spin waves in heisenberg two-dimensional antiferromagnets s = 12 with skyrmions

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Solid State Communications. Vol. 103, No. 4. pp. 249-253, 1997 0 I997 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0038%1098/97 $17.00+.03 PII: SOO38-1098(97)00146-4 SPIN WAVES IN HEISENBERG TWO-DIMENSIONAL ANTIFERROMAGNETS S = l/2 WITH SKYRMIONS S.I. Belov and B.I. Kochelaev Kazan State University, 420008 Kazan, Russia (Received 2 Januar?/ 1997; accepted 17 March 1997 by P. Burlet) Quantum consideration of skyrmions and elementary spin excitations in a two-dimensional Heisenberg antiferromagnet is given. We have shown that spin fluctuations considerably renormalize the local order parameter, skyrmion energy and average radius of thermally excited skyrmions. While this renormalization at low temperatures T e J, where J is the nearest-neighbor exchange constant, results only in a reducing of the exponential divergence of Y,J at T - 0, at high temperatures T - J it leads to a linear temperature dependence of Ilra. 0 1997 Elsevier Science Ltd Keywords: A. high-T, superconductors, magnetically ordered materials, D. spin dynamics, E. neutron scattering. The magnetic behavior and spin dynamics of the cuprate superconductors and their parent compounds have been extensively studied during the last few years. It is well established that their properties are strongly influenced by two-dimensional (2D) critical fluctuations and the undoped materials can be modeled by a nearest- neighbour S = l/2 quantum Heisenberg antiferromagnet (QHAF) on a square lattice with a large isotropic exchange coupling constant (J = 1580 K in LazCuOb [l I). Considerable progress has been made in under- standing of the low temperature properties of 2D QHAF by a mapping to a classical nonlinear sigma model with parameters renormalized by the quantum fluctuations [2-41. The staggered magnetization, spin stiffness constant ps and spin wave velocity c were calculated in a spin wave theory at T = 0. Using this method Chakravarty, Halperin and Nelson (CHN) were able to obtain the correlation length and the spin- correlation functions with no adjustable parameters [2]. The CHN results for the renormalized classical (RC) regime were improved by Hasenfratz and Niedermayer (HN) [5] using the chiral perturbation theory and by Chubukov et al. [6] with the 1lN expansion method on antiferromagnets with an N-component order parameter. The predicted exponential temperature dependence of the spin correlation length 4 was found in a fairly good agreement with neutron-scattering measurements in a pure LazCuOd at T < 600 K [l, 71. At the same time the authors could not avoid a fitting of the ps value. In this paper we want to show that a different approach based on a self-consistent quantum consideration of 2D QHAF with skyrmions can predict similar magnetic properties. In particular, in a simplest local mean field approxima- tion we obtain with no adjustable parameters an average radius of thermally excited skyrmions, which can be identified with the spin correlation length 4. At low temperatures T 6 J its temperature dependence is almost the same as in the CHN-HN theory for the RC regime with the$tted ps. Moreover, at the temperatures T > 0.4J the inverse average radius of skyrmions increases linearly with temperature being rather close to the result of Chubukov et al. [6] obtained for l/t in a quantum critical (QC) region. A general static solution for spin textures called skyrmions in 2D Heisenberg ferromagnets were obtained by Belavin and Polyakov [8] from a classical nonlinear sigma model. A number of qualitative and experimental arguments in favor of an important role of skyrmions in unusual magnetic properties of layered magnets was given later by Waldner [9]. Recent attempts to explain some peculiarities of the spin dynamics in layered anti- ferromagnets, including weekly doped cuprates, by the influence of skyrmions show a renewed interest to this problem [lo]. It seems, however, that an adaptation of the classical description of skyrmions is not sufficient in the case S = l/2. We use from the very beginning a quantum consideration, which gives an opportunity both to take into account the quantum fluctuations to 249

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Page 1: Spin waves in Heisenberg two-dimensional antiferromagnets S = 12 with skyrmions

Solid State Communications. Vol. 103, No. 4. pp. 249-253, 1997 0 I997 Elsevier Science Ltd

