Self-consistent interaction of magnons in a frustrated quantum antiferromagnet

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Self-consistent interaction of magnons in a frustrated quantum antiferromagnet

L. Bergomi, Th. Jolicœur

To cite this version:

L. Bergomi, Th. Jolicœur. Self-consistent interaction of magnons in a frustrated quantum antifer- romagnet. Journal de Physique I, EDP Sciences, 1992, 2 (4), pp.371-377. �10.1051/jp1:1992149�.

�jpa-00246490�

Classification Physics Abstracts 75.10

Short Communication

Self-consistent interaction of _magnons in

_a

ii~ustrated quantum antiferromagnet

L.

Bergomi

and Th.

Jolicceur(*)

Service de Physique

Th60rique(**),

C-E- Saclay, F-91191 Gif-sur-Yvette Cedex, France

(Recdved

19 December 1991, accepted 27 January

1992)

Abstract. We study the magnon interactions by _means of _a self-consistent calculation in a frustrated quantum Heisenberg antiferromagnet with first and second neighbor exchange couplings. We have used the Hartree-Fock-Bogolyubov scheme at _zero temperature. The effect of frustration _on the shape of the spin-wave _spectrum is

investigated:

_we find _a strong decrease of the spin-wave velocity. A clear deviation from spin _wave theory is found when comparing

with finite lattice results.

The

discovery

of the remarkable

magnetic properties

of

high-Tc superconductors

has led to _a reexamination of

antiferromagnetic

quantum

spin

_systems

mainly

in two dimensions.

In the

insulating compounds La2Cu04

and

YBa2Cu306+~

with _{~ <} 0A the _{copper atoms}

are endowed with local moments which order in the N6el pattern at low

enough

temperature.

Inelastic neutron

scattering

studies have been

performed

below and above the Ndel temperature TN for tridimensional

ordering.

When T > TN ^{it is} _very

likely

that the _copper

spins

realize

the square lattice quantum

Heisenberg antiferromagnet (QHAF)

with

spins lI2

and

exchange only

between

nearest-neighbors [I].

One of the crucial issues is the nature of the

ground

state.

Magnetically

disordered

resonating-

valence-bond

(RVB)

states have been

proposed

_{[2] as}

possible

candidates for the

ground

states of the

Heisenberg antiferromagnets (HAF)

in two dimensions. However the

spin lI2

HAF

with nearest

neighbor exchange

has been studied

by

several

techniques

and there is _{now a}

convincing

^evidence

II,

_3] that it has

an ordered

ground

state at least

on the _square lattice.

The situation is far from

being

understood in the _case of frustrated

spin

systems. ^{The addition} of next-to-nearest

neighbor antiferromagnetic couplings certainly

destabilizes N6el order

on a

square lattice and it has been

suggested

_[4] that

beyond

_some critical values of the

couplings

it is

replaced by

_{a new}

magnetically

disordered

phase.

There _are

presently

several candidates for this

phase. Large

^N

techniques

[5, 6] suggest that

spin-Peierls

order _{can occur.} This idea has

(*) C-N-R-S- Research Fellow

(**)

Laboratoire de la Direction des Sdences de la Matibre du Commissariat h

l'Energie

Atomique

372 JOURNAL DE PHYSIQUE I N°4

received _some support from numerical [7] and ^series [8] studies _{on a} S

=

lI2

model with first and second

neighbor couplings (the

so-called Ji J2

model).

Lancz6s studies [7, 9] have mea-

sured _a

susceptibility

suited _to dimerized _states

showing

_an enhancement _near

J2/Ji

"

lI2.

More exotic

proposals

have also been made such _as chiral [10] or

spin

nematic

[Ill _ground

states for

antiferromagnets

with strong

enough

frustration.

