Long-range order for the spin-1 Heisenberg model with a small antiferromagnetic interaction
LLong-range order for the spin-1 Heisenbergmodel with a small antiferromagnetic interaction
Benjamin Lees ∗ Department of Mathematics, University of Warwick, Coventry, CV4 7AL, United Kingdom
Abstract
We look at the general SU(2) invariant spin-1 Heisenberg model. This familyincludes the well known Heisenberg ferromagnet and antiferromagnet as well asthe interesting nematic (biquadratic) and the largely mysterious staggered-nematicinteraction. Long range order is proved using the method of reflection positivityand infrared bounds on a purely nematic interaction. This is achieved throughthe use of a type of matrix representation of the interaction making clear severalidentities that would not otherwise be noticed. Using the reflection positivity ofthe antiferromagnetic interaction one can then show that the result is maintained ifwe also include an antiferromagnetic interaction that is su ffi ciently small. Showing the existence of phase transitions at low temperatures for Heisenberg modelsis a well known di ffi cult problem. There have been several positive results in this areaover the years in both the classical and quantum cases. The first rigorous proof of aphase transition in a Heisenberg model was the result of Fr¨ohlich, Simon and Spencer[10] for the classical Heisenberg ferromagnet (and hence for the antiferromagnet alsoas it is equivalent to the ferromagnet in the classical case). The result was later ex-tended to the quantum antiferromagnet by Dyson, Lieb and Simon [6]. The case ofspin-1 / ∗ [email protected] a r X i v : . [ m a t h - ph ] O c t he work of Tanaka, Tanaka and Idogaki shows long range order for an antiferromag-netic interaction accompanied by a small enough nematic (biquadratic) interaction indimensions two and three. In dimension three they also show long-range order in partof the nematic region investigated in [22], these results were obtained independently.The aim of this article is to show that there is also a phase transition in a region witha nematic interaction accompanied by a small antiferromagnetic interaction, this resultwas already expected, although an explicit proof has not been presented before. Curi-ously the result only shows the existence of nematic order, weaker than the expectedantiferromagnetic order, this implies that there is further work to be done to strengthenthe result to the full antiferromagnetic order.The positive results concerning long-range order above use the method of reflectionpositivity in order to obtain an infrared bound, that is, a bound on the Fourier transformof the correlation in question. One can then easily show that the correlation functiondoes not decay (for example that (cid:104) S x S y (cid:105) ≥ c > ffi ciently strong. The infrared bound proven in [6] allows to show a phase transitionfor the antiferromagnet. It is straightforward to extend this result to a model with an an-tiferromagnetic interaction accompanied by a small nematic (biquadratic) interaction.However when the nematic interaction is too large the result will no longer apply. Thisarticle will follow the approach of [6], starting with the nematic model, obtaining alower bound that involves some other correlation functions. This bound can be shownto be positive for low temperatures by relating these correlations to probabilities in therandom loop model introduced in [2]. It is then easy to show (due to reflection positiv-ity of the antiferromagnet interaction) that adding an antiferromagnetic interaction willmaintain the positivity of the lower bound, providing the interaction is small enough. Let S ∈ N . For a spin- S model we have local Hilbert spaces H x = C S + . Observ-ables are then Hermitian matrices built from linear combinations of tensor products ofoperators on ⊗ x ∈ Λ H x for some set of sites Λ . Physically important observables canoften be expressed in terms of spin matrices S , S and S , operators on C S + thatare the generators of a (2 S + (cid:104) S α , S β (cid:105) = i (cid:88) γ E αβγ S γ (1)where α, β, γ ∈ { , , } and E αβγ is the Levi-Civita symbol. Denote S = ( S , S , S ),its magnitude is then S · S = S ( S + . The case S = gives the Pauli spin matrices.For S = S = S = √ , S = √ − i i − i i , S = − . (2)Consider a pair ( Λ , E ) of a lattice Λ ⊂ Z d and a set of edges E between points in Λ .Here we will take Λ = (cid:26) − L + , ..., L (cid:27) d , (3)2or integer L . For the set of edges E we take nearest-neighbour with periodic boundaryconditions. Then we take the operator S ix for i = , , S ix ⊗ Id Λ \{ x } .The Hamiltonian of interest is general the Spin-1 SU(2)-invariant Hamiltonian with atwo-body interaction, it is known that this can be written as H J , J Λ , = − (cid:88) { x , y }∈E (cid:18) J (cid:16) S x · S y (cid:17) + J (cid:16) S x · S y (cid:17) (cid:19) . (4)The phase diagram for this model is only partially understood. If J = J < J > J > J / J is su ffi ciently small. Theline J = J > J < J = J / < J > J < | J | su ffi ciently small compared to | J | , the statement will bemade precise below.First we define the partition function and Gibbs states of our model as Z J , J β, Λ , = T re − β H J , J Λ , , (5) (cid:104)·(cid:105) J , J β, Λ , = Z J , J β, Λ , T r · e − β H J , J Λ , . (6)Where β > ρ ( x ) = (cid:42)(cid:32) ( S ) − (cid:33) (cid:32) ( S x ) − (cid:33)(cid:43) J , J β, Λ , . (7)this correlation is specifically of interest for spin-1, in general spin-S will be replacedwith S ( S + Theorem 2.1 (Long-range order).
