The nuclear motion induces in the electronic atomic orbitals a nuclear-velocity-dependent phase (also known as electron-translation factor), which modifies the effective Hamiltonians constructed from localised atomic orbitals. In this work, using an Ehrenfest Lagrangian approach for the localised atomic orbitals (LCAO) and tight-binding methods, we determine, at any order in the nuclear velocity, the equations of motion and the vibrational responses within a linear response formalism, focusing on the tight-binding assessment of the Born effective charges and the force-constant matrix. The appearance of nuclear-velocity-dependent Peierls-like phases in the non-local part of the interactions restores the all-electron sum rules for frequency-dependent vibrational responses. In tight-binding models these corrections crucially modify the vibrational response from a qualitative point of view, also yielding contributions required to capture phenomena such as vibrational circular dichroism. We test these corrections in the tight-binding model for metallic gapped graphene - finding excellent agreement with ab initio calculations - and for the topological time-reversal symmetry breaking Haldane model.

Excitons govern the light-matter interaction in 2D gapped materials with intrinsically large binding energies. In spite of plentiful optical measurements in the visible for semiconducting transition-metal dichalcogenides, we still lack optical-absorption studies of the exciton structure of insulating 2D materials that requires ultraviolet (UV) light. Moreover, measurements of the momentum dispersion of excitons in the vicinity of optical limit are rare owing to low resolutions but hold the key to reveal quasiparticle interactions. To close this gap, we employ high momentum resolution electron energy loss spectroscopy (q-EELS) to explore exciton dispersions of mono- and few-layer hexagonal boron nitride. Surprisingly, we reveal a fine structure of the first bright exciton dispersion band composed by two features (A and A ' ), visible only at small momentum, not predicted by frozen-nuclei Bethe-Salpeter calculations. Introducing an optical conductivity approximation (OCA), we extract from the experimental q-EELS spectra deep-UV optical conductivity at zero momentum, σ(ω), and discuss the exciton fine structure in σ(ω), consistent with previous photoluminescence observations. Our findings establish a general methodology to probe the fine structure of exciton dispersions, providing new insights into exciton-phonon sidebands and eventually polarons in low-dimensional materials.

We derive the first-principles Ehrenfest molecular dynamics describing non-adiabatic processes with the inclusion of the nuclear-velocity-dependent phases (also known as electron-translation factors) on the atomic-orbital basis. These phases, appearing when nuclei are treated dynamically, affect effective Hamiltonians constructed from localised orbitals. In this work, we focus on the effects in the first-principles pseudo-potential Hamiltonian, both for the norm-conserving and ultra-soft cases, derived within the Projector-Augmented-Wave (PAW) method framework. Peierls-like phases depending on the nuclear velocities appear in the non-local part of the potential, while additional nuclear velocity and acceleration-dependent corrections appear in the ultra-soft pseudo-potential case. The use of velocity-including atomic orbital basis enables a Galilean-invariant description of the non-adiabatic Ehrenfest molecular dynamics, removing spurious non-adiabatic couplings that arise from neglecting the nuclear velocity phases in the atomic orbitals.

In multilayer graphene, the stacking order of the layers plays a crucial role in the electronic properties and the manifestation of superconductivity. By applying shear stress, it is possible to induce sliding between different layers, altering the stacking order. Here, focusing on bilayer graphene, we analyze how ionic fluctuations alter the free energy barrier between different stacking equilibria. We calculate the free energy barrier through the state-of-the-art self-consistent harmonic approximation, which can be evaluated at unstable configurations. We find that above 100 K there is a large reduction of the barrier of more than 30% due to thermal vibrations, which significantly improves the agreement between previous first-principles calculations and experiments on a single graphite crystal. As the temperature increases, the barrier remains nearly constant up to around 500 K, with a more pronounced decrease only at higher temperatures. Our approach is general and paves the way for systematically accounting for thermal effects in free energy barriers of other macroscopic systems.

Functionalized conjugated polymers are promising materials for electromechanical applications due to predicted giant piezoelectricity, arising from anomalously large dynamical effective charges and an enhanced response in the proximity of the dimerization phase transition. In this work, we assess the impact of quantum ionic fluctuations on piezoelectricity using the stochastic self-consistent harmonic approximation with a Rice-Mele diatomic chain model, parametrized to reproduce hybrid-functional first-principles calculations of prototypical carbyne. The model's accuracy is validated against first-principles calculations both with and without quantum-anharmonic effects. We find that ionic fluctuations strongly impact the structural properties, with the boundary of the dimerization phase transition shifted by 34%. Despite quantum fluctuations in the bond length reaching magnitudes comparable to the average, the strong piezoelectric response persists. The topological enhancement of the effective charges remains robust and is even enhanced by about ∼20% thanks to a quantum-induced shrinking of the electronic gap. The piezoelectric coefficient remains dominated by the internal relaxation and retains a morphotropic-like character, reaching maximum values near the renormalized boundary, with quantum anharmonicity mainly shifting the optimal enhancement window.