In this _{paper we}

study

the Ndel

phase

of the Ji J2 model in detail

by

_means of _a self- consistent

spin-wave

calculation at _zero temperature. ^The

spin-wave velocity

determined in this

approximation

_can thus be

compared

with the finite-size

scaling analysis

of small size clusters [9]. ^{The data show} a clear diference between the semi-classical results

favoring

_a first-order

phase

^transition [12] ^{and the ab iniiio results}

showing

_{a new} behaviour for

J2IJI

_> 0A. We stress that such _a deviation _can be best _seen

by looking

at

quantities

different from the _energy which stays _very ^close to the semi-classical value _as is clear from

previous

works with similar

analytical

methods [13]. ^{It should be clear that} _energy ^{alone is} not

enough

to capture ^the

physics

of such _a system: in the _case of the unfrustrated lattice there _are disordered

RVB-type

variational wavefunctions which _are

extremely

close in energy ^to the

ground

state but with the wrong symmetry.

Recent studies have revealed how close is the unfrustrated lattice from the semiclassical

spin-wave approximation [Ii.

This

phenomenon

however does not survive with frustration: for

J2/Ji

_> 0A there is evidence for _an intermediate

phase

that is not found in the S

- c1o limit.

Let _us focus _on the so-called Ji J2 model:

H ₌

Ji ~j _{S; S;}

₊

_J2 ~ _S;

_Sk

_(I)

n-n- n-n-n-

The first _{sum runs over} nearest

neighbor

links of _{a square} lattice and the second _{over next-to-} nearest

neighbor

links. The S; are

spin lI2

operators and the

couplings

Ji and J2 are both

positive

I-e-

antiferromagnetic.

It is convenient to introduce _a

=

J2/Ji

which is the

unique

dimensionless parameter ^{of this model. We first discuss the}

properties

of the classical model related to

(I)

when the

spin

value S _{- c1o.} For values of _a smaller than

lI2

the classical

ground

state is Ndel ordered I-e- the

ordering

wavevector is Q

"

(x,x).

For _{a >}

lI2

there is

a continuous

degeneracy

at the classical level which is lifted

by

quantum

fluctuations,

the _so- called "order from disorder"

phenomenon [14].

The

ordering

selected

by

quantum fluctuations is defined

by

the wavevector

Q

₌

(0, x)

_or

(x, 0).

It is then

possible

to

study

the model

(I) by

_a

perturbative expansion

in _powers of I

IS starting

from the classical

ground

states [4]: ^{this is the} standard

spin-wave theory

d la Holstein-Primakoff. In the Ndel

phase

_{a <}

lI2

the calculation at

leading

order in the I

IS expansion

^leads to magnon modes with

dispersion

relation

given by:

Wk " Jl

/(I

tY

(1 7[))~ 7(, (1)

where 7k "

lI2 (cosk~

+ cos

kg)

and

7[

₌

cosk~

_cos

kg.

In the collinear

phase

_a >

lI2

_one

finds:

wk " Ji

~/(2a

+ cos kg )~

(2a7[

+ cos

k~)~, (2)

for the _magnon

energies

around the

(0,x)

state. In the classical

description,

there is _no

hysteresis

at the transition between the

(x, x)

and

(x, 0) (or (0, x)) phases.

The energy obtained for the Ji

J2

model when

including zero-point

fluctuations of free magnons is discontinuous at a =

1/2

and the derivatives _are infinite _on both sides of the transition. The

staggered

magnetization

< S~ > when

computed

at the _same order of

approximation

has _a

divergent

correction at _{a =}

1/2.

This has been

interpreted

_as the

signature

of _a

spin-liquid ground

state

[4].

However such behavior is

partly

_an artefact of the

simple approximation

of

non-interacting

magnons. At the classical transition

point

_{a =}

1/2

the

dispersion

^relation for

spin

_waves softens: entire lines of _zero modes _appear in the Brillouin _zone. Since this is _more than _we expect from the Goldstone

theorem,

this

degeneracy

is accidental and should be lifted when the interactions between

spin-waves

_are taken into account. Moreover, the

expansion

parameter of the naive

spin-wave approximation

_grows ^{in the}

neighborhood

of _a

=

1/2.

Its relevance becomes thus doubtful.

To

investigate

the effect of magnon interaction _{we use} the self-consistent Hartree-Fock-

Bogolyubov (HFB)

scheme at _zero temperature. ^{This is}

exactly equivalent

to the so-called modified

spin-wave theory (MSW)

introduced

by

Takahashi [15]. ^{In his scheme} one enforces the

Mermin-Wagner

constraint _< S~ _>

= 0

by

the _use of _a chemical

potential (Lagrange multiplier).

