We present a rigorous result and strong evidence to support existence of antiferromagnetic long-range order on a square lattice for the S = 1/2 antiferromagnetic Heisenberg model.published_or_final_versio

Shen, SQ

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Title Antiferromagnetism in the S=1/2 antiferromagnetic Heisenbergmodel on a two-dimensional square latticeAuthor(s) Shen, SQCitation Physical Review B - Condensed Matter And Materials Physics,2000, v. 62 n. 21, p. 13824-13827Issued Date 2000URL http://hdl.handle.net/10722/43322Rights Creative Commons: Attribution 3.0 Hong Kong LicensePHYSICAL REVIEW B 1 DECEMBER 2000-IVOLUME 62, NUMBER 21Antiferromagnetism in the S˜ 12 antiferromagnetic Heisenberg modelon a two-dimensional square latticeShun-Qing ShenDepartment of Physics, The University of Hong Kong, Pokfulam Road, Hong Kong, Chinaand Institute of Physics, Chinese Academy of Sciences, P.O. Box 603, Beijing 100080, China~Received 4 April 2000; revised manuscript received 12 June 2000!.We present a rigorous result and strong evidence to support existence of antiferromagnetic long-range orderon a square lattice for the S5 12 antiferromagnetic Heisenberg model.Antiferromagnetism in the S5 12 Heisenberg model on asquare lattice has stood as a challenging subject for theoret-ical physicists for a very long time. Rigorous proof of theexistence of antiferromagnetic ~AF! long-range order ~LRO!was given for S>1 by Dyson, Lieb, and Simon.1 It wasgeneralized for S51/2 on a cubic lattice2 and for some XXYmodels in two and three dimensions.3,4 We now know thatAFLRO exists in three dimensions for all spins and in twodimensions for S.1/2. Although there is no rigorous prooffor S51/2 on a square lattice, various numerical and analyti-cal calculations5 suggest that AFLRO exists in the groundstate, which is destroyed by thermodynamic fluctuations atfinite temperatures.6 Another method to prove the existenceof AFLRO was proposed by the author and his co-workers7for some asymmetric bipartite lattices. AFLRO exists even insome one-dimensional systems.8 In this Brief Report, wepresent a rigorous example for AFLRO on a square latticeand a relation for the AF correlation function. Our resultstrongly supports the existence of AFLRO on a square lat-tice.Our starting point is to introduce the model Hamiltonianon a square lattice,H~a!5J (iPA , jPB ,DSiSj1aJ (iPC , jPB ,DSiSj5HA1aHC ,~1!where (i , j) are pairs of the nearest-neighbor sites. a>0 isthe model parameter, and the model is reduced to the usualantiferromagnetic Heisenberg model which we are interestedin when a51. Here J is positive for the antiferromagneticexchange coupling, and for convenience, we take J51 as theunit of energy. We divide the square lattice into four sublat-tice as shown in Fig. 1. Denote the ground state of H(a) byua&. To study AF long-range correlation in the state ua&, weintroduce two correlation functionsFAF~a!5^auO†Oua& ,whereO51ANL(iPLeiQriSi51ANL~SA1SC2SB2SD!andPRB 620163-1829/2000/62~21!/13824~4!/$15.00FNNAF~a!51NL^au(^i , j&eiQ(ri2rj)SiSjua&,where Q5(p ,p), and ^i , j& run over all possible pairs ofnearest-neighbor sites. NL is the total number of lattice sites.SX (X5A ,B ,C ,D) is the total spin on the sublattice X.Mathematically, ifFAF~a!5gNLwith gÞ0, we say that the state ua& possesses AFLRO in thethermodynamic limit. FNNAF(a) can be used to describe thenearest-neighbor antiferromagnetic ~NNAF! correlation inthe state ua& . As the model is defined on a square lattice, thecorrelation functions are invariant under the transformationA→C , which we name the AC symmetry.The antiferromagnetic Heisenberg model was studied ex-tensively over the last decades. Some rigorous results wereestablished for a bipartite lattice by several authors. The bi-partite lattice means that the lattice is divided into two sub-lattices: only spins on difference sublattice sites have AFexchange coupling. According to this definition, A and Cbelong to one sublattice, and B and D belong to anothersublattice. Here we summarize: ~1! The ground state of themodel is nondegenerate apart from the trivial(2Stot11)-fold degeneracy caused by SU~2! symmetry. Thetotal spin Stot is determined by the difference of the twosublattice site numbers NI and NII , i.e.,9,10FIG. 1. The structure of four sublattices on a square lattice inEq. ~1!. The solid line represents that the exchange coupling be-tween spins on the two sites is 1, and the dashed line represents thatthe exchange coupling is a . According to the definition of the bi-partite lattice structure for the Lieb-Mattis theorem, A and C belongto one sublattice, and B and D belong to another sublattice.13 824 ©2000 The American Physical SocietyPRB 62 13 825BRIEF REPORTSStot512 uNI2NIIu.