Introduction to the Hubbard Model
Author: Juan Luis D´ıaz Jim´enez Facultat de F´ısica, Universitat de Barcelona, Diagonal 645, 08028 Barcelona, Spain.
Advisor: Maria dels Angels` Garcia Bach
Abstract: The Hubbard model is very important for the study of of magnetic phenomena and strongly correlated electron systems. This work serves as an introduction to the Hubbard model and a presentation of the elements necessary to reach it. Here it is applied to a simple case to see how you work with it.
I. INTRODUCTION Kn ≈ Ki, Unn ≈ Uii, Uen ≈ Uei 2. Born-Oppenheimer approximation: We assume John Hubbard, at the beginning of 1960, proposed in that the ions in their equilibrium position are at a series of articles a model to describe electrons in transi- rest. We can do this approximation because the tion metal monoxides and electron correlation in narrow mass of the ions is larger than the mass of the elec- energy bands. Due to the complexity of the study of these trons, therefore the speed of the electrons will be systems given the vast amount of bounds and continuum much greater than that of the ions. electron energy levels, Hubbard proposed to reduce the 0 study to a single localized orbital level per site. This Ki ≈ 0, Uii ≈constant (negligible), Uei ≈ Uei model opens the door to study the interactions between 3. Independent electron approximation: We approxi- electrons, giving rise to insulating, magnetic, and even mate the interaction energy as the sum of the in- novel superconducting effects in a solid. teraction energies of the valence electrons. But before talking about the Hubbard model it is nec- essary to introduce Bloch’s theorem used for delocalised X X U + U 0 ≈ (U (r ) + W (r )) = V (r ) (3) studies. ee ei i i i i i i i i
Ui(ri) is the average potential energy created by II. THE BLOCH’S THEOREM the ions on the electron i and Wi(ri) is the inter- action energy of the electron i with the effective A. Schr¨odinger’sequation field created by the rest of electrons, in which each electron moves independently. To describe the steady states of all the particles of a solid we use Schr¨odinger’sequation With all these approaches the Hamiltonian of the solid has been reduced to the Hamiltonian for the valence elec- HΨ = EΨ (1) trons in the solid, as seen as independent particles. From now on, H is understood in this sense. where H is the hamiltonian of the solid, Ψ is the wave X X function of the steady state and E is the energy of the H = Hj = Kj + Vj = solid in this state. j j The Hamiltonian is the sum of the operators of kinetic 2 X ~ 2 X energy and potential energy. − ∇ + (U (r ) + W (r )) = h (4) 2m j j j j j j j j H = K + U, K = Ke + Kn,U = Uee + Unn + Uen (2) The solution to the Schr¨odinger’sequation associated with this Hamiltonian can be writen as an antisym- Ke and Ki are the kinetic operators for the electrons metrized product of monoelectric wave functions ϕi(ri), and ions respectively. Uee, Uii, Uei are the potential oper- where Φ = ϕ1(r1)ϕ2(r2)...ϕN (rN ) ators for interactions between electrons, ions and electron and ion. hiϕi = iϕi (5) Due to the number of variables that we have in the equation, it is necessary to make some approximations. The Hamiltonian is independent of spin. The solutions of H give us the part of the space of the wave function. 1. Valence approximation. We assume that the elec- The function of the electron will be ϕ ⊗ σ where σ is the trons in the inner layers are attached to the nucleus spin. To refer to the spatial part we use ϕ and to refer forming an ion. Therefore we only take into account to the function of the electron considering also the spin the most external electrons, the valence electrons, we use Φ. P to work with the solid Hamiltonian. The energy of the solid is the sum of all i,E = i i. Introduction to the Hubbard Model Juan Luis Dıaz Jimenez
