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Shannon-Rényi entropies and participation spectra across 3d $O(3)$ criticality

Universal features in the scalings of Shannon-Rényi entropies of many-body groundstates are studied for interacting spin-$\frac{1}{2}$ systems across (2+1) dimensional $O(3)$ critical points, using quantum Monte Carlo simulations on dimerized and plaquettized Heisenberg models on the square lattice. Considering both full systems and line shaped subsystems, $SU(2)$ symmetry breaking on the Néel ordered side of the transition is characterized by the presence of a logarithmic term in the scaling of Shannon-Rényi entropies, which is absent in the disordered gapped phase. Such a difference in the scalings allows to capture the quantum critical point using Shannon-Rényi entropies for line shaped subsystems of length $L$ embedded in $L\times L$ tori, as the smaller subsystem entropies are numerically accessible to much higher precision than for the full system. Most interestingly, at the quantum phase transition an additive subleading constant $b_\infty^{*\rm line}=0.41(1)$ emerges in the critical scaling of the line Shannon-Rényi entropy $S_\infty^\text{line}$. This number appears to be universal for 3d $O(3)$ criticality, as confirmed for the finite-temperature transition in the 3d antiferromagnetic spin-$\frac{1}{2}$ Heisenberg model. Additionally, the phases and phase transition can be detected in several features of the participation spectrum, consisting of the diagonal elements of the reduced density matrix of the line subsystem. In particular the Néel ordering transition can be simply understood in the $\{S^z\}$ basis by a confinement mechanism of ferromagnetic domain walls.