Full Counting Statistics using Coherent Potential Approximation

keywords: coherent potential approximation (CPA), full couting statistics (FCS) theory, cumulant generating function (CGF), single site approximation (SSA), matrix inverse average, Broyden's method, self-consistent calculation, recursive Green's function (RGF) algorithm

CPA

Given a complex matrix A of size $N\times N$, and a random diagonal matrix B of the same size with some specified distribution (e.g. uniform, binary) (all diagonal elements of B are drawed independently from the distribution). The CPA algorithm is able to calculate the matrix inverse average $\langle \frac{1}{A-B} \rangle$.

In quantum transport field, CPA algorithm is usually used to calculate the Green's function and related quantities (transmission, shot noise, etc.) (see reference below).

quickstart

install

pip install git+https://github.com/husisy/FCSCPA.git

usage

from FCSCPA import CPA_general, CPA_block_general
from FCSCPA import FCS_CPA_oneband, FCS_CPA_multiband
from FCSCPA import generate_binary_hf_rand, generate_binary_hf_average
from FCSCPA import generate_uniform_hf_rand, generate_uniform_hf_average
from FCSCPA import generate_hf_Bii, Aii_to_matA

all functions provided in FCSCPA module

1. CPA_general(): CPA algorithm
2. CPA_block_general(): CPA algorithm when the random matrix B is Block diagonal
3. broyden_solver(): Broyden' method to solve the self-consistent equation f(x)=x
4. np_inv_along_axis12(): matrix inverse for high-dimensional (np.ndim>2) tensor
5. matA_to_Aii() and Aii_to_matA(): utils for block diagonal matrix
6. generate_binary_hf_rand() and generate_uniform_hf_rand(): (meta function) to generate value with some specified distribution
7. generate_binary_hf_average() and generate_uniform_hf_average(): (meta function) to evaluate the function average with some specified distribution
8. generate_hf_Bii(): (meta function), mapping single random variable to a random matrix
9. FCS_CPA_oneband()
10. FCS_CPA_multiband()
11. ball_in_bowl()
12. Ymn_Leibniz_matAB()
13. Ymn_Leibniz()
14. demo_Ymn_Leibniz()
15. demo_Ymn_Binomial()
16. Ymn_electron_zeroK()

see reference below for the algorithm detail. More examples will be added sooner, currently please see unittest scripts for detailed usage.

for developer

1. clone the entire repository git clone https://github.com/husisy/FCSCPA.git
2. install with editable mode pip install -e .
3. run the unittest pytest -v .
   • take about one minutes
   • sometimes some unittests may fail, since random number is used, it's okay to just run again (smile~)

reference

Broyden's method to solve self-consistent equation

1. A class of methods for solving nonlinear simultaneous equations
2. TODO add more ref

coherent potential approximation

1. Full counting statistics of conductance for disordered systems
2. TODO add more ref

full counting statistics theory

1. Full counting statistics of conductance for disordered systems
2. Full counting statistics of charge transport for disordered systerms

TODO

1. move FCS_CPA folder here
2. replace .CPA with CPA
3. merge CPA with FCSCPA
4. apply FCSCPA to system in finite temperature
5. apply FCSCPA to two-sites system
6. TODO, converge issue, can we start from hermitian point