Symmetry In Real Space Research Articles

Close‐packed layer spacing as a practical guideline for structure symmetry manipulation of IV‐VI/I‐V‐VI2 thermoelectrics

AbstractThe crystal‐structure symmetry in real space can be inherited in the reciprocal space, making high‐symmetry materials the top candidates for thermoelectrics due to their potential for significant electronic band degeneracy. A practical indicator that can quantitatively describe structural changes would help facilitate the advanced thermoelectric material design. In face‐centered cubic structures, the spatial environment of the same crystallographic plane family is isotropic, such that the distances between the close‐packed layers can be derived from the atomic distances within the layers. Inspired by this, the relationship between inter‐ and intra‐layer geometric information can be used to compare crystal structures with their desired cubic symmetry. The close‐packed layer spacing was found to be a practical guideline of crystal structure symmetry in IV‐VI chalcogenides and I‐V‐VI2 ternary semiconductors, both of which are historically important thermoelectrics. The continuous structural evolution toward high symmetry can be described by the layer spacing when temperature or/and composition change, which is demonstrated by a series of pristine and alloyed thermoelectric materials in this work. The layer‐spacing‐based guideline provides a quantitative pathway for manipulating crystal structures to improve the electrical and thermal properties of thermoelectric materials.

Phase diagram of a frustrated Heisenberg model: From disorder to order and back again

We study the effects of bond and site disorder in the classical $J_{1}$-$J_{2}$ Heisenberg model on a square lattice in the order-by-disorder frustrated regime $2J_{2}>\left|J_{1}\right|$. Combining symmetry arguments, numerical energy minimization and large scale Monte Carlo simulations, we establish that the finite temperature Ising-like transition of the clean system is destroyed in the presence of any finite concentration of impurities. We explain this finding via a random-field mechanism which generically emerges in systems where disorder locally breaks the same real-space symmetry spontaneously globally broken by the associated order parameter. We also determine that the phase replacing the clean one is a paramagnet polarized in the nematic glass order with non-trivial magnetic response. This is because disorder also induces non-collinear spin-vortex-crystal order and produces a conjugated transverse dipolar random field. As a result of these many competing effects, the associated magnetic susceptibilities are non-monotonic functions of the temperature. As a further application of our methods, we show the generation of random axes in other frustrated magnets with broken SU(2) symmetry. We also discuss the generality of our findings and their relevance to experiments.

Probing valley population imbalance in transition metal dichalcogenides via temperature-dependent second harmonic generation imaging

Degenerate minima in momentum space—valleys—provide an additional degree of freedom that can be used for information transport and storage. Notably, such minima naturally exist in the band structure of transition metal dichalcogenides (TMDs). When these atomically thin crystals interact with intense laser light, the second harmonic generated (SHG) field inherits special characteristics that reflect not only the broken inversion symmetry in real space but also the valley anisotropy in reciprocal space. The latter is present whenever there exists a valley population imbalance (VPI) between the two valleys and affects the polarization state of the detected SHG. In this work, it is shown that the temperature-induced change of the SHG intensity dependence on the excitation field polarization is a fingerprint of VPI in TMDs. In particular, pixel-by-pixel VPI mapping based on polarization-resolved raster-scanning imaging microscopy was performed inside a cryostat to generate the SHG contrast in the presence of VPI from every point of a TMD flake. The generated contrast is marked by rotation of the SHG intensity polar diagrams at low temperatures and is attributed to the VPI-induced SHG.

Spin-$\frac{1}{2}$ kagome Heisenberg antiferromagnet with strong breathing anisotropy

We study the zero-temperature phase diagram of the spin-1/2 Heisenberg model with breathing anisotropy (i.e., with different coupling strength on the upward and downward triangles) on the kagome lattice. Our study relies on large scale tensor network simulations based on infinite projected entangled-pair state and infinite projected entangled-simplex state methods adapted to the kagome lattice. Our energy analysis suggests that the U(1) algebraic quantum spin-liquid (QSL) ground-state of the isotropic Heisenberg model is stable up to very large breathing anisotropy until it breaks down to a critical lattice-nematic phase that breaks rotational symmetry in real space through a first-order quantum phase transition. Our results also provide further insight into the recent experiment on vanadium oxyfluoride compounds which has been shown to be relevant platforms for realizing QSL in the presence of breathing anisotropy.

