Spin System

Project Goal

The goal of this project is to provide a highly modular, TensorFlow-based framework for simulating the dynamics of classical spin systems. By leveraging TensorFlow, the project aims to utilize vectorization and potential GPU acceleration to efficiently simulate large lattices and complex interactions. The framework is designed to be easily extensible, allowing researchers to define new models, interactions, and dynamic rules with minimal effort.

Current State

The project currently supports the following features:

• Models: unit-vector spin models including Ising (discrete) and Spherical (continuous) systems.
• Interactions: Arbitrary pairwise coupling tensors, with built-in generators for Periodic Nearest-Neighbor, Decaying, Curie-Weiss, and Gaussian random interactions.
• Dynamics: Metropolis-Hastings Monte Carlo dynamics.
• Measurements: Real-time tracking of scalar observables (Energy, Magnetization) and susceptibility.

Main Differences

Compared to other available options for Spin Dynamics:

• TensorFlow Backend: Utilizes modern tensor operations for performance, significantly speeding up updates on large lattices compared to pure Python loops.
• Modular Architecture: Decouples the physical model (spins), the interaction topology (couplings), and the time evolution (dynamics), making it easier to mix and match components.
• Flexible Tracking: Includes a specialized Tracker class that can conceptually handle any observable that can be computed from the system state, recording history efficiently using TensorArrays.

Walkthrough: src/spin_engine

The core logic resides in src/spin_engine, organized into four main modules. Here is how they interact:

1. models

Purpose: Defines the physical system, including the lattice structure, spin dimensionality (discrete vs continuous), and the energy function.

• Interactions: Holds spin_state (the actual configuration) and interaction_matrix (the couplings).
• Key Class: BaseSpinSystem provides the interface. IsingSystem and SphericalSystem are concrete implementations.

2. interactions

Purpose: Generates the coupling tensors ($J_{ij}$) that define the connectivity of the system.

• Interactions: Used by models to initialize the system's Hamiltonian.
• Key Classes: PeriodicNearestNeighborInteraction (standard d-dimensional lattice), DecayingInteraction, CurieWeissInteraction.

3. dynamics

Purpose: Defines the rules for time evolution.

• Interactions: Takes a system and modifies its spin_state over time.
• Key Classes:
  • MetropolisHastings: Performs Monte Carlo sweeps. It proposes changes (flips or rotations), computes energy differences ($\Delta E$), and accepts/rejects based on thermal probabilities.
  • Tracker: Observes the simulation. It is passed to the sweep method to record measurements at specified intervals.

4. measurements

Purpose: Defines observables to be tracked.

• Interactions: Instantiated with a system reference. Called by Tracker during the simulation to compute values like total energy or magnetization.
• Key Classes: Energy, Magnetization, MagneticSusceptibility.

Interaction Flow

1. Define Interaction: Generate an interaction matrix (e.g., Nearest Neighbor).
2. Create Model: Initialize a System (e.g., Ising) with the lattice size and interaction matrix.
3. Setup Logic: Instantiate MetropolisHastings dynamics with the system.
4. Prepare Tracking: Create a Tracker with desired Measurements.
5. Run: Execute dynamics.sweep(), passing the Tracker.

Examples

The examples/ directory contains scripts demonstrating the framework capabilities.

Ising Model Evolution

The script examples/ising.py performs a temperature sweep on a 2D Ising model. It illustrates phase transitions by tracking magnetization across different inverse temperatures (Beta).

Magnetization Evolution

The following plot shows the Monte Carlo time evolution of magnetization for various temperatures. Note the rapid convergence to zero magnetization at high temperatures (low Beta) and stable non-zero magnetization at low temperatures (high Beta).

Magnetic Susceptibility

The project successfully reproduces the critical behavior of the 2D Ising model. The plot below shows the magnetic susceptibility peaking around the critical point ($\beta_c \approx 0.44$), indicating the phase transition.

Contact

Lucas Gomes de Oliveira Corbanez

• Email: lucascorbanez@gmail.com
• Institutional: lucas.gomes.oliveira@uel.br