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Area law violations for entanglement entropy in the form of a square root has recently been studied for one-dimensional frustration-free quantum systems based on the Motzkin walks and their variations.
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S. Bravyi, L. Caha, R. Movassagh, D. Nagaj and P. W. Shor, “Criticality without frustration for quantum spin-1 chains,” Phys. Rev. Lett, 109, 207202 (2012)
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