Anderson model is an important model in the theory of strongly correlated electron system. In this contribution, we introduce this model and the concept of electron correlation by bipartite lattice and prove rigorously theorems leading to the presence of spin singlet in the model. By using the results of Ueda et al (1992) and Tian (1994), we show theoretically that the ground state of the symmetric periodic Anderson model has a short – range order. 1.0.INTRODUCTION The heavy fermion systems are well-known strongly correlated electron systems which exhibit unusual thermodynamic, magnetic, and transport properties.1 microscopic investigations of these systems are generally based on the Periodic Anderson Model (PAM) Take a finite lattice with N❑ lattice points. Neglecting the orbital degeneracy for the localized electrons, the Hamiltonian of the PAM can be written in the form H❑=−t∑σ❑∑ ⟨i,j⟩¿¿ +U∑i∈ni↑ni↓ (1) Where ciσ+¿¿ (ciσ) is the creation (annihilation) of electrons, which creates (annihilates) an itinerant electron of spin σ at site i. Similarly, diσ+¿¿ and diσ are the fermion operators for the localized electrons. 1 niσ=diσ+¿diσ¿ and ⟨i,j⟩ denotes a pair of lattice sites. εd is a local potential and U > 0 represent the on-site coulomb interaction for the localized electrons. Since its introduction , the PAM and its variants constitute an important research topic in theoretical condensed matter physics, particularly in the context of strongly correlated electron system. Most of the many body techniques commonly used in condensed matter physics can be learnt in this context (Jafari, 2008). Also there are some theoretical tools and concepts which apply to this model only [ Mahan, (2000); Enaroseha, (2012); Muhlbacher et al , (2011), Zhu and Zhu (2011)]. The aim of this paper is to provide a smooth introduction in the models and prove that the ground state of the symmetric PAM is a spin singlet. 2.0METHODOLOGY AND MATHEMATICAL FORMULATION. If we assume in particular that, when Ed = Ef = -U2 , the Hamiltonian of the PAM is called Symmetric. Lattice is called bipartite with respect to the Hamiltonian H❑ if it can be divided into two sublattices A and B, such that hopping of electrons does not occur between sites in the same sublattices. For a bipartite lattice, the signs of parameters t and V are not important for the mathematical analysis of this Hamiltonian. They can be changed by a unitary transformation. For definiteness, we shall choose t > o and V > o in the following. To begin with, we first write the Hamiltonian H❑of the symmetric PAM in a generalized Hubbard Hamiltonian. For simplicity, let us consider a specific bipartite lattice: the two-dimensional square lattice with N = L2 lattice points (the lattice constant is taken to be unity). Take two identical copies of this square lattice, ❑1 and ❑2. We make a doubly layered lattice Λ~ by connecting the corresponding lattice points of , ❑1 and ❑2 with bonds of 2 length 1. Now, each point of Λ~ is labelled by r = ( i, m) where m = 1 or 2. Obviously, Λ~ has 2N❑ lattice points. Next, we introduce new fermion operators frσby frσ={Ciσifm=1| (2) With the definitions