Self-consistent Harmonic Approximation Research Articles

H$_{3}$Se in the $Im\overline{3}m$ Phase: A High-Pressure Superconductor with $T_{\text{c}}$ Reaching 200 K at 64 GPa Mediated by Anharmonic Phonons

Abstract Hydrogen-based compounds have attracted significant attention in recent years due to the discovery of conventional superconductivity with high critical temperature under high pressure, rekindling hopes for searching room temperature superconductor. In this work, we investigated systematically the vibrational and superconducting properties of H$_{3}$Se in $Im\overline{3}m$ phase under pressures ranging from 50 to 200 GPa. Our approach combines the stochastic self-consistent harmonic approximation with first-principles calculations to address effects from the quantum and anharmonic vibrations of ions. It turns out that these effects significantly modify the crystal structure, increasing the inner pressure by about 8 GPa compared to situations where they are ignored. The phonon spectra suggest that with these effects included, the crystal can be stabilized at pressures as low as about 61 GPa, much lower than the previously predicted value of over 100 GPa. Our calculations also highlight the critical role of quantum and anharmonic effects on the electron-phonon coupling properties. Neglecting these factors could result in a substantial overestimation of the superconducting critical temperature $T_{\text{c}}$, by approximately 25 K at 125 GPa, for example. With anharmonic phonons, the $T_{\text{c}}$ derived from the Migdal-Eliashberg equations, reaches 200 K ($\mu^\star=0.1, \lambda$=4.1) as the pressure decreases to 64 GPa, making the crystal a rare high-$T_{\text{c}}$ superconductor at moderate pressures.

Fast prediction of anharmonic vibrational spectra for complex organic molecules

Interpreting Raman and IR vibrational spectra in complex organic molecules lacking symmetries poses a formidable challenge. In this study, we propose an innovative approach for simulating vibrational spectra and attributing observed peaks to molecular motions, even when highly anharmonic, without the need for computationally expensive ab initio calculations. Our approach stems from the time-dependent stochastic self-consistent harmonic approximation to capture quantum nuclear fluctuations in atom dynamics while describing interatomic interaction through state-of-the-art reactive machine-learning force fields. Finally, we employ an isotropic charge model and a bond capacitor model trained on ab initio data to predict the intensity of IR and Raman signals.

Unveiling antiferromagnetic resonance: A comprehensive analysis via the self-consistent harmonic approximation

The Self-Consistent Harmonic Approximation (SCHA) has demonstrated efficacy in discerning phase transitions and, more recently, in elucidating coherent phenomena within ferromagnetic systems. However, a notable gap in understanding arises when extending this framework to antiferromagnetic models. In this investigation, we employ the SCHA formalism to conduct an in-depth exploration of the Antiferromagnetic Resonance (AFMR) within both Antiferromagnetic (AF) and Spin-Flop (SF) phases. Our analysis includes thermodynamic considerations from both semiclassical and quantum perspectives, with comparisons drawn against contemporary experimental and theoretical data. By incorporating a treatment utilizing coherent states, we investigate the dynamics of magnetization precession, a fundamental aspect in comprehending various spintronic experiments. Notably, the SCHA demonstrates excellent agreement with existing literature, showcasing its simplicity and efficiency in describing AFMR characteristics, even close to the transition temperature.

Anharmonic and quantum effects in Pm3̄ AlM(M = Hf, Zr)H6 under high pressure: A first-principles study.

First-principles calculations were employed to investigate the impact of quantum ionic fluctuations and lattice anharmonicity on the crystal structure and superconductivity of Pm3̄ AlM(M = Hf, Zr)H6 at pressures of 0.3-21.2GPa (AlHfH6) and 4.7-39.5GPa (AlZrH6) within the stochastic self-consistent harmonic approximation. A correction is predicted for the crystal lattice parameters, phonon spectra, and superconducting critical temperatures, previously estimated without considering ionic fluctuations on the crystal structure and assuming the harmonic approximation for lattice dynamics. The findings suggest that quantum ionic fluctuations have a significant impact on the crystal lattice parameters, phonon spectra, and superconducting critical temperatures. Based on our anharmonic phonon spectra, the structures will be dynamically stable at 0.3GPa for AlHfH6 and 6.2GPa for AlZrH6, ∼6 and 7GPa lower than pressures given by the harmonic approximation, respectively. Due to the anharmonic correction of their frequencies, the electron-phonon coupling constants (λ) are suppressed by 28% at 11GPa for AlHfH6 and 22% at 30GPa for AlZrH6, respectively. The decrease in λ causes Tc to be overestimated by ∼12K at 11GPa for AlHfH6 and 30GPa for AlZrH6. Even if the anharmonic and quantum effects are not as strong as those of Pm3̄n-AlH3, our results also indicate that metal hydrides with hydrogen atoms in interstitial sites are subject to anharmonic effects. Our results will inevitably stimulate future high-pressure experiments on synthesis, structural, and conductivity measurements.

