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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?

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