We study the high-energy phase diagram of a two-dimensional spin-$\frac{1}{2}$ Heisenberg model on a square lattice in the presence of either quenched or quasiperiodic disorder. The use of large-scale tensor network numerics allows us to compute the bipartite entanglement entropy for systems of up to $60\times 7$ lattice sites. We provide evidence for the existence of a many-body localized regime for large disorder strength that features an area law in excited states and that violates the eigenstate thermalization hypothesis. From a finite-size analysis, we determine an estimate for the critical disorder strength where the transition to the ergodic regime occurs in the quenched case.

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• Many-body localization
• 2-dimensional systems
• Density matrix renormalization group
• Heisenberg model
• Matrix product states
• Tensor network methods