1. Introduction

The role of physical models in investigating complicated systems is one of the most significant strategies in Theoretical Condensed Matter Physics. The point is that if the model is able to account for some of the well known properties of the system, then it is expected to account for the more complex properties. Now when one is interested in fundamental properties of the model, exploring the ground-state energy is a feasible starting point [1]. For example, one of us has shown that for any finite value of the physically realistic on-site Coulombic interaction, the ground state of the Hubbard model which is the simplest model for studying strongly correlated systems, is a spin singlet [2]. Thus to use the model to study complicated strongly correlated phenomena like superconductivity and ferromagnetism which require transition from the antiferromagnetic ordering to partial or full spin ordering, other interaction terms needs to be added [3-4]. This is the line of thinking that led to the formation of the Periodic Anderson Model (PAM) [5-7] often used to investigate the heavy fermion systems which are well-known strongly correlated electron systems that exhibit unusual thermodynamic, magnetic, and transport properties [8].

Take a finite lattice with lattice points. Neglecting the orbital degeneracy for the localized electrons, the Hamiltonian of the PAM can be written in the form [9]

(1)

where is the creation (annihilation) of electrons, which creates (annihilates) an itinerant electron of spin σ at site i. Similarly, and are the fermion operators for the localized electrons. and denotes a pair of lattice sites, is a local potential, V is the hybridization matrix element 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. Also there are some theoretical tools and concepts which apply to this model only [10-13]. The aim of this paper is to prove that the ground state of the symmetric PAM is a spin singlet.

2. Methodology and Mathematical Formulation

Lattice is called bipartite with respect to the Hamiltonian 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 > 0 and V > 0 in the following.

To begin with, we first write the Hamiltonian 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, and . We make a doubly layered lattice by connecting the corresponding lattice points of and with bonds of length 1. Now, each point of is labelled by r = ( i, m) where m = 1 or 2. Obviously, has lattice points. Next, we introduce new fermion operators f_ra by

(2)

If we assume in Eq. (1) that, when , the Hamiltonian of the PAM is called Symmetric. With the definitions of and f_ra, the Hamiltonian of the symmetric PAM can be rewritten as the Hamiltonian of a generalized Hubbard model on by ignoring an uninteresting constant .

(3)

where the new hopping constants t_rh are defined by

(4)

It is easy to see is still bipartite with respect to . Furthermore, we observe 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. To account for this interaction, we introduce an auxiliary interaction operator

(5)

to the Hamiltonian and study the ground state of the new Hamiltonian . The in Equation (5) is a generalized interacting potential so that the interaction can be switched off by letting → 0. It is pertinent to point out that this interaction operator in Equation (5) will make it straightforward in future to generalized the proofs here so that by choosing the appropriate interacting elements, we can obtain a generalized Anderson model from which we can in turn also obtain the single site impurity Anderson model and the Kondo model depending on the material system we wish to study.

The purposes of the proofs are to show the existence of a short-range antiferromagnetic d-electron spin correlation in the ground state of the positive-(, 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

(6)

where = 1 if r ∈ A and = -1 if r ∈ B, the Hamiltonian is 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 of the Hamiltonian at half filling and then transform it back to the corresponding ground state (,) of by the inverse of the particle-hole transformation.

We observe that commutes with and respectively. Consequently, the Hilbert space of 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 Ref [14], we proved the following theorem.

Theorem. For any given even integer N, the ground state of is unique and has quantum numbers n₁ = n₂ = ^N/₂. Furthermore, can be written as

(7)

where is an orthogonal real basis for one species of ^N/₂ spinless fermions. Take as a matrix; then W is Hermitian and positive definite.

In Ref. [14], the uniqueness of is the consequence of the positive definiteness of W, which was proved under the condition ≠ 0. Ueda et al [15] 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 is still non-degenerate even if . 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 Ref. [15] guarantees that the ground state → as .

Theorem 1. Let A_r ≡ f_rf_r . Then, for any two distinct lattice points r and h, we have

(8)

Proof. Taking into account the form of Eq. (7), that is, (N/2, N/2), we have

(9)

where

and

So we have

(10)

since is a real basis.

By Lieb’s theorem, W is a positive definite matrix. Consequently, matrix is well defined. Therefore we have

(11)

Theorem 1 is proved.