B. Bloch functions The functions Wn(Rj, r) are the Wannier functions. They can be calculated from the Bloch functions To find the Bloch functions it is necessary to solve (5). 1 X −ik·Rj We must bear in mind that if the ionic nuclei are dis- Wn(Rj, r) = √ ϕn(k, r)e (12) N tributed periodically and regularly we can assume that k the potential V (r) has the periodicity of the Bravais lat- The summation is over all k-vectors in a Brillouin zone. tice. Therefore we can develop V (r) in Fourier series Each Wannier function is centered at the midpoint of the cell. In addition, Wannier functions form an or- X iG·r V (r) = VGe thonormal complete basis G Z Z 1 −iG·r ∗ VG = V (r)e dr Wn (Ri, r)Wm(Rj, r)dr = Vc cell Z 1 X 0 (6) eik·Ri−ik ·Rj ϕ∗ (k, r)ϕ (k0, r)dr = N n m kk0 Where G are the vectors of the reciprocal lattice and V c 1 X is the volume of the cell. eik·(Ri−Rj )δ = δ δ (13) N nm nm ij Since V (r) has the periodicity of the Bravais lattice, k the probability density of the stationary states must also As it is a complete base have this probability. The wave functions associated with these states that meet this condition are of the form: X I = |Wn(Ri, r)ihWn(Ri, r)| (14) ϕ(r) = eif(r)u(r) (7) i Therefore we can say that Wannier functions are lo- u(r) have the periodicity of the lattice and f(r + R) = calized orthonormal functions about individual points in f(r) + f(R), where f(R) = k · R. This the lattice. In other hand, Bloch functions are extended functions. The transformation between both functions is 3 X ni a unitary transformation, i.e. the Hilbert space gener- k = b (8) N i ated by both is the same. Thanks to this we can work i=1 i with one or other according to our needs. The Wannier functions are a great tool to work in sit- is a constant vector where b are the vectors of the base i uations where the spatial position of electrons plays an of the reciprocal lattice and N are the numbers of cells i important role. The most important areas of aplication of the direct lattice in the direction i. are: Bloch’s theorem says that the value of a Bloch func- tion subjected to a translation R in the lattice of Bravais 1. Attempst to derive a transport theory for Bloch differs in a phase eik·R. The probability density is a real electrons. value. 2. Phenomena involving localized electronic levels. if(r+R) ϕ(r + R) = e u(r + R) = 3. Magnetic phenomena. eif(R)eif(r)u(r) = eif(R)ϕ(r) = eik·Rϕ(r) (9) Since the Wannier functions are located on the lat- tice knots, they can be writen as a linear combination The Bloch functions are of atomic functions. As we only take into account the ik·r valence electrons of each atom, we can assume that of- ϕ(k, r) = e u(r) (10) ten each Wannier function can be approximated by an atomic function or an especific linear combination of (quasi-)degenerated atomic functions determined by the III. WANNIER FUCTIONS atoms with which we are working. For example if we are studying a copper lattice the Wannier function could be The Bloch functions are periodic in k-space. Therefore W ∼ χ . 3dx2−y2 they have a Fourier series in plane waves in the reciprocal When approximate Wannier functions by atomic func- of the reciprocal space, that is in the direct space. ϕn can tions we lose the property of orthogonality between Wan- be written like this: nier functions centered on different cells because the atomic functions can overlap with the functions of the neighboring atoms (and the atomic functions are not or- 1 X ik·Rj ϕn(k, r) = √ Wn(Rj, r)e , (11) thogonal unlike the Wannier functions) as we can see in N j Fig.(1).
Treball de Fi de Grau 2 Barcelona, June 2018 Introduction to the Hubbard Model Juan Luis Dıaz Jimenez
they must comply with the anticommutation relations of two operators.
