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 $|x|<sup>{-2Δ}$ for correlation and $|x|<sup>{-2Δ+1}$ 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 $1/L<sup>{18}$ and $1/L<sup>{25}$.

• The paper establishes that static susceptibility can suppress time-derivative fields, causing responses to decay up to 15 additional powers of distance or exponentially compared with correlations.
• The authors apply this framework to gapless symmetry-protected topological phases, designing Ising-critical spin chains with edge-mode splittings scaling as L⁻¹⁸ and L⁻²⁵.
• DMRG results show that causation acts as a primary sieve, identifying previously hidden corner and magnetic-line-defect primaries in the (2+1)D Ising model with dimensions near 8.8 and 4.6.

Overview and main claim

This paper distinguishes two equilibrium probes of quantum critical systems: the familiar correlation function $\langle O_1(x) O_2(0)\rangle$, which passively measures statistical dependence, and the static susceptibility — the response of $\langle O_1(x)\rangle$ to perturbing the Hamiltonian by $O_2(0)$ — which the authors term "causation." While dimensional analysis in a conformal field theory (CFT) predicts that correlation decays as $|x|^{-2\Delta}$ and causation as $|x|^{-2\Delta+1}$, the paper shows that causation can be suppressed far beyond this expectation: up to fifteen additional powers of $|x|^{-1}$ algebraically, or even exponentially. The mechanism is general: operators that are time derivatives of other operators contribute nothing to static response, because inserting the resolvent $G = (E_0 - H)^{-1}$ into a commutator with the Hamiltonian collapses it to an equal-time commutator that vanishes at large separation (2608.12770).

The result has two principal applications. First, finite-size splittings of edge modes in gapless symmetry-protected topological phases (gSPTs) are causation functions of the bulk, which explains previously observed anomalously small edge-mode splittings and guides the construction of spin chains with splittings scaling as $1/L^{18}$ and $1/L^{25}$. Second, in CFTs with three or more spacetime dimensions, causation filters out descendant fields entirely, so the leading causation of a generic lattice operator is set by the lowest-dimension primary. The authors exploit this "primary sieve" with DMRG to identify a corner primary of dimension $\Delta \approx 8.8$ and a magnetic line defect primary of $\langle O_1(x)\rangle$0 in the $\langle O_1(x)\rangle$1D critical Ising model, both invisible to correlation.

The mechanism: time derivatives do not cause

Causation is defined via first-order perturbation theory: for $\langle O_1(x)\rangle$2,

$\langle O_1(x)\rangle$3

with $\langle O_1(x)\rangle$4 the ground-state resolvent. If $\langle O_1(x)\rangle$5 is a time derivative, then using $\langle O_1(x)\rangle$6, the Hermitian combination reduces to $\langle O_1(x)\rangle$7, which vanishes once the supports are disjoint. Moreover, under time-reversal symmetry with $\langle O_1(x)\rangle$8 and $\langle O_1(x)\rangle$9 sharing the same $O_2(0)$0-charge, the individual function $O_2(0)$1 is real yet anti-Hermitian, hence identically zero.

Consequently, if a lattice operator flows to a sum of continuum fields, its correlation is dominated by the lowest-dimension field $O_2(0)$2, decaying as $O_2(0)$3, whereas its causation is controlled by the lowest-dimension field that is not a time derivative, $O_2(0)$4. On $O_2(0)$5D boundaries and defects of a CFT, every descendant is either a time derivative or a quasiprimary, so causation is governed by the lowest-dimension quasiprimary. In higher dimensions, quasiprimaries coincide with primaries, sharpening the sieve further. The authors assume non-degenerate ground states and vanishing expectation values throughout; these are standard but necessary conditions for the resolvent-based definition.

One-dimensional examples: Ising, tricritical Ising, and free fermions

In the critical Ising chain with free boundaries, the authors use Virasoro character counting to locate the first $O_2(0)$6-odd quasiprimary in each tower. For $O_2(0)$7-odd boundary operators flowing to the $O_2(0)$8 tower, the character has coefficients increasing at levels $O_2(0)$9, so the first odd-level quasiprimary sits at level 7 ($|x|^{-2\Delta}$0). Correlation between mirror-image boundary operators therefore decays as $|x|^{-2\Delta}$1 while causation decays as $|x|^{-2\Delta}$2. For $|x|^{-2\Delta}$3-even, $|x|^{-2\Delta}$4-odd operators flowing to the identity tower, the first odd-level descendant appears at level 3 ($|x|^{-2\Delta}$5) but the first odd-level quasiprimary only at level 11 ($|x|^{-2\Delta}$6), giving correlation $|x|^{-2\Delta}$7 versus causation $|x|^{-2\Delta}$8 — the advertised fifteen-order suppression. Both scalings are confirmed numerically via exact free-fermion evaluation of the resolvent after a Jordan-Wigner transformation. The same analysis applied to the tricritical Ising CFT (realized in the Blume-Capel chain) predicts $|x|^{-2\Delta}$9 correlation and $|x|^{-2\Delta+1}$0 causation, also confirmed by direct perturbative measurement.