Based on a many-body formulation, we derive an electron-phonon coupling including excitonic effects that preserves thermodynamic detailed balance between electronic and phononic scattering processes. We start from the microscopic electron-nucleus Hamiltonian, expand around the Born-Oppenheimer equilibrium geometry, and construct an effective action for the electronic and phononic propagators. From the same effective action, we derive both electronic and phononic self-energies in terms of a nonlocal vertex 𝒢^s including excitonic effects, which generalizes the usual local interaction vertex g^s. When electrons and phonons are well-defined quasiparticles in the screened-exchange approximation and 𝒢^s is taken in its static limit, both self-energies reduce to Fermi-golden-rule expressions containing the same 𝒢^s, thereby ensuring detailed balance. As an application, we compute electronic and phononic linewidths in graphene and illustrate this common-vertex construction. We analyze the competition between the reduced scattering phase space induced by the screened-exchange band structure and the enhancement of the electron-phonon vertex due to excitonic effects, finding that the vertex enhancement can compensate for and overcome the phase-space reduction in both electronic and phononic linewidths.

We establish a theoretical framework for the dynamics of a lattice of ions in a mean-field approach, where anharmonicity is included via self-consistency. In this picture, the many-body dynamics of a system of N atoms in three dimensions is mapped onto two kinds of 6N-dimensional vectors: the phonon condensate, describing the evolution of the average atomic positions, and the phonon spinors, describing the evolution of the atomic elastic constants. The phonon spinors are classified by a quantum number that behaves as a spin: the phonon pseudospin. The many-body Liouville equation is replaced by two wave equations equivalent to the time-dependent Schrödinger equation for the electronic wave function in density functional theory. Exploiting this parallelism, we formulate the response of the anharmonic lattice in one-to-one correspondence with time-dependent density functional theory for electrons. In complete analogy with the electronic case, we express the ionic response in terms of matrix elements of operators representing external fields and forces. We show how anharmonicity screens external perturbations through a phonon analogue of the Hartree-exchange-correlation kernel. We provide expressions for the lattice optical and thermal conductivity, showing how thermal conductivity depends on the phonon pseudospin. By approximating the density matrix as a Gaussian, we recover the equations of the time-dependent self-consistent harmonic approximation. In this case, the linear-response equations are formulated in terms of an anharmonic kernel including three- and four-phonon scattering. By translating anharmonic lattice dynamics into the language of density functional theory, this work shows how theoretical and computational advances in modeling the dynamical response of interacting electrons can be directly applied to interacting ions.

Mechanical and elastic properties of materials are among the most fundamental quantities for many engineering and industrial applications. Here, we present a formulation that is efficient and accurate for calculating the elastic and bending rigidity tensors of crystalline solids, leveraging interatomic force constants and long-wavelength perturbation theory. Crucially, in the long-wavelength limit, lattice vibrations induce macroscopic electric fields which further couple with the propagation of elastic waves, and a separate treatment on the long-range electrostatic interactions is thereby required to obtain elastic properties under the appropriate electrical boundary conditions. A cluster expansion of the charge density response and dielectric screening function in the long-wavelength limit has been developed to efficiently extract multipole and dielectric tensors of arbitrarily high order. We implement the proposed method in a first-principles framework and perform extensive validations on silicon, NaCl, GaAs and rhombohedral BaTiO$_3$ as well as monolayer graphene, hexagonal BN, MoS$_2$ and InSe, obtaining good to excellent agreement with other theoretical approaches and experimental measurements. Notably, we establish that multipolar interactions up to at least octupoles are necessary to obtain the accurate short-circuit elastic tensor of bulk materials, while higher orders beyond octupole interactions are required to converge the bending rigidity tensor of 2D crystals. The present approach greatly simplifies the calculations of bending rigidities and will enable the automated characterization of the mechanical properties of novel functional materials.

Using first principles techniques, we show that infrared optical response allows us to discriminate between the topological and the trivial phases of 2D quantum spin Hall insulators (QSHI). We showcase germanene and jacutingaite, of recent experimental realization, as prototypical systems where the infrared spectrum is discontinuous across the transition, due to sudden and large discretized jumps of the Born effective charges (up to ~2). Our results, rationalized thanks to the low-energy Kane–Mele model, are robust with respect to dynamical effects, relevant when the electronic energy gap is comparable with the phonon frequency. In the small gap QSHI germanene, due to dynamical effects, the in-plane phonon resonance in the optical conductivity shows a Fano profile with remarkable differences in the intensity and the shape between different phases. Instead, the large-gap QSHI jacutingaite presents several IR-active phonon modes whose spectral intensities drastically change between different phases.