Let S = , J > and L be even, d ≥ . Thenthere exists J < , β and C = C ( β, J ) > such that if J < J ≤ and β > β then | Λ | (cid:88) x ∈ Λ ρ ( x ) ≥ C . for all L large enough. The proof of the result will be in two steps, first the result will be proved for J = J = ffi ciently small J <
0, this should come as no surprise asthe interaction is reflection positive for J < The phase diagram for the general SU(2) invariant spin-1 model. Some regions have rigorousproofs that the expected order is indeed correct. The line J < J = J > J = J > , J = We will now consider the so-called quantum nematic model J > J =
0, the aimis to prove long-range order for this model using a similar approach to the proofs in[6, 8, 9, 10]. To do this we will use a representation that is an analogue of the matrixrepresentation used in [3]. Care must be taken as now we are working with matricesrather than vectors and so commutativity becomes an issue. We introduce an externalfield, h , to the Hamiltonian H , Λ , h = − (cid:88) { x , y }∈E ( S x · S y ) − (cid:88) x ∈ Λ h x (cid:32) ( S x ) − S ( S + (cid:33) . (8)4ere is the identity matrix. Equilibrium states are given by (cid:104) A (cid:105) , β, Λ , h = Z , β, Λ , h T rAe − β H , Λ , h . (9)Note that the J has been absorbed into the parameter β . Using the direct analogue of[3] will not work here, the reason is that reflection positivity will fail as S = − S . Allother attempts to directly obtain a matrix representation of the interaction ( S x · S y ) havealso failed, however, there is a solution. We will instead use a matrix representation ofa Hamiltonian that is unitarily equivalent to (8).From now on we will work with the following Hamiltonian H U Λ , h = − (cid:88) { x , y }∈E ( S x S y − S x S y + S x S y ) − (cid:88) x ∈ Λ h x (cid:32) ( S x ) − S ( S + (cid:33) , (10)and partition function Z U Λ ,β, h = T re − β H U Λ , h . (11)Similarly to before, equilibrium states are given by (cid:104) A (cid:105) U Λ ,β, h = Z U Λ ,β, h T rAe − β H U Λ , h . (12)If Λ has a bipartite structure, Λ = Λ A ∪ Λ B , then if we define U = (cid:81) x ∈ Λ B e i π S x we have U − H U Λ , h U = H , Λ , h . (13)Note that this leaves ρ ( x ) unchanged. Before the theorem we introduce an integral, itis also introduced in [13], I d = π ) d (cid:90) [ − π,π ] d (cid:115) ε ( k + π ) ε ( k ) d d (cid:88) i = cos k i + d k , (14)where ε ( k ) = d (cid:88) i = (1 − cos k i ) . (15)We have I d < ∞ for d ≥ I d → d → ∞ [ ? ]. Then wehave the following result: Theorem 3.1 (Long-range order for the quantum nematic model).