Large off-diagonal exchange couplings and spin liquid states in C3 -symmetric iridates

Iridate oxides on a honeycomb lattice are considered promising candidates for realization of quantum spin liquid states. We investigate the magnetic couplings in a structural model for a honeycomb iridate K$_2$IrO$_3$, with $C_3$ point group symmetry at the Ir sites, which is an end member of the recently synthesized iridate family K$_x$Ir$_y$O$_2$. Using \textit{ab-initio} quantum chemical methods, we elucidate the subtle relationship between the real space symmetry and magnetic anisotropy and show that the higher point group symmetry leads to high frustration with strong magnetic anisotropy driven by the unusually large off-diagonal exchange couplings ($\Gamma$'s) as opposed to other spin-liquid candidates considered so far. Consequently, large quantum fluctuations imply lack of magnetic ordering consistent with the experiments. Exact diagonalization calculations for the fully anisotropic $K$-$J$-$\Gamma$ Hamiltonian reveal the importance of the off-diagonal anisotropic exchange couplings in stabilizing a spin liquid state and highlight an alternative route to stabilize spin liquid states for ferromagnetic $K$.

Emergence of a real-space symmetry axis in the magnetoresistance of the one-dimensional conductor Li0.9Mo6O17.

We report on an emerging symmetry axis in the magnetoresistance of bulk single crystals of quasi-one-dimensional Li0.9Mo6O17 below T min = 25 K, the temperature at which the electrical resistivity experiences a minimum. Detailed angle-dependent magnetoresistance sweeps reveal that this symmetry axis is induced by the development of a negative magnetoresistance, which is suppressed only for magnetic fields oriented along the poles of the MoO6 octahedra that form the conducting chains. We show that this unusual negative magnetoresistance is consistent with the melting of dark excitons, composed of previously omitted orbitals within the t 2g manifold that order below T min. The unveiled symmetry axis in directional magnetic fields not only provides evidence for the crystallization of these dark excitons but also sheds new light on the long-standing mystery of the metal-insulator transition in Li0.9Mo6O17.

Tuning a random-field mechanism in a frustrated magnet

We study the influence of spinless impurities on a frustrated magnet featuring a spin-density wave (stripe) phase by means of Monte Carlo simulations. We demonstrate that the interplay between the impurities and an order parameter that breaks a real-space symmetry triggers the emergence of a random-field mechanism which destroys the stripe-ordered phase. Importantly, the strength of the emerging random fields can be tuned by the repulsion between the impurity atoms; they vanish for perfect anticorrelations between neighboring impurities. This provides a novel way of controlling the phase diagram of a many-particle system. In addition, we also investigate the effects of the impurities on the character of the phase transitions between the stripe-ordered, ferromagnetic, and paramagnetic phases.

Tunneling anisotropic magnetoresistance via molecular π orbitals of Pb dimers

Spin-orbit coupling (SOC) links spin space and real space and leads to solids with intriguing spin topologies and transport properties, such as anisotropic magnetoresistance (AMR). The orientation of a real-space symmetry axis with respect to the spin direction determines the size of SOC-induced changes of the electronic structure. To show this effect at the single-molecule level, the authors arrange Pb dimers on a ferromagnetic Fe layer and observe that the AMR resulting from their molecular orbitals depends strongly on the dimer orientation.

Complex geometry in the entanglement entropy of fermionic chains

The geometry of Riemann surfaces plays a relevant role in the study of entanglement entropy in the ground state of a free fermionic chain. Recently, a new symmetry for the entropy in non critical theories has been discovered. It is based on the Möbius transformations in a compact Riemann surface associated to the Hamiltonian of the system. Here, we argue how to extend it to critical theories supporting our conjectures with numerical tests. We also highlight the intriguing parallelism that exists with conformal symmetry in real space.

SrAuSi3: A Noncentrosymmetric Superconductor

A novel strontium–gold silicide (SrAuSi3) was successfully synthesized for the first time using a high-pressure technique, at ∼6 GPa. X-ray Rietveld analysis and scanning transmission electron microscopy studies clearly revealed that SrAuSi3 crystallizes in a BaNiSn3-type structure, in space group I4mm with the following lattice parameters: a = 4.4090(1) A, and c = 9.9475(3) A. The structure was found not to have inversion symmetry in real space. From direct measurements of polycrystalline SrAuSi3, the electrical resistivity was zero, the static magnetic susceptibility was almost a full volume Meissner diamagnetic signal, and the specific heat exhibited a sharp jump, all of which were accompanied by a phase transition due to bulk superconductivity at a critical temperature (TC) of 1.54 K. The SrAuSi3 compound is the first noncentrosymmetric superconductor that includes Au as a principal constituent element.

