On the absorption of microwaves by the one-dimensional spin-1/2 Heisenberg-Ising magnet

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by  M. Brockmann

Ontheabsorptionofmicrowavesbytheone-dimensionalspin-1/2Heisenberg-Isingmagnet
Michael Brockmann, Frank Go¨hmann, Michael Karbach, and Andreas Klu¨mper
Fachbereich C – Physik, Bergische Universita¨t Wuppertal, 42097 Wuppertal, Germany
Alexander Weiße
Max-Planck-Institut fu¨r Mathematik, P.O. Box 7280, 53072 Bonn, Germany
WeanalyzetheabsorptionofmicrowavesbytheHeisenberg-Isingchaincombiningexactcalculations,based
on the integrability of the model, with numerical calculations. Within linear response theory the absorbed
intensityisdeterminedbytheimaginarypartofthedynamicalsusceptibility.Themomentsofthenormalized
intensitycanbeusedtodefinetheshiftoftheresonancefrequencyinducedbytheinteractionsandtheline
widthindependentlyoftheshapeofthespectralline.Thesemomentscanbecalculatedexactlyasfunctionsof
2 temperatureandstrengthofanexternalmagneticfield,aslongasthefieldisdirectedalongthesymmetryaxisof
1 thechain.Thisallowsustodiscussthelinewidthandtheresonanceshiftforagivenmagneticfieldinthefull
0 rangeofpossibleanisotropyparameters.Fortheinterpretationofthesedataweneedaqualitativeknowledgeof
2 thelineshapewhichweobtainfromfullynumericalcalculationsforfinitechains.Exactanalyticalresultsonthe
lineshapeareoutofreachofcurrenttheories.Fromournumericalworkwecouldextract,however,anempirical
n
parameter-freemodelofthelineshapeathightemperatureswhichisextremelyaccurateoverawiderangeof
a
J anisotropyparametersandisexactatthefreefermionpointandattheisotropicpoint.Anotherpredictionofthe
lineshapecanbemadeinthezero-temperatureandzeromagneticfieldlimit,wherethesufficientlyanisotropic
1
modelshowsstrongabsorption.Foranisotropyparametersinthemassivephasewederivetheexacttwo-spinon
3
contributiontothespectralline.Fromtheintensitysumruleitcanbeestimatedthatthiscontributionaccounts

formorethan80%ofthespectralweightiftheanisotropyparameterismoderatelyaboveitsvalueattheisotropic
]
l point.
e
-
r
t
s I. INTRODUCTION Inthisworkweshallconcentrateontheone-dimensional
. spin-1 case which is not covered by the more traditional
t 2
a Short-rangeantiferromagneticexchangeinteractionsarethe approaches.1,2 Itisrelevantforthedescriptionofquasione-
m
predominantelectronicinteractionsinMottinsulators. They dimensionalcompounds3,4withstrongexchangeinteractions.
- aremodeledbytheHeisenberg-IsingHamiltonian This case has been successfully studied by field theoretical
d
n methods5whichare,however,restrictedtosmalltemperatures
o H=J∑ sxisxj+syisyj+(1+δ)sziszj , (1) andtoalimitedrangeofmagneticfields. Theyalsoseemto
c ij havebuiltincertainaprioriassumptionsaboutthelineshapes.
(cid:104) (cid:105)(cid:0) (cid:1)
[ Inonedimensionpurelynumericalapproaches6,7areefficient
wherethesumisovernearest-neighborsites,andJmeasures
1 thestrengthoftheexchangeinteraction. Theoperatorssα are aswell. Theyareunbiased,yettheextrapolationofthedatato
v localspin-1 operators,andtheparameterδ takesaccounitofa thethermodynamiclimitoflargechainsmayrequireadditional
3 2 justificationandsupportfromanalyticalcalculations.
possibleexchangeanisotropyandmayincludetheeffectsof
2
Theaimofthepresentworkistoestablishanumberofexact
5 dipolarinteractionsaswell.
resultsforthemicrowaveabsorptionoftheone-dimensional
6 A sensitive experimental probe of magnetic interactions
spin-1 Heisenberg-Isingmagnet,withthehomogeneousmag-
. in solids is the absorption of microwaves, typically in ESR- 2
1 neticfieldalongthemagneticsymmetryaxisofthechain,and
experiments. Inthesimplestexperimentalsetupacircularly
0
tointerprettheseresultsinthelightofnumericalcalculations.
2 polarizedwavetravelsalongthedirectionofahomogeneous
Inturnsthequalityandvalidityofthenumericalcalculations
1 magneticfield. Thefieldisslowlychangedandoneormore
: absorption lines are observed with increasing field, whose canbeestimatedfromtheanalyticalresults.
v
preciselocationandwidthdependsonthetemperature. Ourworkismotivatedbytherecentprogressincalculating
i
X Forexperimentallyaccessiblestrengthsoftheincidentmi- staticshort-rangecorrelationfunctionsoftheintegrablespin-1
2
r crowaveslinearresponsetheory1 providesasatisfactorythe- Heisenberg-Ising chain at finite temperatures and magnetic
a oretical frame for the calculation of the absorbed intensity. fields. This makes it possible to extend a remarkable result
ThenthekeyquantitytobecalculatedforagivenHamiltonian fortheresonanceshiftinone-dimensionalantiferromagnetic
is the (imaginary part of) the dynamical susceptibility. It is chains,whichwasobtainedbyMaedaetal.8andwhichutilizes
theFouriertransformofacertaindynamicalspin-spincorre- theexactnearest-neighborcorrelationfunctionsoftheisotropic
lationfunction(seebelow). Sincesuchaquantitycannotbe spin-1 Heisenbergchain. Wepresentanalternativeframework
2
calculated for an interacting many-body system as the anti- for the derivation of the resonance shift which, in the limit
ferromagnetic Heisenberg-Ising model (1), various kinds of of small anisotropy, reproduces the result of Ref. 8. In our
approximationshavebeentriedinthepast. Mostoftheseap- approachtheanisotropyistreatednon-perturbatively. Itallows
proximationsbreakdown,whenmany-bodycorrelationeffects us,moreover,toderiveanexactformulaforthelinewidthat
arestrong,inparticularinone-andtwo-dimensionalsystems fixedmagneticfield.
