4. Density functional theory 41
4.9. The symmetry-breaking dilemma
4.9.3. Broken-symmetry HF-Slater determinants
s
1 +U 4t
2
+ U 4t
−π
4. (4.82)
The occupations are
fb,σ = 1 2 +1
2cos(2θ), (4.83)
fa,σ = 1 2 − 1
2cos(2θ) (4.84)
for the natural orbitals
|b, σi= √1
2(|χ1,σi+|χ2,σi), (4.85)
|a, σi= √1
2(|χ1,σi − |χ2,σi). (4.86) The dissociation limitd→ ∞of the hydrogen molecule corresponds to theU/t→ ∞-limit of the half-filled Hubbard dimer.
4.9.3. Broken-symmetry HF-Slater determinants
In this section, we investigate the question if a many-particle wave function composed of broken-symmetry Hartree-Fock Slater determinants, i.e., broken-symmetry ground states of the Hartree-Fock approximation, can be constructed and distinguished from the exact ground state. We consider the half-filled Hubbard dimer as an example. In the atomic limit U/t→ ∞, the ground state of the half-filled Hubbard dimer is four-fold degenerate and the ground-state density operator is
ˆ
ρ(N)U/t→∞ =X4
i=1
1
4|Ψi,U/t→∞ihΨi,U/t→∞| (4.87)
|Ψ1,U/t→∞i= √1
2(|1001i − |0110i) (4.88)
|Ψ2,U/t→∞i= √1
2(|1001i+|0110i) (4.89)
|Ψ3,U/t→∞i=|1010i (4.90)
|Ψ4,U/t→∞i=|0101i. (4.91)
The one-particle reduced density matrix is diagonal and the occupations are equal to 1/2.
The symmetry-broken ground states of spin-polarized Hartree-Fock calculations in the limit U/t→ ∞ are
|ΨHF,U/t→∞,1i=|1001i (4.92)
|ΨHF,U/t→∞,2i=|0110i (4.93)
|ΨHF,U/t→∞,3i=|1010i (4.94)
|ΨHF,U/t→∞,4i=|0101i. (4.95)
4.9. The symmetry-breaking dilemma These Slater determinants have diagonal one-particle density matrices with integer occu-pations. The ensemble as a combination of all symmetry-broken Hartree-Fock ground states is equal to the ground-state density operator in Eq. (4.87) and, hence, the two cannot be distinguished by measurements. It should be noted here, that this is no longer the case if we leave away some of the Hartree-Fock states in Eq. (4.96): the density operator
ˆ˜
ρ(N)HF,U/t→∞ =X2
i=1
1
2|ΨHF,U/t→∞,iihΨHF,U/t→∞,i| (4.97) can be distinguished from the exact one in Eq. (4.87) with measurements of the spin-spin correlation function hSˆ1,zSˆ2,zi. The spin-spin correlation functions can also be measured experimentally with neutron scattering experiments, where the spin-structure factor is proportional to the inelastic neutron scattering cross section. The spin-spin correlation functions can be obtained from the spin-structure factor by a Fourier transform. The equivalence of the exact density operator in Eq. (4.87) and the ensemble-HF density operator in Eq. (4.96) suggests to generalize this idea to other situations. We evaluate this approach for the half-filled Hubbard dimer with a large interaction strength U/t >2, i.e., the regime where the Hartree-Fock approximation produces antiferromagnetic symmetry-broken states to investigate the resulting reduced density matrices. A one-particle reduced density matrix that is a minimum of the Hartree-Fock energy-expression in the regime is [Kamil et al., 2016]
The corresponding spin-flipped density matrix is
˜
The one-particle reduced density matrix of the ensemble ofρ(1),HF and the corresponding spin-flipped matrix similar to Eq. (4.96) is then
ρ(1),HF−ensemble= 1
The two-fold degenerate occupations of this one-particle reduced density matrix are (1− cos(2γ))/2 and (1 + cos(2γ))/2. Thus, they show the same qualitative dependence on the interaction strength as the occupations of the exact one-particle reduced density matrix discussed in section 4.9.2. However, the quantitative behaviour is different because the dependence of γ (Eq. 4.99) on the interaction strength is different from the dependence of θ on the interaction strength given in Eq. (4.82). The value of the density-matrix functional in the Hartree-Fock approximation, i.e., the Hartree-Fock interaction energy, is
FHFWˆ [ρ(1),HF−ensemble] = U
2 (4.102)
