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arXiv:2311.01199 · Physical Review Research

Entanglement fractalization

Yao Zhou, Peng Ye

Published Nov 15, 2024• 6 citations
arXivHTMLPDF

Abstract

We numerically explore the interplay of fractal geometry and quantum entanglement by analyzing the von Neumann entropy (known as entanglement entropy) and the entanglement contour in the scaling limit. Adopting quadratic fermionic models on Sierpinski carpet, we uncover intriguing findings. For gapless ground states exhibiting a finite density of states at the chemical potential, we reveal a super-area law characterized by the presence of a logarithmic correction for area law in the scaling of entanglement entropy. This extends the well-established super-area law observed on translationally invariant Euclidean lattices where the Gioev-Klich-Widom conjecture regarding the asymptotic behavior of Toeplitz matrices holds significant influence. Furthermore, different from the fractal structure of the lattice, we observe the emergence of a self-similar and universal pattern termed an “entanglement fractal” in the entanglement contour data as we approach the scaling limit. Remarkably, this pattern bears resemblance to intricate Chinese paper-cutting designs. We provide general rules to artificially generate this fractal, offering insights into the universal scaling of entanglement entropy. Meanwhile, as the direct consequence of the entanglement fractal and beyond a single scaling behavior of entanglement contour in translation-invariant systems, we identify two distinct scaling behaviors in the entanglement contour of fractal systems. For gapped ground states, we observe that the entanglement entropy adheres to a generalized area law, with its dependence on the Hausdorff dimension of the boundary between complementary subsystems. Published by the American Physical Society 2024

