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Finite-Volume Energy Spectrum, Fractionalized Strings, and Low-Energy Effective Field Theory for the Quantum Dimer Model on the Square Lattice

We present detailed analytic calculations of finite-volume energy spectra, mean field theory, as well as a systematic low-energy effective field theory for the square lattice quantum dimer model. The analytic considerations explain why a string connecting two external static charges in the confining columnar phase fractionalizes into eight distinct strands with electric flux $\frac{1}{4}$. An emergent approximate spontaneously broken $SO(2)$ symmetry gives rise to a pseudo-Goldstone boson. Remarkably, this soft phonon-like excitation, which is massless at the Rokhsar-Kivelson (RK) point, exists far beyond this point. The Goldstone physics is captured by a systematic low-energy effective field theory. We determine its low-energy parameters by matching the analytic effective field theory with exact diagonalization results and Monte Carlo data. This confirms that the model exists in the columnar (and not in a plaquette or mixed) phase all the way to the RK point.