Printed in Great Britain. All rights reserved 0038%1098/97 $17.00+.03

PII: SOO38-1098(97)00146-4

SPIN WAVES IN HEISENBERG TWO-DIMENSIONAL ANTIFERROMAGNETS S = l/2 WITH SKYRMIONS

S.I. Belov and B.I. Kochelaev

Kazan State University, 420008 Kazan, Russia

(Received 2 Januar?/ 1997; accepted 17 March 1997 by P. Burlet)

Quantum consideration of skyrmions and elementary spin excitations in a two-dimensional Heisenberg antiferromagnet is given. We have shown that spin fluctuations considerably renormalize the local order parameter, skyrmion energy and average radius of thermally excited skyrmions. While this renormalization at low temperatures T e J, where J is the nearest-neighbor exchange constant, results only in a reducing of the exponential divergence of Y,J at T - 0, at high temperatures T - J it leads to a linear temperature dependence of Ilra. 0 1997 Elsevier Science Ltd

Keywords: A. high-T, superconductors, magnetically ordered materials, D. spin dynamics, E. neutron scattering.

The magnetic behavior and spin dynamics of the cuprate

superconductors and their parent compounds have been extensively studied during the last few years. It is well established that their properties are strongly influenced by two-dimensional (2D) critical fluctuations and

the undoped materials can be modeled by a nearest- neighbour S = l/2 quantum Heisenberg antiferromagnet (QHAF) on a square lattice with a large isotropic exchange coupling constant (J = 1580 K in LazCuOb [l I). Considerable progress has been made in under- standing of the low temperature properties of 2D QHAF by a mapping to a classical nonlinear sigma model with parameters renormalized by the quantum

fluctuations [2-41. The staggered magnetization, spin stiffness constant ps and spin wave velocity c were calculated in a spin wave theory at T = 0. Using this

method Chakravarty, Halperin and Nelson (CHN) were able to obtain the correlation length and the spin- correlation functions with no adjustable parameters [2]. The CHN results for the renormalized classical (RC) regime were improved by Hasenfratz and Niedermayer (HN) [5] using the chiral perturbation theory and by Chubukov et al. [6] with the 1lN expansion method on antiferromagnets with an N-component order parameter. The predicted exponential temperature dependence of the spin correlation length 4 was found in a fairly good agreement with neutron-scattering measurements in a pure LazCuOd at T < 600 K [l, 71. At the same time the authors could not avoid a fitting of the ps value. In this

paper we want to show that a different approach based on a self-consistent quantum consideration of 2D QHAF with skyrmions can predict similar magnetic properties.

In particular, in a simplest local mean field approxima- tion we obtain with no adjustable parameters an average radius of thermally excited skyrmions, which can be identified with the spin correlation length 4. At low temperatures T 6 J its temperature dependence is almost the same as in the CHN-HN theory for the RC regime with the$tted ps. Moreover, at the temperatures T > 0.4J the inverse average radius of skyrmions

increases linearly with temperature being rather close to the result of Chubukov et al. [6] obtained for l/t in a quantum critical (QC) region.

A general static solution for spin textures called skyrmions in 2D Heisenberg ferromagnets were obtained by Belavin and Polyakov [8] from a classical nonlinear

sigma model. A number of qualitative and experimental arguments in favor of an important role of skyrmions in unusual magnetic properties of layered magnets was given later by Waldner [9]. Recent attempts to explain some peculiarities of the spin dynamics in layered anti- ferromagnets, including weekly doped cuprates, by the influence of skyrmions show a renewed interest to this problem [lo]. It seems, however, that an adaptation of the classical description of skyrmions is not sufficient in the case S = l/2. We use from the very beginning a quantum consideration, which gives an opportunity both to take into account the quantum fluctuations to

249

Page 2: Spin waves in Heisenberg two-dimensional antiferromagnets S = 12 with skyrmions

250 SPIN WAVES IN HEISENBERG ANTIFERROMAGNETS Vol. 103, No. 4

renormalize the skyrmion energy and to investigate spin excitations of 2D QHAF above the skyrmion’s back- ground. In the following we calculate by the Green functions method the energy spectrum of the elementary spin excitations, local order parameter, renormalized skyrmion energy and spin correlation length.

