Tensor network simulations of the two-dimensional transverse field Ising (2D-TFI) model using ITensor. This project explores equilibrium properties, quantum quenches, and entanglement dynamics in strongly correlated quantum systems.
This repository contains numerical implementations for studying quantum phase transitions and out-of-equilibrium dynamics in the 2D transverse field Ising model. The simulations use tensor network methods, specifically the Time-Dependent Variational Principle (TDVP), to efficiently simulate quantum many-body systems.
The 2D transverse field Ising Hamiltonian:
H = -J Σ σˣᵢ σˣⱼ - Σ hᵢ σᶻᵢ
where J is the coupling strength, hᵢ is the local transverse field, and the sums run over nearest neighbors and all sites respectively.
Calculates equilibrium properties of the 2D-TFI model, including:
- Energy gap between ground state and first excited state (critical point identification via gap scaling)
- Ground state energy
- Phase transition characterization
Key Features:
- Finite-size scaling analysis
- Critical exponent extraction
- Quantum phase transition mapping
Determines the effective speed of light in the quantum system using entanglement dynamics.
Method:
- Introduces local perturbation to ground state wavefunction
- Evolves disturbed state using 4th order TDVP
- Tracks von Neumann entropy growth
- Extracts propagation velocity from entanglement spreading
Physical Insight: The speed at which entanglement propagates reveals fundamental information about the model's causal structure and emergent light cone.
Simulates uniform (homogeneous) quenches in the 2D-TFI model.
Protocol:
- Initial state: Ground state in gapped phase
- Quench: Instantaneous change to critical point
- Evolution: 4th order TDVP time evolution
Observables:
- Energy density evolution
- Spin correlations
- Von Neumann entanglement entropy dynamics
Purpose: Provides baseline for comparing with inhomogeneous quench dynamics.
The main focus of this project - simulates spatially inhomogeneous quenches with moving quench fronts.
Key Features:
- Quench front moves at constant velocity through the system
- Tracks instantaneous ground state during evolution
- 4th order TDVP for accurate time evolution
- Detailed tracking of non-equilibrium dynamics
Observables:
- Energy density distribution
- Spin correlation functions
- Von Neumann entanglement entropy
- Spatial structure of excitations
Physical Motivation: Studies how quantum information and excitations propagate when parameters change locally and spread through the system - relevant for understanding causality and out-of-equilibirum dynamics in quantum systems.
- criticality: Correlation functions at the critical point
- fidelity: Quantum fidelity calculations for critical point detection
- ising-tebd: Test implementation of 2D Time-Evolving Block Decimation (TEBD) with optimized swap gate ordering
- superluminal: Streamlined version of movingFront without instantaneous ground state tracking (faster but less detailed)
Framework: ITensor - C++ library for tensor network calculations
Key Algorithms:
- 4th order Time-Dependent Variational Principle (TDVP)
- Matrix Product State (MPS) representation
- Finite-size scaling analysis
- Entanglement entropy calculations
Language: C++
This work contributes to understanding:
- Quantum phase transitions in 2D systems
- Non-equilibrium dynamics
- Entanglement propagation and quantum information spreading
- Kibble-Zurek mechanism in quantum quenches
- Emergent causality in quantum many-body systems
This project was developed as part of graduate research in computational quantum many-body physics, focusing on:
- Tensor network methods for 2D systems
- Out-of-equilibrium quantum dynamics
- Quantum criticality and universality
- Numerical methods for strongly correlated systems
This work contributed to the following publication:
Spatiotemporal quenches for efficient critical ground state preparation in the two-dimensional transverse field Ising model
Simon Bernier and Kartiek Agarwal
Physical Review B 111, 054311 (2025)
arXiv:2404.02957
- Quantum Phase Transitions: The 2D-TFI model exhibits a quantum phase transition between ordered and disordered phases
- Entanglement Entropy: Quantifies quantum correlations and serves as a diagnostic for phase transitions
- TDVP: Variational method for time evolution that preserves the tensor network structure
- Kibble-Zurek Dynamics: Universal scaling behavior when crossing phase transitions
- ITensor library
- C++ compiler with C++11 support
- LAPACK/BLAS libraries (typically handled by ITensor)
Each module can be compiled independently. Typical workflow:
- Set physical parameters (system size, coupling strengths, quench protocol)
- Initialize ground state or desired initial state
- Run simulation (equilibrium calculation or time evolution)
- Output observables for analysis
Potential extensions:
- Higher-dimensional generalizations
- Different lattice geometries
- Additional observables (mutual information, correlation length)
- Floquet engineering with periodic driving
- Connections to machine learning and tensor network quantum states
Simon Bernier
- Email: simon.bernier@mail.mcgill.ca
- LinkedIn: simon-bernier-6701a9285
If you use this code or find this work useful, please cite:
@article{bernier2025spatiotemporal,
title={Spatiotemporal quenches for efficient critical ground state preparation in the two-dimensional transverse field Ising model},
author={Bernier, Simon and Agarwal, Kartiek},
journal={Physical Review B},
volume={111},
pages={054311},
year={2025},
publisher={American Physical Society},
doi={10.1103/PhysRevB.111.054311}
}For questions or collaboration ideas, feel free to reach out!
This project demonstrates expertise in: computational physics, tensor networks, C++ programming, quantum many-body systems, and numerical methods for strongly correlated systems.