Quantum Heisenberg antiferromagnets: a survey of the activity in Firenze

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by  Umberto Balucani

Quantum Heisenberg antiferromagnets:
a survey of the activity in Firenze
Umberto Balucani,1 Luca Capriotti,2,3 Alessandro Cuccoli,4,5 Andrea Fubini,4,5
Tommaso Roscilde,6 Valerio Tognetti,4,5 Ruggero Vaia,1,5 and Paola Verrucchi5,1
1Istituto dei Sistemi Complessi, Consiglio Nazionale delle Ricerche,
via Madonna del Piano, I-50019 Sesto Fiorentino (FI), Italy
5 2Valuation Risk Group, Credit Suisse First Boston (Europe) Ltd.,
0 One Cabot Square, London E14 4QJ, United Kingdom
0 3Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA
2 4Dipartimento di Fisica dell’Universit`a di Firenze,

n via G. Sansone 1, I-50019 Sesto Fiorentino (FI), Italy
a 5Istituto Nazionale per la Fisica della Materia, Unit`a di Ricerca di Firenze,
J via G. Sansone 1, I-50019 Sesto Fiorentino (FI), Italy
0 6Department of Physics and Astronomy, University of Southern California, Los Angeles, CA 90089-0484, USA
2 (Dated: February 2, 2008)

Over the years the research group in Firenze has produced a number of theoretical results con-
]
l cerning the statistical mechanics of quantum antiferromagnetic models, which range from the the-
e
ory of two-magnon Raman scattering to the characterization of the phase transitions in quantum
-
r low-dimensional antiferromagnetic models. Our research activity was steadily aimed to the under-
t
s standingof experimental observations.
.
t
a
m I. INTRODUCTION fact that it models the magnetic behavior of the parent
- compounds ofsome high-Tc superconductors2,3. The ex-
d perimental activity on 2D antiferromagnets stems from
n TheHeisenbergmodelmaywellbeconsideredthecor-
the existence of several real compounds whose crystal
o nerstone of the modern theory of magnetic systems; the
structure is such that the magnetic ions form parallel
c reason for such an important role is the simple struc-
[ planesandinteractstronglyonlyifbelongingtothesame
ture of the Hamiltonian, whose symmetries underlie its plane. Asaconsequenceofsuchstructure,theirmagnetic
1 peculiar features. The basic forces which determine the
behavior is indeed 2D down to those low temperatures
v alignment of the spins, are represented by the exchange
where the weak interplane interaction becomes relevant,
6 integrals J’s. At variance with the ferromagnet where
driving the system towards a 3D ordered phase.
9
the parallel alignment is promoted, in the antiferromag-
4 In addition, the 2D Heisenberg model can be enriched
netalotofpeculiararrangementsofthespinscanoccur,
1 trough symmetry-breaking terms – we considered easy-
0 with strong differences between classical and quantum axis (EA) and easy-plane (EP) anisotropy, as well as
5 systems. Asmatteroffact,alsofornearest-neighboranti-
an external uniform magnetic field – which are useful
0 ferromagneticinteractionsthegroundstateoftheHamil-
to reproduce the experimental behavior of many layered
/ tonian is different from the N´eel state with antialigned
t compounds. In the EA case one is left with a discrete
a spins, and the (staggered) magnetization shows the so
m reflection symmetry and the system undergoes an Ising-
calledspin reductionwithrespecttothesaturationvalue
like phase transition. In the EP case or when a mag-
- also at T =0. The linear excitations of an antiferromag-
d neticfieldisappliedtheresidualO(2)symmetryprevents
net can be roughly associated in two families and pair
n finite-temperature ordering4, but vortex excitations are
excitationswith vanishing totalmagnetizationarepossi-
o possibleanddetermine aBerezinskii-Kosterlitz-Thouless
c ble: the fact that the total momentum of these can be (BKT) transition between a paramagnetic and a quasi-
: close to zero allows for their investigation by light scat-
v ordered phase. In spite of the tiny anisotropies of real
Xi tering. systems (usually <∼ 0.01J), it can be shown that they
Whilethesepeculiarfeaturesofantiferromagnetismal- dramatically change the behavior of the spin array al-
r
a ready occur in three-dimensional (3D) compounds, they ready at temperatures of the order of J.
aremorepronouncedinthe low-dimensionalones,where In this paper we report about the progresses in the
othereffects causedbythe enhancedroleofclassicaland theory of Heisenberg antiferromagnets that have been
quantum fluctuations are present and exotic spin config- obtained by our group in Firenze. The early work on
urations associated with field theory models can appear. the theory of two-magnon Raman scattering is summa-
Indeed, the last two decades have seen a renewed inter- rized in Section II, while the following Sections report
est both in the case of the one-dimensional (1D) quan- about the recent activity on low-dimensional antiferro-
tum Heisenberg antiferromagnet (QHAF), for which a magnetism. Section III is devoted to 1D models, and
peculiar behavior of the ground state vs spin value was concernsthestudyofthe effectofsoliton-likeexcitations
predicted1, and of the two-dimensional (2D) QHAF, be- inthecompoundTMMC,aswellastheanisotropicspin-
causeofitstheoreticallychallengingpropertiesandofthe 1 model, for which a reduced description of the ground
2
state allows one to investigate the quantum phase tran- order is still present.
sition in a unitarily transformed representation and to LetusconsiderthefollowingantiferromagneticHamil-
obtain quantitative results for the phase diagram. Sec- tonian with exchange integral J>0, z nearest neighbors
tionIVconcernsthetheoryoftheisotropic2DQHAF,for with displacements labeled by d, and two (a,b) sublat-
whichwereproducedtheexperimentalcorrelationlength tices8:
by means of a semiclassical approach, also deriving the
J
connectionwith(andthelimitationsof)famousquantum H= Si,a·Si+d,b. (1)
2
field theory results. In Section V we summarize several Xid
recentresultsconcerningtheanisotropic2DQHAF,with
The scattering cross section S(ω) turns out9 to be pro-
emphasis onto the different phase diagrams and the ex-
portionaltotheFouriertransformofhM(0)M(t)i,where
perimentallymeasurablesignaturesofXYorIsingbehav-
ior. Eventually,inSectionVIresultsonthe2Dfrustrated
J -J isotropic model are described. M = M S ·S , (2)
1 2 k k −k
Xk
is the effective Raman scattering operator.