+ 0 + + {ciσ, cjσ0 } = δijδσσ , {ciσ, cjσ0 } = 0, {ciσ, cjσ0 } = 0 (16) Let’s see an example of how these operators work: If we have the state |α, β, α, βi and the opretaors + + + c1β, c1α, c1α, c2β, c2α then
+ c1β|α, β, α, βi = |βα, β, α, βi + c1α|α, β, α, βi = 0 + c1βc2β|α, β, α, βi = |αβ, 0, α, βi + c1αc2α|αβ, 0, α, βi = |β, α, α, βi (17)
FIG. 1: Interaction of two atomic functions depending on the B. Representation of the Hamiltonian according to semidistance. r1, r2 and r3 they are the semidistance to the Wannier functions first, second and third neighbor. We can see that the 4s func- tion is much more extended than the 3d function, therefore it will overlap with more functions. Now that we have seen how the creation and destruc- tion operators work and some of their properties, we can use them to rewrite the Hamiltonian using the Wannier If the Wannier functions with which we are going to functions. We want to write the Hamiltonian as a func- work can be approximated by little extended atomic func- tion of the destruction and creation operators. For this tions then we can suppose that the orthonormality is a we will use that the Wannier functions are a complete good approximation, since we are only interested in in- base teractions with first neighbors. Taking the Hamiltonian from (15)
X X 0 IV. THE HUBBARD MODEL hj = I hjI = j j X X Let’s recover the associated Hamiltonian to the solid |W (R , r)ihW (R , r)| h H = K +U +U 0 +U 0 +U −U . Therefore nσ i nσ i j e ee ei ii medium medium niσ j 0 0 X X Ke + Uei + Uii + Umedium ≡ hj |Wmσ0 (Rj, r)ihWmσ0 (Rj, r)| j mjσ0 1 X (18) U − U ≡ g(r , r ) ee medium 2 i j ij X 1 X + H = h + g(r , r ) (15) |Wnσ(Ri, r)i ≡ cniσ j 2 i j j ij hWmσ(Rj, r)| ≡ cmjσ X e2 hWnσ(Ri, r)| hj|Wmσ(Rj, r)i ≡ hnmij (19) Where g(ri, rj) = |ri − rj|. 4π0 Now we can rewrite the Hamiltonian through the Wan- j nier functions and the creation and destruction operators.
X X X X + hj = hnmijcniσ|0ih0|cmjσ = A. Creation and destruction operators j nm ij σ X X X + hnmijc cmjσ (20) + niσ The operator cmiσ creates a electron in a state nm ij σ Wm(Ri, r) with spin σ ∈ {α, β} (where α is for spin up and β is for spin down). This is the creation operator. Since g is a function of two variables, the identities The operator cmiσ destroys a electron in a state that we will use will be W (R , r) with spin σ ∈ {α, β}. This is the destruc- m i X tion operator. I = |Wnσ(Ri, r)Wmσ0 (Rj, r)ihWmσ0 (Rj, r)Wnσ(Ri, r)| Since these operators work with electrons (which are nmijσσ0 fermions) and wave functions must be antisymmetric, (21)
Treball de Fi de Grau 3 Barcelona, June 2018 Introduction to the Hubbard Model Juan Luis Dıaz Jimenez
X X 0 g(ri, rj) = I g(ri, rj)I = ij ij X X X 0 hWn(Ri, r)Wm(Rj, r )| nmn0m0 ijkl σσ0 0 0 0 0 ∗g(r − r )|Wn(Rk, r)Wm(Rl, r )i + + ∗cniσcmjσ0 cm0lσ0 cn0kσ (22) We reduce our study to the case of a single band, there- fore m = n = 1 and FIG. 2: Pictoric representation of the kinetic term t in the Hubbard Hamiltonian. hijnm = hij = hW (Ri, r)|h(r)|W (Rj, r)i (23)
0 0 0 gijkl = hW (Ri, r)W (Rj, r )|g(r−r )|W (Rk, r)W (Rl, r )i (24) Now our Hamiltonian can be written as
X X + 1 X X + + H = h c c + g c c 0 c 0 c FIG. 3: Pictoric representation of the one site repulsion term ij iσ jσ 2 ijkl iσ jσ lσ kσ ij σ ijkl σσ0 U in the Hubbard Hamiltonian. (25) We use Hubbard’s approximations to rewrite hij and gijkl. are first neighbors is due to the wave functions decrease exponentially. If i 6= j then hij = −t when i, j are nearest neighbors, and zero otherwise. This is the hopping term. The term U in absence of the t (U >> t) would favor For gijkl, if i = j = k = l then gijkl = U. For the rest local magnetic moments, since it would suppress the pos- of the cases gijkl = 0. sibility of a second electron at singly occupied sites. In Therefore Hubbard’s Hamiltonian is this case, the energy levels are determined by the number of double occupied sites and the number of electrons in X X + X + + relation to the number of sites. H = −t ciσcjσ + U ciαciβciβciα (26)
