We considered the single site impurity Anderson model with the aid of the application of Exact – diagonalization technique on an open boundary system of two interacting electrons on four sites with twenty eight basis electronic states. The results obtained shows that Increasing Ce content experimentally in CeCu2Si2 is equivalent to increasing the hopping matrix element t in this study; and also increasing the values of the on-site hybridization element, V, the system which was originally AFM becomes FM. We also observed that the cost of performing these theoretical simulations is approximately the same as simulating 2 electrons on 4 sites (1-D) in the Anderson model and 4 electrons on 8 sites (1-D) in the one band Hubbard model. KEYWORDS: CeCu2Si2, ferromagnetism, antiferromagnetism, triplet state, ground state. 1.0 INTRODCTION The problem of two “magnetic” impurities in a non – magnetic metallic environment has attracted attention in the past decades (Zhong, 2022). The wealth of understanding gained in the past fourty years on one – impurity has encouraged the attempt to clarify the rich behaviour of a system in which the Kondo effect (Itai and Fazekas 1996) (low-temperature suppression of the moment) competes against the tendency to magnetic behaviour induced by spin-spin Ruderman-Kittel- 1 Kasuya-Yosida (RKKY) or strong super – exchange interactions. Far from being yet understood, the interplay between Kondo effect and magnetism is indeed one of the key problems in the physics of heavy fermion systems ( Irkhin and Streltsov, 2022). Theoretically, the simplest models which are believed to account for the essential physics of these systems are the single – site impurity Anderson model (Anderson, 1961; Fan et al, 2022) and the analogous version of the Kondo model (Amadon and Hirsch 1996) . The challenges of the theorist are to explore the phase space of the parameters of these models in order to map the possible types of ground states that allow one to describe, and compare the theoretical findings with the rich variety of situation that nature provides (Enaroseha et al, 2021a). In this respect, the study of a simple sigle site – impurity model can be seen as a preliminary step toward the understanding of the physics of metals with lattice of magnetic impurities. 2.0MODEL AND METHODOLOGY We consider the single site – impurity Anderson Hamiltonian ( Anderson 1961; Zhu and Zhu, 2011; Enaroseha et al, 2021b, Enaroseha and Akpojotor, 2013) given by: H=¿❑∑kσ❑ξkCkσ+¿Ckσ+∑iσ❑Edniσ+∑i❑U❑ni↑ni↓+∑kσ❑∑i¿¿¿ ¿ (1) here the Ckσ+¿¿’s are the creation operators associated with a set of itinerant state (conduction electrons) of energy ξk, hybridizing (Vk) with two localized orbitals ( diσ+¿¿, associated with the magnetic impurities) of atomic energy Ed, located at the position R1 and R2. U represents the impurity coulomb repulsion energy, and niσ=diσ+¿diσ¿. H.C. is the Hermitian Conjugation. 