We report intertwined Weyl phases, which come from superposing topological phases by crystalline symmetry. In the intertwined Weyl phases, an unconventional Weyl phase where Weyl points possess a higher charge (monopole charge>1) due to rotation symmetry, and a higher-order topological phase enforced by rotation symmetry, are superposed. The two phases are no longer separable, but intertwine with each other, resulting in the novel phase. Remarkably, the intertwining leads to a prominent characteristic feature of the intertwined Weyl phases: $textit{the change of Fermi-arc topology}$ in a periodic pattern, i.e., the way how Fermi arcs connect to Weyl points changes drastically with respect to surface orientation, which exhibits a periodic pattern. Such a phenomenon is absent in any individual phase alone. Moreover, we elaborate on how to emulate the intertwined double-Weyl phase in cold atoms. Our theory is quite promising for generating new topological phases based on existing ones.
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