TY  - JOUR
PB  - SciPost Foundation
DO  - 10.21468/SciPostPhys.18.3.102
TI  - Conformal geometry from entanglement
PY  - 2025/03/19
UR  - https://scipost.org/SciPostPhys.18.3.102
JF  - SciPost Physics
JA  - SciPost Phys.
VL  - 18
IS  - 3
SP  - 102
A1  - Kim, Isaac H.
AU  - Li, Xiang
AU  - Lin, Ting-Chun
AU  - McGreevy, John
AU  - Shi, Bowen
AB  - 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 $(\mathfrak{c}_{\textrm{tot}}, \eta)$ 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 $\mathfrak{c}_{\textrm{tot}}$ 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) $\mathfrak{c}_{\textrm{tot}}$ 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 $\mathfrak{c}_{\textrm{tot}}$ 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.
ER  -

@Article{10.21468/SciPostPhys.18.3.102,
	title={{Conformal geometry from entanglement}},
	author={Isaac H. Kim and Xiang Li and Ting-Chun Lin and John McGreevy and Bowen Shi},
	journal={SciPost Phys.},
	volume={18},
	pages={102},
	year={2025},
	publisher={SciPost},
	doi={10.21468/SciPostPhys.18.3.102},
	url={https://scipost.org/10.21468/SciPostPhys.18.3.102},
}