Magnetic susceptibility anisotropies in a two-dimensional quantum Heisenberg antiferromagnet with Dzyaloshinskii-Moriya interactions

0.24 MB·English

by  M. B. Silva Neto

Magneti
sus
eptibility anisotropies in a two-dimensional quantum Heisenberg
antiferromagnet with Dzyaloshinskii-Moriya intera
tions
∗
1, 2 1 1
M. B. Silva Neto, L. Benfatto, V. Juri
i
, and C. Morais Smith
1
Institute for Theoreti
al Physi
s, University of Utre
ht,
P.O. Box 80.195, 3508 TD, Utre
ht, The Netherlands
2
SMC-INFM and Department of Physi
s, University of Rome (cid:16)La Sapienza(cid:17),
6
0 Piazzale Aldo Moro 5, 00185, Rome, Italy
0 (Dated: February 2, 2008)
2
The magneti
and thermodynami
properties of the two-dimensional (2D) quantum Heisenberg

n antiferromagnet(QHAF)thatin
orporatesbothaDzyaloshinskii-Moriya (DM)andpseudo-dipolar
a (XY)intera
tionsarestudiedwithintheframeworkofageneralizednonlinearsigmamodel(NLSM).
J We
al
ulate the stati
uniform sus
eptibility and sublatti
e magnetization as a fun
tion of tem-

4 perature and we show that: i) the magneti
-response is anisotropi
and di(cid:27)ers qualitatively from
2 theexpe
tedbehaviorofa
onventionaleaBsy-⊥axisQH2AF;ii)theNéelse
ond-orderphasetransition
be
omes a
rossover, for a magneti
(cid:28)eld CuO layers. We provide a simple and
lear expla-

] nation for all the re
ently reported unusual magneti
anisotropies in the low-(cid:28)eld sus
eptibility of
l 2 4
e La CuO , L. N. Lavrov et al., Phys. Rev. Lett. 87, 017007 (2001), and we demonstrate expli
itly
2 4
- whyLa CuO
annot be
lassi(cid:28)ed as an ordinary easy-axis antiferromagnet.
r
t
s PACSnumbers: 74.25.Ha,75.10.Jm,75.30.Cr
.
t
a
m
I. INTRODUCTION the DM term generates an e(cid:27)e
tive staggered magneti
- (cid:28)eld proportional to the applied uniform (cid:28)eld. In the
d
two-dimensional
ase the
ontinuum (cid:28)eld theory allows
n Theusualstartingpointforthedes
riptionofthemag-
6,7
o netism in undoped
uprates is the 2D isotropi
Heisen- one to reprodu
e straightforwardly the same e(cid:27)e
t, .
7
c berg model In Ref. the attention wasfo
used on the(
lassi
al)spin
[
on(cid:28)gurationsexpe
tedinanexternal(cid:28)eld. Inthispaper

3 H =J Si·Sj, (1) weinvestigateinsteadthethermodynami
alpropertiesof
v <Xij> the system, whi
h enter in a
ru
ial wayχin the evalua-
8 tion of the stati
uniform sus
eptibility, , as a fun
tion
T
8 J of temperature, . In parti
ular, we show that the
ou-
5 withonlytheAF++super-ex
haSnig1e betweenthenearest- plingindu
edbytheDMtermbetweenthemagneti
(cid:28)eld
neighboring Cu spins, . Remarkably, a long-
2
andtheantiferromagneti
orderparameterisparti
ularly
0 wavelength NLSM des
ription of su
h model has been
relevant when the order-parameter (cid:29)u
tuations be
ome
5 shown to
orre
tly
apture many of the features ob-

