To generate data for the microcanonical average in ETH (Eigenstate Thermalization Hypothesis) for a non-integrable Hamiltonian, you can follow these general steps:
1. **Diagonalize the Hamiltonian**: Numerically diagonalize the Hamiltonian of your non-integrable system to obtain its eigenstates and eigenvalues.
2. **Calculate the diagonal elements of the local operator**: Evaluate the diagonal elements of the local operator in the eigenbasis of the Hamiltonian. This will give you the values of the local operator for each eigenstate.
3. **Perform the microcanonical average**: Compute the microcanonical average of the diagonal elements of the local operator. The microcanonical average is defined as:
```
A_mc = (1/n) * sum(A_ii)
```
where `A_ii` are the diagonal elements of the local operator, and `n` is the number of eigenstates within the energy window of interest.
4. **Vary the energy window**: Repeat the microcanonical average calculation for different energy windows to obtain the energy dependence of the microcanonical average.
Here's a Python code snippet that demonstrates the general approach:
```python
import numpy as np
from scipy.linalg import eigh
# 1. Diagonalize the Hamiltonian
H = ... # your non-integrable Hamiltonian
eigenvalues, eigenstates = eigh(H)
# 2. Calculate the diagonal elements of the local operator
A = ... # your local operator
A_diag = [np.abs(np.dot(eigenstates[:, i], np.dot(A, eigenstates[:, i]))) for i in range(len(eigenvalues))]
# 3. Perform the microcanonical average
def microcanonical_average(E_min, E_max):
"""Compute the microcanonical average of the diagonal elements of the local operator."""
indices = np.where((eigenvalues >= E_min) & (eigenvalues
Simulating a Non-integrable Hamiltonian:
Analyzing Diagonal Matrix Elements:
Additional Considerations:
- System Size: ETH predictions become more accurate with increasing system size. Consider increasing the system size in your simulation for a stronger ETH signature.
- Ergodicity: Ergodic systems explore all accessible microstates over time. Ensure your chosen model exhibits ergodic behavior for reliable microcanonical sampling.
- Finite-size effects: For finite systems, there will be deviations from ETH predictions. You might need to analyze the data statistically to account for these effects.
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