II. TWO-MAGNON RAMAN SCATTERING IN Many antiferromagnetic compounds can be mapped
HEISENBERG ANTIFERROMAGNETS
onto this model, even though a small next-nearest-
neighbor exchange interaction without competitive ef-
The scattering of radiation is a very powerful tool fects,aswellasanisotropytermscouldbepresent. Forin-
to study elementary excitations in Condensed Matter stance,thereare3Dperovskiteandrutilestructures(e.g.,
Physics. Any complete experiment gives rise to a quasi- KNiF , NiF ) and 2D layered structures (e.g., K NiF ,
4 2 2 4
elastic component due to non-propagating or diffusive LaCuO ).
2
modes and to symmetrically shifted spectra correspond- Letusrememberthattheexactgroundstateisnotex-
ing to the states of the system under investigation with actlyknown,exceptin1DmodelswithS=1/2orS=∞
an amplitude ratio governed by the detailed balance (i.e., the classical case): in the latter case it coincides
principle. The most sensitive probes for this investi- with the ‘N´eel state’ with antialigned sublattices.
gation are undoubtedly thermal neutrons, because the In the ordered phase the theory can be developed in
characteristicenergiesandwavevectorsfit verywellwith terms of two families of magnon operators (α , β ),
k k
those of the magnetic elementary excitations. However, through the Dyson-Maleev spin-boson transformation
light-scattering experiments can require a simpler appa- and a Bogoliubov transformation:
ratus and offer a better accuracy, although the transfer
H=E +H +V , (3)
wavevector k is much smaller than the size of the Bril- 0 0
louin zone so that usually only the center of this zone where E is the ground state energy in interacting spin-
0
can be directly probed. In spite of this, two-spin Ra- wave approximationand
man scattering involving the creation and destruction of
a pair of elementary excitations can be performed, with H = ω (α†α +β†β ) (4)
0 k k k k k
thecontributionoftwomagnonshavingequalfrequencies Xk
and opposite wavevectors. This two-magnon scattering
is the quadratic part of the Hamiltonian of a magnon
isexpectedtobespreadoverabandoffrequenciesinan-
gas whose frequencies, renormalized by zero-T quantum
tiferromagnets. However, the density of states strongly
fluctuations, are
enhancesthecontributionofzone-boundary(ZB)excita-
tions5, i.e., at k∼k . C 1
ZB ω =JSz 1+ 1−γ2; γ = e−ik·d . (5)
While in ferromagnets the two-spin process is only k (cid:16) 2S(cid:17)q k k z Xd
due to a second order mechanism, orders of magnitude
smallerthanthefirstorderone,inantiferromagnetsadif- The last term in the Hamiltonian, V, represents the
ferent independent process is permitted, stronger than four-magnon interaction, whose most significant term
the corresponding one for single-spin spectra6. Specifi- refers to two magnons of each family and turns out to
cally, an exchange mechanism does not change the total be:
z-component of the spins: exciting two magnons in the Jz
twodifferentsublattices(∆M=0)7 isthedominantscat- V =2N δq+p,q′+p′Iqαqβ′,pp′α†qαq′βp†βp′, (6)
tering process. qqX′pp′
Theone-spinRamanscatteringpeakdisappearsatthe wherethe coefficients Iαβ areknownfunctions of γ .
N´eeltemperaturebecauseitprobesthesmallestwavevec- qq′,pp′ k
In the Hartree-Fock approximation10 the temperature
tors, related with the long-range correlations. In con-
dependentRamanscatteringoperator(2)canbewritten
trast, two-magnon Raman scattering essentially probes
the highest wavevectors, related to short-range correla-
tions. Thereforetwo-spinRamanscatteringfeaturesper- M =α(T) S Φ (α β +α†β†), (7)
sist also in the paramagnetic phase7 where short-range Xk k k k k k
3
FIG. 1: Theoretical two-magnon spectra in KNiF at differ-
3
ent temperatures10. FIG. 2: Experimental two-magnon spectra in KNiF3 at dif-
ferent temperatures5.
with ω (T)=α(T)ω . The two magnons created or de-
k k
stroyed by the operator (7) interact through V as given
by(6),sothatthepeakofthecrosssectionS(ω)appears
at values smaller that 2ω for an amount of the order
ZB
of J. The explicit S(ω) at T =0 was calculated in the
‘ladder approximation’by Elliott and Thorpe and found
in very good agreement with experiments9.
The finite temperature calculation of the two-magnon
Raman scattering cross section in the ordered region,
up to T ∼0.95T was performed by Balucani and
N
Tognetti10, calculating the two-magnon propagator in
the ‘ladder approximation’, taking also into account the
damping and the temperature renormalization of the
magnons at the boundary of the Brillouin zone11. The
calculatedspectraS(ω),atincreasingtemperatures,were
found in very good agreement with the experimental
ones8 andtheircharacteristicparameters(peakandline- FIG. 3: Zone-boundary damping ΓZB vs temperature. The
symbols refer to different experimental approaches: in par-
width) permitted to determine the temperature behav-
ticular the open circles are our light scattering data12. The
ior of the frequency and damping of the ZB magnons12.
dashedlineisanimprovement12 toaprevious(solid)theoret-
In Figs. 1 and 2 we show the excellent agreement of
ical curve10.
our theoretical approachwith the experiments in the or-
dered phase13. The validity of light scattering in prob-
ingthe characteristicofZBmagnonshasbeenconfirmed
character of the ‘excitations’. The calculation of S(ω)
both from the theoretical and the experimental point of
can be instead approached by other more general the-
view12. In Fig. 3 our theoretical ZB magnon damping
oretical methods devoted to the representation of the
calculations are compared with experimental data from
dynamical correlation functions based on the linear re-
different techniques. sponse theory11. Let us consider the ‘Kubo relaxation
In the paramagnetic phase all experimental spectra function’ associated with our scattering process:
show the persistence of a broad inelastic peak up to
T ∼1.4TN. Only at T ≫TN the spectra have a struc- 1 β
tureless shape centered around ω=0. As matter of fact, f0(t)≡ dλ eλHM(0)e−λHM(t) . (8)
hM(0)M(0)iZ
thehighestwavevectorssampleonlythebehaviorofclus- 0 (cid:10) (cid:11)
ters of neighboring spins, thus giving a measure of the
Its Laplace transform f (z) is related to the scattering
short-rangeantiferromagneticorderthatis presentatall 0
cross section:
finite temperatures.
