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Conformal geometry from entanglement
by Isaac H. Kim, Xiang Li, TingChun Lin, John McGreevy, Bowen Shi
Submission summary
Authors (as registered SciPost users):  John McGreevy 
Submission information  

Preprint Link:  https://arxiv.org/abs/2404.03725v1 (pdf) 
Date submitted:  20240704 00:56 
Submitted by:  McGreevy, John 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
In a physical system with conformal symmetry, observables depend on crossratios, measures of distance invariant under global conformal transformations (conformal geometry for short). We identify a quantum informationtheoretic mechanism by which the conformal geometry emerges at the gapless edge of a 2+1D quantum manybody system with a bulk energy gap. We introduce a novel pair of informationtheoretic quantities $(\mathfrak{c}_{\mathrm{tot}}, \eta)$ that can be defined locally on the edge from the wavefunction of the manybody system, without prior knowledge of any distance measure. We posit that, for a topological groundstate, the quantity $\mathfrak{c}_{\mathrm{tot}}$ is stationary under arbitrary variations of the quantum state, and study the logical consequences. We show that stationarity, modulo an entanglementbased assumption about the bulk, implies (i) $\mathfrak{c}_{\mathrm{tot}}$ is a nonnegative constant that can be interpreted as the total central charge of the edge theory. (ii) $\eta$ is a crossratio, obeying the full set of mathematical consistency rules, which further indicates the existence of a distance measure of the edge with global conformal invariance. Thus, the conformal geometry emerges from a simple assumption on groundstate entanglement. We show that stationarity of $\mathfrak{c}_{\mathrm{tot}}$ is equivalent to a vector fixedpoint equation involving $\eta$, making our assumption locally checkable. We also derive similar results for 1+1D systems under a suitable set of assumptions.
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 Provide a novel and synergetic link between different research areas.
 Open a new pathway in an existing or a new research direction, with clear potential for multipronged followup work
 Detail a groundbreaking theoretical/experimental/computational discovery
 Present a breakthrough on a previouslyidentified and longstanding research stumbling block
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Reports on this Submission
Report
This is an interesting paper that attempts to reformulate the emergence of concepts in chiral 2d CFTs on the edge of a gapped 2+1d system purely in informationtheoretic terms. The main results are a definition of the boundary central charge and the geometric crossratio $\frac{\ell_a \cdot \ell_b}{\ell_{ab} \cdot \ell_{bc}}$ for neighboring boundary intervals $(a,b,c)$, which are shown to satisfy consistency conditions for different ways of grouping a larger interval into three subintervals. They hope that this framework may be the starting point for understanding the emergence of conformal properties of chiral edge states more generally.
This is probably an exciting paper for the community of those trying to understand dynamic properties of QFTs using information theory methods. I was confused though how conformal theories could appear so generally in their analysis without having to assume Lorentz invariance from the outset  in the absence of such an assumption, couldn't one easily construct counterexamples with multiple decoupled chiral edge modes that all have different speeds of light, and therefore would not be conformal? The authors talk about potentially deriving all the Virasoro generators of the boundary, but this kind of theory would not be conformal  which of their assumptions eliminates it?
One minor weakness of the work is that their main assumption A1 is a condition that only holds in the infinite IR limit, and they do not have a quantitative description of the corrections to this expression in a lowenergy expansion. Since such corrections are generically present in lattice models, comparisons with computations in such models would benefit from a principled way of parameterizing them. Can they comment on how their formalism might also include such corrections?
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)
Report
I liked this paper very much. It is very original, and despite that, it is very well thought and reasoned. It is about understanding the emergence of conformal symmetry in the boundary of a gapped bulk. Building on previous results in the literature the authors make some natural conjectures about the quantum state in the gapped bulk and a new condition is imposed that has the interpretation of stationarity of the central charge. They derive from these assumptions two quantities with the interpretation of central charge and cross ratio of the emerging CFT. It is interesting that the work only uses very general tools in quantum information to derive the conformal geometry, what makes the scheme suitable to general systems and particularly suitable to computations in concrete lattice models. The work makes patent that there should be a complete and relatively simple understanding of the phenomenon in quantum information theory terms, and I think it is also interesting for developing ideas in quantum information/quantum computation itself. I recommend publication in its preset form.
Recommendation
Publish (surpasses expectations and criteria for this Journal; among top 10%)