of Λ~ and frσ, the Hamiltonian N❑ of the symmetric PAM can be rewritten as the Hamiltonian of a generalized Hubbard model on Λ~ by ignoring an uninteresting constant UH❑4. H~❑=−∑σ❑∑ ⟨rh⟩trh¿¿ (3) where the new hopping constants trh are defined by t if r and h are nearest neighbors in ❑1 (4) trh= - Vif r and h are in different layers and connected by a bond ootherwise. It is easy to see Λ~ is still bipartite with respect to H❑. Furthermore, we notice that the electrons in the second layer (the d – electron layer) do not hop and have an on-site Coulomb repulsion, while the electrons in the first layer are itinerant without interaction. That causes a little technical inconvenience. For this reason, we introduce an auxiliary interaction term H1=∑ rε∧1(nr↑−12)(nr↓−12)❑ (5) to the Hamiltonian H❑ and study the ground state of the new Hamiltonian ~H~❑=H~❑+H1 . In the final step, we let ε  0. The purposes of the proofs are to show the existence of a short-range antiferromagnetic d-electron spin correlation in the ground state Ψ0(ε,U,Λ~ ) of the positive - (ε, U) symmetric PAM at half filling. However, reflection positivity in the spin space does not hold in this case. Fortunately, by the well-known unitary particle-hole transformation ^U0fr↑U0−1=fr↑,^U0fr↓U0−1=ε(r)fr↓ (6) 3 Where ε(r) = 1 if r  A and ε(r)= -1 if r  B, the Hamiltonian of the same form, which has reflection positivity in the spin space. Obviously, the transformed Hamiltonian has the same spectrum. In the following, we shall first study the ground state Ψ0(−ε,−U,Λ~ ) of the Hamiltonian ~H❑(- ε, -U,) at half filling and then transform it back to the corresponding ground state Ψ0( ε ,Λ~) of H❑ (ε, U,) by the inverse of the particle-hole transformation. We notice that ~H~❑ commutes with N↑=∑fr↑fr↑andN↓=∑fr↓fr↓ respectively. Consequently, the Hilbert space of H❑can be divided into numerous subspaces. Each of them is characterized by a pair of integers (N  = n1, N= n2). For this Hamiltonian, following the formalism of Lieb (1989), we proved the following theorem. Theorem. For any given even integer N, the ground state Ψ0 of ~H~❑ is unique and has quantum numbers n 1 =n2 = N/2. Furthermore, Ψ0 can be written as Ψ~ 0¿¿ (7) Where is an orthogonal real basis for one species of N/2 spinless fermions. Take W = ¿¿ as a C2NΛN/2XC2NΛN/2 matrix; then W is a Hermitian and positive definite. In Lieb (1989) proof, the uniqueness of Ψ~ 0¿¿is the consequence of the positive definiteness of W, which was proved under the condition ε ≠ 0. Ueda et al (1992) removed this condition by exploiting a special topology of the PAM in addition to the reflection positivity in the spin space. Consequently the ground state of ~H~❑ is still non-degenerate even if  = 0. With knowledge of this fact, we are able to make our proof simpler by introducing H1 and employing Lieb’s theorem, whose proof is more direct. The theorem of Ueda et al (1992) guarantees that the ground state Ψ0(ε,U,Λ~ ) Ψ0(0,U, Λ~) as ε→0. Theorem 1. Let Ar ≡ frfr . Then, for any two distinct lattice points r and h, we have (8) Proof. By form (7) of Ψ0 (N/2, N/2), we have ⟨Ψ0∨Ar+¿Ah∨Ψ0⟩≡⟨Ψ0∨fr↓+¿fr↑+¿fh↑fh↓∨⟩¿¿¿¿⟨Ψ0∨¿ 4 ¿∑τδ∑αβ¯WτδWαβ⟨Ψ↑τ∨fr↑+¿fh↑∨Ψ↑α⟩⟨Ψ↓δ∨fr↓+¿fh↓∨Ψ↓β⟩¿¿ ¿∑τδ∑αβWδτ+¿Wαβ¿⟨Ψτ∨fr+¿fh∨Ψ↑α⟩⟨Ψδ∨fr+¿fh∨Ψβ⟩¿¿ ¿Tr¿ WhereM≡fr+¿fh⟨Ψδ∨fr+¿fh∨Ψβ⟩=⟨Ψδ∨fr+¿fh∨Ψβ⟩¿¿¿ and W+¿=W¿ (9) ¿∑τδ∑αβWδτ⟨Ψτ∨fr+¿fh∨Ψα⟩Wαβ⟨Ψβ∨fr+¿fh∨Ψδ⟩¿¿ So we have, ⟨Ψ0∨Ar+¿Ah∨Ψ0⟩≡∑τδ∑αβWδτ⟨Ψτ∨M∨Ψα⟩Wαβ⟨Ψβ∨M+¿∨Ψδ⟩¿¿ ¿Tr¿ (10) Since {Ψτ} is a real basis. By Lieb’s theorem, W is a positive definite matrix. Consequently, matrix W1/2 is well defined. Therefore we have Tr¿ (11) Theorem 1 is proved. Q.E.D. Remark 1. it is worthwhile to point out that theorem 1 has a physical implication. We noticed that is , in fact, the reduced on-site two-particle density matrix of Ψ~ 0¿¿. If the largest eigenvalues λmax of this matrix satisfies the condition λmax≥αH❑, where α is a positive constant independent of H❑, then Ψ~ 0¿¿has an off diagonal long-range order (ODLRO), which indicates that Ψ~ 0¿¿ is a superfluid. On the other hand, by theorem 1, we conclude that, if Ψ~ 0¿¿ is a superfluid, it must be a Bose – Einstein condensate, namely, a macroscopic on-site 5 pair of electrons is condensed at p = 0 in Ψ~ 0¿¿. A detailed analysis on this point found in Tian (1992) article. Since theorem 1 hold for any even integer N, in particular, it holds for the ground state Ψ0(−ε,−U,Λ~ ) of the negative – ( ε,U,) symmetric PAM at half filling. Appling the inverse of the particle-hole transformation (6), we immediately obtain the following theorem. Theorem 2: Let Ψ0(ε,U,Λ~ ) be the ground state of the positive-( ε,U) symmetric PAM Hamiltonian at half filling. Define the spin operators of the electron by Si+¿=fi↑+¿fi↑¿¿, Si−=fi↓+¿fi↑¿, Siz=1 2¿(12)  0 if r and h are in the same sublattice ¿ ≤ 0 if r and h are in the different sublattice (13) Proof. By the inverse of the unitary particle-hole transformation, we have Ψ0(−ε,U,~❑)→Ψ0(ε,U,~❑) fr↓+¿fr↑+¿→ε(r)fr↓fr↑+¿¿¿¿, fh↑+¿fh↓→ε(h)fh↑fh↓+¿¿¿ (14) Where ε (r) = 1 if r A, and ε (r) = -1 if r B U❑ 0fr↑U❑ 0−1=fr↑,U❑ 0fr↓U❑ 0−1=ε(r)fr↓ Therefore by theorem 1, 0≤¿ ¿¿ ¿ε(r)ε(h)¿ (15) If r and h belong to the same sublattices, then ε(r)ε(h) = 1 and hence ¿ 6 Otherwise, ε (r) ε (h) = - 1 and ¿ Theorem 2 is proved. Q.E.D. Remark 2. Shen et al (1994) used the same technique to show the existence of ferrimagnetism in some positive-U Hubbard model. Theorem 2 tells us that the short-range transverse spin correlation of d or f electrons in the ground state of the positive-( ε,U,) symmetric PAM is antiferromagnetic. Since the Hamiltonian ~H~❑ has the SU(2) spin symmetry and it ground state Ψ0(ε,U,Λ~ ) at half filling is nondegenerate, one would expect that theorem 2 also holds for the longitudinal spin correlation functions. In fact, we have the following theorem. Theorem 3.Let Ψ0(ε,U,Λ~ ) be the ground state of the positive-( ε,U,) symmetric PAM Hamiltonian at half filling. Let limε→0⟨Ψ0(ε,U,~❑)|SrzShz|Ψ0(ε,U,~❑)⟩⟨Ψ0(ε,U,~❑)|SrzShz|Ψ0(ε,U,~❑)⟩≡C(r,h); (16) then the spin correlation function C (r, h) has satisfy inequality (13). Proof. We first show that for any pair of distinct lattice points r and h, ¿2⟨Ψ0(ε,U,~❑)|SrzShz|Ψ0(ε,U,~❑)⟩ (17) holds. By definition, Sr+¿=Srx+iSry,Sr−=Srx−iSry,Sh−=Shx−iShy¿ . Therefore we have ¿⟨Ψ0(ε,U,~❑)|(Srx+iSry)(Shx−iShy)|Ψ0(ε,U,~❑)⟩ ¿⟨Ψ0(ε,U,~❑)|(SrxShx−iSrxShy+iSryShx+SryShy)|Ψ0(ε,U,~❑)⟩ ¿⟨Ψ0(ε,U,~❑)|(SrxShx+SryShy+iSryShx−iSrxShy)|Ψ0(ε,U,~❑)⟩ ¿⟨Ψ0(ε,U,~❑)|SrxShx+SryShy|Ψ0(ε,U,~❑)⟩+i⟨Ψ0(ε,U,~❑)|SryShx−SrxShy|Ψ0(ε,U,~❑)⟩(18) 7 We first simplify the last term on the right-hand side of (18). since r and h are distinct, [ Srz, Shy] = [ Sry, Shz] = 0. Therefore S ry Shx - Srx Shy is a Hermitian operator. Consequently, it expectation value in any state is a real quantity. On the other hand, since ~H~❑ is a real matrix, its ground state Ψ0(ε,U,Λ~ ) must be chosen as state real vector. Therefore the expectation value F of Sry Shx - Srx Shy in Ψ0(ε,U,Λ~ ) must be a pure imaginary matrix. Consequently F ≡ 0. Next, we apply the unitary operator U❑ 2=exp [(iπ 2)∑ i∈~❑Sis] Which rotates each spin about the S z axis by an angle /2, to the expectation value of S ry Shy in Ψ0(ε,U,Λ~ ) and obtain ⟨Ψ0(ε,U,~❑)|SryShy|Ψ0(ε,U,~❑)⟩¿⟨Ψ0(ε,U,~❑)|SrxShx|Ψ0(ε,U,~❑)⟩(19) Substituting (19) into (18), we obtain the identity (17). That is: ¿ ¿⟨Ψ0(ε,U,~❑)|SrxShx+SryShy|Ψ0(ε,U,~❑)⟩+i⟨Ψ0(ε,U,~❑)|SryShx−SrxShy|Ψ0(ε,U,~❑)⟩ So we have, ¿¿⟨Ψ0(ε,U,~❑)|SrxShx+SryShy|Ψ0(ε,U,~❑)⟩ ¿⟨Ψ0(ε,U,~❑)|SrxShx|Ψ0(ε,U,~❑)⟩+¿⟨Ψ0(ε,U,~❑)|SryShy|Ψ0(ε,U,~❑)⟩ ¿⟨Ψ0(ε,U,~❑)|SrxShx|Ψ0(ε,U,~❑)⟩+¿⟨Ψ0(ε,U,~❑)|SrxShx|Ψ0(ε,U,~❑)⟩ ¿2⟨Ψ0(ε,U,~❑)|SrxShx|Ψ0(ε,U,~❑)⟩ 8 Equation (17) is proved. Similarly, we can show that ⟨Ψ0(ε,U,~❑)|SrxShx|Ψ0(ε,U,~❑)⟩=¿⟨Ψ0(ε,U,~❑)|SrzShz|Ψ0(ε,U,~❑)⟩ (20) By applying the unitary operator U❑ 3=exp [(iπ 2)∑ i∈~❑Siy] To the expectation value of S rz Shz in Ψ0(ε,U,Λ~ ) Combining identities (17) and (20) and inequality (13), we see that the longitudinal spin correlation of Ψ0(ε,U,Λ~ ) is antiferromagnetic. Since this conclusion is true for any ε > 0, it must also hold for in the limit ε 0. Therefore the longitudinal spin correlation of the d or f electrons in the ground state of the positive-U symmetric PAM at half filling is antiferromagnetic. Our proof is accomplished. Q.E D. 3.0SUMMARY AND CONCLUSION This work is an extension and detailed analysis of Tian (1994) article and Ueda (1992) articles in PAM and Hubbard model respectively. We proved the existence of the short – range antiferromagnetic order in the ground state of the symmetric PAM at half filling for arbitrary V and U > 0, we do not claim that the ground state has long – range antiferromagnetic order. In fact Moller and Wolfle (1993) show by using a mean field slave – boson theory that the half – filled ground state may be either antiferromagnetic or paramagnetic. The line separating their region of stability is given by the critical value of their exchange constant. In all the lattice systems studied recently show an antiferromagnetic ground state and the first excited state is always a spin singlet (Enaroseha, 2012). In conclusion, in this paper, we showed rigorously that the spin correlation between (d, d), (f, f), (C, C), (C, d) and (C, f) electrons are antiferromagnetic in the ground state of the PAM at half – fillings. The spin correlations of f electrons in the ground state Ψ0() which is 9 of fundamental importance was not discussed in this article, which hopefully will become possible in the near future.