2D Magnetism Lab: From Quantum DFT to Thermodynamics

An interactive, research-grade computational materials physics laboratory and simulator exploring two-dimensional magnetism in monolayer Chromium Triiodide ($\text{CrI}_3$) and related van der Waals magnets.

Based on the pedagogical framework of Prof. Yen Lee Loh ("Determination of Magnetic Transitions: Monte Carlo Simulations", TÜBİTAK Workshop 2021).

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🌟 Live Interactive Demo

Launch the simulator directly in your browser with zero installation:
👉 Launch 2D Magnetism Lab

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🔬 Scientific Premise & Core Features

1. The Physics Question: Why Does 2D Magnetism Exist?

In 1966, the Mermin-Wagner Theorem proved that continuous $SO(3)$ spin rotational symmetry forbids spontaneous magnetic order in 2D at any finite temperature ($T > 0$) due to divergent long-wavelength Goldstone spin waves (magnons, $\omega \propto k^2$): $$n_{\text{magnon}} \sim \int \frac{k,dk}{e^{D k^2 / k_B T} - 1} \approx \frac{k_B T}{D} \int \frac{dk}{k} \longrightarrow \infty \implies T_c \equiv 0\text{ K}$$

Yet in 2017, physicists discovered that monolayer $\text{CrI}_3$ is ferromagnetic up to $T_c \approx 45\text{ K}$ (Huang et al., Nature). This lab demonstrates how relativistic spin-orbit coupling (SOC) from heavy Iodine ligands ($Z=53$) induces an out-of-plane Magnetic Anisotropy Energy (MAE) barrier ($K \approx 31.4\text{ K}$), opening a spectral gap $\Delta \sim 2K$ in the spin-wave dispersion that cuts off the infrared divergence and rescues finite-temperature ferromagnetism!

2. The 5-Attempt Progression

• Attempt 1: Honeycomb Ising ($J_1$ only) $\to T_c \approx 22\text{ K}$ (minimal cell misses 2nd-neighbor superexchange).
• Attempt 2: Supercell Ising ($J_1, J_2$) $\to T_c \approx 140\text{ K}$ ($J_2 \approx 30.9\text{ K}$ dominates, but 1D Ising overshoots).
• Attempt 3: Classical Isotropic Heisenberg $\to T_c = 0\text{ K}$ (Mermin-Wagner Theorem destroys 2D continuous order).
• Attempt 4: Anisotropic Heisenberg (MAE / SOC) $\to T_c \approx 50\text{ K}$ (easy-axis gap restores order, matching experiment!).
• Attempt 5: Quantum Corrections ($S=3/2$) $\to T_c \approx 45\text{ K}$ (zero-point fluctuations trim $T_c$ to exact match).

3. Interactive Features & Tools

• 3D Crystal Viewport with Orbit Controls: Click & drag to rotate (pitch & yaw), scroll to zoom, right-drag to pan. Presets for 3D Perspective, Top ($c$-axis), Side ($a/b$-profile), and Auto-Rotate.
• Real-Time Metropolis Monte Carlo (60 FPS): High-performance simulation supporting both Ising ($\pm 1$) and 3D Heisenberg continuous vectors ($\vec{S} \in S^2$) using the exact Rodrigues rotation formula for strict detailed balance.
• Materials Project REST API & 2D Library: Connect your free Materials Project API key to fetch live DFT data, or select from pre-loaded 2D magnets: $\text{CrI}_3$, $\text{CrBr}_3$, $\text{CrCl}_3$, $\text{Fe}_3\text{GeTe}_2$, and $\text{Cr}_2\text{Ge}_2\text{Te}_6$.
• High-DPI Scientific Charts: Real-time order parameter $|M(t)|$ and phase diagram $M(T)$ with live pulsing state cursor and theoretical MFT / Onsager curves.
• DFT Linear Regression Lab: Interactive matrix showing feature vectors $X_b$, DFT energies, and live least-squares solver.
• MAE & Anisotropy Lab: Interactive polar angle $\theta$ slider visualizing $E_{\text{MAE}}(\theta) = E_0 + K \sin^2\theta$.
• Binder Cumulant Crossing: Finite-size scaling showing $B(T, L) = \langle M^4 \rangle / \langle M^2 \rangle^2$ intersection at $T_c$.

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🚀 How to Run Locally

Because the application is 100% self-contained in a single file (index.html), simply clone the repository and open index.html in any web browser:

git clone https://github.com/Asif-Iqbal-Bhatti/2d-magnetism-lab.git
cd 2d-magnetism-lab
# Open index.html in your browser:
# On Windows / WSL:
explorer.exe index.html
# On macOS:
open index.html
# On Linux:
xdg-open index.html

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📚 Scientific References & Citations

1. Yen Lee Loh, "Determination of Magnetic Transitions: Monte Carlo Simulations," Magnetic Properties from First Principles, TÜBİTAK Workshop, Nov 24-26 (2021).
2. B. Huang, G. Clark, E. Navarro-Moratalla, et al., "Layer-dependent ferromagnetism in a van der Waals crystal of $\text{CrI}_3$," Nature 546, 270–273 (2017). DOI: 10.1038/nature22391.
3. N. D. Mermin and H. Wagner, "Absence of Ferromagnetism or Antiferromagnetism in One- or Two-Dimensional Isotropic Heisenberg Models," Phys. Rev. Lett. 17, 1133 (1966). DOI: 10.1103/PhysRevLett.17.1133.
4. C. Domb, "On the theory of cooperative phenomena in crystals," Adv. Phys. 9, 149 (1960). DOI: 10.1080/00018736000101199.
5. A. Jain et al., "Commentary: The Materials Project: A materials genome approach to accelerate materials innovation," APL Materials 1, 011002 (2013). Materials Project ID: mp-1213805.

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📄 License

This project is open-source under the MIT License.