riti
al,leadingtoananomalousmagneti
responseofthe
0 served experimentally in the high-temperature param-
4
/ agneti
phase.2,3 At lower temperatures, however, small system, as observed for example in . As an out
ome of
t
a anisotropiesliketheDMandXYintera
tionsbe
omeim- ourstudiesweobservethat,whiletheisotropi
nonlinear
m sigmamodel(NLSM)iswell establishedto giveaproper
portant, and the
ommonly used isotropi
des
ription is
- nolongeradequatetodes
ribethelong-wavelengthmag- des
ription of the maTgnetisTm in the high-temperature
d 2 4 paramagneti
phase, ≫ N,2,3 for the ordered (bro-
neti
properties of La CuO . In a re
ent paper, Lavrov T <T
n N
ken) phase, , we(cid:28)nd moreappropriatetheadop-
o et al. reported on some unusual anisotropies in the
tion of a soft version of su
h (cid:28)xed-length
onstraint, in
c low-(cid:28)eld magneti
sus
eptibility observed in detwinned
: La2CuO4 single
rystals.4 Among the unexpe
ted fea- order to
orre
tly
apture the physi
s of the longitudi-
v
nal (cid:29)u
tuations. For the sake of
larity, we refer expli
-
i tures one
ould mention: i) aχn almost featureless trans- 2 4
X versein-planesus
eptibility, a, foranextendedtemper- itly to the La CuO system throughout the paper, but
theresultspresentedhereapplytoany2Dsquare-latti
e
r ature range; ii) the in
rease of the longitudinal sus
epti-
a χb QHAFwheretheDMand/orXYanisotropyintera
tions
bility, ,asoneapproa
hestheNéeltransitiontempera-
T T =0
N arepresent,asforexampleother
upratesand ni
kelates
ture, ,fromtheorderedside;iii)theanomalous
χ > χ > χ a,b,c 2 4 8
c b a La NiO .
hierar
hy , where represent the low-
temperatureorthorhombi
(LTO)
rystallographi
dire
-
tions of Fig. 1.
Inthisarti
leweinvestigatetheoreti
allythemagneti
response and thermodynami
properties of the square-
latti
e QHAF that in
orporates both the DM and XY
anisotropyterms by investigatingthe generalized NLSM

orresponding to this mi
ros
opi
problem. As it has
5
beenstressedinthe
ontextofone-dimensionalsystems,
2
II. THE LONG-WAVELENGTH LIMIT and
Γ Γ 0 Γ Γ 0
1 2 1 2
S =We1/
2oHnsaimdeirltotnhieanfoflolorwtihnegLsaq2uCaurOe-4lastytis
teemsingle-layer ←→Γ ij=Γ02 Γ01 Γ0 , ←→Γ ik= −0Γ2 −Γ01 Γ0 ,
3 3
   
H =J S S + D (S S )+ S ←→Γ S ,
i j ij i j i ij j
· · × · ·
hXi,ji hXi,ji hXi,ji ij ik ++ d
wherΓe an>d0 label the Cu10−2sites (1s0e−e4Fig. 1), and
(2) and 1,2,3 are of order and , respe
tively,
Dij ←→Γij J 8
where and are, respe
tively, the DM and in units of .
XY anisotropi
intera
tion terms that arise due to To
onstru
t the long wavelength e(cid:27)e
tive theory for
the spin-orbit
oupling and dire
t-ex
hange in the low- theaboveHamiltonianwefollowthestandardpro
edure:
2 4 8
temperature orthorhombi
(LTO) phase of La CuO . we write
Thedire
tionandthealternatingpatternoftheDMve
-
S (τ)
tors, shown in Fig. 1, are a dire
t
onsequen
e of the i =eiQ·xin(x ,τ)+L(x ,τ),
i i
S (4)
tiltingstru
tureoftheoxygeno
tahedraandofthesym-
2 4
metry of the La CuO
rystal. Throughout this work
(bac) n L
we adopt the LTO
oordinate system of Fig. 1, for where and are, respe
tively, the staggered and uni-
Q = (π,π)
both the spin~a=ndkBla=tti1
e degreesof freedom, and we use form
omponeLnts of the spin and . We then
units where . Thus we have that integrate out and we obtain a modi(cid:28)ed NLSM where
1 1 additionalterms appear due to the DM and XY intera
-
D = ( d,d,0), D = (d,d,0), β =1/T
ij ik
√2 − √2 (3) tions ( ) and
1 β
= dτ d2x (∂ n)2+c2( n)2+(D n)2+Γ n2 ,
S 2g c Z Z τ 0 ∇ +· c c (5)
0 0 0 (cid:8) (cid:9)
n2 =1 (cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)
andthe(cid:28)xedlegngth
onstraint isimpli
it(seealso (cid:0)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:1)
6,7 0
Ref. ). Here isthebcare
oupling
onstant,relatedρto (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
0 s (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
tthhreosupgihn-ρwsa=vecv0e(l1o/
Nityg,0−Λ, a/n4dπ)r,e2n,3owrmhearleizΛedisstai(cid:27)
nuetsos(cid:27), for, (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1) D+
a = 1 (cid:0)(cid:1) (cid:0)(cid:1)
momentum integrals (we set the latti
e spa
ing )
N =3 a
and is the number of spin
omponents. Finally, k
D =2S(D +D ), c
+ ij ik
(6)
b
i j
see Fig. 1, and
Γ =32JS2(Γ Γ )>0, −−
c 1− 3 (7) FIG. 1: Left: the hat
hed
ir
les represent the O ions
2
c tiltedabovetheCuO plane+;+theembpatcyonesaretiltedbelow
andwenegle
tedasmallanisotropi