~2! On a chosen basis, the wave function of the ground stateis positive definite, and the static spin-spin correlationssatisfy7,8eiQ(ri2r j)^auSiSjua&.0. ~2!Proof of Eq. ~2!: According to the Lieb-Mattis theorem onEq. ~1!, the ground state of Eq. ~1! for any finite a is non-degenerate and positive definite on a chosen basis. Usuallythe S5 12 operator can be expressed in terms of fermion op-erators ci ,s (s5↑ ,↓),Si15ci ,↑† ci ,↓ ; Si25ci ,↓† ci ,↑ ; Siz5~ci ,↑† ci ,↑2ci ,↓† ci ,↑!/2.The ground state for Eq. ~1! can be expressed asua&5(lClTuf l↑& ^ uf l↓& ,whereT5)iPI~ci ,↓† 2ci ,↓!)iPII~ci ,↓† 1ci ,↓!and uf ls& is an orthogonal basis for N5NL/2 for even NL@and (NL21)/2 for odd NL# fermions with spin s .8 TheLieb-Mattis theorem tells us that the coefficients Cl are posi-tive for all possible configurations l and unique. A simpleevaluation shows thateiQ(ri2r j)^auSi1Sj2ua&5(l ,mClCmu~Vi j! lmu2.0, ~3!where~Vi j! lm5^f lsuci ,s† c j ,sufms &.On the other hand, since the ground state of Eq. ~1! is a spinsinglet for a square lattice, the isotropy of the ground stateleads to^auSixSjxua&5^auSiySjyua&5^auSizSjzua&. ~4!Combination of Eqs. ~3! and ~4! leads to the inequality of Eq.~2!.Based on the two rigorous results, we have our first resultof this paper:lima→0FAF~a!>NL16 . ~5!In other words, in the limit of a→0, it proves rigorously thatAFLRO exists in the ground state of the model.According to the Marshall-Lieb-Mattis theorem,9,10 whena→0, the total spin on sublattices A, B, and D is S5NL/8,since sublattices B and D belong to the same sublattice, andsublattice A belongs to another one, i.e.,^au~SA1SB1SD!2ua&518 NLS 18 NL11 D ,when a→0. Strictly speaking, the ground state of Eq. ~1! ishighly degenerated when a50 as there are no interactionsbetween spins on sublattice C and other sublattices, andspins on sublattice C can orient arbitrarily, which costs noenergy. However, one of the states is semipositive definiteand is spin singlet. The state can be regarded as the groundstate of the limit of a→0 as the state is positive definite andnondegenerate for any finite a . Thus, the total spin in theground state for the whole square lattice is singlet,^au~SA1SB1SC1SD!2ua&50. ~6!According to the theory of angular momentum coupling, thetotal spin SC must be also SC5NL/8, which is maximal. Onthe other hand, from Eq. ~6!, one obtains22^auSC~SB1SD!ua&5^au~SA1SB1SD!2ua&1^auSC2 ua&12^auSASCua&.According to Eq. ~2!,^auSXSY ua&.0,if X , Y belong to A and C or B and D:^auSXSY ua&,0,if X belongs to A and C, and Y belongs to B and D. Combi-nation of these two inequalities leads to^au~SA2SB2SD!2ua&.^au~SA1SB1SD!2ua& .From these relations, one obtains^au~SA2SB1SC2SD!2ua&5^au~SA2SB2SD!2ua&1^auSC2 ua&12^auSASCua&22^auSC~SB1SD!ua&.2^au~SA1SB1SD!2ua&12^auSC2 ua&54NL8 S NL8 11 D .Thus, in the limit of a→0, FAF(a→0) has order of NL .Both FAF(a) and FNNAF(a) are continuous with respect tothe parameter a . This conclusion is drawn from the fact thatthe ground state of Eq. ~1! is nondegenerate for any finite aas shown in Ref. 10. Any discontinuity of the functions mustbe related to the crossover of the ground state from one stateto another with different good quantum numbers. The cross-over can happen only at some point with degeneracy of theground state. The nondegeneracy of the ground state hasruled out this possibility. Therefore we conclude thatAFLRO exists in the ground state of a→0.We have shown the existence of AFLRO at a→0. Theprevious results from numerical and analytical calculationsstrongly support that the correlation function FAF(a) hasorder of NL for a51. From the continuity of