This repository contains the work completed during my Summer 2026 research internship under the supervision of Dr. Panch Ram (Department of Physics, IIT (BHU), Varanasi).
The project investigates the Su–Schrieffer–Heeger (SSH) model, one of the simplest lattice models exhibiting topological phases of matter. Starting from the real-space tight-binding Hamiltonian, the model is developed analytically and explored numerically through Python simulations to study its topological properties, edge states, and the effects of chiral-symmetry breaking.
The primary objectives of this internship were to:
- Develop the SSH model from first principles
- Derive the bulk momentum-space Hamiltonian
- Obtain the dispersion relation
- Study chiral symmetry and its consequences
- Understand the winding number as a topological invariant
- Investigate bulk–boundary correspondence
- Numerically diagonalize finite SSH chains
- Study edge-state localization
- Examine the effect of a staggered on-site potential on the system's topology
ssh-model-study/
│
├── README.md
├── LICENSE
├── requirements.txt
├── .gitignore
│
├── code/
│ ├── ssh_model.py
│ ├── ssh_open_boundary.py
│ ├── ssh_onsite.py
│ └── winding_number.py
│
├── Reading/
│
└── Report/
├── Figures/
├── main.tex
├── preamble.tex
├── References.bib
└── SSH_Model_Internship_Report.pdf
The repository contains four Python programs, located in code/.
Studies the bulk SSH model with periodic boundary conditions.
Features:
- Energy dispersion relation
- Winding trajectories in the
$d_x$ –$d_y$ plane - Visualization of the topological phase transition
- Publication-quality figures using Matplotlib
Numerically diagonalizes finite SSH chains with open boundary conditions.
Features:
- Construction of finite Hamiltonians
- Exact numerical diagonalization
- Complete energy spectrum
- Edge-state probability distributions
- Comparison of trivial and topological phases
Extends the SSH model by introducing a staggered on-site potential.
Features:
- Modified Hamiltonian with on-site energies
- Evolution of edge states
- Energy spectrum
- Localization of positive- and negative-energy edge states
- Investigation of chiral-symmetry breaking
Visualizes the geometric interpretation of topology after adding an on-site potential.
Features:
- Three-dimensional
$d$ -vector trajectories - Projection onto the
$d_x$ –$d_y$ plane - Illustration of the loss of the planar winding number
- Tight-binding approximation
- Bloch's theorem
- Periodic and open boundary conditions
- Band structure
- Topological phase transition
- Chiral symmetry
- Winding number
- Bulk–boundary correspondence
- Edge states
- Symmetry breaking
- Numerical diagonalization
- Python 3.13+
- NumPy
- SciPy
- Matplotlib
- A LaTeX distribution (for compiling the report, with
latexmkrecommended)
Install the Python dependencies with:
pip install numpy scipy matplotlibNote: the plotting scripts set
text.usetex: Truefor publication-quality typesetting, which requires a working LaTeX installation on your system (e.g. TeX Live or MiKTeX).
The complete internship report is included as:
Report/main.pdf
Report/SSH_Model_Internship_Report.pdf
The report contains:
- Analytical derivations (real-space and momentum-space Hamiltonians, chiral symmetry, winding number, bulk–boundary correspondence)
- Numerical methods and Python implementations
- Simulation results (dispersion relations, winding trajectories, energy spectra, edge-state localization)
- Extension of the SSH model with a staggered on-site potential
- Full appendix reproducing all Python source code
To rebuild the report from source:
cd Report
latexmk -pdf main.tex- W. P. Su, J. R. Schrieffer, and A. J. Heeger, Solitons in Polyacetylene, Physical Review Letters 42, 1698 (1979)
- J. K. Asbóth, L. Oroszlány, and A. Pályi, A Short Course on Topological Insulators (Springer, 2015)
- M. Z. Hasan and C. L. Kane, Colloquium: Topological Insulators, Reviews of Modern Physics 82, 3045 (2010)
- X.-L. Qi and S.-C. Zhang, Topological Insulators and Superconductors, Reviews of Modern Physics 83, 1057 (2011)
- N. W. Ashcroft and N. D. Mermin, Solid State Physics (Holt, Rinehart and Winston, 1976)
- David J. Griffiths, Introduction to Quantum Mechanics, 3rd ed.
Full reading material is archived in Reading/.
I sincerely thank Dr. Panch Ram (Department of Physics, IIT (BHU), Varanasi) for his guidance, insightful discussions, and supervision throughout this internship.
Vijay Vedanth Vemula BS–MS Student Indian Institute of Science Education and Research (IISER) Tirupati Summer Research Internship 2026