Maximum entropy method

Introduction

In the Monte Carlo community, the maximum entropy method^[1] is often employed to extract the impurity spectral function $A(\omega)$ from the imaginary-time Green's function $G(\tau)$. Thus, in the HIBISCUS component, we implemented the standard maximum entropy algorithm. In the E-DMFT calculations, sometimes we have to perform analytical continuation for the retarded interaction function $U(i\nu)$ to obtain $U(\nu)$. So we developed a modified version of the maximum entropy method to enable this calculation.

The HIBISCUS/entropy code is often used to perform the analytical continuation to build impurity spectral function from imaginary-time Green's function using the well-known maximum entropy method. In principle, it solves the Laplace transformation

\[ G(\tau) = \int K(\tau,\omega) A(\omega) d\omega\]

where $K(\tau,\omega)$ is the so-called kernel function. Its definition is as follows:

\[ K(\tau,\omega) = \frac{ \exp{(-\tau\omega)} }{1.0+\exp{(-\beta\omega)}}\]

**Anders W. Sandvik**
*Akademi University, Finland*
*email:asandvik@ra.abo.fi*

Usage

$ ./entropy

or

$ mpiexec -n number_of_cores ./entropy

Input

• tau.grn.dat (necessary)
• entropy.in (necessary)

The tau.grn.dat file contains the $G(\tau)$ data. It has to be generated using the HIBISCUS/toolbox/maketau code.

See also toolbox/maketau for more details.

The entropy.in file contains all of the necessary control parameters for the HIBISCUS/entropy code. The syntax of it is the same with the solver.ctqmc.in file. As for the valid control parameters, please see the following text.

See also solver.ctqmc.in for more details.

Output

• mem.dos.dat
• mem.sum.dat

The impurity spectral function $A(\omega)$ is stored in the mem.dos.dat file. In the mem.sum.dat file, the sum-rules for the impurity spectral function are examined.

Parameters

In the following, we will show the original definitions for the control parameters:

!!========================================================================
!!>>> integer variables                                                <<<
!!========================================================================

! number of imaginary time slices sampling by continuous time or hirsh-fye
! quantum Monte Carlo quantum impurity solver
     integer, public, save :: ntime = 129

! number of frequency points on half axis, energy range can be expressed by
! [ -wstep * nwmax, wstep * nwmax ]
     integer, public, save :: nwmax = 200

! maximum number of cycles for classic maximum entropy method
     integer, public, save :: niter = 20

! number of smooth runs for classic maximum entropy method
     integer, public, save :: ntune = 20

! number of annealing steps per classic maximum entropy method cycle
     integer, public, save :: nstep = 4000

! number of bands
     integer, public, save :: nband = 1

! number of orbitals
     integer, public, save :: norbs = 2

! the way the default model function is build
! if ntype == 0, gaussian model
! if ntype == 1, flat model
     integer, public, save :: ntype = 1

!!========================================================================
!!>>> real variables                                                   <<<
!!========================================================================

! initial alpha parameter
     real(dp), public, save :: ainit = 1200._dp

! it is the deviation from the average green's function
     real(dp), public, save :: devia = 0.001_dp

! \beta, inversion of real temperature
     real(dp), public, save :: beta  = 10.00_dp

! gauss broadening parameter, used to build the default model
     real(dp), public, save :: sigma = 1.600_dp

! delta frequency, step of real frequency grid
     real(dp), public, save :: wstep = 0.025_dp

Recipe: how to convert $G(\tau)$ to $A(\omega)$ using the HIBISCUS/entropy code

Step 1:

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

Step 2:

Using the HIBISCUS/toolbox/maketau to post-process the solver.green.dat file or solver.green.dat.````* files. The output should be tau.grn.dat file.

Step 3:

Edit the entropy.in file, setup reasonable control parameters.

Step 4:

Execute the HIBISCUS/entropy code, the tau.grn.dat and entropy.in files are necessary inputs.

Step 5:

Validate the impurity spectral function $A(\omega)$ in the mem.dos.dat file. That is what you need.

Step 6:

Now you can use the data in mem.dos.dat file to generate beautiful figures, or use the other tools to postprocess it again.