Analytical continuation for self-energy

Introduction

The calculated results for the self-energy function on the real axis using Pade approximation strongly depend on the numerical accuracy of the original self-energy data. However, the CT-HYB/DMFT calculations usually yield a Matsubara self-energy function $\Sigma(i\omega_n)$ with numerical noises. In this case, the Pade approximation does not work well. To overcome this problem, Haule et al^[1]. suggested to split the Matsubara self-energy function into a low-frequency part and high-frequency tail. The low-frequency part is fitted by some sort of model functions which depends on whether the system is metallic or insulating, and the high-frequency part is fitted by modified Gaussian polynomials. It was shown that their trick works quite well even when the original self-energy function is noisy, and is superior to the Pade approximation in all cases. Thus, in the HIBISCUS component, we also implemented this algorithm. It has broad applications in the DFT + DMFT calculations.

The HIBISCUS/swing code is used to continue self-energy analytically from imaginary axis to real axis using K. Haule's strategy.

http://hauleweb.rutgers.edu/downloads/

Usage

$ python ./swing_main.py [options]

Input

The options for the HIBISCUS/swing code can be as follows:

-sig filename
  mandatory, input self-energy on imaginary axis (default:Sig.out)

-nom number
  mandatory, number of matsubara points used (default:100)

-beta float
  mandatory, inverse temperature (default:100.0)

-FL bool
  mandatory, low energy expansion of a Fermi liquid of Mott
  insulator (default:True)

-poles [[y0,A0,B0],[y1,A1,B1],...]
  optional, poles of self-energy determined from spectral function,
  poles will be forced at y0,y1,... this is achieved by contribution
  to chi2+= Al/((x_i-yl)**2+w_i*Bl) where x_i are positions and w_i
  weights of lorentzians to be determined

-b float:[0-1]
  optional, basis functions, b parameter to determin family of basis
  functions (default:0.8)

-Ng number
  optional, number of basis functions (default:12)

-wexp float:[1-2]
  optional, wexp^i is position where basis functions are
  peaked (default:1.5)

-ifit number[4-..]
  optional, low energy fit, number of matsubara points used to fit
  the low energy expansion

-alpha3 float[0-1]
  optional, weight for the normalization in functional to be
  minimized (default:0.01)

-alpha4 float[0-1]
  optional, weight for the low energy expansion in the functional
  to be minimized (default:0.001)

-maxsteps number
  optional, maximum number of function evaluations in minimization
  routine (default:400000)

Output

The possible output files are as follows:

sigr.out
  final self-energy function on real axis

sigr_linear.out
  final self-energy function on fine real axis, it is used to
  interface with DFT + DMFT code

siom.nnn
  current function on imaginary axis (nnn counts every 40000 steps)

sres.nnn
  current analytic continuation to real axis (nnn counts every
  40000 steps)

gaus.nnn
  current configuration for gaussians (nnn counts every 40000 steps)

Recipe: how to convert $\Sigma(i\omega)$ to $\Sigma(\omega)$ using the HIBISCUS/swing code

Step 1:

Perform CT-HYB or HF-QMC calculations, generate a solver.sgm.dat file or multiple solver.sgm.dat.````* files.

Step 2:

Using the HIBISCUS/toolbox/makestd to post-process the solver.sgm.dat file or solver.sgm.dat.````* files. The output should be std.sgm.dat file.

Step 3:

Guess whether the system is metallic or insulating. Edit the corresponding Bash shell scripts (metal.sh or insulator.sh). See iqist/working/tools/hibiscus/swing directory for some examples.

Step 4:

Execute the HIBISCUS/swing code via the Bash shell script.

Step 5:

Validate the real-frequency self-energy function $\Sigma(\omega)$ in the sigr.dat file or sigr_linear.dat file. That is what you need.