2 We defined the Hilbert space for many-electron problems using the fermion operator formalism: the formalism is called “second quantization”. For each lattice site x  , and spin index we associate a fermion operator C x, One can freely take the conjugate, products, and linear combinations (with complex coefficients) of these operators (and the identity operator) to get new operators. We require that these operators satisfy the anticommutation relations {C+x, σ, Cy, τ} = δ x, y, δ στ, (2) And {C+x, σ, C+y, τ} = {Cx, σ, Cy, τ} = 0 (3) For any x, y  , and σ, τ =, where {A, B} = AB + BA. Note that equations (2) and (3) implies (C x, σ)2 = (C+x,σ)2 = 0. Physically, Cx,σ and C+x,σ are interpreted as the operators which respectively annihilate and create an electron at site x and spin σ. 3.0 THEORETICAL CALCALATIONS 3.1 CALCULATION FOR A SYSTEM OF 2-ELECTRONS ON A 4-SITE (1- D) USING THE SINGLE-SITE IMPURITY ANDERSON HAMILTONIAN AT SITE 1 Consider a system of two interacting electrons on four-sites with open boundary conditions using the single site impurity Anderson model (1), shown in Fig. 3.1. 1 2 3 4 Fig 3.1: Four interacting electrons on sites 1, 2, 3 and 4 in 1-D (SIAM). 3 The Single Site impurity Anderson model for two interacting electrons on four sites has twenty eight basis electronic states. At site 1 the Hamiltonian matrix given in (1), leads to an extended equation given in (4) H=¿❑t¿¿¿Ci+2↓+¿C1+1↓+Ci+2↑+¿Ci+3↑+Ci+3↑+¿Ci+2↑+Ci+2↓+¿Ci+3↓+Ci+3↓+¿Ci+2↓¿❑+¿¿¿¿¿¿Ef¿¿Ufi↑+¿fi↑fi↓+¿fi↓+¿¿¿V¿¿ (4) Expanding completely we have: H=¿❑t¿¿¿ C3↓+¿C21↓+¿¿C3↑+¿C4↑+C4↑+¿C3↑+C3↓+¿C4↓+C4↓+¿C3↓¿❑+¿¿¿¿¿ Ef¿¿Uf1↑+¿f1↑f1↓+¿f1↓+¿¿¿V¿¿ (5) Using the Hamiltonian (5) to act on the 28 basis electronic states, the results obtained were simplified in the Hamiltonian matrix. For simplicity, a 16 x 16 matrix was formed representing the singlet states matrix for the net spin 0 states (the antiferromagnetic states), see equation (6) and a 12 x 12 matrix was also formed representing the triplet states matrix for the net spin ±1 (the ferromagnetic states), see (9). Because of the complex nature of the matrices, the eigenvalues, eigenvectors and the wave function for a particular case was considered. Consider the case where Ef =1, u = 1, v = 2 and t =1. For the singlet states, the Hamiltonian matrix is: 4 2Efu4v000tt0000000000 0000tt0000tt0000 0000000000tt00tt 00000000000000tt tt00Ef2v0t000000000 tt000Ef2v0t00000000 0000t0Ef2v0t0t00000 00000t0Ef2v0t0t0000 000000t0Ef2v000t000 0000000t0Ef2v000t00 0tt000t00000t000 0tt0000t00000t00 00000000t0t000t0 000000000t0t000t 00tt00000000t000 00tt000000000t00 (6) The ground state energy is given by Es = -2.91137 (7) The corresponding ground state amplitudes are ¿ϕ⟩s=¿{-0.00632628 ¿1↑1↓⟩-0.28448¿2↑2↓⟩ -0.481464¿3↑3↓⟩-0.227213¿4↑4↓⟩ +0.0440036¿1↑2↓⟩-0.0440036¿1↓2↑⟩ -0.0573223¿1↑3↓⟩+0.0573223¿1↓3↑⟩ (8) +0.0393841¿1↑4↓⟩-0.0393841¿1↓4↑⟩ +0.37011¿2↑3↓⟩-0.37011¿2↓3↑⟩ -0.25426¿2↑4↓⟩+0.25426¿2↓4↑⟩ +0.330751¿3↑4↓⟩-0.330751¿3↓4↑⟩} The matrix of the Hamiltonian for the triplet states is given by equation (9) 5 Ef2v0t000000000 0Ef2v0t00000000 t0Ef2v0t0t00000 0t0Ef2v0t0t0000 00t0Ef2v000t000 000t0Ef2v000t00 00t00000t000 000t00000t00 0000t0t000t0 00000t0t000t 00000000t000 000000000t00 (9) The ground state energy of the triplet states is given by: Et = -1.55332 The corresponding ground state amplitude is ¿ϕ⟩t=¿{-0.0150754 ¿1↑2↑⟩+0.00266558 ¿1↓2↓⟩ +0.0987942¿1↑3↑⟩-0.0174684¿1↓3↓⟩ -0.121166¿1↑4↑⟩+0.0214241¿1↓4↓⟩ (10) -0.511189¿2↑3↑⟩+0.0903863¿2↓3↓⟩ +0.695246¿2↑4↑⟩-0.122931¿2↓4↓⟩ -0.447587¿3↑4↑⟩+0.0791404¿3↓4↓⟩} 