Transport and local spectroscopy measurements have revealed that monolayers of tungsten ditelluride ($1T^{\prime }{\mathrm{WTe}}_2$) display a quantum spin Hall effect and an excitonic gap at neutrality, besides becoming superconducting at low electron concentrations. With the aim of studying the competition between different broken-symmetry phases upon electron doping, we perform extensive Hartree-Fock calculations as a function of electron density and Coulomb interaction strength. At charge neutrality, we reproduce the emergence of a spin density wave and a spin spiral state surrounding a quantum spin Hall insulator at intermediate interaction strengths. For stronger interactions, the spin spiral is disrupted by a state breaking both inversion and time-reversal symmetries (but not their product) before the system becomes a trivial band insulator. With electron doping, the quantum spin Hall insulator evolves into an easy-plane ferromagnet due to a Stoner-like instability of the conduction band. This phase competes energetically with the spin spiral state. We discuss how our results may help to interpret past and future measurements.

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• Berry curvature
• Charge density waves
• Fermi surface
• Phase transitions
• Quantum anomalous Hall effect
• Spin-orbit coupling
• Topological insulators
• Valley degrees of freedom
• Transition metal dichalcogenides
• Hartree-Fock methods