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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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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.
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Publish (surpasses expectations and criteria for this Journal; among top 10%)