| Summary: | We develop a theory of the excitonic phase recently proposed as the
zero-field insulating state observed near charge neutrality in monolayer
WTe$_2$. Using a Hartree-Fock approximation, we numerically identify two
distinct gapped excitonic phases: a spin density wave state for weak but
non-zero interaction strength $U_0$, and spin spiral order at larger $U_0$,
separated by a narrow window of trivial insulator. We introduce a simplified
model capturing essential features of the WTe$_2$ band structure, in which the
two phases may be viewed as distinct valley ferromagnetic orders. We link the
competition between the two phases to the orbital structure of the electronic
wavefunctions at the Fermi surface and hence its proximity to the underlying
gapped Dirac point in WTe$_2$. We briefly discuss collective modes of the two
excitonic states, and comment on implications for experiments.
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