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 . If the largest eigenvalues of this matrix satisfies the condition , where is a positive constant independent of , then has an off diagonal long-range order (ODLRO), which indicates that is a superfluid. On the other hand, by Theorem 1, we conclude that, if is a superfluid, it must be a Bose – Einstein condensate, namely, a macroscopic on-site pair of electrons that is condensed at p = 0 in . A detailed analysis on this point is found in Ref. [16].

Since Theorem 1 hold for any even integer N, in particular, it holds for the ground state of the negative – (,U,) symmetric PAM at half filling. Appling the inverse of the particle-hole transformation (6), we immediately obtain the following theorem.

Theorem 2: Let be the ground state of the positive -() symmetric PAM Hamiltonian at half filling. Define the spin operators of the electron by

(12)

Thus

(13)

Proof. By the inverse of the unitary particle-hole transformation, we have

,

where (r) = 1 if r A, and (r) = -1 if r ∈B

Therefore by Theorem 1,

(15)

If r and h belong to the same sublattices, then (r) (h) = 1 and hence

Otherwise, (r) (h) = -1 and

Theorem 2 is proved.

Remark 2. The workers in Ref. [17] 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-(,U,) symmetric PAM is antiferromagnetic. Since the Hamiltonian has the SU(2) spin symmetry and its ground state at half filling is nondegenerate, one would expect that Theorem 2 also holds for the longitudinal spin correlation functions.

Theorem 3. Let be the ground state of the positive-(,U,) symmetric PAM Hamiltonian at half filling. Let

(16)

then the spin correlation function C (r, h) has satisfied inequality (13).

Proof. We first show that for any pair of distinct lattice points r and h,

(17)

holds.

By definition,

.

Therefore we have

(18)

We first simplify the last term on the right-hand side of (18). Since r and h are distinct, [ S_rz, S_hy] = [ S_ry, S_hz] = 0. Therefore S_ryS_hx - S_rx S_hy is a Hermitian operator. Consequently, its expectation value in any state is a real quantity. On the other hand, since is a real matrix, its ground state must be chosen as state real vector. Therefore the expectation value F of S_ryS_hx - S_rx S_hy in must be a pure imaginary matrix. Consequently F ≡ 0.

Next, we apply the unitary operator

which rotates each spin about the S_z axis by an angle π/₂, to the expectation value of S_ry S_hy in and obtain

(19)

Substituting Eq.(19) into Eq.(18), we obtain the identity in Eq.(17), that is:

So we have,

Eq.(17) is proved.

Similarly, we can show that

(20)

By applying the unitary operator

to the expectation value of S_rzS_hz in .

Combining the identities in Eq.(17) and Eq.(20) and the inequality Eq.(13), we see that the longitudinal spin correlation of is antiferromagnetic. Since this conclusion is true for any > 0, it must also hold for the limit→ 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.

3. Summary and Conclusions

This work is an extension and detailed analysis of Tian article [16] and Ueda et. al. [15] for PAM. 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 it has been shown, by using a mean field slave – boson theory that at the half – filled, the ground state may be either antiferromagnetic or paramagnetic [18]. The line separating their region of stability is given by the critical value of their exchange constant. In all the lattice systems studied recently, it was observed that they have an antiferromagnetic ground state and the first excited state is always a spin singlet [8].

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. It is often assume that the low-temperature physics of heavy fermion systems is driven solely by AFM interactions. Therefore the heavy-fermion metals offer an interesting playground where magnetism and superconductivity can both compete and coexist [5-6, 19]. The theoretical prove here that the PAM ground state is antiferromagnetic means the model is a good candidate to investigate this competition and co-existence. The will involve investigating the spin correlations of f electrons in the ground state of the PAM which is of fundamental importance but was not discussed in this article. It is hoped that such investigation will be done to ascertain if we can observe in the PAM the growing consensus that the ground state of heavy fermion systems can be either normalconducting, superconducting or magnetically ordered with or without superconductivity. There are currently six heavy fermions compounds having superconducting ground state [20]. In general, the ground state properties of a system are of fundamental importance and the starting point for considering the excitations that enliven real systems [19].

ACKNOWLEDGEMENTS

We appreciate the very useful discussions with J. O. A. Idiodi, B. Iyorzor and S. I. Okunzuwa. EGA acknowledges partial funding by ICBR.