orre
tionto
om- it;smallbla
k
ir
lesareCu ions; orthorhombi

oordi-
Γ J
ing from the XY term, sin
e 1,2 ≪ . natesystem. Right: S
hemati
arrangementof thestaggered
ObservethatintheNLSM(5)onlythesumoftheDM magnetization (small bla
k arrows) andDDM ve
tors (open
(ij) (ik) +
arrows). Center:
ontinuumde(cid:28)nitionof .
and XY terms on the and bonds appears. This
has several
onsequen
es: (i) the XYΓternm2 gαen=eraat,ebs,ca

ontribution to (5) proportional to − αα α, , bc
Γ =(1/2)(Γ +Γ ) = (Γ ,Γ ,Γ )
αβ ij ik αβ 1 1 3 unit ve
tors), rendering a se
ond easy plane. When
wn2he=re1 diag . Sin
e
the system orders we (cid:28)nd the staggered magnetization
the
las,ssi
ua
lhe
noenrgtryibpulutisonthreeΓduc
ne2cstteorma
inonEstqa.n(t5s)h.iTfthiins athni=σ0xˆb, asxˆobserxvˆedexperimentally,10 beD
a2usetΓhe
ab orientationalong aor cwould
ostanenergy +or c,
allows us to identify the as the (cid:28)rst easy plane. The
Γ Γ
1 3 respe
tively. Moreover, sin
e the uniform magnetization
property that only the
ombination − a(cid:27)e
ts the
6,7
is given by
physi
s of the model (2) was an out
ome of the numer-
1
i
al results of the RPA improved spin-wave analysis of L = ( n D ),
Ref.9, but no explanation for this e(cid:27)e
t was provided. h i 2J h i× + (8)
(ii)TheDMdistortionentersthea
tion(5)onlythrough
D = D xˆ xˆ ,xˆ xˆ L xˆ
+ + a a b c c
the ve
tor ( , and are the LTO we (cid:28)nd that, in the AF phase, h i, is dire
ted along ,
3
++ ab D
+
so that the Cu spins are
anted out of the plane depending on the dire
tions of and of the applied
bc
and
on(cid:28)ned to the plane. (cid:28)eld,therewillbe(ornot)additionaltermsproportional
G
α
to in Eq. (11), responsible for the unusual magneti
response. Se
ond, this
oupling is generi
to any system
III. MAGNETIC SUSCEPTIBILITY where the la
k of inversion-
enter symmetry allows for
anos
illatingDMintera
tionbetweenneighboringspins.
χ
To evaluate the stati
uniform sus
eptibility, α, we Third,Eqs.(11)areexa
t, within linearresponsetheory,
rederivefrom themi
ros
opi
Hamiltonian(2) theqBuan- even though the σev0a(Tlu)ation Mofαt(hTe)zero-(cid:28)eld thermody-
tum a
tigonµ(5)/~in=th1e presen
ge of a2magneti
(cid:28)eld (in nami
quantities and will require, in gen-
units of S B , where S ≈ is the gyromagneti
eral,a
ertaindegreeofapproximation. Nevertheless,we
ratio and µB is the Bohr magneton)6,7 show now that all the qualitative features observed in4
(B) = (∂ n ∂ n+iB n) are already present at the mean-(cid:28)eld level.
τ τ
S S → × The thermodynami
properties of the model (5)
an
1 β N 3
+ g0c0 Z0 dτZ d2xB·(D+×n). (9) Nbe=in∞ve,staignadtefodrbTy>mTeaNn,swoef(cid:28)andlaσrg0e=- 0 eaxnpdansion. At
M2 =c2ξ−2, M2 =D2+c2ξ−2, M2 =Γ +c2ξ−2,
The last term in the above equation, whi
h
ouples the b 0 a + 0 c c 0 (14)
D n
+
DM ve
tor and the staggered magnetization , is ξ
responsible for many of the unusual features observed where the
orrelation length, 1, i=s
Nal
Iul(aξt)ed througIh a=n
experimentally in La2CuO4. However, this
oupling is a(1v/e2ra)(gIed+
oIns)traint equation ⊥ . Here ⊥
a c
generi