Inthedisorderedphaseconventionalmany-bodymeth- ω
S(ω)∝ ℜf (z=iω). (9)
odsareoflittleuseforaquantitativeinterpretationofthe 1−e−βω 0
observed largely spread spectra. The concept of quasi-
particle loses its meaning because of the overdamped Mori14 has given the following continued fraction repre-
4
celebratedrealizationsof these models occur in 1D mag-
nets. An original suggestion by Mikeska18 was that the
antiferromagneticchainTMMC[(CH ) NMnCl ]canbe
3 4 3
mapped onto a sine-Gordon classical 1D model. The el-
ementary excitations of the sine-Gordon field are given
in terms of linear small-amplitude spin-waves and non-
linear breathers and kink-solitons. The non-linear ele-
mentaryexcitationsgiveadetectablecontributiontothe
magnetic specific heat.
TMMCiscomposedofHeisenberg(S=5/2)antiferro-
magnetic chains along the z-axis:
FIG.4: Two-spin Stokesspectrumin KNiF3 atT≃1.02TN. H=J S ·S −δSzSz , (13)
The line reports the theoretical shape15, compared with ex- i i+1 i i+1
Xi (cid:0) (cid:1)
perimental data.
with a very small easy-plane anisotropy (δ=0.0086).
Amagneticfieldoftheorderof1÷10Tcanbeapplied
sentation of the relaxation function11:
perpendicularly (y-axis) or along the chain . In the first
1 1 case, with approximations the more valid the lower the
f (z)= ; f (z)= ,
0 n magnetic field (H<5T), in the continuumlimit TMMC
z+∆ f (z) z+∆ f (z)
1 1 n+1 n+1
canberepresentedbytheclassicalsine-GordonHamilto-
(10)
nian:
whichis formally exact,but allowus to do someapprox-
imationsabouttheleveloftheterminationf (z). The
n+1 A
quantities∆ canbeexpressedintermsoffrequencymo- H= dx Φ˙2+c2Φ2 +2ω2(1−cosΦ) , (14)
n 2 Z 0 x 0
ments: (cid:2) (cid:3)
∞ whose parameters are related with the magnetic Hamil-
hω2ni=Z dωω2nf0(ω). (11) tonian (13), the reduced magnetic field h=gµBH, and
−∞ the lattice spacing a as follows:
In our calculations of f (z) in the entire paramagnetic
0
phase15,16, the coefficients ∆ e ∆ have been approx- 1 δ δ
1 2 A= , c =aJS 1− , ω =h 1− . (15)
imately evaluated by means of a decoupling procedure. 8Ja 0 r 2 0 r 2
Moreover, the third stage of the continued fraction (10)
The energy of kink-soliton turns out to be
is evaluated assuming that
∆ f (z)∼∆ [f (0)+zf′(0)]. (12) Es =8Aω0c0 ≃hS , (16)
3 3 3 3 3
and depends on the applied field. At difference with the
Theparametersinvolvedin(12)canbe estimatedbythe
ferromagnetic solitons, these solitons can be easily ex-
knowledgeoftheshorttimebehavioroff (t)determined
0
by the first moments, hω2i and hω4i. cited at lowest temperatures and can give a significant
contribution to the thermodynamics19. When the field
Theresultsofourapproachintheparamagneticregion
isappliedlongitudinallyalongthez-axisonlyspin-waves
arecomparedwiththe experimentinFig.4,showingthe
are present: therefore, the specific-heat measurements
persistence of the peak of the ZB magnetic excitations
were performedin the two configurations. The contribu-
above the critical temperature.
tion from the nonlinear excitations was obtained as the
difference ∆C between the two experiments.
The thermodynamic quantities were calculated by the
III. THE ONE-DIMENSIONAL
classical transfer-matrix method20 for the sine-Gordon
ANTIFERROMAGNET
model (14). We then used a classical discrete planar
model21:
A. Solitons in the antiferromagnet TMMC
H= 2JS2cos(Φ −Φ )+hS(1−cosΦ ) , (17)
i i+1 i
Interest in low-dimensional systems is motivated by Xi (cid:2) (cid:3)
the much greater simplicity of calculation as compared
with the 3D ones. The powerful mathematical approach verifying that it is qualitatively similar to the sine-
based on the inverse-scattering and Bethe Ansatz tech- Gordon. The comparison21 is shownin Fig.5, where the
niques permits to exactly solve some 1D models, calcu- linear spin-wave specific heat was subtracted to empha-
lating thermodynamic and sometimes transport quanti- sizethenonlinearcontribution,togetherwiththepredic-
ties both in classical and quantum cases17. The most tion of the ‘classical soliton gas phenomenology’19.
5
Lieb-Schultz-Mattis theorem23. Despite the generaliza-
tion of such theorem to integer-spin systems being im-
possible, its general validity has been taken for granted
till 1983, when Haldane1 suggested, for the integer-
spin Heisenberg chain, an unexpected T =0 behavior:
a unique and genuinely disordered ground state, mean-
ing exponentially decaying correlation functions and a
finite gap in the excitation spectrum. After more than
twodecadesHaldane’sideathatinteger-spinsystemscan
have a genuinely disordered ground state still stands as
a conjecture. However, theoretical24,25,26,27, experimen-
tal28,29,30,31,32 andnumerical33,34,35,36,37,38,39 workshave
definitely confirmed its validity.