We start with a unitary transformation of the Heisen- berg Hamiltonian to new local quantization axes at every site of a square lattice of spins S = l/2

~ = U-‘HU = J~Si~j. (1) (!j)

Here J is an antiferromagnetic exchange integral, (zj) means the sum over the nearest neighbors of two sublattices and U has the form

U = fl exp (iLS;pi + iS;pj) exp (i$‘ei + iSTf3j). ij

(2)

Having in mind an existence of the local antiferro- magnetic order, we define the local quantization axes in such a way that the transverse spin components, averaged with an equilibrium density matrix, turn to

zero and the longitudinal spin components of the nearest neighbors i and j will have an opposite sign: (S,) = (S,) = 0, (sf) = - (Sj) = 0. We expect that the local order parameter u does not depend on the site number. In the case S = l/2 it is convenient to introduce

different Pauli operators for two sublattices instead of spin operators S’ = Y + 9

Si- = Cr+, So = pi, So = l/2 - a+cui;

Sj- = pj, Si’ = pi, So = - l/2 + P’Pi, (3)

yy+ + y+y = 1, yy = 7+7+ = 0, y = (Y, p.

All the Pauli operators for different sites commute with

each other. Then the Hamiltonian takes the form

E. = - Ja( 1 - a)Nz, Esk = Ja( 1 - a) x( 1 - Aii), (4) (ij)

Ho = J x {uAij(a’ai + b;bj) + Bijaibj + Bia+bT). (ii)

Here E. + Esr, represents the ground energy of the metastable state, including the inhomogeneous part Esk

and Ho + HI describes the spin fluctuations. We did not write down explicitly the term HI since it gives no contribution to equations of motion in the mean field approximation and continuum limit. N is a full number of spins, z is a number of the nearest neighbors (z = 4). coefficients A, and B, are rather cumbersome functions of angles 8 and cp for the particular sites i and j. The structure of the metastable inhomogeneous state we find

from the minima conditions for the functional Esk( (61,

((0)). For a single skyrmion we obtain

tan(e/2) = T/To, cp = 9, Esk = 47rJu( 1 - 0) = f,k

(5)

where r and 6 are the polar coordinates and ro has a meaning of the skyrmion size (at r = r.

($) = u cos 8 = 0). This form of the skyrmion coincides with the result of Skyrme Ill], derived in context of topological solitons of the bosonic pion fields. The skyrmion shape can be visualized by the stereographic projection of the sphere points onto the plane, with an

additional reflection in a mirror through the sphere center in the case of antiskyrmions. In a general case the energy

Esk = &k, where e iS the hrgeSt from the numbers of skyrmions and antiskyrmions [8]. We have to point out, however, that the skyrmion energy in the classical sigma model c,k = 47rJS2.

The next step is to derive an equation for the local order parameter u. Using the Hamiltonian (4), the mean field approximation and the continuum limit, it is easy to obtain the following equations of motion for the Pauli

operators CY and /3

ifi&(r, t) = Jua*/iicu(r, t) + ?/3(r, t)),

Z&r, r) = - Jua*[ii/S(r, t) + i%r(r, r)],

where we have introduced the operators

ii = .7,la* - 2 sin28(Vp)*,

(6)

0 = z/a2 + V* - 2i cos e(vg) - (VP)*. (7)

Straightforward calculations by the Green functions

method on the basis of the equations (6) and (7) give the result

Here ef = 2&uJakf is the energy of elementary

excitations and #i(r) and kj are the eigenfunction and eigenvalue of the operator ii - 0

(2 - %+(r) = k&(r). (9)

In the case of a single skyrmion a solution of this equation has a cylindrical symmetry

$,k(r) = --teimQ&k(r), 6

(10)

where m is an azimuthal quantum number. An approximate solution of equation (10) is found to be

&k(r)‘” CmkJ,,, - 1 W, r/r0 -% 1;

&k(r)= C’,,,kJm+l (kr),

(11) r/r0 S 1.

