Extracting Scaling Dimensions via Operator Covariance in an S = 1/2 Quantum Spin Chain

In critical lattice models, the real-space two-point function of a local lattice operator consists of a sum of power laws, ∑A_ir^(-2Δ_i), where each Δ_i is the scaling dimension of a continuum field operator. When several fields contribute within a given symmetry sector, single-operator fits become unstable. Here we implement a covariance-matrix-based approach, previously used for classical models [A. W. Sandvik, arXiv:2406.12681], in which the distance-dependent eigenvalues of a covariance matrix built from multiple symmetry-adapted lattice operators disentangle individual scaling dimensions, including primaries and low-lying descendants. We apply this framework to the spin-1/2 JQ₂ chain, which adds four-spin interactions to the SU(2)-symmetric Heisenberg model and can drive the system to a dimerized state. Using Quantum Monte Carlo simulations at the critical dimerization point, we can extract primary as well as descendant scaling dimensions that are difficult to resolve with single-operator analysis. The same framework applies to imaginary-time correlations, which exhibit crossover between short-τ power-law decay and long-τ exponential decay set by the finite-size gap. We can extract scaling dimensions from either the power-law window or the exponential tail. These results demonstrate covariance-diagonalization as an enhancement of standard correlation functions.