In the Comment by Bordag et al. [Phys. Rev. B 113, 207401 (2026)], concerns are raised about the validity of the results presented by Rodriguez-Lopez et al. [Phys. Rev. B 111, 115428 (2025)], where the theoretical descriptions of the electric conductivity of graphene obtained from the Kubo formula and from quantum field theory via the polarization tensor are compared. In this Reply, we show that these concerns arise from misinterpretations of our work, in which the results are either inaccurately represented or applied outside the domain of validity of the model. We address the comments concerning the derivation of the Luttinger formula for the electric conductivity from the Kubo formula and clarify why the results of our work cannot be arbitrarily extended to make claims on the gauge invariance. We further demonstrate that our findings are fully consistent with the established and widely accepted literature cited in the Comment. We confirm that the model for electric conductivity discussed in our work correctly predicts a vanishing electric current in the absence of an external electric field, as physically required, and in contrast to the model advocated by the Authors of the Comment. We also show that the electric permittivity does not exhibit a double pole in $\omega$, contrary to the claim made in the Comment. Finally, we emphasize that the inclusion of losses is a standard and well-established approach in the study of transport properties of materials, including graphene, and we take the opportunity to correct a few minor typographical errors in our work. We show and maintain that all results derived in our work are fully valid and correct.

${\mathrm{PbTiO}}_3/{\mathrm{SrTiO}}_3$ artificial superlattices recently emerged as a prototypical platform for the emergence and study of polar topologies. While previous studies mainly focused on the polar textures inherent to the ferroelectric ${\mathrm{PbTiO}}_3$ layers, the oxygen octahedra rotations inherent to the paraelectric ${\mathrm{SrTiO}}_3$ layers have attracted little attention. Here, we highlight a biunivocal relationship between distinct polar topologies—including $a_1/a_2$ domains, polar vortices, and skyrmions—within the ${\mathrm{PbTiO}}_3$ layers and specific patterns of oxygen octahedra rotations in the ${\mathrm{SrTiO}}_3$ layers. This relationship arises from a coupling mediated by inhomogeneous strains between the two materials and is shown to be reciprocal. Through second-principles atomistic simulations, we demonstrate that each polar texture imposes a corresponding rotation pattern, while conversely, a frozen oxygen octahedra rotation dictates the emergence of the associated polar state. This confirms the strong coupling between oxygen octahedra rotations in ${\mathrm{SrTiO}}_3$ and polarization in ${\mathrm{PbTiO}}_3$, highlighting their cooperative role in stabilizing complex polar textures in related superlattices.

Understanding the mechanism of structural phase transitions in rare-earth elements is a fundamental challenge in condensed matter physics, with significant implications for materials science applications. In this study, we present a systematic investigation of the phase transitions of yttrium under pressure conditions ($<30\mathrm{GPa}$) focusing on the hcp, Sm-type, and double hcp phases. A comparative study between the generalized gradient approximation (GGA) and meta-GGA functionals reveals that the Perdew-Burke-Ernzerhof functional significantly underestimates the phase transition pressures, whereas the restored regularized strongly constrained and appropriately normed functional (${\mathrm{r}}^2\mathrm{SCAN}$) provides accurate predictions of phase transition pressures which are in excellent agreement with experimental data. The results confirm that the experimentally observed phase transitions in yttrium are driven by vibrational instabilities, as evidenced by the emergence of soft acoustic modes in the phonon dispersion curves for both hcp and Sm-type phases. The analysis of the soft mode further reveals a different intermediate phase of $P\overline{6}2c$ symmetry occurring during the transition between the hcp and Sm-type phases. A transition from the centrosymmetric space group $P{6}_3/mmc$ (hcp) to the $t$-type subgroup $P\overline{6}2c$ involves loss of inversion symmetry and a reduction in point group symmetry from $D_{6h}$ to $D_{3h}$. Additionally, calculations of elastic properties confirm mechanical softening at the phase boundaries, particularly in the hcp phase, suggesting a strong correlation between elastic softening and structural transitions. These findings indicate that the pressure-induced modifications of electronic bonding, reflected in gradual $s$ to $d$ charge transfer and accompanied by phonon softening and mechanical weakening near the transition pressures, play a central role in driving the structural phase transitions in yttrium.

