Entropy is a fundamental thermodynamic quantity that is a measure of the accessible microstates available to a system, with the stability of a system determined by the magnitude of the total entropy of the system. This is valid across truly mind boggling length scales, from nanoparticles to galaxies. However, quantitative measurements of entropy change using calorimetry are predominantly macroscopic, with direct atomic-scale measurements being exceedingly rare. Here, we experimentally quantify the polar configurational entropy (in meV/K) using sub-angstrom resolution aberration corrected scanning transmission electron microscopy in a single crystal of the prototypical ferroelectric ${\mathrm{LiNbO}}_3$ through the quantification of the niobium and oxygen atom column deviations from their paraelectric positions. Significant excursions of the niobium-oxygen polar displacement away from its symmetry-constrained direction are seen in single domain regions which increase in the proximity of domain walls. Combined with first-principles theory plus mean field effective Hamiltonian methods, we demonstrate the variability in the polar order parameter, which is stabilized by an increase in the magnitude of the configurational entropy. This study presents a powerful tool to quantify entropy from atomic displacements and demonstrates its dominant role in local symmetry breaking at finite temperatures in classic, nominally Ising ferroelectrics.

We study theoretically Anderson localization of two-dimensional massless pseudospin-1 Dirac particles in a random one-dimensional scalar potential. We focus explicitly on the effect of disorder correlations, considering a short-range correlated dichotomous random potential at all strengths of disorder. We also consider a $\delta$-function correlated random potential at weak disorder. Using the invariant imbedding method, we calculate the localization length in a numerically precise way and analyze its dependencies on incident angle, disorder correlation length, disorder strength, energy, wavelength, and average potential over a wide range of parameter values. In addition, we derive analytical formulas for the localization length, which are very accurate in the weak and strong disorder regimes. From the Dirac equation, we obtain an expression for the effective wave impedance, using which we explain several conditions for delocalization. We also deduce a condition under which the localization length vanishes. For all cases considered, the localization length depends nonmonotonically on the disorder correlation length and diverges as ${\theta }^{-4}$ as the incident angle $\theta$ goes to zero. As the disorder strength is varied from zero to infinity, we find that there appear three different scaling regimes. As the energy or wavelength is varied from zero to infinity, there appear three or four different scaling regimes with different exponents, depending on the value of the average potential. The crossovers between different scaling regimes are explained in terms of the disorder correlation effect.

We report the discovery and direct visualization of skyrmion clusters with mutually orthogonal orientations of constituent isolated skyrmions in chiral liquid crystals and ferromagnets. We show that the nascent conical state underlies the attracting inter-skyrmion potential, whereas an encompassing homogeneous state leads to the repulsive skyrmion-skyrmion interaction. The crossover between different regimes of skyrmion interaction could be identified upon changing layer thickness and/or the surface anchoring. We develop a phenomenological theory describing two types of skyrmions and the underlying mechanisms of their interaction. We show that isolated horizontal skyrmions with the same polarity may approach a vertical isolated skyrmion from both sides and thus constitute two energetically different configurations which are also observed experimentally. In an extreme regime of mutual attraction, the skyrmions wind around each other forming compact superstructures with undulations. We also indicate that our numerical simulations on skyrmion clusters are valid in a parameter range corresponding to the A-phase region of cubic helimagnets.

We demonstrate the existence of the giant proximity magnetoresistance (PMR) effect in a graphene spin valve where spin polarization is induced by a nearby magnetic insulator. PMR calculations were performed for yttrium iron garnet (YIG), cobalt ferrite (CFO), and two europium chalcogenides EuO and EuS. We find significant PMR (up to 100%) values defined as a relative change of graphene conductance with respect to parallel and antiparallel alignment of two proximity-induced magnetic regions within graphene. Namely, for high Curie temperature ($T_C$) CFO and YIG insulators, which are particularly important for applications, we obtain 22% and 77% at room temperature, respectively. For low $T_C$ chalcogenides, EuO and EuS, the PMR is 100% in both cases. Furthermore, the PMR is robust with respect to system dimensions and edge-type termination, and it even maintains significant values (around 50% for YIG) in the presence of considerable spin-orbit coupling strength. Our findings show that it is possible to induce spin-polarized currents in graphene with no direct injection through magnetic materials.

