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We investigate the quantum phases of hard-core bosonic atoms in an extendedHubbard model where particles interact via soft-shoulder potentials in onedimension. Using a combination of field-theoretical methods and strong-couplingperturbation theory, we demonstrate that the low-energy phase can be aconformal cluster Luttinger liquid (CLL) phase with central charge $c=1$, wherethe microscopic degrees of freedom correspond to mesoscopic ensembles ofparticles. Using numerical density-matrix-renormalization-group methods, wedemonstrate that the CLL phase, first predicted in [Phys. Rev. Lett. 111,165302 (2013)], is separated from a conventional Tomonaga-Luttinger liquid byan exotic critical point with central charge $c=3/2$. The latter is expressionof an emergent conformal supersymmetry, which is not present in the originalHamiltonian. We discuss the observability of the CLL phase in realisticexperimental settings with weakly-dressed Rydberg atoms confined to opticallattices. Using quantum Monte-Carlo simulations, we show that the typicalfeatures of CLLs are stable up to comparatively high temperatures. Using exactdiagonalizations and quantum trajectory methods, we provide a protocol foradiabatic state preparation as well as quantitative estimates on the effects ofparticle losses.

Cluster Luttinger liquids and emergent supersymmetric conformal critical points in the one-dimensional soft-shoulder Hubbard model