Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Failure of the Goldstone Theorem for Vector Fields and Boundary-Mode Proliferation in Hyperbolic Lattices

Daniel Sela, Nan Cheng, Kai Sun

TL;DR

Hyperbolic lattices, with their negative curvature and extensive boundaries, qualitatively alter vector-field phonons compared with Euclidean crystals. The authors show that the Goldstone theorem breaks down for vector fields on hyperbolic lattices with $z>2d$, yielding a finite bulk gap in the phonon spectrum despite spontaneous translational symmetry breaking, because Goldstone modes reside in $3$-component nonunitary representations of the translation group (via $SO(1,2)$) and do not populate bulk bands. When boundaries are included, an extensive set of low-frequency boundary modes fills the bulk gap, producing an insulating bulk with conducting boundary behavior and a finite boundary-mode fraction at $\, extomega=0$. The analysis combines a moment-based spectral approach with recursive lattice counting, enabling gap characterization without full high-dimensional representation theory and suggesting broader applicability to other homogeneous, locally connected systems on curved spaces.

Abstract

Hyperbolic lattices extend crystallinity into curved space, where negative curvature and exponentially large boundaries reshape collective excitations beyond Euclidean intuition. In this Letter, we push the study beyond scalar fields by exploring vector fields on hyperbolic lattices. Using phonons as an example, we show that the Goldstone theorem breaks down for vector fields in hyperbolic lattices. In stark contrast to Euclidean crystals, where the Goldstone theorem ensures that acoustic phonon modes are gapless, hyperbolic lattices with coordination number $z > 2d$ exhibit a finite bulk phonon gap. We identify the origin of this breakdown: the Goldstone modes here belong to nonunitary representations of the translation group and therefore cannot form gapless excitation branches. We further show that when boundaries are included, this bulk spetrum gap is filled by an extensive number of low-frequency boundary modes.

Failure of the Goldstone Theorem for Vector Fields and Boundary-Mode Proliferation in Hyperbolic Lattices

TL;DR

Hyperbolic lattices, with their negative curvature and extensive boundaries, qualitatively alter vector-field phonons compared with Euclidean crystals. The authors show that the Goldstone theorem breaks down for vector fields on hyperbolic lattices with , yielding a finite bulk gap in the phonon spectrum despite spontaneous translational symmetry breaking, because Goldstone modes reside in -component nonunitary representations of the translation group (via ) and do not populate bulk bands. When boundaries are included, an extensive set of low-frequency boundary modes fills the bulk gap, producing an insulating bulk with conducting boundary behavior and a finite boundary-mode fraction at . The analysis combines a moment-based spectral approach with recursive lattice counting, enabling gap characterization without full high-dimensional representation theory and suggesting broader applicability to other homogeneous, locally connected systems on curved spaces.

Abstract

Hyperbolic lattices extend crystallinity into curved space, where negative curvature and exponentially large boundaries reshape collective excitations beyond Euclidean intuition. In this Letter, we push the study beyond scalar fields by exploring vector fields on hyperbolic lattices. Using phonons as an example, we show that the Goldstone theorem breaks down for vector fields in hyperbolic lattices. In stark contrast to Euclidean crystals, where the Goldstone theorem ensures that acoustic phonon modes are gapless, hyperbolic lattices with coordination number exhibit a finite bulk phonon gap. We identify the origin of this breakdown: the Goldstone modes here belong to nonunitary representations of the translation group and therefore cannot form gapless excitation branches. We further show that when boundaries are included, this bulk spetrum gap is filled by an extensive number of low-frequency boundary modes.

Paper Structure

This paper contains 8 sections, 2 theorems, 45 equations, 4 figures.

Table of Contents

  1. Supplementary Materials
  2. Goldstone Modes Carry $3$D Non-unitary Representations
  3. An Estimate for $E_{0}$
  4. Estimating $\mathcal{F}_{\infty}$ from Samples $\{\mathcal{F}_{n}\}_{n=N_{1}}^{N_{2}}$: Analytic Framework, Weighting, and Model Selection
  5. Moment Computation in the $\{8, 8\}$ Lattice
  6. Recursive Counting Formula
  7. Non-zero Fraction of Modes Below the Gap
  8. Non-rigidity of Finite Graphs

Key Result

Theorem 1.1

Let $G = (V, E)$ be a finite planar graph. Suppose one face of $G$ consists of $\geq 3$ edges and every other face consists of $\geq 4$ edges. Then, $G$ is not $2$-rigid.

Figures (4)

  • Figure 1: (a) The figure of $f_{n}(x)=(1-x)^{n}$ for $n=1,5,10$. As $n$ increase, $f_{n}(x)$ are more and more localized. (b) The triangular lattice is fitted with Eq. \ref{['eq:fittingformula']} by $\Delta=0$, $r=0.0411\pm0.0557$. The theoretical result is $\Delta=0$ and $r=0$. (c) The square lattice is fitted with Eq. \ref{['eq:fittingformula']} by $\Delta=0$, $r=-0.5018\pm 0.0018$. The theoretical result is $\Delta=0$ and $r=-0.5$.
  • Figure 2: The hyperbolic $\{3,7\}$ and $\{7,3\}$ lattices and their low frequency phonon density of states property. (a) The elastic hyperbolic $\{7,3\}$ lattice in the Poincaré disk model. (b) Fitting $\{\mathcal{F}_{n}\}_{n=8}^{15}$ with respect to $1/n$ using Eq. \ref{['eq:fittingformula']} for the hyperbolic $\{3,7\}$ lattice. (c) The elastic hyperbolic $\{7,3\}$ lattice in the Poincaré disk model. (b) Fitting $\{\mathcal{F}_{n}\}_{n=7}^{13}$ with respect to $1/n$ using Eq. \ref{['eq:fittingformula']} for the hyperbolic $\{7,3\}$ lattice, $\omega_{g}=0$.
  • Figure 3: (a) The lower bound on the fraction of floppy modes for hyperbolic $\{7,3\}$, $\{7,4\}$ and $\{8,8\}$ lattices. (b) Fraction of phonon eigenmodes below the band gap as a function of $1/N$ in the $\{3, 7\}$ lattice, where $N$ is the number of nodes in the network. (c) The shape of a random eigenmode of the $\{3, 7\}$ lattice in the phonon band gap at $\omega=0$. The figure is the plot of the norm-squared of the amplitude verses distance averaged over bins of size $0.2 R$, where $R$ is the radius of curvature of the hyperbolic space.
  • Figure 4: Plot of $f \log(N)$ as a function of $1/N$.

Theorems & Definitions (4)

  • Theorem 1.1
  • Lemma 1.1
  • proof
  • proof : Proof of Theorem S1