F-band condensates in exciton-polariton lattice systems

We report the condensation of microcavity exciton-polaritons at $\ensuremath{\Gamma}$ points located on the boundary between the third and higher Brillouin zones in hexagonal lattices: triangular and honeycomb geometries. We collect experimental evidence that supports the finite momentum condensation: (1) the coherent Bragg peaks formed at nonzero $\ensuremath{\Gamma}$ points; (2) the nonlinear intensity increase in the exciton-polariton emission at quantum degeneracy threshold; (3) the spectral linewidth behavior: narrowing near threshold and broadening above threshold; and (4) the equivalent 4${f}_{{y}^{3}\ensuremath{-}3{x}^{2}y}$-like orbital symmetry in real space. The $f$-orbital state at $\ensuremath{\Gamma}$ points appears as a metastable momentum valley to trap exciton-polaritons, which is explained by single-particle band-structure calculations.

Real-Space Observation of Skyrmion Lattice in Helimagnet MnSi Thin Samples

Observing and characterizing the spin distributions on a nanometer scale are of vital importance for understanding nanomagnetism and its application to spintronics. The magnetic structure in MnSi thin samples prepared from a bulk, which undergoes a transition from a helix to a skyrmion lattice, was investigated by in situ observation using Lorentz microscopy. Stripe domains were observed at zero applied field below 22.5 K. A skyrmion lattice with 6-fold symmetry in real space appeared when a field of 0.18 T was applied normal to the film plane. The lattice constant was estimated to be 18 nm, almost identical to the helical period. In comparison with the marginally stable skyrmion phase in a bulk sample, the skyrmion phase was stable over a wide range of temperatures and magnetic fields in the thin samples.

Predicting the Structure of Alloys Using Genetic Algorithms

We discuss a novel genetic algorithm that can be used to find global minima on the potential energy surface of disordered ceramics and alloys using a real-space symmetry adapted crossover. Due to a high number of symmetrically equivalent solutions of many alloys, conventional genetic algorithms using reasonable population sizes are unable to locate the global minima for even the smallest systems. We demonstrate the superior performance of the use of symmetry adapted crossover by the comparison of that of a conventional GA for finding global minima of two binary Ising-type alloys that either order or phase separate at low temperature. Comparison of different representations and crossover operations show that the use of real-space crossover outperforms crossover operators working on binary representations by several orders of magnitude.

Unusual States of Vortex Matter in Mixtures of Bose-Einstein Condensates on Rotating Optical Lattices

In a single-component superfluid under rotation a broken symmetry in the order parameter space results in a broken translational symmetry in real space: a vortex lattice. If translational symmetry is restored, the phase of the order parameter disorders and thus the broken symmetry in the order parameter space is also restored. We show that for Bose-Einstein condensate mixtures in optical lattices with negative dissipationless drag, a new situation arises. This state is a modulated vortex liquid which breaks translational symmetry in the direction transverse to the rotation vector.

Three-dimensional silicon inverse photonic quasicrystals for infrared wavelengths

Quasicrystals are a class of lattices characterized by a lack of translational symmetry. Nevertheless, the points of the lattice are deterministically arranged, obeying rotational symmetry. Thus, we expect properties that are different from both crystals and glasses. Indeed, naturally occurring electronic quasicrystals (for example, AlPdMn metal alloys) show peculiar electronic, vibrational and physico-chemical properties. Regarding artificial quasicrystals for electromagnetic waves, three-dimensional (3D) structures have recently been realized at GHz frequencies and 2D structures have been reported for the near-infrared region. Here, we report on the first fabrication and characterization of 3D quasicrystals for infrared frequencies. Using direct laser writing combined with a silicon inversion procedure, we achieve high-quality silicon inverse icosahedral structures. Both polymeric and silicon quasicrystals are characterized by means of electron microscopy and visible-light Laue diffraction. The diffraction patterns of structures with a local five-fold real-space symmetry axis reveal a ten-fold symmetry as required by theory for 3D structures.

