Variational Monte Carlo (VMC) Method Discussion

1. Introduction to Antiferromagnetic Model

The antiferromagnetic (AFM) model describes a system of spins where adjacent spins are aligned oppositely. This leads to a ground state with zero net magnetization, and it can be studied using various techniques including the Heisenberg model.

2. Hamiltonian Matrix for Heisenberg Model with 2 Spins

The matrix representation for the Hamiltonian with two spins can be constructed as follows:

3. Hamiltonian Matrix for 3 Spins in 1D

The matrix representation for the Hamiltonian with three spins:

4. Hamiltonian Matrix for 4 Spins in 1D with PBC

The matrix representation for the Hamiltonian with four spins with periodic boundary conditions:

5. Variational Approach to Ground State

To calculate the variational energy of a Gutzwiller wave function for a chain of 16 spins, we can implement a Variational Monte Carlo (VMC) approach.

6. Python Code for VMC Calculation

import numpy as np

# Define constants
L = 16  # Number of spins
J = 1.0  # Coupling constant

def hamiltonian_element(state, i, j):
    """Calculate the Hamiltonian matrix element for spins i and j."""
    if abs(i - j) == 1 or (i == 0 and j == L - 1) or (i == L - 1 and j == 0):
        return -J * 0.5  # Nearest neighbor interaction
    return 0.0

def gutzwiller_wave_function(state, g):
    """Calculate the Gutzwiller wave function."""
    return np.exp(-g * np.sum(state))  # Example functional form

def variational_energy(g):
    """Calculate the variational energy given the Gutzwiller parameter g."""
    energy = 0.0
    num_states = 2**L
    
    for i in range(num_states):
        state = np.array([int(x) for x in f"{i:0{L}b}"])  # Binary representation of state
        wf = gutzwiller_wave_function(state, g)

        # Calculate the energy contribution for this state
        state_energy = 0.0
        for a in range(L):
            for b in range(L):
                state_energy += hamiltonian_element(state, a, b) * wf

        energy += np.real(np.conj(wf) * state_energy)

    return energy / num_states  # Normalize by the number of states

# Main optimization loop
g_start = 0.5  # Initial guess for Gutzwiller parameter
learning_rate = 0.01
num_iterations = 100
g = g_start

for iteration in range(num_iterations):
    energy = variational_energy(g)
    print(f"Iteration {iteration}: g = {g:.3f}, Variational Energy = {energy:.3f}")

    # Update g using a simple gradient descent (placeholder)
    g -= learning_rate * (energy - np.mean(energy))  # A simplistic update rule

print(f"Optimized Gutzwiller parameter: {g:.3f}")
    

7. Efficient Memory Usage in VMC

To handle larger systems, we can compute matrix elements on-the-fly instead of storing the entire Hamiltonian matrix.

import numpy as np

# Define constants
L = 16  # Number of spins
J = 1.0  # Coupling constant

def hamiltonian_element(state, i, j):
    """Calculate the Hamiltonian matrix element for spins i and j."""
    if abs(i - j) == 1 or (i == 0 and j == L - 1) or (i == L - 1 and j == 0):
        return -J * 0.5  # Nearest neighbor interaction
    return 0.0

def gutzwiller_wave_function(state, g):
    """Calculate the Gutzwiller wave function."""
    return np.exp(-g * np.sum(state))  # Example functional form

def variational_energy(g):
    """Calculate the variational energy given the Gutzwiller parameter g."""
    energy = 0.0
    num_states = 2**L
    
    for i in range(num_states):
        state = np.array([int(x) for x in f"{i:0{L}b}"])  # Binary representation of state
        wf = gutzwiller_wave_function(state, g)

        # Calculate the energy contribution for this state
        state_energy = 0.0
        for a in range(L):
            for b in range(L):
                state_energy += hamiltonian_element(state, a, b) * wf

        energy += np.real(np.conj(wf) * state_energy)

    return energy / num_states  # Normalize by the number of states

# Main optimization loop
g_start = 0.5  # Initial guess for Gutzwiller parameter
learning_rate = 0.01
num_iterations = 100
g = g_start

for iteration in range(num_iterations):
    energy = variational_energy(g)
    print(f"Iteration {iteration}: g = {g:.3f}, Variational Energy = {energy:.3f}")

    # Update g using a simple gradient descent (placeholder)
    g -= learning_rate * (energy - np.mean(energy))  # A simplistic update rule

print(f"Optimized Gutzwiller parameter: {g:.3f}")