withstrongexchangeinteractions. We utilize the fact that the absorbed intensity I(ω,h) is a
2
positive function, whose integral over ω exists. The field- II. THEMETHODOFMOMENTS
dependentmomentsofthecorrespondingnormalizedintensity
turnouttobestaticshort-rangecorrelationfunctionswhichcan For any spin system with Hamiltonian H linear response
becalculateddirectlyfortheinfinitechain. Thefirstmoment theoryrelatestheintensityabsorbedfromacircularlypolarized
is the average absorption frequency. In case that there is a electro-magneticwave,whosewavelengthislargecompared
singlepronouncedabsorptionlineitgivesameasureforthe tothedistancebetweenthespins,tothe(imaginarypartofthe)
shiftoftheresonancecomparedtotheparamagneticabsorp- dynamicalsusceptibility1
tion frequency ω =h (in the units used in this work). This
measureiscompletelyindependentoftheactualshapeofthe 1 ∞
χ (ω,h)= dteiωt [S+(t),S ] . (2)
spectralline. Similarly,thesecondmomentprovidesashape- +(cid:48)(cid:48) 2L − T
independent measure of the line width. The idea of using − (cid:90)−∞
(cid:10) (cid:11)
momentswasintroducedbyvanVleck9evenbeforethelinear Here S = Sx iSy, and the Sα = ∑L sα, α = x,y,z, are
response theory was created. Here we combine it with the the com±ponent±s of the total spin. Lj=is1thje number of lat-
finite-temperaturelinearresponsetheory. Theresultsofvan tice sites, stands for the canonical average at tempera-
T
Vleckarethenrecoveredintheinfinite-temperaturelimit. ture T calc(cid:104)·u(cid:105)lated by means of the statistical operator ρ =
Alternatively we may normalize the intensity by its inte- e−(H−hSz)/T/tr e−(H−hSz)/T . Through this average the dy-
{ }
gral over h. In this case the moments cannot be expressed namical susceptibility depends on T and on an external ho-
by finite-range correlation functions. Still, these frequency- mogeneousmagneticfieldhwhichisusuallyappliedinESR
dependentmomentscanbeexpandedintoaninfiniteseriesof experiments. ThetimeevolutionofS+ in(2)mustbecalcu-
field-dependentmoments10whichisausefulstartingpointfor latedwithH hSz. Theabsorbedintensityperspin,normal-
−
approximationssuchasthehigh-temperatureexpansionoran izedbytheintensityoftheincidentwaveandaveragedovera
expansionforsmallanisotropy. Interestingly,thelinewidth half-periodπ/ω ofthemicrowavefield,is
definedintermsofthefrequency-dependentmomentsshowsa
ω
temperaturebehaviorratherdifferentfromthatdeterminedby I(ω,h)= χ (ω,h). (3)
2 +(cid:48)(cid:48)
thefield-dependentmoments. −
Whenthereismorethanasingleresonance,theinterpreta- Inordertokeepthispaperself-containedweincludedaderiva-
tionofthemomentsislessintuitive. Incaseoftwopeaks,for tionof(2)andof(3)inApp.A.
instance,thefirstmomentwouldbesomethingliketheaverage In this work we shall exclusively concentrate on the one-
location of the two peaks. For this reason it is desirable to dimensionalversionoftheHeisenberg-Ising(orXXZ)Hamilto-
havesomeknowledgeaboutthefullabsorptionspectrum(‘the nian(1).ThisHamiltonianisintheclassofintegrableHamilto-
lineshape’). Hence,wecomplementedourexactcalculation nians,butsofarthisdoesnotmeanthatdynamicalcorrelation
ofthemomentswithnumericalcalculationsofthedynamical function such as the expectation value [S+(t),S ] in (2)
− T
susceptibility on finite lattices up to 32 sites. The combina- couldbecalculatedanalytically. Weareonlyawareofthree
tion of both, the exact calculation of the moments and the veryspecialcaseswherethisispossible(cid:10). Theseare(cid:11)the‘free
numerics,allowsustoproposeamodelforthelineshapein Fermion case’ δ = 1 at T ∞, the isotropic limit δ 0
thehigh-temperaturelimitwhichhasnofreeparameters. The and the Ising limit δ− ∞. W→e shall discuss these cases→be-
actualparametersofthecorrespondingdistributionfunction low. In the general ca→se so far only the short time behavior
(anormal-inverseGaussian)aredeterminedfromthefirstfour of [S+(t),S ] can be accessed by directly calculating
− T ∞
exactlycalculatedmoments. thecommutatorsi→nvolvedinthetimeevolutionuptoacertain
Anunbiasedbutapproximatecalculationofthelineshape pow(cid:10)er. Wegene(cid:11)ratedthisseriesuptotheordert38(cf.App.C).
is possible in the massive ground state phase of the model Still,theresultsfortheshort-timebehavioralonearenothelp-
atvanishingmagneticfield. Foranisotropicsystemthereis fulforcalculatingtherighthandsideof(2).
noabsorptionwithoutanexternalfield. Forsufficientlyhigh Interestinglyenough, aswehaveshown,10 somemoreel-
anisotropy, however, the absorption becomes large. In the ementaryspectralcharacteristics,suchasthepositionofthe
massivephasethematrixelementsofthelocalspinoperators resonanceorthelinewidth,areeasiertocalculate. Theymay
betweenthegroundstatesandexcitedstates(‘formfactors’) be expressed in terms of certain static correlation functions
areexactlyknown11andgenerallynon-vanishinginthethermo- thatdeterminethemomentsofanormalizedintensityfunction
dynamiclimit. Theyareclassifiedas2n-spinoncontributions inonevariable. SinceI(ω,h)isafunctionofthefrequencyω
accordingtothe(even)numberofelementaryexcitationsin- andofthemagneticfieldhwemaynormalizeitbydividing
volved. Herewecalculatethetwo-spinoncontributiontothe eitherbytheintegraloverω orbytheintegraloverh.
