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Beyond the Li-Haldane conjecture, we find the entanglement Hamiltonian (EH)is actually not closely similar to the original Hamiltonian on the virtualedge. Unexpectedly, the EH has some relevant long-range interacting terms whichhugely affect the physics. Without loss of generality, we study a spin-1/2Heisenberg bilayer to obtain the entanglement information between the twolayers. Although the entanglement spectrum carrying Goldstone mode seems like aHeisenberg model on a single layer which is consistent with Li-Haldaneconjecture, we demonstrate there actually is a finite temperature phasetransition for the EH. The results violate the Mermin-Wagner theorem whichmeans there should be relevant long-range terms in the EH. It reveals that theLi-Haldane conjecture ignores necessary corrections for the EH which may leadtotally different physics.

Relevant long-range interaction of the entanglement Hamiltonian emerges from a short-range system