Two interacting spins¶

Combine electronic spins $S_0=1$ and $S_1=1/2$:

$$ H=J\,\mathbf{S}_0\cdot\mathbf{S}_1+D S_{0z}^{2}-g\mu_B B_z(S_{0z}+S_{1z}). $$

Use $J=0.1$ meV, $D=0.02$ meV and $B_z=0.5$ T. Positive Hamiltonian exchange is antiferromagnetic; positive $D$ is easy-plane anisotropy. The external field acts on both spins. Dipolar and transverse-anisotropy terms are omitted.

In [1]:

import numpy as np
import plotly.io as pio

from spinfinity import SpinSys, hamil, plotting

pio.renderers.default = "plotly_mimetype+notebook"
# Use "fixed" for compact plots, or "no" to suppress display.
plotting.set_display_mode("responsive")

import numpy as np import plotly.io as pio from spinfinity import SpinSys, hamil, plotting pio.renderers.default = "plotly_mimetype+notebook" # Use "fixed" for compact plots, or "no" to suppress display. plotting.set_display_mode("responsive")

In [2]:

spin = SpinSys([1.0, 0.5])
spin.B = [0.0, 0.0, 0.5]  # T
spin.BTip = [0.0, 0.0, 0.0]  # T
spin.gfactorvector = [2.0, 2.0]
spin.Dvector[0] = 0.02  # meV

hamil.assign_ExchangeInteraction(spin, [(0, 1, [0.1, 0.1, 0.1])])

spin = SpinSys([1.0, 0.5]) spin.B = [0.0, 0.0, 0.5] # T spin.BTip = [0.0, 0.0, 0.0] # T spin.gfactorvector = [2.0, 2.0] spin.Dvector[0] = 0.02 # meV hamil.assign_ExchangeInteraction(spin, [(0, 1, [0.1, 0.1, 0.1])])

Solve the coupled system¶

The Hilbert-space dimension is $(2S_0+1)(2S_1+1)=6$. The exchange helper assigns [Jxx, Jyy, Jzz] in meV; equal components give isotropic exchange.

These coefficients are stored in Jvector, not the separate dimensionless tunneling parameter J. Rebuild after changing an interaction.

In [3]:

hamil.calc_EigStates(spin, rebuildH=True)

print("Hilbert-space dimension:", spin.dimensionOfMatrix)
print("Relative energies (meV):", np.round(spin.E_All, 6))
print("Product basis:", spin.basisStates)

for state in spin.states:
    print(state)

hamil.calc_EigStates(spin, rebuildH=True) print("Hilbert-space dimension:", spin.dimensionOfMatrix) print("Relative energies (meV):", np.round(spin.E_All, 6)) print("Product basis:", spin.basisStates) for state in spin.states: print(state)

Hilbert-space dimension: 6
Relative energies (meV): [0.       0.057884 0.0994   0.144568 0.202452 0.273052]
Product basis: ['|1.0,0.5>' '|1.0,-0.5>' '|-0.0,0.5>' '|-0.0,-0.5>' '|-1.0,0.5>'
 '|-1.0,-0.5>']
|0> (E = 0.0) = -0.777+0.000j|1.0,-0.5>+0.629+0.000j|-0.0,0.5>
|1> (E = 0.0579) = -0.777+0.000j|-1.0,0.5>+0.629+0.000j|-0.0,-0.5>
|2> (E = 0.0994) = +1.000+0.000j|1.0,0.5>
|3> (E = 0.1446) = -0.777+0.000j|-0.0,0.5>-0.629+0.000j|1.0,-0.5>
|4> (E = 0.2025) = -0.777+0.000j|-0.0,-0.5>-0.629+0.000j|-1.0,0.5>
|5> (E = 0.2731) = +1.000+0.000j|-1.0,-0.5>

Energy levels and eigenstate composition¶

The first plot places energies against $\langle S_{0z}+S_{1z}\rangle$. A product state in the second plot is written $|m_0,m_1\rangle$. Exchange mixes states with the same total $m_z$.

Each probability-map row sums to one; it is not a population distribution over energy eigenstates. Both figures are resized before their first display.

In [4]:

hamil.plot_EnergyVsS(spin, type="z", energyscale="meV");
hamil.plot_EigenMatrix(spin);

hamil.plot_EnergyVsS(spin, type="z", energyscale="meV"); hamil.plot_EigenMatrix(spin);