Johan Félisaz


The semiclassical Monte-Carlo method

This is an overview of the semiclassical Monte-Carlo (SC-MC) method,
as used in the paper [[https://arxiv.org/abs/1403.7903]["Semi-classical spin dynamics of the
antiferromagnetic Heisenberg model on the kagome lattice"]]. It uses the
method on a Kagome lattice, but it can be applied to any lattice. It
relies on the Heisenberg model, that is the $$S \rightarrow \infty$$
limit, with quantum spins replaced by unit vectors in
$$\R^3$$.

Goal

This is used to compute the dynamical structure factor, also called scattering function:

\begin{equation*} S(\vec Q, t) = \langle s_{-\vec Q}(0) \cdot s_{\vec Q}(t) \rangle, \end{equation*} with \begin{equation*} s_{\vec Q}(t) = \sum_{i, \alpha} s_{i, \alpha}(t) e^{-i (\vec R_i + \vec r_\alpha) \cdot \vec Q} \end{equation*}

Method

The problem is twofold. First, it is a dynamical problem, that is it requires to solve the equations of motion of the system to see how each spin evolve with time. Second, it is a statistical problem, in the sense that the scattering function is defined as an average over the canonical ensemble (fixed temperature $$T$$). This is why it will be solve in a two steps process:

1. Sample from the canonical ensemble using MC methods
2. Evolve each sample in time using semi-classical methods

The Monte-Carlo step

The single spin-flip Metropolis Hastings algorithm is used. However, it can be quite inefficient at low temperature due to the number of rejected samples. To reduce this effect, the solid angle for each spin flip is reduced to keep the acceptance rate above 0.4. A pseudocode implementation is given below:

Let S[i] the initial system state.

Repeat N_steps times:
While not accepted:
Choose a random site i.
Choose a random 3D unit vector S'.
Let E the energy of the state S[i].
Let E' the energy of the state S[i], with S[i] replaced with S'.

If E' < E:
Accept.
Else:
Accept with probability exp((E - E') / kT)
End If
End While
End Repeat

/How is actually implemented the solid angle constrained practically ?/ /Having a running estimator of the acceptance rate, and updating the solid angle accordingly ?/

Choice of parameters

• Number of samples: 1000
• Stride between two samples: enough such that the *stochastic

correlation* is less than 0.1.

/how is actually defined the stochastic correlation ? $$Corr(\{\vec S_i\}, \{\vec S'_i\})$$ ?/

Overrelaxation

At low temperature, the system has trouble to get out of the local minimum of energy to explore other regions of the phase space. To mitigate this issue, an overrelaxation scheme is used. Recall that the hamiltonian can be written in terms of the local field $$h_\alpha(\vec R)$$ at each site $$(\vec R, \alpha)$$: \begin{equation*} H = \sum_{\vec R \vec R', \alpha, \beta} J_{\alpha, \beta}(\vec R' - \vec R) \vec S_\alpha(\vec R) \cdot \vec S_\beta(\vec R') = \sum_{\vec R, \alpha} \vec h_\alpha(\vec R) \cdot \vec S_\alpha(\vec R) \end{equation*} One can see that the energy does not change if a spin is rotated, while keeping the angle with its local field fixed. This is exploited by the overrelaxation method. Each time a spin is selected, it is first rotated around its local field, then assigned a random value. If it is rejected, the rotation is still kept.

/Is it correct ? Especially, is it applied to each selected spin, or only to failing ones ?/

Semiclasical spin dynamics

Using the Heisenberg picture equations of motion $$\pdiff{A}{t} = i/\hbar [H, A]$$, and the spin 1/2 commutation relation $$[\hv S^i, \hv S^j] = \varepsilon_{ijk} \hv S^k$$, one can obtain the (semiclassical) non-linear Bloch equations: \begin{equation*} \diff{s_i(t)}{t} = - s_i(t) \times \left( \sum_j J_{ij} s_j(t) \right) \end{equation*} This equation is simply numerically integrated. An 8th order explicit Runge-Kutta scheme is used.

Choice of parameters

The RK error parameter as well as the RK order have been fixed in order to preserve the Euclidean distance $$d = [\sum_i (\vec s_i^{RK} - s_j^{BS})^2]^{1/2}$$, i.e. the distance between time trajectories obtained with the RK method and with the more robust but time consuming Burlisch-Stoer (BS) algorithm.

/I could not find how the $$\frac{8(8-1)}2 = 28$$ RK8 params are defined, along with the timestep./

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