand, strong exchange coupling between neighboring spins in cuprates prevents any significant distortion of the AFM order by available magnetic fields. Therefore, most theoretical works on a square lattice AFM ~SAFM! have been focused on the zero-field properties and the limit H→0. Recent progress in synthesis of molecular based magnets has provided a new family of spin-1/2 square lattice antiferromagnets: ~5CAP! 2 CuBr~Cl! 4 , ~5MAP! 2 CuBr~Cl! 4 , which have extremely small in-plane exchange constants in the range J.1 –8 K.2 Moderate magnetic fields 4–20 T are sufficient to destroy the AFM order in these materials and to saturate magnetic moments. Theoretical results for this newly achievable regime of the quantum SAFM are quite restricted. They include calculations of the linear susceptibility,3–5 numerical simulations of the magnetization curve by exact diagonalization on finite spin clusters,6 and the hard-core boson study of the region close to the saturation field H c . 7 Our aim is to advance the theory of SAFM in strong magnetic fields at T50 by calculating the whole magnetization curve up to the second order in 1/S expansion. We consider the Heisenberg Hamiltonian in a uniform field:

ˆ5 H

z Si •S j 2H ( S i ( i i, j ^ &

0

,

~1!

where summation over all pairs of nearest-neighbor spins is assumed. Since the crystal anisotropy is not included in Eq. ~1!, the only low-field phase at zero temperature is a state with the AFM vector oriented perpendicular to H and sublattices canted towards the field direction by the angle u : u 50 at H50, u 590° at the saturation field H c 58S. Spin components in the laboratory frame are related with those in the rotating local coordinate system by

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z

S i 0 5S zi e iQ–ri cos u 2S xi sin u ,

~2!

x

S i 0 5S zi sin u 1S xi e iQ–ri cos u , y

S i 0 5S iy and Q5 ~ p , p ! . Our spin-wave calculations are based on the standard procedure, which transforms spin operators into bosons by the Holstein-Primakoff transformation expanding further square roots in powers of 1/S. Aiming to calculate the magnetization up to the second order in 1/S we use S zi 5S2n i ,

n i 5a 1 i ai ,

S1 i 5 A2S2n i ,

x y S1 i 5S i 1iS i ,

S

a i ' A2S 12

D

~3!

ni a . 4S i

Substituting Eqs. ~2! and ~3! into Eq. ~1! we obtain for the Hamiltonian ˆ ~ 1 ! 1H ˆ ~ 2 ! 1H ˆ ~ 3 ! 1H ˆ ~ 4 ! 1¯, ˆ 5H ˆ ~ 0 ! 1H H

~4!

ˆ (n) denotes terms having a factor S 22n/2 and consistwhere H ˆ (0) , H ˆ (2) , H ˆ (4) ing of products of n boson operators. Terms H ˆ (1) and H ˆ (3) appear only for exist even at H50, whereas H the noncollinear phase, when spins are tilted by applied field. It is instructive to look first at the results of the harmonic ~or linear spin-wave! approximation, which amounts to keepˆ . Minimiing only the first three terms in the expansion of H zation of the classical energy ~per one spin! ˆ ~ 0 ! 522S 2 cos 2u 2HS sin u H

~5!

with respect to the tilting angle u yields sin u5H/Hc , which ˆ (1) [0. The remaining quadratic form of is equivalent to H ˆ (2) becomes after Fourier transformation boson operators H ˆ ~ 2 !5 H

(k @ A ka †ka k2

1 2

A k54S ~ 11sin2 ug k! ,

5013

B k~ a ka 2k1H.c.!# , ~6! B k54S cos2 ug k ,

© 1998 The American Physical Society

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ˆ (2) can be easily diagonalized where g k5 21 (cos kx1cos ky). H by the Bogoliubov transformation † a k5u kb k1 v kb 2k ,

~7!

u 2k511 v 2k5 21 (A k / v k11), u kv k5B k/2v k , v k5 AA 2k2B 2k. Eigenfrequencies give the classical spin-wave spectrum

v k52S A~ 11 g k!~ 12cos 2ug k! 52S n k ,

~8!

whereas quantum effects add a zero-point contribution to the ground-state energy ˆ ~ 0 ! 1 21 E g.s.5H

(k ~ v k2A k! .

~9!

For the further consideration it is convenient to define the following averages:

(k v 2k ,

m5 ^ a †i a j & 5

d 5 ^ a i a i & 5 ( u kv k ,

D5 ^ a i a j & 5

n5 ^ a †i a i & 5

(k v 2kg k ,

~10!

FIG. 1. Magnetization curve for spin-1/2 SAFM. Circles are numerical data of Ref. 5. Dashed and solid lines are the first- and second-order spin-wave results, respectively. Field is measured in units of J.

Zero-temperature magnetization is calculated by taking the derivative of E g.s. with respect to the magnetic field

which should be compared to the expansion in even powers of H for the energy of the classical AFM. The linear dependence of the total susceptibility x (H)5dM /dH on magnetic field following from Eq. ~13! has recently been observed in an antiferromagnetic material Ba 2 CuGe 2 O 7 .10 In the opposite limit, for fields close to the saturation field Eq. ~11! gives

k

M5

F

H 1 12 8 2SN

(k u kv kg k .

