Analytical continuation

Below we describe how to used the analytical continuation program. Because of non-analyticity of QMC results one cannot make direct (like Pade) analytical continuation from Matsubara to real axis. What we do in practice we make the analytical continuation of Green function from imaginary time axis to real axis using Maximum Entropy (ME) method. Then using this GF and Hilbert transformation formulae one solve this equation (Hilbert) for the self-energy.

Below we describe input and output parameter for analytical continuation program.

File inpmax:

 2           64            1 0.003        16.0        300            0.02         3000          1200             1          1234          0  0.040     -8.7 4. !  Ns                                        !  L                                         ! idg, delta-G (0 - propotional G, 1 = const)  !  Beta                                     !  Ne                                        !  De                                         !  Nmc                                                                                                                                                                              ! iflat,eim                                  !mu  U

The first line number of bands 1,2,3 … In our case bands are degenerate one band is enough. The second line is number of time slices used (it is defined in QMC program). Idg is the way the model function for ME method is build. From our experience follows that constant model function is the best one. Delta-G is deviation from GF (0.001-0.005 is usual interval we worked with). The forth line is clear – inverse temperature. Ne is number of points on real axis (maximum 600) and de is frequency step on real axis: de*Ne=energy window on real axis. Nmc is number of annealing steps. 1200 is alpha coefficient (recall exponent in ME method). The ninth line is number of smoothing runs. Tenth line is random seed number. Iflat is flat model which is used in ME method (zero corresponds to constant and 1 corresponds to other model which was based on Pade approximation, but it us better always to keep zero). Eim is connected to the first model described and if one takes zero then eim is irrelevant. The last line is the chemical potential, Coulomb repulsion.

Another input file called Gtau1.dat contains GF on imaginary axis. Output files are “dos” which contains DOS.