Magnetic fields#

In flatspin, external fields and temperature are modeled as a combination of magnetic fields. The total magnetic field affecting each spin is the sum of three types of fields:

  1. Dipolar fields from neighboring magnets (h_dip)

  2. An external field (h_ext)

  3. A stochastic thermal field (h_therm)

Magnetic field vectors can be viewed from two frames of reference:

  1. The global reference frame [h_x, h_y] is the most intuitive, where h_x is the horizontal component (from left to right) and h_y is the the vertical component (going upwards).

  2. In the local reference frame, the field is projected along the magnetization direction of each spin, resulting in a vector [h_par, h_perp] where h_par is the field component which is parallel to the magnetization, and h_perp is the perpendicular component. In other words, the local reference frame represents the magnetic field “felt” by each spin. When h_par is positive, the field is pushing in the same direction as the magnetization, and when h_par is negative, the field is pushing in the opposite direction of the magnetization.

In general, field attributes (such as h_ext) are always stored in the global reference frame, while the fields returned by field methods (such as external_fields()) are in the local reference frame.

(fields-dipolar)

Dipolar fields#

Spins are coupled through dipole-dipole interactions, and each spin is subject to a magnetic field from all neighboring spins. These dipolar fields can be calculated using dipolar_fields(), which returns an array of vectors [h_par, h_perp] where h_par is the parallel component of the dipolar field and h_perp is the perpendicular component (local reference frame).

Below the spins are colorized according their h_par values.

from flatspin.model import SquareSpinIceClosed

model = SquareSpinIceClosed()
model.spin[[0, 4, 39]] = -1 # flip spins 0, 4 and 39

h_dip = model.dipolar_fields()
quiv = model.plot(C=h_dip[:,0], cmap='coolwarm_r')
plt.colorbar(quiv);
_images/fields_3_0.svg

The strength of the dipolar fields scale with the parameter alpha (the coupling strength): a higher alpha results in stronger dipolar fields.

Because the dipole-dipole interaction quickly fades over long distances, flatspin only considers a finite neighborhood when computing dipolar fields. The radius of this neighborhood can be configured with the neighbor_distance model parameter. Setting neighbor_distance=np.inf results in a global neighborhood, but is computationally costly for large systems.

Below we illustrate how the neighborhood (light red) of the center spin (dark red) changes for different values of neighbor_distance. See also Neighbor distance for a discussion on the effects of neighbor distance.

plt.figure(figsize=(8,4))
for nd in [1, 2, 3]:
    plt.subplot(1,3,nd)
    plt.title(f'neighbor_distance={nd}')
    m = SquareSpinIceClosed(size=(6,6), neighbor_distance=nd)
    i = m.spin_count//2 - 1
    neighs = np.zeros(m.spin_count)
    neighs[i] = 1
    neighs[m.neighbors(i)] = 0.8
    m.plot(C=neighs, cmap='coolwarm', clim=(0,1))
_images/fields_5_0.svg

External fields#

The external magnetic field is the primary mechanism for altering the state of an ASI in a controlled manner. To set the external field, use set_h_ext(). The external field can be set either globally for the entire system, or locally on a per spin basis. The h_ext attribute stores the external fields for all the spins (global reference frame).

As with the dipolar fields, the parallel and perpendicular components of the external field acting on each spin can be calculated using external_fields().

Passing a single vector to set_h_ext() sets a global field:

from flatspin.plotting import plot_vectors

# Global external field
model.set_h_ext([0.1, 0.1])

plt.figure()
plt.title("h_ext")
plot_vectors(model.pos, model.h_ext, normalize=True)

plt.figure()
plt.title("h_ext_par")

h_ext = model.external_fields()

# Colorize spins by the parallel component of the external field
quiv = model.plot(C=h_ext[:,0], cmap='coolwarm_r')
plt.colorbar(quiv);
_images/fields_7_0.svg_images/fields_7_1.svg

Passing an array of vectors to set_h_ext() sets a local external field, i.e., an individual external field for each spin. The length of the array must match the number of spins in the system.

