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arXiv:2601.12365

Universal and non-universal contributions of entanglement under different bipartitions

Zhe Wang, Chunhao Guo, Bin-Bin Mao, Zheng Yan

Published Jan 18, 2026• 0 citations
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Abstract

Entanglement entropy (EE) is a fundamental probe of quantum phases and critical phenomena, which was thought to reflect only bulk universality for a long time. Very recently, people realized that the microscopic geometry of the entanglement cut can induce distinct entanglement-edge modes, whose coupling to bulk critical fluctuations may alter the scaling of the EE. However, this perception is very qualitative and lacks quantitative consideration. Here, we investigate this problem through high-precision quantum Monte Carlo simulations combined with the analysis of scaling theory to build a quantitative understanding. By considering three distinct bipartitions corresponding to three surface criticality types, we reveal a striking dependence of the constant term γ on the microscopic cut at the quantum critical point. Notably, cuts that generate extra gapless edge modes yield a sign reversal in γ compared to those producing gapped edges. We explain this behavior via a modified scaling form that incorporates contributions from both bulk and surface critical modes. Furthermore, we demonstrate that the derivative of EE robustly extracts the bulk critical point and exponent ν regardless of the cut geometry, providing a reliable diagnostic of bulk universality in the presence of strong surface effects. Our work for the first time establishes a direct quantitative connection between surface criticality and entanglement scaling, challenging the conventional view that EE solely reflects bulk properties and offering a refined framework for interpreting entanglement in systems with boundary-sensitive criticality.

