Spin System Simulation

A flexible and extensible framework for simulating classical spin systems on arbitrary lattices using the Metropolis-Hastings Monte Carlo algorithm. The code supports a variety of models and interaction structures, enabling the study of statistical mechanics phenomena in both standard and exotic settings.

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Hamiltonians and Supported Models

General Hamiltonian

The energy of a spin configuration is defined by a general quadratic form: $$ \mathcal{H} = \sum_{pq} J_{pq} \sigma_p \sigma_q $$ where $\sigma_p$ are spin variables and $J_{pq}$ is the user-supplied interaction matrix. This formulation allows for arbitrary lattice geometries and interaction patterns.

Available Models

• Ising Model
  Spins $\sigma_i = \pm 1$ on a lattice, with Hamiltonian: $$ \mathcal{H}{\text{Ising}} = -\sum{ij} J_{ij} \sigma_i \sigma_j - \sum_i h_i \sigma_i $$ Supports arbitrary interaction matrices and external fields.

• Spherical Model
  Continuous spins constrained by $\sum_i \sigma_i^2 = N$, with Hamiltonian: $$ \mathcal{H}{\text{Spherical}} = -\sum{ij} J_{ij} \sigma_i \sigma_j - \sum_i h_i \sigma_i $$ The spherical constraint can be enforced optionally.

• $\mathbb{Z}_2$ Gauge Model
  Defined on 2D lattices with link variables and plaquette interactions. The Hamiltonian is constructed from products of link variables around plaquettes, supporting arbitrary coupling matrices. $$ \mathcal{H}{\text{Z2 Gauge}} = -\sum{p} \prod_{\ell \in p} \sigma_\ell $$

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Monte Carlo Method

The simulation uses the Metropolis-Hastings algorithm to sample spin configurations according to the Boltzmann distribution at inverse temperature $\beta$. The update scheme is model-dependent:

• Ising Model: Random single-spin flips.
• Spherical Model: Random pairwise spin rotations, optionally enforcing the spherical constraint.
• $\mathbb{Z}_2$ Gauge Model: Random flips of gauge links.

Energy changes are computed efficiently using local field updates and explicit matrix operations.

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Features

• Arbitrary lattice dimension and size.
• Multiple replicas for parallel simulation.
• User-defined interaction matrices and external fields.
• Observable tracking: energy, magnetization, overlap matrix, plaquette values, Wilson loops (for gauge models).
• Multi-temperature sweeps for studying phase transitions.
• Parallel programming and GPU acceleration via TensorFlow for high-performance simulations.
• TensorFlow-based for efficient computation and automatic differentiation.

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Project Status

This project is a work in progress.
Features, APIs, and documentation may change. Contributions and feedback are welcome.

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Maintainer

• Lucas Gomes de Oliveira Corbanez

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Usage

1. Define the interaction matrix $J_{pq}$ and (optionally) the external field.
2. Initialize a SpinSystem with desired parameters (lattice size, model type, etc.).
3. Run simulations using metropolis_sweep or multi_temperature_sweep.
4. Analyze observables such as energy, magnetization, and overlaps.

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References

• Sandvik, A. W. (2010). Computational Studies of Quantum Spin Systems.
• Altieri, A. (2024). Introduction to the Theory of Spin Glasses.
• Bienzobaz, P. (2012). Quantização Canônica e Integração Funcional.