Off-diagonal Components Research Articles

Longitudinal and transverse magnetoresistance of SrTiO3 with a single closed Fermi surface

The magnetoresistance (MR) of SrTiO3 is theoretically investigated based on the Boltzmann equation by considering its detailed band structure. The formula for MR proposed by Mackey and Sybert is extended to be applicable to a system with an arbitrarily shaped Fermi surface. It is shown that the angular dependence of the diagonal component of the mass tensor causes transverse MR, whereas that of the off-diagonal component causes longitudinal MR with only a single closed Fermi surface, which overturns the textbook understanding of MR. The calculated MR (300% at 10 T) quantitatively agrees with the experimental results for SrTiO3 including the behavior of the linear MR. The negative Gaussian curvature of the Fermi surface of SrTiO3 and its resulting negative longitudinal and transverse MR are also discussed.

Numerical study of plane Couette flow: turbulence statistics and the structure of pressure–strain correlations

AbstractThe paper presents the results of direct numerical simulation of turbulent plane Couette flow. The calculations were performed for Reynolds numbers Re =U0H/ν(His the height of the channel, ±U0/2 is the motion velocities of the lower and upper walls, respectively,νis the kinematic viscosity) from 5200 (where viscous effects significantly affect the flow structure) to 80000 (where a logarithmic layer is clearly observed). Estimates of terms of the equation for the balance of turbulent Reynolds stresses are obtained, which indicate the importance of the kinetic energy transport by velocity fluctuations. The vertical transport of the turbulent momentum flux is less important and partly compensated by the transport of pressure fluctuations. It is shown that in the logarithmic layer the normal components of the ‘pressure–strain rate’ correlation tensor are described in the framework of the ‘isotropization of production’ model, and in the central part of the channel they are described by the linear Rotta model [29]. The additive model considering both the interaction of the velocity field fluctuation and the influence of the mean velocity gradient is a good approximation only for the off-diagonal component of the tensor entering the balance equation for the turbulent momentum flux.

Dynamical spin susceptibility of silicene

We present a detailed study of the imaginary and real parts of the spin-susceptibility of silicene which can be generalized to other buckled honeycomb structure. We find that while the off-diagonal components are non-zero in individual valleys, they add up to zero upon including contributions from both the valleys. We investigate the interplay of the spin-orbit interaction and an external electric field applied perpendicular to the substrate and find that although the xx and yy components of the susceptibility are identical, they differ from the zz-component. The external electric field plays an important role in modifying the allowed inter-subband regions. In the dynamic limit, the real part of the susceptibility exhibits log-divergence, position of which can be tuned by the electric field and therefore has implications for spin-collective excitations. The effect of the electric field on the static part of the susceptibility and its consequence for the long distance decay of the spin-susceptibility have been explored.

Accuracy of tomographic particle image velocimetry data on a turbulent round jet

Tomographic particle image velocimetry (Tomo-PIV) was applied on a turbulent round air jet to quantitatively assess the accuracy of velocity gradients obtained in the self-similar turbulent region. The jet Reynolds number based on the nozzle diameter (d) was Red = 3000. Mean velocity, turbulent intensities, and Reynolds shear stress at the center plane of the jet were measured. In addition, statistical results of Tomo-PIV along the axial direction were assessed by performing a separate set of two-dimensional two-component PIV experiments on a “side view” plane along the jet axis. Moreover, the probability distribution functions of four components of the measured velocity gradients in the axial and radial directions were validated by these “side view” planar PIV data. The root mean square of the velocity divergence values relative to the norm of the velocity gradient tensor was 0.36. Furthermore, the on- and off-diagonal components of the velocity gradients satisfied the axisymmetric isotropy conditions. The divergence error in the data affected only areas with low gradient magnitude. Therefore, turbulent structures in the regions with intense vorticity and dissipation can be closely monitored. On this basis, the joint pdfs of the invariants of the velocity gradient and strain and rotation tensor rates were produced and compared well with those in isotropic turbulence studies.