FAF(a) weanticipate that FAF(a) has order of NL for small finite a .However, we encounter some difficulty in generalizing theresult in Eq. ~5! to the case of finite a . Instead of FAF(a),we have a weaker conclusion on FNNAF(a):FNNAF~a! has its maximum at a51.13 826 PRB 62BRIEF REPORTSThe result tells us that the nearest-neighbor antiferromag-netic long-range correlation is strongest at a51, which isthe case we are mostly interested in. This result stronglysupports the existence of AFLRO at a51, as the nearest-neighbor correlation should become dominate for a systemwith the nearest-neighbor interaction.Denote the ground state and ground-state energy of theHamiltonian in Eq. ~1! by ua& and E(a), respectively. As theground state is spin singlet and nondegenerate for any finitea , E(a) must be continuous with respect to a . Due to theAC symmetry, when we exchange the position of A and Csublattice sites, the ground-state energy should remain un-changed. However, asHA1aHC5aS 1a HA1HCD ,we obtain a relationE~a!5aES 1a D .The Hamiltonian can be reorganized asS b1 1b D ~HA1aHC!5bS HA1 ab2 HCD11b~HA1ab2HC!.Multiplying the bra ^au from the left hand side and the ketua& from the right hand side, respectively,S b1 1b D ^au~HA1aHC!ua&5b^auS HA1 ab2 HCD ua&11b^au~HA1ab2HC!ua&.Since ua& may not be the ground state for the HamiltonianHA1(a/b2)HC , from the variational principle, we have^auS HA1 ab2HCD ua&>K ab2US HA1 ab2HCD U ab2L .Therefore,S b1 1b DE~a!>bES ab2D 1 1b E~ab2!.The energy E(a) is continuous with respect to a . We ex-pand E(a/b2) and E(ab2) near b51, and obtaind2da2E~a!<0. ~7!On the other hand, from the definition of FNNAF , we ob-tainFNNAF~a!521NL^au~HA1HC!ua&.The differentiation of ^au(HA1HC)ua& with respect to a isdda ^au~HA1HC!ua&5dda ^au~HA1aHC!ua&1dda @~12a!^auHCua&#5dda E~a!2^auHCua&1~12a!dda ^auHCua&.~8!Furthermore,dda E~a!5^auF dda ~HA1aHC!G ua&5^auHCua&. ~9!The combination of Eqs. ~9! and ~8! leads todda ^au~HA1HC!ua&5~12a!d2da2E~a!. ~10!In other words, by using the inequality ~7!, one obtainsdda FNNAF~a!5H >0 for a,1,50 for a51,<0 for a.1.In fact, from the AC symmetry,FNNAF~a!5FNNAFS 1a D .Therefore we conclude that the nearest-neighbor antiferro-magnetic correlation has its maximum at a51. Notice thefact that FNNAF(a) is always positive. The maximal value ofFNNAF(a) indicates that the nearest-neighbor antiferromag-netic correlation is the strongest at a51. As long-range an-tiferromagnetism exists at the small a limit as we just provedin the preceding paragraphs, we anticipate that the strongershort-range antiferromagnetic correlation will enhance thelong-range correlation, and the long-range correlation for a51 is stronger than that for small a .In short, we present a rigorous example with AFLRO on asquare lattice for small a . By utilizing the symmetry of thesystem and the variational principle, we show that thenearest-neighbor AF correlation becomes the strongest at a51; i.e., it is stronger than the correlation at small a .This work was supported by a RGC grant from HongKong.PRB 62 13 827BRIEF REPORTS1 F. Dyson, E. H. Lieb, and B. Simon, J. Stat. Phys. 18, 335 ~1978!.2 T. Kennedy, E. H. Lieb, and B. S. Shastry, J. Stat. Phys. 53, 1019~1988!.3 T. Kennedy, E. H. Lieb, and B. S. Shastry, Phys. Rev. Lett. 61,2582 ~1988!.4 K. Kubo and T. Kishi, Phys. Rev. Lett. 61, 2585 ~1988!.5 S. D. Liang, Phys. Rev. B 42, 6555 ~1990!; M. Gross, E.Sanchez-Velasco, and E. Siggia, ibid. 39, 2484 ~1989!; J. D.Reger and A. P. Young, ibid. 37, 5978 ~1989!; R. R. P. Singh,ibid. 39, 9760 ~1989!.6 M. D. Mermin and H. Wagner, Phys. Rev. Lett. 17, 1133 ~1966!.7 S. Q. Shen, Z. M. Qiu, and G. S. Tian, Phys. Rev. Lett. 72, 1280~1994!.8 S. Q. Shen, Int. J. Mod. Phys. B 12, 709 ~1998!; Phys. Rev. B 53,14252 ~1996!.9 W. Marshall, Proc. R. Soc. London, Ser. A 232, 48 ~1955!.10 E. Lieb and D. Mattis, J. Math. Phys. 3, 749 ~1962!.

Antiferromagnetism in the S=1/2 antiferromagnetic Heisenberg model on a two-dimensional square lattice

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Antiferromagnetism in the S=1/2 antiferromagnetic Heisenberg model on a two-dimensional square lattice

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