Gap and spectral function

functional by using the decomposition of the interaction Hamiltonian of Eq. (5.33) as

F_β^Wˆ[ρ⁽¹⁾]≈F_β,loc^Wˆ [ρ⁽¹⁾] =^X

R

F_β^WˆR[ρ⁽¹⁾] +F_β^Wˆrest[ρ⁽¹⁾]. (5.34) Thus, the approximation assumes linearity of the density-matrix functional with the in-teraction Hamiltonian. The local approximation is a lower bound for the exact functional, i.e.

F_β^Wˆ[ρ⁽¹⁾]≥F_β,loc^Wˆ [ρ⁽¹⁾]. (5.35) This can easily be seen when considering that the minimization problem of Eq. (5.25) has been approximated by a sum of several independent minimization problems. As a consequence also the total energy obtained with the local approximation is a lower bound of the exact ground-state energy.

Results of the local approximation for small Hubbard chains at zero temperature have been reported [Blöchl et al., 2011]. Blöchl et al. have used individual sites as local clus-ters R and the numerically exact density-matrix functionals for the local contributions F_β^WˆR[ρ⁽¹⁾]. The remaining term F_β^Wˆrest[ρ⁽¹⁾] vanishes because ˆW_rest = 0 for the Hubbard model. The results show that the ground-state energy, double occupancies, and spin-spin correlation functions are well reproduced. However, there is no derivative discontinu-ity of the energy at integer particle numbers and, hence, a vanishing fundamental gap.

Approximations of the gaps with Eq. (2.50) show good agreement with the exact gaps.

The unique feature of the local approximation compared to parametrized approxima-tions of the density-matrix functional is that it can be systematically converged to the exact result by subdividing the interaction into fewer terms in Eq. (5.33).

5.6. Gap and spectral function

Reduced density-matrix functional theory with the exact density-matrix functional gives the exact ground-state energy and one-particle reduced density matrix of a system. Thus, for the exact ground state of a particle-number conserving Hamiltonian the energy will consist of linear segments between integer particle numbers and the fundamental gap can be evaluated with Eq. (2.48), Eq. (2.49) or Eq. (2.50). All three equations are equivalent in this case.

However, we are not aware of an approximate parametrized density-matrix functional or practical approximation scheme that leads to fractional occupation numbers at zero temperature ¹, is applicable to a general system and reproduces the derivative disconti-nuities of the exact total energy at integer particle numbers. Thus, the characterization of a material based on the definition of the fundamental gap in Eq. (2.48) is not appli-cable because the differences in the derivatives of the total energy approaching integer particle numbers from above and from below are equal. The approximation of the gap by Eq. (2.50) is applicable in the case of a smoothened derivative discontinuity but system-atically underestimates the gap [Sharma et al., 2008].

1This excludes the HF-approximation.

5.6.1. Local spectral function

The one-particle spectral function defined in Eq. (2.151) is not directly accessible in RDMFT. Several methods for the approximation of the spectral function in an RDMFT calculation have been proposed and analyzed [Sharma et al., 2013; Di Sabatino et al., 2015, 2016]. However, these approximations are not suitable for the application within a hybrid method combining DFT and RDMFT. For example, the approximation proposed by Sharma et al. [Sharma et al., 2013] requires many evaluations of the derivatives of the density-matrix functional and is tailored for parametrized density-matrix functionals that can be evaluated with a rather small computational effort. This approach is not suitable for density-matrix functionals that are evaluated from a numerically demanding constrained minimization problem. The evaluation of the density-matrix functional from a constrained minimization over many-particle wave functions, that will be discussed in detail in section 9, opens another route for the approximation of the spectral function.