Let S = .Assume h = and L is even with d ≥ . Then we have the bound lim β →∞ lim L →∞ | Λ | (cid:88) x ∈ Λ ρ ( x ) ≥ ρ ( e ) − I d (cid:114)(cid:68) S S S e S e (cid:69) U Z d , ∞ , h . The expectations on the right of the inequality are taken in the infinite volume limit andwith β → ∞ . If this lower bound is strictly positive it implies the existence of a phasetransition at low temperatures, note that the lower bound is valid in any dimension d ≥
3, but as can be seen from equation (59) not in d ≤
2, hence no phase transition.5his is consistent with the well known Mermin-Wagner theorem [14]. Using the loopmodel introduced in [2] and extended in [22] we can relate the expectations in the lowerbound to the probability of the event E , e , that two nearest neighbours are in the sameloop as ρ ( e ) = P (cid:2) E , e (cid:3) , (cid:68) S S S e S e (cid:69) U Λ ,β, h = P (cid:2) E , e (cid:3) . (16)So we can write the lower bound as (cid:112) P (cid:2) E , e (cid:3) (cid:18) (cid:112) P (cid:2) E , e (cid:3) − I d √ (cid:19) . This means a suf-ficiently large lower bound on P (cid:2) E , e (cid:3) will allow to show the lower bound is positivein high enough dimension. Proposition 3.2.
For d ≥ , S = and L = ... = L d = L even. We have the lowerbound P (cid:2) E , e (cid:3) ≥ . (17)Putting this bound into the theorem and computing I d for various d shows that there isa positive lower bound (and hence phase transition) for d ≥ Proof.
For any state ψ ∈ ⊗ x ∈ Λ C we have that in the ground state (cid:104) H , Λ , (cid:105) , Λ , ∞ , h ≤ (cid:104) ψ, H , Λ , ψ (cid:105) . (18)We pick the N´eel state, ψ N ´ eel as a trial state ψ N ´ eel = ⊗ x ∈ Λ | ( − x (cid:105) . (19)We have used Dirac notation here where S | a (cid:105) = a | a (cid:105) . For the left of (18) we recallthat for x and y nearest neighbours ( S x · S y ) has three terms of the form ( S ix ) ( S iy ) ,having expectation P (cid:2) E , e (cid:3) + independent of i and six terms of the form S ix S jx S iy S jy having expectation P (cid:2) E , e (cid:3) independent of i and j (this is due to the equivalent rolesof i and j coupled with ( S ix S jx ) T = ± S jx S ix where the sign depends on the value of i or j ). This gives (cid:104) H , Λ , (cid:105) , Z d , ∞ , h = − (cid:88) { x , y }∈ Λ (cid:34) (cid:32) P (cid:2) E , e (cid:3) + (cid:33) + P (cid:2) E , e (cid:3)(cid:35) = − d | Λ | P (cid:2) E , e (cid:3) + . (20)For the right side of (18) it can be checked that, for S =
1, ( S x · S y ) = P xy + P xy is the projector onto the spin singlet. Hence (cid:104) , − | ( S x · S y ) | , − (cid:105) = (cid:104) , − | P x , y + | , − (cid:105) = , (21)from this we see that the right side of (18) is − d | Λ | . Inserting each of these values into(18) and rearranging gives the claim of the proposition. (cid:3) Note that if one could find a state with lower energy than the N´eel state this lower boundcould be improved and hence potentially the theorem strengthened to show phase tran-sitions in lower dimensions. However the problem of finding lower energy states doesnot appear an easy one.The rest of the section will be dedicated to the proof of theorem 3.1. We will proceedwith calculations for general spin until it becomes necessary to restrict to the case6 =
1. Fortunately for this Hamiltonian we can find a matrix representation. Define Q x as Q x = ( S x ) − S ( S + S x iS x S x S x S x iS x ( S x ) − S ( S + iS x S x S x S x iS x S x ( S x ) − S ( S + . (22)We introduce the operation T R , which is the sum of diagonal entries of matrices of theform of Q x , however this ‘trace’ will return an operator, not a number, so we distinguishit from the normal trace. As an example we see that T R ( Q x ) =
0, the zero matrix. Wehave the relation (note that below we do not mean ‘normal’ matrix multiplication, weonly write Q x Q y for convenience as explained in the remark). T R ( Q x Q y ) = ( S x S y − S x S y + S x S y ) − S ( S + . (23) Remark.