Let us consider Eq. (18) for integer spin: in the (d,λ)
plane one may identify different quantum phases, corre-
FIG. 5: Experimental contribution of nonlinear excitations
sponding to models whose ground states share common
to the specific heat of TMMC, ∆C=C(H)−C(0)−∆C .
SW features. For λ>0 three phases are singled out: the
The field values are H=5.39T (•) and H=2.5T (◦). The
N´eel phase (λ≫d), where the ground state has a N´eel-
dash-dottedlinereportstheresultofthefreesolitongasphe-
like structure, the so-called large-d phase (d≫λ), where
nomenology. Theplanarmodel(interpolatedcrosses)appears
to quantitatively explain the behaviorof TMMC. the ground state is characterized by a large majority of
siteswhereSz=0,andtheHaldanephase,whichextends
around the isotropic point (d=0, λ=1), and is charac-
This provedthe presence of nonlinear excitations sim- terized by disordered ground states.
ilar to sine-Gordon solitons, but the peak of the spe- WefirstdealwiththeIsinglimit,H/J=λ SzSz :
i i i+1
cific heat occurs at temperatures where solitons can- Uponits groundstate,the antiferromagneticPallyordered
not be considered to be non interacting and the ‘classi- N´eel state, one may construct three types of excitations:
cal soliton-gas phenomenology’ breaks down. When the a single deviation, a direct soliton, an indirect soliton,
magnetic field is increased up to 9.98T the model (17) where direct (indirect) refers to the fact that the excita-
is no more able to describe the experiments. A quasi- tionbe generatedby flipping allthe spins onthe rightof
uniaxialmodel21 was proposedandfound in goodagree- a given site while keeping the z component of the spin
ment. For general reference on the subject see22. on such site unchanged (setting it to zero). The above
configurations have all energy +2λ with respect to that
ofthegroundstate,anddogenerate,whenproperlycom-
B. The S=1 quantum antiferromagnet bined, all the excited states; amongst them, we concen-
trateuponthose containinga coupleofadjacentindirect
solitonsandnoticethattheirenergyis+3λ,whileexcited
We here deal with quantum antiferromagnetic spin
states containing two separate indirect solitons have en-
chains, focusing our attention on the class of models de-
ergy +4λ. Therefore, indirect solitons are characterized
fined by the Hamiltonian
by a bounding energy λ; moreover, one may easily see
H that isolated solitons may effectively introduce disorder
= (SxSx +SySy )+λSzSz +d(Sz)2 (18)
J i i+1 i i+1 i i+1 i in the global configuration of the system, while coupled
Xi h i solitons do only reduce the magnetization of each of the
two antiferromagneticsublattices40. In fact, strings con-
withexchangeintegralJ>0andsingle-ionanisotropyd.
taining any odd (even) number of adjacent solitons act
One of the most surprising evidence of the differ-
on the order of the global configuration as if they were
ence between ferro- and antiferromagnetic systems is re-
isolated (coupled) solitons.
lated with the so-called Haldane conjecture, i.e. with
As we move from the Ising limit, the transverse in-
T =0propertiesofinteger-spinantiferromagneticchains.
teraction (SxSx +SySy ) comes into play, and is
In general, we expect three possible situations for the i i i+1 i i+1
ground state of a magnetic system: either it is or- seen41 to mPore efficiently lower the energy of the system
dered (with finitely constant correlation functions), or by delocalizing indirect solitons rather than single devi-
quasi-ordered(withpower-lawdecayingcorrelationfunc- ations or direct solitons, thus indicating configurations
tions), or completely disordered (with exponentially de- whichuniquely containindirectsolitons as crucialin un-
caying correlation functions). One could intuitively ex- derstanding how the system evolves from the Ising limit
pect the third option to be possibly dismissed, based on (N´eel phase) to the isotropic case(Haldane phase).
the idea that, when thermal fluctuations are completely From the above ideas we may draw a simple but sug-
suppressed,thesystembeinanorderedoratleastquasi- gestive scheme for such evolution:
ordered ground state. This idea is in fact proved cor- - in the Ising limit (λ→∞) the groundstate is the anti-
rectfor half-integerspinsystems,thanks to the so called ferromagnetically ordered N´eel state;
6
- as λ decreases,indirect solitons appear along the chain Our work developed as follows: one first assumes that
in pairs, thus keeping the antiferromagnetic order; the relevant configurations, as far as the N´eel-Haldane
- as λ is further lowered, indirect soliton pairs dissoci- transition is concerned, belong to the reduced Hilbert
ateduetothetransverseinteractionwhich,byspreading space; this permits, by the KT transformation, to re-
solitons along the chain, can cause the ground state to strict the analysis to the subspace of states with either
be disordered. Sz=1 or Sz=0, ∀i. Then the expectation value of the
i i
Due to the privileged role of indirect solitons in the transformed Hamiltonian H Eq.(21) is minimized on a
abovescheme,weconcentrateonconfigurationswhichdo trial ground state whose structure takes into account
onlycontainindirectsolitons. Suchconfigurationsgener- at least short-range correlaetions between spins. By this
ateasubspacefortheHilbertspaceofthesystem,which procedure, we aim at following the effective dissociation
is referred to, in the literature, as the reduced Hilbert of soliton pairs, in order to clarify the connection be-
space42. StatesbelongingtothereducedHilbertspaceare tween the occurrence of isolated solitons in the ground
strongly characterized by the fact that if one eliminates state, and the transition towards the completely disor-
allsiteswithSz=0,aperfectlyantiferromagneticallyor- dered Haldane-phase39,41,42,45.
deredchainisleft. Remarkably,thistypeoforder,which In the framework of a standard variational approach,
is called hidden order in the literature, is not destroyed we should minimize hΦ |H|Φ i with respect to a cer-
0 0
by soliton pairs dissociation, and it actually character- tain number of variational parameters entering the ex-
izes the disorderedground state of a Haldane system, as pression of the normalized trial ground-state |Φ i. By
0
discussed below. applying the non-local unitary transformation U we in-
In 1992Kennedy and Tasaki(KT) defined a non-local stead minimize hΨ |UHU−1|Ψ i with |Ψ i ≡ U|Φ i.
