Results of the Q-approach

When defined according to the interacting branch cut geometry, we will refer toQ_r,ϑ simply by the symbol Q. In order to obtain estimates for the finite-U Anderson impurity model out of equilibrium, we determine a most probable default model

7.3. Results of the Q-approach

(D.3),

D˜_σdef(x_ϕ, x_ω) = 1 2π

X

α=±1

σ_def

(x_ω− ^α₂(x_ϕ−Φ))²+σ²_def, (7.22) first and, in a second step, refine the knowledge about its low-frequency region, with an extra low-energy width, and a transient behaviour as in equation (D.4),

σ_def(x_ϕ) = Γ + (˜σ_def−Γ) R²

x²_ϕ+R², (7.23)

where we set R = 5Γ. Due to the approximative nature of Q, also the range of the dataG(iϕm,iωn) has to be constrained in order to obtain spectra with a normalization

≈ 1. One finds empirically that a major constraint is ω_n > |ϕ_m/2| for finite values of U. By this, the data space is in fact originated in the neighbouring wedges of the retarded Green’s function’s wedge. It is however very large as compared to the data space of the single-wedge approach as used in section 6.2.1. Furthermore, as many-body correlations increase, small values of ω_n must be neglected in order to apply Q. This is in accordance with the observation that Q is only perturbatively correct.

It will be shown explicitly that the extended data range makes the inverse problem and also the MaxEnt procedure more well-behaved as compared to the single-wedge approach. Results of spectral functions and transport properties are presented in the weak- to intermediate coupling regime.

7.3.1. Weak-Coupling Regime

In the weak-coupling regime, U ≤ πΓ, the second-order perturbation theory in U/Γ is still expected to be rather accurate. A useful formula for the calculation of second-order perturbation spectra atT = 0 is provided by Ref. [126] for comparison purposes.

Already in the weak-coupling regime, U = 2Γ, βΓ = 5, the applicability of the Q approach is limited a posteriori by bad behaviour of the inferred spectral functions precisely to input data with ω_n > |ϕ_m/2|. If any data point G(iϕ_m,iω_n) with ω_n <

|ϕ_m/2|is included, the MaxEnt procedure ceases to converge to a reasonable solution.

This corresponds to not crossing the principal branch cutγ =±1 in figure 3.4, when coming from the retarded Green’s function. Apart from this, there appears to be no further constraint. We use all Matsubara data ω_n, extending from n = 1 to n = 8, whilem =±1, . . . ,±5 in ϕ_m, with the ω_n>|ϕ_m/2| constraint for ω_n.

As a first test, figure 7.5 displays the performance of the MaxEnt method for both, the wedge and the multiple-wedge approach for the given data set. The single-wedge approach implicitly assumes the interacting Green’s function to be analytic for Imz_ω > |Imz_ϕ/2|. Apparently, this wrong assumption makes no fit with a positive definite ˜A(x) possible. Instead, the χ² value does not drop below ≈ 10⁶, and the

10⁰ 10⁴ _α 10⁸ 10^-6

10⁰ 10⁶

χ2 /Ndata

single analytic wedge (r=2, ϑ=0) Q-approach

Figure 7.5: Comparison ofχ² in the MaxEnt for single-wedge kernel P_r,ϑ and multi-wedge kernel Q atU = 2Γ, β = 5Γ⁻¹, and eΦ = Γ as a function of the regularization parameter α. For the same input set, the single-wedge approach already clearly fails to converge due to the wrong assumption of negligible branch cuts |γ| ≥3.

-10

(a) a-posteriori identified best default model

-10

Figure 7.6: Application of the MaxEnt procedure for theQ-mapping to the nonequilibrium weak-coupling caseU = 2Γ, β = 5Γ⁻¹, eΦ = Γ, with CT-QMC data as input. The default model has been identified via its maximal posterior probability.

MaxEnt does not converge. In sharp contrast, values ofχ²/N_data≈1 may be reached with the MaxEnt with respect toQ. Also, controls such as the MaxEnt error rescaling merit do not indicate the presence of any abnormalities.

For the Q-mapping, a well-behaved MaxEnt solution is obtained. Applying the procedure described in appendix D, the edge function displayed in figure 7.6a is identified as a-posteriori most probable default model. Using this default model, the edge function in panel 7.6b is obtained. An overall moderate sharpening of the edge function along the cross-like structure is observed.

The resulting spectral function is displayed in figure 7.7 together with the spectral function of the default model which has been identified as most probable a posteriori.

