Many-body localization (MBL) provides a mechanism to avoid thermalization in many-body quantum systems. Here, we show that an emergent symmetry can protect a state from MBL. Specifically, we propose a Z2 symmetric model with nonlocal interactions, which has an analytically known, SU(2) invariant, critical ground state. At large disorder strength all states at finite energy density are in a glassy MBL phase, while the lowest energy states are not. These do, however, localize when a perturbation destroys the emergent SU( 2) symmetry. The model also provides an example of MBL in the presence of nonlocal, disordered interactions that are more structured than a power law. The presented ideas raise the possibility of an 'inverted quantum scar', in which a state that does not exhibit area law entanglement is embedded in an MBL spectrum, which does.

Conformal field theory has recently been applied to derive few-body Hamiltonians whose ground states are lattice versions of fractional quantum Hall states. The exact lattice models involve interactions over long distances, which is difficult to realize in experiments. It seems, however, that such long-range interactions should not be necessary, as the correlations decay exponentially in the bulk. This poses the question, whether the Hamiltonians can be truncated to contain only local interactions without changing the physics of the ground state. Previous studies have in a couple of cases with particularly much symmetry obtained such local Hamiltonians by keeping only a few local terms and numerically optimizing the coefficients. Here, we investigate a different strategy to construct truncated Hamiltonians, which does not rely on optimization, and which can be applied independent of the choice of lattice. We test the approach on two models with bosonic Laughlin-like ground states with filling factor 1/2 and 1/4, respectively. We first investigate how the coupling strengths in the exact Hamiltonians depend on distance, and then we study the truncated models. For the case of 1/2 filling, we find that the truncated model with truncation radius √ 2 lattice constants on the square lattice and 1 lattice constant on the triangular lattice has an approximate twofold ground state degeneracy on the torus, and the overlap per site between these states and the states constructed from conformal field theory is higher than 0.99 for the lattices considered. For the model at 1/4 filling, our results give some hints that a truncation radius of √ 5 on the square lattice and √ 7 on the triangular lattice might be enough, but the finite size effects are too large to judge whether the topology is, indeed, present in the thermodynamic limit. The states with high overlap also have the expected topological entanglement entropies.

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