Page 3: Spin waves in Heisenberg two-dimensional antiferromagnets S = 12 with skyrmions

Vol. 103, No. 4 SPIN WAVES IN HEISENBERG ANTIFERROMAGNETS 251

Here J,(kr) is the Bessel function, C,,,k is a constant and a

possible spectrum of k depends on boundary conditions.

For a sample with infinite sizes this spectrum is con- 0.6.

tinuous and does not depend on m and C,, = 1. 1

Substituting equations (10) and (11) into equation (8) 20.5.

and performing the sum over m, we are arriving to the 5

conclusion that the dependence on r disappears from the 20.4.

equation for the local order parameter in accordance with Z our expectations. For the anti-skyrmions the functions in g 0.3

equation (11) have to be interchanged. 8

In the case of a large number of the topological t 0.2 .

excitations it is natural to expect that the nearest 5

neighbors of every skyrmion will be antiskyrmions and 0.1

vice versa; it follows that the full staggered magnetiza-

tion equals to zero (the Mermin-Wagner-Hohenberg 0”

theorem). The probability of creation of a skyrmion- 0.2 0.4 0.6 0.6 1

T/J antiskyrmions pair in the thermodynamic equilibrium is proportional to exp (-E&‘), what gives an opportunity to Fig. 1. Inverse average skyrmion radius as a function of

estimate an average radius of the skyrmion (antiskyrmion) x = TM. The solid lines are a/r,, in the RC and QC

2rri = Na*/(Q) = a2[exp (e,JT) - 11. regions. The dotted line is the numerical solution of a

(12) set of equations (12) and (16). The dashed line is the

Evidently, the condition Q > 1 is fulfilled for a certain CHN-HN result for a/f with 27rp, = 0.945; the dash-

relation between the linear size of a sample and the dotted line is a/[ of [6] for the QC region.

temperature with a,, = 0.443. In the case (I = a,, it would be

T > e&n N = e,J2 In (Ua) = T*. (13) a/r0 = 0.7x. Such a linear temperature dependence of

Although the long range order is absent, the local order the inverse correlation length in the CHN theory was

still takes place. To solve the equation for u [equation 8)] related to the quantum-critical regime [2]. However,

we use the skyrmion shape [equation (5)] for r < r. and an inspection of a set of equations (12) and (16) shows

the corresponding approximation for R&r) in equation that d has a maximum at x = 1, a,,, = 0.3317 < CJ,,.

(11). Since this solution is exact at r - 0, it is reasonable Nevertheless the system at x near to 1 is being close

to put in equation (8) r = 0. Then the sum over m is enough to the critical point. At the temperatures of the

reduced to the single term with m = 1. The spectrum of k order of J we can put (T = a,,, in equation (16) and expand it near x = 1. The result is a universal linear function

is determined by the boundary condition for R&r) at r = r-0. For m = 1 we have found an exact solution of

a/r0 = 0.867x - 0.245. (17)

equation (9)

Cd-i Clkr* Rdr) = ,_z + 4 J&r> - r2 + ri Jz(kr). (14)

For a large skyrmion r. P a the particular form of the boundary condition is not important for quantities like a density of states, which is actually needed in

equation (8). In the case of Rlk(rO) = 0 we have for a sequence of k,

Evidently, this expression is valid for r. > a only. In

Fig. 1 it is plotted in the region 0.4 < x < 1. An approximate solution of equations (12) and (16) for the temperature region T* < T 4 J is found to be

(1% where nmax is determined by the completeness condition of the $,(r> basis for a square lattice. Using equation (15) we can reduce equation (8) to the form

a

G = 2&a ex exp[ 2r(ur0c’)], x = TLi, (16)

(18)

(19)

__ln 4+vo 0,83x v=va+ 2?r

( )T

: = ~~4~i/r.ljlxp~2~uo(:..)l,

with a0 = 0.181 and (Y = 0.53 1. In Fig. 1 aho is plotted according to equation (19) at x < 0.4. It is shown, also, a numerical solution of the set of equations (12) and (16). One can see that, while aho at x > 0.4 is the linear function, ro exponentially diverges as x- 0. Such a behavior of the spin correlation length in 2D QHAF was recognized first by Chakravarty et al. [2]; the corresponding ranges of temperatures were referred as

