Phonon spectra, quantum geometry, and the Goldstone theorem

TL;DR

The paper investigates how quantum geometry of electronic bands, encoded in the quantum geometric tensor, shapes phonon dispersions via the electronic contribution to the dynamical matrix. Using a Gaussian-hopping approximation and graphene as a case study, it decomposes the electronic DM into non-geometric and geometric parts, revealing that the geometric term induces a non-analytic $\tilde{\omega}_{\rm LA/TA}(\bm q)\propto \sqrt{|\bm q|}$ near $\Gamma$ in the massless Dirac limit. Introducing a finite gap $\Delta$ regularizes this to linear behavior below $q_{\rm thr}=\Delta/(\hbar v_{\rm F})$, demonstrating a purely geometric mechanism behind the long-wavelength phonon anomalies. The work provides a general framework to quantify quantum-geometric effects on phonons and highlights how Dirac-like electronic structure can leave robust fingerprints in phonon spectra, with implications for 2D materials and beyond.

Abstract

Phonons are essential quasi-particles of all crystals and play a key role in fundamental properties such as thermal transport and superconductivity. In particular, acoustic phonons can be interpreted as Goldstone modes that emerge due to the spontaneous breaking of translational symmetry. In this article, we investigate the quantum geometric contribution to the phonon spectrum in the absence of Holstein phonons. Using graphene as a case study, we decompose the dynamical matrix into distinct terms that exhibit different dependencies on the electron energy and wavefunction. We then examine the role of quantum geometry in shaping the material's phonon spectrum, and we find that removing the nontrivial quantum geometric contribution from the dynamical matrix causes the acoustic phonon modes to behave in a non-analytic fashion.

Figures (5)

