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Conformal geometry from entanglement
by Isaac H. Kim, Xiang Li, Ting-Chun Lin, John McGreevy, Bowen Shi
Submission summary
| Authors (as registered SciPost users): | John McGreevy |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2404.03725v1 (pdf) |
| Date submitted: | 2024-07-04 00:56 |
| Submitted by: | McGreevy, John |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
In a physical system with conformal symmetry, observables depend on cross-ratios, measures of distance invariant under global conformal transformations (conformal geometry for short). We identify a quantum information-theoretic mechanism by which the conformal geometry emerges at the gapless edge of a 2+1D quantum many-body system with a bulk energy gap. We introduce a novel pair of information-theoretic quantities $(\mathfrak{c}_{\mathrm{tot}}, \eta)$ that can be defined locally on the edge from the wavefunction of the many-body 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 entanglement-based assumption about the bulk, implies (i) $\mathfrak{c}_{\mathrm{tot}}$ is a non-negative constant that can be interpreted as the total central charge of the edge theory. (ii) $\eta$ is a cross-ratio, 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 fixed-point equation involving $\eta$, making our assumption locally checkable. We also derive similar results for 1+1D systems under a suitable set of assumptions.
Author indications on fulfilling journal expectations
- 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 multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing 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.
Recommendation
Publish (surpasses expectations and criteria for this Journal; among top 10%)