Jordan-Wigner fermions and the spin-12 anisotropic XY model on a square lattice

Jordan-Wignerfermions

arXiv:cond-mat/0111203v1 11 Nov 2001anisotropicXY(XZ)model.Wecompare

theresultsofdifferentapproachesfortheground-stateandthermodynamicpropertiesofthemodel.

2

PACSnumber(s):75.10.-bKeywords:square-latticespin-1

anisotropicXYmodelonaspatiallyanisotropicsquarelatticewithintheframeworkofthescheme

basedonthe2DJordan-Wignerfermionizationandcomparetheresultsderivedfortheground-stateandthermodynamicquantitieswiththeexactones(1Dlimit,square-latticeIsingmodel)andthepredictionsofotherapproximatetheories.Theperformedcalculationsyieldanimpressionabouttheregionofvalidityofsomeapproachesusuallyappliedforastudyofthermodynamicsof2Dquantumspinmodels.

WestartfromamodelofN→∞spins1

2

HereJandJ⊥=RJaretheexchangeinteractionsbetweentheneighbouringsitesinarowandacolumn,respectively(forconcretenessweassumebothtobepositive),andtheparameterγcontrolstheanisotropyoftheexchangeinteraction.Makinguseofthe2DJordan-Wignerfermionizationandadoptingamean-fieldtreatmentofthephasefactorswhichappearafterthefermionization[2,3]weperformconsequentlytheFourierandBogolyubovtransformationstoarriveatthefollowingHamiltonianofnoninteractingspinlessfermionswhichrepresenttheinitialspinmodel(1)

H=

2󰀃′󰀃k

+

Λα(k)ηk,αηk,α−

α=1

󰀁

1

Λ2(k)=

(J⊥cosky+γJcoskx)2+(Jsinkx+γJ⊥sinky)2,

󰀇

2β

󰀅

π

dkx

2π

−π

󰀁󰀁

βΛ1(k)

ln2cosh

2

󰀂󰀂

(3)

yieldsthethermodynamicpropertiesofthespinmodel(1).InFig.1weplottheground-stateenergypersiteofthespinmodel(1),(2)(dottedcurves)incomparisonwiththeexactresultsifR=0(1DXYmodel)orγ=1(square-latticeIsingmodel)andthespin-wavetheoryresultforγ=0,R=1(spatiallyisotropicsquare-latticeisotropicXYmodel).Eq.(3)containstheexactresultin1Dlimit(Fig.1b),however,deviatesnoticeablyfromtheexactresultforγ=1(comparethecurves3inFig.1a).Forγ=0,R=1Eq.(3)yieldstheresultwhichdiffersfromthespin-wavetheorypredictiondenotedbythefullcircles.(Theoutcomesofdifferentnumericalapproaches(see[4])liewithinthefullcircles.)Fromtheexactcalculationforγ=1[5]weknowthatthetemperaturedependenceofthespecificheatexhibitsalogarithmicsingularity.Obviously,theJordan-Wignerfermions(2),(3)cannotreproducethispeculiarityinherentinthespinmodel.

Itisworthtoremindherethattheconventionalspin-wavetheorywasoriginallythoughttobeun-satisfactoryforquantumXYmodels[6].However,theauthorsofthepaper[7]showedthatconsideringtheXZratherthantheXYHamiltonianonegetswithinthespin-wavetheorysatisfactoryresultsofthesamequalityasfortheHeisenbergHamiltonian.Followingthisideaweperformtherotationofthespinaxessx→−sz,sy→sx,sz→−syandconsiderinsteadof(1)thefollowingHamiltonian

∞󰀃∞󰀁󰀆󰀈󰀃

xxzz

H=J(1−γ)si,jsi+1,j+(1+γ)si,jsi+1,j

i=0j=0

+J⊥

󰀆

󰀈󰀂

xzz

(1−γ)sx.i,jsi,j+1+(1+γ)si,jsi,j+1

(4)

Proceedingfurtherwith(4)inthedescribedabovemannerandassuming(forconcreteness)antiferromag-neticlong-rangeorderwhiledecouplingthequarticfermionicterms[8]wegetinsteadof(2)thefollowing

Hamiltonian

󰀁2󰀃′󰀃1+

Λα(k)ηkH=,αηk,α−

kα=1

A2+B2+C2+D2+M2±2

󰀄

44

J⊥cosky,B=

1−γ1−γ

Jsinkx,D=

wheremisdeterminedself-consistentlybyminimizingtheHelmholtzfreeenergypersite

󰀁

1∂Λ1(k)∂Λ2(k)

2(1+γ)(J+J⊥)m=+222(2π)

󰀂

.(6)

InFig.2weplottheground-stateenergyofthespinmodel(4),(5),(6)(dashedcurves).TheresultsbasedonEqs.(5),(6)forγ=1reproducetheexactresultforsquare-latticeIsingmodel(curve3inFig.2a)aswellasthespin-wavetheorypredictionforγ=0,R=1.However,theresultbasedonEqs.(5),(6)forR=0doesnotcoincidewiththeexactonein1Dlimit(curve1inFig.2b).

Tosummarize,wehavecalculatedthethermodynamicquantitiesforthespin-1

anisotropicXYmodel(1)

e0vs.R(a)(1–γ=0,2–γ=0.5,3–γ=1)ande0vs.γ(b)(1–R=0,2–R=0.5,3–R=1);exactresults(solidcurves)andtheapproximateresultsobtainedonthebasisof(2)(dottedcurves);thefullcirclescorrespondtothespin-waveresultforγ=0,R=1.

2

FIGURE2.ThesameasinFig.1fortheXZHamiltonian(4);theapproximateresultsobtainedonthebasisof(5),(6)areshownbydashedcurves.

3

e0

-0.5a1233-1.00.40.8R e0

-0.5b123-1.00.40.8γFigure1:

4

e0

-0.5a123123-1.00.40.8Rb11223e0

-0.53-1.00.40.8γFigure2:

5