Correlation versus Causation in Quantum Criticality

Abstract

Correlation functions $\langle O_1(x) O_2(0) \rangle$ reveal scaling dimensions through spatial decay. We instead consider static susceptibility, the change in $\langle O_1(x) \rangle$ from perturbing the Hamiltonian by $O_2(0)$, 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 $Δ\approx 8.8$ and a heavy magnetic line defect primary of $Δ\approx 4.6$ 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 $1/L^{18}$ and $1/L^{25}$.