johanschott / impuritymodel Goto Github PK

To make things more readable, perhaps this could help?

At the moment use bitarray from the bitarray package to represent a product state.

Store Hamiltonian matrix elements in sparse format in a dict.
The product states are dict-keys so also need an immutable representation of a product state.
Get this by using bitarray's .tobytes() method.

Instead of using bitarray one could use one Python built-in integer to represent a product state.
It is immutable so no need to convert between two representations during calculations.
It should be roughly as quick to change a state (removing or adding an electron) using the integer representation as using bitarray, since bit-shifts are used modify the integer representation (since #47).

If use the integer representation in the calculations could remove bitarray as a representation of a product state.

uint in create.py and remove.py convert the product state from the integer representation to a string representation.
This is not needed. One can stay in the integer representation by using bit-operations: <<, >>, &, |, ^ and python3.10 method https://docs.python.org/3.10/library/stdtypes.html#int.bit_count

There is a small bug in get_spectrum and get_spectrum_using_CF. The magnetic field hx is overwritten by hy in the following lines:

    hHfieldOperator[(((l, 1, m), 'c'), ((l, 0, m), 'a'))] = hx*1/2.
    hHfieldOperator[(((l, 0, m), 'c'), ((l, 1, m), 'a'))] = hx*1/2.
    hHfieldOperator[(((l, 1, m), 'c'), ((l, 0, m), 'a'))] = -hy*1/2.*1j
    hHfieldOperator[(((l, 0, m), 'c'), ((l, 1, m), 'a'))] = hy*1/2.*1j

/Patrik

The Lanczos algorithm may suffer from numerical instability, see discussion in e.g. https://en.wikipedia.org/wiki/Lanczos_algorithm.
The generous unit-test tolerances indicate numerical round-off errors are indeed present (compared the same code on Mac OS X and Ubuntu).
Until now, the errors I have seen have been smaller than the physics of interest.
But it is good to be aware of this numerical instability issue, in particular if very subtle features are of interest.

In https://en.wikipedia.org/wiki/Lanczos_algorithm, a few ideas and references are provided of how to improve the numerical stability of the Lanczos algorithm, that might be relevant for this repository.