Data-efficient reconstruction of critical quantum dynamics via blind fractional-envelope extrapolation

Simulating real-time dynamics of quantum systems is often limited to short times by entanglement growth. Finite-pole reconstructions such as linear prediction and related machineries extrapolate such data reliably when the spectrum is a finite set of excitations, but at criticality the low-energy spectrum is a power-law continuum $A(\omega)\sim|\omega|^{\alpha-1}$ -- a branch cut whose real-time tail $G(t)\sim t^{-\alpha}$ finitely many poles cannot represent. Here we develop a fractional-calculus-motivated envelope extrapolation for such data. Its structure is motivated by a fractional form of Schwinger--Dyson (fSD) equation, in which the Laplace symbol $s^{\alpha}$ carries the branch cut analytically while the residual self-energy remains meromorphic. On real data we employ the corresponding operational alternative -- the exponent $\alpha$ is selected blindly inside the fit window, the signal is detrended by $t^{\alpha}$, the residual is fitted by a stabilized finite-pole model, and the algebraic envelope is restored. On the critical XXZ chain this blind fractional-envelope method (fSD for short) extrapolates short-time data typically several-fold more accurately than finite-pole methods, with the exponent $\alpha$ identified blindly from the fit window alone and bracketing the closed-form Luttinger value at weak coupling. The same blind search finds the $z=2$ dilute-magnon exponent $\alpha=1/2$ at the $\Delta=1$ saturation transition, and the advantage persists in the gapped free-magnon phase with its sharp band edges. On noncritical dynamical mean-field spectra, whose low-frequency response is regular, fSD by contrast fails to select any stable fractional envelope and reduces to the standard pole result rather than manufacturing a spurious power law, making it an efficient and accurate route to quantum critical dynamics when only short simulation times are accessible.