absorbed intensity exactly. From the intensity sum rule we Inthefirstcaseweinterprettheresultingnormalizedfunc-
inferthatforanisotropiesmoderatelyabovetheisotropicpoint tionasadistributionfunctionoffrequencieswhichdepends
the two-spin contribution is dominant and amounts to more parametricallyonthemagneticfield. Thenitsmomentsm are
n
than80%oftheabsorbedintensity. Forgrowinganisotropyit fielddependent. Thiscorrespondstoanexperimentalsituation
rapidlyapproaches100%. Butasopposedtothesituationwith wherethefieldiskeptfixedandthefrequencyisvaried. We
thedynamicstructurefactorforwhichtherelativecontribution shallseethat,fromatheoreticalperspective,thiscaseiscom-
ofthetwo-spinonexcitationisstilldominantintheisotropic paratively simple, since the moments depend only on static
limit,12–14itdropsoffrapidlyforthedynamicalsusceptibility. correlation functions of finite range. The lowest moments
3
whichdeterminethepositionoftheresonanceanditswidth Thefirstfewofthemcanbeeasilycalculated. Inparticular,
canbeexpressedbycorrelationfunctionsrangingoverupto
fourlatticesites,whichcanbecalculatedexactly.15,16 m = 1 [S+,S ] = 1 Sz (9)
In the second case, when the intensity is normalized as a 0 2L − T L T
functionofthemagneticfield,thecorrespondingmomentsM (cid:10) (cid:11) (cid:10) (cid:11)
n whichisthemagnetizationperlatticesite. Forthesubsequent
dependonthefrequency.ThiscorrespondstothestandardESR
moments,whichdonothavesuchanimmediateandsimple
setupinwhichthemagneticfieldisslowlysweptatfixedfre-
interpretation,weobtain
quency. Asweshallseebelowthiscaseismoresophisticated
foratheoreticalanalysis,sincestaticcorrelationfunctionsfor m =δ s+s 2szsz , (10a)
arbitrarydistancesarealreadyinvolvedinthecalculationofthe 1 (cid:104) 1 −2 − 1 2(cid:105)T
1
lsoewrieesstinmtoemrmenstos.fSthtiellm,tohmeeMnntscmanbaendexthpeainrddeedriivnattoivaens,inwfihniicthe m2= 2δ2(cid:104)sz1+4sz1sz2sz3−4sz1s+2s−3(cid:105)T, (10b)
n
1
mayserveasastartingpointforsystematicapproximations. m = δ2 2s+s+s s +4s+s s+s 2s+s s s+
3 4 1 2 −3 −4 1 −2 3 −4 − 1 −2 −3 4
8(cid:10)szszs+s 4szs+szs +8szs+s sz 4s+s
− 1 2 3 −4 − 1 2 3 −4 1 2 −3 4− 1 −2
A. Field-dependentmoments s+s +8szszszsz+2szsz 4szsz
− 1 −3 1 2 3 4 1 3− 1 2
+δ(8szs+s sz+2s+s 8szsz) . (10c)
We temporarily assume that our chain is large but finite. 1 2 −3 4 1 −2 − 1 2 T
Thenthespectrumisboundedandtheintegrals
Thesemomentsarecertaincombinationsof(cid:11)staticshort-range
∞ correlationfunctionswhichimplies,inparticular,thattheyall
In= dωωnI(ω,h) (4) existinthethermodynamiclimitL ∞. Hence,wemayrelax
(cid:90)−∞ our restriction that we are dealing→with a finite chain at this
exist for all non-negative integers n. Since I(ω,h) is non- point. An interesting conclusion which can be drawn from
negative everywhere and since I > 0, we may interpret theexistenceofthemomentsisthatthefield-dependentline
0
I(ω,h)/I asaprobabilitydistributionwithmomentsI . As shapecannotbeLorentzian,asissometimesassumedinthe
0 n
weshallsee,inourcaseitisconvenienttoexpresstheI in literature,sincethesecondmomentofaLorentziandoesnot
n
termsofanothercloselyrelatedsequenceofintegrals exist. Infact,inournumericaldataforfinitechainsweseean
exponentialdecayawayfromtheresonance(seebelow). Note
∞ dω
mn(T,h)=J−n(cid:90)−∞ 2π(ω−h)nχ+(cid:48)(cid:48)−(ω,h) (5) tmh2atisfoorffionridteermδa2g,nbeutitcafillelhdigmh0erismoofmorednetrs1a,rme1ofisoordfeorrdδe2raδs,
well. Thisisclearfrom(8)andwillberelevantbelow.
which, by slight abuse of language, we shall call (shifted)
The formulae (10) are appealing from a theoretical per-
momentsaswell. Theexistenceoftheintegralsisobviousfor
spective, since, due to recent progress in the theory of in-
everyfinitechain.
tegrable systems, static short-range correlation functions of
Nowbydefinitiontheshiftoftheresonanceforfixedhis
theHeisenberg-Isingchaincanbecalculatedexactly. Ithas
thedeviationoftheaveragefrequencyfromtheparamagnetic
been shown that all static correlation functions of the one-
resonanceatω =h,
dimensionalHeisenberg-Isingmodelarepolynomialsinthe
δω = I1 h=JJm2+hm1. (6) derivativesofthreefunctions17ϕ,ω,andω(cid:48)which,asiscom-
I0 − Jm1+hm0 mon in integrable models, can be expressed in terms of the
solutionsofcertainnumericallywellbehavedlinearandnon-
Ameasureforthelinewidthisthemeansquaredeviationfrom linearintegralequations.18Weprovidethedefinitionofthese
theaveragefrequency functionsinthecriticalcase( 1<δ <0)inApp.D.Forthe
−
massive case (δ >0) the definitions are similar and can be
∆ω2= I2 I12 =J2Jm3+hm2 δω2. (7) foundinRef.16.
I0 −I02 Jm1+hm0 − Although, in principle, all static correlation functions of
the Heisenberg-Ising chain in the thermodynamic limit are
Weseethat,inordertocalculatetheresonanceshiftandthe
knownexactly,theirexplicitcalculationworksoutonlyatshort
linewidth,weneedtoknowthefirstfourshiftedmomentsm ,
0 distances. Atlargerdistancestheamountofcomputeralgebra
m ,m ,m ofthedynamicsusceptibilityχ .
1 2 3 +(cid:48)(cid:48) involvedinthecalculationsgrowsexcessively. InRefs.19,15,
Inthefollowingweshallemploythenota−tionadX =[X, ] and16weobtainedallcorrelationfunctionsrangingoverat
for the adjoint action of an operator X. Then ·S+(t) =· most four lattice sites. This is just enough to calculate the
e−ihteitadHS+, since [H,Sz]=0 and [Sz,S+]=S+, and it fol- momentsm0,m1,m2,m3. Forthesimplercaseoftheisotropic
lowswith(2)and(5)that
modelinvanishingmagneticfieldweobtainedthecorrelation
functionsrangingoveruptosevenlatticesites.20
1
mn= 2L [S+,adnH/JS−] T. (8) WhenconsideringtheHeisenberg-IsingHamiltonianasan
integrable model it is customary to parameterize all func-
(cid:10) (cid:11)
The latter formula shows that the moments m are static tions by a deformation parameter q in terms of which the
n
correlationfunctionswhosecomplexitygrowswithgrowingn. anisotropy is δ = (q 1)2/2q. Employing the shorthand
−
4
notationsϕ =∂nϕ(x) , f =∂m∂nf(x,y) , for f =ω,ω ,weobtain10