We develop an ab initio framework that captures the impact of electron-electron and electron-hole interactions on phonon properties. This enables the inclusion of excitonic effects in the optical phonon dispersions and lifetimes of graphene, both near the center (Γ) and at the border (K) of the Brillouin zone, at phonon-momenta relevant for Raman scattering and for the onset of the intrinsic electrical resistivity. Near K, we find a phonon redshift of ∼150  cm^{-1} and a 10× enhancement of the group velocity, together with a 5× increase in linewidths due to a 26× increase of the electron-phonon matrix elements. These effects persist for doping 2E_{F}<ℏω_{ph} and are quenched at higher dopings. Near Γ, the excitonic effects are minor because of the gauge field nature of the electron-phonon coupling at small phonon momentum.

In transmission electron energy loss (EEL) spectroscopy, the cross section in two dimensions (2D) is quenched by kinematic effects once the momentum transfer becomes smaller than a critical value set by qz, the momentum loss parallel to the beam. Our highly momentum (Aq = 0.02 & Aring;-1) and energy (AE = 45 meV) resolved setup is instrumental in delivering the experimental verification of quenched 2D EEL spectra on freestanding graphene at momentum transfers q below 0.06 & Aring;-1. We retrieve the intrinsic uniform dielectric response of graphene from measured spectra by quantifying the kinematic suppression.

Excitons govern the light-matter interaction in 2D gapped materials with intrinsically large binding energies. In spite of plentiful optical measurements in the visible for semiconducting transition-metal dichalcogenides, we still lack optical-absorption studies of the exciton structure of insulating 2D materials that requires UV light. Moreover, measurements of the momentum dispersion of excitons in the vicinity of optical limit are rare owing to low resolutions but hold the key to reveal quasiparticle interactions. To close this gap, we employ high momentum resolution electron energy loss spectroscopy (q-EELS) to explore exciton dispersions of mono- and few-layer hexagonal boron nitride. Surprisingly, we reveal a fine structure of the first bright exciton dispersion band composed by two features (A and A'), visible only at small momentum, not predicted by Bethe-Salpeter calculations. Introducing an optical conductivity approximation (OCA), we extract from the experimental q-EELS spectra the ultraviolet (UV) optical conductivity at zero momentum, σ(ω), and discuss the exciton fine structure in σ(ω), consistent with previous photoluminescence observations. Our findings establish a general methodology to probe the fine structure of exciton dispersions, providing new insights into exciton-phonon sidebands and eventually polarons in low-dimensional materials.

We show that the many-body features of graphene band structure and electronic response can be accurately evaluated by applying many-body perturbation theory to a tight-binding (TB) model. In particular, we compare TB results for the optical conductivity with previous ab initio calculations, showing nearly perfect agreement both in the low-energy region near the Dirac cone ('100 meV) and at the higher energies of the pi plasmon ('5 eV). A reasonable agreement is reached also for the density-density response at the Brillouin zone corner. With the help of the reduced computational cost of the TB model, we study the effect of self-consistency on the screened interaction (W) and on the quasiparticle corrections, a task that is not yet achievable in ab initio frameworks. We find that self-consistency is important to reproduce the experimental results on the divergence of the Fermi velocity, as confirmed by previous studies, while it marginally affects the optical conductivity. Finally, we study the robustness of our results against doping or the introduction of a uniform dielectric environment.

Little is known about the quantum effects of friction, and first-principles studies are virtually non-existent. In multilayer graphene, the order of the stacking of the layers plays a crucial role in the electronic properties and the manifestation of superconductivity, and friction affects how layers slide to change the stacking. Here, we study the effects of friction in bilayer graphene by focusing on quantum fluctuations in the free energy barrier that layers have to overcome to slide with respect to each other. We evaluate the free energy at the barrier in bilayer graphene through the self-consistent harmonic approximation, accounting for quantum and thermal ionic fluctuations. We obtain a very large reduction of the barrier of more than 30%. Temperature effects are weaker, although high temperatures induce a significant reduction of the barrier as well. Our approach is general and paves the way for systematically accounting for nuclear quantum and thermal effects in free energy barriers of other macroscopic systems.