to any square-latti
e system, as other
uprates with
andni
kelates,wheresymmetryguaranteesthattheDM g c g T sinh(c Λ/2T)
8 I = 0 0 G (k,ω )= 0 log 0 .
ve
tors alternate in sign between neighboring bonds. α βV α n 2πc (cid:26)sinh(M /2T)(cid:27)
The zero-(cid:28)eld magneti
sus
eptibility is
al
ulated in kX,ωn 0 α
linear response theory as
(15)
a c
1 ∂2logZ Sin
e both transverse spin-wave modes, and , are
χ = ,
α βV ∂B2 (cid:12) (10) gapped, the 2D system orders at a (cid:28)nite Néel temper-
α (cid:12)B=0 TN 1=NI⊥(ξ = )12
(cid:12) ature, , de(cid:28)ned by ∞ .
Z(B) = nexp (B) (cid:12) For the ordered phase, T < TN, instead of repla
ing
where D {−S } is the Eu
lidean par-
tition fun
tion fRor the a
tion (9), and we obtain the lo
al
onstraint by(ua0/n2)a(vne2rage1)o2ne, we introdu
e a
potential of the type − in the a
tion (5),
σ2
χ = χu+ 0 , makingthe
onstraintlo
allysoft. Inthis
aseweobtain
a a g0c0 a generalized linear sigma model (LSM), whi
h leads to
D2 an ordered phase des
ribed by:
χ = χu+ + G (0,0),
b b g c c σ2 =1 NI , M2 =2σ2u , M2 =D2, M2 =Γ .
0 0 0 − ⊥ b 0 0 a + c c
σ2 D2
χ = χu+ 0 + + G (0,0), (16)
c c g0c0 g0c0 b (11) We should emphasizue t=ha(tγtJh)e2only newγparameter here
0
is the energy s
ale , where is a (cid:28)tting pa-
where rameterthatwillbe(cid:28)xed laterthrough
omparisonwith
G−α1(k,ωn)=c20k2+ωn2 +Mα2 (12) dexi
paetreismtehnatst.neHitohweervTerN, annorintshpee
vtailounesofofEσq0. a(n1d6)M, ian,c-
u
0
a,b,c depend on . The only di(cid:27)erTen<
eTis the physi
s of the
is the inverMse propagator for the magneti
modes , longitudinal (cid:29)u
tuations, for N. While the hard
α
with gaps and
onstraint yields a longitudinal mode always gapless,13
χu = 1 G (q)+G (q) 4ω2 G (q)G (q) , the sToft one gives a mass for the longitudinal mode be-
α βV β γ − n β γ low N that is proportional to the strength of the order
q=X(k,ωn)(cid:8) (cid:9)
parameter.
N
2 4
(13) We (cid:28)x the large- parameters for La CuO by
om-
ξ
is the traditional (cid:16)uniform(cid:17)
ontribution to the sus
epti- paring the
orrelation length, , with the experimental
(αβγ)=(abc) M (T =
a
bility, with and its permutations. data in the paramagneti
phase, and we use
0) D = 2.5 M (T = 0) √Γ = 5.0
+ c c
The set of Eqs. (11), following from the quantum a
- ≡ meV and ≡ meV,
tions (5) and (9), is the main results of this arti
le. A for the zone
enter in-plane and out-of-plane spin-wave
14
few remarks are in order now
on
erning the usefulness gaps as determined from neuξtr−o1n s
attering. ρIn=th0e.1iJn-
s
of the NLSM approa
h to the model (2) with respe
t to set (a) of Fig. 2 we
ompare obtained for ,
c = 1.3J J = 100 ξ
8,9,11 0 exp
otherapproa
hes. First, thea
tion(9)allowsoneto and meV with the
urve , whi
h
T >T
N
easily tra
k down the sour
e of theBani(sDotropin
)magneti
representsthebest(cid:28)toftheexperimentaldaξta−1at
response,as
omingfrom theterm · +× . Infa
t, obtained in the isotropi
NLSM15 (so that exp deviates
4
0.7 χu
0.06 tσw2/eegncthe de
rease of a and the in
rease of the term
0.6 cc a −1Å) 0.04 x−1xe−x1p 0We0t0u.rn now to χc and χb. As T →TN,