Two-dimensional layered itinerant ferromagnetic ${\mathrm{Fe}}_{3-x}{\mathrm{GeTe}}_2$ is considered as a candidate for applications in heterostructure-based spintronics because of its near-room-temperature Curie temperature and highly tunable characteristic of ferromagnetism. Moreover, the strong anisotropic magnetism of ${\mathrm{Fe}}_{3-x}{\mathrm{GeTe}}_2$ is another great advantage for fabricating magnetic storage devices. However, many relevant properties of its anisotropy still remain poorly understood, especially the basic mechanism of anisotropic magnetic interaction. In this work, we focus on the study of magnetic coupling in single-crystal ${\mathrm{Fe}}_{3-x}{\mathrm{GeTe}}_2$ ($x\approx 0.28$) by the anisotropic magnetization, magnetic entropy change, and critical behavior. Our results confirm that the magnetization is angle dependent $\left[M\right(\phi \left)\right]$, in which the easy magnetic axis is along the $c$ axis while it exhibits absolute isotropic property in the $ab$ plane. The magnetic entropy change $\left[\mathrm{\Delta }{S}_M\right]$ also reveals an anisotropic feature between $H//c$ and $H//ab$. By fitting the field-dependent parameters of $\mathrm{\Delta }{S}_M(T,H)$, it gives the critical exponents $\beta =0.361\left(3\right), \gamma =1.736\left(7\right)$, and $\delta =5.806\left(8\right)$ for $H//c$, while $\beta =0.714\left(3\right), \gamma =1.243\left(7\right)$, and $\delta =2.741\left(1\right)$ for $H//ab$. The critical exponents with $H//c$ belong to the theoretical prediction of three-dimensional Heisenberg model, which suggest a short-range magnetic coupling. However, the critical exponents with $H//ab$ are close to those of mean-field model, which indicates a long-range magnetic coupling. The determined critical exponents suggest that the anisotropic magnetic coupling of ${\mathrm{Fe}}_{3-x}{\mathrm{GeTe}}_2$ is strongly dependent on orientations of the applied magnetic field.

Metallic nonlocal spin valves (NLSVs) are important in modern spintronics due to their ability to separate pure spin current from charge current. These metallic nanostructures, often constructed from features with widths in the deep submicron regime, generate significant thermal gradients in operation, and the heat generated has important consequences for spin injection and transport. We use e-beam nanolithography to manufacture NLSVs with Ni-Fe alloy ferromagnetic nanowires and aluminum spin channels on 500-nm silicon nitride (Si-N) membranes to lower the thermal conductance of the substrate dramatically. While this extreme example of thermal engineering in a spintronic system increases the background nonlocal signals in ways expected based on earlier work, it also enhances thermoelectric effects, including the anomalous Nernst effect, and reveals a previously unknown thermally assisted electrical spin injection that results from a purely in-plane thermal gradient. We examine these effects as a function of temperature and, by careful comparison with 2D finite element models of the thermal gradients calculated at a single temperature, demonstrate that the anomalous Nernst coefficient of the 35-nm-thick Ni-Fe alloy, $R_{\mathrm{N}}=0.17$ at $T=200\mathrm{K}$, is in line with the few previous measurements of this effect for thin films.