3.3bCALCULATION FOR A SYSTEM OF 2-ELECTRONS ON A 4-SITE (1-D) USING THE SINGLE-SITE IMPURITY ANDERSON HAMILTONIAN AT SITE 2 6 Consider a system of two interacting electrons at four-sites with open boundary conditions using the single site impurity Anderson model (2), shown in Fig. 3.1. At site 2, the Single Site impurity Anderson model for two interacting electrons on four sites, given in (2), leads to an extended equation given in (11) H=¿❑t¿¿¿Ci+2↓+¿C1+1↓+Ci+2↑+¿Ci+3↑+Ci+3↑+¿Ci+2↑+Ci+2↓+¿Ci+3↓+Ci+3↓+¿Ci+2↓¿❑+¿¿¿¿¿¿Ef¿¿Ufi+1↑+¿fi+1↑fi+↓+¿fi+1↓+¿¿¿V¿¿ (11) Expanding completely we have: H=¿❑t¿¿¿C3↑+¿C4↑+C4↑+¿C3↑+C3↓+¿C4↓+C4↓+¿C3↓¿❑+¿¿¿¿¿Ef¿¿Uf2↑+¿f2↑f2↓+¿f2↓+¿¿¿V¿¿ (12) As usual we simplify the results obtained by using a 16 x 16 matrix to represent the singlet states matrix for the net spin 0 states (the antiferromagnetic states) see (13), and a 12 x 12 matrix was used to represent the triplet states matrix for the net spin ±1 (the ferromagnetic states) see (16). Because of the complex nature of the matrices, the eigenvalues, eigenvectors and the wave function for a particular case was considered. Consider the case where Ef =1, u = 1, v = 2 and t =1. The Hamiltonian matrix for the singlet states is given by (13): 7 m0000tt0000000000 02Efu4v00tt0000tt0000 0000000000tt00tt 00000000000000tt tt00Ef2v0t000000000 tt000Ef2v0t00000000 0000t000t0t00000 00000t000t0t0000 000000t00000t000 0000000t00000t00 0tt000t000Ef2v0t000 0tt0000t000Ef2v0t00 00000000t0t0Ef2v0t0 000000000t0t0Ef2v0t 00tt00000000t000 00tt000000000t00 (13) The ground state energy is given below: Es = -2.20148 (14) The corresponding ground state amplitude is ¿ϕ⟩s=¿{0.0159005¿1↑1↓⟩+0.0180259¿2↑2↓⟩ +0.52829¿3↑3↓⟩+0.436096¿4↑4↓⟩ -0.0175024¿1↑2↓⟩+0.0175024¿1↓2↑⟩ -0.0921165¿1↑3↓⟩+0.0921165¿1↓3↑⟩ (15) -0.0838086¿1↑4↓⟩+0.0838086¿1↓4↑⟩ -0.101482¿2↑3↓⟩+0.101482¿2↓3↑⟩ +0.0923863¿2↑4↓⟩-0.0923863¿2↓4↑⟩ -0.480028¿3↑4↓⟩+0.480028¿3↓4↑⟩} The matrix of the Hamiltonian for the triplet state is given by (16) 8 Ef2v0t000000000 0Ef2v0t00000000 t000t0t00000 0t000t0t0000 00t00000t000 000t00000t00 00t000Ef2v0t000 000t000Ef2v0t00 0000t0t0Ef2v0t0 00000t0t0Ef2v0t 00000000t000 000000000t00 (16) The ground state energy of the triplet states is given by: Et = -1.28684 (17) The corresponding ground state amplitude is ¿ϕ⟩t=¿{0.112253¿1↑2↑⟩+0.00377247 ¿1↓2↓⟩ -0.705719¿1↑3↑⟩-0.0237169¿1↓3↓⟩ +0.660665¿1↑4↑⟩+0.0222028¿1↓4↓⟩ (18) +0.13523¿2↑3↑⟩+0.00454464 ¿2↓3↓⟩ -0.144452¿2↑4↑⟩-0.00485457 ¿2↓4↓⟩ +0.112253¿3↑4↑⟩+0.00377247 ¿3↓4↓⟩ 4. 0RESULTS AND DISCUSSION. 4.1RESULTS FOR A SYSTEM OF 2 ELECTRONS ON A 4 SITES (1-D) USING THE SIAM HAMILTONIAN AT SITE 1 AND 4. The Hamiltonian matrix form is given by (6) for the singlet state energy and (9) for the triplet states energy respectively at site 1 and 4. Due to edge effect in the interactions, site 1 and 4 are treated together, as open boundary condition is considered here. Hence sites 1 and 4, shown in Fig. 3.1 are having the same ground 9 state energies for both the singlet states and triplet states. Numerical analyses carried out to obtain the series coefficients for the ground state energies in order to determine the single and multi – phase dynamics are given in Fig. 4.1, 4.2 and 4.3 respectively. -20-1001020304050 -25-20-15-10-50 Es Et UEs, Et Fig. 4.1:Graph of Singlet (E s) and Triplet (Et) state energies plotted against U for a system of 2 electrons on 4 lattice sites at sites 1 and 4 Using the SIAM (1-D). -25-20-15-10-50510152025 -80-60-40-20020406080 E s tEs, Et Fig 4.2:Graph of Singlet (E s) and Triplet (Et) state energies plotted against t for a system of 2 electrons on 4 lattice sites at site 1 and 4 using SIAM (1-D). 