1. Diagrammatic auxiliary particle impurity solvers - SUNCA Diagrammatic auxiliary particle impurity solvers - SUNCA Auxiliary particle method How to set up perturbation theory? How to choose diagrams? Where Luttinger Ward functional comes in? How to write down end evaluate Bethe- Salpeter equation? How to solve SUNCA integral equations? Some results and comparison Summary

2. Advantages/Disadvantages Advantages: Fast (compared to QMC) IS with no additional time cost for large NFast (compared to QMC) IS with no additional time cost for large N Defined and numerically solved on real axis (more information)Defined and numerically solved on real axis (more information)Disadvantages: Not exact and needs to be carefully tested and benchmarkedNot exact and needs to be carefully tested and benchmarked (breaks down at low temperature T<<Tk – but gets (breaks down at low temperature T<<Tk – but gets better with increasing N) No straightforward extension to a non-degenerate AIM (relays onNo straightforward extension to a non-degenerate AIM (relays on degeneracy of local states) – but straightforward extension to out-of-equilibrium AIM

3. Problems solving SUNCA Usual perturbation theory not applicable (no conservation of fermions)Usual perturbation theory not applicable (no conservation of fermions) slightly modified perturbation slightly modified perturbation carefully determine sign and prefactor Two projections:Two projections: exact projection on the pysical hilbert spaceexact projection on the pysical hilbert space need to project out local states with energies far fromneed to project out local states with energies far from the chemical potential Numerics!Numerics! Solve Bethe-Salpeter equation with singular kernelSolve Bethe-Salpeter equation with singular kernel Pseudo particles with threshold divergence –Pseudo particles with threshold divergence – need of non-equidistant meshes need of non-equidistant meshes Integration over T-matrices that are defined on non-equidistant meshIntegration over T-matrices that are defined on non-equidistant mesh

4. Diagrammatic auxiliary particle impurity solvers Exact diagonalization of the interacting region (impurity site or cluster) Introduction of auxiliary particles  # electrons in even is boson  # electrons in odd is fermion Local constraint (completeness relation) for pseudo particles:

5. Local constraint and Hamiltonian Hamiltonian in auxiliary representation Representation of local operators local hamiltonian quadratic (solved exactly) bath hamiltonian is quadratic perturbation theory in coupling between both possible

6. Remarks Why do we introduce “unnecessary” new degrees of freedom?Why do we introduce “unnecessary” new degrees of freedom? (auxiliary particles)  Interaction is transferred from term U to term V Coulomb repulsion U is usually large and V is small But the perturbation in V is singular!  Unlike the Hubbard operator, the auxiliary particles are fermions and bosons and the Wick’s theorem is valid Perturbation expansion is possible

7. Example: 1band AIM

8. Diagrammatic solutions Since the perturbation expansion in V is singular it is desirable toSince the perturbation expansion in V is singular it is desirable to sum infinite infinite number of diagrams (certain subclass). This is sum infinite infinite number of diagrams (certain subclass). This is necessary to get correct low energy manybody scale T K. necessary to get correct low energy manybody scale T K. Definition of the approximation is done by defining the Luttinger-WardDefinition of the approximation is done by defining the Luttinger-Ward functional : functional : Fully dressed pseudoparticle Green’s functions Procedure guarantees that the approximation is conservingProcedure guarantees that the approximation is conserving for example:

9. Infinite U AIM within NCA  Gives correct energy scale  Works for T>0.2 T K  Below this temperature Abrikosov-Suhl resonance exceeds unitarity limit exceeds unitarity limit  Gives exact non-Fermi liquid exponents in the case of 2CKM  Naive extension to finite U very badly fails  T K several orders of magnitude too small

10. How to extend to finite U? Schrieffer-Wolff transformation

11. Luttinger-Ward functional for SUNCA

12. Self-energies and Green’s function

13. Bethe-Salpeter equations

14. Pseudo-fermion self-energy

15. Light pseudo-boson self-energy

16. Heavy pseudo-boson self-energy

17. Physical spectral function (bath self- energy)

18. Physical spectral function

19. Scaling of T K

20. Comparison with NRG

21. Comparison with QMC and IPT

22. Comparison with QMC T=0.5

23. Comparison with QMC T=0.0625

25. T-dependence for t2g DOS

26. Doping dep. for t2g DOS

27. Summary To get correct energy scale for infinite U AIM, self-consistent method is needed Infinite series of skeleton diagrams is needed to recover correct low energy scale of the AIM at finite Coulomb interaction U The method can be extended to multiband case (with no additional effort) Diagrammatic method can be used to solve the cluster DMFT equations.

28. Exact projection onto Q=1 subspace Hamiltonian commutes with Q Q constant in timeHamiltonian commutes with Q Q constant in time Q takes only integer values (Q=0,1,2,3,...)Q takes only integer values (Q=0,1,2,3,...) How to project out only Q=1?How to project out only Q=1?  Add Lagrange multiplier If then Proof:

29. Exact projection in practice How can we impose limit analytically? Only integral around branch-cut of bath Green’s function survives (bath=Green’s functions of quantities with nonzero expectation value in Q=0 subspace) Exact projection is done analytically!

30. Physical quantities Exact relation: Dyson equation: In grand-canonical ensemble

31. Comparison of various approximations