Recently, antiferroelectric-like behavior has been unexpectedly observed in several studies on ${\mathrm{BiFeO}}_3$ (BFO) thin films and superlattices; however, this phenomenon has still received limited attention. In this work, we investigate $n/n {\mathrm{BiFeO}}_3/{\mathrm{SrRuO}}_3$ (BFO/SRO) multiferroic-metal superlattices based on density functional theory (DFT) calculations. Two complex polar ordering states were identified in the BFO/SRO superlattices, characterized by an in-plane polar order (IP phase) and an in-plane antipolar order (IAP phase). Meanwhile, both phases exhibit out-of-plane polar displacements, even in ultrashort-period superlattices. As the layer thickness increases up to $n=5$, the ground state transitions from the IP phase to the IAP phase. This process is analogous to a ferroelectric-to-antiferroelectric phase transition along the in-plane direction. The emergence of the antiferroelectric-like IAP phase is closely linked to the modulation of electrostatic boundary conditions. Furthermore, our analysis of the depolarization field with increasing layer thickness reveals a competition between the metallic screening effect and the depolarization field. This study provides insights for modulating the potential antiferroelectric-like phase in BFO-based electronic nanodevices.

Minimally stable site (MSS) clusters play a dominant role in shaping avalanches in the self-organized critical (SOC) systems. A local smoothing (diffusion) mechanism reshapes the MSS landscape, delaying rare avalanches until the emergence of spanning avalanches, thereby inducing self-organized bistability. We find bistability-induced self-similarity in MSS clusters, which is suppressed in the thermodynamic limit. The inverse Ising problem technique is employed to track the dynamical transition to this point at which the spanning avalanches form. An emergent magnetic instability in the dual Ising model is observed, marking a transition to a correlated percolation regime. At the bistability-induced self-similarity point, the effective pairwise interactions in the dual Ising system vanish—signaling a magnetic transition characterized by abrupt changes in magnetization and spin susceptibility. Crucially, we show that diffusion fundamentally reshapes avalanche dynamics: the spatial anticorrelations of MSSs in standard SOC systems transform into positive correlations when diffusion is introduced.

Symmetries strongly influence transport properties of quantum many-body systems and can lead to deviations from the generic case of diffusion. In this work, we study the impact of time-reversal symmetry breaking on the transport and its universal aspects in integrable chiral spin ladders. We observe that the infinite-temperature spin transport is superdiffusive with a dynamical critical exponent $z=3/2$ matching the one of the Kardar-Parisi-Zhang (KPZ) universality class, which also lacks the time reversal symmetry. However, we find that fluctuations of the net magnetization transfer deviate from the KPZ predictions. Moreover, the full probability distribution of the associated spin current obeys fluctuation symmetry despite broken time-reversal and space-reflection symmetries. To further investigate the role of conserved quantities, we introduce an integrable quantum circuit that shares the essential symmetries with the chiral ladder and which exhibits analogous dynamical behavior in the absence of energy conservation. Our work shows that time-reversal symmetry breaking is compatible with superdiffusion but insufficient to stabilize the KPZ universality in integrable systems. This suggests that additional fundamental features are missing in order to identify the emergence of such dynamics in quantum matter.