At _zero temperature this chemical

potential

for bosons goes ^to zero and _one is left

with _an

approximation

which is

simply

the HFB scheme. We have used the Holstein-Primakov

(HP) representation

of

spin

operators

leading

to _a bosonic Hamiltonian. It is convenient _{to use}

a

quantization

^{axis for} the

spin

operators which is

aligned

with the classical minimum at each site _so that there is

only

_one kind of boson in the

problem.

The

expansion

of the _{square roots} in the HP

representation

is

performed keeping

four-bosons terms in the

resulting

Hamiltonian:

H ₌

NS~J(Q)

₊ ^~

~j ^Bka)ak _{Ah (aka-k}

₊

a)at _~)

2

~

2

+

~ _{~ (2J(k4) J(k4}

+

Q) J(k4 Q))

6

(ki

k2

k3 k4) (a)a2a3a4

⁺

h-c-)

i~ _(2J(ki)

+

J(ki

+

Q)

+

J(ki Q))

6

(ki

+ k2 k3

k4) (a)a)a3a4

⁺

h-c-)

~~~

~~

k,

+

j _{~ (J(k~}

k~

Q)

+

J(k~

k~ +

Q))

6

(ki

_{+ k~ k~}

_k~) all

a~a~, k,

where h. _{c. means} hermitian

conjugate.

In this

equation

the

spin-wave

operator _ah

satisfy

the bosonic commutation

relations,

the

ordering

wavevector is denoted

by Q

_so that

equation (3)

is true for both N6el and collinear

phase.

There _{are no} three-bosons interactions since _we restricted ourselves to these two commensurate structures. The k _{sums run over} the two- dimensional Brillouin _zone and

J(Q)

is the Fourier transform of the

exchange coupling.

The

quadratic

part contains the two

quantities:

Ah ₌

2J(k) J(k

+

Q) J(k Q)

and Bk ₌

2J(k)

+

J(k

+

Q)

+

J(k Q) 4J(Q). (4)

We then search _a variational _vacuum (0) ^defined

by

the

density

matrix elements _{pk #}

(0(a)

_ah(0)

and _Kk

"

(0(aka-k(0).

These matrix elements _are

conveniently parametrized

in _terms of _an

angle bkl

p~ =

(cash

_R~

i)

and _~~

=

)sinh

^R~.

⁽⁵⁾

Minimization of the _energy leads to a set of self-consistent

equations

for the unknown

ik

Direct iteration _can then be

implemented numerically.

This

reproduces

the

findings

of reference [13]

using

the formalism of Takahashi. It is

amusing

to note that in the _zero temperature ^limit

(as

ⁱⁿ

Ref.

[13])

^{there is} no need of such _a formalism since it reduces _to the well-known self-consistent scheme for bosons.

The Ndel state is stabilized

beyond

the classical

instability point.

This

interesting phe-

nomenon of quantum stabilization of _a state is also found in the

Schwinger

boson

approach

374 JOURNAL DE PHYSIQUE I N°4

to the J2 Ji model

[16, Ii].

It also

happens

in the _case of helical

ordering [18].

Both N4el and collinear

phase

have metastable domains of existence. The

crossing

of

energies happens

for _{ac m} 0.62 and at this

point

there is _no

divergence

of the

staggered magnetization.

The collinear

phase disappears

with

diverging

< S~ _{>- -c1o}

right

at the classical transition

point

a =

1/2.

However this

phenomenon happens only

within the metastable

region

where the Ndel

phase

is lower in energy. In the HFB scheme there is thus _a first-order transition between the two ordered states. There

findings

_can be

qualitatively

^obtained from _a

simple

treatment of the HFB

equations

_{[12]: one uses} the zeroth order contractions of the Bose operators in the

quartic

terms of

equation (4)

to find the correction _to the

Bogolyubov angles

ik. In such _{a treatment}

one expects to

get

rid of the accidental

degeneracies

_seen in the lowest order calculation. Our calculation confirms that the behavior found in this _way is

generic.

The finite size

scaling analysis

for the

staggered magnetization performed

in reference [9]

shows _a behavior

quite

different from that obtained in the HFB

analysis.

The value _< S~ _>

goes to _zero

already

at o m 0A well before the semiclassical value.

We have thus studied the

shape

of the _one-magnon excitation in the IIFB scheme. This self-consistent method allows to find non-trivial modifications of the

spin-wave

spectrum _wk.

We have concentrated _on the Ndel

phase

of the Ji J2 model where the

discrepancy

between small-cluster results and the semiclassical results is

stronger.

In

figure I,

_we have

plotted

the

spin-wave velocity

C _{as a} function of _{a up} to the first-order transition

point

_{ac m} 0.6. There is _an almost linear decrease from the unfrustrated value to a nonzero value of C _{at ac.} This is

very different from the finite-size

scaling

results which show _an almost constant value of C _up to a transition

point

located around _a m 0A.

18

16

/

~

~~

~ 14

/

~

~

~

~ ]~~

~

Q

j

© 10~

j

z, u

~

T