c bc −1x ( 0.02 D2 D2 1
(a) χ + G (0,0)= +
0.5 0 c ≈ g0c0 b g0c0Mb2 (17)
c(1/J) 0.4 M/Ja 0 0.1.15 MMMbac divergesbe
auseMof(tThe vaTn+is)hing0of the mass of the lon-
0.05 (b) gitudinalmode, b → N → ,andthisisasso
iated
11
0.3 0 0 0.1 0.2 0.3 0.4 0.5 0.6 withtheferromagneti
orderiMngbofthe
TaNntedmoments.
T/J Itisessentialtohavea(cid:28)nite below inorderto
or-
χ
c
re
tlyreprodu
ethebehaviorof intheorderedphase.
0.2
So, although theMhard-sTpin mTo+del
orre
tly reprodu
es
the vanishing of b as → N, it is not adequate for
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 M = 0
T/J the ordered phase, where b leading to a diverging
χ T <T
c N
forall . Thatiswhy wehaveadoptedthesoft
T
N
FIG. 2: (Color online) Temperature dependen
e of the mag- version of the
onstraint below .
tneemti
pesruast
uerpetdibeiplietniedsenχ
ae,oχfbMaan,db,cχacndfro(bm) ξ(1−11).. Insets: (a) TNFbinuatlleyx,hwibeiotsbsaerwveelDlinp2rFGoing(o.0u2,n0t
)he/adgtpχcebaki=s.nTDoht2ids/ii(vsgearcg
eoMnnts2ae)t-
quen
e of the term + c 0 0 + 0 0 c .
T
N
from the experimental data near TN). When 1/N
or- Apa
r
aomrdaignngettio
Esiqd.e ξ(−1D24)2,g/oa(egss wctoeΓzae)prporo1aa/n
(d4hgDc+2/)(fgr0ocm0Mthc2e)
re
tionstoour
al
ulationsarein
ludedoneexpe
tsthat approa
hesthe value + 0 0 c ≈ 0 0 , whi
h is
T /J
N 13
the ratio is redu
ed, whi
h allows for larger and
ontrolledby the ratiobetweenthe DM and XY intera
-
ρ c J
s 0 2,3
more realisti
values for , , and . Nevertheless, tionterms. Observethattheabsen
eofasimilarfeature
with the above parameters we (cid:28)nd σ0(T = 0+)+≈ 0.46 in χχa isdue tothefa
tthat,forBka,B·χ(Du +×n)=0,
ewµxhp=i
ehrgimSleSeanσdt0ssµ.t1Bo5 Ta≈nhe0e.r(cid:27)4ee6s
uµtlBitvi,engimnMogmao,oeb,ndctaaosgfaretfehumene
Cntiutoγnw=oitfshp0te.it1nmh5se-, iisntos
rueanaTishfeoarisnmoχnpblayarstt,hTχe→ubu,nTwiDfN−ohr2i
mi/sh(
g
ilsoecanmrtMlroyinb2aou)ttroieonsnui
latlaoly.ftiThneh
ruseusa,mstinhogef
peratureareshownintheinset(b)ofFig.2for , with T, and the term + 0 0 c ,ξw2hi
h isT
onstant
N N
a
hoi
e that will be justi(cid:28)ed below. below and de
reases instead as ∼ above .
A. Temperature dependen
e of χa,χb and χc B. Unusual χa(0)<χb(0)<χc(0) hierar
hy
Let us now dis
uss how all the unusual features of AT
o=m0mentisinordernow
on
erningthehierar
hyof
the sus
eptibilities reported in4 follow straightforwardly the sus
eptibilitiesinFig.2. Foranordinaryeasy-
lferovmel (Eseqesχ.Fig(1.12)).alrAeatdylaragte ttheme pmereaatnu-Tr(cid:28)eesl,dT(la≫rgeTNN), vaexrisseAgFapws,ithMdai(cid:27)<ereMntc,init-pilsaneexpaen
dteoduχttuh-o(a0ft-)ptl<haneχeTut(r0=a)n<s0-
all three a,b,c exhibit the usual linear- behavior ob- uχnui(f0o)rm sus
eptibilities should satisfy b a
served experimentally16 and expe
ted for the uniform c , with the lowest value for the easy axis and with
T 0
sus
eptibility ofχtuhe is0otropi
NLSMχ.u3 As χu→ , on a transversesus
eptibility smallerin the dire
tion of the
the other hand, b → while both c and a saturate smaller gap. This is in fa
t the result thDa2t one obtains
