Topology in the Hofstadter Model

Hofstadter Model on a square lattice loaded with interacting bosons.

Table of Contents

• General Info
• Features
• Usage
• Room for Improvement
• Publications
• Acknowledgements
• Contact

General Information

This project is related to my research activity as a PhD student in the LPMMC laboratory, CNRS (Grenoble), under the supervision of C. Repellin.

It provides an exact diagonalization of the Hofstadter model for interacting bosons on a square lattice: $$H = -J\sum_{m,n} \bigg ( b^\dagger_{m+1,n} b_{m,n}e^{i2\pi\alpha n} + b^\dagger_{m,n+1} b_{m,n} + \text{h.c.} \bigg ) + \frac{U}{2}\sum_{m,n} b^\dagger_{m,n} b_{m,n} \left( b^\dagger_{m,n} b_{m,n} - 1 \right) + \frac{U_3}{6}\sum_{m,n} b^\dagger_{m,n} b_{m,n} \left( b^\dagger_{m,n} b_{m,n} - 1 \right)\left( b^\dagger_{m,n} b_{m,n} - 2 \right) + \sum_{m,n} V(m,n) b^\dagger_{m,n} b_{m,n}$$ It also gives a practical way to perform edge state spectroscopy via light-matter interaction, by mean of a Laguerre-Gauss beam (periodic perturbation of the Hamiltonian above). The spatial modes of the LG laser are: $$f_{n,l}(r,\theta) = L_n^l(r)\big( \frac{r}{r_0} \big)^l e^{\frac{r^2}{2r_0^2}} e^{i\theta l}$$ This gives a way to study the rich topological features of this model, that has already been implemented in experiments and can be useful for future applications in quantum computing.

Features

• Exact diagonalization of the Hofstadter Model with open boundary conditions
• Energy spectrum resolved in the C4 rotation symmetry
• Absorption spectra calculated via transition matrix elements (from the ground state) $\bra{\psi_n}f_{n,l}\ket{\psi_0}$
• Local particle density of a generic eigenstate of the Hamiltonian: $\bra{\psi_n}\hat n_i\ket{\psi_n}$
• Time evolution protocol: ground state driven in time by a Laguerre-Gauss beam
• Entanglement spectrum for a target state calculated with particle partition in real space

Usage

For info about usage just type 'python [name of the program].py -h'. Example of output below:

usage: HofstadterThreeBody.py [-h] [-N N] [-L L] [-J J] [-U U] [-U3 U3] [--conf CONF] [--alpha ALPHA] [--hardcore [HARDCORE]]
                              [--savestates [SAVESTATES]] [--nbreigenstates NBREIGENSTATES] [--c4symmetry [C4SYMMETRY]]

options:
  -h, --help            show this help message and exit
  -N                   number of particles
  -L                   side of the square lattice of size LxL
  -J                   tunneling energy
  -U                   two-body onsite interaction (only in softcore mode)
  -U3                  three-body onsite interaction (only in softcore mode)
  --conf CONF           harmonic trap confinement strength (v0) as v0 * r^2
  --alpha ALPHA         magnetic flux density as alpha=p/q
  --hardcore [HARDCORE]
                        hardcore bosons mode
  --savestates [SAVESTATES]
                        save eigenvectors
  --nbreigenstates NBREIGENSTATES
                        number of eigenstates to be saved
  --c4symmetry [C4SYMMETRY]
                        use the c4 rotation symmetry in the exact diagonalization

Room for Improvement

Here some inspiration for improvement/features that might be done in the future, and an imminent to-do list.

Room for improvement/new features:

• Optimization for the building of the C4 symmetric Hamiltonian
• Add periodic boundary conditions (cylinder and torus geometries)

Publications

F. Binanti, N. Goldman, C. Repellin, ArXiv:2306.01624

In this paper I used the code to benchmark an experimental protocol to detect topological edge states.

Acknowledgements

Many features of this project were inspired by Cecile Repellin (LPMMC Grenoble, CNRS).

Contact

Francesco Binanti (francesco.binanti@lpmmc.cnrs.fr) - feel free to contact me!