Perspective: Exact Q-approach

The underlying continuity assumption of the Q-approach is only approximate, be-cause in higher order of perturbation theory, terms (B.1) which do not characterize the collection of wedges, but merely just isolated wedges, are generated. These terms are manifested in discontinuities of the real part of the Green’s function at the branch point Imz = 0.

In order to extend the Q-approach to the full nonequilibrium Kondo regime U ≥ 2πΓ, eΦ ∼ TK, β⁻¹ ∼ TK, one has to take these contributions into account. This

4SNRG data were provided by Sebastian Schmitt and Frithjof Anders. Used with permission.

7.4. Perspective: Exact Q-approach default model [P_max]

2nd order pert. theory, T=0

(a)U = 4Γ,eΦ = 0.125Γ default model [P_max] 2nd order pert. theory, T=0

(b)U = 4Γ,eΦ = 0.25Γ default model [P_max] 2nd order pert. theory, T=0

(c)U = 4Γ,eΦ = 0.5Γ default model [P_max] 2nd order pert. theory, T=0

(d)U = 4Γ,eΦ = 1.0Γ default model [P_max] 2nd order pert. theory, T=0

(e) U = 6Γ,eΦ = 0.25Γ default model [P_max] 2nd order pert. theory, T=0

(f)U = 6Γ,eΦ = 1.0Γ

Figure 7.8: Intermediate-coupling nonequilibrium spectral functions at inverse temperature β= 5Γ⁻¹, as compared to zero-temperature second-order perturbation theory.

-10

(a) a-posteriori identified best default model

-10 2nd order pert. theory, T=0 SNRG 2nd order pert. theory, T=0 SNRG 2nd order pert. theory, T=0 SNRG 2nd order pert. theory, T=0 SNRG

(d)U= 8Γ,eΦ = 1.0Γ,β = 5Γ⁻¹ Figure 7.10: Comparison to the scattering-states numerical renormalization group

7.4. Perspective: Exact Q-approach

0 0.2 0.4

Voltage eΦ/Γ 0

0.05 0.1

Current

Q-Approach, βΓ=5 Q-Approach, βΓ=10 rt-QMC, βΓ=10 U=0

0 0.2 0.4

Voltage eΦ/Γ

U=4Γ U=6Γ

Figure 7.11: Currents forU = 4Γ and U = 6Γ as compared to real-time QMC [32] and the noninteracting case.

0 0.2 0.4 0.6 0.8 1

Voltage eΦ/Γ 0

0.05 0.1 0.15 0.2 0.25

Current

U=4Γ, βΓ=10 U=6Γ, βΓ=10 U=8Γ, βΓ=10

Figure 7.12: Current as a function of voltage atβ = 10Γ⁻¹for several interaction strengths.

requires the consideration of the full analytic structure of the theory as illustrated in figure 7.13b. As a consequence, extra terms have to be added to the representation of G(zϕ, zω) within the MaxEnt procedure. Natural candidates for such degrees of freedom are the residual imaginary parts of edge functions

R˜_ν(x) := ImG(x+ i0^ϑν)−π·(Q^(edge)_ϑ

ν

A)(x),˜ (7.24) for the edge of the ν-th wedge with orientation ϑν. Due to the properties of the residual terms (B.1), ˜R_ν(x) is expected to vanish as x → ∞. I.e. the functions are essentially localized within a finite radius around 0 which can be expected to be of the order of magnitude of the energy scales Γ, U, εd, and eΦ. One can therefore hope that although one formally introduces infinitely many two-dimensional variable vectors to take into account the exact analytic structure, in the actual discretized version of the problem, the growth in variable space is not so dramatic. This must be seen in relation to the ˜A function, which has to be discretized over a very wide range in order to pay tribute to the strong intertwining of edge function structure and branch cut structure.

For data in the ν-th wedge, one then has theexact equation

ImG(iϕ_m,iω_n) = (P_rν,ϑνR˜_ν)(iϕ_m,iω_n) +π·(QA)(iϕ˜ _m,iω_n), (7.25) wherer_ν is the opening ratio of the wedge.⁵ The inverse problem to be solved by the MaxEnt is to determine ˜R_ν and ˜A simultaneously. Practically, the terms ˜R_ν would act as “valves” for the conceptual imperfection of theQ-mapping within the Bayesian information flow.

Because the functions ˜R_ν(x) cannot be expected to be positive, it is necessary to introduce a shift to a positive function, such as for the spectral functions of the static observables in section 6.1. The terms ˜R_ν(x) are most dominant for wedges next to the noninteracting Green’s function’s branch cuts. A very careful Bayesian analysis, including an appropriate set of choosable default models constructed from a-priori information, is probably required for a successful application of the exact approach.⁶

7.5. Summary

In this chapter, the MaxEnt approach to the double analytic continuation was sys-tematically extended to a substantially larger range of QMC simulation data. At the cost of an approximation to the structure at the branch point,⁷ it enabled to estimate dynamical properties up to the intermediate coupling regime which are in reasonable agreement with other computational techniques.

5For the definitions see figure 5.4.

6Some a-priori information is provided in appendix B.

7We refer to the point where all wedges join as branch point.

7.5. Summary

Imzϕ

Imzω

(a) imaginary-voltage formalism data

A(x˜ ϕ, xω) Imzϕ

Imzω

(b)Q-simplified set of edge functions

Figure 7.13: Effective simplification of the analytic structure as founda posteriori for the Q-mapping. It is valid up to intermediate many-body correlation strengths. Compare to the general structure depicted in figure 7.1. The inaccessibility of the upper and lower wedges toQ is presumably due to the strong violation of the shared-real-part assumption within the multiplicative structure of the Dyson series, as predicted as an a-priori constraint by section 7.2.2. In strong contrast to the single-wedge approach, the ϕ_m = 0 data are not used, because they are located on the accumulation axis of interacting branch cuts and also on the Imzϕ = 0 branch cut.

Resulting from an a-posteriori constraint, the accessible data range was extended to the one symbolized by the set of green crosses in figure 7.13. The mathemati-cal setup is much simpler than the nonapproximate analytic structure sketched in figure 7.1. The technical realization of a matrix representation for the resulting map-pingQand the Bayesian inference procedure, however, required a carefully prepared computational procedure.

As a future application, an exact generalization of the approach was suggested. It introduces additional degrees of freedom from several distinct edge functions which have to be determined sufficiently accurately by the MaxEnt procedure. Along the route to the ultimate goal of providing estimates on non-equilibrium strong-coupling spectra, in particular the identification of appropriate a-priori information for incor-poration into the default model are expected to be a non-trivial task.