Phases that exhibit topological order cannot be identified with a local order parameter; they are characterized by a global parameter, making their identification difficult. Their topological nature is encoded in the entanglement between its constituents. In that spirit, the entanglement spectrum introduced by Li and Haldane [1] in the context of the Fractional Hall effect. The latter corresponds to the spectrum of the reduced density matrix $\rho_\mathrm{A} \propto e^{-K_\mathrm{A}}$ when we divide the system into two parts: A and its complement B. The entanglement Hamiltonian's ($K_\mathrm{A}$) spectrum enveil a unique feature: its low-lying levels can be brought into direct correspondence with the excitation spectrum at a physical edge of the system. Studying the entanglement spectrum has proven to be a powerful tool for probing topological order in correlated quantum Hall phases and revealing symmetry-protected edge mode in topological insulators. Generally speaking, determining the entanglement spectrum has proven challenging, especially for large system sizes. Here, we map entanglement to spectral properties [2] by physically realizing a system whose single-particle dynamics is governed by the entanglement Hamiltonian of a quantum Hall system. We use a synthetic dimension, encoded in the electronic spin of dysprosium atoms, to implement spatially deformed dynamics, as suggested by the Bisognano-Wichmann theorem [3]. The realized Hamiltonian, probed with bosonic atoms with negligible interactions, exhibits a chiral dispersion akin to a topological edge mode, revealing the fundamental link between bulk entanglement and boundary physics. Additionally, we numerically show that our protocol could be extended to interacting systems in fractional quantum Hall states.

[1] Li, H. and Haldane, F. D. M. Phys. Rev. Lett. 101(1), 010504 July (2008). arXiv:0805.0332 [cond-mat].

[2] Dalmonte, M., Vermersch, B., and Zoller, P. Nature Phys 14(8), 827–831 August (2018). arXiv:1707.04455 [cond-mat, physics:quant-ph].

[3] Bisognano, J. J. and Wichmann, E. H. J. Math. Phys. 17, 303–321 (1976).