G

(k g kA12cos2 kug k . 11 g

~11!

Although very simple, the above expression has not yet been discussed in the literature on SAFM. For H→0 one can substitute u '0 into Eq. ~11!. Then, an expression in the square brackets gives renormalization of the classical susceptibility x cl51/8. For S51/2 one finds the well-known linear spinwave result x 50.05611.8 At finite fields u Þ0 and the magnetization of the quantum AFM varies nonlinearly with H. The upward curvature of the magnetization curve ~plotted for S51/2 in Fig. 1! is a result of gradual suppression of zero-point oscillations by external field. Spin reduction decreases continuously in magnetic field,

^ S & ' ^ S & 01

1 H , p Hc

for H!H c ,

~12!

where ^ S & 0 50.3034,8 contributing to the nonlinear growth of M (H). Expanding Eq. ~11! in powers of H/H c we obtain M ~ H ! ' x H1

2 H2 . p H 2c

~13!

Note that the second nonlinear term in M (H) characteristic to a two-dimensional ~2D! spin system9 is quite unusual, since it implies nonanalytic field correction to the ground state energy of SAFM from quantum fluctuations. Direct expansion of Eq. ~9!, indeed, yields

x 2 2 uHu3 E g.s.~ H ! 5E g.s.~ 0 ! 2 H 2 , 2 3 p H 2c

~14!

F

M ~ H ! 5S 12

G

p p2 ln , pS 4p

~15!

where p5(H c 2H)/H c . The magnetization curve approaches a plateau in saturated phase with an infinite slope at H5H c . An analogous result for the singularity in the field derivative of M (H) at H5H c with exactly the same logarithmic prefactor was obtained in Ref. 7. The above derivation in the framework of the linear spin-wave theory is, however, much simpler and Eq. ~11! for the magnetization holds at the entire region 0,H,H c . Note that the actual singularity of susceptibility near H c ~Fig. 1! is somewhat larger than predicted by ~15!. As we can see from Fig. 1, already the harmonic approximation for the magnetization of SAFM contains qualitative differences with respect to the classical result and satisfactory fits numerical data. We study convergence of the 1/S expansion in magnetic field by calculating the next order correction to Eq. ~11!. For that we find contributions of the order of O(1/S 0 ) to the ground-state energy. But first we briefly discuss how to guess the behavior of the corrected curve before doing any calculations. The starting point is that at T50 the magnetization is determined by the derivative of the field-dependent ground-state energy. Hence, E g.s.~ 0 ! 2E g.s.~ H c ! 5

E

Hc

0

M ~ H ! dH.

~16!

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This formula is valid either for classical or for quantum AFM’s, for the latter both sides have to be calculated to the same order in 1/S. At H5H c , all spins become ferromagnetically aligned along H. At zero temperature quantum fluctuations disappear in the saturated phase and the exact ground-state energy of the quantum system coincides with the classical result. Now let us apply Eq. ~16! to the first and second order results of the 1/S expansion. As is well known, the nonlinear corrections decrease further the ground-state energy at H50.8 Consequently, the integral on the right hand side also decreases, what implies that the corrected magnetization curve lies ‘‘in a whole’’ below the harmonic result. In magnetic field nonlinear corrections to E g.s. come from (4) ˆ H ~first-order contribution! ˆ ~ 4 !5 H

(

^ i, j &

5015

E 83 524 sin2 u ~ D1m2n ! 2 .

Finally, there is one more contribution to E g.s. from the ˆ (3) determined by three-magnon processes. cubic term H They appear when all three bosons in the Fourier transform of Eq. ~18! are taken with different momenta. Keeping only those terms that give a nonvanishing second-order contribution we find ˆ ~23 ! 54 H

2 ~ 12cos

2u% ~17!

ˆ (3) ~second-order contribution! and from H ˆ ~ 3 ! 52 H

A

S sin 2u 2

e iQ–r ~ a †i 1a i ! n j . ( i, j

2sin2 u ~ 2nm1D d !# .

~19!

ˆ (3) leads to the term that is The same procedure applied to H linear in boson operators

A

SN sin 2u ~ D1m2n !~ a †Q1a Q! . 2

~20!

Its role is to renormalize the canting angle from the classical u value u to ˜

F

1 2SN

g 2k1 g k cos 2 u 21

(k A~ 11 g !~ 12cos 2ug ! 11 k

~23!

H 2 cos2 u 48S 2 N 2

F ~ k,q! 2

, ( k,q n k1 n q1 n k1q2Q

~24!

where symmetric vertex F(k,q) is defined as F ~ k,q! 5 g k~ u k1 v k!~ v qu k1q2Q1u qv k1q2Q! 1 g q~ u q1 v q! 1 v k1q2Q!~ v ku q1u kv q! .