Impact of quantum fluctuations thermally renormalize lattice vibrations on superconducting state of transition metal functionalized two-dimensional hydrides monolayer

Exploring superconducting materials stands as a pivotal pursuit in condensed matter physics, particularly, the investigation of superconductivity in two-dimensional metal hydrides is of paramount importance due to its intriguing nature. Our study focuses on elucidating the metallic state of van der Waals layered TM-H (TM = Ti, Zr, Hf) compounds, a key factor in predicting their superconducting (SC) characteristics. Leveraging an evolutionary algorithm rooted in density functional theory, we predicted the structures of hydrides, including Ti2H2, Ti2H4, Zr2H2, Zr2H4, Zr2H5, Hf2H2, Hf2H4, Hf2H5, and determined their energetically stable structures. Along with exploring the potential for SC, we conducted a comprehensive examination of relevant electronic properties. A significant aspect addressed was the influence of anharmonic phonon properties in determining the stochastic self-consistent harmonic approximation. These findings underscore the pivotal role of anharmonicity in determining the SC of two-dimensional metal hydrides.

First-Order Rhombohedral-to-Cubic Phase Transition in Photoexcited GeTe.

Photoexcited GeTe undergoes a nonthermal phase transition from a rhombohedral to a rocksalt crystalline phase. The microscopic mechanism and the nature of the transition are unclear. By using constrained density functional perturbation theory and by accounting for quantum anharmonicity within the stochastic self-consistent harmonic approximation, we show that the nonthermal phase transition is strongly first order and does not involve phonon softening, at odds with the thermal one. The transition is driven by the closure of the single particle gap in the photoexcited rhombohedral phase. Finally, we show that ultrafast x-ray diffraction data are consistent with a coexistence of the two phases, as expected in a first order transition. Our results are relevant for the understanding of phase transitions and bonding in phase change materials.

Kapitza Stabilization of Quantum Critical Order

Dynamical perturbations modify the states of classical systems in surprising ways and give rise to important applications in science and technology. For example, Floquet engineering exploits the possibility of band formation in the frequency domain when a strong, periodic variation is imposed on parameters such as spring constants. We describe here Kapitza engineering, where a drive field oscillating at a frequency much higher than the characteristic frequencies for the linear response of a system changes the potential energy surface so much that maxima found at equilibrium become local minima, in precise analogy to the celebrated Kapitza pendulum where the unstable inverted configuration, with the mass above rather than below the fulcrum, actually becomes stable. Our starting point is a quantum field theory of the Ginzburg-Devonshire type, suitable for many condensed matter systems, including particularly ferroelectrics and quantum paralectrics. We show that an off-resonance oscillatory electric field generated by a laser-driven terahertz source can induce ferroelectric order in the quantum-critical limit. Heating effects are estimated to be manageable using pulsed radiation; “hidden” radiation-induced order can persist to low temperatures without further pumping due to stabilization by strain. We estimate the Ginzburg-Devonshire free-energy coefficients in SrTiO3 using density-functional theory and the stochastic self-consistent harmonic approximation accelerated by a machine-learned force field. Although we find that SrTiO3 is not an optimal choice for Kapitza stabilization, we show that scanning for further candidate materials can be performed at the computationally convenient density-functional theory level. We suggest second harmonic generation, soft-mode spectroscopy, and x-ray diffraction experiments to characterize the induced order. Published by the American Physical Society 2024

Quantum symmetrization transition in superconducting sulfur hydride from quantum Monte Carlo and path integral molecular dynamics