at very small, but nonzero, values. The smooth evolu- afterdroppingthetermsproportionalto B+ in Enq. (11).
tχiaonisthdruoeugtohtthheeftar
atntshitaiotnχeuax
poenritmaiennstatellrymosblsiekreveIαd iinn gχHiov(we0se)vriesre, ttohethuenutesurmals
poruopploinrgtiobneatwl teoenD+2 Tainn=dχb0(0i)na(n9d)
c
Eq. (15), whi
h areregularfor the transversemodes (al- , whi
h, in turn, lead to the following values
ways gapped by the DM and XY intera
tions) and di- for the three sus
eptibilities
T
N
avesrgMeb(loTga→rithTmNi)
a→lly 0a.t Thisfordivtheerχgelonn
geitaupdpineaarlsmaosdae χa ≈ gσc02 , χb = g1c MD+22, χc ≈χa+ g1c MD+22,
humpinthenumeri
alevaluationof a presentedinFig. 0 0 0 0 c 0 0 b
2 and is an artifa
t of both the low dimensionality and (18)
N
2
of the large- limit (a
oupling between CuO layers or resulting in the unusual hierar
hy of the zero-
η
2 4
a nonzero anomalous dimension for the propagator of temperature values and rendering La CuO an example
thelongitudinalmoderegulatessu
hlogarithmi
infrared ofanun
onventionaleasy-axisantiferromagnet. Observe
I T χ χ (0)/χ (0) > 1 T = 0
b N a c a
divergen
e of at and wipes out the feature in ). that always, and we use the in-
χ T 0 χ γ = 0.15
a c
Finally, the weak in
rease observed in as → , also ter
ept of to (cid:28)x . Finally, for our
hoi
e
χ (0)/χ (0) 1.1
b a
observed experimentally, is due to the
ompensation be- of parameters the ratio ∼ , in relatively
5
4
goodagreementwiththeexperiments, butitdependsin proper des
ription of the longitudinal-(cid:28)eld (cid:29)u
tuations
17
general on the values of the transversemasses. below the
rossovertemperature
an be done within the
LSM,wheretheorder-parameterequationandthelongi-
0.6 0.4
tudinal mass read:
6
00..45 -710 emu/g) 345 cEccexcx pp 00..335 u0σ0(σ02−1+NI⊥(M⊥2+h/σ0,T))=h, Mb2 =2u0σ02+(2σh10).
c ( 2 0.25 Observe that the
rossover temperature obtained from
0.3 1 0.2 Eq. (19) and Eq. (21) is almost the same.(MToexpin)2te=r-
0.2 MMbebx/pJs/ J0 0 100 20 0T(K )3 00 400 500 00..115 pm
uoeoarslraxbet(leyha/tb(cid:28)iσote0tnt,iwn2leeguen0nMgσt02tbhehx+aepbhttoow/vσoet0hT)seoNaltnu.hdteAioewsnxefspa(cid:28)erwrxaeimstdhteehen(cid:28)etpnacael-radaxamitsaebtsoeurns
vteahple--
0.1 H=1 T tibility is
on
erned it is now evident from the inset of
0.05 χ M
c b
Fig.3that nolongerdiverges( isalways(cid:28)nite)but
M
b
0 0 be
omes peaked at a
rossovertemperatureat whi
h
0 0.1 0.2 0.3 0.4 χa
T/J hχasaminimum(seeFig.3). Observealsothatfor and
c
the e(cid:27)e
ts of a (cid:28)nite magneti
(cid:28)eld are instead negli-
FIG. 3: (ColorMonexlipne) Plot of σ0, Mb, and Mbexp from Eqs. gible, leading only to small quantitative di(cid:27)ereBn
e=s0with
(21)and(20). b is(cid:28)ttedtotheexperimentaldataforthe respe
t to the
al
ulation presented before for .
TN
inverse
orrelationlengthabove ,seedis
ussioninthetext
χc
belowEq. (21).χeIxnpset: inunitsofemu/gandexperimMenextapl
points from4. c is obtained from Eqs. (11) using b .
ρs =0.07J γ =0.5 a=5.3 V. CONCLUSIONS
Here , , Å, the out-of-plane latti
e
d = 13.5 T
parχamete≈r 1.0×10−Å7, and we added a -independent shift
of emu/g to a

