Ising Model

Beginner Explanation

Imagine a group of tiny magnets that can either point up or down. The Ising Model helps us understand how these magnets interact with each other. When most magnets point in the same direction, they create a strong magnetic field, like how a group of friends can make a loud cheer together. But if they point in different directions, the cheer gets quiet. The Ising Model shows how temperature affects this cheer: when it’s hot, the magnets get restless and point in different directions, but when it’s cold, they settle down and align together.

Technical Explanation

The Ising Model is a mathematical framework used to study phase transitions, particularly in ferromagnetic systems. It consists of discrete variables (spins) that can take values of +1 or -1, representing the two possible orientations of a magnetic dipole. The energy of a configuration is given by the Hamiltonian: H = -J Σ σ_i σ_j – h Σ_i σ_i, where J is the interaction strength, h is the external magnetic field, and σ_i indicates the spin at site i. The model can be simulated using Monte Carlo methods, such as the Metropolis algorithm. Here is an example of a simple 1D Ising model simulation in Python: “`python import numpy as np import matplotlib.pyplot as plt # Parameters L = 100 # Length of the lattice J = 1.0 # Interaction strength T = 2.0 # Temperature # Initialize spins spins = np.random.choice([-1, 1], size=L) # Monte Carlo step for _ in range(10000): i = np.random.randint(0, L) delta_E = 2 * J * spins[i] * (spins[(i-1) % L] + spins[(i+1) % L]) if delta_E < 0 or np.random.rand() < np.exp(-delta_E / T): spins[i] *= -1 # Plotting the results plt.plot(spins) plt.title('1D Ising Model') plt.show() ```

Academic Context

The Ising Model, introduced by Ernst Ising in 1925, is a cornerstone of statistical mechanics and condensed matter physics. It serves as a simplified representation of ferromagnetism and has been extensively studied for its critical phenomena and phase transitions. The model is mathematically grounded in the theory of Markov chains and can be analyzed using tools from statistical physics, such as the partition function and mean-field theory. Key papers include the original work by Ising (1925) and subsequent analyses by Lars Onsager (1944), who solved the 2D Ising model exactly. The model has applications beyond magnetism, influencing fields such as neural networks and social dynamics.

Code Examples

Example 1:

import numpy as np
import matplotlib.pyplot as plt

# Parameters
L = 100  # Length of the lattice
J = 1.0  # Interaction strength
T = 2.0  # Temperature

# Initialize spins
spins = np.random.choice([-1, 1], size=L)

# Monte Carlo step
for _ in range(10000):
    i = np.random.randint(0, L)
    delta_E = 2 * J * spins[i] * (spins[(i-1) % L] + spins[(i+1) % L])
    if delta_E < 0 or np.random.rand() < np.exp(-delta_E / T):
        spins[i] *= -1

# Plotting the results
plt.plot(spins)
plt.title('1D Ising Model')
plt.show()

Example 2:

i = np.random.randint(0, L)
    delta_E = 2 * J * spins[i] * (spins[(i-1) % L] + spins[(i+1) % L])
    if delta_E < 0 or np.random.rand() < np.exp(-delta_E / T):
        spins[i] *= -1

Example 3:

import numpy as np
import matplotlib.pyplot as plt

# Parameters
L = 100  # Length of the lattice

Example 4:

import matplotlib.pyplot as plt

# Parameters
L = 100  # Length of the lattice
J = 1.0  # Interaction strength

View Source: https://arxiv.org/abs/2511.15377v1

Ising Model

Beginner Explanation

Technical Explanation

Academic Context

Code Examples

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Pre-trained Models

nqs-models/ising_fnqs

nqs-models/ising_disorder_fnqs

Relevant Datasets

SecureFinAI-Lab/Ising_Model_Instances

External References

Beginner Explanation

Technical Explanation

Academic Context

Code Examples

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Like this:

Pre-trained Models

nqs-models/ising_fnqs

nqs-models/ising_disorder_fnqs

Relevant Datasets

SecureFinAI-Lab/Ising_Model_Instances

External References

Related Concepts