Consider the one-particle reduced density matrix ρ⁽¹⁾ and the corresponding density-matrix functional F^Wˆ[ρ⁽¹⁾]. In section 5.2 we have shown that the derivative of the density-matrix functional with respect to the elements of the one-particle reduced density matrix is

∂F_β^Wˆ[ρ⁽¹⁾]

∂ρ⁽¹⁾_α,β =−˜h_β,α, (5.36)

where ˜h_β,α are the Lagrange multipliers of the constraints of the one-particle reduced density matrix. The minimization problem of the ground-state energy in Eq. (5.3) has the minimum condition

0 =h_β,α−˜h_β,α−µ1 (5.37)

with the matrix elements h of the one-particle Hamiltonian of the system and the La-grange multiplierµof the particle-number constraint. This minimum condition only holds for a one-particle reduced density matrix ρ⁽¹⁾ that does not lie at the boundary of the subspace of ensemble N-representable one-particle reduced density matrices. So far we have not obtained anything useful, because if we could compute the spectral function of the Hamiltonian

Hˆ˜ =^X

α,β

(˜h_β,α−µδ_α,β)ˆc^†_αcˆ_β+ ˆW (5.38) we could certainly also compute the spectral function of the original Hamiltonian

Hˆ =^X

α,β

h_α,βˆc^†_αcˆ_β+ ˆW . (5.39) We include the local approximation introduced in section 5.5 and a simple truncation of the one-particle basis² into our considerations. We assume that the interaction ˆW can be split up into local contributions, i.e., ˆW = ^PRWˆ_R. Thus, we approximate the density-matrix functional as

F^Wˆ[ρ⁽¹⁾]≈^X

R

F^WˆR[ρ⁽¹⁾]≈^X

R

F^WˆR[ρ⁽¹⁾_R ], (5.40)

2A systematic truncation of the one-particle basis is introduced in section 8 in the form of the adaptive cluster approximation.

5.6. Gap and spectral function where ρ⁽¹⁾_R is the one-particle density matrix of the truncated one-particle basis that only contains states in the vicinity of the site R. F^WˆR[ρ⁽¹⁾_R ] is evaluated with a constrained minimization over many-particle wave functions. The derivatives are

∂F_β^WˆR[ρ⁽¹⁾_R ]

∂ρ⁽¹⁾_R,α,β =−˜h_R,β,α (5.41)

and the minimum condition of the total-energy minimization reads 0 = hβ,α−^X

R

˜hR,β,α−µ1. (5.42)

Thus, the Hamiltonian

Hˆ˜_R=^X

α,β

˜h_R,α,βˆc^†_αˆc_β + ˆW_R (5.43)

describes the system in the vicinity of site R. From the constrained minimization we also know a many-particle wave function |ΨRi that is an eigenstate of ˆ˜H_R and has the correct one-particle reduced density matrix ρ⁽¹⁾_R . We propose to consider this state as an approximation of the ground state in the vicinity of the siteR and use this state together with the Hamiltonian ˆ˜H_Rto construct the local spectral functionA_local,α,β(). The spectral function can be computed with the Lanczos method for spectral functions [Koch, 2011].

5.6.2. Kohn-Sham-like spectral function

In DFT, the derivatives of the total energy with respect to the occupations are related by Janak’s theorem [Janak, 1978],

_n= ∂E_DFT[{f_n},{|ψ_ni}]

∂f_n , (5.44)

to the Kohn-Sham energy eigenvalues. In RDMFT we define Kohn-Sham-like energy eigenvalues

˜

_n= ∂E_RDMFT[{f_n},{|ψ_ni}]

∂f_n (5.45)

as derivatives of the RDMFT total energyE_RDMFT defined in Eq. (5.15). Equation (5.45) does not define the Kohn-Sham-like energy eigenvalues uniquely: the subspace of natural orbitals |ψ_ni with identical occupations can be rotated with a unitary transformation without changing the one-particle reduced density matrix. Thus, we additionally require that the matrix

H˜_i,j =hψ_i|∂E_RDMFT[{f_n},{|ψ_ni}]

∂hψ_j| (5.46)

is diagonal for all indexes i and j that have identical occupations, i.e.,f_i =f_j. With this diagonality requirement the Kohn-Sham-like energy eigenvalues ˜_n are uniquely defined.

In RDMFT, states with fractional occupations 0< f_n <1 have derivatives

∂ERDMFT[{fn},{|ψni}]

∂f_n =µ. (5.47)

In other words, if we assign Kohn-Sham-like energy eigenvalues to the natural orbitals in RDMFT, natural orbitals with fractional occupations lie at the chemical potential.

Thus, a physical interpretation of the Kohn-Sham-like density of states estimated from the natural orbitals and Kohn-Sham-like energy eigenvalues ˜_nwill tend to falsely indicate a metallic state because fractionally occupied states lie by construction at the chemical potential.