List of Figures

• 1.1. A physical interpretation of the underlying coupling in the Kondo model.
• 1.2. A physical depiction of the the Hubbard model allowing for electrons to hop with transfer integral $t$ and experience a local Coulomb repulsion $U$.
• . The dimensionless effective hybridization function $\tilde\Delta_{\rm eff}(x)$ evaluated for $\rho _0 J=0.1$ in the SIKM. The resulting function is symmetric and is only plotted for $x>0$. It coincides with the universal form for $x\lesssim 30$. The dashed line represents the fit (2.14) and is nearly indistinguishable from the actual data. Notice especially the appearance of logarithmic behavior in the crossover region.
• 2.2. Universal curve for $S(0)$ as function of the local magnetic field $h$.
• 2.3. $\chi _{zz}''(\omega )$ for three characteristic local magnetic fields $h=0$, $g\mu _B h=T_{\rm K}$ and $g\mu _B h=5T_{\rm K}$.
• 2.4. The magnetic susceptibility $\chi _0(h)$ from equ. (2.23) (circles) and the same quantity obtained from an NRG calculation.
• 2.5. Results for the Shiba ratio $R_{\rm S}$ (full line) and the Wilson ratio $R_{\rm W}$ (dashed line) as a function of a local magnetic field. The correct limiting values at $h\to 0$ are missed by approximately 5%.
• 3.1. In the DMFA all electron propagators are averaged over the entire Brillouin Zone (BZ); effectively mapping the lattice onto a single point (top). In the DCA, we break the BZ into several sub-cells which are now in turn averaged over mapping the lattice onto a finite sized cluster (bottom).
• 3.2. Self-consistency loop for the DCA.
• 3.3. Diagrams relating the self-energy to the reducible two-particle longitudinal spin and charge (top) and transverse spin (middle) vertices. A similar relation is obtained for the particle-particle channel (bottom).
• 3.4. Bethe-Salpeter equation relating the reducible two-particle vertex $F$ to the irreducible vertex function $\Gamma$.
• 3.5. Flow chart for the Ansatz self-consistent implementation of the MSMB/DCA.
• . Second order diagrams for the vertex functions $\Gamma^2_{\uparrow\uparrow}$ - a.) and $\Gamma^2_{\uparrow\downarrow}$ - b.) for an external momentum transfer $q=0$.
• 3.7. Imaginary part of the self-energy at lowest Matsubara frequency as obtained by various cluster solvers and FLEX/MSMB method at $\beta =8$, $U=W=1.0$, and $n=0.75$. The MSMB cluster sizes are $N_c^{(1)}=8$ and $N_c^{(2)}=32$.
• 3.8. Flow chart depicting two independent self-consistent DCA calculations combined via an Ansatz to construct a MSMB self-energy.
• 3.9. Imaginary part of the self-energy at lowest Matsubara frequency as obtained by the MSMB method using first (FLEX) and second order approximated irreducible vertices $\Gamma$ in comparison to the large single-cluster QMC results at $\beta =31$, $U=W=1.0$, and $n=0.75$. Multi-scale results are for cluster sizes $N_c^{(1)}=8$ and $N_c^{(2)}=32$.
• 3.10. Imaginary part of the $\lambda$-approximated self-energy at lowest Matsubara frequency for the large cluster ( $N_c^{(2)}=32$) using various cluster solvers in comparison to the single cluster QMC result at $\beta =31$, $U=W=1.0$, and $n=0.75$. Also shown is the overestimated Ansatz self-consistent self-energy of the MSMB method for the second order approximated $\Gamma$.
• 3.11. Imaginary part of the self-energy at lowest Matsubara frequency as obtained by the MSMB approach using the second order $\lambda$-approximation and large single-cluster QMC. Results are for various small cluster sizes $N_c^{(1)}$ at $\beta =31$, $U=W=1.0$, and $n=0.75$.
• 3.12. Spin and charge velocities ($v1$ and $v2$ respectively) obtained by fitting the different results with the Luttinger Green's functions (Eq. 3.26) about cluster momentum $k=\pi /2$ for $\beta =31$, $U=W=1.0$, and $n=0.75$. Multi-scale results are for a small cluster size $N_c^{(1)}=8$ and different large cluster sizes, and the QMC velocities were obtained from a single 32 site cluster calculation.
• 3.13. Imaginary part of the self-energy at lowest Matsubara frequency as obtained by the MSMB method using second order approximated and full QMC evaluated $\Gamma$s in comparison to a large single cluster QMC results at $\beta =31$, $U=\frac {3}{2}W=1.5$, and $n=0.75$. Multi-scale results are for cluster sizes $N_c^{(1)}=8$ and $N_c^{(2)}=32$.
