How to run the NCA program

The program can be compiled on linux PC by typing "make PLATFORM=PC RELEASE=1". The second option produces the optimized code.

A short help can always be obtained by executing the program and giving no arguments. Thus typing nca gives the following output

Sig = sig.inp	# The name of the input Self-energies
Ac = Ac.inp	# The name of the input bath function
cix = cix.dat	# The name of the input file containing information about bands and their degeneracy
out = .	# The name of the output directory
gloc = gloc.out	# The name of the output Green's function
sig = sig.out	# The name of the output Self-energy
Ed = -2	# Energy level
U = 4	# Coulomb repulsion
T = 0.2	# Temperature
Q = 1	# Default average Q in grand-canonical ansamble
alpha = 0.5	# The fraction of the new self energy to be used in the next iteration
max_diff = 1e-06	# Criterium to finish the procedure
max_steps = 300	# Criterium to finish the procedure
StartLambda = -1	# Where to start searching for the lambda0
EndLambda = 1	# Where to stop searching for the lambda0
dLambda = 0.1	# Step in searching for the lambda
followPeak = -3	# Wheather to determin zero frequency from the diverging pseudo-particle
	(-2: lamdba0==StartLambda, -1: Q==Q0, 0: follow b, 1: follow f, 2: foolow a)
prt = 1	# Weather to print intermediate results
MissDopSt = -100	# Missing doping due to projection and finite mesh starting at MissDopSt

All control parameters are printed with there default values. After the symbol # a short description of each parameter is also given. Below we will give some more description to each parameter but let us run an example program first.

The necessary files can be found in the subdirectory PC/work and PC/work/start. The example is for the f-electrons with around 6 electrons on the impurity. The crystal field splits the 7 band problem into 3+3+1 band problem (To generate the input file one thus needs to use 3 bands and degeneracy 6,6,2). All the impurity states with 4,5,6 and 7 electrons on the impurity are keept which results in a 64 nonequivalent pseudoparticle (atomic) states. The necessary coefficients $F$'s to run the program are already prepared in the file PC/work/start/cix4567.cix.

To run the program, one also needs a bath spectral function (it is in PC/work/start/Ac.12) and starting guess for the pseudo self-energies (can be found in PC/work/start/Sigma.000). The input bath spectral function contain four columns, frequency and three nonequivalent bath spectral functions. The pseudo self-energy files contains 65 columns, in addition to frequency, a guess self-energy for all 64 pseudo particles.

An example of how to run the program can be found in the file history.nca in a subdirectory PC/work and reads

../nca Sig=start/Sigma.000 Ac=start/Ac.12 cix=start/cix4567.cix U=4 T=0.1 "Ed={-20.67,-20.58,-20.64}"

Each time the program is executed, the command line is saved (appended) to the file history.nca for easier restart of a job at a later time.

In the example, the self-consistency is reached after about 30 iterations (the difference between the steps below $10^{-6}$) and the following results are ready to be plotted

• Impurity Green's function is in the file gloc.out (the filename can be changed in the command line)
• The physical self-energy is in the file sig.out (the filename can also be chnaged if desired)
• the latest pseudo self-energies are in the file Sigma.000
• the latest pseudo spectral functions are in the file Spec.000
• pseudo self energies for each iteration are saved in a file Sigma.xxx
• pseudo spectral functions for each iteration are saved in a file Spec.xxx

The spectral function obtained in this example is plotted in figure 1. At each iteration, the following information is written in the standard output

Figure 1: The spectral function in the example.

• The value of $\lambda$ discussed in the chapter 1. (lambda=-64.8101064793649)
• The value of $Q$ that was used (Q=9438 which is total number of states here)
• The occupancy of each atomic state (the sum of all numbers is unity)
• Total impurity occupancy (nd: 5.73398078352997)
• Norm of every pseudo spectral function (should be close to one if mesh resolves the structure and $\lambda$ was chosen correctly).
• Impurity occupancy per each band (0: 2.5582 1: 2.4337 2: 0.7420)
• The difference between last two steps (Difference between steps: 6.768e-07).

Finally, let me give some more details of every parameter of the program

• Sig: The filename of the input guess for the pseudo self-energies
• Ac: The filename of the input bath spectral function
• cix: The input file that was generated by the program generate and discussed in chapter 2. Note that the number of columns in the Sig file should match the number of pseudoparticle states from cix-file and the number of columns in Ac file should match the number of baths in the cix-file.
• out: The name of the output directory. Default is the current directory.
• gloc : Filename of the output local Green's function
• sig : Filename of the output local self-energy
• Ed : Impurity energies. The number of impurity energies should match the number of baths (one impurity energy for each nonequivalent bath). If not, the program assumes that the rest of the energies are equal to the last specified energy. They have to be given in curly brackets and in the shell, additional quotation marks should be used, for example "Ed={-3,-3,-3}".
• U : The Coulomb repulsion
• T : Temperature
• Q : The desired charge $Q$ explained in chapter 1. This is used only when followPeak is set to -1.
• alpha : The mixing parameter for the pseudo self-energy
• max_diff : The difference between successive self-energies has to be smaller than this number to stop iterating.
• max_steps : The number of iterations is always less than this number (regardless of max_diff) to avoid infinite loops.
• StartLambda : If $Q$ is fixed number, one needs to find the corresponding $\lambda$ as explained in chapter 1. For that, one needs to solve a nonlinear equation and the solution is always looked for in the interval StartLambda and EndLambda. StartLambda should be enough negative to catch the solution. If the solution is not found, tray to decrease this number (more negative) and if the solution is found, keep the new value. However, very small StartLambda can result in worse performance.
• EndLambda: Explained above
• dLambda : If one looks for the $\lambda$ allover the interval [StartLambda,EndLambda], the solution is very unlikely to be cached. The reason is that when a guess $\lambda$ is larger than the $\lambda$ that corresponds to the solution, the spectral functions are more diverging and therefore the spectral functions are much worse resolved. The strategy is to start looking for solution at most negative lambda (StartLambda) and slowly move up until the solution is bracket and then use the Newton procedure. The small step when bracketing the solution is dLambda. Can be much decreased if the solution is hard to catch (very close to the Mott transition).
• followPeak: There are four strategies implemented to fix zero frequency of the pseudoparticle quantities. When followPeak is -3, the $Q$ is fixed at the total number of all local states and $\lambda$ is then looked for. If the followPeak is -2, $\lambda$ is chosen to be equal to $StartLambda$ and $Q$ is calculated from the equation (16). If followPeak is -1, $Q$ is fixed to the input $Q$ (specified in the command line, the parameter Q above) and the corresponding $\lambda$ is looked for. If follow peak is a non-negative number $a$, the zero frequency (and $\lambda$) is chosen such that the maximum of the pseudoparticle spectral function with index $a$ is at zero frequency.
• prt : If prt is 1, pseudo self-energy and spectral functions are printed at each iteration. It can also be set to 0.
• MissDopSt: Sometimes the mesh on which the electron spectral function is calculated, does not catch all the spectral weight at negative frequencies, or alternatively, there is some weight of spectral function lost due to projection (if we take occupancies n-1,n,n+1,... and n-2 was still having some nonzero probability). In this case, we want to calculate how much weight is missing when integrating spectral function on the output mesh. This is very important in the real material calculation, because the missing weight should be added to the total number of electrons below the Fermi level, to get the right occupancy and right chemical potential. The parameter should be set to the point below which the weight is considered to be lost.