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arXiv:2512.00382 · Reports on Progress in Physics

Precise computation of universal corner entanglement entropy at 2+1 dimension: From Ising to Gaussian quantum critical points

Ben Lee-Yeung Ngai, Justin Tim-Lok Chau, Junchen Rong, Yuan Da Liao, Zi Yang Meng

cond-mat.str-elcond-mat.stat-mechquant-phSubmitted Nov 29, 2025• Announced Dec 2, 2025• 0 citations
v1 · Nov 29, 2025v2 · Jan 4, 2026v3 · Jun 22, 2026
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Abstract

Computing the subleading logarithmic term in the entanglement entropy (EE) of (2+1)d quantum many-body systems remains a significant challenge, despite its critical role in revealing universal information about quantum states and quantum critical points (QCPs). Capitalizing on recent algorithmic advances that enable the stable calculation of EE as an exponential observable, we develop a {\it bubble basis} projector quantum Monte Carlo (QMC) algorithm to precisely and efficiently compute the universal corner EE along a parameter trajectory connecting the Ising and Gaussian QCPs; where in the latter case there exists an analytic value to benchmark with. Our approach eliminates the dominant area-law term directly within the QMC sampling, thereby promoting the coveted subleading logarithmic term to the leading contribution. We apply this technique to calculate the EE in a (2+1)d square-lattice transverse-field Ising model augmented with a four-body interaction, tracing a path from the Ising to the Gaussian QCPs and then to the first-order transition. The consistent values obtained for the universal corner EE along the path validate the reliability and precision of our approach for extracting challenging entanglement properties. These results 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.

Journal reference
Rep. Prog. Phys. 89 068006 (2026)
DOI
10.1088/1361-6633/ae78fe
Comments
11+5 pages, 8+5 figures