We have investigated ${\mathrm{NdCo}}_{2-x}{\mathrm{Ni}}_x$ cubic Laves compounds with 0 $\le x\le$ 1 using neutron diffraction and bulk magnetization measurements to study the influence of partial Ni substitutions of Co on the phase transitions and the magnetocaloric effect (MCE). Upon cooling, ${\mathrm{NdCo}}_2$ undergoes a cubic-to-tetragonal transition at 100 K and a tetragonal-to-orthorhombic transition at 42 K. The transitions are associated with long-range ferromagnetic ordering of the magnetic moments along the $c$ axis and spin reorientation into the $ab$ plane, respectively. Both transitions shift to lower temperatures as the Ni content $x$ increases. For $x\ge 0.5$, the orthorhombic phase is suppressed. Additionally, there was a reduction in the magnetic moment upon increasing the Ni substitution of Co. The MCE was determined both indirectly and directly, with good agreement between the methods. ${\mathrm{NdCo}}_2$ exhibits an adiabatic temperature change of 6.3 K for a field of 20 T, which is decreased to 4.9 K for NdCoNi for the same field strength due to the reduced magnetic moment upon Ni substitution.

Hopfions, as three-dimensional topological solitons, have recently attracted extensive interest due to their unique properties and potential applications in spintronic devices. While the dynamics of magnetic hopfions under electrical and magnetic stimuli have been well established, their behavior in response to mechanical excitation remains elusive. In this paper, we demonstrate numerically and analytically that magnetic hopfions undergo rectilinear motion in chiral magnetic trilayers when subjected to a strain gradient. To identify the strain range under which hopfions remain stable during motion, a phase diagram of strain versus the Dzyaloshinskii-Moriya interaction (DMI) is constructed. When a strain gradient is applied within the identified strain range, the hopfion moves sustainably along the gradient direction without exhibiting a Hall effect. Moreover, the hopfion velocity can be effectively tuned by adjusting the strain gradient, magnetic damping, and intrinsic magnetoelastic parameters. This work demonstrates an energy-efficient approach to manipulating magnetic hopfions and establishes a theoretical foundation for the mechanical control of three-dimensional magnetic solitons.

${\mathrm{MnBi}}_2{\mathrm{Te}}_4$ (MBT) is the first intrinsic magnetic topological insulator, combining a topologically protected surface metallic state and intrinsic magnetic order. A structural compatibility with the nonmagnetic ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ (BT) parent compound gives a possibility to create MBT/BT heterostructures and manipulate their magnetic state in view of optimizing the quantum anomalous Hall effect (QAHE). In this work an extensive nuclear magnetic resonance (NMR) study, supported by the bulk magnetization measurements, has been performed at 4.2 K on a self-organized single crystal ${\mathrm{MnBi}}_2{\mathrm{Te}}_4/{\left({\mathrm{Bi}}_2{\mathrm{Te}}_3\right)}_n$ heterostructure, obtained from the ${\mathrm{Mn}}_{0.81}{\mathrm{Bi}}_{2.06}{\mathrm{Te}}_{4.13}$ melt. ${}^{55}\mathrm{Mn}$ and ${}^{209}\mathrm{Bi}$ NMR signals have been recorded as a function of the out-of-plane magnetic field up to 6 T, covering a spin-flop transition from the antiferromagnetic (AFM) to the canted antiferromagnetic configuration of the Mn layers. The canting angle at different external field values has been estimated based on NMR data. Presence of the AFM-coupled Mn antisites has been evidenced and shown to induce an antiparallel magnetic moment on Bi atoms within the host Bi layer. Detection of the induced magnetic moment on bismuth which contributes a new ferromagnetic component is of utmost importance for understanding the magnetic interactions in the MBT/BT system. These findings have potentially important implications for engineering the QAHE devices.

Magnetoelectric coupling in multiferroics provides a low-energy means of manipulating magnetic spin textures, including topological ones such as skyrmions, using an electric field instead of a current. However, existing demonstrations were realized in multiferroic heterostructures, relying on ferroelectric switching of one of the constituents and the interface effects. Here, by combining first-principles calculations, symmetry analysis, Monte Carlo and spin-dynamics simulations, we predict a two-dimensional multiferroic material with multiple mechanisms of spin-induced ferroelectricity, and then demonstrate the electric-field control of skyrmions. We further identify the mechanism of spin-induced ferroelectricity most relevant to skyrmion manipulation, providing a more general design strategy for realizing electric-field control of skyrmions in single-phase multiferroic materials.

