SciPost Submission Page
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
Current status:
Reports on this Submission
Strengths
The logic are well-explained and easy follow. This work brings a number of ideas from CFT and entanglement together in a novel way.
Weaknesses
Some of the critical assumptions in this work may seem opaque and confusing if the reader is not familiar with some of the background and motivations relying on previous (recent) work.
Report
In this paper, the authors argues that several aspects of conformal field theory and geometry can be extracted from a single wavefunction on the disk.
This work is part of the so-called "entanglement bootstrap" program that studies states satisfying a number of locally checkable conditions, extracting universal properties of such states.
As I understand it, the motivation for the work is as follows:
For three contaguous partitions A,B,C in a CFT, using the standard entropy formula:
I = (ctot/6) ln(1/(1-eta)),
Delta = (ctot/6) ln(eta),
where eta = |A| |C| / |AB| |BC| is the cross-ratio.
Here the authors inverts this idea. From I and Delta, the authors define ctot and eta, which effective defines the edge geometry up to a mobius transformation.
Then they show consistency of the definitions, such as cross-ratio via different definitions (5.1), and different rulers (5.2).
I have a number of questions/issues with the manuscript.
- Ref [25] argues that |psi> is stationary when K[D] is applied. The discussion on the stationary condition is motivated/relies heavily on previous work. This manuscript can be substantially improved by providing CFT motivations for the assumption.
- It is possible for ctot to change along the boundary by adding domain walls or modifying boundary Hamiltonian. Trivially, any pair of opposite chiral theories can be gapped out within a region. I would guess that the staionary assumption is broken on the edge. Is this true and how does such assumption fail?
- What happens to with a non-conformal boundary to a chiral state? For instance, the p+ip superconductor can be modified such that its edge has a p^3 dispersion. Again, what assumptions are violated with such edge?
Overall the paper appears technically sound and advances the entanglement bootstrap program to include critical states. The presentation is good overall, but some of the technical formula / assumptions can use more explanations to motivate their form. Perhaps the authors can provide a few more counterexamples of how the assumptions can fail for physical bulk/edges.
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Recommendation
Ask for minor revision
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 information-theoretic terms. The main results are a definition of the boundary central charge and the geometric cross-ratio $\frac{\ell_a \cdot \ell_b}{\ell_{ab} \cdot \ell_{bc}}$ for neighboring boundary intervals $(a,b,c)$, which are shown to satisfy consistency conditions for different ways of grouping a larger interval into three sub-intervals. 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 counter-examples 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 low-energy 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%)