10 -2.5-2-1.5-1-0.500.511.522.5 -10-8-6-4-20246810 E s Es, EtV Fig 4.3:Graph of Singlet (E s) and Triplet (Et) state energies plotted against V for a system of 2 electrons on 4 lattice sites at site 1 and 4 using the SIAM (1-D). 4.3bRESULTS FOR A SYSTEM OF 2 ELECTRONS ON A 4 SITE (1-D) USING THE SIAM HAMILTONIAN AT SITES 2 AND 3. The Hamiltonian matrix form for the SIAM is given by (13) for the singlet state energies at site 2 and site 3 while (16) provides the Hamiltonian matrix for the triplet states energies at site 2 and 3. There is edge effect in the interactions, as open boundary condition is used here. Hence site 2 will interact with both sites 1 and 3, while site 3 will interacts with sites 2 and 4, as shown in Fig 3.1. As a result of the edge effects, sites 2 and 3 will provide the same ground state energies for the singlet and triplet states. Numerical analyses carried out to obtain the series coefficients for the ground state energies in order to determine the magnetic phase diagram are given in Fig 4.4, 4.5 and 4.6 respectively. 11 -20-1001020304050 -25-20-15-10-505 Es Et UEs, Et Fig 4.4:Graph of Singlet (E s) and Triplet (Et) state energies plotted against U for a system of 2 electrons on 4 lattice sites at site 2 and 3 Using the SIAM (1-D). -25-20-15-10-50510152025 -80-60-40-20020406080 E s tEs , Et Fig. 4.5:Graph of Singlet (E s) and Triplet (Et) state energies plotted against t for a system of 2 electrons on 4 lattice sites at site 2 and 3 using SIAM (1-D). 12 -2.5-2-1.5-1-0.500.511.522.5 -10-8-6-4-202468 E s VEs, Et Fig 4.6:Graph of Singlet (E s) and Triplet (Et) state energies plotted against V for a system of 2 electrons on 4 lattice sites at sites 2 and 3 using the SIAM (1-D). 4.2 DISCUSSION OF RESULTS FOR THE 2 ELECTRONS ON 4-SITES (1-D). Inspection of Fig. 4.1 and 4.4, shows that, as the values of the on-site Coulomb repulsion of the f electrons, U increases, the lattice system that was originally AFM becomes unstable as U→∞, the instability is in agreement with the result obtained by Batista et al (2001). Batista et al (2001) applied constrained-path QMC method to the SIAM at half fillings and obtained magnetic instability as U→∞, and the system has antiferromagnetic groundstate. Increasing U slightly, produces a FM groundstate. The behaviuor with doping of one and two electrons is reminiscent of Nagaoka FM and its instability as U→∞ in the nearest-neighbour hopping hubbard model. While our model (Anderson and Hubbard) are not the same, the proposed mechanism for FM is very similar: An electron lower it kinetic energy by moving through a FM groundstate which agree qualitatively with results obtained by Zhong (2022).. In this study the electron lower the systems kinetic energy by inter band processes 13 enabling its hopping between two occupied f-state