Entanglement corner dependence in two-dimensional systems: A tensor network perspective

TL;DR

The paper addresses how universal corner-dependent entanglement terms from continuum 2D quantum field theories arise when representing states with discrete tensor networks, specifically iPEPS. It develops a counting approach that relates the corner term to the number of crossed bonds and lattice corners and shows that averaging over lattice orientations recovers a continuum-like $b_A(\theta)$ with $f(\theta)=\alpha+\beta(\pi-\theta)\cot(\theta)$. The main result is a corner contribution $b_A(\theta)= -(1 + (\pi-\theta)\cot(\theta)) \cdot F$, where $F$ is independent of $|A|$ and $| abla A|$ for gapped systems, with possible logarithmic scaling for critical systems; gauge-invariant systems add an extra lattice-corner term. This work connects continuum entanglement corner laws to discrete tensor-network geometry and emphasizes the role of isotropy averaging in discretizations, offering a framework to study entanglement across dimensions and extensions to gauge theories.

Abstract

In continuous quantum field theories, the entanglement entropy of a subsystem with sharp corners on its boundary exhibits a universal corner-dependent contribution. We study this contribution through the lens of lattice discretization, and demonstrate that this corner dependence emerges naturally from the geometric structure of infinite projected entangled pair states (iPEPS) on discrete lattices. Using a rigorous counting argument, we show that the bond dimension of an iPEPS representation exhibits a corner-dependent term that matches the predicted term in gapped continuous systems. Crucially, we find that this correspondence only emerges when averaging over all possible lattice orientations and origin positions, revealing a fundamental requirement for properly discretizing continuous systems. Our results provide a geometric understanding of entanglement corner laws and establish a direct connection between continuum field theory predictions and the structure of discrete tensor network representations. We extend our analysis to gauge-invariant systems, where lattice corners crossed by the bipartition boundary contribute an additional corner-dependent term. These findings offer new insights into the relationship between entanglement in continuous and discrete quantum systems.

Figures (6)

• Figure 1: (a) A TN is composed of tensors, each corresponding to a lattice site. The leg colored in pink of each tensor is referred to as the physical leg, and corresponds to an index varying over a basis of the Hilbert space of a single lattice site, while legs colored in blue are referred to as the virtual legs, corresponding to summed-over indices corresponding to a virtual Hilbert space with dimension $\chi$, referred to as the bond dimension. A two- (or higher) dimensional TN is referred to as a PEPS, and a one-dimensional TN is referred to as an MPS. (b) The number of nonzero RDM eigenvalues is bounded from above by the total bond dimension across the boundary between the two subsystems, here depicted in yellow. (c) The bond indices on the boundary may be merged into a single index, whose dimension is the product of all boundary bond dimensions. This dimension bounds the rank of the Schmidt decomposition, Eq. (\ref{['eq:schmidt']}).
• Figure 2: (a) The studied bipartition between a subsystem $A$ and its complement, where the bipartition boundary contains a sharp corner $\theta$. (b) Following Ref. corner_general_Estienne_2022, the results are obtained for a circular system of radius $r$ with subsystem $A$ being a circular section with central angle $\theta$. The results are assumed to hold for an infinite system, that is, $r\rightarrow\infty$.
• Figure 3: (a) The gray grid indicates the discretized lattice, with the system's degrees of freedom residing on the lattice sites. The virtual legs crossed by the bipartition are accented in pink, their total number is denoted by $n_\text{legs}$. (b) Lattice corners crossed by the bipartition are accented in pink, their total number is denoted by $n_\text{corners}$. (c) In order to conserve the rotational invariance of the system, we average $n_\text{legs},n_\text{corners}$ over different orientations of subsystem $A$. The orientation angle is denoted throughout the paper by $\varphi$.
• Figure 4: (a) The number of virtual legs $n_\text{legs}$ crossed by the boundary of $A$ when $\theta\rightarrow 0$ . Legs crossed by the boundary are colored in pink, while legs"skipped" by the boundary are colored in green. The number of legs skipped depends on the orientation and vertex position. (b) When $\theta > 2\tan^{-1}(1/2)$, the only legs that may be "skipped" are the ones closest to the apex, as depicted in the illustration. For a given orientation $\varphi$, the apex positions for which the leg will be "skipped" are colored in blue. In the \ref{['sec:appendix_large_theta']} we integrate over this blue region and over $\varphi$ to obtain $p_\text{skip}(\theta)$.
• Figure 5: (a) Numerical counting of $n_\text{legs}$ for various values of $\theta$, fitted to the expected $\theta$-dependence in Eq. (\ref{['eq:fit_function']}), for a system with radius $r=4$. (b) The extracted coefficient $\beta$ of Eq. (\ref{['eq:fit_function']}) as a function of $r$, is found to remain relatively independent of $r$ (up to a relative error of $\sim10^{-2}$ ), that is, independent of $|\partial A|$, both for the number of virtual legs and for the number of lattice corners crossed by the bipartition boundary. (c) Fitting quality of $n_\text{legs},n_\text{corners}$ to the expected $\theta$-dependence as in Eq. (\ref{['eq:fit_function']}). In all graphs, the results were obtained by averaging over different orientations of $A$ with resolution $2\pi\cdot10^{-2}$ and different apex positions with resolution $10^{-1}a$ in both dimensions.