(n) x x=0 (m,n) x y x=y=0 (cid:48)
| |
1
m = ϕ ,
0 −2 (0)
m = (q−1)2(q2+4q+1)ω((cid:48)0,1) (q3−1)ω(0,0),
1 16q2 − 4q(q+1)
(q 1)2
m2= 25−6q4 4q(q+1)(q3−1)(ω(0,2)ϕ(0)−2ω(1,1)ϕ(0)−ω(0,0)ϕ(2))
(cid:2) +(q2 1)2(q2+4q+1)(ω ϕ +ω ϕ ) 16q2(q 1)2ϕ ,
− ((cid:48)1,2) (0) ((cid:48)0,1) (2) − − (0)
m3= 98304q8((qq4−1)14)(q6 1) 16q2(q2−1)3(q4−1)(q6−1)(q2+4q+1)(ω(0,2)ω((cid:48)0,1)+ω(0,0)ω(cid:3)((cid:48)1,2))
− −
+64q2(q2 1)4(2q10 q9+(cid:2)4q8 4q7 12q6 14q5 12q4 4q3+4q2 q+2)ω ω
(0,0) (1,1)
− − − − − − − −
16q2(q2 1)2(q4 1)(q6 1)(3q4+14q2+3)ω +8q2(q2 1)(q4 1)2(q6 1)(q+1)2(8ω ω )
− − − − ((cid:48)0,3) − − − ((cid:48)1,2)− ((cid:48)2,3)
+192q4(q2 1)2(q4 1)(q6+18q4+8q3+18q2+1)ω
(0,2)
− −
+64q2(q2 1)2(q4 1)(q+1)2(2q8 5q7+26q6 49q5+28q4 49q3+26q2 5q+2)ω
(1,1)
− − − − − −
16q2(q2 1)2(q4 1)(q+1)2(q8 q7+q6+q5+2q4+q3+q2 q+1)(2ω 3ω )
(1,3) (2,2)
− − − − − −
+64q2(q4 1)(q6 1)(3q8+2q6+24q5 130q4+24q3+2q2+3)ω
− − − ((cid:48)0,1)
+(q4 1)(q6 1)(q+1)2(q10 2q9+25q8+16q7+118q6+164q5
− − −
+118q4+16q3+25q2 2q+1)(ω ω +ω ω )
− ((cid:48)0,3) ((cid:48)1,2) ((cid:48)0,1) ((cid:48)2,3)
1536q5(q4 1)(4q8 9q7 2q6 6q5+8q4 6q3 2q2 9q+4)ω
(0,0)
− − − − − − − −
+4q2(q6 1)(q+1)2(q2+1)(5q8 2q7+32q6+50q5+70q4+50q3+32q2 2q+5)
− − −
(2ω ω 3ω ω +ω ω 2ω ω 3ω ω ω ω )
(1,3) ((cid:48)0,1)− (2,2) ((cid:48)0,1) (0,2) ((cid:48)0,3)− (1,1) ((cid:48)0,3)− (0,2) ((cid:48)1,2)− (0,0) ((cid:48)2,3)
16q2(q+1)2(q16 q15+8q14+9q13+47q12+45q11+96q10+91q9+128q8+91q7+96q6
− −
+45q5+47q4+9q3+8q2 q+1)(3ω2 6ω ω +2ω ω 3ω ω ) . (11)
− (0,2)− (1,1) (0,2) (0,0) (1,3)− (0,0) (2,2)
(cid:3)
Thesearethemomentsinthethermodynamiclimit. Wecan e.g. the distance between the inflection points right and left
calculatethemwithhighnumericalaccuracyoverthewhole tothemaximumoftheintensity(‘peak-to-peakwidth’). This
rangeofthephasediagram,foralltemperaturesandmagnetic discrepancywasalreadynotedbyvanVleck.9Oneadvantage
fields as well as for arbitrary anisotropy δ. In Figs. 1-3 we ofthemeansquaredeviationfromtheresonancefrequencyas
showtwoexamplesnottoofarawayfromtheisotropicpoint, ameasureofthelinewidthisthatitisdefinedindependentlyof
namely δ = 0.1 in the critical phase and δ =0.25 in the thelineshape. Inprincipleitshouldbenoproblemtoextractit
−
massivephase.Valuesnottoofarawayfromtheisotropicpoint fromexperimentaldata. Yet,weexpectthatincases,wherethe
aremostrelevantforrealmaterials. Inbothcasesweobserve contributionsfromthetailsofthespectrallineareimportant,a
anincreaseoftheresonanceshiftδω andabroadeningofthe problemmightbetoresolvethesetailsfromthe‘background’.
spectrallines,measuredasanincreaseof∆ω,fordecreasing
temperatures. Thedifferencebetweendifferentmeasuresofthelinewidth
This is interesting as it seems to contradict experimental becomesratherclearfromournumericalanalysisbelow(com-
results4whichclaimanarrowing. Thisdiscrepancyisdueto pare also Ref. 7). For high temperatures, where we could
thedifferentmeasuresforthelinewidthhereandintheexperi- extract a model for the line width from our numerical data,
mentalliterature. ThemeansquaredeviationusedinFigs.1-3 the‘peak-to-peakwidth’ismuchsmallerthan∆ω. Thiscan
isacustomarymeasureforthewidthofwavefunctionsinquan- beattributedtotheshallowtailsoftheabsorbedintensity. In
tummechanics. ForGaussiansitisoftheorderofmagnitude experiments these tails may be misinterpreted as stemming
ofanintuitivelinewidthdrawnbyeye,butfordistributions fromcouplingsofthespinchaintootherdegreesoffreedom
whichhavelongandshallowtailsthisisnolongerthecase. andmayleadtoanoverestimationofthebackground. Onthe
Hence,∆ω maystronglydeviatefromatypicalmeasureofthe other hand, tails are expected to have less influence on the
linewidthusedintheinterpretationofexperimentaldataas resonanceshift. Aslongastheyarenottooasymmetricthe
5
0.30
0.10 +L=16 h/J =0.1
h/J =0.3
L=24
× 0.20 h/J =0.5
h/J =0.7
J J
/ /
ω ω
δ 0.05 δ
0.10
0.00 0.00
2.0
h/J =0.1
+L=16 T/J =0.01 2.0 h/J =0.3
L=24 T/J =0.30
1.5 × T/J =1.00 h/J =0.5
1.5 h/J =0.7
J T/J =3.00 J
/ /
ω 1.0 T/J =100. ω
∆ ∆ 1.0
0.5 0.5
0.0 0.0
0.0 0.5 1.0 1.5 2.0 10 2 10 1 100 101 102
− −
h/J T/J
FIG.1. Resonanceshiftδω/Jandlinewidth∆ω/Jinthecritical FIG.3. Resonanceshiftδω/Jandlinewidth∆ω/Jinthemassive
regimeatδ = 0.1asfunctionofthemagneticfield.Crossesfrom regimeatδ =0.25asfunctionofthetemperature.
−
fullynumericalcalculationforfinitechainHamiltoniansof16and24
sites.
inexperiments.
Forothervaluesoftheanisotropyparameterstheresonance
0.10 +L=16 shiftandthelinewidthinthecriticalregimefor, 1 δ <0,
L=24 showaqualitativelysimilarbehaviorasinFigs.1−and≤2. Both,
×
δω and ∆ω, increase with decreasing temperature at fixed
J magneticfield. Notethatinthemassiveregime,exemplified
/
ω 0.05 withFig.3,theresonanceshiftmaybehavenon-monotonically
δ h/J =0.1
asafunctionoftemperature.
h/J =0.2
h/J =0.3
h/J =0.4
0.00 B. Frequency-dependentmoments
1.5
In order to obtain the resonance shift and the line width
+L=16
L=24 asdefinedintheprevioussectionexperimentallyonewould
×
have to measure the microwave absorption at fixed Zeeman
1.0
J fieldhforvariousvaluesofthefrequencyω andthencalculate
/
ω the required averages as integrals over ω. In current ESR
∆
0.5 experimentsdifferentdatasetsarerecorded. Themicrowave
frequencyω iskeptfixedandtheabsorbedintensityI(ω,h)=
ωχ (ω,h)/2isdeterminedasafunctionofh. Thisintensity
+(cid:48)(cid:48)
0.0 funct−ioncanbenormalizedbydividingbyitsh-integral,and
10−2 10−1 100 101 102 thecorrespondingfrequency-dependingmomentsdefinethe
T/J resonance shift and line width in ‘h-direction’. Away from
theisotropicpoint(δ =0),whereχ (ω,h)issymmetricand
+(cid:48)(cid:48)
FIG.2. Resonanceshiftδω/Jandlinewidth∆ω/Jinthecritical theabsorptionlineisextremelynarr−ow,thesemayberather
regimeatδ= 0.1asfunctionofthetemperature.Crossesfromfully unrelatedtotheirfield-dependentcounterpartsoftheprevious
−
numericalcalculationforfinitechainHamiltoniansof16and24sites. section.