Decorrelating the errors of the galaxy correlation function with compact transformation matrices

Abstract Covariance matrix estimation is a persistent challenge for cosmology, often requiring a large number of synthetic mock catalogues. The off-diagonal components of the covariance matrix also make it difficult to show representative error bars on the 2-point correlation function (2PCF) since errors computed from the diagonal values of the covariance matrix greatly underestimate the uncertainties. We develop a routine for decorrelating the projected and anisotropic 2PCF with simple and scale-compact transformations on the 2PCF. These transformation matrices are modelled after the Cholesky decomposition and the symmetric square root of the Fisher matrix. Using mock catalogues, we show that the transformed projected and anisotropic 2PCF recover the same structure as the original 2PCF while producing largely decorrelated error bars. Specifically, we propose simple Cholesky-based transformation matrices that suppress the off-diagonal covariances on the projected 2PCF by ${\sim } 95{{\ \rm per\ cent}}$ and that on the anisotropic 2PCF by ${\sim } 87{{\ \rm per\ cent}}$. These transformations also serve as highly regularized models of the Fisher matrix, compressing the degrees of freedom so that one can fit for the Fisher matrix with a much smaller number of mocks.

Thermal conductivity tensor of NbO2

Among several transition metal oxides showing metal to insulator transition, NbO2 has attracted attention as the active material of memory devices due to its high thermal stability. Although clarification of thermal conductivity (κ) is essentially important for optimizing the NbO2 device performance, it has not been reported thus far probably due to the anisotropic crystal structure. Here we show the thermal conductivity tensor of NbO2 crystal. We measured the thermal conductivities of (1 1 0)-, (0 0 1)-, and (1 1 1)-oriented NbO2 films and found 〈1 1 1〉 -direction shows ∼1 W m−1 K−1 higher thermal conductivity compared with the other directions (∼3 W m−1 K−1). Using the measured κ-values, we successfully determined thermal conductivity tensor components κ11, κ13, and κ33. The diagonal components of the thermal conductivity tensor were ∼2.5 W m−1 K−1 while the nonzero off-diagonal component was ∼1 W m−1 K−1. We believe that the present results are of great value in optimizing the performance of NbO2 based devices.

Complete characterization of GaAs gradient-elastic tensors and reconstruction of internal strain in GaAs/AlGaAs quantum dots using nuclear magnetic resonance

Recently, we have calibrated the diagonal components ${S}_{11}={S}_{xxxx}$ of the GaAs gradient elastic tensor ${S}_{ijkl}$ using nuclear magnetic resonance (NMR) and photoluminescence spectroscopy of GaAs/AlGaAs quantum dot structures in Faraday geometry [E. A. Chekhovich et al., Phys. Rev. B 97, 235311 (2018)]. Here we measure quantum dot NMR spectra in oblique magnetic fields giving access to the off-diagonal components ${S}_{yzyz}$. We find the ratios ${S}_{yzyz}/{S}_{xxxx}\ensuremath{\approx}\phantom{\rule{0.16em}{0ex}}+1.{98}_{\ensuremath{-}0.27}^{+0.21}$ for $^{75}\mathrm{As}$ and $\ensuremath{\approx}\ensuremath{-}0.{40}_{\ensuremath{-}0.31}^{+0.23}$ for $^{69}\mathrm{Ga}$. Combined with our previous results, we find all independent nonzero components of ${S}_{ijkl}$: $Q{S}_{xxxx}\ensuremath{\approx}+0.758\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}$ V, $Q{S}_{yzyz}\ensuremath{\approx}+1.51\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}$ V for $^{75}\mathrm{As}$ and $Q{S}_{xxxx}\ensuremath{\approx}\ensuremath{-}0.377\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}$ V, $Q{S}_{yzyz}\ensuremath{\approx}+0.151\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}$ V for $^{69}\mathrm{Ga}$, where $Q$ is the corresponding nuclear quadrupolar moment. Our results show that earlier nuclear acoustic resonance experiments [R. K. Sundfors, Phys. Rev. B 10, 4244 (1974)] most likely overestimated the GaAs gradient elastic tensors, especially for $^{69}\mathrm{Ga}$. We further use NMR spectroscopy in oblique fields for assumption-free measurement of the intrinsic strain in GaAs/AlGaAs quantum dots. We find deviations of the strain principle directions from the sample growth axis, which vary between individual quantum dots.

