Diagrammatic auxiliary particle impurity solvers SUNCA Auxiliary particle

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

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

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

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

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

Remarks • Why do we introduce “unnecessary” new degrees of freedom? (auxiliary particles) ØInteraction 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! ØUnlike the Hubbard operator, the auxiliary particles are fermions and bosons and the Wick’s theorem is valid Perturbation expansion is possible

Example: 1 band AIM

Diagrammatic solutions • Since the perturbation expansion in V is singular it is desirable to sum infinite number of diagrams (certain subclass). This is necessary to get correct low energy manybody scale TK. • Definition of the approximation is done by defining the Luttinger-Ward functional : Fully dressed pseudoparticle Green’s functions • Procedure guarantees that the approximation is conserving for example:

Infinite U AIM within NCA ØGives correct energy scale ØWorks for T>0. 2 TK ØBelow this temperature Abrikosov-Suhl resonance exceeds unitarity limit ØGives exact non-Fermi liquid exponents in the case of 2 CKM ØNaive extension to finite U very badly fails ØTK several orders of magnitude too small

How to extend to finite U? Schrieffer-Wolff transformation

Luttinger-Ward functional for SUNCA

Bethe-Salpeter equations

Pseudo-fermion self-energy

Light pseudo-boson self-energy

Heavy pseudo-boson self-energy

Physical spectral function (bath selfenergy)

Physical spectral function

Scaling of TK

Comparison with NRG

Comparison with QMC and IPT

Comparison with QMC T=0. 5

Comparison with QMC T=0. 0625

Comparison with QMC T=0. 0625

T-dependence for t 2 g DOS

Doping dep. for t 2 g DOS

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.

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

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!

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