~18!

i

E 4 52 @ 2cos 2u ~ n 2 1m 2 1D 2 ! 1cos2 u ~ 2nD1m d !

sin ˜ u 5sin u 12

(

3~ v ku k1q2Q1u kv k1q2Q! 2 g k1q~ u k1q2Q

ˆ (4) & has to be found by normal The expectation value E 4 5 ^ H ordering of boson operators b k after applying a Bogoliubov transformation to Eq. ~17!. It can be more easily calculated by contracting various four-boson terms in pairs using Eq. ~10!:

ˆ ~13 ! 54 H

S sin 2u g k~ u k1 v k! 2N k,q

Corresponding expression for the second-order energy shift is E 39 52

2 u !@ a †i ~ n i 1n j ! a j 1H.c.# 2n i n j cos

A

1 1 3 v qu k1q2Q~ b 1 k b q b Q2k2q1H.c. ! .

$ 81 ~ 11cos 2u !@~ n i 1n j ! a i a j 1H.c.#

1 8

~22!

k

G

.

~21!

The quantum correction to the angle vanishes at H5H c , indicating that transition into the saturated phase occurs at the same critical field in both classical and quantum models. If this expression for the angle is combined with the renormalization of spin length to get the uniform magnetization M 5 ^ S & sin ˜ u , we again arrive at the above formula ~11!. The correction to the ground-state energy associated with du5˜ u 2 u consists of the linear contribution in d u obtained by variation of the zero-point energy 21 ( kv k and the quadratic contribution obtained by variation of the classical enˆ (0) . Their sum is ergy H

~25!

In the above expression for E 93 wave vectors k and q are taken independently in the first Brillouin zone. This leads to an additional factor 1/3!, because the same set of three magnons @ k,q,(Q2k2q)# appear 3! times in the sum. All contributions of the order of O(1/S 0 ) must be summed up to get d E g.s.5E 83 1E 93 1E 4 , which, then, has to be differentiated numerically to obtain d M 52 ] ( d E g.s.)/ ] H. In the linear regime at H→0 differentiation can be done analytically. The corresponding expression for the susceptibility of SAFM is

x5

H

F

1 1 1 4 12 c 21 ~ c 1 2c 21 ! 1 I ~ c 21 2c 1 ! 1 2 8 2S 3 ~ 2S !

GJ

,

~26!

8

where we used the standard notations c n5

1 N

(k @~ 12 g k! n/221 # ,

~27!

c 1 520.15795, c 21 50.39320, and I stands for the 4D integral appearing in Eq. ~24!, which is taken at H50. The expression ~26! for susceptibility of SAFM is much simpler than the previously obtained second-order spin-wave results,4,5 e.g., in that it contains only one 4D integration. We have also been able to do numerical integration with higher accuracy, thus, getting

x 51/820.06889~ 2S ! 21 10.00815~ 2S ! 22 .

~28!

For S51/2, the susceptibility is x 50.0642660.00001. This second-order spin-wave result for x is close to the result 0.0659 obtained by series expansion around the Ising limit.3 Because of the positive second-order correction to x , the renormalized magnetization curve lies above the first order curve at low fields. This has to be reversed at higher fields, in

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particular near H c , in order to have smaller value for H * 0 c M dH. We plot the first and the second spin-wave results for the magnetization curve of spin-1/2 SAFM in Fig. 1 together with numerical data obtained by exact diagonalization of small clusters up to 50 spins.6 The second spin-wave correction to the magnetization is of the order of few percent in the whole field range 0,H,H c and improves an agreement with numerical data near singular point H5H c . At the same time, our results show a limitation of the numerical data,6

which predict the magnetization at low fields below both the first and the second order results of the spin-wave theory. In conclusion, we have demonstrated that the spin-wave theory continues to give consistent results at T50 for the Heisenberg square-lattice AFM in external field. The second order 1/S correction is small and yields improved agreement with numerical data for the magnetization curve.

*On leave of absence from L. D. Landau Institute for Theoretical

J. Igarashi, Phys. Rev. B 46, 10 763 ~1992!. M. S. Yang and K. Mu¨tter, Z. Phys. B 104, 117 ~1997!. 7 S. Gluzman, Z. Phys. B 90, 313 ~1993!. 8 R. Kubo, Phys. Rev. 87, 568 ~1952!; T. Oguchi, Phys. Rev. 117, 117 ~1960!. 9 For a 3D AFM at zero temperature quantum fluctuations yield nonlinear correction to the magnetization of the form (H/H c ) 3 ln(Hc /H). 10 A. Zheludev et al., Phys. Rev. Lett. 78, 4857 ~1997!.

Physics, Moscow, Russia. 1 E. Manousakis, Rev. Mod. Phys. 63, 1 ~1991!. 2 P. R. Hammer et al., J. Appl. Phys. 81, 4615 ~1997!; A. S. Albrecht et al. ~unpublished!. 3 Z. Weihong, J. Oitmaa, and C. J. Hamer, Phys. Rev. B 43, 8321 ~1991!. 4 C. J. Hamer, Z. Weihong, and P. Arndt, Phys. Rev. B 46, 6276 ~1992!.

The work of T. N. was supported by the Japan Society for the Promotion of Science.

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