The polarization of $\mathrm{PbZ}{\mathrm{r}}_{0.52}\mathrm{T}{\mathrm{i}}_{0.48}{\mathrm{O}}_3$ thin film is switched by changing film thickness through the competition between the strain relaxation-induced flexoelectric fields and the interfacial effects. Without an applied electric field, this reversal of polarization is exploited to control the magnetic properties of $\mathrm{L}{\mathrm{a}}_{0.67}\mathrm{S}{\mathrm{r}}_{0.33}\mathrm{Mn}{\mathrm{O}}_3$ by the competition/cooperation between the charge-mediated and the strain-mediated effects. Scanning transmission electron microscopy, polarized near-edge x-ray absorption spectroscopy, and half-integer diffraction measurements are employed to decode the intrinsic magnetoelectric effects in the $\mathrm{L}{\mathrm{a}}_{0.67}\mathrm{S}{\mathrm{r}}_{0.33}\mathrm{Mn}{\mathrm{O}}_3/\mathrm{PbZ}{\mathrm{r}}_{0.52}\mathrm{T}{\mathrm{i}}_{0.48}{\mathrm{O}}_3$ heterostructures. With $\mathrm{PbZ}{\mathrm{r}}_{0.52}\mathrm{T}{\mathrm{i}}_{0.48}{\mathrm{O}}_3$ films $<48\mathrm{nm}$, the polarization-driven carrier density modulation around the $\mathrm{L}{\mathrm{a}}_{0.67}\mathrm{S}{\mathrm{r}}_{0.33}\mathrm{Mn}{\mathrm{O}}_3/\mathrm{PbZ}{\mathrm{r}}_{0.52}\mathrm{T}{\mathrm{i}}_{0.48}{\mathrm{O}}_3$ interface and the strain-mediated Mn $3d$ orbital occupancy work together to enhance the magnetism of 14-unit-cell $\mathrm{L}{\mathrm{a}}_{0.67}\mathrm{S}{\mathrm{r}}_{0.33}\mathrm{Mn}{\mathrm{O}}_3$ film; with $\mathrm{PbZ}{\mathrm{r}}_{0.52}\mathrm{T}{\mathrm{i}}_{0.48}{\mathrm{O}}_3$ layers $>48\mathrm{nm}$, the strain-induced change of bond length/angle of $\mathrm{Mn}{\mathrm{O}}_6$ accompanied with a modified spin configuration are responsible for the decrease in the Curie temperature and magnetization of 14-unit-cell $\mathrm{L}{\mathrm{a}}_{0.67}\mathrm{S}{\mathrm{r}}_{0.33}\mathrm{Mn}{\mathrm{O}}_3$ film.

Spin waves propagation in ferromagnetic films, several tens of nanometers thick, have recently received increasing attention, in view of the development of magnonic devices operating in the GHz range of frequencies. A detailed knowledge of the dispersion curves and of the spatial characteristics of the spin-wave modes is preliminary to any technological application, particularly in the “mesoscopic range” (50–200 nm) of film thickness, where several dipole-exchange modes may appear in the spectrum, exhibiting frequency crossing and hybridization as a function of their wave number. In this work, the mutual interaction and the hybridization of the dipole-exchange spin-wave modes was investigated in a nitrogen-implanted iron (Fe-N) film, 78-nm-thick, in-plane magnetized. The spin-wave dispersion curves were measured by using Brillouin light scattering, and the experimental results were interpreted combining micromagnetic simulations and theoretical calculations in the framework of a dipole-exchange spin-wave mode approach. A noticeable hybridization between the spin-wave modes was observed, due to the simultaneous presence of a marked perpendicular magnetic anisotropy and a rather high saturation magnetization. The hybridization was found to induce a very large gap ($\mathrm{\Delta }\nu \approx 5$ GHz) between the low-frequency spin waves at high wave vector ($k\approx {10}^5$ rad/cm). Consequently, in such a $k$ range the simultaneous presence of two spin-wave modes with sizeable ($v_g\approx 1.5$ km/s) but opposite group velocity was observed, opening a way for the potential use of Fe-N films in magnon spintronics.

