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arXiv:2605.00104 · Physical Review Letters

Criticality on Rényi defects at (2+1)ddd O(3) quantum critical points

Yanzhang Zhu, Zhe Wang, Meng Cheng, Zheng Yan

cond-mat.str-elcond-mat.stat-mechhep-thquant-phSubmitted Apr 30, 2026• Announced May 4, 2026• 0 citations• Revised Jul 16, 2026 (v2)
v1 · Apr 30, 2026v2 · Jul 16, 2026v3 · Oct 8, 2026
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Abstract

At a quantum critical point, the universal scaling behavior of Rényi entanglement entropy is controlled by the universality class of the codimension-two Rényi (or conical) defects in the infrared theory. In this work we perform a systematic study of critical correlations along Rényi defect lines in (2+1)d quantum spin models realizing quantum phase transitions described by the O(3) Wilson-Fisher universality class, using large-scale quantum Monte Carlo simulations. We present numerical evidence that, for a fixed Rényi index $n$, there exist multiple Rényi defect universality classes, with distinct critical exponents for the O(3) order parameter on the defect. These universality classes are realized by choosing microscopically different entanglement cuts in lattice models, which we classify as ordinary, special and extraordinary according to their relation to surface criticality. For the extraordinary entanglement cut, we further find evidence for a phase transition on the defect as a function of the Rényi index. Our results highlight the key role of defect universality classes in determining the universal scaling of Rényi entropy, and provide a framework for understanding the previously observed dependence of Rényi entropy scaling on microscopic lattice details.

Journal reference
Phys. Rev. Lett. 137, 156501 (2026)
DOI
10.1103/7zzj-78fv
Comments
7+4 pages; 4+6 figures