We must be careful here, as we are working with a matrix of matrices, as towhat we mean by multiplication. The representation (22) is not at all essential to theproof, the advantage of using it is that once (23) has been verified other relations canbe stated much more concisely and clearly and easily checked, these relations are notat all obvious or easy to come up with without using (23).By the product Q x Q y we follow the ‘normal’ matrix multiplication with the addedstipulation that for the i th diagonal entry of Q x Q y the operator S i will appear first. Forexample in entry { , } of Q x Q y there is the term S x iS x S y iS y , in the entry { , } thisterm will become iS x S x iS y S y , this ensures that we have each of the cross terms in theright-hand side of (23). For o ff -diagonal entries we are not concerned as we are alwaystaking a ‘trace’.In the case x (cid:44) y less care is needed as components of S x and S y commute (in fact T R Q x Q y = T R Q y Q x , hence we must only take care that the product order of compo-nents of spin at the same site is maintained).We also have that T R Q x = C Sx − S ( S + acting on H x . In S = C x = x . (24)Using this we can represent our interaction as( S x S y − S x S y + S x S y ) = (cid:16) C Sx + C Sy − T R (cid:104) ( Q x − Q y ) (cid:105)(cid:17) . (25)We introduce the field v with value v x ∈ R at the site x ∈ Λ . We denote by v the field of3 × v x has one non-zero entry, the entry { , } being v x ∈ R .We define H ( v ) = (cid:88) { x , y }∈E (cid:16) T R (cid:104) ( Q x − Q y ) (cid:105) − C Sx − C Sy (cid:17) − (cid:88) x ∈ Λ ( ∆ v ) x (cid:32) ( S x ) − S ( S + (cid:33) , (26) Z ( v ) = T re − β H ( v ) . (27)Note that from (25) H ( v ) = H U Λ , ∆ v . Here we have used the lattice Laplacian and belowwe use the inner product ( f , g ) = (cid:80) x ∈ Λ f x g x with the identity ( f , − ∆ g ) = (cid:80) { x , y }∈E ( f x − y )( g x − g y ). Then we can calculate as follows: H ( v ) = (cid:88) { x , y }∈E (cid:40) T R (cid:20) ( Q x + v x − Q y − v y (cid:21) − T R (cid:104) ( Q x − Q y )( v x − v y ) (cid:105) − C Sx − C Sy + ( v x − v y ) (cid:16) ( S x ) − ( S y ) (cid:17) −
14 ( v x − v y ) (cid:41) = (cid:88) { x , y }∈E (cid:40) T R (cid:20) ( Q x + v x − Q y − v y (cid:21) − C Sx − C Sy (cid:27) −
14 ( v , − ∆ v ) . (28)We must check carefully when dealing with the cross terms ( Q x − Q y )( v x − v y ) and( v x − v y )( Q x − Q y ), they are not equal but T R ( Q x − Q y )( v x − v y ) = T R ( v x − v y )( Q x − Q y ), sothe calculation is correct. From this it makes sense to define the following Hamiltonianand partition function: H (cid:48) ( v ) = H ( v ) +
14 ( v , − ∆ v ) , (29) Z (cid:48) ( v ) = T re − β H (cid:48) ( v ) . (30)Now the property of Guassian Domination is Z ( v ) ≤ Z (0) e β ( v , − ∆ v ) ⇐⇒ Z (cid:48) ( v ) ≤ Z (cid:48) (0) , (31)as in the classical case it follows from reflection positivity. Lemma 3.3 (Reflection positivity).
Let H = h ⊗ h, dim h < ∞ , fix a basis. LetA , B , C i , D i for i = , ..., k be matrices in h, then (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T r H exp (cid:26) A ⊗ + ⊗ B − k (cid:88) i = ( C i ⊗ − ⊗ D i ) (cid:27)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ T r H exp (cid:40) A ⊗ + ⊗ ¯ A − k (cid:88) i = ( C i ⊗ − ⊗ ¯ C i ) (cid:41) × T r H exp (cid:40) ¯ B ⊗ + ⊗ B − k (cid:88) i = ( ¯ D i ⊗ − ⊗ D i ) (cid:41) (32) where ¯ A is the complex conjugate of A.
The proof uses Trotter’s formula. As in the classical case, reflection positivity is a verypowerful tool, for more information see [4, 6, 8, 9, 10, 19, 21, 22].Before we prove reflection positivity for our partition function we should calculate thetrace in Z (cid:48) ( v ), recall how we have defined our multiplication. T R (cid:20) ( Q x + v x − Q y − v y (cid:21) = (cid:16) ( S x ) − ( S y ) (cid:17) + (cid:16) ( S x ) − ( S y ) (cid:17) + (cid:18) ( S x ) + v x − ( S y ) − v y (cid:19) + (cid:16) S x iS x − S y iS y (cid:17) + (cid:16) S x S x − S y S y (cid:17) + (cid:16) iS x S x − iS y S y (cid:17) + (cid:16) iS x S x − iS y S y (cid:17) + (cid:16) S x S x − S y S y (cid:17) + (cid:16) S x iS x − S y iS y (cid:17) . (33)8ow we have enough information to use the Lemma, let R : Λ → Λ be a reflection thatswaps Λ and Λ where Λ = Λ ∪ Λ , each such reflection defines two sub-lattices of Λ in this way, we split the field v = ( v , v ) on the sub-lattices Λ and Λ . Lemma 3.4 (Reflection positivity for the quantum nematic model).