AbstractWe study the structural phase transition, originally associated with the highest superconducting critical temperature Tc measured in high-pressure sulfur hydride. A quantitative description of its pressure dependence has been elusive for any ab initio theory attempted so far, raising questions on the actual mechanism leading to the maximum of Tc. Here, we estimate the critical pressure of the hydrogen bond symmetrization in the Im$$\bar{3}$$ 3 ¯ m structure, by combining density functional theory and quantum Monte Carlo simulations for electrons with path integral molecular dynamics for quantum nuclei. We find that the Tc maximum corresponds to pressures where local dipole moments dynamically form on the hydrogen sites, as precursors of the ferroelectric Im$$\bar{3}$$ 3 ¯ m-R3m transition, happening at lower pressures. For comparison, we also apply the self-consistent harmonic approximation, whose ferroelectric critical pressure lies in between the ferroelectric transition estimated by path integral molecular dynamics and the local dipole formation. Nuclear quantum effects play a major role in a significant reduction (≈50 GPa) of the classical ferroelectric transition pressure at 200 K and in a large isotope shift (≈25 GPa) upon hydrogen-to-deuterium substitution of the local dipole formation pressure, in agreement with the corresponding change in the Tc maximum location.

Observation of the most H2-dense filled ice under high pressure.

Hydrogen hydrates are among the basic constituents of our solar system's outer planets, some of their moons, as well Neptune-like exo-planets. The details of their high-pressure phases and their thermodynamic conditions of formation and stability are fundamental information for establishing the presence of hydrogen hydrates in the interior of those celestial bodies, for example, against the presence of the pure components (water ice and molecular hydrogen). Here, we report a synthesis path and experimental observation, by X-ray diffraction and Raman spectroscopy measurements, of the most H[Formula: see text]-dense phase of hydrogen hydrate so far reported, namely the compound 3 (or C[Formula: see text]). The detailed characterisation of this hydrogen-filled ice, based on the crystal structure of cubic ice I (ice I[Formula: see text]), is performed by comparing the experimental observations with first-principles calculations based on density functional theory and the stochastic self-consistent harmonic approximation. We observe that the extreme (up to 90 GPa and likely beyond) pressure stability of this hydrate phase is due to the close-packed geometry of the hydrogen molecules caged in the ice I[Formula: see text] skeleton.

Epiq: An open-source software for the calculation of electron-phonon interaction related properties

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 include: 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. Program summaryProgram Title:epiqCPC Library link to program files:https://doi.org/10.17632/f2syws66d7.1Developer's repository link:https://gitlab.com/the-epiq-team/epiqLicensing provisions: GPLv3Programming language: FORTRAN95External routines: BLAS (http://www/netlib.org/blas), LAPACK (http://www.netlib.org/lapack), Quantum ESPRESSO (https://www.quantum-espresso.org/), wannier90 (https://wannier.org/)Nature of problem: Direct first principles calculation of quantities obtained via linear response methods in solid-state systems, such as the deformation potential, can be computationally demanding, hindering proper convergence.Solution method: An interpolation scheme exploiting 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 is implemented in epiq. Within this approach it is possible to calculate many electron-phonon related properties with high accuracy and a low computational effort.

Theoretical determination of Ising-type transition by using the Self-Consistent Harmonic Approximation

Over the years, the Self-Consistent Harmonic Approximation (SCHA) has been successfully utilized to determine the transition temperature of many different magnetic models, particularly the Berezinskii–Thouless–Kosterlitz transition in two-dimensional ferromagnets. More recently, the SCHA has found application in describing ferromagnetic samples in spintronic experiments. In such a case, the SCHA has proven to be an efficient formalism for representing the coherent state in the ferromagnetic resonance state. One of the main advantages of using the SCHA is the quadratic Hamiltonian, which incorporates thermal spin fluctuations through renormalization parameters, keeping the description simple while providing excellent agreement with experimental data. In this article, we investigate the SCHA application in easy-axis magnetic models, a subject that has not been adequately explored to date. We obtain both semiclassical and quantum approaches of the SCHA for a general anisotropic magnetic model and employ them to determine various quantities such as the transition temperature, spin-wave energy spectrum, magnetization, and critical exponents. To verify the accuracy of the method, we compare the SCHA results with experimental and Monte Carlo simulation data for many distinct well-known magnetic materials.

