Physics and Equations¶

The spindynam package implements the coupled spin-lattice dynamics formalism described in Tranchida et al., J. Comput. Phys. 372 (2018) 406-425.

Hamiltonian¶

The total Hamiltonian of the coupled spin-lattice system is given by Equation 4:

(1)¶\[\mathcal{H}_{sl} = \mathcal{H}_{mag} + \sum_{i=1}^N \frac{|\mathbf{p}_i|^2}{2m_i} + \sum_{i<j} V(r_{ij})\]

Where: - \(\mathbf{p}_i\) is the momentum of atom \(i\). - \(m_i\) is the mass of atom \(i\). - \(V(r_{ij})\) is the interatomic potential (mechanical part).

The magnetic Hamiltonian \(\mathcal{H}_{mag}\) includes Zeeman and Exchange interactions (Equation 3):

(2)¶\[\mathcal{H}_{mag} = - \mu_B \sum_{i=1}^N g_i \mathbf{s}_i \cdot \mathbf{H}_{ext} - \sum_{i<j} J(r_{ij}) \mathbf{s}_i \cdot \mathbf{s}_j\]

Where: - \(\mathbf{s}_i\) is the atomic spin (unit vector). - \(J(r_{ij})\) is the distance-dependent exchange interaction. - \(\mathbf{H}_{ext}\) is the external magnetic field. - \(\mu_B\) is the Bohr magneton and \(g_i\) is the Landé g-factor.

Functional Forms¶

The SpinLatticeHeisenberg calculator utilizes the following functional forms for the interactions:

Exchange Interaction \(J(r_{ij})\)¶

The exchange coupling follows an exponential decay with distance:

(3)¶\[J(r_{ij}) = J_0 \exp\left[-\alpha \left(\frac{r_{ij}}{r_0} - 1.0\right)\right]\]

Where: - \(J_0\) is the exchange constant at the equilibrium distance \(r_0\). - \(\alpha\) is the dimensionless decay parameter.

Atomic Potential \(V(r_{ij})\)¶

The interatomic mechanical potential is modeled as a harmonic well:

(4)¶\[V(r_{ij}) = \frac{1}{2} K (r_{ij} - r_0)^2\]

Where: - \(K\) is the spring constant. - \(r_0\) is the equilibrium distance.

Equations of Motion¶

The time evolution of any observable \(A\) in the coupled spin-lattice phase space is governed by the Poisson bracket (Equation 5):

(5)¶\[\{F, G\} = \sum_{i=1}^N \left( \frac{\partial F}{\partial \mathbf{r}_i} \cdot \frac{\partial G}{\partial \mathbf{p}_i} - \frac{\partial F}{\partial \mathbf{p}_i} \cdot \frac{\partial G}{\partial \mathbf{r}_i} \right) + \sum_{i=1}^N \frac{\partial F}{\partial \mathbf{s}_i} \cdot \left( \frac{\mathbf{s}_i}{\hbar} \times \frac{\partial G}{\partial \mathbf{s}_i} \right)\]

The resulting equations of motion for the positions, momenta, and spins are given by (Equation 12):

(6)¶\[\begin{split}\dot{\mathbf{r}}_i &= \{\mathbf{r}_i, \mathcal{H}_{sl}\} = \frac{\mathbf{p}_i}{m_i} \\ \dot{\mathbf{p}}_i &= \{\mathbf{p}_i, \mathcal{H}_{sl}\} = \mathbf{F}_i \\ \dot{\mathbf{s}}_i &= \{\mathbf{s}_i, \mathcal{H}_{sl}\} = \boldsymbol{\omega}_i \times \mathbf{s}_i\end{split}\]

Forces and Effective Fields¶

The total force \(\mathbf{F}_i\) acting on atom \(i\) consists of mechanical and magnetic contributions (Equation 7):

(7)¶\[\mathbf{F}_i = - \frac{\partial \mathcal{H}_{sl}}{\partial \mathbf{r}_i} = \sum_{j \neq i} \left[ -\frac{dV(r_{ij})}{dr_{ij}} + \frac{dJ(r_{ij})}{dr_{ij}} (\mathbf{s}_i \cdot \mathbf{s}_j) \right] \mathbf{e}_{ij}\]

Where \(\mathbf{e}_{ij} = \frac{\mathbf{r}_i - \mathbf{r}_j}{r_{ij}}\) is the unit vector pointing from atom \(j\) to atom \(i\).

The magnetic effective field \(\boldsymbol{\omega}_i\) (or torque) acting on spin \(i\) is defined as (Equation 8):

(8)¶\[\boldsymbol{\omega}_i = \frac{1}{\hbar} \left( g_i \mu_B \mathbf{H}_{ext} + \sum_{j \neq i} J(r_{ij}) \mathbf{s}_j \right)\]

Symplectic Integrator¶

To preserve the geometric structure of the phase space (including the spin norm and total energy for NVE), we use a second-order symmetric Suzuki-Trotter decomposition (Equation 13):

(9)¶\[e^{\mathcal{L} \Delta t} \approx e^{\mathcal{L}_p \frac{\Delta t}{2}} e^{\mathcal{L}_s \frac{\Delta t}{2}} e^{\mathcal{L}_r \Delta t} e^{\mathcal{L}_s \frac{\Delta t}{2}} e^{\mathcal{L}_p \frac{\Delta t}{2}}\]

Where: - \(\mathcal{L}_p\) updates momenta using forces. - \(\mathcal{L}_r\) updates positions using momenta. - \(\mathcal{L}_s\) is the magnetic Liouvillian, updated via a sequential rotation of each spin.

Spin Rotation (Equation 15)¶

Individual spin updates are performed using a norm-preserving rational propagator:

(10)¶\[\mathbf{s}_i(t+\Delta t) = \mathbf{s}_i(t) + \frac{\Delta t (\boldsymbol{\omega}_i \times \mathbf{s}_i) + \frac{\Delta t^2}{2} \boldsymbol{\omega}_i \times (\boldsymbol{\omega}_i \times \mathbf{s}_i)}{1 + \frac{\Delta t^2}{4} |\boldsymbol{\omega}_i|^2}\]

This formulation avoids the need for ad-hoc normalization and strictly preserves the spin magnitude throughout the simulation.

References¶

Tranchida et al., “A symplectic algorithm for coupled atomistic spin-lattice dynamics”, Journal of Computational Physics, 372 (2018) 406-425. DOI: 10.1016/j.jcp.2018.06.042