ount for the van Vle
k In
on
lusion, we have derived the uniform magneti
vV
4
paramagneti
sus
eptibility. sus
eptibilitywithinthelong-wavelengthe(cid:27)e
tivetheory
(5)-(9)forthesingle-layersquare-latti
eQHAFwithDM
andXY intera
tions(2). Weobtainedthatthemagneti
responseisanisotropi
,inremarkableagreementwiththe
4
IV. EFFECS OBF⊥A FINITE MAGNETIC FIELD experiments of , and di(cid:27)ers from the expe
ted behavior
CuO2 LAYERS foramore
onventionaleasy-axisQHAF.Duetothepres-
en
e of the DM term, the uniform magneti
(cid:28)eld gener-
χc ates an e(cid:27)e
tive staggered (cid:28)eld proportional to both the
The result for plotted in Fig. 2
an be further im-
DMintera
tionandtotheapplied(cid:28)eld. Weshowedthat
proved by
onsidering expli
itly the nontrivial e(cid:27)e
ts of
B ab D+ =0 the
oupling between the magneti
(cid:28)eld and the stag-
the
ombination ⊥ plane and 6 on the ther-
gered order parameter is parti
ularly relevant when the
modynami
properties of the theory (9). In fa
t, in this
(1/g0c0) hnb order-parameter (cid:29)u
tuations be
ome
riti
al, leading to