The character of the superconducting phase of ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$ is the topic of a longstanding discussion. The classification of the symmetry allowed order parameters has relied on the tetragonal symmetry of the lattice and on cylindrical Fermi surfaces, usually taken to be featureless, not including the nontrivial symmetry aspects related to their orbital content. Here we show how the careful account of the orbital degree of freedom and a three-dimensional description lead to a much richer classification of order parameters. We analyze the stability and degeneracy of these new order parameters from the perspective of the concept of superconducting fitness and propose new order parameter candidates which can systematically account for the observed phenomenology in this material.

We investigate the behavior of the critical temperature $T_c$ in superconductor/ferromagnet/superconductor (S/F/S) trilayers in the dirty limit as a function of the ferromagnetic layer thickness $d_f$ and the S/F interface transparency. We perform $T_c$ calculations using the general self-consistent multimode approach based on the Usadel equations in Matsubara Green's functions technique, and compare the results with the single-mode approximation, widely used in literature. Both methods produce similar results for sufficiently low interface transparency. For transparent interfaces, we obtain a qualitatively different $T_c\left(d_f\right)$ behavior. Using the multimode approach, we observe multiple 0-$\pi$ transitions in critical temperature, which cannot be resolved by the single-mode approximation. We also calculate the critical S layer thickness at given $d_f$ when an S/F/S trilayer still has a nonzero critical temperature. Finally, we establish the limits of applicability of the single-mode approximation.

Nodal line semimetals with symmetry protected band structure offer a platform for the investigation of a number of emerging quantum phenomena. Using first-principles calculation combined with symmetry analysis, we show that ${\mathrm{Ba}}_3{\mathrm{TiIr}}_2{\mathrm{O}}_9$ hosts a Dirac nodal line (DNL) along the $A\text{−}L$ direction of the Brillouin zone, protected by the glide reflection symmetry. In the presence of the spin-orbit coupling, even though the DNL along the $A\text{−}L$ direction is protected, DNLs along other directions as well as multiple nodal loops present in the system gap out. The gapped out Dirac nodal loops act as a source of spin Berry curvature, resulting in a large spin Hall conductivity ($\approx 300$ $\frac{\hslash }{e}{\mathrm{\Omega }}^{-1}{\mathrm{cm}}^{-1}$). This suggests the possible application of the material as a spin current detector.

Electrons on a two-dimensional lattice which is exposed to a strong uniform magnetic field show intriguing physical phenomena. The spectrum of such systems exhibits a complex (multi)band structure known as Hofstadter's butterfly. For fillings at which the system is a band insulator, one observes a quantized integer-valued Hall conductivity ${\sigma }^{xy}$ corresponding to a topological invariant, the first Chern number ${\mathcal{C}}_1$. This is robust against many-body interactions as long as no changes in the gap structure occur. Strictly speaking, this stability holds only at zero temperatures $T$, while for $T>0$ correlation effects have to be taken into account. In this paper, we address this question by presenting a dynamical mean-field theory (DMFT) study of the Hubbard model in a uniform magnetic field. The inclusion of local correlations at finite temperature leads to (i) a shrinking of the integer plateaus of ${\sigma }^{xy}$ as a function of the chemical potential and (ii) eventually to a deviation from these integer values. We demonstrate that these effects can be related to a correlation-driven narrowing and filling of the band gap, respectively.

We revisit the issue of the resistivity of a two-dimensional electronic system tuned to a nematic quantum critical point (QCP), focusing on the nontrivial impact of the coupling to the acoustic phonons. Due to the unavoidable linear coupling between the electronic nematic order parameter and the lattice strain fields, long-range nematic interactions mediated by the phonons emerge in the problem. By solving the semiclassical Boltzmann equation in the presence of scattering by impurities and nematic fluctuations, we determine the temperature dependence of the resistivity as the nematic QCP is approached. One of the main effects of the nematoelastic coupling is to smooth the electronic nonequilibrium distribution function, making it approach the simple cosine angular dependence even when the impurity scattering is not too strong. We find that at temperatures lower than a temperature scale set by the nematoelastic coupling, the resistivity shows the $T^2$ behavior characteristic of a Fermi liquid. This is in contrast to the $T^{4/3}$ low-temperature behavior expected for a lattice-free nematic quantum critical point. More importantly, we show that the effective resistivity exponent ${\alpha }_{\text{eff}}\left(T\right)$ in $\rho \left(T\right)-{\rho }_0\sim T^{{\alpha }_{\text{eff}}\left(T\right)}$ displays a pronounced temperature dependence, implying that a nematic QCP cannot generally be characterized by a simple resistivity exponent. We discuss the implications of our results to the interpretation of experimental data, particularly in the nematic superconductor ${\mathrm{FeSe}}_{1-x}{\mathrm{S}}_x$.