We develop a fully first-principles approach for topological magnons within the framework of time-dependent density functional perturbation theory. Spin dynamics equations are accurately solved to obtain magnon energies and corresponding density fluctuations. We demonstrate that magnon wave function can be expressed by a collection of perturbed electronic wave functions obtained from the decomposition of density fluctuation, enabling direct calculation of magnonic quantities including Berry curvature and Chern number. As a concrete example, we show that monolayer ${\mathrm{CrI}}_3$ can host topological magnons driven by spin-orbit coupling. Our model-free approach paves the way for comprehensive studies of topological magnons in real materials.

We study the magnetic-field response of interacting electron systems within mean-field theory using perturbation theory. We show that the linear response of the mean-field density matrix to a weak magnetic field is purely geometric: it depends only on wave function derivatives, the Berry connections linking the occupied and unoccupied subspaces, and does not explicitly depend on the interaction potential and the quasiparticle dispersion. This leads to compact, gauge-invariant projector expressions for both the Středa formula and the formula for orbital magnetization. Our calculation explicitly elucidates the role of exchange and self-consistency in defining current vertices for orbital magnetization calculations. Our work establishes a direct connection between mean-field theory, quantum geometry, and the noninteracting topological band theory.

The possibility of a zero-temperature altermagnetic instability in anisotropic two-dimensional electron systems in the diffusive regime is analyzed in the presence and absence of spin-orbit coupling. Allowing for ferromagnetism, a phase diagram is built as a function of the parameter that controls anisotropy and the strength of the interactions. It is found that at zero spin-orbit coupling ferromagnetism only dominates at small values of the anisotropy and the coupling constant. Larger values of these parameters favor the formation of altermagnetism. At finite spin-orbit coupling, a paramagnetic phase competes with the other two, and a quantum critical point appears. The phase transition from the paramagnetic to the magnetically ordered phases is of second order, while the phase transition between ferromagnet and altermagnet states is first order. The altermagnetic phase is not sensitive to small magnetic fields, coexisting with a field-induced magnetization.

We present a first-principles framework for the calculation of phonons in nanostructures with cyclic and/or helical symmetry. In particular, we derive a cyclic- and helical-symmetry-adapted representation of the dynamical matrix at arbitrary phonon wave vectors within a variationally formulated, symmetry-adapted density functional perturbation theory framework. In so doing, we also derive the acoustic sum rules for cylindrical geometries, which include a rigid-body rotational mode in addition to the three translational modes. We implement the cyclic- and helical-symmetry-adapted formalism within a high-order finite-difference discretization. Using carbon nanotubes as representative systems, we demonstrate the accuracy of the framework through excellent agreement with periodic plane-wave results. We further apply the framework to compute the Young's and shear moduli of carbon nanotubes, as well as the scaling laws governing the dependence of ring and radial breathing mode phonon frequencies on nanotube diameter. The elastic moduli are found to be in agreement with previous density functional theory and experimental results, while the phonon scaling laws show qualitative agreement with previous atomistic simulations.

Motivated by recent experimental advancements in scanning optical stroboscopic confocal microscopy and spectroscopy measurements, which have facilitated exceptional energy-space-time resolution for investigating edge and bulk dynamics in fractional quantum Hall systems, we formulated a model for the pump-probe process on the edge. Starting with a ground state, we applied a tip potential near the fractional quantum Hall liquid edge, which was subsequently turned off after a defined time duration. By examining how the specific nature of the tip potential influences the evolution of the wave function and its distribution in energy spectrum, we identify that quench dynamics of the edge pulse leads to excitations that spread both along the edge and perpendicularly into the bulk. Moreover, magnetoroton excitations are predominant among the bulk excitations. These results align well with the experimental observations. Furthermore, we analyzed the effects of the tip's position, intensity, and duration on the dynamics.