Inanalogywith(5)wedefinethefrequency-dependentmo-
ments
shiftδω oftheaverageoftheabsorbedintensityshouldagree ∞ dh
M (T,ω)=J n (h ω)nχ (ω,h). (12)
withtheshiftofitsmaximum,whichisthecommonmeasure n − (cid:90)−∞2π − +(cid:48)(cid:48)−
6
Thesecanbeexpressedintermsofthem andtheirderivatives. whereω isthemicrowavefrequency. Thisformulaprovides
n
Denotingthekthderivativewithrespecttothesecondargument asimplemeanstodirectlymeasuretheanisotropyparameter
byasuperscript(k)weobtaintherepresentation δ. ForT ∞itturnsintoEq.(10)ofRef.9uponaproper
→
identificationofparameters.
Mn(T,ω)=(−1)n∑∞ (−kJ!)km(kk+)n(T,ω), (13) iniItnialtlhyisslciagshetlaysinwcerlelatshinegmforommenittsumin-fibnaistee-dtelminpeewraitdutrhe∆lihmiist
k=0
δ /√2 when the temperature is reduced. But our numeri-
for the frequency-dependent moments. This representation | |
caldata(seecrossesinFig.4)andthesecondordertermof
involves static correlation functions for arbitrarily large dis-
the high-temperature expansion show that ∆h behaves non-
tances. For this reason the M cannot be calculated by our
n monotonically and decreases again for temperatures lower
exactmethodabove. Yet,incertaincasesfinitelymanyterms
thanJ. Forsmalltemperaturesitseemstoapproachzerolin-
oftheseriesaresufficientforagoodapproximation.
early. Thelattertypeofbehaviorisinaccordancewithfield
Wefirstofallexpresstheresonanceshiftδh=(cid:104)h(cid:105)−ω and theoreticalpredictions5andexperimentalresults.3,4Wewould
themeansquaredeviationfromthecenteroftheabsorption
liketostress,however,thatthereisnocontradictionbetween
peak∆h2= h2 h 2intermsoftheM ,
(cid:104) (cid:105)−(cid:104) (cid:105) n thebroadeningshownbytheuppercurveinthefirstpanelof
δh M ∆h2 M M2 Fig.4andthenarrowingshownbythelowercurve. Infact,
= 1, = 2 1 . (14) bothcurveswereobtainedfromthesamenumericaldatasetfor
J M0 J2 M0 −M02 thedynamicalsusceptibility. Theuppercurvewascalculated
Thereareatleasttwocases,wheretheseformulaesimplifyand with(7),whereasthelowercurvewascalculatedbymeansof
finitelymanyofthem areenoughtodetermineasystematic (14). What is important is that the upper curve can be com-
n
approximationtoδhand∆h. paredwithourexactresults(solidlineintheupperpanel). The
The equation for the resonance shift simplifies for small goodagreementofthecrosseswiththeexactcurvecreatescon-
anisotropy, δ 1. Since M =m +O(δ), M = m + fidenceinournumericaldata. Itshowsthattheyarereliable
0 0 1 1
O(δ2)and,g|en|e(cid:28)rically,m itselfisoforderδ (see(10)−,(13)) whenusedinintegrations. Thisisanon-trivialstatement,since
1
weobtaintolinearorderinδ ournumericaldatahappentobenoisyandfinite-sizeaffected
atlowtemperatures(seeSec.VE).
δh m
= 1. (15) Wewouldliketopointoutthattheresonanceshiftδω/J
J −m0 orδh/Jandthelinewidth∆ω/Jor∆h/Jdefinedintermsof
Inpreviouswork2,8thesameequationwasobtainedbyamore momentsshowasimplescalingbehavior. Theydependonthe
intuitivereasoning. Itleadstoresultswhichcomparerather exchangeinteractiononlythroughtheratiosT/Jandh/J. In
wellwithexperiments.8However,somecareisnecessarywith thissensethecurvesinFigs.1-4areuniversal.
theinterpretationof(15). Sincem /δ vanishesatδ =h=0, Themethodofmomentsisnotonlyusefulfortheintegrable
1
itfollowsthatm =δ(ah+bδ+...)withsomecoefficients Heisenberg-Isingchain. Itmaybeappliedtonon-integrable
1
a,b,whenceh/J mustbelargecomparedtoδ for(15)tobe spinchainsandtotwo-andthree-dimensionalmodelsaswell.
applicable. Thefield-dependentmomentsandthecorrespondingshiftsand
Recall that the higher moments m , n 2, are of order widths may be accurately calculated by approximate meth-
n
δ2. Hence, for the line width there is no≥simplification for ods,sincetheyaredeterminedbystaticshort-rangecorrelation
smallanisotropy,likein(15). Butthereisanothermeasurable functions. For a discussion of the numerical calculation of
quantitywhichdoesallowforasystematicsmall-δ expansion the moments in one dimension see below. In any case, the
tofirstorder,namelytheintegratedintensityπωM ,since frequency-dependentmomentsarehardertoobtain,sincethey
0
requirethecalculationofaninfinitenumberofstaticcorrela-
M0=m0−Jm(cid:48)1. (16) tionfunctions.
Fortheresonanceshiftitfollowstolinearorderinδ from(6)
and(15)thatδh(T,ω)= δω(T,h) . Forthelinewidth
h=ω
thereisnosuchsimplerel−ationbetwe|en∆ω and∆h,noteven C. Integratedintensity
forsmallδ.
Therepresentation(13)isaseriesinascendingpowersof Animportantquantityinexperimentsofelectronspinreso-
J/T (withstilltemperature-dependentcoefficients). Thiscan nanceistheintegratedintensity.21 Asinthedefinitionofthe
beusedtoevaluate(14)asymptoticallyforhightemperatures. momentswemayeitherintegrateoverthefrequencyorover
It turns out that the leading terms in the J/T expansion of themagneticfield. Forfixedmagneticfieldhourdefinition(5)
m1 and Jm(cid:48)2 cancel each other (δh∼ 2hTδ →0 in the high- ofthemomentsm0andm1impliesthat
temperaturelimitT J)and
(cid:29) ∞ ω
dω χ (ω,h)=π(Jm +hm ). (18)
∆h δ (1+δ)J 2 +(cid:48)(cid:48) 1 0
= | | 1+ (cid:90)−∞ −
J √2 4T
(cid:18) Asexplainedintheprevioussection,inusualESRexperi-
(6δ2+10δ+9)J2+4ω2
+... , (17) mentstheabsorbedintensityismeasuredasfunctionofhfor
− 32T2 fixedmicrowavefrequencyω. Bydefinitionthecorresponding
(cid:19)
7
Intheremainderofthisworkweshalltrytodrawatleasta
L=24
× qualitativepictureofhowthelinesareshapedbyconsidering
0.4
allavailablelimitingcases,whereexactresultsareknown,and
J
/ bycomplementingthesewithnumericaldata. Inthissection
ω
∆ ω integration(h/J =0.4) wereviewthoselimitingcaseswhereexactresultsareknown.
d
n Inthefollowingsectionwedescribeournumericalcalculations.
a
J 0.2 Finally,inSec.V,weshallconsiderthelineshapesforδ >0
/
h andT =h=0intwo-spinonapproximation.
