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How universal is the entanglement spectrum?

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It is now commonly believed that the ground state entanglement spectrum (ES) exhibits universal features characteristic of a given phase. In this talk, I will present evidence to the contrary. I will show that the entanglement Hamiltonian can undergo quantum phase transitions in which its ground state and low energy spectrum exhibit singular changes, even when the physical system remains in the same phase. For broken symmetry problems, this implies that the ES and the Renyi entropies can mislead entirely, while for quantum Hall systems the ES has much less universal content than assumed to date. I will also discuss the consequences of the eigenstate thermalization hypothesis for the entanglement Hamiltonian, showing that a pure state in a sub-system can capture the properties of the reduced density matrix.