Numerical study of spontaneous SUSY breaking

Overview

This repository contains a comp. physics project which demonstrates the spontaneous breaking of Supersymmetry (SUSY) in Quantum Mechanics. This was done by applying the Finite Difference Method (FDM), where we solved the stationary Schrödingers eq. using a modified asymmetric Pöschl-teller potential (Rosen-Morse II potential) and analizing the energy spectra of the partner hamiltonians.

Theoretical background

In unbroken SUSY QM, the ground state of enegery of the first hamiltonian is zero $E_0^{(1)}$, whereas the spectra of the partner Hamiltonian $H_1$ and $H_2$ are completely degenerate. To induce the spontaneous breaking we introduce an assymetric parameter $\lambda$ to the superpotential: $$W(x) = A \tanh(\alpha x) + \lambda$$

The theoretical constrains dictate that when $\lambda$ crosses the threshold $\lambda \ge A$ the groundstate is no longer normalizable, which forces the $E_0^{(1)} > 0$, breaking the suppersimetry.

Methodology

• Numerical Method: Matrix diagonalization via FDM using a discrete spatial grid.
• Tools: Python, numpy, scipy.sparse, and matplotlib.
• Simulation Parameters: The spatial domain is truncated using Dirichlet boundary conditions at $x=\pm30$ with a grid resolution of $N = 3000$ points to minimize the finite-size effects and truncation errors.

Key results

1. Critial Treshold: The numerical rupture occurs at the theoretical threshold $\lambda = A = 2.0$. The slight divergence prior to this value is caused by the finite-size effect, causde by the wavefunction's tail colliding with the Dirichlet walls of the grid.
2. Asymptotic Degeneracy: Under a strong perturbation $\lambda \gg A$, the potencials $V_{1,2}$ become spatially identical within the confined region, leading to mathematical degenerate states of energy despite the broken symmetry.

Validation

The simulation was validated by setting the asymmetry parameter to zero ($\lambda = 0$), this way using the standard symmetric Pöschl-Teller potential. The numerical ground state energy obtained was $E_0 \approx -1.0000e-04$, consistent with the theoretical value of exactly zero, confirming the accuracy of the finite difference discretization.

How to run the simulation

To reproduce the numerical results and generate the bifurcation and potential geometry plots do:

git clone [https://github.com/Raisuky/spontaneous-susyqm-breaking-fdm.git](https://github.com/Raisuky/spontaneous-susy-breaking-fdm.git)
cd spontaneous-susyqm-breaking-fdm
python susy_solver.py