- 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 and experience a local Coulomb repulsion .
- . The dimensionless effective hybridization function evaluated for in the SIKM. The resulting function is symmetric and is only plotted for . It coincides with the universal form for . 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 as function of the local magnetic field .
- 2.3. for three characteristic local magnetic fields , and .
- 2.4. The magnetic susceptibility from equ. (2.23) (circles) and the same quantity obtained from an NRG calculation.
- 2.5. Results for the Shiba ratio (full line) and the Wilson ratio (dashed line) as a function of a local magnetic field. The correct limiting values at 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 to the irreducible vertex function .
- 3.5. Flow chart for the
*Ansatz*self-consistent implementation of the MSMB/DCA. - . Second order diagrams for the vertex functions - a.) and - b.) for an external momentum transfer .
- 3.7. Imaginary part of the self-energy at lowest Matsubara frequency as obtained by various cluster solvers and FLEX/MSMB method at , , and . The MSMB cluster sizes are and .
- 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 in comparison to the large single-cluster QMC results at , , and . Multi-scale results are for cluster sizes and .
- 3.10. Imaginary part of the -approximated self-energy at
lowest Matsubara frequency for the large cluster (
)
using various cluster solvers in comparison to the single cluster
QMC result at , , and . Also shown is the
overestimated
*Ansatz*self-consistent self-energy of the MSMB method for the second order approximated . - 3.11. Imaginary part of the self-energy at lowest Matsubara frequency as obtained by the MSMB approach using the second order -approximation and large single-cluster QMC. Results are for various small cluster sizes at , , and .
- 3.12. Spin and charge velocities ( and respectively) obtained by fitting the different results with the Luttinger Green's functions (Eq. 3.26) about cluster momentum for , , and . Multi-scale results are for a small cluster size 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 s in comparison to a large single cluster QMC results at , , and . Multi-scale results are for cluster sizes and .
- 3.14. Imaginary part of the self-energy at lowest Matsubara
frequency as obtained by the MSMB method using the second order
approximated in conjunction with the real-space and
momentum-space based
*Ansatz*at , , and . Also shown is the large single-cluster QMC result. Multi-scale results are for cluster sizes and . - 3.15. Parquet equation relating the transverse particle-hole irreducible vertex to the fully irreducible vertex 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 (left) and the vertex (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 (a) and quasiparticle weight (b) versus the dimensionless electron-phonon coupling (see Eq. 4.7) at . Holstein model with and and .
- 4.3. Holstein polaron dispersion (a) and quasiparticle weight (b) calculated with SCBA (dotted line) and QMC (circles). c) Spectral representation obtained with SCBA. Everywhere and .
- 4.4. H (circles) and B (diamonds) polaron energy (a) and quasiparticle weight (b) versus at and . The open symbols are the results obtained after only one iteration (first order in ). The dashed line in (b) corresponds to (see Eq. 4.16).
- 4.5. The quasiparticle weight (a) and the polaron energy (b) at zero momentum versus phonon frequency for the H and B models at and respectively .
- 4.6. Quasiparticle weight (a) and the polaron energy (b) for the H and B model for at coupling and respectively. The dotted horizontal lines in (b) marks the first phonon threshold energy .
- 4.7. Spectral representation for:
a) H model,
, tight binding dispersion,
b) B model,
, tight binding dispersion,
c) B model,
, tight binding dispersion,
d) B model,
, linear dispersion with (Eq. 4.25),
e) H model,
, linear dispersion with ,
f) H and B model,
, with dispersion
.
Everywhere
.
The arrows indicate the first (
*lower*) and the second (*upper*) phonon threshold energies corresponding to and respectively . - 4.8. Spectral function (energy distribution curves) for a) H model with and b) B model with . Free electron tight binding dispersion and is considered.
- 4.9. Spectral function (energy distribution curves) for a) H model with and b) B model with . Free electron linear dispersion () and is considered.
- 4.10. a) Energy levels for zero coupling (). 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. is the momentum where the electron dispersion reaches and has the most significant contribution to the scattering at the first phonon threshold. is the value of momentum with significant contribution to the scattering at the second phonon threshold. b) For the polaron with momentum the scattering with momentum has a significant contribution. Although the polaron momentum is close to , thus is small, is large because which connects two points on the Fermi surface can be large.