Stochastic analytical continuation

Introduction

An alternative way to extract the $A(\omega)$ from $G(\tau)$ is using the stochastic analytical continuation^[1]. Unlike the maximum entropy method, the stochastic analytical continuation does not depend on any a priori parameters. It has been argued that the stochastic analytical continuation can produce more accurate spectral functions with more subtle structures.

In the HIBISCUS component, we also implemented the stochastic analytical continuation which can be viewed as a useful complementary procedure to the maximum entropy method. Since the stochastic analytical continuation is computationally much heavier than the maximum entropy method, we parallelized it with MPI and OpenMP.

The HIBISCUS/stoch code is often used to perform the analytical continuation to build impurity spectral function from imaginary-time Green's function using the modern stochastic analytic continuation 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:

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

Usage

$ ./sac

or

$ mpiexec -n number_of_cores ./sac

Input

• tau.grn.dat (necessary)
• sac.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 sac.in file contains all of the necessary control parameters for the HIBISCUS/stoch 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

• sac.image.dat
• sac.imsum.dat
• sac.move.dat
• sac.swap.dat

The $\alpha$-resolved and $\alpha$-averaged impurity spectral functions are stored in the sac.image.dat and sac.imsum.dat files, respectively. In the sac.move.dat and sac.swap.dat files, the statistics for the Monte Carlo move/swap operations are recorded.

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 = 1024

! number of frequency points on half axis, energy range can be expressed by
! [ -wstep * nwmax, wstep * nwmax ]
     integer, public, save :: nwmax = 128

! number of slices of x in [0,1]
     integer, public, save :: ngrid = 10001

! number of configurations, dimension for r_{\gamma} and a_{\gamma}
     integer, public, save :: ngamm = 1024

! number of alpha parameters used in parallel tempering
! note: it must be an even number, since we need to exchange configurations
! between different alpha channel
     integer, public, save :: nalph = 10

! maximum number of thermalization steps
     integer, public, save :: nwarm = 4000

! maximum number of quantum Monte Carlo sampling steps
     integer, public, save :: nstep = 4000000

! output period for stochastic analytic continuation code
     integer, public, save :: ndump = 40000

! measurement scheme
! if ltype == 1, normal measurement
! if ltype == 2, using legendre polynomial representation
     integer, public, save :: ltype = 1

! maximum order for legendre polynomial
     integer, public, save :: lemax = 64

! number of mesh points for legendre polynomial in [-1,1] range
     integer, public, save :: legrd = 20001

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

! initial alpha parameter
     real(dp), public, save :: ainit = 1.00_dp

! \alpha_(p+1) / \alpha_p = R, used to build alpha parameter list
     real(dp), public, save :: ratio = 2.00_dp

! \beta, inversion of real temperature
     real(dp), public, save :: beta  = 10.0_dp

! lorentz broadening parameter \eta, used to represent delta function
     real(dp), public, save :: eta1  = 0.02_dp
     real(dp), public, save :: eta2  = 4E-4_dp

! gauss broadening parameter, used to build the default model
     real(dp), public, save :: sigma = 1.00_dp

! frequency step, used to build the frequency mesh
     real(dp), public, save :: wstep = 0.05_dp

Recipe: how to convert $G(\tau)$ to $A(\omega)$ using the HIBISCUS/stoch 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 sac.in file, setup reasonable control parameters.

Step 4:

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

Step 5:

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

Step 6:

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