Volume61A, number7 PHYSICSLETTERS 27 June1977

LIFSHITZ POINTS OF HIGHER CHARACTER

WalterSELKEFachrichtungTheoretischePhysik,UniversitätdesSaarlandes,6600Saarbrilcken,WestGermany

Received20 April 1977

Lifshitzpoints of higher characterarefound for the sphericalmodelwith multi-neighbourinteractions.TheLifshitz point of characterthreeis a point of intersectionof lines of first and secondordertransitions.

Recently,Hornreichet al. introduceda newcritical point, the Lifshitz point (LP) [1, 2], associatedwith the onsetof helical order in magneticsystems.Amoregeneralsituationhasbeendiscussedby Stanley 2

et al. definingin a formal mannera “Lifshitz point ofcharacterL” [3—51;seealso ref. [6], wherethis pointhasbeencalleda “LP of order(L—1)”.

In this note I shalldiscussthecritical propertiesof ~~TC FERROMAGNETIC

a concretemodelexhibiting LP’s of highercharacter.I shall considerthesphericalmodel [7] with more- 1than-nearestneighbourinteractionsalongm(m~ d)axesof ad-dimensionallattice; theFourier transformof the exchangeintegralsis givenby

~. LPOF2ND ORDER

J(q) = 2(J1�I~cos(q,)+ ~ Ji cos(iqj)), (1) ~j=1

~Jl/J2

the latticeconstantis takento be one. Fig. 1. Phasediagramof thegroundstateof the sphericalmodelwith nearest(Ji > 0 ferromagnetic)nextnearest

Dependingon the valuesofJ. the absolutemaxi- .(~12< 0; antiferromagnetic)and third nearestneighbourmumof J(q)is at q = 0, at q = ir or at q ~�r 0, ir leading (J3 > 0)interactionsdisplayinga Lifshitzpoint of second

to a ferromagnetic,antiferromagneticor helical ground order(character3). The stability boundarybetweenthe hell-

state.ChoosingJ~/J1(i ~ L) suchthat the leading andferromagneticstateis givenby —J3/J2 = (4 + Ji 1.12)/9termsof the Taylorexpansionof (I) are (full line) and —J3/J2 = 1/(4(—J1/J2— 1)) (dashedline).

J(q)= + c2 �~q?L + c3 ~ q? + (2) also ref. [8]. The Lifshitz point of character3 is the11 / m+i ~ ‘ pointof intersectionof a line of “secondordertransi-

tions” (full line; LP’s of character2),wherethehelixa “rn-axial Lifshitz pointof characterL” will be ex- decreasescontinuouslyto zero by goingfrom thehibited; c~are someconstants.Forexample,the helical to the ferromagneticstate,anda line of “first“isotropic” (rn = d) LPcanbe achievedfor .J2/J1 = order transitions”(dashedline),wheretheangleof—1/4; a LP of characterthree(ordertwo) will bedis- thehelix jumps from a finite valueto zeroby crossingplayedforJ1/J2= —5/2, J3/J2 = —1/6; the LPof thestability boundaryto theferromagneticstate.Thuscharacter4 canbe realizedforJ2/J1 = —29/90, J3/J1 thereis a certainsimilarity to a tricritical point.= 1/135andJ4/J1 = 1/72 etc.In fig. 1 thephasedia- Thecritical propertiesfor the sphericalmodel cangram for the groundstateof the sphericalmodelwith be obtainedwithout difficulty. The borderdimensioninteractionsup to thethird neighboursis shown;see d.1. (abovewhich meanfield exponentshold) for a LP

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Volume61A, number7 PHYSICSLETTERS 27June1977

of characterL is thecritical exponentstoo. For isotropicLP’s themodified scalinglaws [9] are fulfilled. Someexponents

(4+rn—m/L for rn<d ared~=~ (3)

~4L for rn=d.y=2L/(2L—d),

(8)Infrared divergencies,i.e. the critical temperature a= —d/(2L — d) ; ~j= 2 — d

Tc = 0, set in for lattice dimensionsd ~ dIf rn <d alterationshaveto be madesimilar to the

(2 + rn — rn/L for m <d onesford <d ~ d~.Onefinds(4) -

12L for rnd. y2L/((2+rn—d)L—rn),

Ford <d ~ d.~.the criticalexponentsfor the iso- a= (—rn + L (m — d))/(m — L(2 + rn — d)) (9)tropic LP’s are relatedto oneanotherby scalinglaws

= —L/(rn — L(2 + rn — d)); 1’m =[5].One finds e.g.

17d = rim = 2 — d.a(d—4L)/(d—2L),

(5) Thebestcandidatesfor systemsdisplayingLP’s of7 = 2L/(d — 2L). highercharactershouldbe alloys [1], wherethe corn-

Forthe rn-axial (rn <d) LP onehasto distinguishthe positionplaysthe role of theexchangeintegrals.correlationsalongthe axeswith more-than-nearestneighbourinteractions(indexm) andthe onesalong I wishto thank I. Peschelfor the usefuldiscussions.the axeswith only nearestneighbourinteractions(indexd); see [1,2,4,5]. This leadsto newscalinglaws for threeindependentexponents;for example References

2 — a= rnvm + (d — m)vd . (6) [1] R.M. Hornreich,M. Lubanand S. Shtrikman,Phys.Rev.

Someexponentsfor the sphericalmodel are Lett. 35 (1975) 1678.[21 R.M. Hornreich,M. Lubanand S. Shtrikman,Phys.Lett.

a= (rn + L(d — 4 — rn))/(rn +L(d — 2 — rn)) 55A (1975) 269.[3] J.F.Nicoll, T.S. ChangandH.E. Stanley,Phys.Rev. A13

7 = 2L/((d —2 — rn)L + rn), (7) (1976) 1251.[4JJ.F.Nicoll, G.F. Tuthill, T.S. ChangandH.E. Stanley,

~d ~m 0; Vd =y/2, vm =y/(2L). Phys.Lett. 58A (1976)1.

Eqs.(3)—(7) agreewith theresultsin refs. [1,2, 5] for [5] J.F.Nicoll, G.F. Tuthill, T.S. ChangandH.E. Stanley,Physica86-88B(1977)618.

rn = d andrn = 1. The critical temperatureTc decreases [61 W. Selke,Z. PhysikB (1977)in press.

with increasingL - Forexample,for d = 3 andrn = 1 171 G.S.Joyce,in: Phasetransitionsandcritical phenomena,onegets Tc i/L for largeL. eds.C. Domb andM.S. Green(AcademicPress,New

Ford ~ d the phasetransitionsfor the models York, 1972)Vol. 2, p. 375.[8] T. Nagamiya,K. NagataandY. Kitano,Prog.Theor.exhibitingLP’s occur at zero temperature.Following Phys.27 (1962) 1253.

the notationof BakerandBonner [9] we cancalculate [9] G.A.Baker andJ.C.Bonner,Phys.Rev. B12 (1975)3741.

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