• Figure 1: (Color online) Dispersion of the four in-plane phonon branches (LA, TA, LO, and TO) in graphene. Results in this figure are plotted along the $\Gamma$KM$\Gamma$ high-symmetry path in the first Brillouin zone illustrated in the inset. Black lines: Phonon dispersion $\omega_{\ell}(\bm q)$ calculated analytically within a next-nearest neighbor Born-von Karman framework Falkovsky2007. The relevant parameters needed to produce these plots have been determined by fitting ab initio density functional theory results. Blue lines: Phonon dispersion $\Tilde{\omega}_{\ell}(\bm q)$ as calculated after removing from the full dynamical matrix ${\cal D}({\bm q})$ the quantum geometric contribution ${\cal D}_{\rm g}({\bm q})$---see Eq. \ref{['eq:dm-decomposition2']}. A zoom near $\Gamma$ is reported in Fig. \ref{['fig:2']}, showing that both the LA and TA blue curves exhibit a non-analytic dependence on the wavevector $\bm q$ in a neighborhood of $\Gamma$.
• Figure 1: (Color online) (a) Plot of the $\pi$ bands of graphene along the high-symmetry $\Gamma{\rm KM}\Gamma$ path in the first Brillouin zone. Red dots: Results of the ab initio DFT calculations. Black line: best fit based on our analytical overlap-inclusive NNNN model. Inset shows the low-energy electron dispersion near the Dirac point for $\Delta = 0$ (solid black line) and $\Delta = 10$ meV (dotted black line), corresponding respectively to massless and massive Dirac fermions. (b) Hopping integrals generating the graphene's $\pi$ bands, obtained via the strained-lattice method. Red circles: inter-sublattice ($AB$) hopping integrals. Blue dots: intra-sublattice ($AA$) hopping integrals. Both are obtained as optimal parameters from a least-square best-fit procedure on the ab initio bands of strained lattices. Black crosses: hopping integrals in a relaxed graphene lattice. Red (blue) lines: Gaussian best fit to the inter- (intra-) sublattice hopping integral as a function of the inter-atomic distance $r$. The inset shows the hierarchy of nearest neighbors in a honeycomb lattice. (c) Black lines: Phonon dispersions in graphene along $\Gamma$KM$\Gamma$. Red lines: eigenvalues of ${\cal D}(\bm q) - {\cal D}^ {{\rm (el},\, \pi )}(\bm q)$, where ${\cal D}^ {{\rm (el},\, \pi )}(\bm q)$ is the electronic contribution of the $\pi$ bands. The acoustic modes still behave linearly in $|\bm q|$ for $\bm q \to \bm \Gamma$ after the removal of the entire electronic contribution of the $\pi$ bands to the dynamical matrix, thus proving the non-analytic behavior of $\tilde{\omega}_{\rm LA/TA}(\bm q \to \bm \Gamma)$ to be of purely geometric nature. (d) Behavior of the acoustic modes near $\Gamma$ for $|\bm q|\gtrsim q_{\rm thr}$, as illustrated in Fig. 2 of the main text; here, $\bm q$ is on the M$\to\Gamma$ portion of the high-symmetry contour.
• Figure 2: (Color online) (a) Zoom-in of Fig. \ref{['fig:1']} ($\Delta=10~{\rm meV}$) near the $\Gamma$ point. Data for the LA (TA) branch are denoted by magenta (blue) filled circles (squares). The two lowest-lying eigenfrequencies of ${\cal D}_{\rm ng}(\bm q)$, i.e. $\tilde{\omega}_{\rm LA/TA}(\bm q)$, follow a $\propto\sqrt{|\bm q|}$ trend for $\bm q$ near $\Gamma$ as long as $q$ is larger than the threshold $q_{\rm thr} = \Delta/\hbar v_{\rm F}$ (vertical dash-dotted line). This behavior is well-described by a massless Dirac fermion model. Dashed lines serve as guides to the eye, illustrating linear and $\sqrt{|\bm q|}$ trends. (b) For a relatively large fermion gap $\Delta = 500$ meV, the eigenfrequencies $\tilde{\omega}_{\rm LA/TA}(\bm q)$ behave linearly in $\bm q$ for $|\bm q| < q_{\rm thr}$, as the quantum metric is regularized by $\Delta$. This is explained by a massive Dirac fermion model.
• Figure 2: (Color online) Regularized QGT for gapped graphene. For better illustration, results in this figure have been calculated with a large gap of $\Delta = 20~{\rm meV}$. (a) Trace of the Fubini-Study metric $g^{(\pm)}_{ij}(\bm k)$--- which is the same for both conduction and valence bands due to the quasi-exact particle-hole symmetry in the vicinity of the $\bm K$ point---plotted along the path in the FBZ shown as a red arrow in the inset. Cyan circles: numerical results obtained within the NN tight-binding model. Magenta squares: numerical results obtained within the NNNN model. Solid blue line: analytical results for massive Dirac fermions. (b) The only non-zero component of the conduction-band Berry curvature, ${\cal F}_{xy}^ {(+)}(\bm k) \equiv -2\operatorname{Im} {\cal Q}_{xy}^ {(+)}(\bm k) = {\cal F}_{yx}^ {(-)}(\bm k)$, plotted along the same path as in panel (a). The valence-band curvature is identical apart from a sign. Legend is the same as in panel (a). (c)-(d) 2D color plots of $g^{(\pm)}_{ij}(\bm k)$ and ${\cal F}_{xy}^ {(+)}(\bm k)$ calculated within the NNNN model as functions of $\bm k$. The insets provide a zoom-in view of the $\bm K$ point, where the singularity discussed in the text has been cured.
• Figure 3: (Color online) Workflow followed for the case of graphene. The same method can be applied to any material for which the relevant hopping integrals are well-described by the GA. Fitting the BvK model to ab initio data on the high-symmetry line $\Gamma$KM$\Gamma$ allows us to extract the force constants and thus calculate the DM on the entire FBZ. The same fitting procedure is followed in order to obtain all the relevant information on electrons, thus allowing us to calculate the geometric contribution to the DM ${\cal D}_{\rm g}(\bm q)$, from which we obtained the phonon dispersion lines $\Tilde{\omega}_{\ell}(\bm q)$ illustrated in Figs. \ref{['fig:1']} and \ref{['fig:2']} in the main text.