Electronic Hamiltonian in Second Quantization

M. A. Gusm˜ao– IF-UFRGS 1

FIP10601 – Text 10

Electronic Hamiltonian in second quantization

When discussing the Hartree and Hartree-Fock approximations as well as DFT, we used wavefunctions that depended on coordinates of all the electrons in the system. Since solids are macroscopic, such wave functions, in general, cannot be determined. Even in the independent-electron approximation the functions would be Slater determinants of dimension equal to the number of electrons. In the usual formalism of many-body theory, it is convenient to avoid the coordinate representation, as we will see in the following.

Occupation number representation

Although Slater determinants are not solutions of the interacting-electron problem, they provide a basis for a useful representation. In such a basis, the noninteracting part of the Hamiltonian is diagonal. We will use this basis to rewrite the complete Hamiltonian, but in doing so we will avoid to work explicitly with functions of coordinates. Taking into account that the single-particle states appearing in Slater determinants are known (Bloch states for a given lattice potential), one just needs to inform which ones are involved, i.e. occupied by electrons, in a given global state of the system. The individual states are characterized by a wavevector k and a spin quantum number σ, being denoted by |kσi in the corresponding (abstract) Hilbert space of a single electron. In coordinate/spin representation, we have hr|kσi = ψk(r)χσ, which can be generalized to include a band index (for simplicity, we will restrict this discussion to a single band). Then, the basis vectors of an abstract space with states of arbitrary electron numbers, known as Fock space, may be represented as

|n n n ...i , (1) k1σ1 k2σ2 k3σ3 where nkσ is the occupation number of the state |kσi, i.e., the number of electrons in that state. Due to the fermionic nature of electrons, this number can only assume the values 0 (zero) or 1 (one). It is important to point out that, by construction, there is a one-to-one correspondence between a particular basis vector of the Fock space and a Slater determinant built with the individual Bloch states |k σ i for which n = 1. i i kiσi

Second-quantization formalism

The Fock space contains states with any number of particles. Physical states of a system of N electrons are vectors belonging to an N-particle subspace, i.e., spanned by basis 2 M. A. Gusm˜ao– IF-UFRGS

vectors containing N occupation numbers equal to 1 and all the others equal to zero. However, it is convenient to deﬁne operators that allow us to go to subspaces with diﬀerent particle numbers. Such operators (for obvious reasons) are called single-particle creation and annihilation operators. We can view a general N-particle state as being obtained from the vacuum (zero-particle state) by successive application of N creation operators. † Denoting by ckσ the creation operator of an electron in the single-particle basis state |kσi, we have, for a generic basis state in Fock space,

nk σ nk σ nk σ |n n n ...i = c† 1 1 c† 2 2 c† 3 3 ... |0i , (2) k1σ1 k2σ2 k3σ3 k1σ1 k2σ2 k3σ3 where |0i ≡ |000 ...i is a simpliﬁed notation for the vacuum. The order of creation operators in Eq. (2) is relevant. This is due to the aforementioned correspondence with Slater determinants. Each pair of indices kσ is associated with a row of the Slater determinant. So, an exchange of two creation operators corresponds to an exchange of two rows of the determinant, with a consequent sign change. We say, then, that two creation operators anticommute, which means that their anticommutator is null:

† † † † † † {ckσ, ck0σ0 } ≡ ckσck0σ0 + ck0σ0 ckσ = 0 . (3) An immediate consequence of this relationship is that 2 † ckσ = 0 , (4) expressing Pauli’s Exclusion Principle. † It is easy to see that the Hermitian conjugate ckσ of the creation operator ckσ is the corresponding annihilation operator. It suﬃces to observe what happens to a single particle. In this case we have † ckσ|0i = |kσi , (5) and the dual relation is

hkσ| = h0|ckσ , (6)

which means that ckσ is a creation operator to the left. Now, taking into account the normalization of vectors, we have

† hkσ|kσi = h0|ckσckσ|0i = 1 . (7) This may be viewed in two ways:

† † hkσ|ckσ|0i = h0|0i = 1 ⇒ hkσ|ckσ = h0| , (8)

h0|ckσ|kσi = h0|0i = 1 ⇒ ckσ|kσi = |0i . (9) † So, we see that ckσ annihilates on the left while ckσ annihilates on the right. On the other hand, orthogonality between the vacuum and single-particle states allows us to write

hkσ|0i = h0|ckσ|0i = 0 ⇒ ckσ|0i = 0 , † † h0|kσi = h0|ckσ|0i = 0 ⇒ h0|ckσ = 0 . (10) M. A. Gusm˜ao– IF-UFRGS 3

We see that an annihilation operator annihilates the vacuum on the right while a creation operator annihilates the vacuum on the left. † The conjugation relationship between ckσ and ckσ allows to write, from Eq. (3), the anticommutation relation for annihilation operators,