0 0 0 0
unitary transformation43 which makes the hidden order and the transformed hamiltonian H ≡ UHU−1 defined
visible, meanwhile clarifying its meaning. The transfor-
byEq.(21); if|Φ ibelongstothereducedHilbertspace,
mation is defined by U=(−1)N0+[N/2] kUk with it is 0 e
Q
Uk = 1 eiπ kp=−11Spz−1 eiπSkx + 1 eiπ kp=−11Spz+1 , |Ψ0i≡U|Φ0i= c{s}|s1s2 ··· sNi (22)
2(cid:16) P (cid:17) 2(cid:16) P (cid:17) X{s}
where N is the number of sites of the chain, [N/2] is with {s}≡(s ,s ,s ...s ), and s ≡hΨ |Sz|Ψ i=+1,0.
1 2 3 N i 0 i 0
the integer part of N/2, and N0 is the number of odd The simplest trial ground state allowing the de-
sites where Sz=0. If the pure state |Ψi has hidden or- scription of soliton pairs dissociation is that defined
der, meaningthat it only containsindirect solitons,then by Eq. (22) with c =t t ··· t .
U|Ψi has spins with Sz6=0 all parallel to each other. The variational par{asm}eters1ss2sa3res2st3hse4 sixsNa−m2psNli−tu1sdNes
This point is made transparent by the introduction of t , t =t , t , t , t =t , t ,
+++ ++0 0++ +0+ 0+0 00+ +00 000
the string order parameter44 where |t |2 represents the probability for
si−1sisi+1
(Sz ,Sz,Sz )tobeequal(s ,s ,s );acommonar-
j−1 i−1 i i+1 i−1 i i+1
bitrary factor may be used for normalizing |Ψ i. We
Oα (H)≡ lim Sα exp iπ Sα Sα , (19) 0
string |i−j|→∞(cid:28) i (cid:20) Xl=i l (cid:21) j(cid:29)H notice that the chosen form for c{s} is such that the
probability for |Ψ i to contain coupled solitons is finite
0
where α=x,y,z, and h···i indicates the expectation independently of that relative to the occurrence of iso-
H
valueoverthegroundstateoftheHamiltonianH. Itmay lated solitons, whose presence is unambiguously marked
be shown that Osztring(H)6=0 if and only if the ground by t+0+6=0.
state belongs to the reduced Hilbert space. In other Without going into the details of the variational cal-
terms,whileferromagneticorderisrevealedbytheferro- culations, reported in Ref. 46, we here discuss our final
magnetic order parameter Oα ≡ lim hSαSαi , results. Due to the normalization condition, the num-
ferro |i−j|→∞ i j H
the hidden order is revealed by the string order param- ber of variationalparameters is reduced from six to five;
eter Eq. (19). In fact, the non local transformation U moreover,theenergyhΨ |UHU−1|Ψ iisfoundtodepend
0 0
relates the above order parameters through the relation just on four precise combinations of the original param-
e
eters,
Oη =Oη (UHU−1) , (20)
string ferro γ ≡|t |2|t |2 ⇒ (··· ++00++ ···)
++0 00+
forη=x,z,meaningthattheanalysisofthehiddenorder π ≡|t |2|t | ⇒ (··· ++0++ ···)
++0 +0+
inasystemdescribedbyHmaybedevelopedbystudying
χ≡|t |2|t | ⇒ (··· 00+00 ···)
the ferromagnetic order in the system described by the 00+ 0+0
transformed Hamiltonian H≡UHU−1, which reads, for ρ≡|t000| ⇒ (··· 000 ···) , (23)
H defined by Eq. (18),
whosesquaremoduliarerelatedtotheprobabilitiesthat
e
the corresponding strings ( ⇒ ) be contained in |Ψ i; in
H 0
= −SxSx +SyeSiz+Six+1Sy −λSzSz +d(Sz)2 . particular,γ2andπ2refertotheprobabilitiesforcoupled
Je Xi h i i+1 i i+1 i i+1 i i andisolatedsolitons,respectively,tooccurintheground
(21) state.
7
FIG. 6: Phase diagram in the λ>0 half-plane: our results
FIG. 7: Parameters w (squares), w (circles), w (up-
(squares) are shown together with those of Ref. 43 (dotted (1) (2) (3)
ward triangles), and w (downward triangles), for d=0.
lines); the Haldane phase should correspond to the shaded (2,2)
area, according to the best available numerical data33 (solid
lines).
to which we have actually minimized the energy,the fol-
lowing combinations are considered:
Both the analytical expression for the energy and the
w =π2/t2 ; w =γ ;
numericalminimizationshowthatitexistsacriticalvalue (1) +++ (2)
λc=λc(d)>dsuchthat,forλ>λc theminimalenergyis w(2,2) =(χ γ t+++)1/2 ; w(3) =t2+++(ρ γ)2/3 .(25)
attained for π=ρ=0; the condition λ=λ (d) can hence
c
define a curve of phase separation. We therefore single The above quantities have a straightforward physical
out three different phases, characterizedby meaning, as they are directly related with the probabili-
ties fora solitonto appearalongthe chainasanisolated
(a) π=ρ=0, (b) all parameters6=0, (c) χ=t+++=0 excitation(w(1)),aspartofasolitonpair(2w(22)),aspart
(24) of a string made of three adjacent solitons (3w3 ), and
in the ground state. The corresponding phase diagram (3)
finally as part ofa string made of two solitonpairs sepa-
is shown in Fig. 6, together with that obtained with a
rated by one site (4w4 ). From Fig. 7 it turns evident
factorizedtrialgroundstate43, andby numericalsimula- (2,2)
tions33. that the Haldane phase is featured by the occurrence of
isolated solitons (w 6=0), as well as of strings made of
The (a)−(b) transition is seen to quite precisely de- (1)
three adjacent solitons (w 6=0).