Kondo effect in triple quantum dots: interplay between continuous and discrete symmetries

The physics of Kondo effect and related phenomena in a triangular triple quantum dot (TTQD) is studied. A fascinating property of TTQD is the interplay between continuous SU ( 2 ) symmetry in spin space and discrete C 3 v symmetry in real space. We show that this interplay is manifested in strong oscillations of conductance as a function of magnetic flux through TTQD due to interplay between Kondo and Aharonov–Bohm effect.

Open-shell localized Hartree–Fock approach for an efficient effective exact-exchange Kohn–Sham treatment of open-shell atoms and molecules

Within the framework of the symmetrized Kohn–Sham (KS) formalism an effective exact-exchange approach, the open-shell localized Hartree–Fock (OSLHF) method, is derived. The derivation relies on an approximation of energy denominators in KS one-particle Greens functions, the approximation of average magnitudes of eigenvalue differences, which is introduced. With the OSLHF method open-shell molecules can be treated efficiently and numerically stable without running into the symmetry problem encountered in conventional KS methods. The self-interaction free Hamiltonian operator of the OSLHF approach is in all cases non-spin-polarized and exhibits the full molecular point group symmetry in real space. Results for atoms as well as small and medium size molecules are presented. A procedure is introduced which yields special basis sets required for an efficient construction of the Slater potential which enters the effective KS potential of localized Hartree–Fock methods.

Symmetry and spectral statistics of the magnetic band structure of a one-dimensional surface superlattice

We report a theoretical study of the spectral statistics of a quasi-one-dimensional surface superlattice in perpendicularly applied magnetic fields. The energy-level-spacing distribution and the Dyson–Mehta Δ3 statistic of the magnetic band structure of the system are calculated. The calculations show that for the system with inversion symmetry, the magnetic band structure at the wave vector k = 0 is well described by the statistic derived by a superposition of two independent Gaussian orthogonal ensemble (GOE) statistics. This result is consistent with the fact that the system shows a false time-reversal violation and a real-space symmetry. The calculations show also that when the wave vector k is moved away from the k = 0 point, the statistical properties of the magnetic band structure are excellently described by the GOE statistics. The GOE statistics are also found in the magnetic band structure when the inversion symmetry is removed from the system.

Hydrodynamic parameters and correlation functions of the polar phase in superfluid 3He

In the main part of this manuscript we present the correlation function description of the polar phase in superfluid 3He in the bulk in the hydrodynamic regime. This is done using the formalism introduced by Mori and Zwanzig and exploited in the hydrodynamic regime by Forster. The present investigation was triggered by recent NMR experiments on superfluid 3He in aerogels of very high porosity which suggested that the superfluid phase found might be different from the A and B phases present in the bulk. It appears from more microscopic calculations that the polar phase is one of the leading alternative candidates. As one of the results, which can only be obtained in the framework of the projector formalism, we find that the characteristic contributions in the hydrodynamic equations which guarantee the correct transformation behavior under rigid rotations are exclusively due to the non-instantaneous reversible contributions from the memory matrix. This result for the polar phase contrasts with that for other systems with spontaneously broken rotational symmetry in real space – which have the same type of contributions in the phenomenological hydrodynamic equations – such as nematic liquid crystals and the superfluid A phase of 3He. In addition to the truly hydrodynamic regime, we study the influence of the magnetic dipole interaction and of a weak external magnetic field thus including the k=0 dynamics relevant for NMR. For the first time we examine the influence of the presence of the aerogel on the reversible and dissipative macroscopic dynamics in the framework of the projector formalism.

Methodology employed for the structure determination of tumour necrosis factor, a case of high non-crystallographic symmetry

The structure of the protein tumour necrosis factor (TNF) was determined from crystals of space group P3(1)21 which contain six copies of the TNF monomer per crystallographic asymmetric unit [Jones, Stuart & Walker (1989). Nature (London), 338, 225-228]. The nature of these crystals (relatively high crystallographic symmetry coupled with multiple copies of the protein in the asymmetric unit) led to some peculiarly challenging problems at several points in the structure determination. In particular, (1) self-rotation function calculations failed to yield clearly interpretable solutions, (2) the analysis of difference Patterson maps for heavy-atom derivatives required the development of a Patterson search program suite GROPAT. The redundancy in the asymmetric unit allowed refinement of poor-quality isomorphous phases at 4 A resolution and phase extension from 4 to 2.9 A resolution using real-space symmetry averaging and solvent flattening in the absence of any isomorphous phase information. Despite further difficulties caused by structural differences between the six independent copies of the monomer the resultant electron density map was of high quality and proved to be easily interpretable.