Application to Anderson models

We study the results of the adaptive cluster approximation for the single-impurity An-derson model defined in section 3.2 by Eq. (3.2)-Eq. (3.7). We compare numerically exact results for the zero-temperature ground state obtained from exact diagonalization with results from reduced density-matrix functional theory with the (corrected) adaptive clus-ter approximation withM effective bath levels. The results presented in this section have been published in [Schade and Blöchl, 2018]. Thus, we solve

E(N) = min

ρ⁽¹⁾:0≤ρ⁽¹⁾≤1,Trρ⁽¹⁾=N

nTr[ρ⁽¹⁾h] +F_β,(c)ACA(M^Wˆ ₎[ρ⁽¹⁾]^o, (8.22) where h are the matrix elements of the one-particle Hamiltonian and N the particle number. We have not restricted the symmetry of the one-particle reduced density ma-trix and, hence, also allow ground states with collinear or non-collinear magnetization.

We have used the numerically exact evaluations of the density-matrix functional of the impurity and the effective bath levels proposed in section 9.4 without additional approx-imations. Figure 8.2 shows the deviation of the total energy ∆E = E_(c)ACA(M)−E_exact,

8.6. Application to Anderson models the interaction energy ∆W = W_(c)ACA(M)−W_exact and the impurity occupation ∆n_f = nf,(c)ACA(M)−n_f,exact from the exact results shown in figure 3.2 for the interaction-strength dependence. The parameters are described in detail in section 3.2.1. The ACA(M) and the corrected ACA(M) with a correction from the Müller functional correctly predicts a vanishing impurity magnetization in this parameter range. Within the ACA with M = 1 for this model, we have to evaluate the density-matrix functional for four one-particle states, two impurity states and two effective bath states. The results are equivalent to the results of the two-level approximation of Töws and Pastor [Töws and Pastor, 2011].

The uncorrected ACA tends to overestimate the impurity occupation n_f because the in-teraction energy is underestimated. As a consequence, the total energy is underestimated.

Figure 8.1 compares the discarded weight defined in Eq. (8.17) of the one-particle reduced density matrix of the exact ground state, the ACA(M=1) and the corrected ACA(M=1).

The discarded weight of ACA(M=1) is much larger than the discarded weight of the exact one-particle reduced density matrix. This is due to the missing force on these matrix el-ements. The corrected ACA(M=1) proposed in section 8.5 with the Müller functional as the correction prevents the growth of the truncated matrix elements during the minimiza-tion and leads to a lower discarded weight, that agrees better with the discarded weight of the exact solution. Thus, the correction step serves its purpose and greatly improves the results of the ACA. Larger numbers of effective bath levels, i.e., M = 2 and M = 3, converge the results towards the exact result. The ACA with three effective bath sites is visually indistinguishable from the exact results. This is a consequence of the fact that the bath occupations, i.e., the eigenvalues of the one-particle reduced density matrix of the bath states, are contained in only three clusters with a negligible spread as shown in Figure 3.4. As discussed in detail in section 8.4 and appendix B.3 the ACA(M) is exact if the one-particle reduced density matrix of the non-interacting states containsM or less distinct eigenvalues.

The deviations of the (c)ACA(M) from the exact results for single-impurity Anderson model defined in section 3.2.1 for different impurity on-site energies _f is shown in the left columns of figure 8.3. The ACA(M) describes the three regimes without breaking the spin-symmetry. Thus, it describes the Kondo-regime physically correctly. The deviation of the interaction energy and impurity occupation are largest during the transition from the doubly occupied impurity to the singly occupied impurity and from the singly occupied impurity to the empty impurity. The ACA with one effective bath level,M = 1, is exact in the limit of a doubly occupied impurity and the limit of an empty impurity. The Müller-corrected ACA drastically improves the results for impurity on-site energy dependence, especially close to the mixed-valence transition points.

Finally, the right column of figure 8.3 presents the results of the ACA with M = 1 and M = 2 for the bandwidth dependence of the single-impurity Anderson model. The ACA with one effective bath level, i.e., M = 1, becomes exact in the limit of widely separated bath energy levels t → ∞ and in the limit of a vanishing bath bandwidth t → 0 as discussed in section 8.4. In all parameter ranges studied, the ACA with M = 3 is visually identical to the exact results and the ACA reproduces the correct non-spin-polarized ground state. In conclusion, we have shown that the adaptive cluster approximation converges rapidly to the exact results for single-impurity Anderson models, does not break the spin-symmetry, and that the correction with an approximate parametrized functional serves its purpose of preventing an uncontrolled growth of the discarded weight.

0.0 0.1 0.2 0.3

0 2 4 6 8

ACA(M=1)

cACA(M=1) exact

σ

(˜ ρ

) ,σ

(˜ ρ

)

U/t

Figure 8.1.: Comparison of the discarded weight σ_M=1 defined Eq. (8.17) for the one-particle reduced density matrix of the exact ground state, the ACA(M=1) and the corrected ACA(M=1). The model is the half-filled single-impurity Anderson model defined by Eq. (3.2)-Eq. (3.7) withL_bath = 11, N_e =L_bath+ 1 = 12, _f = 0, V /t = 0.4 and t > 0. The parameters of the model are described in detail in section 3.2.1.

8.6. Application to Anderson models

-0.02 -0.01 0.00

ACA(M=1)

cACA(M=1) ACA(M=2) cACA(M=2) ACA(M=3)

-0.010 -0.005 0.000 0.005

0.00 0.01 0.02 0.03

0 2 4 6 8

∆E/t∆W/t ∆nf

U/t

Figure 8.2.: Deviation from exact results of the (corrected) ACA(M) for the interaction-strength dependence of the ground-state of a half-filled single-impurity Ander-son model defined by Eq. (3.2)-Eq. (3.7) withL_bath = 11,N_e =L_bath+1 = 12, _f = 0, V /t = 0.4 and t > 0. The black lines show results for the ACA with M = 1, red lines for M = 2 and blue lines for M = 3. The solid lines show results for the uncorrected ACA and dashed lines the corresponding results for the uncorrected ACA. The parameters of the model are described in detail in section 3.2.1 and the exact results are shown in figure 3.2.

-0.03 -0.02 -0.01 0.00

ACA(M=1)

cACA(M=1) ACA(M=2)

cACA(M=2)

-0.3 -0.2 -0.1 0.0

-0.025 0.000 0.025

−10 −5 0 5

∆E/t ∆W/t∆nf

ǫf/t

-0.08 -0.06 -0.04 -0.02 0.00

ACA(M=1) cACA(M=1) ACA(M=2)

cACA(M=2)

-0.04 -0.02 0.00

0.00 0.01 0.02 0.03 0.04

0 10 20

∆E/V∆W/V ∆nf

t/V

Figure 8.3.: Deviation from exact results of the (corrected) ACA(M) for the impurity on-site-energy dependence and the bandwidth dependence of the ground-state of a half-filled single-impurity Anderson model defined by Eq. (3.2)-Eq. (3.7) with L_bath = 11, N_e = L_bath + 1 = 12. The black lines show results for the ACA with M = 1 and the red lines for M = 2. Results for M = 3 are not chosen, because they are visually identical with the exact results.

The solid lines show results for the uncorrected ACA and dashed lines the corresponding results for the uncorrected ACA. The parameters of the model are described in detail in section 3.2.1. The corresponding exact results are shown in figure 3.3.