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At a quantum critical point, the universal scaling behavior of Rényi entanglement entropy is controlled by the universality class of the codimension-two Rényi (or conical) defects in the infrared theory. In this work we perform a systematic study of critical correlations along Rényi defect lines in (2+1)d quantum spin models realizing quantum phase transitions described by the O(3) Wilson-Fisher universality class, using large-scale quantum Monte Carlo simulations. We present numerical evidence that, for a fixed Rényi index nnn, there exist multiple Rényi defect universality classes, with distinct critical exponents for the O(3) order parameter on the defect. These universality classes are realized by choosing microscopically different entanglement cuts in lattice models, which we classify as ordinary, special and extraordinary according to their relation to surface criticality. For the extraordinary entanglement cut, we further find evidence for a phase transition on the defect as a function of the Rényi index. Our results highlight the key role of defect universality classes in determining the universal scaling of Rényi entropy, and provide a framework for understanding the previously observed dependence of Rényi entropy scaling on microscopic lattice details.
At a quantum critical point, bipartite entanglement entropies have universal quantities which are subleading to the ubiquitous area law. For Renyi entropies, these terms are known to be similar to the von Neumann entropy, while being much more amenable to numerical and even experimental measurement. We show here that when calculating universal properties of Renyi entropies, it is important to account for unusual corrections to scaling that arise from relevant local operators present at the conical singularity in the multisheeted Riemann surface. These corrections grow in importance with increasing Renyi index. We present studies of Renyi correlation functions in the 1+11+11+1 transverse-field Ising model (TFIM) using conformal field theory, mapping to free fermions, and series expansions, and the logarithmic entropy singularity at a corner in 2+12+12+1 for both free bosonic field theory and the TFIM, using numerical linked cluster expansions. In all numerical studies, accurate results are only obtained when unusual corrections to scaling are taken into account. In the worst case, an analysis ignoring these corrections can get qualitatively incorrect answers, such as predicting a decrease in critical exponents with the Renyi index, when they are actually increasing. We discuss a two-step extrapolation procedure that can be used to account for the unusual corrections to scaling.
We investigate the second-order Rényi entanglement entropy at the quantum critical point of a spin-1/2 antiferromagnetic Heisenberg model on a columnar dimerized square lattice. The universal constant γγγ in the area-law scaling S2(ℓ)=αℓ−γS_{2}(\ell) = α\ell - γS2​(ℓ)=αℓ−γ is found to be sensitive to the entangling surface configurations, with γsp>0γ_{\text{sp}} > 0γsp​>0 for strong-bond-cut (special) surfaces and γord<0γ_{\text{ord}} < 0γord​<0 for weak-bond-cut (ordinary) surfaces, which is attributed to the distinct conformal boundary conditions. Introducing boundary dimerization drives a renormalization group (RG) flow from the special to the ordinary boundary criticality, and the constant γγγ decreases monotonically with increasing dimerization strength, demonstrating irreversible evolution under the boundary RG flow. These results provide numerical evidence for a higher-dimensional analog of the ggg theorem, and suggest γγγ as a possible characteristic function for boundary RG flow in (2+1)(2+1)(2+1)-dimensional conformal field theory.
Computing the subleading logarithmic term in the entanglement entropy (EE) of (2+1)d quantum many-body systems remains a significant challenge, despite its central role in revealing universal information about quantum states and quantum critical points (QCPs). Building on recent algorithmic advances that enable the stable calculation of EE as an exponential observable~\cite{zhouIncremental2024,zhangIntegral2024,liaoExtracting2024}, we develop a {\it bubble basis} projector quantum Monte Carlo (QMC) algorithm to precisely and efficiently compute the universal corner of EE at QCPs in a (2+1)d square-lattice transverse-field Ising model augmented with a four-body interaction. Turning on this interaction allows us to trace an Ising critical line, reaching the tricritical point, and then a line of first-order phase transition. In (2+1)d, the tricritical point is described by the Gaussian theory, where a theoretical calculation of the corner logarithmic term in the 2nd Rényi entropy term is available~\cite{UniversalCasini2007}. Our QMC results are in quantitative agreement with this theoretical value, providing a highly nontrivial benchmark of the algorithm. Furthermore, we also study the Rényi EE at the Ising critical line and on the first-order transition line, obtaining results consistent with theoretical expectations. These findings establish the long-sought connection between the universal values of an exactly solvable limit and those of a strongly correlated regime at (2+1)d.
Continuous phase transitions exhibit richer critical phenomena on the surface than in the bulk, because distinct surface universality classes can be realized for the same bulk critical point by tuning the surface interactions. The exploration of surface critical behavior provides a window looking into the higher-dimensional boundary conformal field theories. In this work, we discover a direct special transition from the ordinary phase into an extraordinary phase on the surface of a two-dimensional (2D) quantum critical Heisenberg model by tuning the surface coupling strength. The extraordinary phase has a long-range antiferromagnetic order on the surface, in sharp contrast to the logarithmically decaying spin correlations in the 3D classical O(2) and O(3) models. The anomalous dimension of the surface antiferromagnetic order in the extraordinary phase η∥=−0.29(1)\eta_{\parallel}=-0.29(1)η∥​=−0.29(1) is independent of the surface coupling strength. The special transition point also has a new set of critical exponents, ys=0.82(4)y_{s}=0.82(4)ys​=0.82(4) and η∥=−0.304(5)\eta_{\parallel}=-0.304(5)η∥​=−0.304(5), which are distinct from the special transition of the classical O(3) model and indicates a new surface universality class of the 3D O(3) Wilson-Fisher theory.
Universal information extracted from entanglement at a critical point is commonly associated with the long-distance properties of the bulk state, while microscopic details of a lattice bipartition are expected to be washed out under coarse graining. Counterintuitively, we show directly that such microscopic information can survive in the statistical ensemble that determines the second Rényi entropy and obviously change the physics there. Using quantum Monte Carlo simulations of the square-lattice JJJ-Q3Q_3Q3​ model at the Néel--valence-bond-solid (VBS) transition, we measure magnetic and dimer order parameters in the Rényi-2 ensemble. For a standard axis-aligned bipartition, the VBS component perpendicular to the entanglement boundary remains finite in the thermodynamic extrapolation and is inequivalent to the parallel component, while the Néel order vanishes. For a 45∘45^\circ45∘ tilted bipartition, in contrast, the two VBS components remain equivalent within numerical accuracy. We interpret this contrast in terms of the Rényi twist defect: the two bipartitions have the same bulk theory but realize different microscopic defect symmetries. Our results demonstrate that lattice information invisible to the bulk continuum theory alone can remain active in the long-distance Rényi ensemble, providing a microscopic explanation for the strong bipartition dependence previously observed in entanglement studies of deconfined criticality.
We develop a nonequilibrium increment method to compute the Rényi entanglement entropy and investigate its scaling behavior at the deconfined critical (DQC) point via large-scale quantum Monte Carlo simulations. To benchmark the method, we first show that, at a conformally invariant critical point of O(3) transition, the entanglement entropy exhibits universal scaling behavior of area law with logarithmic corner corrections, and the obtained correction exponent represents the current central charge of the critical theory. Then we move on to the deconfined quantum critical point, where we still observe similar scaling behavior, but with a very different exponent. Namely, the corner correction exponent is found to be negative. Such a negative exponent is in sharp contrast with the positivity condition of the Rényi entanglement entropy, which holds for unitary conformal field theories (CFTs). Our results unambiguously reveal fundamental differences between DQC and quantum critical points described by unitary CFTs.