scribe the N´eel−Haldaneone, and this leads us to define (3)
This resultconfirmsthat, aselicitedby the analysisof
the condition (a), meaning the occurrence of exclusively
the phase-diagram, the Haldane phase is characterized
coupled solitons, as typical of the N´eel phase. As for
by our condition (b).
the (c)−(b) transition, it is to be noticed that the use
Given their essential role, we have also studied the x
of the reduced Hilbert space is not fully justified in the
andzcomponentofthestringorderparameter,aswellas
λ<d region, where we in fact do not expect quantita-
thesolitonsdensityn =1− (Sz)2 . AfterKTweexpect
tively precise results. 0
Oz (H)6= 0 in both the N(cid:10)´eel an(cid:11)d the Haldane phase,
As foracomparisonbetweenourresultsandthe exact string
and Ox (H) 6= 0 just in the Haldane phase. In fact,
numerical data available, we have considered, along the string
d=0 axis,two specific quantities: the criticalanisotropy analytical expressions for Ox and Oz may be written46
λ (d), where the N´eel phase becomes unstable with re- in terms of four of the five variational parameters (25),
c
spect to the Haldane one, and the ground-state energy and show that
E0(d,λ) at the isotropic point λ=1. For the critical -Osxtring(H)=0ifπ=ρ=0orχ=t+++=0,i.e. inphase
anisotropywefindλ (0)=1.2044(5)tobecomparedwith (a) and (c);
c
the value obtained by exact diagonalization37, λ (0) ≈ - Oz (H)>0 in all phases, asymptotically vanishing
c string
1.19; for the energy we find E (0,1)= −1.3663(5)to be as ρ→1, i.e. in the far large-d phase.
0
compared with E (0,1)= −1.4014(5), again from exact In more details, we notice that Ox =0 whenever
0 string
diagonalizationtechnique47; the value obtained with the the ground state does not contain strings made of an
factorized trial ground state is43 E (0,1)= −4/3. odd number of adjacent spins; as soon as the shortest
0
In Fig. 7 we show the variational parameters as λ is string of such type, namely the isolated soliton, appears
varied with d=0, i.e. along the y axis of the phase- along the chain, then Ox gets finite. The unphysi-
string
diagram;infact,ratherthantheparameterswithrespect cal result Oz >0 in the (c) phase, vanishing only as
string
8
universality class the Haldane transition is suggested to
belong to42. Atthe isotropicpoint(d=0, λ=1)we find
Ox =0.3700(5) in full agreement with the value ob-
string
tained by exact diagonalization48.
The overall good agreement between our results and
the numerical available data, allows us to conclude that
the N´eel-Haldane transition is a second-order one, and
that the string order parameter Ox , revealing hidden
string
order along the x direction, is the appropriate order pa-
rameter for the Haldane phase. The disordered ground
state featuring the Haldane phase is seen to originateby
solitonpairsdissociation,accordingtothispath: Solitons
occur just in pairs in the antiferromagnetically ordered
N´eel phase; at the N´eel-Haldane transition soliton pairs
dissociate and the byproducts rearrange in strings made
of an odd number of solitons. These strings are ulti-
FIG. 8: String order parameters Ox (squares), Oz mately responsible for the disorder of the ground state.
string string
(circles), and solitons density n (triangles), for d=0.
0
IV. TWO-DIMENSIONAL ISOTROPIC
HEISENBERG MODEL
The 2DisotropicQHAFonthe squarelattice is oneof
the magneticmodelsmostintensivelyinvestigatedinthe
last two decades. This is due both to its theoretically
challenging properties and to its being considered the
bestcandidateformodelingthemagneticbehaviorofthe
parent compounds of some high-T superconductors2,3.
c
From a theoretical point of view the fully isotropic
Heisenberg model in d dimensions, thanks to the simple
structureofitsHamiltonian(whosehighsymmetryisre-
sponsible for most of its peculiar features), may well be
consideredacornerstoneofthe moderntheoryofcritical
phenomena,withitsrelevanceextendingwellbeyondthe
onlymagneticsystems. Thed=2caseearnedadditional
FIG. 9: Critical behavior of Ox for d=0: squares interest,representingtheboundarydimensionseparating
string
are results; curves are obtained by best-fit procedure from systemswithandwithoutlong-rangeorderatfinite tem-
Osxtring∼(λ−λc)β with β fixed to 0.125 (dashed curve), and perature4. The antiferromagnetic coupling adds further
asfittingparameter,resultinginβ=0.217(solidcurve). Both
appeal, as the classical-like N´eel state is made unstable
procedures give λ =1.2044(5), marked by a circle in figure.
c byquantumfluctuationsandthegroundstateofthesys-
tem is not exactly known. It can be rigorously proven49
to be ordered for S≥1; for S=1/2 there is no rigorous
d→∞ rather than everywhere in the large-d phase, is proof,althoughevidencesforanorderedgroundstatecan
due to our assuming the ground state to belong to the be drawn from many different studies (for a review, see,
reduced Hilbert space, which is actually licit just in the for instance, Ref. 50).
