Superluminal and Hyperbolic Quenches in Quantum Spin Models

Master's thesis research investigating spatiotemporal quenches in the 1D transverse field Ising (TFI) and Heisenberg XXX models. This foundational work established numerical methods and explored quench protocols that directly led to our subsequent publications on long-range and 2D systems.

🔬 Overview

This repository contains all simulation code used in my Master's thesis at McGill University. The work explores whether superluminal (constant velocity) and hyperbolic (accelerating) quench fronts can efficiently prepare critical ground states in 1D quantum spin models.

Thesis: Superluminal and Hyperbolic Quenches in the Transverse Field Ising and XXX Heisenberg Model
Simon Bernier, McGill University (2022)

Research Questions

1. Can superluminal quenches prepare critical ground states in the 1D TFI model?
2. Do hyperbolic quenches exhibit the predicted energy scaling?
3. Can we improve upon previous numerical methods for superlumincal quenches?

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🎯 Key Findings and Impact

What This Work Achieved

✅ Validated and improved methods for Heisenberg XXX model

• Confirmed previous theoretical predictions with higher accuracy
• Extended tensor network techniques to this challenging interacting model
• Established reliable benchmarks for future work

✅ First systematic study of TFI spatiotemporal quenches

• Extended quench protocols to free-fermion critical systems
• Demonstrated feasibility of moving front simulations in TFI
• Critical foundation that enabled our PRB papers

✅ Identified key limitations and next directions

• Discovered that simple conformal field theory scaling miss important physics
• Found that finite-size effects dominate hyperbolic quench scaling
• Recognized need for correlation function and entanglement analysis

What We Learned (Candidly)

❌ Superluminal quenches in 1D TFI were uninsightful

• Missed the crucial scaling behavior in energy and correlations
• These insights directly motivated our long-range study where we found the key signatures Doppler cooling

❌ Hyperbolic quenches didn't show expected scaling

• Quench front smoothing effects dominated
• Finite system size limitations
• Gap between conformal field theory predictions and numerical reality

Why This Matters

This thesis is a perfect example of valuable exploratory research. The "negative" results and methodological challenges directly led to:

1. PRB 108, 024310 (2023) - Long-range paper where we found the scaling we missed here
2. PRB 111, 054311 (2025) - 2D paper building on methods developed here
3. Established numerical framework used in both subsequent publications

Research insight: Sometimes the most valuable work is figuring out what doesn't work and why, then using those lessons to succeed.

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📂 Project Structure

The repository is organized into two main folders:

TFI (Transverse Field Ising Model)

Physical Model:

H = -J Σᵢ σˣᵢ σˣᵢ₊₁ - Σᵢ hᵢ σᶻᵢ

At criticality (h = J), the model maps to free fermions with linear dispersion - a key simplification that makes it an ideal test case.

Modules:

critical

• Calculates equilibrium critical ground state properties
• Reference state for quench protocols
• Correlation functions and entanglement

superlum / movingFront

• Inhomogeneous quenches with quench front at constant velocity v
• Tests whether v ≈ c (speed of excitations) is optimal
• Measures energy, correlations

hyperbolic

• Quench front follows hyperbolic trajectory in spacetime
• Designed to test conformal field theory scaling predictions
• Explores variable-velocity protocols

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Heisenberg (XXX Model)

Physical Model:

H = J Σᵢ (σˣᵢ σˣᵢ₊₁ + σʸᵢ σʸᵢ₊₁ + σᶻᵢ σᶻᵢ₊₁) + Σᵢ hᵢ σᶻᵢ

An interacting integrable model - much more challenging than TFI but still tractable with tensor networks.

Modules:

critical

• Critical ground state preparation
• Spin-1/2 XXX chain at criticality
• Benchmark calculations

superlum / movingFront

• Moving quench fronts in interacting model
• Key achievement: Verified previous theoretical predictions
• Improved numerical accuracy over prior work

hyperbolic

• Variable-velocity quenches
• Tests of conformal field theory prediction in interacting systems

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🛠️ Technical Implementation

Framework

ITensor (C++) - Matrix Product State (MPS) tensor network library

Key Algorithms

• 4th order Time-Dependent Variational Principle (TDVP) for time evolution
• MPS ground state optimization
• Entanglement entropy calculations
• Finite-size scaling analysis

Numerical Challenges Addressed

1. Time evolution stability - TDVP maintains MPS structure
2. Entanglement growth - Adaptive bond dimension
3. Quench front implementation - Spatially varying Hamiltonian
4. Long-time dynamics - Careful truncation strategies

Language: C++

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📊 Main Observables

Throughout both models, we track:

• Energy density - Local and total energy evolution
• Spin correlations - ⟨σᶻᵢ σᶻⱼ⟩ to detect order
• Von Neumann entropy - Entanglement spreading and structure
• Defect density - Residual excitations after quench

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🎓 Academic Context

Master's Thesis (2022)
McGill University, Department of Physics
Supervisor: Prof. Kartiek Agarwal

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🔗 How This Led to Published Work

Methodological Foundation

• Established TDVP framework for moving quenches
• Developed observables and analysis pipeline
• Built intuition about finite-size effects

Critical Insights

What we missed in the thesis:

• Need to look at energy scaling, not just absolute values
• Correlation function behavior reveals the critical scaling of the final state
• Importance of implementing the models in realistic quantum simulators α-dependent dynamics (long-range paper)

What we learned:

• Free-fermion models (TFI) are simpler starting points
• Need multi-scale analysis (1D → long-range → 2D progression)
• Finite-size effects can mask key physics

Direct Impact on Publications

Long-Range Paper (2023):

• Used identical TDVP methods
• Added the crucial scaling analysis we missed
• Discovered α-dependent optimal velocity
• Found clear scaling behaviour in correlations

2D Paper (2025):

• Extended quench protocols to 2D
• Built on TFI understanding from thesis
• Showed efficiency in higher dimensions

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🔗 Related Publications

This thesis work directly enabled:

1. Spatiotemporal Quenches in Long-Range Hamiltonians
   Simon Bernier and Kartiek Agarwal
   Physical Review B 108, 024310 (2023)

2. 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)

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🔗 Dependencies

• ITensor C++ library
• C++11 compatible compiler
• LAPACK/BLAS libraries
• HDF5 (for data storage)

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📧 Contact

Simon Bernier

• Email: simon.bernier@mail.mcgill.ca
• LinkedIn: simon-bernier-6701a9285

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📝 Citation

If you use methods or build upon this work, please cite the thesis:

@mastersthesis{bernier2022superluminal,
  title={Superluminal and Hyperbolic Quenches in the Transverse Field Ising and XXX Heisenberg Model},
  author={Bernier, Simon},
  year={2022},
  school={McGill University},
  url={https://escholarship.mcgill.ca/concern/theses/br86b846h}
}

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🎯 Bottom Line

This thesis is valuable not despite its challenges, but because of them. It shows:

• Rigorous research methodology
• Ability to learn from unexpected results
• Path from exploration to breakthrough
• Foundation for two PRB publications

Research is iterative. This was step one. The PRB papers were steps two and three.

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This project demonstrates: exploratory computational physics, tensor network methods, critical self-evaluation, turning challenges into insights, and building a systematic research program from foundational work.