anti-aligned with its spin. The need for inter band processes differentiate our mechanism from Nagaoka’s. The cost of performing these simulations is approximately the same as simulating 2 electrons on 4 sites (1-D) in the Anderson model and 4 electrons on 8 sites (1-D) one band Hubbard model. Observations from Fig. 4.2 and 4.5 shows that, as the values of the hopping matrix element, t is increased, the system which was originally AFM (metal) becomes FM (Insulator), in agreement with theoretical and experimental results obtained by Streltsov et al (2011). Streltsov et al (2011) applied Green’s function technique with the use of Linear Muffin-Tin Orbital (LMTO) and the local density approximately methods to show the ce 4f for Ce-α, Ce-β, Ce-γ and CeCu 2Si2 can produce a metal to insulator transition which they confirmed on experimentally increasing the Cerium content of CeCu2Si2 using different types of spectra (Photoemission Bremsstrahlung isochromatic and Electron Energy Loss) experiments. Increasing Ce contents in CeCu2Si2 is equivalent to increasing t in this study. Inspection of Fig. 4.3 and 4.6 shows that, as the values of the on-site hybridization element, V is increased, the system which was originally AFM becomes FM. 5.0CONCLUSION In this research work, the Exact-Diagonalization technique has been employed to provide information on the behavior of two interacting electrons in quarter filled band on 1-D lattices. The lattices considered are small in size. It is not obvious that the results obtained for larger lattices in the literature are applicable here. In 1-D, the lattice systems considered are 2 electrons on 4 sites with open boundary condition. Finite sized lattices with open boundary condition 14 were specifically considered and the dynamics of the interacting electrons were described by the single site – impurity Anderson Hamiltonians used. The good agreement of the results in this work with the results of workers like Streltsov et al (2011) who applied Green’s function technique with the use of Linear Muffin-Tin Orbital (LMTO) and the local density approximately methods to show the ce 4f for Ce-α, Ce-β, Ce-γ and CeCu 2Si2 can produce a metal to insulator transition which they confirmed on experimentally increasing the Cerium content of CeCu 2Si2 using different types of spectra (Photoemission Bremsstrahlung isochromatic and Electron Energy Loss) experiments. Increasing Ce contents in CeCu 2Si2 is equivalent to increasing t in this study. The theoretical treatment covers the SIAM at quarter fillings in 1-D. The maximum number of electrons encountered in this work are two, keeping up to twenty eight basis electronic states. In all these systems, even though the underlying solid state chemistry are rather different, the resulting phase diagrams are strikingly similar and robust. This similarity suggests that the overall feature of these phase diagrams is controlled by a single energy scale. The results obtained reveal that the ED technique represents a suitable approach for a deeper understanding of the rich single and multi - phase dynamics of CeCu2Si2. The results obtained here should also be of relevance for actual Heavy Femion systems like CeCu2Si2 or related compounds.