Exercise 7.1

Classification of magnetic phases using fully-connected networks

Imagine a 2-dimensional lattice arrangement of $n \times n$ magnetic dipole moments (spins) that can be in one of two states ($+1$ or $−1$, Ising model). With interactions between spins being short ranged, each spin interacts only with its four neighbors. The probability to find a spin in one of the orientations is a function of temperature $T$ according to $p \sim e^{−a/T},\;a = \mathrm{const.}$).

At extremely low temperatures $T \rightarrow 0$, neighboring spins have a very low probability of different orientations, so that a uniform overall state (ferromagnetic state) is adopted, characterized by $+1$ or $−1$. At very high temperatures $T \rightarrow \infty$, a paramagnetic phase with random spin alignment results, yielding $50\%$ of $+1$ and $0%$ of $−1$ orientations. Below a critical temperature $0 < T < T_c$, stable ferromagnetic domains emerge, with both orientations being equally probable in the absence of an external magnetic field. The spin-spin correlation function diverges at $T_c$, whereas the correlation decays for $T > T_c$.

The data for this task contain the $n \times n$ dipole orientations on the lattice for different temperatures $T$. Classify the two magnetic phases (paramagnetic/ferromagnetic)!

In [6]:
from tensorflow import keras
import numpy as np
callbacks = keras.callbacks
layers = keras.layers

print("keras", keras.__version__)
keras 2.4.0

Load and prepare dataset

See https://doi.org/10.1038/nphys4035 for more information

In [2]:
import gdown
url = "https://drive.google.com/u/0/uc?export=download&confirm=HgGH&id=1Ihxt1hb3Kyv0IrjHlsYb9x9QY7l7n2Sl"
output = 'ising_data.npz'
gdown.download(url, output, quiet=True)

f = np.load(output, allow_pickle=True)
n_train = 20000

x_train, x_test = f["C"][:n_train], f["C"][n_train:]
T_train, T_test = f["T"][:n_train], f["T"][n_train:]
In [3]:
import matplotlib.pyplot as plt

for i,j in enumerate(np.random.choice(n_train, 6)):
    image = x_train[j]
    plot = plt.imshow(image)
    plt.title("T: %.2f" % T_train[j])

In [4]:
Text(0, 0.5, 'frequency')

Set up training data - define magnetic phases

In [5]:
Tc = 2.27
y_train, y_test = T_train > Tc, T_test > Tc


  • evaluate the test accuracy for a fully-connected network,
  • plot the test accuracy vs. temperature.