DOI
10.48550/arxiv.2601.12365

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Entanglement entropy (EE) is a fundamental probe of quantum phases and critical phenomena, which was thought to reflect only bulk universality for a long time. Very recently, people realized that the microscopic geometry of the entanglement cut can induce distinct entanglement-edge modes, whose coupling to bulk critical fluctuations may alter the scaling of the EE. However, this perception is very qualitative and lacks quantitative consideration. Here, we investigate this problem through high-precision quantum Monte Carlo simulations combined with the analysis of scaling theory to build a quantitative understanding. By considering three distinct bipartitions corresponding to three surface criticality types, we reveal a striking dependence of the constant term γ on the microscopic cut at the quantum critical point. Notably, cuts that generate extra gapless edge modes yield a sign reversal in γ compared to those producing gapped edges. We explain this behavior via a modified scaling form that incorporates contributions from both bulk and surface critical modes. Furthermore, we demonstrate that the derivative of EE robustly extracts the bulk critical point and exponent ν regardless of the cut geometry, providing a reliable diagnostic of bulk universality in the presence of strong surface effects. Our work for the first time establishes a direct quantitative connection between surface criticality and entanglement scaling, challenging the conventional view that EE solely reflects bulk properties and offering a refined framework for interpreting entanglement in systems with boundary-sensitive criticality.
Recent works on the scaling behaviors of entanglement entropy at the SO(5) deconfined quantum critical point (DQCP) sparked a huge controversy. Different bipartitions gave out totally different conclusions for whether the DQCP is consistent with a unitary conformal field theory. In this Letter, we connect two previously disconnected fields-the many-body entanglement and the surface criticality-to reveal the behaviors of entanglement entropy in various bipartite scenarios, and point out that only the ordinary bipartition purely reflects the criticality of the bulk; otherwise, the extra gapless edge mode will also contribute to the entanglement. We have demonstrated that the correspondence between the entanglement spectrum and the edge energy spectrum still approximately persists even at a bulk-gapless point, thereby influencing the behavior of entanglement entropy. Our results establish that boundary conditions induced by the cut are decisive for entanglement-based probes and provide practical protocols to separate bulk from boundary contributions.
Generally speaking, entanglement entropy (EE) between two subregions of a gapped quantum many-body state is proportional to the area/length of their interface due to the short-range quantum correlation. However, the so-called area law is violated logarithmically in a quantum critical phase. Moreover, the subleading correction exists in long-range entangled topological phases. It is referred to as topological EE which is related to the quantum dimension of the collective excitation in the bulk. Furthermore, if a non-smooth sharp angle is in the presence of the subsystem boundary, a universal angle dependent geometric contribution is expected to appear in the subleading correction. In this work, we simultaneously explore the geometric and edge contributions in the integer quantum Hall (IQH) state and its edge reconstruction in a unified bipartite method. Their scaling is found to be consistent with conformal field theory (CFT) predictions and recent results of particle number fluctuation calculations.
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.
The entanglement entropy is a unique probe to reveal universal features of strongly interacting many-body systems. In two or more dimensions these features are subtle, and detecting them numerically requires extreme precision, a notoriously difficult task. This is especially challenging in models of interacting fermions, where many such universal features have yet to be observed. In this Letter we tackle this challenge by introducing a new method to compute the Rényi entanglement entropy in auxiliary-field quantum Monte Carlo simulations, where we treat the entangling region itself as a stochastic variable. We demonstrate the efficiency of this method by extracting, for the first time, universal subleading logarithmic terms in a two-dimensional model of interacting fermions, focusing on the half-filled honeycomb Hubbard model at T=0. We detect the universal corner contribution due to gapless fermions throughout the Dirac semi-metal phase and at the Gross-Neveu-Yukawa critical point, where the latter shows a pronounced enhancement depending on the type of entangling cut. Finally, we observe the universal Goldstone mode contribution in the antiferromagnetic Mott insulating phase.
The fermion sign problem is often viewed as a sheer inconvenience that plagues numerical studies of strongly interacting electron systems. Only recently has it been suggested that fermion signs are fundamental for the universal behavior of critical metallic systems and crucially enhance their degree of quantum entanglement. In this work we explore potential connections between emergent scale invariance of fermion sign structures and scaling properties of bipartite entanglement entropies. Our analysis is based on a wave-function Ansatz that incorporates collective, long-range backflow correlations into fermionic Slater determinants. Such wave functions mimic the collapse of a Fermi liquid at a quantum critical point. Their nodal surfaces, a representation of the fermion sign structure in many-particle configurations space, show fractal behavior up to a length scale $\ensuremath{\xi}$ that diverges at a critical backflow strength. We show that the Hausdorff dimension of the fractal nodal surface depends on $\ensuremath{\xi}$, the number of fermions and the exponent of the backflow. For the same wave functions we numerically calculate the second R\'enyi entanglement entropy S2{S}_{2}S2​. Our results show a crossover from volume scaling, ${S}_{2}\ensuremath{\sim}{\ensuremath{\ell}}^{\ensuremath{\theta}}$ $(\ensuremath{\theta}=2$ in d=2d=2d=2 dimensions), to the characteristic Fermi-liquid behavior ${S}_{2}\ensuremath{\sim}\ensuremath{\ell}ln\ensuremath{\ell}$ on scales larger than $\ensuremath{\xi}$. We find that volume scaling of the entanglement entropy is a robust feature of critical backflow fermions, independent of the backflow exponent and hence the fractal dimension of the scale invariant sign structure.
The interplay of the constituents of interacting many-body systems may reveal emergent properties on the macroscopic scale which are not inherent to the individual constituents. These properties are expressed in macroscopic observables \ndescribing the state — denoted as the phase of the system. Continuous phase transition between phases are generically manifested in critical behavior, for example, a divergence of a macroscopic observable. The identification of the present phase of a system and the classification of critical phenomena into universality classes are exciting challenges of condensed-matter physics. \nIn this thesis, we use entanglement entropies as macroscopic quantities for the characterization of phases of quantum matter and critical theories. For groundstates of quantum many-body systems the entanglement entropy is a measure of \nthe amount of entanglement between two subsystems. The generic dependence of the entanglement entropy on the size and shape of the subsystems is contained in the well-known boundary law — stating a scaling of the entanglement entropy with the boundary between the two subsystems. We numerically investigate how the coefficient of this dependence reflects quantum phase transitions in simple spin-half bilayer models. \nSubleading terms to the boundary law such as a logarithmic contribution provide universal numbers for the criticality of field theories. We examine free and interacting theories from an entanglement entropy perspective in order to assess the role of the coefficient of the logarithmic correction induced by corners in the subsystems. Beyond its universality, this coefficient also quantifies degrees of freedom of low-lying excitations in the conformal field theory describing a critical point. \nA constant contribution to the boundary law indicates the presence of so-called topological order in the ground state of a many-body system. This extremely useful property can also be identified in classical counterparts of entanglement entropy which we study at the example of various toric code models. To this endeavor, we have designed Monte Carlo techniques which allow for an efficient numerical computation of the constant contribution. In particular, we analyze via entanglement entropies under which conditions remnants of topological order are present in the quantum system at finite temperature and at perturbations from a magnetic field. The major motivation behind this effort is to use topological order for the robust storage of a quantum information — a basic need for the construction of quantum \ncomputers.