As electronic devices continue to advance, accurate thermal management at the nanoscale is becoming increasingly crucial in engineering design. While the phonon Boltzmann transport equation (BTE) is commonly used for this purpose, conventional boundary conditions, such as the acoustic mismatch model (AMM) and diffuse mismatch model (DMM), fail to accurately predict nanoscale heat transport. In this work, we evaluate an interfacial boundary condition based on the scattering matrix (S-matrix) formalism within a low-variance direct simulation Monte Carlo (DSMC) solution of the BTE. The S-matrix framework resolves frequency- and polarization-dependent phonon transmission and reflection, enabling us to examine how microscopic details, including anisotropic scattering neglected by traditional models, lead to large differences in interfacial thermal resistance at ideal and disordered silicon/germanium interfaces. We further demonstrate that while S-matrix data can be used to accurately tune DMM for larger systems of known materials, DMM remains unsuitable for simulating nanoscale devices. Our predictions show good agreement with experimental measurements, and can be used for more accurate nanoscale thermal design of real-world devices.

A general symmetry-controlled mechanism is proposed for realizing dual flat bands of bound state in the continuum (BIC) and its radiative counterpart in photonic crystal slabs. By breaking the vertical mirror symmetry of a slab, interpolarization coupling between TE-like and TM-like modes is activated, while intrapolarization coupling among modes within the same polarization class is simultaneously preserved. The cooperative action of these two coupling channels gives rise to the concurrent flattening of both the BIC-hosting band and the radiative band, resulting in a dual flat-band system with strongly contrasting quality (Q) factors. An effective two-step coupling model is constructed to capture the essential physics and show that the emergence of the flat bands is governed by geometric tuning rather than accidental degeneracies. The mechanism is shown to be generic with respect to polarization and material platforms, enabling dual flat-band states in both low- and high-index systems, with substantially enhanced angular bandwidths in the latter. These findings establish a unified route for flat-band photonic engineering and provide a robust platform for angle-tolerant resonant photonic functionalities.

Defects in two-dimensional (2D) semiconductors play a decisive role in determining their electronic, optical, catalytic and quantum properties. Understanding how defect energy levels respond to variations in layer thickness is essential for achieving reproducible and scalable device performance. We report the persistence of layer-tolerant defect levels in rhenium disulfide $\left({\mathrm{ReS}}_2\right)$, where both donor- and acceptor-type charge transition levels remain nearly unchanged from monolayer to bulk in both AA and AB stacking. The associated two-level quantum system also retains its character across thicknesses, enabling ${\mathrm{ReS}}_2$ to serve as a platform for layer-tolerant single-photon emitters. The invariance arises from the interplay between electronic energy minimization and structural relaxation, which together counteract quantum confinement and reduced dielectric screening. Additionally, the intrinsically weak interlayer coupling in ${\mathrm{ReS}}_2$ plays a crucial role. Our findings uncover the microscopic origin of this unique behavior, distinguishing ${\mathrm{ReS}}_2$ from other transition-metal dichalcogenides and highlighting its potential for thickness-independent optoelectronic and quantum photonic applications.

A recent STM experiment in two-dimensional bilayer graphene [Y.-C. Tsui et al., Nature (London) 628, 287 (2024)], under a strong perpendicular magnetic field, has made a direct observation of the existence of three distinct filling-factor-dependent quantum phases in the lowest Landau level: the incompressible fractional quantum Hall liquid, a crystalline compressible hexagonal Wigner crystal with long-range order and rotational symmetry breaking, and a random localized solid phase with no spatial order. We argue that the spatially random localized phase at low filling is the recently proposed disorder-dominated strongly localized amorphous “Anderson solid” phase (A. Babbar et al., arXiv:2601.03521), which appears generically at a sample-dependent filling factor.