ase the last term in Eq. (9) be
omes − ,
h= B D+ R an anomalous magneti
response of the system, as ob-
where | × |. Within the NLSM formulation the 4
served for example in . Moreover, this same
oupling
averaged
onstraint equation de(cid:28)nes a self-
onsisten
y
is responsible for the unusual zero temperature hiera
hy
equation for the order parameter given by: 2 4
of the sus
eptibilities, rendering La CuO an example
σ2 =1 NI (M2 +h/σ )
0 − ⊥ ⊥ 0 (19) of an un
onventional easy-axis antiferromagnet. Finally,
we
onsidered the e(cid:27)e
t of a (cid:28)nite magneti
(cid:28)eld on the
χ B ab
c
where the masses appearing in the transverse spin (cid:29)u
- sus
eptibility. We found that, for a nonzero ⊥
σ
tuations (15) are themselves a fun
tion of 0: plane, the Néel se
ond order phase transition be
omes a

rossover, see Fig. 3, in analogy with the beahvior of a
h h
M2 =D2 + , M2 =Γ + +B2. ferromagnetin a(cid:28)nite magneti
(cid:28)eld. The (cid:28)nite-(cid:28)eld ef-
a + σ c c σ (20)
0 0 fe
tsareinsteadirrelevantasfarasthesus
eptibilitiesin
a b
the and dire
tionare
onsidered. However,the
aseof
B b
As one
an see, besieds a (quantitatively small) hard- k presentsinterestingout
omesasfarasthesele
tion
c
ening of the mode due to the applied (cid:28)eld, we (cid:28)nd rules for one-magnon Raman s
attering are
on
erned,
h/σ
0 18
that the termc2ξ−2 plays a a role similar to the
orrσe- as it has been dis
ussed re
ently in Ref. . While Eq.
0 2 4
lation length, , in Eq. (14). The result is that (9) and (11) have been derived for the
ase of La CuO
h=0
never vanishes for 6 (see Fig. 3), and be
ause of the materials, the present analysis
an be extended to other
B ab
DMintera
tion,anonvanishing ⊥ planetransforms QHAF systems where the la
k of inversion-
enter sym-
the Néel se
ond order phase transition into a
rossover. metry allows for an os
illating DM intera
tion between
B = 0
In analogy with the dis
ussion of the
ase , the neighboring spins.
6
VI. ACKNOWLEDGEMENTS troNeto, and R. Gooding. We alsothankA. Lavrov,for
fruitfull dis
ussionsand for providing us with the exper-
χ
c 4
The authors have bene(cid:28)tted from dis
ussions with imental data for from , shown in Fig. 3 (inset).
A. Aharony,Y. Ando, B.Binz, C. Castellani, A. H.Cas-
∗
Ele
troni
address: barbosaphys.uu.nl M. A. Kastner, H. P. Jenssen, D. R. Gabbe, C. Y. Chen
1
M.A.Kastner,R.J.Birgeneau,G.Shirane,andY.Endoh, and R. J. Birgeneau, and A. Aharony, Phys. Rev. B 38,
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S. Chakravarty, B. I. Halperin, and D. R. Nelson, Phys. 73, 894 (1994).
12 J⊥
Rev. B 39, 2344 (1989). The interlayer
oupling, , plays also an important role
3 2 4
A. V. Chubukov,S. Sa
hdev,andJ. Ye,Phys.Rev.B 49, for the Néel transition in La CuO . We will nevertheless
11919 (1994). pro
eed with our 2D model sin
e it already
aptures the
4 4
A. N. Lavrov,Y. Ando,S. Komiya,and I. Tsukada, Phys. essential physi
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13
Rev. Lett. 87, 017007 (2001). V. Yu. Irkhin, and A. A. Katanin, Phys. Rev. B 57, 379
5
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e on Strongly Correlated Ele
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8 16
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9 1/g0c0 χb(0)>χa(0)
K.V. Tabunsh
hyk,and R. J.Gooding, Phys.Rev.B 71, andasa
onsequen
e isonlyobtained
Ma >Mc 14
214418 (2005). if , in
ontrast to the experimentalsituation .
10 18
D. Vaknin, S. K. Sinha, D. E. Mon
ton, D. C. Johnston, M. B. Silva Neto and L. Benfatto,
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T. Thio, T. R. Thurston, N. W. Preyer, P. J. Pi
one,