The discovery of high-temperature superconductivity in hydrogen-rich compounds under extreme pressures has prompted great excitement, intense research, but also debate over the past decade. Electrical transport has been the primary diagnostic tool for identifying superconductivity in these systems, whereas complementary probes, including magnetic, spectroscopic, tunnelling and ultrafast methods, remain mostly qualitative due to experimental constraints and sample heterogeneity. Recent concerns over their reliability have fuelled controversy, leading to scepticism and pointing out the need for alternative, quantitative approaches. In this study, we acquired unprecedented high-quality Raman spectra of hexagonal LaH10 at approximately 145 GPa and low temperatures, in conjunction with electrical transport measurements. Upon cooling, we observe a drop of resistivity and simultaneous remarkable variations of phonon frequencies and linewidths. These effects are interpreted and perfectly reproduced by the Migdal-Eliashberg theory, providing a definitive proof of phonon-mediated superconductivity and enabling a quantitative determination of the superconducting energy gap. Our results establish Raman spectroscopy as a robust, contact-free probe with micrometric resolution for studying high temperature superconductivity, opening a powerful route to its discovery and characterization.

We consider the dynamical electronic response function in theoretical frameworks that include nonlocal exchange interactions, such as the Bethe-Salpeter equation with the frequency independent approximation of the screened interaction, Hartree-Fock, and range-separated Hybrid DFT approaches. Within these pictures, we demonstrate that any time-dependent electronic linear response function allows for a formulation which is variational in the electronic density matrix. To achieve our goal, we consider the usual form of a response function, written in terms of a screened and a bare electronic vertices (`bare-screen'), and perform an exact rewriting in terms of purely screened electronic vertices (`screen-screen'). Within the `screen-screen' formulation, the response function can be written as a stationary point of a functional of the exact density matrix. Further, we show that the imaginary part of any electronic response can be written in the form of a generalized Fermi Golden Rule, by introducing an exact complementary rewriting in terms of vertices related by complex conjugation (`screen*-screen'). The screen-screen formulation can be further extended partitioning the electronic interaction in separate contributions, expressing the response in terms of partially screened electronic vertices (`partial screen-partial screen'), preserving the stationary properties. We numerically validate the effectiveness of our formalism by calculating the optical conductivity of graphene, which exhibits strong excitonic effects. To do so, we solve the Bethe-Salpeter Equation on a tight-binding model, including exchange effects in the response of graphene. Our findings show the advantages of the variationality of the screen-screen formulation over the others both in convergence properties and robustness with density-matrix approximations.

We propose a new approach to study the transition between different topological states, based on the assessment of the vibrational resonances in infrared spectra. We consider the Haldane and Kane-Mele models finding that Born effective charges are nearly quantized, with a discontinuous jump concomitant with the topological phase transition. In particular, Born effective charges display a finite value in the trivial phase and a null one in the nontrivial one. This is rooted in the connection between Born effective charges and electronic Berry curvature at the band edges. Finally, at the topological phase transition of the Haldane model, we also observe a nearly quantized jump of the chiral splitting of the zone-center phonon frequencies, induced by time-reversal symmetry breaking.

In transmission electron energy-loss spectroscopy, the cross section in 2D is quenched by kinematic effects once the momentum transfer becomes smaller than a critical value set by q_z, the momentum loss parallel to the beam. Our highly momentum (Δ q = 0.02 Å^-1) and energy (Δ E = 45 meV) resolved setup is instrumental on delivering the unprecedented experimental verification of quenched 2D EEL spectra on freestanding graphene at momentum transfers q below 0.06Å^-1. We retrieve the intrinsic uniform dielectric response of graphene from measured spectra by quantifying the kinematic suppression.

EPIq (Electron-Phonon wannier Interpolation over k and q-points) is an open-source software for the calculation of electron-phonon interaction related properties from first principles.Acting as a post-processing tool for a density-functional perturbation theory code ( Quantum ESPRESSO ) and wannier90, EPIq exploits the localization of the deformation potential in the Wannier function basis and the stationary properties of a force-constant functional with respect to the first-order perturbation of the electronic charge density to calculate many electron-phonon related properties with high accuracy and free from convergence issues related to Brillouin zone sampling. EPIq features includes: the adiabatic and non-adiabatic phonon dispersion, superconducting properties (including the superconducting band gap in the Migdal-Eliashberg formulation), double-resonant Raman spectra and lifetime of excited carriers. The possibility to customize most of its input makes EPIq a versatile and interoperable tool. Particularly relevant is the interaction with the Stochastic Self-Consistent Harmonic Approximation (SSCHA) allowing anharmonic effects to be included in the calculation of electron-properties. The scalability offered by the Wannier representation combined with a straightforward workflow and easy-to-read input and output files make EPIq accessible to the wide condensed matter and material science communities.