Temperature effect on GMI sensor: Comparison between diagonal and off-diagonal response

In this paper, the effect of temperature on Giant Magneto-Impedance (GMI) sensors is studied in amorphous wire Co-Fe-Si-B from Unitika. Both diagonal, Z11, and the off-diagonal, Z21, components of the impedance tensor are considered, with a primary focus on the changes of intrinsic sensitivity and offset, which are the most crucial relevant quantities for practical sensor implementation. Experiments are conducted in the almost industrial temperature range of [−20 °C, 120 °C]. For the component Z11, the results show a maximum change of the offset of −0.13 (A/m)/°C (170 nT/°C), at a 33 A/m bias field. The sensitivity at this field increases for temperatures between −20 °C and + 40 °C, and then it decreases (a maximum change of −0.5%/°C). For the off-diagonal component, Z21, the change of the offset near the zero-field is smaller (−0.015 (A/m)/°C or −19 nT/°C), whereas the sensitivity increases with temperature by about +0.4%/°C.

3D Inversion of Audio Magnetotelluric Data for Sulphide Mineralization over Dhanjori Basin

Abstract Quartz-pebble conglomerate (QPC) in the Dhanjori basin, a rifted greenstone belt, is known for its gold mineralization. Audio-magnetotelluric (AMT) soundings have been acquired for 32 sites spread in grid fashion for frequency 10kHz to 10 Hz along four parallel profiles. Due to noise conditions in the data only 26 sites have been used for further studies. The site spacing between adjacent AMT sites were ~250 m. The grid spreads over QPCD and Mayurbhang Granite (MG) of rocks. The dimensionality and the directionality analysis hinted towards a complex structure which tempted to perform a 3D inversion of AMT data for full impedance data. To obtain the optimum value of initial uniform half space resistivity, 3D inversion of off-diagonal component of inversion data was performed with various values of resistivity. The optimized value of resistivity was then used to perform 3D inversion for full impedance component with different set of error floor for the impedance component. The final 3D model showed the presence of a continuous conducting feature all along the three profile. Whereas, the southernmost profile showed three well developed conductors. The model corroborates well with electrical resistivity tomography (ERT) sections and GSI borehole which lies within the same geological domain. The borehole intercepted sulphide mineralization with varying concentration from the near surface to ~ 100m. The near surface conducting heterogeneity thus is attributed to sulphide mineralization.

Imaging Regional Geology and Au – Sulphide mineralization over Dhanjori greenstone belt: Implications from 3-D Inversion of Audio Magnetotelluric data and Petrophysical Characterization

