Tight Binding and Anderson Localization on Arbitrary Graphs

Created by Ravi Koka, Abby Perryman, Anish Pandya, Patrick Devaney

Getting Started

To use the notebooks here, create a conda environment using qgraph_env.yml.

Instructions for creating a conda environment from a yaml file are located here. This will automatically create an conda environment named qgraph, which you should use as your Jupyter kernel.

Introduction

This project implements the tight binding model for an electron on an arbitrary graph structure.

By varying disorder and random parameters, Anderson Localization can be demonstrated.

There are interfaces for generating graphs that represent 1D and 2D lattices and for generating Erdős–Rényi random graphs.

There is direct support for calculating the inverse participation ratio (IPR), a measure of locality.

Numerical Approach

The Tight Binding Model

An electron can exist in finitely many discrete sites numbered $1 , 2 , \ldots , N$ with corresponding states $| 1 \rangle , | 2 \rangle , \ldots , | N \rangle$ (where $N$ is the total number of sites). Different sites have arbitrary connections to each other, with the connections forming a simple graph. This graph can then be represented by an $N \times N$ adjacency matrix.

The state of the electron is denoted $| \psi \rangle$ and is a normalized linear combination of the site states.

The Tight Binding Hamiltonian

The Hamiltonian for the tight binding model for a single electron can be written as follows:

$$\langle n | H | m \rangle = H_{n,m} = \underbrace{\epsilon \delta_{n,m}}_\textrm{binding} \underbrace{- t(\delta_{n+1,m} + \delta_{n-1,m} )}_\textrm{hopping}$$

where $H$ is the electron Hamiltonian, $\epsilon$ is the binding energy, $\delta$ is the Kronecker delta, and $t$ is the hopping parameter. $n$ and $m$ refer to states associated with a particular site as in the tight binding formalism.

Anderson localization can be demonstrated on this model when the diagonal of the Hamiltonian $H$ is randomized and if values of $\epsilon$ are randomly sampled from a uniform distribution on $[-W, , W]$ where $W$ is known as the disorder parameter. In this case, larger $W$ correlates with increased localization.

Time Evolution

The system can be time evolved using the following equation:

$$ | \psi(t) \rangle = \exp(-i H t / \hbar) | \psi(0) \rangle $$

Inverse Participation Ratio (IPR)

The inverse participation ratio is a measure of locality defined as the sum of the fourth power of the magnitude of the wave function of each site in the lattice. Explicitly,

$$ \textrm{IPR} = \sum_n |\psi(n)|^4 $$

IPR takes a range of $1 / N$ to $1$, with the former being the least localized and the latter the most.

Basic Usage

AndersonGraph.py contains the AndersonGraph class which implements the tight binding model on a graph, with the input graph being a NetworkX graph object.

Multiple code examples can be found in summary.ipynb.

plotting.py has functions for plotting states of the graph. animations.py has functions for generating animations graphs. plot_ipr.py has functions for plotting the IPR of a graph. Most of these functions take an AndersonGraph object and a few visual parameters.

To create an instance of an AndersonGraph object, the constructor takes as arguments a NetworkX graph object (a simple graph), an array of wave functions at each site, a range of values on which $\epsilon$ can be sampled from, and the hopping parameter $t$.

Here is a basic example creating an 80-site 1D ring with $| \psi(0) \rangle$ fully localized at a single site and $W = 1$.

import AndersonGraph as ag # The exact form of this may change based on the relative location of AndersonGraph.py

N = 80 # Define number of sites
psi_0 = np.zeros(N)
psi_0[N//2] = 1 # Create wave function

ring = nx.grid_graph(dim=[N], periodic=True) # Create ring

anderson_ring = ag.AndersonGraph(graph=ring, psi_0=psi_0, eps_range=[-1, 1], t_hop=1) # Construct AndersonGraph object

anderson_ring.plot_density(t=17) # Plot the time evolution of the system

And here is the plot output:

Plot and Animation Examples

1D Ring

Diffusive Case

$W = 0 $

Localized Case

$W = 10$

IPR

Periodic 2D Lattice

Diffusive Case

$W = 0$

Localized Case

$W = 10$

IPR

Random Graph

Diffusive Case

$W = 0$

Localized Case

$W = 100$

IPR

Many example plots and animations can be found in the "plots" folder here.

Potential Applications

The basic setup here of allowing an electron to exist in discrete, finite spaces can be applied to some interesting quantum mechanical systems. A non-exhaustive list of applications is given below.

1. Anderson Localization: This project can be used to show Anderson localization on graphs.
2. Quantum Dot Cellular Automata: A computing paradigm that is an alternative to standard transistor-based computing, based on cellular automata. Encodes information in charges localized in quantum dots.
3. Many-Body Localization: Anderson localization can be extended to the localization of many bodies, something thermodynamically interesting.

Resources

This started as a project for PHY 381C (Computational Physics) at UT Austin (class website). Here is a link to a presentation given in-class on this repository, which is provided in case anyone finds it useful. It is also available as a PDF (which precludes animationes) here.

Future Directions

1. Generalize to Many-Body Case: Our single-electron approach allows for a closed-form solution for the wave function at any point in time. Introducing multiple electrons would necessitate making a choice about iterative and approximate solution methods and would introduce more complex quantum mechanical behaviors.
2. Introduce Different Boundary Conditions: Every connection on the graph here behaves as a periodic boundary condition — that is, the electron is free to travel between any node unhindered by reflective, absorptive, or other effects.
3. Add Graph Directionality and Weight: Extensions past simple graph structures may be able to realize more complex physical phenomena, inclding transport phenomena and variable site spacing.

Major Dependencies

This project makes extensive use of NetworkX for representing graph structures.

Credits

Special thanks to Dr. William Gilpin ([email protected]), Edoardo Luna ([email protected]), and Anthony Bao ([email protected]) for such good course instructors.