Wigner Gaussian dynamics: Simulating the anharmonic and quantum ionic motion

The atomic motion controls important features of materials, such as thermal transport, phase transitions, and vibrational spectra. However, the simulation of ionic dynamics is exceptionally challenging when quantum fluctuations are relevant (e.g., at low temperatures or with light atoms) and the energy landscape is anharmonic. In this work, we formulate the Time-Dependent Self-Consistent Harmonic Approximation (TDSCHA) in the Wigner framework, paving the way for the efficient computation of the nuclear motion in systems with sizable quantum and thermal anharmonic fluctuations. Besides the improved numerical efficiency, the Wigner formalism unveils the classical limit of TDSCHA and provides a link with the many-body perturbation theory of Feynman diagrams. We further extend the method to account for the non-linear couplings between phonons and photons, responsible, e.g., for a nonvanishing Raman signal in high-symmetry Raman inactive crystals, firstly discussed by Rasetti and Fermi. We benchmark the method in phase III of high-pressure hydrogen ab initio. The nonlinear photon-phonon coupling reshapes the IR spectra and explains the high-frequency shoulder of the H2 vibron observed in experiments. The Wigner TDSCHA is computationally cheap and derived from first principles: it is unbiased by assumptions on the phonon-phonon and phonon-photon scattering and does not depend on empirical parameters. Therefore, the method can be adopted in unsupervised high-throughput calculations.

Quantum paraelectricity and structural phase transitions in strontium titanate beyond density functional theory

We demonstrate an approach for calculating temperature-dependent quantum and anharmonic effects with beyond density-functional theory accuracy. By combining machine-learned potentials and the stochastic self-consistent harmonic approximation, we investigate the cubic to tetragonal transition in strontium titanate and show that the paraelectric phase is stabilized by anharmonic quantum fluctuations. We find that a quantitative understanding of the quantum paraelectric behavior requires a higher-level treatment of electronic correlation effects via the random phase approximation. This approach enables detailed studies of emergent properties in strongly anharmonic materials beyond density-functional theory.

Stochastic path-integral approach for predicting the superconducting temperatures of anharmonic solids

We develop a stochastic path-integral approach for predicting the superconducting transition temperatures of anharmonic solids. By defining generalized Bloch basis, we generalize the formalism of the stochastic path-integral approach, which is originally developed for liquid systems. We implement the formalism for ab initio calculations using the projector augmented-wave method, and apply the implementation to estimate the superconducting transition temperatures of metallic deuterium and hydrogen sulfide. For metallic deuterium, which is approximately harmonic, our result coincides well with that obtained from the standard approach based on the harmonic approximation and the density functional perturbation theory. For hydrogen sulfide, we find that anharmonicity strongly suppresses the predicted superconducting transition temperature. Compared to the self-consistent harmonic approximation approach, our approach yields a transition temperature closer to the experimentally observed one.

Impact of ionic quantum fluctuations on the thermodynamic stability and superconductivity of LaBH8

The recent prediction of a metastable high-symmetry $Fm\overline{3}m$ phase of ${\mathrm{LaBH}}_{8}$ gives hopes to reach high superconducting critical temperatures at affordable pressures among ternary hydrogen-rich compounds. Making use of first-principles calculations within density functional theory and the stochastic self-consistent harmonic approximation, we determine that ionic quantum fluctuations drive the system dynamically unstable below 77 GPa, a much higher pressure than the 45 GPa expected classically. Quantum anharmonic effects stretch the covalent B-H bond in the ${\mathrm{BH}}_{8}$ units of the structure, and consequently, soften all hydrogen-character modes. Above 77 GPa $Fm\overline{3}m\phantom{\rule{4pt}{0ex}}{\mathrm{LaBH}}_{8}$ remains metastable, and interestingly, its superconducting critical temperature is largely enhanced by quantum anharmonic effects, reaching critical temperatures around 160 K at the verge of the dynamical instability. Our results suggest that low-pressure metastable phases with covalently bonded symmetric ${\mathrm{XH}}_{8}$ units will be destabilized by ionic quantum fluctuations.

