Swap operation with the Heisenberg Hamiltonian

Question

According to REF 1 equation 3, a SWAP operation can be achieved via the Heisenberg Hamiltonian for spins $H=J(t)\mathbf{S}_1\mathbf{S_2}$

$U^{1/2}_{SWAP}=e^{-i\frac{\pi}{8}}\exp\left(i\frac{\pi}{2}\mathbf{S}_1\mathbf{S}_2\right)$

where $\int_0^td\tau J(\tau)=\pi/2$

I wrote out the expression of $U^{1/2}_{SWAP}$ and it indeed results into the matrix $U^{1/2}_{SWAP} =\left(\begin{array}{cccc}{1} & {0} & {0} & {0} \\ {0} & {\frac{1}{2}(1+i)} & {\frac{1}{2}(1-i)} & {0} \\ {0} & {\frac{1}{2}(1-i)} & {\frac{1}{2}(1+i)} & {0} \\ {0} & {0} & {0} & {1}\end{array}\right)$

The physical interpretation of the term $\exp\left(i\frac{\pi}{2}\mathbf{S}_1\mathbf{S}_2\right)$ is by coupling two electron spins in quantum dots via a tunnel coupling $t_0(t)$, $J=4t_0(t)^2/U$, where $U$ is the coulomb interaction between the electrons, for a time such that the SWAP operation is achieved and then turning off the coupling.

But how do I interpret the prefactor $e^{-i\pi/8}$ in physical terms?

REF 1: https://arxiv.org/abs/cond-mat/9905230

Answer 1

The prefactor is there to get the global phase of the SWAP right, likely for aesthetic reasons. It bears no physical relevance. You can interpret it as shifting the overall energy of the system, but this again has no physical relevance, as you can define energy offsets as will.