# Pymablock #

## What is Pymablock ?#

Pymablock (Python matrix block-diagonalization) is a Python package that constructs effective models using quasi-degenerate perturbation theory. It handles both numerical and symbolic inputs, and it efficiently block-diagonalizes Hamiltonians with multivariate perturbations to arbitrary order.

Building an effective model using Pymablock is a three step process:

• Define a Hamiltonian

• Request the desired order of the effective Hamiltonian

from pymablock import block_diagonalize

# Define perturbation theory
H_tilde, *_ = block_diagonalize([h_0, h_p], subspace_eigenvectors=[vecs_A, vecs_B])

# Request correction to the effective Hamiltonian
H_AA_4 = H_tilde[0, 0, 4]


## Why Pymablock ?#

Here is why you should use Pymablock :

• Do not reinvent the wheel

Pymablock provides a tested reference implementation

• Apply to any problem

Pymablock supports numpy arrays, scipy sparse arrays, sympy matrices and quantum operators

Due to several optimizations, Pymablock can reliably handle both higher orders and large Hamiltonians

## How does Pymablock work?#

Pymablock considers a Hamiltonian as a series of $$2\times 2$$ block operators with the zeroth order block-diagonal. To carry out the block-diagonalization procedure, Pymablock finds a minimal unitary transformation $$U$$ that cancels the off-diagonal block of the Hamiltonian order by order.

(1)#$\begin{gather} H = \begin{pmatrix}H_0^{AA} & 0 \\ 0 & H_0^{BB}\end{pmatrix} + \sum_{i\geq 1} H_i,\quad U = \sum_{i=0}^\infty U_n \end{gather}$

The result of this procedure is a perturbative series of the transformed block-diagonal Hamiltonian.

(2)#$\begin{gather} \tilde{H} = U^\dagger H U=\sum_{i=0} \begin{pmatrix} \tilde{H}_i^{AA} & 0 \\ 0 & \tilde{H}_i^{BB} \end{pmatrix}. \end{gather}$

Similar to Lowdin perturbation theory or the Schrieffer–Wolff transformation, Pymablock solves Sylvester’s equation and imposes unitarity at every order. However, differently from other approaches, Pymablock uses efficient algorithms by choosing an appropriate parametrization of the series of the unitary transformation. As a consequence, the computational cost of every order scales linearly with the order, while the algorithms are still mathematically equivalent.

To see Pymablock in action, check out the tutorial. See its algorithms to learn about the underlying ideas, or read the reference documentation for the package API.

## What does Pymablock not do yet?#

• Pymablock is not able to treat time-dependent perturbations yet

• Pymablock does not block diagonalize more than two subspaces simultaneously

## Installation#

The preferred way of installing pymablock is to use mamba/conda:

mamba install pymablock -c conda-forge


Or use pip

pip install pymablock


Important

Be aware that the using pymablock on large Hamiltonians requires Kwant with MUMPS support. For this purpose, install Kwant via conda in Linux and MAC OS. Unfortunately, MUMPS support in Kwant is not available for Windows. If you need it, try Windows Subsystem for Linux (WSL).

## Citing#

If you have used Pymablock for work that has lead to a scientific publication, please cite it as

@misc{Pymablock,
author = {{Araya Day}, Isidora and Miles, Sebastian and Varjas, Daniel and Akhmerov, Anton R.},
doi = {10.5281/zenodo.7995684},
month = {6},
title = {Pymablock},
year = {2023}
}


## Contributing#

Pymablock is an open source package, and we invite you to contribute! You contribute by opening issues, fixing them, and spreading the word about pymablock.