{ckσ, ck0σ0 } ≡ ckσck0σ0 + ck0σ0 ckσ = 0 , (11) also showing that 2 (ckσ) = 0 , (12) as should be expected. The above derivations can be easily extended to independent-particle basis vectors of a many-particle Fock space. We just replace what we called “vacuum” by a basis vector with zero particles in the individual state corresponding to the operator subscript. The complete set of anticommutation relations for creation and annihilation of electrons in Bloch states with speciﬁed spin is

† † † {ckσ, ck0σ0 } = 0 , {ckσ, ck0σ0 } = 0 , {ckσ, ck0σ0 } = δkk0 δσσ0 , (13) which agrees with the generic algebra of fermion operators. The last relationship above can be veriﬁed by evaluating matrix elements of the anticommutator between states of zero or one particle. We deﬁne the occupation-number operator as

† nˆkσ ≡ ckσckσ . (14) It is easy to check that its eigenvalues are indeed 0 and 1, the only possible occupation numbers of a single-fermion state.

Operator representation in Fock space

It is convenient to express any operator in terms of creation and annihilation operators, since we know how the latter act on Fock-space vectors.. We begin by noticing that a † product like ck0σ0 ckσ has the same eﬀect of a projector within a single-particle subspace, since applying it to any state of a single electron, written as a linear combination of Bloch states, yields the same result as applying the projector |k0σ0ihkσ|. So, given that a generic single-particle operator A(1) may be written as

(1) X X (1) † A = Ak0σ0,kσ ck0σ0 ckσ . (16) kk0 σσ0 4 M. A. Gusm˜ao– IF-UFRGS

Note that, with the same meaning for A(1), Eq. (15) refers to a single-electron Hilbert space while Eq. (16) deﬁnes an operator in the Fock space. For operators that do not depend on spin, Eq. (16) simpliﬁes to

(1) X (1) † A = Ak0k ck0σ ckσ . (17) kk0σ The matrix elements can be evaluated in coordinate representation: Z (1) 3 ∗ (1) Ak0k = d r ψk0 (r)A ψk(r) . (18)

Similarly, let us consider a generic two-particle operator B(2) (also spin-independent, for simplicity). The projectors now involve two-particle states, written as direct products of single-particle ones, and the development is similar:

(2) X X 0 0 0 0 (2) B = |k1σ1i|k2σ2i hk1σ1|hk2σ2|B |k2σ2i|k1σ1i hk2σ2|hk1σ1| 0 0 k1k1σ1 k2k2σ2 X X (2) 0 0 = B 0 0 |k1σ1i|k2σ2ihk2σ2|hk1σ1| , (19) k1k2,k2k1 0 0 k1k1σ1 k2k2σ2 with Z Z (2) 3 3 0 ∗ ∗ 0 (2) 0 B 0 0 = d r d r ψ 0 (r)ψ 0 (r )B ψk (r )ψk (r) . (20) k1k2,k2k1 k1 k2 2 1 Comparing with the one-particle case, we see that a projector in a two-particle space is equivalent to the action of two annihilation and two creation operators. Thus, the Fock- space representation of the operator B(2) is

(2) X X (2) † † B = B 0 0 c 0 c 0 c c . (21) k1k2,k2k1 k1σ1 k2σ2 k2σ2 k1σ1 0 0 k1k1σ1 k2k2σ2

Electronic Hamiltonian

The general rules derived above to express one- and two-particle operators in Fock space allow to rewrite the electronic Hamiltonian [Eq. (1) of Text 9] in the form

X † 1 X † † H = εk c c + Uk0 k0 ;k k c 0 c 0 0 c 0 c , (22) kσ kσ 2 1 2 2 1 k1σ k2σ k2σ k1σ 0 kσ k1k1σ 0 0 k2k2σ where the matrix elements of the e-e interaction are given by Z Z 3 3 0 ∗ ∗ 0 0 0 Uk0 k0 ;k k = d r d r ψ 0 (r)ψ 0 (r )U(r − r )ψ (r )ψ (r) . (23) 1 2 2 1 k1 k2 k2 k1 M. A. Gusm˜ao– IF-UFRGS 5

With our choice of basis, the single-particle part of the Hamiltonian has a diagonal representation, involving the energies of Bloch states. Denoting by H(1) the one-electron 0 Hamiltonian, i.e., kinetic energy and lattice periodic potential (called Hl in the beginning of Text 9), we have (1) εk = hkσ|H |kσi , (24) which is spin-independent and can be evaluated in the coordinate representation as

Z h¯2 ε = d3r ψ∗(r) − ∇2 + V (r) ψ (r) . (25) k k 2m k

As remarked before, we are using, for simplicity, a notation without band indices, which is applicable when the relevant physical processes occur in a single band. This can be easily generalized to take into account other bands, with the biggest complication arising in the e-e interaction term, which may involve interactions between electrons in diﬀerent bands.

