Ab initio Complex Langevin computation of the roton gap for a dipolar Bose condensate

We compute from first principles the dispersion relation $\omega (k)$ of a dipolar Bose gas of erbium atoms close to the roton instability by employing the Complex Langevin (CL) algorithm. Other than the path integral Monte Carlo algorithm, which samples the quantum mechanical path integral in the $N$-particle basis, CL samples the field-theoretic path integral of interacting bosons and can be evaluated for experimentally realistic atom numbers. We extract the energy of roton excitations as a function of the s-wave scattering length, and compare our results to those from Gross-Pitaevskii theory, with and without quantum fluctuation corrections.