For free Majorana chains in the BDI class, the suppression becomes exponential. Because lattice fermions Wick-contract, any continuum field appearing in a lattice Majorana's expansion must be a single fermion; on the boundary these are exhausted by $|x|^{-2\Delta+1}$1 and its time derivatives, so all $|x|^{-2\Delta+1}$2-odd lattice operators are time derivatives and causation vanishes exactly. Adding irrelevant perturbations replaces strictly local generators with exponentially localized ones, yielding causation $|x|^{-2\Delta+1}$3. This is formalized in a theorem: for any BDI chain with winding number $|x|^{-2\Delta+1}$4, end-to-end causation either vanishes (when the Laurent polynomial $|x|^{-2\Delta+1}$5 has no poles) or decays as $|x|^{-2\Delta+1}$6 with $|x|^{-2\Delta+1}$7 set by the innermost zero of $|x|^{-2\Delta+1}$8. The proof constructs an exponentially localized operator whose commutator with $|x|^{-2\Delta+1}$9 reproduces the edge Majorana up to exponentially small errors.

Application: edge-mode localization in gapless SPTs

The second-order energy splitting induced by bulk-edge couplings $|x|^{-1}$0 is proportional to $|x|^{-1}$1 — precisely a bulk causation function. Edge-mode localization in gSPTs is thus governed by causal rather than correlational decay, providing a unified explanation for the exponentially small splittings found in free-fermion chains and the $|x|^{-1}$2 splitting found previously in interacting models.

Guided by this framework, the authors construct two new interacting gSPTs at Ising criticality protected by $|x|^{-1}$3. The first model, built from cluster-like terms, maps under a local unitary to the critical Ising chain with decoupled edge qubits; symmetry-allowed bulk-edge couplings involve $|x|^{-1}$4-odd, $|x|^{-1}$5-odd operators flowing to the $|x|^{-1}$6 tower, producing splittings of $|x|^{-1}$7 (from the level-7 quasiprimary) and $|x|^{-1}$8 (from level 9, which first contributes through the determinant of the $|x|^{-1}$9 causation matrix). The second model involves $G = (E_0 - H)^{-1}$0-even, $G = (E_0 - H)^{-1}$1-odd couplings flowing to the identity tower, giving splittings of $G = (E_0 - H)^{-1}$2 and $G = (E_0 - H)^{-1}$3. These are, to the authors' knowledge, the smallest power-law edge-mode splittings reported, and both scalings are confirmed numerically. A caveat noted in the paper: the second-order formula assumes symmetry forbids $G = (E_0 - H)^{-1}$4 contributions, which can fail when the two ends carry distinct boundary conditions.

Application: the primary sieve in higher-dimensional CFTs

In CFTs with $G = (E_0 - H)^{-1}$5 spacetime dimensions, the conformal algebra is finite-dimensional and all descendants are spatial and temporal derivatives of primaries. Temporal derivatives drop out of causation entirely; spatial derivatives survive but cannot mask their own primary, since a generic lattice operator without definite spatial parity flows to fields of all parities, and the primary has strictly lower dimension than any of its descendants. Causation therefore always resolves the lowest-dimension primary, even when correlation is dominated by a light descendant.

The authors demonstrate this on the $G = (E_0 - H)^{-1}$6D transverse-field Ising model using DMRG (bond dimension up to $G = (E_0 - H)^{-1}$7). Corner-to-corner correlation of $G = (E_0 - H)^{-1}$8-odd operators resolves only the descendant $G = (E_0 - H)^{-1}$9 with $1/L^{18}$0, consistent with the known corner magnetization exponent. Causation instead decays as $1/L^{18}$1, identifying a new $1/L^{18}$2-odd, $1/L^{18}$3-odd corner primary with $1/L^{18}$4, consistent with a mean-field estimate of a composite of three $1/L^{18}$5's and three time derivatives ($1/L^{18}$6). Similarly, with a pinned center spin realizing a magnetic line defect, causation decays as $1/L^{18}$7, yielding a defect primary of dimension $1/L^{18}$8 — in excellent agreement with the value $1/L^{18}$9 obtained independently by fuzzy-sphere methods. The agreement with an established bootstrap-adjacent technique lends credibility to causation as a spectroscopic tool, though the extracted exponents carry finite-size uncertainty not fully quantified in the paper.

Limitations and open questions

The suppression mechanism relies essentially on antiunitary time-reversal symmetry ($1/L^{25}$0) to isolate time-derivative contributions; whether other symmetries can play an analogous role is left open. The definition is restricted to the static ($1/L^{25}$1) limit of the Kubo response, and the authors note that at finite frequency descendants may reappear, leaving the frequency dependence of the suppression unexplored. Numerically, the DMRG extraction of heavy primaries rests on power-law fits over limited system-size ranges, and the discrepancy between the measured corner dimension ($1/L^{25}$2) and the Monte Carlo hinge exponent ($1/L^{25}$3) is attributed to finite-size effects without systematic error analysis. Finally, a systematic survey of causation across symmetry sectors of higher-dimensional lattice CFTs — which could uncover further heavy bulk, defect, and boundary primaries — is deferred to future work.

Conclusion

This paper establishes static susceptibility ("causation") as a probe of critical systems whose scaling can differ dramatically from, and often complement, that of correlation functions. Time-derivative fields are invisible to static response, causing causation to be governed by the lowest non-time-derivative quasiprimary — or, in $1/L^{25}$4, the lowest primary. This single mechanism explains anomalously sharp edge-mode localization in gapless SPTs, enables the design of spin chains with splittings as small as $1/L^{25}$5, and provides a practical numerical route to heavy primary operators in higher-dimensional CFTs that are inaccessible to conventional correlation measurements.