Diagrammatic auxiliary particle impurity solvers SUNCA

1
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 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 TltltTk 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

3
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

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
Representation of local operators
Hamiltonian in auxiliary representation
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?
• (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 Wicks 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 to
• sum infinite 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 Greens functions

• Procedure guarantees that the approximation is
  conserving
• for example

9
Infinite U AIM within NCA

• Gives correct energy scale
• Works for Tgt0.2 TK
• Below this temperature Abrikosov-Suhl resonance
• exceeds unitarity limit
• Gives exact non-Fermi liquid exponents in the
  case
• of 2CKM

• Naive extension to finite U very badly fails
• TK 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 Greens 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 TK
20
Comparison with NRG
21
Comparison with QMC and IPT
22
Comparison with QMC T0.5
23
Comparison with QMC T0.0625
24
Comparison with QMC T0.0625
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 Q1 subspace

• Hamiltonian commutes with Q Q constant
  in time
• Q takes only integer values (Q0,1,2,3,...)
• How to project out only Q1?
• Add Lagrange multiplier

If
then
Proof
29
Exact projection in practice
How can we impose limit
analytically?
Only integral around branch-cut of bath Greens
function survives (bathGreens functions of
quantities with nonzero expectation value in Q0
subspace)
Exact projection is done analytically!
30
Physical quantities
Exact relation
Dyson equation
In grand-canonical ensemble
31
Comparison of various approximations