qojulia / collectivespins.jl Goto Github PK

Dear CollectiveSpins developers,

I've noticed that in the function Omega_k_chain the following expression is calculated:
$$\Re(\text{Li}_3(e^{i(k_0+k)a}) + \text{Li}_3(e^{i(k_0-k)a}) - ik_0a[\text{Li}_2(e^{i(k_0+k)a}) + \text{Li}_2(e^{i(k_0-k)a})])$$
This expression uses (time-costly) complex polylogarithms. However, the result is purely real. I believe this expression can be expressed in terms of simpler real Clausen functions, due to the following relations:
$$\Re[\text{Li}_3(e^{i\theta})] = \text{Cl}_3(\theta),\qquad\Im[\text{Li}_2(e^{i\theta})] = \text{Cl}_2(\theta)$$
Using these relations I believe the first expression can be written as:
$$\text{Cl}_3[(k_0+k)a] + \text{Cl}_3[(k_0-k)a] + k_0a(\text{Cl}_2[(k_0+k)a] + \text{Cl}_2[(k_0-k)a])$$
which can be significantly faster when a fast implementation of the Clausen functions is used.

In Julia one could use the package ClausenFunctions.jl which provide very fast implementations of $\text{Cl}_2$ and $\text{Cl}_3$:

using ClausenFunctions
cl2(1.0)
cl3(1.0)

Best regards
Alexander Voigt

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