Page 4: Spin waves in Heisenberg two-dimensional antiferromagnets S = 12 with skyrmions

252 SPIN WAVES IN HEISENBERG ANTIFERROMAGNETS Vol. 103, No. 4

the quantum-critical and renormalized classical regions. For the RC regime the CHN-HN theory predicted [5]

The renormalized value Ma = 2&SZ, = 1.645,

2ap, = 0.943. Since our result in the exponent for

27rJaa( 1 - era) = 0.93J practically coincides with this value, equations (19) and (20) differ in the pre-

exponential factor only. Recently Chubukov et al. 161 have calculated 5 in the whole region 0 < T < J by the l/N expansion method on antiferromagnets with an N-component order parameter. Their formula for the RC regime coincides with the CHN result (the pre- exponential factor is OS), but 27rp, = 1.135 due to the next order spin wave theory corrections. This value is very close to 27rp, = 1 .l lJ, which was obtained

from the best fit of equation (20) to the neutron scattering data in La2Cu04 at x < 0.35 [7]. Deep in the QC region their prediction for a/( differs from our

equation (17) only slightly by the temperature slope

value

a - = 1.039 t

= 0.63x - 0.24. (21)

Here 27rp, = 1.133, N = 3 and y = 1 were used. To compare our results for a/r,, with these findings, they are shown, also, in Fig. 1. At x < 0.4 we have plotted a/[

from equation (20) with 27rp, = 0.945 and at x > 0.4, the

linear function (21). The RC region in a larger scale is presented in Fig. 2 together with the experimental data [7] and with an additional curve for a/( with

2xp, = 1.13J. One can see that the latter is very close to our result [equation (19)]. It is worth to mention, also, that the RC curve a/[ of [6] with 27rp, = 1.13J is undistinguishable from ours [equation (19)] and it is

not shown. Regarding a qualitative and even quantitative

agreement between our results for the temperature dependence of the average skyrmion radius ro and the corresponding results for the spin correlation length 4 calculated in the most advanced theories of the 2D QHAF homogeneous state, one can ask whether the skyrmion theory is just an alternative method of calcula- tions, or there are alternative testing predictions behind it. We believe that in the case of a long enough life time of skyrmions in comparison with a characteristic time scale of measurements one can reveal the spin texture of the skyrmion background and peculiarities of the spin waves inside the skyrmions. In particular, the spin texture leads to additional “tails” in the static magnetic structure factor observable by elastic diffuse neutron

0.02

?OO 200 300 400 !500 800 Temperature M

Fig. 2. Inverse correlation length. The same curves as in Fig. 1 for RC region are plotted against T plus the dashed-dotted line for a/,$ with 2?rp, = 1.135. Experimental points for La2Cu04 with TN = 325 K are taken from [7].

scattering and to an additional asymmetric broadening of the NMR line [ 121.

In conclusion, we have studied the spin wave excitations in a two-dimensional Heisenberg antiferro- magnet S = l/2 with thermally excited skyrmions. By the quantum Green functions method we have calculated in the simplest mean field approximation the energy spectrum, local order parameter and the average radius

r. of the skyrmions. We have shown that ro has almost the same temperature dependence as the spin correlation

length t calculated by different rather sophisticated methods for 2D QHAF in a homogeneous state. This mapping concerns both an exponential divergence of ro

at T - 0, which is characteristic for the renormalized classical regime for .$ and the linear temperature dependence of alro at T of the order of J, what can be recognized as the quantum critical region.

Acknowledgements-One of the authors (BIK) is grateful to Franz Waldner for drawing attention to this problem and fruitful discussions. He also thanks Zurich University, where part of this work was carried out, for its hospitality. This work was supported in part by the Russian Foundation for Basic Research under Grant No. 96-02-02725.

1.

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Vol. 103, No. 4 SPIN WAVES IN HEISENBERG ANTIFERROMAGNETS 253

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