λ>d region. On the experimental side the attention on the proper-
In Fig. 8 we show Ox , Oz , and n as λ varies ties of 2D QHAF was mainly triggered by the fact that
string string 0
with d=0: We underline that Ox gets finite contin- among the best experimental realizations of this model
string
uously but with discontinuous derivative at the transi- we find several parent compounds of high-T supercon-
c
tion (reflecting the behavior of w and w shown in ductors, as, e.g., La CuO or Sr CuO Cl 51,52,53, both
(1) (3) 2 4 2 2 2
Fig. 7), so that the N´eel−Haldane quantum phase tran- having spin S=1/2. In such materials, as well as in
sition is recognized as a second order one. In Fig. 9 other magnetic compounds with a layered crystal struc-
we zoom the order parameter Ox around the critical ture as La NiO 54 and K NiF 52,53 (S=1), Rb MnF 55
string 2 4 2 4 2 4
point: itsbehaviorisseentobedescribedbyapowerlaw and KFeF 56 (S=5/2) or copper formate tetradeuter-
4
Ox ∼ (λ −λ)β, as expected for a continuous phase ate (CFTD, S=1/2)57 the magnetic ions form parallel
string c
transition;ourestimatedvalueforthecriticalexponentis planesandinteractstronglyonlyifbelongingtothesame
β=0.217(5)to be comparedwith β=0.125,correspond- plane. The interplane interaction in these compounds
ing to the Ising model in a transverse field, to whose is several orders of magnitude smaller than the intra-
9
planeone,thusofferingalargetemperatureregionwhere cuprous oxides by Chakravarty, Halperin, and Nelson58
their magnetic behavior is indeed 2D down to those low (CHN) who used symmetry arguments to show that the
temperatures where the weak interplane interaction be- long-wavelengthphysicsoftheQHAFisthesameofthat
comesrelevant,drivingthe systemtowardsa3Dordered of the QNLσM; in other words the physical observables
phase: an antiferromagnetic Heisenberg interaction and of the two models show the same functional dependence
the small spin value make these compounds behave as upon T, if the long-wavelength excitations are assumed
2D QHAFs. Even the onset of 3D magnetic long-range to be the only relevant ones, as one expects to be at low
orderishoweverstronglyaffectedbythe2Dpropertiesof temperature.
the system: indeed, the observed3Dmagnetictransition The analysiscarriedout by CHN on the QNLσM lead
temperature is comparable with the intraplane interac- to single out three different regimes,called quantum dis-
tion energy, i.e., several order of magnitude larger than ordered, quantum critical (QCR) and renormalized clas-
that one can expect only on the basis of the value of sical (RCR), the most striking difference amongst them
the interplane coupling. Such apparently odd behavior being the temperature dependence of the spin correla-
canbeeasilyunderstoodbyobservingthattheestablish- tions. If g is such as to guarantee LRO at T=0, the
ing of in-plane correlationsona characteristicdistanceξ QNLσM is in the RCR at very low-temperature and the
effectively enhances the interplane coupling by a factor correlationlength ξ behaves as60:
(ξ/a)2, a being the lattice constant. The latter consid-
e c 2πρ T
eration is one of the reasons explaining why most of the
ξ = exp 1− . (28)
3l
attention, both from the experimental and theoretical 8(cid:16)2πρ(cid:17) (cid:16) T (cid:17) h 4πρi
point of view, was devoted to the low-temperature be-
CHN found also that by raising the temperature any
havior of the correlation length ξ of the 2D QHAF (in
2D QNLσM with an ordered ground state crosses over
thefollowingξ willbealwaysgiveninunitsofthelattice
from the RCR to the QCR, characterized by a correla-
constant a).
tion length ξ ∝α(T)=c/T.
The 2D QHAF is described by the Hamiltonian
Thefirstdirectcomparisonbetweenexperimentaldata
J onspin1/2compoundsandthepredictionoftheQNLσM
H= S ·S , (26)
i i+d field theory in the RCR gave surprisingly good agree-
2
Xi,d mentandcausedanintenseactivity,boththeoreticaland
experimental, in the subsequent years. However, with
where J is positive and the quantum spin operators S
i
the accumulation of new experimental data on higher
satisfy |S |2=S(S+1). The index i ≡ (i ,i ) runs over
i 1 2
spin compounds it clearly emerged that the experimen-
thesitesofasquarelattice,anddrepresentsthedisplace-
tally observed behavior of ξ(T) for larger spin could not
mentsofthe4nearest-neighborsofeachsite,(±1,0)and
be reproduced neither by the originalsimplified (2-loop)
(0,±1).
form of Eq. (28) given by CHN (which does not contain
In addition to the first approximations usually em-
the term in square brackets), nor by the three-loops re-
ployed to investigate the low temperature properties of
sult(28)derivedbyHasenfratzandNiedermayer60(HN);
magnetic systems as, e.g., mean-field and (modified)
moreover no trace of QCR behavior was found in pure
spin-wave theory, the critical behavior of the 2D QHAF
compounds. The discrepancies observed could be due
was commonly interpreted on the basis of the results
to the fact that the real compounds do not behave like
obtained by field theory starting from the so-called 2D
2D QHAF or to an actual inadequacy of the theory.
quantum nonlinear σ model(QNLσM)58,whoseactionis
In particular the CHN-HN scheme introduces two pos-
given by
sible reasons for such inadequacy to occur: the physics
1 u ofthe 2DQHAFis notproperlydescribedbythatofthe
S = dx dτ |∇n|2+|∂ n|2 ; |n|2 =1. (27)
τ 2DQNLσMand/orthetwo(three)-looprenormalization-
2g Z Z
0 (cid:0) (cid:1) group expressions derived by CHN-HN do hold at tem-
In the last Equation n(x) is a unitary 3D vector peratureslowerthanthoseexperimentallyaccessible. Af-
field, g=cΛ/ρ and u=cΛ/T are the coupling and the teranalmosttenyearslongdebate,thelatterpossibility
imaginary-time cut-off respectively, and the two param- hasfinallyemergedasthecorrectone,beingstronglysup-
eters ρ and c are usually referredto as spin stiffness and portednotonlybyourownwork,butalsobyotherinde-
spin-wave velocity. Despite their names, the two param- pendent theoretical approaches61, joined with the anal-
eters ρ and c are however just phenomenological fitting ysis of the experimental and the most recent quantum
constants which can be rigorously related to the proper Monte Carlo (QMC) data for the 2D QHAF.
