Multiple-Basis Representation States for the Transverse Field Ising Model

This repository contains the code to simulate a two-dimensional transverse field Ising model with multiple-basis representation states. The Hamiltonian of the model is given by

$$ H = J \sum_{\langle i,j \rangle} X_{i} X_{j} + h \sum_i Z_{i}, $$

where $J$ is the Ising coupling and $h$ is the transverse field. The matrices $X_i$ and $Z_i$ are the respective Pauli matrices acting on site $i$.

Additionally, the repository contains benchmarking code in the form of matrix-product state (MPS), projected entangled pair state (PEPS) and exact diagonalization (ED) simulations.

Installation

The code is written for Python 3 and tested to work with Python 3.8.

To make sure that all requirements of the package are fulfilled, the easiest way to use the code is to create a virtual environment and install the mbrsim package.

1. Creation of a virtual environment
   You can create the environment in a folder of your choice. For the rest of the tutorial, we assume it to be in ~/.pyenv/

cd ~/.pyenv
python -m venv mbrsimenv

Assuming you are using bash or zsh, you can activate the environment with source ~/.pyenv/mbrsimenv/bin/activate. Upon activation, you will notice that your prompt changes. As long as it is prefixed with (mbrsimenv) the virtual environment is active. The virtual environment can be deactivated with deactivate.

2. Cloning the code
   You can obtain the code by cloning the repo with

git clone git@github.com:patrick.emonts/multiple-basis-representation

3. Install the mbrsim package The mbrsim package is still in the development phase. We recommend an editable install, since the main reason to install the package is development. Execute the following command while standing in the root directory of the repository

pip install -e .

Pip installs all dependencies as specified in the pyproject.toml and yields a working environment.

Structure for the Code

The repository is split into two main parts: the package mbrsim (at src/mbrsim) and utility scripts in the main folder.

The package mbrsim contains the simulation code, i.e. the actual implementation of the physical problem, the transverse field Ising model. All scripts in the main folder call parts of the package and provide the infrastructure to manage the simulations.

The package mbrsim is divided into several modules:

• graph.py: Build a graph representation of a given lattice
• hamiltonian.py: Simulation tools for the TFIM with different techniques (exact diagonalization, PEPS, MPS)
• mbr.py: Simulation tools for the TFIM with multi-basis representation states
• utils.py: Utility functions for data management and debugging

The Manager

The script manager.py is the main script that manages all simulations. The manager.py can start different simulations. Currently, there are four supported simulations: MBR, MPS, PEPS and exact diagonalization (ED).

An example call for a 2x2 lattice with open boundary conditions, solved with ED looks like

python manager.py --nx 2 --ny 2 --hmin 0.2 --hmax 1.0 --nsteps 9 --type ed

The call executes 9 simulations from $h=0.2$ until $h=1.0$ with an Ising coupling of $J=1$.

Options

A full list of all options can be found with

python manager.py --help

Future directions and known issues

• The magnetization observables have to be tested more in a system where the ground state is not degenerate, e.g. the ferromagnetic case with a small longitudinal field.
• The staggered magnetization in the MBR case has some issue