~~~

o~i

,

j

-0 O 0 2 0 4 0 6

J z Ii

Fig. I. The spin _wave velocity _{as a} ^function of J2 from the self-consistent calculation. The points

are the results of the finite size scaling of reference [9]

To

gain

further

understanding

in the behavior of the model under

increasing

frustration _we have

computed

the

dispersion

relation _wk for various values of the

coupling

a

(see Figs. 2,3).

At

-2

2

Fig. 2. The dispersion relation of _magnons from the self-consistent calculation when there is _no frustration I-e- J2 _" 0. The energy along the vertical axh is plotted throughout the Brillouin _zone

[-x,

x]~ in the basal plane.

J2 = 0

(Fig.2)

the

only

remarkable

phenomenon

is the _presence of the usual Goldstone modes of the

antiferromagnet

^and

nothing peculiar happens

at the _zone boundaries. However with

increasing J2

^the

(0, x)

and

(x, 0)

"collinear" modes

began

to lower their energy. This trend shows up even more

clearly

_very close to the

point

ac G# 0.6

(Fig.3)

where the "collinear modes"

have smallest _energy in the Ndel

phase.

It is

interesting

to note that _up to the transition

point

no

complete softening

_appears. There is thus _no

divergence

in the

staggered magnetization

from the Ndel side of the transition. This behavior of the

dispersion

relation is in

complete

agreement with the fact that the transition of the Ji J2 ^model is first-order.

Let _us

emphasize

that the limit S

- c1o of the self-consistent IIFB calculation

(or MSW)

is identical to the naive

spin-wave theory.

If _one

ignores

_magnon interactions then HFB

simply

reduces to the standard linear

spin-wave theory.

For

example

^{the Ndel}

phase

has _a metastable domain _{up to a} critical value [12] o =

1/2

+

O(logs/S) (m

0.7 for

spin 1/2).

In the limit S - c1o this metastable domain

disappears

_as is

correctly

found within the

Schwinger

boson

approach [16, Ii].

The

phase diagram given by

Xu and

Ting

[13] ^is on the contrary

incqrrect.

The quantum stabilization of the N4el state is thus

intrinsically

_a

finite

S

phenomenon.

In

conclusion,

_we have studied the

dispersion

relation of _magnons in the Ji J2 model

by

_means of _a self-consistent calculation. The

spin

_wave

velocity

^{and the} _{energy wk} of the

one-magnon states has been obtained in the Ndel

phase.

We find the first-order transition

quoted

in

previous

works [6,

12, 13, 16, Ii].

We

display explicitly

the effect of

increasing

frustration _on the spectrum. ^lvhen

compared

with finite-size results [9] a clear deviation is found in the behavior of the

spin-wave velocity (Fig.I).

For

J2/Ji

_> 0A the

comparison

of the self-consistent results with Lancz6s data is

strongly suggestive

^of an intermediate

phase,

whose nature remains _up to now unsettled.

376 JOURNAL DE PHYSIQUE I N°4

-2

o

Fig.