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Computing the subleading logarithmic term in the entanglement entropy (EE) of (2+1)d quantum many-body systems remains a significant challenge, despite its critical role in revealing universal information about quantum states and quantum critical points (QCPs). Capitalizing on recent algorithmic advances that enable the stable calculation of EE as an exponential observable, we develop a {\it bubble basis} projector quantum Monte Carlo (QMC) algorithm to precisely and efficiently compute the universal corner EE along a parameter trajectory connecting the Ising and Gaussian QCPs; where in the latter case there exists an analytic value to benchmark with. Our approach eliminates the dominant area-law term directly within the QMC sampling, thereby promoting the coveted subleading logarithmic term to the leading contribution. We apply this technique to calculate the EE in a (2+1)d square-lattice transverse-field Ising model augmented with a four-body interaction, tracing a path from the Ising to the Gaussian QCPs and then to the first-order transition. The consistent values obtained for the universal corner EE along the path validate the reliability and precision of our approach for extracting challenging entanglement properties. These results 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.
The subleading corner logarithmic corrections in entanglement entropy (EE) are crucial for revealing the universal characteristics of quantum critical points (QCPs), but they are challenging to detect. Motivated by recent developments in the stable computation of EE in (2+12+12+1)-dimensional [(2+12+12+1)D] quantum many-body systems, we have developed a method for directly measuring the corner contribution in EE with lower computational cost. The cornerstone of our approach is to measure the subtracted corner entanglement entropy (SCEE), defined as the difference between the EEs of subregions with the same boundary length for smooth and cornered boundaries during a sign-problem-free quantum Monte Carlo simulation. Our improved method inherently eliminates not only the area-law term of EE but also the subleading log corrections arising from Goldstone modes, leaving the universal corner contribution as the leading term of the SCEE with greatly improved data quality. Utilizing this advanced approach, we calculate the SCEE of the bilayer Heisenberg model on both square and honeycomb lattices across their (2+12+12+1)D OOO(3) QCPs with different opening angles on the entanglement boundary and obtain accurate values of the corresponding universal corner log coefficients. These findings will encourage further theoretical investigations to access controlled universal information for interacting conformal field theories at (2+12+12+1) dimensions.
Although the leading-order scaling of entanglement entropy is non-universal at a quantum critical point (QCP), sub-leading scaling can contain universal behaviour. Such universal quantities are commonly studied in non-interacting field theories, however it typically requires numerical calculation to access them in interacting theories. In this paper, we use large-scale T = 0 quantum Monte Carlo simulations to examine in detail the second Rényi entropy of entangled regions at the QCP in the transverse-field Ising model in 2 + 1 space–time dimensions—a fixed point for which there is no exact result for the scaling of entanglement entropy. We calculate a universal coefficient of a vertex-induced logarithmic scaling for a polygonal entangled subregion, and compare the result to interacting and non-interacting theories. We also examine the shape-dependence of the Rényi entropy for finite-size toroidal lattices divided into two entangled cylinders by smooth boundaries. Remarkably, we find that the dependence on cylinder length follows a shape-dependent function calculated previously by Stephan et al (2013 New J. Phys. 15 015004) at the QCP corresponding to the 2 + 1 dimensional quantum Lifshitz free scalar field theory. The quality of the fit of our data to this scaling function, as well as the apparent cutoff-independent coefficient that results, presents tantalizing evidence that this function may reflect universal behaviour across these and other very disparate QCPs in 2 + 1 dimensional systems.
In a D=2+1D=2+1D=2+1 quantum critical system, the entanglement entropy across a boundary with a corner contains a subleading logarithmic scaling term with a universal coefficient. It has been conjectured that this coefficient is, to leading order, proportional to the number of field components NNN in the associated O(N)\text{O}(N)O(N) continuum ${\ensuremath{\phi}}^{4}$ field theory. Using density matrix renormalization group calculations combined with the powerful numerical linked cluster expansion technique, we confirm this scenario for the O(2)\text{O}(2)O(2) Wilson-Fisher fixed point in a striking way, through direct calculation at the quantum critical points of two very different microscopic models. The value of this corner coefficient is, to within our numerical precision, twice the coefficient of the Ising fixed point. Our results add to the growing body of evidence that this universal term in the R\'enyi entanglement entropy reflects the number of low-energy degrees of freedom in a system, even for strongly interacting theories.
A quantum critical (QC) fluid exhibits universal subleading corrections to the area law of its entanglement entropies. In two dimensions, when the partition involves a corner of angle $\ensuremath{\theta}$, the subleading term is logarithmic with coefficient ${a}_{\ensuremath{\alpha}}(\ensuremath{\theta})$ for the $\ensuremath{\alpha}$-R\'enyi entropy. In the smooth limit $\ensuremath{\theta}\ensuremath{\rightarrow}\ensuremath{\pi},{a}_{1}(\ensuremath{\theta})$ yields the central charge of the stress tensor when the QC point is described by a conformal field theory (CFT). For general R\'enyi indices and angles, ${a}_{\ensuremath{\alpha}}(\ensuremath{\theta})$ is richer and few general results exist. We study ${a}_{\ensuremath{\alpha}}(\ensuremath{\theta})$ focusing on two benchmark CFTs, the free Dirac fermion and boson. We perform numerical lattice calculations to obtain high precision results in $\ensuremath{\theta},\ensuremath{\alpha}$ regimes hitherto unexplored. We derive field theory estimates for ${a}_{\ensuremath{\alpha}}(\ensuremath{\theta})$, including exact results, and demonstrate an excellent quantitative match with our numerical calculations. We also develop and test strong lower bounds, which apply to both free and interacting QC systems. Finally, we comment on the near collapse of ${a}_{\ensuremath{\alpha}}(\ensuremath{\theta})$ for various theories, including interacting O(N)O(N)O(N) models.
The entanglement entropy for a quantum critical system across a boundary with a corner exhibits a subleading logarithmic scaling term with a scale-invariant coefficient. Using a Numerical Linked Cluster Expansion, we calculate this universal quantity for a square-lattice bilayer Heisenberg model at its quantum critical point. We find, for this 2 + 1 dimensional O (3) universality class, that it is thrice the value calculated previously for the Ising universality class. This relation gives substantial evidence that this coefficient provides a measure of the number of degrees of freedom of the theory, analogous to the central charge in a 1 + 1 dimensional conformal field theory.
The entanglement entropy in many gapless quantum systems receives a contribution from the corners in the entangling surface in 2+1d, which is characterized by a universal function $a(\ensuremath{\theta})$ depending on the opening angle $\ensuremath{\theta}$, and contains pertinent low energy information. For conformal field theories (CFTs), the leading expansion coefficient in the smooth limit $\ensuremath{\theta}\ensuremath{\rightarrow}\ensuremath{\pi}$ yields the stress tensor two-point function coefficient CT{C}_{T}CT​. Little is known about $a(\ensuremath{\theta})$ beyond that limit. Here, we show that the next term in the smooth limit expansion contains information beyond the two- and three-point correlators of the stress tensor. We conjecture that it encodes four-point data, making it much richer. Further, we establish strong constraints on this and higher-order smooth-limit coefficients. We also show that $a(\ensuremath{\theta})$ is lower-bounded by a nontrivial function multiplied by the central charge CT{C}_{T}CT​, e.g., $a(\ensuremath{\pi}/2)\ensuremath{\ge}({\ensuremath{\pi}}^{2}ln2){C}_{T}/6$. This bound for 90-degree corners is nearly saturated by all known results, including recent numerics for the interacting Wilson-Fisher quantum critical points (QCPs). A bound is also given for the R\'enyi entropies. We illustrate our findings using O(N)\text{O}(N)O(N) QCPs, free boson and Dirac fermion CFTs, strongly coupled holographic ones, and other models. Exact results are also given for Lifshitz quantum critical points, and for conical singularities in 3+1d.