for the one-particle reduced density matrix ρ(1),HF−ensemble in Eq. (4.101). The Hartree-Fock density-matrix functional for the one-particle reduced density matrices of the broken-symmetry states are
FHFWˆ [ρ(1)A ] =FHFWˆ [ρ(1)B ] = 2t2
U . (4.103)
and different from the value of the ensemble-state. This is a direct consequence of con-cavity,
FHFWˆ [ρ(1)A /2 +ρ(1)B /2] =FHFWˆ [ρ(1),HF−ensemble]≥FHFWˆ [ρ(1)A ]/2 +FHFWˆ [ρ(1)B ]/2, (4.104) of the density-matrix functional in the Hartree-Fock approximation. In contrast, a convex density-matrix functional would obey
FWˆ[ρ(1),HF/2 + ˜ρ(1),HF/2]≤FWˆ[ρ(1),HF]/2 +FWˆ[˜ρ(1),HF]/2. (4.105) Thus, a convex density-matrix functional would not have produced the symmetry-broken states in the first place, but rather a spin-symmetric ground state. An alternative estimate of the interaction energy can be obtained by constructing many-particle wave functions that are Slater determinants and have the one-particle reduced density matrix in Eq. (4.98) respectively its spin-flipped modification in Eq. (4.100). We obtain the many-particle wave functions
|ΨHF,1i=ˆc†1,↑cos(γ−π/4) + ˆc†2,↑cos(γ+π/4) ˆc†1,↓cos(γ+π/4) + ˆc†2,↓cos(γ−π/4)|Oi (4.106)
|ΨHF,2i=ˆc†1,↓cos(γ−π/4) + ˆc†2,↓cos(γ+π/4) ˆc†1,↑cos(γ+π/4) + ˆc†2,↑cos(γ−π/4)|Oi.
(4.107) The density operator
ˆ
ρ(N)HF =X2
i=1
1
2|ΨHF,iihΨHF,i| (4.108) of these two many-particle wave functions has the one-particle reduced density matrix in Eq. (4.101) and the interaction energy U/2 cos2(2γ). This density operator ˆρ(N)HF can be distinguished from the exact ground-state density operator by measuring the spin-spin
4.9. The symmetry-breaking dilemma correlation functions. Results for the spin-spin correlation function hS~ˆ1·S~ˆ2iof the density operator ˆρ(N)HF and the exact density operator are shown in figure 4.2.
The characteristic difference in the spin-spin correlation functions also persist if an effective interactions strength Uef f is used in the Hartree-Fock approximation that is chosen such that the reconstructed one-particle reduced density matrix in Eq. (4.101) is identical to the exact ground-state density matrix6. We conclude that the reconstruc-tion of a spin-symmetric state from broken-symmetric states gives a qualitatively correct one-particle reduced density matrix. If the interaction energy is calculated from Slater determinants and not from the density-matrix functional in the Hartree-Fock approxima-tion, this scheme yields the same interaction energy as for a symmetry-broken state. Other shortcomings such as the systematic overestimation of the total energy in the Hartree-Fock approximation or over-localization of electrons are not affected by this symmetry reconstruction scheme.
A related method that reconstructs symmetric states from broken-symmetry Slater determinants is the variational symmetry-projected Hartree-Fock method [Schmid, 2004;
Schmid et al., 2005; Scuseria et al., 2011; Rodríguez-Guzmán et al., 2012, 2013]. This method has only recently made its way from nuclear physics to solid-state theory and is based on the idea to restore the symmetry of a Slater determinant with projections.
Firstly, any Slater determinant with M ∈ N particles in a d-dimensional one-particle basis can be parametrized by a d-by-d unitary matrix U ∈ U(d), |Ψ(U)i. Then required physical symmetries of the system can be restored with applications of the corresponding projection operators ˆPi. The projected determinant
|ΨU,Pi=
P
iPˆi|Ψ(U)i
P
i,jhΨ(U)|Pˆj†Pˆi|Ψ(U)i (4.109) is then used as a variational wave function. The approximate ground-state energy is obtained from the minimization
E0 ≈ min
U∈U(d)hΨU,P|H|ˆ ΨU,Pi. (4.110)
The approximation can be improved systematically by using multiple non-orthogonal determinants instead of one [Rodríguez-Guzmán et al., 2012, 2013]. Investigation for the one-dimensional and two-dimensional Hubbard models [Schmid et al., 2005; Rodríguez-Guzmán et al., 2012, 2013] as well as for molecules [Scuseria et al., 2011] have shown promising results.