Energy-saving spintronics are believed to be implementable on systems hosting the persistent spin helix (PSH) since they support an extraordinarily long spin lifetime of carriers. However, achieving the PSH requires a unidirectional spin configuration in the momentum space, which is practically nontrivial due to the stringent conditions for fine-tuning the Rashba and Dresselhaus spin-orbit couplings. Here, we predict that the PSH can be intrinsically achieved on a two-dimensional (2D) group-IV monochalcogenide $MX$ monolayer, a new class of the noncentrosymmetric 2D materials having in-plane ferroelectricity. Due to the $C_{2v}$ point-group symmetry in the $MX$ monolayer, a unidirectional spin configuration is preserved in the out-of-plane direction and thus maintains the PSH that is similar to the [110] Dresselhaus model in the [110]-oriented quantum well. Our first-principle calculations on various $MX$ ($M=$ Sn, Ge; $X=$ S, Se, Te) monolayers confirmed that such typical spin configuration is observed, in particular, at near the valence-band maximum where a sizable spin splitting and a substantially small wavelength of the spin polarization are achieved. Importantly, we observe reversible out-of-plane spin orientation under opposite in-plane ferroelectric polarization, indicating that an electrically controllable PSH for spintronic applications is plausible.

In Weyl semimetals, there is an intriguing possibility of realizing a pseudo-magnetic field in the presence of small strain due to certain special cases of static deformations. This pseudo-magnetic field can be large enough to form quantized Landau levels and thus become observable in Weyl semimetals. In this paper we experimentally show the emergence of a pseudo-magnetic field ($\sim 3$ T) by scanning tunneling spectroscopy on the doped Weyl semimetal Re-${\mathrm{MoTe}}_2$, where distinct Landau-level oscillations in the tunneling conductance are clearly resolved. The crystal lattice is intrinsically strained where large area scanning tunneling microscopy imaging of the surface reveals differently strained domains where atomic scale deformations exist forming topographic ripples with varying periodicity in the real space. The effect of the pseudo-magnetic field is clearly resolved in areas under maximum strain.

We study the spin-boson model without the counterrotating terms by a numerically exact method based on variational matrix product states. Surprisingly, the second-order quantum phase transition (QPT) is observed for the sub-Ohmic bath in the rotating-wave approximations. Moreover, first-order QPTs can also appear before the critical points. With the decrease of the bath exponents, these first-order QPTs disappear successively, while the second-order QPT remains robust. The second-order QPT is further confirmed by multi-coherent-state variational studies, while the first-order QPT is corroborated with the exact diagonalization in the truncated Hilbert space. Extension to the Ohmic bath is also performed, and many first-order QPTs appear successively in a wide coupling regime, in contrast to previous findings. The previous pictures for many physical phenomena for the spin-boson model in the rotating-wave approximation have to be modified at least at strong coupling.