Fermion operators in real space

It is sometimes convenient to define annihilation and creation operators in real space (i.e., coordinate space). The usual notation for fermions (e.g., in Quantum Field Theory) is ˆ ˆ† ψσ(r) and ψσ(r), respectively. Since spatial coordinates are continuous variables, these operators are interpreted as representing a fermionic field, and are called field operators. Our notation includes a hat to avoid confusion with wavefunctions. In analogy to what we did in k-space, we can define the field operators by their action on the vacuum, i.e, ˆ† ψσ(r)|0i = |rσi , h0|ψσ(r) = hrσ|. (26) The one-particle state |rσi is a direct product |ri|σi, where |ri is a single-particle position eigenvector. So, these operators can be viewed as creation and annihilation operators of fermions at well defined points in space. It is straightforward to relate them to the previously defined k-space operators. For instance,

which means that ˆ† X ∗ † ψσ(r) = ψk(r)ckσ . (28) k Hermitian conjugation then yields

ˆ X ψσ(r) = ψk(r)ckσ , (29) k 6 M. A. Gusm˜ao– IF-UFRGS and the inverse forms are Z 3 ∗ ˆ ckσ = d r ψk(r) ψσ(r) , (30) Z † 3 ˆ† ckσ = d r ψk(r) ψσ(r) . (31)

These relationships are consistent with

X ∗ 0 X 0 0 0 ψk(r)ψk(r ) = hr|kihk|r i = hr|r i = δ(r − r ) , (32) k k which is the expected “normalization” for eigenvectors of an operator with a continuous spectrum. It implies that the anticommutator for the ﬁeld operators is

n ˆ ˆ† 0 o 0 ψσ(r), ψσ0 (r ) = δ(r − r )δσσ0 . (33)

Finally, the occupation-number operator is replaced by an operator describing the spatial density of electrons, ˆ† ˆ ρˆσ(r) = ψσ(r)ψσ(r) . (34)

Here we see a possible justiﬁcation for the expression “second quantization”: by comparison with the position probability density in Quantum Mechanics, the wave function appears “promoted” to an operator. Equation (34), after summation over the spin index, provides a density operator, whose existence was crucial to the formal construction of density functionals in our discussion of DFT (Text 9).

Wannier representation in Fock space

The electronic Hamiltonian written as in Eq. (22) corresponds to a Bloch representation in Fock space. In Text 4 we discussed the Wannier representation, which uses basis states associated to lattice sites instead of wavevectors. The Hamiltonian written in that Text can be extended to include the e-e interaction. Again we write it for the case of a single band (although it can be easily generalized to include a band index):

X 1 X X H = |iσihiσ|H(1)|jσihjσ| + |i0σi|j0σ0ihi0σ|hj0σ0|U|jσ0i|iσihjσ0|hiσ| . (35) 2 ijσ ii0σ jj0σ0

In analogy to what was done in Bloch representation, the projectors can be replaced by products of creation and annihilation operators. The Hamiltonian then takes the form

X † X † 1 X X † † H = ε c c − t c c + U 0 0 c 0 c 0 0 c 0 c , (36) 0 iσ iσ ij iσ jσ 2 i j ,ji i σ j σ jσ iσ iσ ijσ ii0σ jj0σ0 M. A. Gusm˜ao– IF-UFRGS 7

† where ciσ creates an electron of spin σ in the Wannier state associated to lattice site i (it is implicitly assumed that tii = 0). In addition, we use the notation

(1) ε0 ≡ hiσ|H |iσi , (37) (1) tij ≡ −hiσ|H |jσi , (38) 0 0 0 0 Ui0j0,ji ≡ hi σ|hj σ |U|jσ i|iσi , (39)

observing that the local energy ε0, the hopping integral tij, and the matrix elements of the e-e interaction do not depend on spin. The Coulomb-interaction matrix elements (39) generate same-site and inter-site terms (also inter-orbital if extended to more orbitals per site). The number of interaction parameters is usually reduced in speciﬁc models, based on physical arguments of relative importance. The simplest form for a single orbital per site keeps only the fully local interaction (i = i0 = j = j0). It is known as Hubbard model, and the Hamiltonian can be written as

X † X † X H = ε0 ciσciσ − tijciσcjσ + U ni↑ni↓ , (40) iσ ijσ i

† where niσ = ciσciσ is the number operator associated to the spin-state σ at site i. This model is widely employed in the study of strongly correlated electron systems. ˆ ˆ† It is worth remarking that the ﬁeld operators ψσ(r) and ψσ(r) can also be written in the Wannier representation. One just has to replace wavevectors by lattice sites, and Bloch by Wannier states or functions in Eqs. (28–32).

The second-quantized Hamiltonians introduced here will be starting points for our discussion of interactions in the remaining of this course.