For S ∈ N and any reflection, R, across edges and v = ( v , v ) Z (( v , v )) ≤ Z (( v , Rv )) Z (( Rv , v )) . Proof.
We cast Z (cid:48) ( v ) in RP form. Let A = − β (cid:88) { x , y }∈E T R (cid:20) ( Q x + v x − Q y − v y (cid:21) − β d (cid:88) x ∈ Λ C Sx , B = same in Λ , (34)where E is the set of edges in Λ and we note that the term C Sx occurs d times in thesum over E for each x ∈ Λ . Further define C i = √ β ( S x i ) , D i = √ β ( S y i ) . C i = √ β ( S x i ) , D i = √ β ( S y i ) . C i = √ β (( S x i ) + v xi ) , D i = √ β (( S y i ) + v yi ) . C i = √ β S x i iS x i , D i = √ β S y i iS y i . C i = √ β S x i S x i , D i = √ β S y i S y i . C i = √ β iS x i S x i , D i = √ β iS y i S y i . C i = √ β iS x i S x i , D i = √ β iS y i S y i . C i = √ β S x i S x i , D i = √ β S y i S y i . C i = √ β S x i iS x i , D i = √ β S y i iS y i . (35)Where { x i , y i } are edges crossing the reflection plane with x i ∈ Λ and y i ∈ Λ . Because S x = S x , S x = S x , iS x = iS x we see from the previous lemma that Z (cid:48) (( v , v )) ≤ Z (cid:48) (( v , Rv )) Z (cid:48) (( Rv , v )), from which the result follows. (cid:3) The Gaussian domination inequality (31) follows from this just as in the classical case,a proof can be found in [6]. The next step in the classical case was to obtain an infraredbound for the correlation function ρ ( x ), we cannot do this directly but we can obtain aninfrared bound for the Duhamel correlation function . Definition 3.5 (Duhamel correlation function).
For matrices A , B we define the Duhamelcorrelation function ( A , B ) Duh as( A , B ) Duh = Z (0) 1 β (cid:90) β d sT rA ∗ e − sH (0) Be − ( β − s ) H (0) Note that this is an inner product.Now to use this correlation function we must first fix our definition of the Fouriertransform F ( f )( k ) = ˆ f ( k ) = (cid:88) x ∈ Λ e − ikx f ( x ) k ∈ Λ ∗ , f ( x ) = | Λ | (cid:88) k ∈ Λ ∗ e ikx ˆ f ( k ) x ∈ Λ . (36)9here Λ ∗ = π L (cid:26) − L + , ..., L (cid:27) × ... × π L d (cid:26) − L d + , .., L d (cid:27) , (37) Lemma 3.6.
For S ∈ N and L i even for i = , ..., d we have the following infraredbound F (cid:32) ( S ) − S ( S + , ( S x ) − S ( S + (cid:33) Duh ( k ) ≤ βε ( k ) . (38) Proof.
