Long-Range Transverse Field Ising Model - Moving Front Quenches

Tensor network simulations of the transverse field Ising model with long-range algebraic interactions using ITensor. This project investigates spatiotemporal quenches and quantum dynamics in systems with power-law decaying interactions J(r) ~ 1/r^α, relevant to experimental quantum simulators based on Rydberg atoms and trapped ions.

🔬 Overview

This repository contains the numerical implementations used to generate all figures in our Physical Review B publication on spatiotemporal quenches in long-range Hamiltonians. The work extends previous studies on short-range models to realistic experimental systems where interactions decay algebraically with distance.

Physical Model

The long-range transverse field Ising Hamiltonian:

H = -Σᵢⱼ J(rᵢⱼ) σˣᵢ σˣⱼ - h Σᵢ σᶻᵢ

where J(r) ~ 1/r^α is the long-range coupling strength, h is the transverse field, and α controls the range of interactions.

Key Physics

Long-range interactions fundamentally change:

• Critical behavior and universality classes
• Dynamical critical exponents (z ≠ 1 for α < 3)
• Speed of information propagation
• Efficiency of spatiotemporal quenches for ground state preparation

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📂 Project Structure

Core Modules

1. gapLR - Energy Gap Calculations ⭐

Calculates the energy gap between ground and first excited states in the long-range TFI model.

Key Applications:

• Determining critical points via gap scaling
• Extracting critical exponents
• Validating exponential sum approximation of algebraic interactions
• Testing the accuracy of the sum-of-exponentials representation

Technical Note: The gap calculations verify that representing 1/r^α as a sum of exponentials (using optimized parameters from input_alpha_... files) accurately reproduces the physics of true algebraic interactions.

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2. vCrit - Speed of Excitations

Determines the effective speed of light in the long-range quantum system.

Method:

• Introduces local perturbation to ground state
• Evolves using 4th order TDVP
• Tracks von Neumann entropy spreading
• Extracts propagation velocity from entanglement light cone

Physical Significance: In long-range models, the effective speed of excitations depends on α and can differ from the nearest-neighbor case, affecting the optimal quench protocol.

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3. superluminal - Moving Front Quenches (Main Focus) 🌟

Simulates inhomogeneous quenches with moving quench fronts in the long-range TFI model.

Protocol:

• Quench front propagates at constant velocity v
• Initial state: Ground state in gapped phase
• Final state: Critical ground state
• Evolution: 4th order Time-Dependent Variational Principle (TDVP)

Observables:

• Local and global energy density evolution
• Spin correlation functions
• Von Neumann entanglement entropy dynamics

Key Results:

• For α ≳ 3 (where z = 1), optimal cooling occurs when v ≈ c
• For α < 3, the long-range nature modifies optimal quench protocols
• Spatiotemporal quenches remain efficient for ground state preparation

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4. critical - Critical Ground State Properties

Calculates equilibrium properties at the critical point, including:

• Ground state correlations
• Critical exponents
• Scaling behavior

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5. staticFront - Static Front Analysis

Investigates ground state properties with a static (non-moving) quench front.

Purpose: Provides baseline comparison for moving front dynamics.

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🛠️ Technical Implementation

Framework

ITensor C++ Library - Tensor network calculations optimized for 1D systems

Key Innovation: Sum-of-Exponentials Representation

Challenge: True algebraic interactions J(r) ~ 1/r^α are computationally expensive in tensor network methods.