• 3.14. Imaginary part of the self-energy at lowest Matsubara frequency as obtained by the MSMB method using the second order approximated $\Gamma$ in conjunction with the real-space and momentum-space based Ansatz at $\beta =31$, $U=W=1.0$, and $n=0.75$. Also shown is the large single-cluster QMC result. Multi-scale results are for cluster sizes $N_c^{(1)}=8$ and $N_c^{(2)}=32$.
• 3.15. Parquet equation relating the transverse particle-hole irreducible vertex $\Gamma ^{p-h}_{trans}$ to the fully irreducible vertex $\Lambda$ plus contributions from the longitudinal and particle-particle cross channels. Similar relations apply for the remaining channels (not shown).
• 3.16. Lowest order non-local corrections to the fully-irreducible vertex $\Lambda$ (left) and the vertex $\Gamma$ (right).
• 4.1. Schematic representation of the Dyson equation (upper). In the SCBA approximation the self-energy is a summation of all non-crossing diagrams (middle) and can be determined self-consistently using Eq. 4.20 (bottom).
• 4.2. Numerically exact QMC (diamonds) and SCBA (squares) results for polaron energy $E_0$ (a) and quasiparticle weight $Z_0$ (b) versus the dimensionless electron-phonon coupling $\lambda$ (see Eq. 4.7) at $k=0$. Holstein model with $t=1$ and $\omega_0=0.1~t$ and $2M=1$.
• 4.3. Holstein polaron dispersion $E(k)$ (a) and quasiparticle weight $Z(k)$ (b) calculated with SCBA (dotted line) and QMC (circles). c) Spectral representation obtained with SCBA. Everywhere $\lambda _H=0.25$ and $\omega_0=0.5~t$.
• 4.4. H (circles) and B (diamonds) polaron energy (a) and quasiparticle weight (b) versus $\lambda$ at $k=0$ and $\omega _0=0.2 t$. The open symbols are the results obtained after only one iteration (first order in $D$). The dashed line in (b) corresponds to $\lambda ^{ME}=\frac {1}{Z_0}-1$ (see Eq. 4.16).
• 4.5. The quasiparticle weight $Z_0$ (a) and the polaron energy $E_0$ (b) at zero momentum versus phonon frequency $\omega _0$ for the H and B models at $\lambda _H =0.2$ and respectively $\lambda _B =1.3$.
• 4.6. Quasiparticle weight $Z(k)$ (a) and the polaron energy $E(k)$ (b) for the H and B model for $\omega_0=0.2~t$ at coupling $\lambda _H =0.2$ and $\lambda _B =1.3$ respectively. The dotted horizontal lines in (b) marks the first phonon threshold energy $E_0+\omega _0$.
• 4.7. Spectral representation for: a) H model, $\lambda _H =0.2$, tight binding dispersion, b) B model, $\lambda _B =1.3$, tight binding dispersion, c) B model, $\lambda _B=0.5$, tight binding dispersion, d) B model, $\lambda ^{lin}_B=1.6$, linear dispersion with $v_F=t$ (Eq. 4.25), e) H model, $\lambda ^{lin}_H=0.9$, linear dispersion with $v_F=t$, f) H and B model, $\lambda _H=\lambda _B=0.2$, with dispersion $\epsilon(k)=-2t \cos(2k)$. Everywhere $\omega _0=0.2 t$. The arrows indicate the first (lower) and the second (upper) phonon threshold energies corresponding to $E_0+\omega _0$ and respectively $E_0+2\omega _0$.
• 4.8. Spectral function (energy distribution curves) for a) H model with $\lambda _H =0.2$ and b) B model with $\lambda _B=0.5$. Free electron tight binding dispersion and $\omega _0=0.2 t$ is considered.
• 4.9. Spectral function (energy distribution curves) for a) H model with $\lambda _H^{lin}=0.9$ and b) B model with $\lambda _B^{lin}=1.6$. Free electron linear dispersion ($v_F=t$) and $\omega _0=0.2 t$ is considered.
• 4.10. a) Energy levels for zero coupling ($g=0$). The solid line shows the non-interacting electron dispersion, the dashed ones are one phonon + one electron states and the dotted one two phonons + one electron, etc. $k_1$ is the momentum where the electron dispersion reaches $\omega _0$ and has the most significant contribution to the scattering at the first phonon threshold. $k_2$ is the value of momentum with significant contribution to the scattering at the second phonon threshold. b) For the polaron with momentum $k$ the scattering with momentum $q$ has a significant contribution. Although the polaron momentum is close to $k_F$, thus $k_a=k-k_F$ is small, $q=k_a+k_b$ is large because $k_b$ which connects two points on the Fermi surface can be large.