Li–Haldane conjecture on universal entanglement spectra

Let $|\Psi\rangle$ be a ground state in a two-dimensional topological phase, let $A$ be a spatial subsystem, and let $\rho_A=\operatorname{Tr}_{\bar A}|\Psi\rangle\langle\Psi|$ be its reduced density matrix. Define the entanglement energies by $\xi_i=-\log\lambda_i$ , where $\lambda_i$ are the eigenvalues of $\rho_A$ . After resolving the entanglement spectrum by the conserved quantum numbers $q$ associated with the boundary, the multiplicities of its universal low-lying levels should equal the excitation multiplicities of the corresponding gapless edge conformal field theory: $m_{\mathrm{ES}}(q,\Delta)=m_{\mathrm{edge}}(q,\Delta)$ for every quantum-number sector $q$ and every level $\Delta$ in the universal low-lying regime.

References

Primary source

arXiv

Additional references

Unravelling the Li-Haldane Conjecture with the Projected Ensemble — arXiv

condensed-matter-physics conformal-field-theory quantum-information

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed progress

The conjecture remains unproved in general, while a new preprint reports a sharper version of its predicted signal in several quantum Hall systems.

The Li–Haldane conjecture proposes that the low-lying entanglement spectrum of a topological state reproduces the excitation spectrum of its gapless edge theory. Li and Haldane introduced this proposal in 2008 using Moore–Read and realistic fractional quantum Hall states.

Known results

A 2011 analysis proved an upper bound on edge-mode counting for several clustering fractional quantum Hall states, but not equality in full generality.
A 2017 study reported correspondence of universal entanglement-spectrum eigenstates with projected physical edge excitations in selected ideal and Coulomb states.
A 2022 study numerically confirmed the correspondence in selected integer quantum Hall and one-dimensional symmetry-protected states.
A 2023 study found numerical evidence for a local Bisognano–Wichmann entanglement Hamiltonian in selected Laughlin and Moore–Read states.

September 3, 2026 measurement-resolved refinement

On September 3, 2026, the unrefereed preprint Unravelling the Li-Haldane Conjecture with the Projected Ensemble reported a measurement-resolved Li–Haldane structure for fractional quantum Hall states, including the Moore–Read state and realistic Coulomb ground states. This is claimed progress in special cases, not a proof of the conjecture; the result is unverified.

Current status (as of September 2026): The conjecture has substantial analytical and numerical support in selected models, but no general proof or counterexample is recorded; the new measurement-resolved result remains an unverified special-case claim.

Sources

No solutions have been posted yet.