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We carry out an analytical study of quantum spin ice, a U$(1)$ quantum spin liquid close to the classical spin ice solution for an effective spin $1/2$ model with anisotropic exchange couplings $J_{zz}$, $J_{pm}$ and $J_{zpm}$ on the pyrochlore lattice. Starting from the quantum rotor model introduced by Savary and Balents in Phys. Rev. Lett. 108, 037202 (2012), we retain the dynamics of both the spinons and gauge field sectors in our treatment. The spinons are described by a bosonic representation of quantum XY rotors while the dynamics of the gauge field is captured by a phenomenological Hamiltonian. By calculating the one-loop spinon self-energy, which is proportional to $J_{zpm}^2$, we determine the stability region of the U$(1)$ quantum spin liquid phase in the $J_{pm}/J_{zz}$ vs $J_{zpm}/J_{zz}$ zero temperature phase diagram. From these results, we estimate the location of the boundaries between the spin liquid phase and classical long-range ordered phases.
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