∆
hintegration(ω/J =0.4)
0.0 A. Heisenberglimit
0.0 1.0 2.0 3.0
The simplest case where we know the line shape exactly
istheisotropiccaseδ =0.5 ItmaybecalledtheHeisenberg
0.09 high-T expansions: L=24
× limit of the Heisenberg-Ising chain. In this case H is still a
√|δ2| 1+(1+4Tδ)J ω/J =0.4 complicatedmanybodyHamiltonian,butS+commuteswithH
∆h/J 0.08 √|δ2| (cid:0)(cid:0)1+(1+4Tδ)J(cid:1)−(6δ2+103δ2+T92)J2+4ω2(cid:1) Tanhdusth,eS+ti(mt)e=evoe−luihtitoSn+o,fanSd+isdrivenbySzalone(seeApp.A).
I(ω,h)=πδ(ω h)hm(T,h). (21)
−
Thismeansthatthereisasinglesharppeak,andtheabsorbed
0.07 intensityisproportionaltothemagneticenergyhm(T,h)per
latticesite. Thiscaseincludesthefamiliarparamagneticreso-
1 3 5 7 9
nance(Zeemaneffect)forwhichthemagnetizationisknown
T/J explicitly,namelym(T,h)= 1th h forJ=0. Inthegeneral
2 2T
casethemagnetizationmustbecalculatedfromsolutionsof
FIG.4. Linewidths∆ω/J (green)and∆h/J (red)inthecritical linearandnon-linearintegraleq(cid:0)uati(cid:1)ons.22 Inourcontextwe
rneugmimereicaatlδc=alc−ul0a.t1ioansffourncatifionnisteocfhthaeinteHmapmeirlattounriea.nDoafta24frosimtesa.fTulhlye infer from (9) and (11) that m(T,h)=m0=−12ϕ(0), i.e. the
lowestmomentm alonecharacterizesthelineshape.
blackcrossinthelowerpanelmarkstheinfinite-temperaturelimit 0
δ /√2. Thesolidorangelinesarehigh-temperatureexpansionsof
| |
∆h/Jaccordingto(17)uptofirstandsecondorderinJ/T.
B. Isinglimit
Theonlyotherlimitingcaseinwhichthelineshapeisknown
integratedintensityis foralltemperaturesandmagneticfieldsistheIsinglimit.23For
∞ ω theIsinglimitwereplacetheHeisenberg-IsingHamiltonian
I+(int)(ω)= dh 2χ+(cid:48)(cid:48) (ω,h)=πωM0(T,ω). (19) H H/δ andsendδ ∞. Then
− (cid:90)−∞ − → →
L
Intheparamagneticregime,wherehislargecomparedtoJ,
H H =J∑sz sz. (22)
the integrated intensity is proportional to the magnetization → I j 1 j
j=1 −
m(T,ω). The high-temperature expansion of the frequency-
dependentmomentM0, Dδunemto(f8o)rtδhisim∞p.liesthatthemomentsarereplacedasmn→
− n
M (T,ω)= h + Jh (δ 1)+..., (20) In the Isin→g limit the time evolution of S+ in (2) can be
0 4T 8T2 − calculatedexplicitly. AsweshowinApp.A6thisleadstothe
followingfrom(13),providesanothermeanstodeterminethe formula
anisotropyδ fromexperimentaldata.
1 1
χ (ω,h)= (m m )δ(ω h+J)
2π +(cid:48)(cid:48)− 2 2− 1 −
1
III. EXACTLINESHAPES +(m m )δ(ω h)+ (m +m )δ(ω h J) (23)
0 2 2 1
− − 2 − −
Aswehaveseenthemethodofmomentsallowsus,atleast forthedynamicalsusceptibility.
inthefield-dependentcase,toobtainexactcharacterizationsof Using this formula we can calculate the moments mn by
theresonanceshiftandthelinewidthforarbitrarytemperatures, meansof(5)andverifyitsconsistency. Integratingoverω we
magneticfieldsandanisotropyparametersfortheinfinitechain. obtainindeedm0. Theintegralsforthehighermomentsyield
Unfortunately,itdoesnotteachusmuchabouttheactualline
m ifn Nisodd
sthheapliense.cEovmenprtihseesm,roesmtealienmsgenentaerryalqlyueusntiaonns,wheorwedm. anypeaks mn=(cid:40)m21 ifn∈Niseven, (24)
∈
8
i.e. thereareno‘newmoments’forn>2. Thethreeindepen- T/J =0.1 T/J =5
dentmomentsm ,m andm correspondtothethreeδ-peaks 9
0 1 2 0.4
in the dynamical susceptibility. They can be calculated by
6
meansofthe2 2transfermatrix24oftheIsingchain.Explicit
× 0.2
expressionsareshowninApp.A6. 3
TheIsinglimitisimportantforus,sinceitiseasytointer-
0 0.0
pret and since itprovides aphysicalpicture for the massive
phase. TheeigenstatesoftheIsingchainHamiltonianareten- T/J =0.5 T/J =10
sorproductsoflocalszeigenstates. Weinferfromthespectral 6
0.2
representation, App. A7, that transitions can occur only be-
4
tweenstateswhichdifferbyasingleflippedspin. Thus,the
0.1
followingtransitionsinwhichthechainabsorbstheenergy∆E 2
arepossible:
0 0.0
, ∆E=J h T/J =1.0 T/J =50
···↑↓↑···→···↑↑↑··· − 3
, ∆E=J+h 0.04
···↓↑↓···→···↓↓↓···
, ∆E=h 2
···↑↑↓···→···↑↓↓···
0.02
1
Thefirsttwocorrespondtothecreationofapairofdomain
walls(orspinons)inoneoftheNe´elgroundstates. Thethird
0 0.00
oneisimpossibleinthegroundstates,whencethecoefficient -1 0 1 2 3 4 -1 0 1 2 3 4
m m infrontofthecorrespondingtermin(23)mustvanish
0 2 h/J h/J
−
atzerotemperature.
Atfiniteδ >0theoperatorS−inthespectralrepresentation FIG.5. DynamicalsusceptibilityJχ+(cid:48)(cid:48) from(23)asfunctionofh/J
(A35)doesnolongerinducetransitionsbetweeneigenstates. forω=J. Theδ-peaksareconvolve−dwithaLorentzian(25)with
Thethreeδ-peaksin(23)broadenwhichreflectstheonsetof parameterε=0.12J.
interactionsbetweenthespinons. Still,webelievethatthree
peaksarecharacteristicofthemassivephase,δ >0,atleast
at not too large magnetic fields. This is in accordance with
ournumericaldataandwithpreviousnumericalwork.6,7The Ifweuse(23)andtheformulae(A32)forthemomentsm1
relativeheight ofthe three peaksis stillapproximately well andm2,wecanexplicitlyperformtheinfinite-temperaturelimit
describedbytherelativeprefactorsoftheδ-peaksin(23). At andobtainthefunctionφ(ω h)definedinthenextsection
−
T =0inthetwo-spinonapproximation(seebelow)thecentral in(26)and(27). Wefindthatitsδ-peaksareweightedby1/2
peak vanishes due to the same intuitive argument as given forthecentralpeakandby1/4forthetwoside-peaks.
above.