We use a family of critical spin chain models discovered recently by one of us [M. Greiter, Mapping of Parent Hamiltonians (Springer, Berlin, 2011)] to propose and elaborate that non-Abelian, SU(2) level-$k=2S$ anyon statistics manifests itself in one dimension through topological selection rules for fractional shifts in the spacings of linear momenta, which yield an internal Hilbert space of $2^n$ (in the thermodynamic-limit) degenerate states. These shifts constitute the equivalent to the fractional shifts in the relative angular momenta of anyons in two dimensions. We derive the rules first for Ising anyons, and then generalize them to SU(2) level-$k$ anyons. We establish a one-to-one correspondence between the topological choices for the momentum spacings and the fusion rules of spin-$\frac{1}{2}$ spinons in the SU(2) level-$k$ Wess-Zumino-Witten model, where the internal Hilbert space is spanned by the manifold of allowed fusion trees in the Bratteli diagrams. Finally, we show that the choices in the fusion trees may be interpreted as the choices of different domain walls between the $2S+1$ possible, degenerate dimer configurations of the spin-$S$ chains at the multicritical point.

We predict a mechanism of spontaneous stabilization of a uniaxial density wave in a two-dimensional metal with an isotropic Fermi surface in the presence of external magnetic field. The topological transformation of a closed Fermi surface into an open one decreases the electron band energy due to delocalization of electrons initially localized by magnetic field, additionally affected by the magnetic breakdown effect. The driving mechanism of such reconstruction is a periodic potential due to the self-consistently formed electron density wave. It is accompanied with quantum oscillations periodic in inverse magnetic field, similar to the standard de Haas-van Alphen effect, due to Landau-level filling. The phase transition appears as a quantum one at $T=0$, provided the relevant coupling constant is above the critical one. This critical value rapidly decreases, and finally saturates toward zero on the scale of tens of Tesla. Thus, a strong enough magnetic field can induce the density wave in the system in which it was absent in zero field.

This work reports a fundamental study on the electronic structure, optical properties, and defect chemistry of a series of Co-based spinel oxide (${\mathrm{Co}}_3{\mathrm{O}}_4, {\mathrm{ZnCo}}_2{\mathrm{O}}_4$, and ${\mathrm{CoAl}}_2{\mathrm{O}}_4$) epitaxial thin films using x-ray photoemission and absorption spectroscopies, optical spectroscopy, transport measurements, and density functional theory. We demonstrate that ${\mathrm{ZnCo}}_2{\mathrm{O}}_4$ has a fundamental bandgap of 1.3 eV, much smaller than the generally accepted values, which range from 2.26 to 2.8 eV. The valence band edge mainly consists of occupied $\mathrm{Co}3d{t}_{2\mathrm{g}}^6$ with some hybridization with O $2p$/Zn $3d$, and the conduction band edge of unoccupied $e_g^{*}$ state. However, optical transition between the two band edges is dipole forbidden. Strong absorption occurs at photon energies above 2.6 eV, explaining the reasonable transparency of ${\mathrm{ZnCo}}_2{\mathrm{O}}_4$. A detailed defect chemistry study indicates that Zn vacancies formed at high oxygen pressure are the origin of a high $p$-type conductivity of ${\mathrm{ZnCo}}_2{\mathrm{O}}_4$, and the hole conduction mechanism is described by small-polaron hoping model. The high $p$-type conductivity, reasonable transparency, and large work function make ${\mathrm{ZnCo}}_2{\mathrm{O}}_4$ a desirable $p$-type transparent semiconductor for various optoelectronic applications. Using the same method, the bandgap of ${\mathrm{Co}}_3{\mathrm{O}}_4$ is further proved to be ∼0.8 eV arising from the tetrahedrally coordinated ${\mathrm{Co}}^{2+}$ cations. Our work advances the fundamental understanding of these materials and provides significant guidance for their use in catalysis, electronic, and solar applications.

A charged vacancy on the Si surface, which generates deep levels within the band gap to accommodate multiple charge states, becomes increasingly important with the device scaling. Yet its characteristics have not been addressed as comprehensively as the bulk vacancy in Si. Here we generate the adatom vacancies on a heavily B doped Si(111) surface and measure their structural and electronic properties by using scanning tunneling microscopy. We find that they are Jahn-Teller distorted in the ground state but undergo the symmetry-restoring transition when gated by the external bias. We also determine the stabilization energy of the Jahn-Teller transition as well as the hopping energy between the dangling bonds of the adatom vacancy quantitatively.