parameter J and S of the original magnetic Hamilto- The theoretical approach we employed to inves-
nian (26) only in the large-S limit1,59. The source of tigate the 2D QHAF is the effective Hamiltonian
non-linearity in the model Eq. (27), which is seemingly method62,63,64, developed within the framework of the
quadraticinthefieldvariables,istheconstraintimposed pure-quantum self-consistent harmonic approximation
onto the length of the field n. (PQSCHA) we introduced at the beginning of the
The relation between the 2D QHAF and the QNLσM 90’s65,66. The PQSCHA starts from the Hamiltonian
was first exploited to interpret the experimental data on path-integral formulation of statistical mechanics which
10
allows one to separate in a natural way classical and tion can be employed finally obtaining63:
quantumfluctuations: onlythe latterarethentreatedin
H θ4
a self-consistent harmonic approximation,finally getting eff = s ·s +NG(t) , (29)
i i+d
an effective classical Hamiltonian, whose properties can JS2 2 Xi,d
thereafter be investigatedby all the techniques available
forclassicalsystems. Theideaofseparatingclassicaland G(et) = t lnsinhfk −2κ2D , (30)
N θ2f
quantumfluctuationsturnedouttobefruitfulnotonlyin Xk k
view of the implementation of the PQSCHA, but also in
with the temperature and spin dependent parameters
the final understanding of the connection between semi-
classicalapproachesand quantum field theories67, which θ2 = 1− D , (31)
could be possible also thanks to the paper by Hasen- 2
fdruaetzt6o8caubtooufftecffoercrtesc.tions to the field-theoretical results D = S1N Xk (1−γk2)12Lk , (32)
The PQSCHA naturally applies to bosonic systems, f = eωk , L =cothf − 1 . (33)
whose Hamiltonian is written in terms of conjugate k 2St k k fk
operators qˆ ≡ (qˆ ,...qˆ ), pˆ ≡ (pˆ ,...pˆ ) such that
1 N 1 N In the previous Equations γ =(cosk + cosk )/2, N is
[qˆ ,pˆ ]=iδ ; the method, however, does not require e k 1 2
m n mn the number of sites of the lattice and k ≡ (k ,k ) is
H(pˆ,qˆ) to be standard, i.e., with separate quadratic ki- 1 2
thewave-vectorinthe firstBrillouinzone;S≡S+1/2is
netic pˆ-dependent and potential qˆ-dependent terms, and
the effective classical spin length, which naturally fol-
its application may be extended also to magnetic sys-
e
tems, according to the following scheme65: The spin lows from the renormalization scheme, and t≡T/JS2
Hamiltonian H(S) is mapped to H(pˆ,qˆ) by a suitable is the reduced temperature defined in terms of the en-
e
spin-bosontransformation;once the correspondingWeyl ergy scale JS2. The renormalizationscheme is closedby
symbol H(p,q), with p ≡ (p ,...p ) and q ≡ (q ,...q ) the self-consistent solution of the two coupled equations
classicalphase-spacevariable1s,hasNbeen determin1ed, tNhe ωk=4κ2(1 −eγk2)1/2 and κ2=θ2 −t/(2κ2). The pure-
PQSCHArenormalizationsmaybe evaluatedandtheef- quantumrenormalizationcoefficientD=D(S,t)takesthe
fective classicalHamiltonianH (p,q)andeffectiveclas- main contribution from the high-frequency part (short-
eff
sical function O (p,q) corresponding to the observable wavelength) of the spin-wave spectrum, because of the
eff
O of interest follow. Finally the effective functions appearanceoftheLangevinfunctionLk. Dmeasuresthe
H (s) and O (s), both depending on classical spin strengthofthepure-quantumfluctuations,whosecontri-
eff eff
variabless with |s|=1 andcontaining temperature- and bution to the thermodynamics of the system is the only
spin-dependent quantum renormalized parameters, are approximated one in the PQSCHA scheme. The theory
reconstructed by the inverse of the classical analogue of is hence quantitatively meaningful as far as D is small
the spin-boson transformation used at the beginning. enough to justify the self-consistent harmonic treatment
ofthepure-quantumeffects. Inparticular,thesimplecri-
In order to successfully carry out such renormaliza-
terion D<0.5 is a reasonable one to assess the validity
tion scheme, the Weyl symbol of the bosonic Hamilto-
of the final results.
nianmustbe awell-behavedfunctioninthe whole phase
ThemostrelevantinformationwegetfromEq. (29)is
space. Spin-boson transformations, on the other hand,
that the symmetry of the Hamiltonian is left unchanged
can introduce singularities as a consequence of the topo-
so that from a macroscopic point of view the quantum
logical impossibility of a global mapping of a spherical
systemessentiallybehaves,atanactualtemperaturet,as
phase space into a flat one. The choice of the trans-
its classicalcounterpartdoes atan effective temperature
formation must then be such that the singularities oc-
t =t/θ4(S,t) . This allows us to deduce the behavior
eff
cur for configurations which are not thermodynamically
of many observables (but not all!, see Refs. 62,63,64 for
relevant, and whose contribution may be hence approxi-
details) directly from the behavior of the corresponding
mated. Most of the methods for studying magnetic sys-
classical quantities. This is the case of the correlation
tems do in fact share this problem with the PQSCHA;
length, which turns out to be given simply by:
whatmakes the difference is that by using the PQSCHA
one separates the classical from the pure-quantum con- ξ(t)=ξ (t ) (34)
cl eff
tribution to the thermal fluctuations, and the approxi-
so that once θ4(S,t) has been evaluated, the only addi-
mation only regards the latter, being the former exactly
tional information we need is the classical ξ (t), which
taken into account when the effective Hamiltonian is re- cl
is available from classical Monte Carlo simulation and
cast in the form of a classical spin Hamiltonian.
analytical asymptotic expressions69 as t→0.
The spin-boson transformation which constitutes the SampleresultsobtainedbyPQSCHAareshowninthe
first step of the magnetic PQSCHA is chosen according figures. In Fig. 10 the correlationlength for S=1/2 and
to the symmetry properties of the original Hamiltonian S=1iscomparedwithexperimentaldata;asimilarcom-
and of its ground state. In the case of the 2D QHAF parison, including MC data for S=1/2 and experimen-
both Dyson-Maleev and Holstein-Primakoff transforma- tal data on S=5/2 compounds KFeF and Rb MnF is
4 2 4