Peierls Instability in Hexagonal Lattices

TL;DR

This work investigates a Peierls-type distortion in graphene by coupling a tight-binding electronic energy $-\underline{\mathrm{Tr}}|T(\mathbf{t})|$ to a quadratic elastic penalty on bond distortions, reduced to a single lattice-rigidity parameter $\mu$. The authors show that minimizers among 3-periodic, Kekulé-symmetric configurations exhibit two equal hopping amplitudes (Kekulé $O$-type) for any $\mu>0$, and identify a phase transition: for small $\mu$ the ground state breaks translation symmetry (gapped, insulating Kekulé phase), while for large $\mu$ the unique minimizer is the pristine, translation-invariant graphene (gapless, metallic). The analysis combines trace-per-atom formalisms, spectral study of the Kekulé tight-binding operator, perturbative Hessian arguments around pristine graphene, and convexity methods to establish sharp thresholds $\mu_c \approx 0.888$ and $\mu_c' \le 1.114$, illustrating a 2D Peierls-type transition with practical implications for strain-driven band-gap engineering in graphene. Overall, the results connect lattice rigidity to symmetry-breaking distortions and electronic phase behavior, highlighting a mechanism by which Kekulé textures can emerge and vanish in graphene-like materials.

Abstract

We investigate a conventional tight-binding model for graphene, where distortion of the honeycomb lattice is allowed, but penalized by a quadratic energy. We prove that the optimal 3-periodic lattice configuration has Kekulé O-type symmetry, and that for a sufficiently small elasticity parameter, the minimizer is not translation-invariant. Conversely, we prove that for a large elasticity parameter the translation-invariant configuration is the unique minimizer.

Figures (5)

• Figure 1: (Above, left) The standard honeycomb lattice, with unit vectors ${\mathbf a}_1, {\mathbf a}_2$ and the unit cell $\Gamma_2$. The blue and red atoms make up its two triangular sublattices. (Above, right) The larger unit cell $\Gamma_6$ defined with the new vectors ${\mathbf b}_1, {\mathbf b}_2$, containing six atoms and nine bonds. (Below) A minimizing configuration following frank2011possible, with our conventions for the atom and bond labels.
• Figure 2: The Brillouin zones $B_2$ and $B_6$ are the unit cells of the reciprocal lattices ${\mathbb L}_2^*$ and ${\mathbb L}_6^*$, represented in rhombic (left) and Voronoi or hexagonal forms (right). Note that $B_6$ is exactly one-third of $B_2$, and the two Dirac points $\mathbf{K}, \mathbf{K}' \in B_2$ are "folded" onto central point $(\Gamma)$ when enlarging unit cell $\Gamma_2$ to $\Gamma_6$. The high-symmetry points $(K)$ and $(M)$ are used for the band diagram in Figure \ref{['fig:bdiagram']} below.
• Figure 3: Band diagram cut of graphene over $B_6$ with six surfaces for each eigenvalue of $T(t,u,v,{\mathbf k})$, in the undistorted case (left, with two Dirac cones at central point $(\Gamma)$), and in the distorted case (right, with a bandgap).
• Figure 4: Exaggerated representation of the Kekulé O-type distortion, with a characteristic spread of both regular and benzene-like hexagons.
• Figure 5: The Kagome lattice (squares and full lines) appears as the line graph of the honeycomb (in dotted lines), with the same lattice vectors $\textbf{a}_1, \textbf{a}_2$ and three sites per $\Gamma_2$ cell.

Theorems & Definitions (15)

• Theorem 1.1: Kekulé symmetry
• Theorem 1.2: Phase transition
• Lemma 2.1
• proof
• Lemma 3.1
• proof : Proof of Lemma \ref{['thm:Ksym_long']}
• Lemma 3.2
• proof : Proof of Lemma \ref{['lem:sphere_bound']}
• Lemma 3.3
• proof : Proof of Lemma \ref{['lem:maxtuv']}