Conformal geometry from entanglement

TL;DR

This paper demonstrates how conformal geometry naturally emerges at the edge of 2+1D quantum systems through entanglement, introducing new information-theoretic quantities that reveal the underlying conformal structure.

Contribution

It introduces a novel pair of local information-theoretic quantities that characterize emergent conformal geometry at the edge of gapped quantum systems.

Findings

Total central charge is a constant derived from entanglement.

Cross-ratio quantity obeys conformal invariance rules.

Emergent conformal geometry is linked to groundstate 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 $(\mathfrak{c}_{\mathrm{tot}}, \eta)$ 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.