DOI
10.1103/physrevresearch.6.043145

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We numerically explore the interplay of fractal geometry and quantum entanglement by analyzing the von Neumann entropy (known as entanglement entropy) and the entanglement contour in the scaling limit. Adopting quadratic fermionic models on Sierpinski carpet, we uncover intriguing findings. For gapless ground states exhibiting a finite density of states at the chemical potential, we reveal a super-area law characterized by the presence of a logarithmic correction for area law in the scaling of entanglement entropy. This extends the well-established super-area law observed on translationally invariant Euclidean lattices where the Gioev-Klich-Widom conjecture regarding the asymptotic behavior of Toeplitz matrices holds significant influence. Furthermore, different from the fractal structure of the lattice, we observe the emergence of a self-similar and universal pattern termed an “entanglement fractal” in the entanglement contour data as we approach the scaling limit. Remarkably, this pattern bears resemblance to intricate Chinese paper-cutting designs. We provide general rules to artificially generate this fractal, offering insights into the universal scaling of entanglement entropy. Meanwhile, as the direct consequence of the entanglement fractal and beyond a single scaling behavior of entanglement contour in translation-invariant systems, we identify two distinct scaling behaviors in the entanglement contour of fractal systems. For gapped ground states, we observe that the entanglement entropy adheres to a generalized area law, with its dependence on the Hausdorff dimension of the boundary between complementary subsystems. Published by the American Physical Society 2024
Entanglement entropy scaling links geometry to quantum correlations and diagnoses quantum criticality. Fractal lattices extend this link to noninteger dimensions, where boundary and bulk dimensions are independent. It remains open which dimension governs entropy scaling and how strongly gap closing can enhance it. Nonlocal occupied-state projectors and the lack of translation symmetry obstruct standard methods. We overcome these obstacles and classify free-fermion entropy scaling with bounds set by the hopping across the subsystem boundary: an upper bound from its rank and the spectral gap, and a lower bound from how much it lowers the ground-state energy. With only negative-energy states filled and this energy lowering proportional to the boundary, the entanglement entropy obeys the boundary law SX=Θ(LXdb)S_X=Θ\left(L_X^{d_{\mathrm b}}\right)SX​=Θ(LXdb​​) under a uniform gap and satisfies LXdb≲SX≲LXdbln⁡LXL_X^{d_{\mathrm b}}\lesssim S_X\lesssim L_X^{d_{\mathrm b}}\ln L_XLXdb​​≲SX​≲LXdb​​lnLX​ when the gap closes no faster than algebraically. Here LXL_XLX​ is the subsystem linear size and dbd_{\mathrm b}db​ the boundary dimension. Half occupation of the chiral zero modes gives a mixed-state entropy SXmixS_X^{\mathrm{mix}}SXmix​ with max⁡{SX,zXln⁡2}⩽SXmix⩽SX+zXln⁡2\max\left\{S_X,z_X\ln2\right\}\leqslant S_X^{\mathrm{mix}}\leqslant S_X+z_X\ln2max{SX​,zX​ln2}⩽SXmix​⩽SX​+zX​ln2, where zXz_XzX​ is the local zero-mode weight; an extensive zXz_XzX​ thus makes SXmixS_X^{\mathrm{mix}}SXmix​ follow the bulk dimension.
In the context of characterizing the structure of quantum entanglement in many-body systems, we introduce the entanglement contour, a tool to identify which real-space degrees of freedom contribute, and how much, to the entanglement of a region A with the rest of the system B . The entanglement contour provides a complementary, more refined approach to characterizing entanglement than just considering the entanglement entropy between A and B , with several concrete advantages. We illustrate this in the context of ground states and quantum quenches in fermionic quadratic systems. For instance, in a quantum critical system in D = 1 spatial dimensions, the entanglement contour allows us to determine the central charge of the underlying conformal field theory from just a single partition of the system into regions A and B (using the entanglement entropy for the same task requires considering several partitions). In D ⩾ 2 dimensions, the entanglement contour can distinguish between gapped and gapless phases that obey the same boundary law for entanglement entropy. During a local or global quantum quench, the time-dependent contour provides a detailed account of the dynamics of entanglement, including propagating entanglement waves, which offers a microscopic explanation of the behavior of the entanglement entropy as a function of time.
We study the scaling of the ground-state entanglement entropy of various free fermionic models on one-dimensional lattices, where the hopping and pairing terms decay as a power law. We seek to understand the scaling of entanglement entropy in generic models as the exponent of the power law $\ensuremath{\alpha}$ is varied. We ask if a common ${\ensuremath{\alpha}}_{c}$ exists across different systems governing the transition to area-law scaling found in local systems. We explore several examples numerically and argue that when applicable, the scaling of entanglement entropy in long-range models is constrained by predictions from the low-energy theory. In contrast, disordered models and models without a continuum limit show fractal scaling of entanglement approaching volume-law behavior as $\ensuremath{\alpha}$ approaches zero. These general features are expected to persist on turning on interactions.
Traditional holographic duality usually assumes a smooth and continuous spacetime background, where boundary entanglement strictly follows the area law. However, the real microscopic quantum gravity background may possess a non-trivial fractal structure. Within the framework of connectivity engineering, this paper investigates the regulatory mechanism of fractal networks (Sierpinski gasket) on quantum entanglement structures. Numerical results show that, unlike a one-dimensional gapped chain restricted to an extremely low area law constant, the entanglement capacity of the fractal network is significantly enhanced due to the presence of multiple branches and geometric shortcuts, approaching the upper bound of the Page curve at the bipartite symmetry point for this finite size. This topological resonance brought by self-similar hierarchical levels of the fractal demonstrates a generalized area law dominated by fractal dimension. This result proves that the fractal dimension of the underlying spacetime is the key geometric origin determining the long-range distribution of quantum information and the upper bound of entanglement capacity.
This paper provides an in-depth analysis of entanglement entropy (EE) in quantum field theory (QFT), with a particular focus on its computation using the replica trick and its applications to both conformal and non-conformal systems. Beginning with an introduction to the basics of QFT, the study explains how entanglement entropy quantifies the quantum correlations between subsystems in a pure state, represented by the von Neumann entropy of the reduced density matrix. The replica trick is employed to derive the entanglement entropy, involving path integrals over n-sheeted Riemann surfaces. The paper demonstrates the method’s utility in 1+1-dimensional conformal field theory (CFT), where the central charge governs universal properties of entropy. It further extends the analysis to massive field theories, finite systems, and topological phases, exploring how deviations from conformal symmetry affect entanglement. The work also delves into the AdS/CFT correspondence, showcasing how holographic techniques facilitate entanglement entropy calculations in higher-dimensional systems via the Ryu-Takayanagi formula. In addition, the study investigates entanglement entropy in quantum lattice systems, considering the effects of spatial and thermal fluctuations. Numerical methods are used to compute the scaling of entanglement entropy with interval length for varying system masses and temperatures, highlighting the influence of these parameters on entropy behavior. Fractal lattice structures are explored to uncover unique entropy scaling laws and self-similar entanglement patterns. This study advances the under-standing of EE in diverse quantum systems and establishes a foundation for exploring its role in quantum criticality and fractal geometries. The paper also explores multiscale entanglement entropy (MSE), offering a comprehensive framework for future research in quantum critical phenomena and non-equilibrium systems, shedding light on the complex relationship between quantum entanglement and thermal effects. This work serves as a foundational reference for future theoretical and computational studies in quantum entanglement
Random fractal lattices provide a geometrically disordered setting in which quantum correlations can be shaped by noninteger dimensionality rather than onsite randomness. We investigate the entanglement properties of noninteracting fermions on random fractal lattices generated by a stochastic growth algorithm. By varying the growth parameter and adding missing links with probability ppp, we tune the Hausdorff and spectral dimensions while keeping the system free of onsite disorder. For ground states at different fillings, we compute the bipartite entanglement entropy of subregions defined by graph distance and analyze its scaling with subsystem size. Over a broad parameter range, we find robust power-law behavior governed primarily by the Hausdorff dimension, consistent with a generalized area law and without the logarithmic enhancement familiar from Euclidean free fermions. We also study entanglement growth following a global quench from an uncorrelated checkerboard state and uncover an asymptotic scaling collapse in which the subsystem-size dependence is governed by the Hausdorff dimension, while the temporal evolution is governed by the spectral dimension. The resulting dynamics are logarithmically slow over an extended intermediate-time window. These results show that geometric randomness alone can generate both nontrivial ground-state entanglement structure and slow quantum-information spreading in free-fermion systems.