Phonon dynamics and chiral modes in the two-dimensional square-octagon lattice

TL;DR

This study extends the physics of chiral phonons to the two-dimensional square-octagon lattice using a classical spring-mass model with central-force interactions among nearest, next-nearest, and third-nearest neighbors. The authors construct an 8×8 dynamical matrix D(k) for a four-atom unit cell and analyze phonon bands, density of states, and mode patterns, highlighting how longer-range couplings and sublattice mass contrast shape dispersion, gaps, and anisotropy in phase and group velocities. By introducing a weak time-reversal-symmetry breaking term, they demonstrate the emergence of chiral phonons with finite L_z, lifting degeneracies at high-symmetry points and yielding helically polarized modes; they further connect these modes to infrared circular dichroism via a minimal optical-model framework showing derivative-like CD spectra that scale with the TRS-breaking strength and phonon angular momentum. The work thus provides a tunable route to phonon band engineering, anisotropic transport, and optical signatures of lattice chirality in square-octagon networks, with potential implications for phononic devices and flat-band physics in mechanical systems.

Abstract

Chiral phonons, originally identified in two-dimensional hexagonal lattices and later extended to kagome, square, and other lattices, have been extensively studied as manifestations of broken inversion and time-reversal symmetries in vibrational dynamics. In this work, we investigate the vibrational dynamics of the two-dimensional square-octagon lattice using a spring-mass model with central-force interactions. The model incorporates mass contrast and variable coupling strengths among nearest, next-nearest, and third-nearest neighbors. From the dynamical matrix, we obtain the phonon dispersion relations and identify tunable phononic band gaps governed by both mass and spring-constant ratios. The angular dependence of phase and group velocities is analyzed to reveal the pronounced anisotropy inherent to this lattice geometry. We also examine the distinctive features of the square-octagon geometry, including flat-band anomalies in the density of states and anisotropic sound propagation induced by longer-range couplings. In addition, we explore the emergence of chiral phonons by introducing a time reversal symmetry-breaking term in the dynamical matrix, and to elucidate their optical signatures, we construct a minimal model to study infrared circular dichroism arising from chiral phonon modes.

Figures (21)

• Figure 1: Schematic representation of a 2D square-octagon lattice model, with a repeating unit cell outlined by a black dashed square containing four atomic sites. Nearest neighbor (NN) springs are shown as solid blue sinusoidal lines, next nearest neighbor (NNN) springs as dashed purple sinusoidal lines, and 3rd-NN connections as straight colored bonds extending from the central cell. The coupling constants $K_1$, $K_2$, and $K_3$ correspond to representative NN, NNN, and 3rd-NN interactions, respectively.
• Figure 2: Phonon dispersions along the $X$--$\Gamma$--$M$--$X$ path and the corresponding atom-projected density of states for the uniform-mass case ($m_\alpha = 1$) with fixed NNN coupling $K_{2} = 0.5$ and increasing third-neighbor coupling, (a) $K_{3} = 0.05$, (b) $K_{3} = 0.1$, (c) $K_{3} = 0.25$, and (d) $K_{3} = 0.5$. The color shading in the DOS indicates the relative contributions of individual sublattices to each vibrational mode.
• Figure 3: Phonon dispersions along the $X$--$\Gamma$--$M$--$X$ path and the corresponding atom-projected density of states for the square-octagon lattice with fixed couplings $K_{1} = 1.0$ and $K_{2} = 0.5$, and varying 3rd-NN coupling $K_{3}$. (a) Weak long-range coupling with small mass contrast ($m_A = m_C = 1.0$, $m_B = m_D = 1.5$, $K_{3} = 0.05$); (b) moderate coupling with the same mass configuration ($K_{3} = 0.25$); (c) large mass contrast ($m_A = m_C = 1.0$, $m_B = m_D = 5.0$, $K_{3} = 0.25$); and (d) a light impurity on the $B$ sublattice ($m_B = 0.3$, others unity, $K_{3} = 0.25$). The color shading in the DOS indicates the relative contributions of individual sublattices to each vibrational mode.
• Figure 4: Representative phonon eigenmodes of the square-octagon lattice at high-symmetry point $\Gamma$. (a) and (b) correspond to acoustic-like translations, with all atoms moving in phase along orthogonal directions. (c), (d), (e), and (f) represent optical shear or stretching distortions where sublattices move out of phase and (g) and (h) show breathing-type modes. Together these eight patterns illustrate the characteristic vibrational motifs of the lattice.
• Figure 5: Phonon eigenmodes of the square-octagon lattice at high-symmetry points $X$ (a-d) and $M$ (e-h). At $X$, the modes are predominantly shear and stretching type. At $M$, the modes appear as rotated or hybridized versions of the $\Gamma$-point motifs, combining shear and breathing distortions along diagonal directions.