Figure5showsthedynamicalsusceptibility(23)intheIsing
limitasafunctionofh/J. Wevisualizetheδ-peaksbycon- C. High-temperaturelimit
volvingthemwithaLorentzianoftheform
Intheinfinite-temperaturelimitthedynamicalsusceptibility
J ε
L(h)= π h2+ε2. (25) Eχ+q(cid:48)(cid:48).−(Ava3n7i)s.heFsroidmenthtiecaslalym.eTehqiusaftioolnlowansd, fforormintshtaenscuem, frroumle
(A38)weobtaintheleadinghigh-temperaturecontributionto
Whileinthelow-temperaturelimit(T/J=0.1)therightpeak
χ ,
at h=ω+J is the highest, in the intermediate-temperature +(cid:48)(cid:48)
−
regime (T/J =0.5,1.0,5) the relative height of the central
ωπ
peakath=ω increasesrapidlywithincreasingtemperatures. χ (ω,h)= φ(ω h)+O(T 2), (26)
For even higher temperatures (T/J =5,10,50) the relative +(cid:48)(cid:48)− 2T − −
height of the left peak at ω =h J increases as well and where
−
reaches values comparable to those of the right peak. The
relativeheightsofthecentralpeakandtherightpeakarecom- 2 L ∞
φ(ω)= − dteiωttr (eitadHS+)S . (27)
pRaetfi.b6le.wTihthetahbesonluumteevriaclauledoaftathfeorhδeig=h1t osfhothwencienntFriagl.p8eaokf πL (cid:90)−∞ −
(cid:8) (cid:9)
differsapproximatelybyafactorof8,becausetheauthorsof Thefunctionφ iseven,non-negativeandnormalized. Hence,
Ref.6plotthefunctionχ =(χ +χ )/4insteadofJχ itmaybeinterpretedagainasadistributionfunction.
x(cid:48)(cid:48)x +(cid:48)(cid:48) (cid:48)(cid:48)+ +(cid:48)(cid:48)
andscaleitwithafactorof1+δ−=2−. Concerningtheloca−- The function φ is simpler as compared to χ . Still, in
+(cid:48)(cid:48)
tionofthethreepeaksaswellastherelativeandtheabsolute general,weareunabletocalculateitexactly. In−App.Cwe
heights of the central peak and the right peak, the curves in commentonthesmall-t expansionoftheintegrand,whichwe
Fig. 8 of Ref. 6 can be qualitatively explained by the exact havecalculateduptotheordert38,anddrawsomeconclusions.
result(23)forthedynamicalsusceptibilityintheIsinglimit. InthefreeFermioncaseδ = 1itispossibletocalculateitto
−
9
allorders. ThetermssumuptoaGaussian,25and δ= 1.0
−
100 0.6
e (ω/J)2
φ(ω)= − . (28)
J√π
10 2 0.3
−
From this explicit result we can calculate the field- and
frequency-dependentlinewidthsoftheprevioussection,
10 4 0.0
−
∆ω2 3/2+2(h/J)4 ∆h 1 -2 0 2 -2 0 2
= , = . (29)
J2 1+2(h/J)2 2 J √2 δ= 0.7
−
100
Thesecondequatio(cid:0)nisinaccord(cid:1)ancewiththehigh-tempera-
1.0
tureresult(17). Ourexplicitexampleclearlyshowsthatχ
+(cid:48)(cid:48)
isasymmetricinhandω andthatthetwolinewidths∆ω an−d 10−2 0.5
∆hareratherdifferentquantities.
Welearnfromtheabovediscussionthat2Tχ+(cid:48)(cid:48) (ω,h)/πω 10 4 0.0
is a ‘good function’. It converges to a normaliz−ed function −
-2 0 2 -2 0 2
which depends only on the difference ω h for T ∞. In
the free Fermion case δ = 1 and in the−isotropic→case this δ= 0.4
functionhasasinglepeakat−ω =h. Fromournumericaldata − 3.0
(see Fig. 6) we see that this seems to be true for all values 100
of δ between 1 and 0 and even for small positive δ. The
1.5
Gaussiandeca−yforlargeω seemstobepeculiarofthefree 10−2
Fermionpoint. Atallvaluesofδ whicharelargerthan 1our
−
logarithmicplotsinFig.6indicateanexponentialdecay. 10 4 0.0
−
Whenlookingforasimplemodelforsuchtypeoflineshape
-2 0 2 -2 0 2
wefoundtheso-called‘normal-inverseGaussian’,
δ= 0.1
−
αβeαβK (α x2+β2)
N(xα,β)= 1 , α,β >0, (30) 101 40
| π x2+(cid:112)β2
10 1
−
whereK isamodifiedB(cid:112)esselfunction. ItbecomesaGaussian 20
1 10 3
−
inthelimitα ∞,aLorentzianforα 0andaδ-function
for β 0. It→s moments can be easil→y calculated from its 10−5 0
→
characteristicfunction -2 0 2 -0.1 0 0.1
∞
N(kα,β)= dω eikxN(xα,β)=eβ(α √k2+α2). (31) δ=0.25
−
| (cid:90)−∞ | 8
For(cid:101)instance, 100
x2 N= β , x4 N=3 β2 + β . (32) 10−2 4
(cid:104) (cid:105) α (cid:104) (cid:105) α2 α3
(cid:18) (cid:19) 10 4 0
−
Thiscanbecomparedwiththedimensionlessmomentsof
-2 0 2 -2 0 2
thedistributionfunctionφ,whichfollowfrom(5)and(26),
h/J h/J
ω2n 4T
φ
(cid:104) (cid:105) = lim m (T,h). (33)
J2n T ∞ J 2n−1 FIG.6. Normal-inverseGaussian(blacklines)asamodelforthe
→ high-temperaturelineshape,comparisonwithnumericaldata(red
Heretherighthandsidecanbeeasilycalculated.Ifwedemand
lines). Parametersofthenormal-inverseGaussianascalculatedin
that (35). AllpanelsshowJχ (ω,h)asafunctionofh/J,leftpanels
+(cid:48)(cid:48)
logarithmicscale,rightpane−lslinearscale.Notethedifferentscaleon
ω2 δ2
(cid:104)x2(cid:105)N= (cid:104) J2(cid:105)φ = 2 , (34a) tdhaetaxw-aexriesoibntathineerdigfohrtTpa/nJe=lfo10r0δ,=ω/−J0=.10..I4n,Lge=ne1r6a,lathnednMum=e1r0ic2a4l
ω4 δ2 3 (seebelow).Forδ = 0.1thechainlengthwasincreasedtoL=20
(cid:104)x4(cid:105)N= (cid:104) J4(cid:105)φ = 2 2+δ+δ2 , (34b) andtheresolutiontoM−=4096.
