Citation: Li, Hui, and F. Duncan M. Haldane. "Entanglement Spectrum as a Generalization of Entanglement Entropy: Identification of Topological Order in Non-Abelian Fractional Quantum Hall Effect States." Physical review letters 101.1 (2008): 010504.
Web: https://arxiv.org/abs/0805.0332
Tags: Physical, Information-theory, Statistical-mechanics, Conformal-field-theory, TQFT
In this paper argue a specific incarnation of the CFT/TQFT correspondence, Suppose that \psi is a ground state of a topologically ordered system. Suppose that \rho is the density matrix obtained by taking \rho and tracing out all the degrees of freedom outside a large disk. They write \rho = e^{H} for some matrix H. Because \rho is a density matrix it is Hermitian, so H is Hermitian as well. The authors observe that H behaves like the Hamiltonian of a conformal field theory on the boundary of the disk. They make this observation in special cases related to the factional Hall effect, where they see that this conformal field theory is exactly the conformal field theory we expect to correspond to the TQFT. In this way, the CFT/TQFT correspondence can be seen in the entanglement spectrum.
When the TQFT Hamiltonian is not chosen to be a toy model (it is a generic Hamiltonian in the topological phase), the spectrum of the CFT Hamiltonian H behaves as follows. It has a collection of low-lying gapless CFT-like states, then it has a finite gap independent of system size, and then it has extra level which seem to depend on the specific choice of Hamiltonian. These low-lying CFT-like states are independent of the specific Hamiltonian up to exponential corrections, and are thus invariants of the topological phase.
The state e^H is the (temperature = 1) Gibbs ensemble of the CFT. Hence, understanding the relationship between the CFT and TQFT relies on the theory of quantum statistical mechanics.
This sort of correspondence was already discussed in the final section of
> Kitaev, Alexei, and John Preskill. "Topological entanglement entropy." Physical review letters 96.11 (2006): 110404.
From what I understand, Kitaev-Preskill's paper was the first recording of this correspondence in literature. However, it was not widely appreciated until Li-Haldane's paper. There have since been several proofs os Li-Haldane's observation. The most notable are
> Swingle, Brian, and T. Senthil. "Geometric proof of the equality between entanglement and edge spectra." Physical Review B—Condensed Matter and Materials Physics 86.4 (2012): 045117.and
> Fidkowski, Lukasz. "Entanglement spectrum of topological insulators and superconductors." Physical review letters 104.13 (2010): 130502.
In the case that the TQFT is doubled the theory has boundary excitations which changes the story a bit, though the Levin-Wen model gives an explicit Hamiltonian making the study much easier. This was done in
> Luo, Zhu-Xi, et al. "Correspondence between bulk entanglement and boundary excitation spectra in two-dimensional gapped topological phases." Physical Review B 99.20 (2019): 205137.