How to understand frustration phase factor?

For a system with basis $$|m\rangle$$, where $$m=1,2,3...$$ The frustration phase factor is defined in any path $$C=\{m_1m_2...m_\alpha\}$$ as: $$\Phi_C=\arg[(-1)^\alpha H_{m_1m_2} H_{m_2m_3}\dotsb H_{m_\alpha m_1}]$$ The product of elements of Hamiltonian can be understood directly from the view of path integral, but what's the origin of $$(-1)^\alpha$$ ? And why the physical phenomenon will be so different if $$\Phi_C$$ is non-zero?

In the other words, I am confused that why the sign of elements of Hamiltonian is so important?