Conformal geometry from entanglement

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 $({\mathfrak{c}}_{\mathrm{tot}},\hat{I}\cdot )$ 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) $\hat{I}\cdot$ 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 $\hat{I}\cdot$, making our assumption locally checkable. We also derive similar results for 1+1D systems under a suitable set of assumptions.

48+31 pages, 25 figures