Universal and non-universal contributions of entanglement under different bipartitions

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.

Figures (3)

• Figure 1: Columnar and staggered dimerized spin-1/2 Heisenberg models on the square lattice, with couplings $J$ and $J_D > J$ on alternating bonds. (a,d) Entanglement/physical boundary cuts weak bonds (CD-W). (b,e) Cuts strong bonds (CD-S). (c,f) Cuts staggered bonds in the staggered model (SD). The system size is $L \times L$; subsystem $A$ is a $(L/2) \times L$ cylinder, separated from environment $B$ by dashed lines. Physical boundaries (open circles) denote true open edges.
• Figure 2: Second Rényi EE $S^{(2)}$ under three bipartition schemes versus system size $L$, at the QCP $J_c$ (a) or in the gapped dimer phase at $J_c/2$ (b). The fitting results are listed in Table \ref{['ext1']}.
• Figure 3: The derivative of the second Rényi EE and the scaling function $\tilde{S}_{0}'(x)$ obtained from the data collapse analysis of the DEE near the transition. (a,d) Data of the columnar model with entanglement boundary cuts weak bonds (CD-W). (b,e) Data of the columnar model with entanglement boundary cuts strong bonds (CD-S). For columnar model $J_{D}=1.0$ is fixed, while $J$ is tuned around the QCP $J_{c}=0.52337(3)$. (c,f) Data of the staggered model with entanglement boundary cuts staggered bonds (SD). For staggered model, $J_{D}=1.0$ is fixed, while $J$ is tuned around the QCP $J_{c}=0.39692(1)$.