Entanglement Spectroscopy and probing the Li-Haldane Conjecture in Topological Quantum Matter

Torsten V. Zache1,2, Christian Kokail1,2, Bhuvanesh Sundar1,3, and Peter Zoller1,2

1Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, Innsbruck 6020, Austria
2Center for Quantum Physics, University of Innsbruck, Innsbruck 6020, Austria
3JILA, Department of Physics, University of Colorado, Boulder CO 80309, USA

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Topological phases are characterized by their entanglement properties, which is manifest in a direct relation between entanglement spectra and edge states discovered by Li and Haldane. We propose to leverage the power of synthetic quantum systems for measuring entanglement via the Entanglement Hamiltonian to probe this relationship experimentally. This is made possible by exploiting the quasi-local structure of Entanglement Hamiltonians. The feasibility of this proposal is illustrated for two paradigmatic examples realizable with current technology, an integer quantum Hall state of non-interacting fermions on a 2D lattice and a symmetry protected topological state of interacting fermions on a 1D chain. Our results pave the road towards an experimental identification of topological order in strongly correlated quantum many-body systems.

Topological Quantum Matter — in contrast to ordinary phases of matter — can not be detected by probing local observables, such as the magnetisation of a magnet. Instead, topological phases are characterised by their quantum correlations, as well as excitations supported at the boundary of the system. The spectrum of these edge excitations is directly related to the structure of the entanglement in the bulk, a relation known as the Li-Haldane conjecture. In this article, we propose to leverage the power of quantum simulators to probe the Li-Haldane conjecture experimentally. Our approach is based on the quasi-locality of the Entanglement Hamiltonian, which enables the application of recently developed protocols to measure the entanglement spectrum. We demonstrate the feasibility of our proposal with numerical simulations of examples in one and two spatial dimensions.

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