Theoretical analysis of FMR-driven spin pumping current and its properties via the self-consistent harmonic approximation

We applied the self-consistent harmonic approximation (SCHA), combined with coherent states formalism, to study the ferromagnetic resonance (FMR) in a ferromagentic/normal metal junction. Due to the interface interaction, the FMR-generated spin current is injected from the magnetic insulator to the normal metal, the so-called spin pumping. Ordinarily, ferromagnetic models are described by bosonic representation or phenomenological theories; however, in a coherent magnetization state, the SCHA is the more natural choice to treat FMR problems. Over the years, the SCHA has successfully applied to investigate ferro- and antiferromagnetism in a wide range of scenarios. The main point of the SCHA formalism involves the adoption of a quadratic model for which corrections are included through temperature-dependent renormalization parameters. Therefore the SCHA is an efficient method for determining the properties of magnetically ordered phases. Using the SCHA, we obtained the temperature dependence of FMR-driven spin pumping. In addition, we found the spin-mix conductance, the additional damping from the angular momentum injection into the normal metal side, and the magnetic susceptibility. The SCHA outcomes are in remarkable agreement with the results of the literature.

Effectiveness of the Self-Consistent Harmonic Approximation in ferromagnets with dipolar interactions

Among the various methods for treating magnetic models, the Self-Consistent Harmonic Approximation (SCHA) has successfully described ferro and antiferromagnetism in many different scenarios. In particular, the SCHA is a valuable and easy formalism for determining transition temperatures as, for example, the Berezinskii–Kosterlitz–Thouless. The heart of the method includes thermal fluctuations through of a renormalization parameter depending on temperature. Nevertheless, most of the work has been done considering only short-range interactions, which results in an incomplete description of actual magnetic samples. Here, we generalize the SCHA to include the dipolar interaction in the thermodynamic analysis. The method is applied to analyze the well-known Europium Chalcogenides EuO and EuS. The SCHA results are in good agreement with the experimental measurements.

Berezinskii-Kosterlitz-Thouless transitions in classical and quantum long-range systems

In the past decades, considerable efforts have been made to understand the critical features of both classical and quantum long-range (LR) interacting models. The case of the Berezinskii-Kosterlitz-Thouless (BKT) universality class, as in the two-dimensional (2D) classical $XY$ model, is considerably complicated by the presence, for short-range interactions, of a line of renormalization group fixed points. In this paper, we discuss a field-theoretical treatment of the 2D $XY$ model with LR couplings, and we compare it with results from the self-consistent harmonic approximation. These methods lead to a rich phase diagram, where both power law BKT scaling and spontaneous symmetry breaking appear for the same (intermediate) decay rates of LR interactions. We also discuss the Villain approximation for the 2D $XY$ model with power law couplings, providing hints that, in the LR regime, it fails to reproduce the correct critical behavior. The obtained results are then applied to the LR quantum $XXZ$ spin chain at zero temperature. We discuss the relation between the phase diagrams of the two models, and we give predictions about the scaling of the order parameter of the quantum chain close to the transition.

LO-mode phonon of KCl and NaCl at 300 K by inelastic x-ray scattering measurements and first principles calculations

Longitudinal-optical (LO) mode phonon branches of KCl and NaCl were measured using inelastic x-ray scattering (IXS) at 300 K and calculated by the first-principles phonon calculation with the stochastic self-consistent harmonic approximation. Spectral shapes of the IXS measurements and calculated spectral functions agreed well. We analyzed the calculated spectral functions that provide higher resolutions of the spectra than the IXS measurements. Due to strong anharmonicity, the spectral functions of these phonon branches have several peaks and the LO modes along Γ–L paths are disconnected.

Effectiveness of the Self-Consistent Harmonic Approximation in Ferromagnets with Dipolar Interactions

Among the various methods for treating magnetic models, the Self-Consistent Harmonic Approximation (SCHA) has successfully described ferro and antiferromagnetism in many different scenarios. In particular, the SCHA is a valuable and easy formalism for determining transition temperatures as, for example, the Berezinskii-Kosterlitz-Thouless. The heart of the method includes thermal fluctuations through of a renormalization parameter depending on temperature. Nevertheless, most of the work has been done considering only short-range interactions, which results in an incomplete description of actual magnetic samples. Here, we generalize the SCHA to include the dipolar interaction in the thermodynamic analysis. The method is applied to analyze the well-known Europium Chalcogenides EuO and EuS. The SCHA results are in good agreement with the experimental measurements.