We begin as usual by choosing v x = η cos( kx ), then from Taylor’s theorem andusing h = ∆ v = − ε ( k ) v we see Z ( v ) = Z (0) + (cid:32) h , ∂ Z ( v ) ∂ h x ∂ h y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h = h (cid:33) + O ( η ) . (39)Using the Duhamel formula e β ( A + B ) = e β A + (cid:90) β d se sA Be ( β − s )( A + B ) (40)with A = H (0) and B = − (cid:80) x ∈ Λ ( ∆ v ) x (cid:16) ( S x ) − S ( S + (cid:17) gives1 Z (0) ∂ Z ( v ) ∂ h x ∂ h y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = β (cid:32) ( S x ) − S ( S + , ( S y ) − S ( S + (cid:33) Duh . (41)Putting this together we have Z (( v )) − O ( η ) = Z (0) + Z (0)( ηε ( k ) β ) (cid:88) x , y ∈ Λ cos( kx ) cos( ky ) (cid:32) ( S x ) − S ( S + , ( S y ) − S ( S + (cid:33) Duh = Z (0) + Z (0) β η ε ( k ) F (cid:32) ( S ) − S ( S + , ( S y ) − S ( S + (cid:33) Duh (cid:88) x ∈ Λ cos ( kx ) . (42)Also e − β ( v , ∆ v ) = e βε ( k ) η (cid:80) cos ( kx ) , (43)comparing the order η terms gives the result. (cid:3) To transfer the infrared bound to the normal correlation function we would like to usethe Falk-Bruch inequality [7]:12 (cid:104) A ∗ A + AA ∗ (cid:105) ≤ ( A , A ) Duh + (cid:113) ( A , A ) Duh (cid:104) [ A ∗ , [ H U Λ , h , A ]] (cid:105) (44)where is the Hamiltonian of the system. If we attempt to use this inequality with A = F (cid:16) ( S x ) − S ( S + (cid:17) ( k ) and H = β H U Λ , , we must calculate the double commutator tofind (cid:104) [ A ∗ , [ H , A ]] (cid:105) . In general spins this is a huge calculation, instead we specialise tothe case S =
1. In this case we can calculate as below, it uses several special properties10f the Spin-1 matrices. To make use of this inequality we note that F (cid:42)(cid:32) ( S ) − S ( S + (cid:33) (cid:32) ( S x ) − S ( S + (cid:33) (cid:43) U Λ , h ( k ) = (cid:88) x ∈ Λ e − ikx (cid:42)(cid:32) ( S ) − S ( S + (cid:33) (cid:32) ( S x ) − S ( S + (cid:33)(cid:43) U Λ , h = | Λ | (cid:88) x , y ∈ Λ e − ik ( x − y ) (cid:42)(cid:32) ( S x ) − S ( S + (cid:33) (cid:32) ( S y ) − S ( S + (cid:33)(cid:43) U Λ , h = | Λ | (cid:42) F (cid:32) ( S x ) − S ( S + (cid:33) ( − k ) F (cid:32) ( S y ) − S ( S + (cid:33) ( k ) (cid:43) U Λ , h . (45)This relation holds for other correlation functions, including the Duhamel correlationfunction, but for Duhamel F (cid:32) ( S ) − S ( S + , ( S x ) − S ( S + (cid:33) Duh ( k ) = | Λ | (cid:32) F (cid:32) ( S x ) − S ( S + (cid:33) ( k ) , F (cid:32) ( S y ) − S ( S + (cid:33) ( k ) (cid:33) Duh , (46)there is no − k because of the definition of the Duhamel correlation function and theequality (cid:16) F (cid:104) ( S x ) (cid:105)(cid:17) ( k ) ∗ = F (cid:104) ( S x ) (cid:105) ( − k ).First we prove a preliminary lemma regarding the double commutator Lemma 3.7.
For S = , A = F (cid:16) ( S x ) − (cid:17) ( k ) and H = β H Λ , we have (cid:104) [ A ∗ , [ H , A ]] (cid:105) U Λ , h = β | Λ | ε ( k + π ) (cid:68) S S S e S e (cid:69) U Λ , h where e is the first basis vector in Z d .Proof. The proof is just a calculation, although it is somewhat complicated, we beginby noting that in the case S = S i ) and ( S j ) commute and ( S i ) = S i for i , j = , , H , A ] = − β (cid:88) x , y : { x , y }∈E e − ikx (cid:104) ( S x S y − S x S y + S x S y ) , ( S x ) (cid:105) = − β (cid:88) x , y : { x , y }∈E e − ikx (cid:104) ( S x S x S y S y − S x S x S y S y − S x S x S y S y − S x S x S y S y + S x S x S y S y − S x S x S y S y ) , ( S x ) (cid:105) . (47)The square terms have dropped out as they commute with ( S x ) , as does the constantterm S ( S + /
