[Paper Review] Entanglement Spectroscopy and probing the Li-Haldane Conjecture in Topological Quantum Matter

Torsten V. Zache, Christian Kokail+2 more|arXiv (Cornell University)|Oct 8, 2021

Quantum many-body systemsPhysics and Astronomy49 references28 citations

TL;DR

This paper proposes an experimentally feasible protocol to probe the Li-Haldane conjecture in topological quantum matter using synthetic quantum systems. By leveraging entanglement Hamiltonian tomography (EHT) and quantum variational learning (QVL) on ultracold atoms or trapped ions, it demonstrates that the entanglement spectrum of topological states—such as integer quantum Hall states and symmetry-protected topological phases—exhibits a one-to-one correspondence with the conformal field theory (CFT) edge modes, confirming the conjecture in both non-interacting and interacting systems with high fidelity.

ABSTRACT

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.

Motivation & Objective

• To provide an experimental framework for identifying topological order in strongly correlated quantum many-body systems, which cannot be detected via local order parameters.
• To test the Li-Haldane conjecture experimentally by linking the entanglement spectrum of a subsystem to the edge state excitations of a topological phase.
• To demonstrate that the entanglement Hamiltonian (EH) of topological states can be efficiently reconstructed using quasi-local parametrizations based on the Bisognano-Wichmann theorem.
• To validate the feasibility of this approach in current quantum simulation platforms, such as ultracold fermions in optical lattices and trapped ions.
• To extend the applicability of entanglement spectroscopy beyond zero temperature, showing robustness of the EH reconstruction under finite-temperature conditions.

Proposed method

• Utilizes Entanglement Hamiltonian Tomography (EHT) and Quantum Variational Learning (QVL) to reconstruct the entanglement Hamiltonian (EH) from measured observables in a subsystem.
• Applies the Bisognano-Wichmann (BW) theorem to model the EH as a local deformation of the system Hamiltonian, parametrized by spatially varying coupling strengths (e.g., hopping amplitudes) that scale linearly with distance from the entanglement cut.
• Employs a quasi-local ansatz for the EH: ˜Hdef_A(g) = ∑_{j∈A} g_j h_j + const, where g_j are tunable parameters learned from experimental data.
• For non-interacting systems (e.g., integer quantum Hall state), uses EHT with exact diagonalization to reconstruct the EH and compute the entanglement spectrum (ES) efficiently.
• For interacting systems (e.g., 1D SPT phase), applies QVL to variational optimization of the EH ansatz, enabling ES spectroscopy in strongly correlated regimes.
• Compares the reconstructed ES to the theoretical CFT spectrum of edge modes to test the Li-Haldane correspondence.

Experimental results

Research questions

• RQ1 Can the Li-Haldane conjecture—that the low-lying entanglement spectrum of a topological state matches the conformal field theory (CFT) of its edge modes—be experimentally verified in synthetic quantum systems?
• RQ2 To what extent can the entanglement Hamiltonian (EH) of a topological many-body state be reconstructed using quasi-local parametrizations based on the Bisognano-Wichmann theorem?
• RQ3 How robust is the EH reconstruction and the resulting entanglement spectrum under finite-temperature effects in current quantum simulation platforms?
• RQ4 Can the proposed protocol be applied to both non-interacting and strongly correlated topological phases, such as the integer quantum Hall state and symmetry-protected topological phases?
• RQ5 What experimental resources (e.g., measurement budget, system size) are required to resolve the characteristic degeneracy pattern of the entanglement spectrum in real experiments?

Key findings

• The entanglement spectrum (ES) reconstructed via EHT for a 2D integer quantum Hall state shows excellent agreement with the exact CFT edge mode spectrum, including the characteristic linear dispersion and degeneracy pattern (1, 1, 2, ...).
• The EH for the integer quantum Hall state is well approximated by a quasi-local deformation of the system Hamiltonian, with hopping amplitudes showing a linear dependence on distance from the entanglement cut, consistent with the Bisognano-Wichmann theorem.
• For the 1D symmetry-protected topological phase, QVL successfully reconstructs the EH and reproduces the ES with high fidelity, confirming the Li-Haldane correspondence in an interacting system.
• The quasi-local EH ansatz remains valid at finite temperatures (T ≲ Δ, where Δ is the bulk energy gap), with the von Neumann entropy of the reduced state accurately reproduced by the learned EH.
• At high temperatures (T ≫ Δ), the reduced state approaches a thermal state, and the ES becomes increasingly mixed, but the low-lying spectrum still shows qualitative agreement with the CFT, indicating robustness of the protocol.
• Finite-temperature effects slightly smear the degeneracy pattern of the ES, but the pattern remains resolvable at low temperatures, suggesting that the Li-Haldane conjecture can be probed experimentally in current platforms with realistic thermal noise levels.