Correlation versus Causation in Quantum Criticality

Correlation functions langle O₁(x) O₂(0) rangle reveal scaling dimensions through spatial decay. We instead consider static susceptibility, the change in langle O₁(x) rangle from perturbing the Hamiltonian by O₂(0), which we term causation for short. In a conformal field theory (CFT), dimensional analysis predicts decay of |x|^-2Δ for correlation and |x|^-2Δ+1 for causation. Yet we find causation can decay up to fifteen additional orders in x through a general mechanism, which we trace to time-derivative fields being unable to contribute to static response. In higher-dimensional CFTs, this mechanism ensures leading causation arises from primaries, even when descendants dominate correlation, which we leverage with DMRG to identify a previously unresolved corner primary of Δapprox 8.8 and a heavy magnetic line defect primary of Δapprox 4.6 in the (2+1)D critical Ising model. Moreover, the same mechanism governs edge-mode localization in (1+1)D gapless symmetry-protected topological phases, explaining previously observed anomalously small edge-mode splittings and guiding our construction of spin chains with splittings as small as 1/L¹⁸ and 1/L²⁵.