Discrete holography and density of states in the crossover from hyperbolic to Euclidean lattices

TL;DR

The paper addresses how holographic boundary physics and bulk spectral properties evolve when Euclidean defects are introduced into hyperbolic lattices. It develops defected hyperbolic-to-Euclidean flakes via tile-by-tile inflation and analyzes a quadratic tight-binding model to extract boundary two-point functions and bulk density of states across the crossover parameter $\rho$. The key finding is that boundary observables preserve the hyperbolic scaling up to $\rho \lesssim 0.9$, while the bulk DOS gradually loses hyperbolic features, with genuine Euclidean characteristics emerging only near $\rho \sim 0.95$; this implies boundary physics can be realized with defective lattices, reducing resource requirements for experiments and simulations. Overall, the work demonstrates that essential boundary physics associated with holographic duality in discrete systems is robust to substantial Euclidean defect fractions, enabling scalable experimental and numerical exploration of AdS/CFT-like phenomena on non-ideal lattices.

Abstract

We study tight-binding models in the crossover from hyperbolic to Euclidean lattices, realized through the successive insertion of Euclidean defects into hyperbolic lattices. We analyze how the holographic two-point boundary correlation function and bulk density of states evolve as defects are gradually introduced. We find that bulk properties are strongly affected by the presence of Euclidean defects, whereas boundary observables remain remarkably robust even at high defect fractions. This robustness indicates that essential features of boundary physics on hyperbolic lattices, which capture aspects of AdS/CFT-like dualities in discrete systems, can be reproduced both experimentally and numerically without requiring perfectly hyperbolic lattices, thereby reducing the system size needed for implementation.

Figures (8)

• Figure 1: We show some realizations of the hyperbolic-to-Euclidean lattice crossover, where hexagonal faces are randomly inserted as defects into a hyperbolic $\{7,3\}$ graph, eventually producing a Euclidean $\{6,3\}$ honeycomb lattice. The fraction of hexagons is denoted by $\rho$. Note that the properties of tight-binding models on these graphs are independent of the physical coordinates of the vertices in the plane, which may thus be chosen arbitrarily. The connectivity of the graph, on the other hand, has crucial influence on the physical properties.
• Figure 2: Construction of a defective $\{7,3\}$-flake with three concentric rings through tile-by-tile inflation. The graph features three defective tiles, one pentagon and two hexagons on the second ring, as given in the example in Eq. \ref{['eq:ConstructionGraph']}. The purple vertices labelled "$\hbox{u}$" correspond to sites pointing "outward" or "up", while the green vertices labelled "$\hbox{d}$" correspond to sites pointing "inward" or "down". The arrow indicates that the rings are created counterclockwise by starting on the first ring, then proceeding to the second ring, and finally to the third ring. The dark red and the orange envelopes indicate that each entry "$\hbox{u}$" of the $(l-1)$th ring has been replaced according to Eq. \ref{['eq:uuRule']} and Eq. \ref{['eq:udRule']}, respectively, to create the $l$th ring
• Figure 3: a) Graph $\mathcal{G}$ divided into bulk, $\mathring{\mathcal{G}}$, depicted by blue vertices and boundary, $\partial \mathcal{G}$, depicted by orange vertices. b) The graph distance $D_{ab}$ between two points $a$ and $b$ on the boundary is indicated by a blue line, while the boundary distance $d_{ab}$ between the same two points is indicated by an orange line. c) The large orange vertices constitute boundary sites $a$ with coordination number $q=3$, i.e., the down-sites on the boundary, while the large blue vertices constitute the corresponding bulk sites $i_0(a)$ that connect to them, i.e., the up-sites of the last interior ring.
• Figure 4: Boundary distance $d_{ab}$ as a function of the graph distance $D_{ab}$ in the hyperbolic-to-Euclidean lattice crossover ${\{7,3\}\to \{6,3\}}$, shown for several values of $\rho$. For $\rho<1$ we apply the fit from Eq. \ref{['holo5']}, while for $\rho =1.00$, we apply the fit from Eq. \ref{['holo4']}. The inset for $\rho =1.00$ displays the data on a linear scale, highlighting the linear Euclidean scaling behavior. The error bars indicate the standard deviation of the mean boundary distance computed for each graph distance. The deviations result from the discreteness of the flakes and their rugged boundaries. The specific colors used for various $\rho$ in this plot are also employed in the remaining figures of the work.
• Figure 5: Boundary two-point correlation functions $\langle\mathcal{O}_a \mathcal{O}_b \rangle$ in the hyperbolic-to-Euclidean lattice crossover $\{7,3\}\to \{6,3\}$ as a function of graph distance $D_{ab}$ (upper row) and boundary distance $d_{ab}$ (lower row), shown for several values of $\rho$. Note that the upper row are log-linear plots, whereas the lower row are log-log plots. The color gradient for each value of $\rho$ indicates the change of the bulk mass $m^2\ell^2$: darker tones correspond to smaller values of the bulk mass $m^2\ell^2$. The bulk masses range from $0.00$ to $1.00$ in increments of $0.20$ from top to bottom. Dashed lines indicate fits according to Eq. \ref{['eq:GabDgraph']} in the upper row, and according to Eq. \ref{['eq:GabDbound']} in the lower row.