This project implements a from-scratch Matrix Product State (MPS) simulation of the 1D Transverse-Field Ising Model (TFIM), one of the most fundamental quantum many-body systems in condensed matter physics.
The ground state energy is computed using variational MPS optimization and validated against exact diagonalization, demonstrating the power and accuracy of tensor network methods for simulating quantum systems classically.
The Transverse-Field Ising Model Hamiltonian is:
H = -J Σᵢ Zᵢ Zᵢ₊₁ - h Σᵢ Xᵢ
where:
Jis the nearest-neighbor coupling strengthhis the transverse magnetic fieldZᵢ,Xᵢare Pauli operators on sitei
This model exhibits a quantum phase transition at h/J = 1.0 between a ferromagnetic phase (h < J) and a paramagnetic phase (h > J).
- From-scratch MPS implementation using only NumPy (no external tensor network libraries)
- Exact diagonalization for validation on small system sizes
- Ground state energy computation via variational optimization
- Quantum phase transition visualization as a function of
h/J - Entanglement entropy calculation from the MPS bond structure
- Publication-quality plots saved automatically
pip install -r requirements.txtpython main.pyThis will:
- Compute the exact ground state energy using exact diagonalization (up to N=12 spins)
- Compute the MPS-approximated ground state energy
- Plot the ground state energy and magnetization across the phase transition
- Save the plot as
results/phase_transition.png
The script generates:
- Console output comparing exact vs MPS energies
- A plot showing the quantum phase transition
This project demonstrates core competencies in:
- Tensor Network methods (MPS representation, SVD truncation)
- Quantum many-body physics (Ising model, phase transitions)
- Numerical simulation of quantum systems using classical methods
MIT License