The spectrum is in good agreement with the zero-temperature second-order pertur-bation theory. Presumably due to the finite temperature, the quasi-particle weight is slightly smaller than the perturbative one.

7.3. Results of the Q-approach

-8 -4 0 4 8

ω/Γ 0

0.05 0.1 0.15 0.2 0.25

A(ω)Γ

MaxEnt, Q-Approach default model [P

max] 2nd order pert. theory, T=0

Figure 7.7: Spectral function as inferred for the nonequilibrium weak-coupling caseU = 2Γ, eΦ = Γ, β = 5Γ⁻¹, compared to zero-temperature second-order perturbation theory.

7.3.2. Intermediate-Coupling Regime

Using the same range within Matsubara space,ω_n>|ϕ_m/2|, the method can well be applied to intermediate coupling strengths. At the comparably small inverse temper-ature β = 5Γ⁻¹, calculations are most readily done, mainly due to the comparably small QMC data space of approximately 50 imaginary-time-theory data points. In general, the amount of data will grow quadratically as a function of inverse tem-perature, due to the simultaneous presence of Matsubara voltage and Matsubara frequency. Additionally, at low temperatures, sharp features in the spectral function and hence the ˜A will have to be resolved, requiring an enhanced grid refinement.

Altogether, matrix sizes in the MaxEnt are blown up substantially when the temper-ature is decreased. In particular, the computational effort for the generation of an appropriate kernel matrix (cf. appendix E) is increased by this. Apart from this pre-sumably, in principle, minor technical issue, the memory consumption of the MaxEnt itself poses a limitation at lower temperatures at the current stage of development.

Additionally, it is well-known that the resolution of low-temperature features with the MaxEnt method requires a careful Bayesian analysis based on higher-temperature data, i.e. an “annealing procedure” involving many higher-temperature data [35; 127].

Figure 7.8 shows some spectral functions obtained within the intermediate-coupling regime at β = 5Γ⁻¹, but without imposing any annealing procedure. In parts, the spectral function has not been modified by the MaxEnt, but remains attached to the default model. The agreement in shape and height with second-order zero-temperature perturbation theory seems satisfactory.

However, in particular in the cases (U = 4Γ, eΦ = 0.125Γ), (U = 4Γ, eΦ = 0.25Γ), and (U = 6Γ, eΦ = 0.25Γ), the formation of peculiar side-bands can be observed, coming along with an unphysical increase in the spectral sum. They are clearly an artifact of the continuation procedure. The resulting unphysical region however seems

to be limited to a certain energy range which is not necessarily of physical interest. It can often be traced back to certain structures which seem to form along well-defined angular directions in the function ˜A(xϕ, xω). For the case (U = 6Γ, eΦ = 0.25Γ), the MaxEnt inference process is displayed in figure 7.9. The quasi-particle resonance is generated by the peak at x ≈ 0. The formation of broad Hubbard shoulders is accompanied by the structures producing the unphysical sidebands.

A comparison to scattering-states numerical renormalization group (SNRG) spec-tra is shown in figure 7.10.⁴ While data seem to be compatible with a splitting of the quasi-particle resonance, i.e. peaks at ±eΦ/2, the SNRG does not exhibit this feature. The double-peak structure is found to be generated by the Matsubara data points in the vicinity of the edge, i.e. the low-energy imaginary-time theory. Due to the assumption which the Q-approach relies on, it cannot yet be decided whether the prediction of the quasi-particle peak to split is a prediction of Matsubara voltage theory or is an artifact of theQ-approach. The peak-splitting could also be generated as an artifact of the MaxEnt procedure – however, the rather systematic positions at

±eΦ/2 oppose against this hypothesis.

7.3.3. Transport properties

Due to the increased quality of spectral functions, the calculation of transport proper-ties is much more reliable with the multi-wedge approach than with the single-wedge continuation. In figure 7.11, resulting electric currents are compared to real-time quantum Monte-Carlo data by Werner et al. [32]. The results for β = 5Γ⁻¹ are presumably more reliable than at lower temperature β = 10Γ⁻¹. A good agreement with the real-time QMC data is found. The extraordinarily high value atβ = 10Γ⁻¹ and U = 6Γ for voltage eΦ = 0.0675Γ can be traced back to an unphysically high quasi-particle peak, which may however still be resulting from stochastic noise or imperfections of the MaxEnt procedure. The latter is again more demanding for β = 10Γ⁻¹ than for β = 5Γ⁻¹. Figure 7.12 shows the current as a function of the voltage for several values of U.