Treball de Fi de Grau 4 Barcelona, June 2018 Introduction to the Hubbard Model Juan Luis Dıaz Jimenez imation, we will also assume that there can only be one To calculate the energy we must calculate the eigen- site doubly occupied. values of the matrix. For this we take t = λU and U = 1. The first thing we will do is create a base in the Hilbert We vary the value of λ. space with the occupation numbers. The first 6 elements Note that in small t, the fundamental state is domi- of the base that we take are the neutral states nated by the neutral states (states in which there is only one electron in each knot). When growing t the weight |α, β, α, βi = e1, |α, α, β, βi = e2, of the ionic states (states in which one knot is empty |β, α, β, αi = e3, |α, β, β, αi = e4, and another has two electrons) is increased, indicating a
|β, β, α, αi = e5, |β, α, α, βi = e6, mobility of the electrons and the possible transition to a conducting state (Mott transition), with a state domi- Applying the Hamiltonian to ei we find other possible nated by delocalized electrons. combinations of these four electrons. t E0 neutral weight ionic weight The total number of combinations (with the above re- 0.1 -0.088742 0.921662 0.078338 strictions) between the four electrons is 30, that is, our 0.5 -1.230997 0.568828 0.431172 Hilbert space has dimension 30 and is made up of 30 0.7 -1.907582 0.509556 0.490444 components. We apply H to e : 1 1.1 -3.293969 0.452992 0.547008 H|α, β, α, βi = −t|0, αβ, α, βi − t|αβ, 0, α, βi TABLE I: Energy values of the ground state, the weight of +t|α, 0, αβ, βi + t|α, αβ, 0, βi − t|α, β, 0, αβi − neutral states and ionic states for different λs. t|α, β, αβ, 0i =
−te7 − te8 + te9 + te10 − te11 − te12 (27) V. CONCLUSIONS
Realizing this same calculation for the remaining ele- In this paper we have seen how to introduce the Wan- ments of the base we obtain the matrix of the Fig. (4). nier functions as a unitary transformation of the Bloch functions. We have also seen how to rewrite the Hamilto- nian of the solid using the Wannier functions getting to express the Hamiltonian using the creation destruction operators. This Hamiltonian is called Hubbard’s Hamil- tonian. Finally, we have calculated its associated matrix for the Hamiltonian Hubbard case of 4 sites. This model allows us to study the passage of the insulating material to the conductive material (Mott transition) or to use the perturbation theory for small t to find the Hamiltonian of Heisenberg. These are some examples of the many utilities that the Hubbard model has.
Acknowledgments FIG. 4: Matrix for the 4-site Hubbard Hamiltonian. I would like to thank my adviser, Maria Angels,` who If we allow two doubly occupied sites, we just have to guided and helped me, and my parents and my sister add five more elements to the base, the 2U terms would who have always supported me with my studies. Thanks appear on the diagonal and the terms t would have to be also to all the researchers who have preceded me, without calculated. them this work would not be possible.
[1] Neil W. Ashcroft and N. David Mermin, Solid State George Sawatzky (Eds.),Quantum Materials: Experiments Physics, (W.B. Saunders Company, 1976). and Theory,(Forschungszentrum Jlich, 2016) [2] Stanley Raimes, Many-Electron Theory, (North-Holland [5] S Akbar Jafari, Introduction to Hubbard model and exact Publishing Company, Amsterdam London 1972) diagonalization, (Iranian Journal of Physics Research, Vol. [3] O. Madelung, Introduction to solid-state theory, (Springer, 8, No. 2, 2008). Berlin, 1978) [4] Eva Pavarini, Erik Koch, Jeroen van den Brink, and
Treball de Fi de Grau 5 Barcelona, June 2018