Pseudo-magnetic Fields and Effective Dynamics in Strained Honeycomb Structures

TL;DR

The paper develops a rigorous framework to justify wave-packet dynamics near Dirac points in strained honeycomb structures, revealing that envelope dynamics are governed by a two-dimensional Dirac equation with gauge fields induced by strain. Using a novel resolvent-based spectral analysis, it controls the impact of second-order perturbations and establishes error bounds showing the true solution remains close to the Dirac-ensemble ansatz for times up to O(ε^{-1}). It handles both positive V and the singular V ≡ 0 case via near/far-energy decompositions, and demonstrates the emergence of pseudo-magnetic effects, including strain-induced Landau levels, with supporting numerical simulations. The results provide a rigorous mathematical understanding of pseudo-magnetic phenomena in honeycomb lattices and offer a versatile method for higher-order perturbations in similar periodic systems.

Abstract

Strain offers a straightforward and effective method for generating pseudo-magnetic fields in optical and acoustic materials, thereby enabling precise manipulation of wave propagation. In this article, we investigate and justify wave packet dynamics localized near Dirac points in strained honeycomb-structured media. We develop a novel approach based on spectral analysis to control the error from second-order differential residue terms caused by the strain. The analysis yields a two-dimensional Dirac equation with nontrivial gauge fields governing the envelope dynamics, which is proved to well approximate the true solution in a long but finite time. These results contribute to the mathematical understanding of pseudo-magnetic effects in strained honeycomb structures and pave the way to systems with general higher-order perturbation terms.

Figures (6)

• Figure 1: A honeycomb structure. A fundamental cell of honeycomb lattice is shaded in yellow.
• Figure 2: A strained honeycomb structure, where we used deformation $\mathbf{u}(\mathbf{X}) = (0,\beta X_1^2)$.
• Figure 3: A schematic of Landau levels $E_{n,\pm}=\pm v\sqrt{2n |B_0|}$ for $v=B_0=1$ and $|n|\le 6$. The blue lines are the energy-momentum curves $E(k)$. Draw a horizontal line for a fixed $E$. If $E=E_{n,\pm}$, then it meets uncountably many points on the blue lines. So the density of state is highly concentrated on these Landau levels.
• Figure 4: Plots of absolute values of eigenmodes corresponding to Landau levels when $B_0=1$. (a) is for the upper component of $\Psi_{0,k}$, which is $\psi_{0,k}$, while its lower one is $0$. The momentum $k$ only affects the center where it concentrates, and which is labeled in dashed line. (b) and (c) are the upper and lower components of $\Psi_{2,k}$, respectively.
• Figure 5: Density distribution of the solution $\boldsymbol{\alpha}$ to (\ref{['StandardDiracDynamics']}) with prescribed $A_1,A_2$, subjected to the initial value (\ref{['LandauInitialData']}) with $k_0 = 0, w= 0.3$. We plot the density of $\boldsymbol{\alpha}(t)$ at $t=0,2,4$ in (a), (b) and (c), respectively. We can see that the wave-packet hardly moves, which agrees with the fact that the initial value we choose is approximately an eigenfunction of the operator $\mathcal{D}$.

Theorems & Definitions (29)

• Proposition 2.1
• Proposition 2.2
• Definition 1
• Definition 2
• Proposition 2.3
• Proposition 2.4
• Proposition 2.5
• proof
• Lemma 2.6
• Proposition 2.7