(cid:18) (cid:19)
weobtain
6 δ2 6 Figure6comparesthenormal-inverseGaussianwithparame-
α = , β = . (35)
(cid:115)(1+δ)(3 δ) 2 (cid:115)(1+δ)(3 δ) ters(35)withournumericalhigh-temperaturelineshapes.
− −
10
Forδ = 1themodellineshapeisexact. Thisisnolonger theresonanceshiftbasedonthedataofRef.26.Theauthorsof
−
thecaseforδ > 1whichcanbeseenbycomparingthesixth Ref.27obtainforthesamematerialasinRef.26(LiCuVO )
4
normalizedmom−entofφ withthesixthmomentofN. Still, theexchangeintegralJ=30K. FromFig.4thereinwecan
wefinditremarkablehowwellthemodellineshapefitsour readoff∆ h=1.5kOe,andobtaintogetherwithEq.(39)the
pp
numericaldata. Especiallytheexponentialtailsvisibleinthe anisotropy δ 0.088 which is compatible with the value
≈−
leftpanelsofFig.6havenotbeenfittedtothenumericaldata. J /J 2K/30K 0.067ofRef.27.
zz
≈− ≈−
Thegoodagreementcomesoutautomatically. Ontheother
hand,thedeviationofthecenterofthepeakintherightpanel
forδ = 0.1inFig.6doesnotseemtobeduetoaresolution IV. NUMERICALLINESHAPES
−
problemofournumericalcalculation. Weratherattributeit
toaslightmismatchofthenormalinverseGaussianatsmall The numerical approach we are using for the calculation
anisotropy. ofthedynamicalsusceptibility χ (ω,h),Eq.(2),hasbeen
Ifwefixtheparametersα andβ ofN(xα,β)accordingto described in detail elsewhere.28 W+(cid:48)(cid:48)−e will therefore give only
|
(35),weseefrom(34a)thatthewidth,calculatedbyitssecond
ashortoutlineofthemethodanddiscussafewspecialtricks
moment,behavesas x2 N δ . Ontheotherhand,itis beneficialforthepresentproject.
(cid:104) (cid:105) ∼| |
easytocalculatethepeak-to-peakwidthofthenormal-inverse Startingpointofthenumericsforfinitechainsisthespectral
(cid:112)
Gaussian, which is the distance of its two inflection points. representation(seealsoApp.A7),
Settingthesecondderivative∂2N(xα,β)tozeroandusing
x
thedifferentialequationdefiningK1,| χ (ω,h)= π ∑ e En/T e Em/T
+(cid:48)(cid:48)− LZm,n − − −
y2K1(cid:48)(cid:48)(y)=(y2+1)K1(y)−yK1(cid:48)(y), y=α x2+β2, (36) m(cid:0) S− n 2δ(ω E(cid:1)m+En) (41)
×|(cid:104) | | (cid:105)| −
weseethatthepositiveinflectionpointy isth(cid:112)esolutionofthe π ∞
0 = dys(y+ω,y) e y/T e (y+ω)/T ,
equation LZ(cid:90)−∞ − − −
(cid:0) (cid:1)
1 y(y2 (αβ)2) which can be written as an integral over thermal weighting
y+ 3y2 −4(αβ)2 =∂yln(K1(y)). (37) factorsandthetemperature-independentfunction
−
In principle this algebraic equation can be solved numeri- s(x,y)=∑ mS− n 2δ(x Em)δ(y En). (42)
|(cid:104) | | (cid:105)| − −
cally, but for our purposes the following argument leading m,n
to an estimation for the inflection point is sufficient. Since
At first sight the calculation of this function seems to
∂ ln(K (y))< 1forally 0andtheleft-handsidebecomes
y 1 − ≥ require knowledge of all eigenvectors and eigenvalues of
largeandpositivefory 0,∞,thesolutiony hastobelocated
→ 0 the Heisenberg-Ising Hamiltonian on a finite lattice. How-
closetotheleftofthepole. Hence,weobtain
ever, it is significantly more efficient to rescale all energies
E E=aE+b,suchthatE [ 1,1],andtoexpands(x,y)
4 β
y0<∼(cid:114)3αβ ⇒ x0<∼ √3. (38) in→term(cid:101)sofChebyshevpolyn(cid:101)om∈ia−lsofthefirstkindTi, (cid:101)(cid:101)
Anupperlimitofthepeak-to-peakwidthisthereforegivenby s(x,y)=M∑−1 µijgigj(2−δi0)(2−δj0)Ti(x)Tj(y). (43)
π2 (1 x2)(1 y2)
i,j=0
− −
∆ x 2 (cid:101) (cid:101)
pp =2x <δ2 δ2. (39) (cid:101)(cid:101) (cid:112)
0 Theproblemthenreducestothecalculationoftheexpansion
J ∼ (cid:115)(1+δ)(3+δ) ∼ (cid:101) (cid:101)
coefficientsµ whicharegivenbytraces,
ij
Accordingly, the normal-inverse Gaussian is an example of
1
a distribution for which the width calculated by its second µ = dxdys(x,y)T(x)T (y)=tr[S+T(H)S T (H)].
ij i j i − j
moment ( δ ) and the peak-to-peak width ( δ2) behave 1
∼| | ∼ (cid:90)(cid:90)− (44)
asymptotically differently for small anisotropies δ and can
InsteadofsummingoverthewholeHilbertspa(cid:101)ce,these(cid:101)traces
thereforedifferstronglyinvalue. Fittingexperimentaldataat
arewellapproximatedbyaveragesoverafewrandomstates.
hightemperaturestoanormal-inverseGaussian,thisoffersa
waytodetermineδ independentlyoftheexchangeintegralJ TheactionofTk(H)onanarbitrarystatecanbequicklyeval-
fromtheratioofthetwolinewidths, uatedwith therecursion relationsof theChebyshevpolyno-
mials. Taking int(cid:101)o account symmetries of the Hamiltonian
∆ppx 2δ (Sz-conservationandtranslation),itisthusfeasibletocalculate
| | . (40)
x2 N ≈ (1+δ)(3+δ) theµij forsystemsofuptoL=32latticesitesandexpansion
(cid:104) (cid:105) ordersuptoM=4096onaveragehardware.
Ifweestimat(cid:112)ethepeak-t(cid:112)o-peakwidthofFig.6ofRef.26 Oncewehaveacompletesetofexpansioncoefficientsµ
ij
to∆ h 2kOe=0.27K,wecansolve(39)numerically(J= foragivenlatticesizeL,anisotropyδ andallSz sectors,we
pp
≈
22K)andobtainδ 0.12. Thisvalueiscompatiblewiththe obtain s(x,y) from Eq. (43) using fast Fourier methods. In
predictionofMaeda≈e−tal.8(δ = 0.15)obtainedfromafitfor Eq. (43), the damping factors g cure the Gibbs oscillations
(cid:98) − k