Quantum Entanglement of Anyonic Charges and Emergent Spacetime Geometry

TL;DR

This paper argues that mutual entanglement between anyonic edge charges in disordered zigzag graphene nanoribbons can give rise to an emergent AdS-like geometry, even without conformal symmetry. Using a Hartree-Fock Hubbard model, it analyzes how disorder-induced instantons generate fractional charges $e/2$ on opposite edges and how their non-local entanglement encodes bulk geometric structure. Mutual information between edge regions reveals scale-dependent, long-range correlations that map to geodesic distances in a hyperbolic-like space, which the authors implement via a soft-wall embedding with a negative-curvature metric. The resulting framework supports a holographic interpretation of edge entanglement as geometry, providing a concrete realization of the idea that quantum correlations can glue together emergent spacetime in non-CFT condensed-matter systems.

Abstract

Intrinsically topologically ordered phases can host anyons. Here, we take the view that entanglement between anyons can give rise to an emergent geometry resembling Anti-de Sitter (AdS) space. We analyze the entanglement structure of fractionalized anyons using mutual information and interpret the results within this emergent geometric framework. As a concrete example, we consider pairs of $e/2$-charged semions that arise from instanton configurations in a disordered zigzag graphene nanoribbon. These fractional charges, located on opposite zigzag edges, show long-range quantum entanglement despite being spatially separated. We analyze the scale dependence of their entanglement and embed the ribbon into an AdS-like bulk geometry. In this setup, the entanglement structure defines minimal surfaces in the bulk, providing a geometric view of the edge correlations. This gives a holographic picture of fractionalized degrees of freedom in quasi-one-dimensional systems and shows how quantum entanglement can generate emergent geometry even without conformal symmetry.

Figures (5)

• Figure 1: In the disordered ground state of a zigzag ribbon, fractional charge pairs are localized at opposite edges. Yellow dots schematically represent the site-resolved probability density of a fractional charge. It decays rapidly from the edges. While each $e/2$ fractional charge obeys semionic statistics, the composite object formed by a pair, highlighted by the red curve, obeys Fermi statistics Yang2025. The number of composite objects scales linearly with the ribbon length.
• Figure 2: (a) Site index for a ZGNR with length $L$ and width $W$. Each carbon site is labeled by $m=(x, y)$, where $x$ denotes the row index and $y$ the column index. Note that each site labeled by $x$ may correspond to different physical positions across horizontal carbon lines, due to their relative displacement within the graphene lattice. Red (blue) sites are labeled by chirality A (B). The opposite zigzag edge sites that form upper and lower boundaries have different chiralities. (b) Left: Each panel shows $\delta n_{m, \uparrow}$, the change in spin-up occupation at site $m$ between disordered and clean cases, plotted along two opposite horizontal carbon chains. Right: Pairs of opposite chains are shown as horizontal dashed lines. The vertical green line corresponds to a vertical dotted line in the left panels; that is, all sites along the green line share the same horizontal index $x$. Fractional charges appear near the two ends of the green line, where $\delta n_{m, \uparrow}$ changes sharply. The Hamiltonian's parameters are $(U, \Gamma) = (2t, 0.1t)$ and the ZGNR geometry is $(L, W) = (120, 8)$.
• Figure 3: (a) Mutual information between the two yellow boxes is plotted as a function of their position $x$ along the nanoribbon, measured from the left end. The size of each box is $(\ell, w) = (5, 1)$, where $\ell$ is the length (number of sites along a zigzag edge) and $w$ is the width (number of carbon rows). (b) Mutual information as a function of the separation $\Delta y$ between two boxes, each with $(\ell, w) = (5, 1)$. The maximal separation between two boxes is $\Delta y = W = 16$. Results are presented for two scenarios: when each box contains a fractional charge, and when they do not. (c) Mutual information between two boxes, each with $(\ell, w) = (5, w)$, is plotted as a function of the width $w$. Here, the boxes are positioned to enclose a localized fractional charge at $x = 120$. All three panels share the same parameters and disorder realization, with $(U, \Gamma) = (t, 0.01t)$ and $(L, W) = (300, 16)$. The horizontal dashed lines in lower row figures indicate zigzag edges.
• Figure 4: Schematic illustration of the multiscale mutual entanglement pattern induced by a pair of fractional charges located on opposite zigzag edges. Grey (white) dots represent carbon sites on the A (B) sublattice. Each closed curve defines a region used to compute the mutual information. Each curve color corresponds to a scale of entanglement (e.g., blue = short range, red = longer range). When two closed curves of the same color appear near opposite edges of the ribbon, they enclose regions that are strongly entangled with each other. For instance, a pair of blue circles on opposing sides indicates entanglement at the shortest scale. The scale increases with the color gradient, progressing from blue to red. Only the strongest mutual information links are shown. Color legend: blue indicates 1 layer, green indicates 2 layers, and red indicates 3 layers. Links involving more than 3 layers are omitted for clarity. Also, pairs of opposite sites with weak mutual information are not enclosed by curves.
• Figure 5: (a) "Hyperbolic geometry" of a ZGNR with width $W = 32$ and characteristic curvature scale $y_0 = 9$. A and B carbon sites are represented by red and blue colors, respectively. The ribbon comprises two interpenetrating triangular sublattices, shown in red and blue. The red and blue zigzag edge sites lie directly opposite each other across the ribbon. (b) The coarse-grained (disorder-averaged) mutual information $\log M(y)$ is computed with a transverse width $w = 1$, for typical separations between fractional charges $\ell_{\text{frac}} = 10,\,15$. The fitted slope yields an inverse decay length $\alpha = 0.09$ (see Eq. (\ref{['eq:MI_decay']})), with the curvature scale parameter fixed at $y_0 = 10$. Parameters for Hamiltonian are $(U, \Gamma) = (t, 0.1 t)$. The geometries of ribbon for $\ell_\text{frac}= 10$ and $\ell_\text{frac}= 15$ are $(L, W) = (200, 48)$ and $(L, W) = (150, 72)$, respectively. The results show little dependence on $\ell_{\text{frac}}$: the vertical separation between the two $\log M(y)$ curves corresponds to only a tiny relative difference, $\Delta M(y) \sim 10^{-3}$.