Citation: Kim, Isaac H., et al. "Chiral central charge from a single bulk wave function." Physical Review Letters 128.17 (2022): 176402.
Web: https://arxiv.org/abs/2110.06932
Tags: MTC-reconstruction, Conformal-field-theory
In this paper the authors give a definition of chiral central charge from a single bulk wavefunction, in quite a similar fashion to the way TEE is defined. The authors give a non-rigorous justification for why this formula should hold. This justification is written in the language of quantum Markov states and entropic conditions, in the general style of entanglement bootstrap. Thus, it can be said that this paper includes the chiral central charge as part of the entanglement bootstrap programme. This is very nice, because all of the data of a topological order is (conjecturally) contained in the MTC and the chiral central charge. Entanglement bootstrap was making good progress on the MTC, but the chiral central charge feels qualitatively different. In this paper, they bridge the gap.
The formula is very clean - it is in terms of the expectation value of a certain commutator of modular Hamiltonians. This is expected - the expectation of the commutator of two modular Hamiltonians is intuitively related to the flow of energy between those two terms at T=1.
The authors justify their formula as follows. First, they break the modular Hamiltonian into a sum of many small pieces. This is done via a formula of Petz, which says that the modular Hamiltonian of a quantum Markov state on ABC can be decomposed as K_(ABC) = K_(AB) + K_(BC) - K_B. After breaking up the Hamiltonian into small pieces, the flow between local terms can be computed. Adding up these contributions leads to an amazing phenomenon where a whole bunch of complicated non-universal fractions add up to 1/2. This is great justification for why this paper is on to something - such cancellation would not happen by chance. This final computed value is then compared to the chiral central charge by a heuristic argument, completing the justification.
More details about this proposal, surrounding work, and numerical verification are found in the companion article:
> Kim, Isaac H., et al. "Modular commutator in gapped quantum many-body systems." Physical Review B 106.7 (2022): 075147.
In the case of non-interacting fermions, a proof of this modular commutator formulas is now known:
> Fan, Ruihua, Pengfei Zhang, and Yingfei Gu. "Generalized real-space Chern number formula and entanglement hamiltonian." SciPost Physics 15.6 (2023): 249.
Just like how cluster states can give spurious TEE, they can give spurious contributions to the modular commutator as well. This comes in part from the fact that the modular Hamiltonians do NOT need to be the sum of local terms. That is: the modular Hamiltonians of ground states of local Hamiltonians are not necessarily themselves local Hamiltonians. Strange! This seems to be the root of a lot of the counterexamples of strange properties of cluster states and related counterexamples. The reference is here:
> Gass, Julian, and Michael Levin. "Many-body systems with spurious modular commutators." arXiv preprint arXiv:2405.15892 (2024).
A formula inspired by the one in this article has since been introduced for the computation of quantum Hall conductance:
> Fan, Ruihua, Rahul Sahay, and Ashvin Vishwanath. "Extracting the quantum Hall conductance from a single bulk wave function." Physical Review Letters 131.18 (2023): 186301.