3. The dispersion relation _at the boundary of the N6el state

J2/Ji

₌ 0.6 _near the first-order transition. There is partial softening at the _zone boundary of the (o, K), (K,

0)

modes.

Acknowledgements.

We thank A. V.

Chubukov,

T. Garel for various discussions and II. J. Schulz for

informing

us

about reference [9] ^before

publication.

References

[I] For general reviews _see: Barnes T., Int. J. Mod. Phys. C2

(1991)

659;

Manousakis E., Rev. Mod. Phys. 63

(1991)

1.

[2] ^Anderson P.W., Science 235

(1987)

1196;

Fazekas P. and Anderson P.W., Philos. Mag. 30

(1974)

423.

[3] Reger D. and Young A-P-, Phys. Rev. 837

(1988)

5978;

Liang S., Phys. Rev. 842

(1990)

6555;

Dagotto

E. and Moreo A., Phys. ^Rev. Lett. 63

(1989)

2148;

Ruse D. and Elser V., Phys. Rev. Lett. 60

(1988)

2531.

[4] Iofle ^L-B- and Larkin A-I-, Int. J. Mod. Phys. 82

(1988)

203;

Chandra P. and Doucot B., Phys. Rev. 838

(1988)

9335;

Einarsson T. and Johannesson H., Phys. Rev. 843

(1991)

5867.

[5] ^{Read N. and Sachdev} S., Nucl. Phys. 8316

(1989)

609, Phys. Rev. Lett. 59

(1989)

799.

[6] ^{Read N. and Sachdev} S., Phys. ^{Rev. Left.} 66

(1991)

17?3.

[?] Dagotto E. and Morec A., Phys. ^{Rev. Lett.} 63

(1989)

2148;

Singh R.R.P. and Narayan R., Phys. Rev. Lett. 65

(1990)

1072.

For _a review _see Dagotto E., Int. J. Mod. Phys. 85

(1991)

907.

[8] ^Gelfand M-P-, Singh R-R-P- and Ruse D.A., Phys. Rev. B40

(1989)

10801;

Sachdev S. and Bhatt R.N., Phys. Rev. 841

(1990)

4502.

[9] Schulz H.J. and Ziman T-A-L-, Finite-size scaling study ^of a quantum frustrated

antiferromagnet,

Orsay preprint, to appear in Europhys. Lett.;

See also Poilblanc D. et al., Phys. Rev. 843

(1991)

10970.

[10] Kalmeyer V. and Laughlin R.B., Phys. Rev. Lent. 59

(1987)

2095;

Wen X.-G., Wilczek F. and Zee A., Phys. Rev. 839

(1989)

11413.

[11] ^Chandra P., Coleman P. and Larkin A-I-, Phys. Rev. Lett. 64

(1990)

88;

Chandra P. and Coleman P., Int. J. Mod. Phys. 83

(1989)

1729.

[12] ^Chubukov A-V-, Phys. Rev. 844

(1991)

392.

[13] ^{Barabanov A.F, and} Starykh O-A-, JETP Lett. 51

(1990)

311;

Xu J.H. and Ting C.S., Phys. Rev. 842

(1990)

6861;

Nishimori H. and Saika Y., J. Phys. Soc. Jpn 59

(1990)

4454.

[14] ^Shender E., Sov. Phys. ^JETP 56

(1982)

178;

Chandra P., Coleman P. and Larkin A.I., Phys. Rev. Lent. 64

(1990)

88; ^J. Phys. Cond. Matter 2

(1990)

7933;

Pimpinelli A., Rastelli E. and Tassi A., J. Phys. Cond. Matter1

(1989)

2131;

Moreo A., Dagotto E., Jolicoeur Th. and Riera J., Phys. Rev. 842

(1990)

6283.

[Is]

Takahashi M., Phys. Rev. B40

(1989)

2494.

[16] Mila F., Poilblanc D. and Bruder C., Phys. Rev. 843

(1991)

7891.

[17] ^{Bruder C. and Mila} F., Europhys. Lent, to appear

(1991).

[18] ^{Harris A-B- and Rastelli} E., J. Appl. Phys. 63

(1988)

3083.