Conformal geometry from entanglement

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) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:msub> <mml:mstyle mathvariant="fraktur"> <mml:mi>𝔠</mml:mi> </mml:mstyle> <mml:mtext mathvariant="normal">tot</mml:mtext> </mml:msub> <mml:mo>,</mml:mo> <mml:mi>η</mml:mi> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> 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}} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mstyle mathvariant="fraktur"> <mml:mi>𝔠</mml:mi> </mml:mstyle> <mml:mtext mathvariant="normal">tot</mml:mtext> </mml:msub> </mml:math> 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}} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mstyle mathvariant="fraktur"> <mml:mi>𝔠</mml:mi> </mml:mstyle> <mml:mtext mathvariant="normal">tot</mml:mtext> </mml:msub> </mml:math> is a non-negative constant that can be interpreted as the total central charge of the edge theory. (ii) \eta <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>η</mml:mi> </mml:math> 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}} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mstyle mathvariant="fraktur"> <mml:mi>𝔠</mml:mi> </mml:mstyle> <mml:mtext mathvariant="normal">tot</mml:mtext> </mml:msub> </mml:math> is equivalent to a vector fixed-point equation involving \eta <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>η</mml:mi> </mml:math> , making our assumption locally checkable. We also derive similar results for 1+1D systems under a suitable set of assumptions.