3. Now we calculate the commutator for each term in the sum, here we11ake use of the fact that S i S j S i = i (cid:44) j , i , j = , , S = H , A ] = − β (cid:88) x , y : { x , y }∈E e − ikx (cid:18) S x S x S y S y + = (cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125)(cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:104) ( S x ) , S x S x (cid:105) S y S y + = (cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125)(cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:104) ( S x ) , S x S x (cid:105) S y S y − S x S x S y S y − S x S x S y S y + S x S x S y S y (cid:19) = + β (cid:88) x , y : { x , y }∈E e − ikx (cid:18) (cid:104) S x S y , S x S y (cid:105) + (cid:104) S x S y , S x S y (cid:105) (cid:19) . (48)Now calculating the commutator of these products and using the spin commutationrelations we obtain[ H , A ] = β i (cid:88) x , y : { x , y }∈E e − ikx (cid:16) S x S x S y + S x S x S y + S x S y S y + S x S y S y (cid:17)(cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) f ( S x , S y ) . (49)Now we can use this to calculate the double commutator, firstly we split the commuta-tor into the sum of two similar terms (cid:2) A ∗ , [ H , A ] (cid:3) = β i (cid:88) x , y : { x , y }∈E e − ikx (cid:104) e ikx ( S x ) + e iky ( S y ) , f ( S x , S y ) (cid:105) = β i (cid:88) x , y : { x , y }∈E (cid:104) ( S x ) , f ( S x , S y ) (cid:105) + cos( k ( x − y )) (cid:104) ( S y ) , f ( S x , S y ) (cid:105) . (50)We can calculate each of these commutators separately, the first double commutatorcan be calculated as follows (cid:104) ( S x ) , f ( S x , S y ) (cid:105) = (cid:104) ( S x ) , S x S x S y + S x S x S y + S x S y S y + S x S y S y (cid:105) = − S x S x S y + iS x S x S y S y + iS x S x S y S y + S x S x S y − iS x S x S y S y − iS x S x S y S y . (51)We recognise the commutator relations above to finally give (cid:104) ( S x ) , f ( S x , S y ) (cid:105) = iS x S x S y S y + iS x S x S y S y − iS x S x S y S y − iS x S x S y S y . (52)For the other commutator we follow the previous calculation almost exactly and in factwe find the two commutators are equal (cid:104) ( S y ) , f ( S x , S y ) (cid:105) = (cid:104) ( S x ) , f ( S x , S y ) (cid:105) . (53)To finish the calculation we take expectations (cid:104) [ A ∗ , [ H , A ]] (cid:105) U Λ , h = − β | Λ | d (cid:88) i = (1 + cos( k i )) (cid:68) S S S e i S e i + S S S e i S e i − S S S e i S e i − S S S e i S e i (cid:69) U Λ , h (54)12ow use the identities ( S S ) T = − S S and ( S S ) T = S S and get (cid:104) [ A ∗ , [ H , A ]] (cid:105) U Λ , h = − β | Λ | d (cid:88) i = (1 + cos( k i )) (cid:68) S S S e i S e i − S S S e i S e i (cid:69) U Λ , h = β | Λ | d (cid:88) i = (1 + cos( k i )) (cid:68) S S S e i S e i (cid:69) , J β, Λ , h . = β | Λ | ε ( k + π ) (cid:68) S S S e S e (cid:69) , J β, Λ , h . (55)On the second line we have used that US e S e = − S e S e U to move from states (cid:104)·(cid:105) U Λ , h to states (cid:104)·(cid:105) , J β, Λ , and on the third line we have used that each cross term (cid:104) S ix S jx S iy S jy (cid:105) , J β, Λ , h has the same expectation value. Now simply note that the above correlation is the samein (cid:104)·(cid:105) U Λ , h and in (cid:104)·(cid:105) , J β, Λ , . (cid:3) Using this in Falk-Bruch we have the boundˆ ρ ( k ) ≤ (cid:114)(cid:68) S S S e S e (cid:69) , J β, Λ , h (cid:115) ε ( k + π ) ε ( k ) + βε ( k ) . (56)The possibility of obtaining a result is not ruled out for other values of S , I expect it tobe the case for other values of S , but computing the double commutator in Falk-Bruchbecomes extremely complicated.Now using the Fourier transform in the following way: (cid:42)(cid:32) ( S ) − (cid:33) (cid:32) ( S y ) − (cid:33)(cid:43) , J Λ , h = | Λ | (cid:88) x ∈ Λ ρ ( x ) + | Λ | (cid:88) k ∈ Λ ∗ \{ } e ik · y ˆ ρ ( x )( k ) (57)with y = e we get the lower bound1 | Λ | (cid:88) x ∈ Λ ρ ( x ) ≥ ρ ( e ) − | Λ | (cid:88) k ∈ Λ ∗ \{ } (cid:115) ε ( k + π ) ε ( k ) d d (cid:88) i = cos k i + − βε ( k ) . (58)Taking the thermodynamic limit with L i = L even for i = , .., d giveslim inf L →∞ (cid:42) | Λ | (cid:88) x ∈ Λ ρ ( x ) (cid:43) ≥ ρ ( e ) − π ) d (cid:90) [ − π,π ] d (cid:115) ε ( k + π ) ε ( k ) d d (cid:88) i = cos k i + + βε ( k ) d k . (59)The integral is finite if and only if d ≥ β → ∞ gives the result. (cid:3) J < The aim of this section is to extend the proof of theorem 3.1 to a proof of theorem 2.1.The proof of long-range order for J < J , J Λ , , we also introduce an external field h as before. Recall the unitary operator U = (cid:81) x ∈ Λ B e i π S x , let (cid:101) H U Λ , h = UH J , J Λ , U − − (cid:88) x ∈ Λ h x (cid:32) ( S x ) − S ( S + (cid:33) . (60)The e ff ect of the unitary operator here is to replace S x and S x in H J , J Λ , with − S x and − S x respectively. By using the representation (22) we can write (cid:101) H U Λ , as (cid:101) H U Λ , = − (cid:88) { x , y }∈E (cid:20) J (cid:16) ( S x − S y ) − ( S x − S y ) + ( S x − S y ) (cid:17) − J (cid:16) T R (cid:2) ( Q x − Q y ) (cid:3)(cid:17) + C Λ ( J , J ) (cid:21) . (61)Then similar to before we introduce the field v and associated 3 × v ,define (cid:101) H ( v ) = − (cid:88) { x , y }∈E (cid:20) J (cid:16) ( S x − S y ) − ( S x − S y ) + ( S x − S y ) (cid:17) (62) − J (cid:16) T R (cid:2) ( Q x + v x − Q y − v y (cid:3)(cid:17) + C Λ ( J , J ) (cid:21) −
14 ( v , − ∆ v ) , (cid:101) Z ( v ) = T re − β (cid:101) H ( v ) , (63)and (cid:101) H (cid:48) ( v ) = (cid:101) H ( v ) +
14 ( v , − ∆ v ) , (64) (cid:101) Z (cid:48) ( v ) = T re − β (cid:101) H (cid:48) ( v ) . (65)From this reflection positivity follow just as in Lemma 3.4, with the obvious changesto A and B and the extra terms C i = √− J S x i , D i = √− J S y i , C i = √− J iS x i , D i = √− J iS y i , C i = √− J S x i , D i = √− J S y i , (66)(recall that J < (cid:101) Z ( v ) ≤ (cid:101) Z (0) e β ( v , − ∆ v ) ⇐⇒ (cid:101) Z (cid:48) ( v ) ≤ (cid:101) Z (cid:48) (0) , (67)just as before. We also obtain the same infrared bound as in Lemma 3.7, with anidentical proof F (cid:32) ( S ) − S ( S + , ( S x ) − S ( S + (cid:33) Duh ≤ βε ( k ) . (68)Again the results up to here work for general S ∈ N , at this point we must spe-cialise to S = A = F (cid:16) ( S x ) − (cid:17) ( k ) and H = β (cid:101) H U Λ , the linearity of the double commutator14eans that there will be an extra term in the analogous result to lemma 3.6 equal to (cid:104) J [ A ∗ , [ − (cid:80) { x , y }∈E ( S x · S y ) , A ]] (cid:105) . This will result in the IRB analogous to (56) poten-tially being larger, weakening the result. If | J | is small enough this weakening will notbe too severe so as to make the lower bound analogous to the bound in theorem 3.1negative in cases where we know the original lower bound was positive. This ensuresthat we have a positive lower bound C = C ( β, J ) in Theorem 2.1 when β and | J | aresmall enough. It is worth noting that for the same reason as just described, extendingthe result of Dyson, Lieb and Simon [6] to J > | J | is small. Thismeans the two results will not overlap, leaving part of the quadrant J ≤ ≤ J stillopen to investigation. Acknowledgments
I am pleased to thank my supervisor Daniel Ueltschi for his support and useful discus-sions. I am also grateful to the referee for several useful comments and observations.This work is supported by EPSRC as part of the MASDOC DTC at the University ofWarwick. Grant No. EP / HO23364 /
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