# A local field which increases in strength with spin index
H = np.linspace(0, 0.1, model.spin_count)
phi = np.deg2rad(10)
h_ext = np.column_stack([H * np.cos(phi), H * np.sin(phi)])
print(f'h_ext.shape: {h_ext.shape}')
model.set_h_ext(h_ext)

plt.figure()
plt.title("h_ext")
plot_vectors(model.pos, model.h_ext, normalize=True)

plt.figure()
plt.title("h_ext_par")

h_ext = model.external_fields()

# Colorize spins by the parallel component of the external field
quiv = model.plot(C=h_ext[:,0], cmap='coolwarm_r')
plt.colorbar(quiv);
h_ext.shape: (40, 2)
_images/fields_9_1.svg_images/fields_9_2.svg

It is also possible to map a spatial vector field defined on a grid, onto the spins with set_h_ext_grid():

# A spatial vector field defined on a grid
x = np.linspace(0, 2*np.pi, 9, endpoint=True)
y = np.linspace(0, 2*np.pi, 9, endpoint=True)
xx, yy = np.meshgrid(x, y)
H = np.cos(xx) + np.cos(yy)
h_ext = np.stack([0.01*H, 0.1*H], axis=-1)
print(f'h_ext.shape: {h_ext.shape}')

model.set_h_ext_grid(h_ext)

plt.figure()
plt.title("H")
plt.imshow(H, cmap='coolwarm_r')
plt.colorbar()

plt.figure()
plt.title("h_ext")
plot_vectors(model.pos, model.h_ext, normalize=True);
h_ext.shape: (9, 9, 2)
_images/fields_11_1.svg_images/fields_11_2.svg

Thermal fields#

The thermal field is a stochastic field which represents thermal fluctuations acting independently on each spin. In most cases, the thermal fields will be directed antiparallel to the spin magnetization, i.e., towards the easiest switching direction (see also Switching).

After the temperature is set with set_temperature(), the thermal field must be re-sampled using update_thermal_noise():

# Set temperature to 300 K, and re-sample h_therm
model.set_temperature(300)
model.update_thermal_noise()

plt.title("h_therm @ 300 K")
h_therm_magnitude = norm(model.h_therm, axis=-1)
# Colorize vectors by their magnitude
quiv = plot_vectors(model.pos, model.h_therm,
                    C=h_therm_magnitude, cmap='coolwarm', normalize=True)
plt.colorbar(quiv);
_images/fields_13_0.svg

Increasing the temperature increases the magnitude of h_therm (but does not change the direction):

model.set_temperature(600)
model.update_thermal_noise()

plt.title("h_therm @ 600 K")
h_therm_magnitude = norm(model.h_therm, axis=-1)
# Colorize vectors by their magnitude
quiv = plot_vectors(model.pos, model.h_therm,
                    C=h_therm_magnitude, cmap='coolwarm', normalize=True)
plt.colorbar(quiv);
_images/fields_15_0.svg

Total field#

Finally, the total magnetic fields are the sum of the dipolar, external and thermal fields. The total field for each spin can be computed directly using total_fields():

plt.figure()
plt.title("h_tot_par")

h_tot = model.total_fields()

# Colorize spins by the parallel component of the total field
quiv = model.plot(C=h_tot[:,0], cmap='coolwarm_r')
plt.colorbar(quiv);
_images/fields_17_0.svg

GPU acceleration#

flatspin provides GPU acceleration to speed up calculations of the magnetic fields. Enabling GPU acceleration with use_opencl=True can greatly speed up calculations, especially for large systems:

model_cpu = SquareSpinIceClosed(size=(100,100))
%timeit model_cpu.dipolar_fields()
651 ms ± 15 ms per loop (mean ± std. dev. of 7 runs, 1 loop each)
model_gpu = SquareSpinIceClosed(size=(100,100), use_opencl=True)
%timeit model_gpu.dipolar_fields()
246 µs ± 25.4 µs per loop (mean ± std. dev. of 7 runs, 1 loop each)