Solution: Approximate power-law decay as sum of exponentials:

J(r) ~ 1/r^α  ≈  Σᵢ Aᵢ exp(-r/ξᵢ)

Implementation:

• Parameters {Aᵢ, ξᵢ} obtained via nonlinear optimization
• Stored in input_alpha_... files for different α values
• TFIsingLR.h header file implements the long-range Hamiltonian using these optimized parameters
• Validation: gapLR module confirms approximation accuracy

Why This Matters:

• Enables efficient Matrix Product State (MPS) representation
• Preserves physical properties (critical points, exponents, dynamics)
• Makes long-range simulations tractable while maintaining accuracy

Algorithms

• 4th order Time-Dependent Variational Principle (TDVP)
• Matrix Product State (MPS) evolution
• Finite-size scaling
• Entanglement entropy calculations

Language: C++

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📊 Physical Insights

This work demonstrates:

• Spatiotemporal quenches work efficiently even with long-range interactions
• Optimal quench velocity depends on the dynamical critical exponent z
• For α ≳ 3: Lorentz-like behavior (z = 1), optimal v ≈ c
• For α < 3: Modified dynamics, non-Lorentzian behavior
• Quantum simulators (Rydberg atoms, trapped ions) naturally realize these long-range models

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🔬 Experimental Relevance

This work directly applies to:

• Rydberg atom arrays: Van der Waals interactions (α = 6) or dipolar (α = 3)
• Trapped ion systems: Coulomb interactions with tunable range
• Polar molecules: Electric dipole-dipole interactions
• Quantum simulation of many-body dynamics

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📄 Publication

This code was used to generate all figures in:

Spatiotemporal Quenches in Long-Range Hamiltonians
Simon Bernier and Kartiek Agarwal
Physical Review B 108, 024310 (2023)
arXiv:2212.07499

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🎓 Academic Context

This project extends our previous work on 2D nearest-neighbor models to experimentally relevant systems with long-range interactions, bridging:

• Tensor network methods
• Quantum simulation platforms
• Non-equilibrium quantum dynamics
• Critical phenomena beyond nearest neighbors

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🔗 Dependencies

• ITensor library (C++)
• C++11 compatible compiler
• LAPACK/BLAS libraries
• Optimized input_alpha_... parameter files (included in repository)

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💡 Key Files

TFIsingLR.h

Core header file implementing the long-range transverse field Ising Hamiltonian using the sum-of-exponentials approximation.

Usage: This file must be properly configured with the appropriate input_alpha_... parameter file for your chosen interaction range α.

input_alpha_... Files

Pre-computed optimization results for different values of α, containing:

• Exponential amplitudes {Aᵢ}
• Decay lengths {ξᵢ}
• Obtained via nonlinear least-squares fitting to 1/r^α

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📈 Computational Strategy

1. Choose interaction range α (determines physics regime)
2. Load optimized parameters from corresponding input_alpha_... file
3. Configure TFIsingLR.h with these parameters
4. Run simulations (equilibrium or time evolution)
5. Validate approximation using gapLR module if needed

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🔗 Related Work

This repository complements our 2D work:

• ising2d - Short-range 2D TFI model
• Both projects study spatiotemporal quenches but in different geometries and interaction ranges

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📚 Physics Background

Dynamical Critical Exponent (z):

• Controls how time and space scale at criticality: τ ~ ξ^z
• Short-range models: z = 1 (Lorentz invariance)
• Long-range models: z depends on α
  • α > 3: z = 1 (short-range-like)
  • α < 3: z < 1 (true long-range behavior)

Kibble-Zurek Mechanism: Predicts defect scaling during quenches. Spatiotemporal quenches can outperform uniform quenches by respecting causal structure.

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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 this code or build upon this work, please cite:

@article{bernier2023spatiotemporal,
  title={Spatiotemporal Quenches in Long-Range Hamiltonians},
  author={Bernier, Simon and Agarwal, Kartiek},
  journal={Physical Review B},
  volume={108},
  pages={024310},
  year={2023},
  publisher={American Physical Society},
  doi={10.1103/PhysRevB.108.024310}
}

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🎯 Future Directions

• Extension to 2D long-range models
• Floquet engineering with periodic driving
• Machine learning for parameter optimization
• Connections to near-term quantum devices

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This project demonstrates expertise in: computational quantum many-body physics, tensor networks, long-range interactions, nonlinear optimization, C++ programming, and quantum simulation relevant to experimental platforms.