The 2.1-Ga-old Paleoproterozoic Dhanjori Group of rocks covers 800 sq. km area within the Singhbhum Crustal province of eastern India, and is composed of siliciclastic metasedimentary sequence interlayered with mafic to ultramafic rocks. The meta-volcanosedimentary sequence is deformed and is of low-grade greenschist metamorphic facies. The gold occurrences within the study area are confined to the sulphide hosted mafic/ultramafic volcanic rocks and also in the basal quartz pebble conglomerates (QPC) within the Dhanjori Formation. Gold mineralization within the mafic/ultramafic meta-volcanics formations occurs within the quartz ± carbonate vein i.e. lode gold and as detrital gold within the QPC (0.5−7 ppm) of Phuljhari Formations. Regional Audio Magnetotelluric (AMT) measurements across the geological domains of Dhanjori Group and Iron Ore Group (IOG) were carried out in the frequency range of 10 kHz–10 Hz. This work discusses the conductivity model from 3-D inversion along with geology and petrophysical properties in mapping the regional geological features, associated sulphide mineralization and influence of texture, grain and bulk properties on mineralization. Dimensionality and strike analysis of the AMT data indicate complex conductivity structures. 3-D inversion of off-diagonal component of impedance was performed for obtaining an optimum value of uniform half space initial resistivity and error floor for off-diagonal component of impedance data. These were used to perform 3-D inversion of full impedance to produce a conductivity model of the study area. The conductivity model delineated major geological boundaries and the sulphide bearing mineralized zone. A major conductor is observed which extends to a depth of ∼250 m within meta-volcanics. On comparison with the available borehole adjacent to this, it is confirmed that the conductor corresponds to sulphide mineralization. The resistivity low at a depth of 150 m coincides with metabasalts (pyrite present which extends to 159 m with arsenopyrite). Another mineralized zone is observed at the boundary of iron ore group volcanics and Proterozoic-gabbro-anorthosite-ultramafics (PGAU), it starts appearing from a depth of ∼400 m and can be seen which extends to a depth of ∼800 m. It is worth mentioning that there lies Kundarkocha gold mine in the same geological domain. In addition, near surface conductor to a maximum depth of ∼50 m is observed all along the AMT profile. The sediments and fluids could account for this conducting feature. This is in agreement with available Electrical Resistivity Tomography section. The borehole within Dhanjori Volcanics also detected sulphide mineralization for ∼50 m. Thus, it can be inferred that the near surface conductor indicates the presence of gold in association with other sulphide minerals. The petrophysical studies helped to estimate the concentration of minerals within the rocks. High density and low susceptibilities in the area is an indicative of low-grade metamorphism and low level of serpentenization. Combined density and susceptibility scatter diagram and component proportion phase diagram suggests 20 vol%, 78 vol% and 2 vol% of haematite/sulphides, barren rocks and magnetite respectively. The high value of chargeability and negligible value of susceptibility indicates the presence of disseminated sulphides rather than haematites and/or massive sulphides in the rocks. Ternary plot suggest that the mineralization is controlled by texture and grain properties. Petrophysical analysis suggests that the lithotypes of the study area are similar to those of greenstone terrain in the Abitibi Subprovince. Integrating inverted section, geology and petrophysical properties has a substantial impact on the study area in terms of predicting the mineralization prospect and at the same time substantially reduces the drilling cost. Thus, the study provides a new orientation and advancement to the exploration strategy for deeper deposits.

Laser-stimulated elastoopticity in nanocrystalline CdCl0.5J0.5 layers

A significant laser-induced piezooptical response in novel CdCl0.5J 0.5 nanolayers is obtained under the influence of laser illumination. The maximal piezo-optic response is observed for off-diagonal piezooptical tensor components. The layered structure allowed to obtain the thin specimens of thickness up to 1 nm with mirror-like surfaces. The observed studies show huge dependence of the piezooptics on the nanolayer thickness and the photoinduced beam power density. The effect is completely reversible. This fact allows proposing a new type of nanomaterials, which have significant benefits with respect to the other types of piezooptical materials (i.e. a possibility to use them in the laser operated devices).

An efficient general curvilinear coordinates finite element method for the linear dynamic study of thickness-independent shells

Abstract yyTo date, a large number of finite element methods have been developed to study the dynamics of shell structures. Most of them are generally based on the degenerated solid approach and other less in shell theories, but introducing, in this last case, some assumptions to analyze this problem: some of them refer to shallow shells (slightly curved shells), or consider thin shells neglecting shear deformation, or dispense some terms in their stress-strain developments like the off-diagonal components of the curvature or metric tensors (orthogonal coordinates). In the present work, we present an improved finite element method for the linear dynamic analysis of shells, from thin to moderately thick and thick shells, developed in general curvilinear coordinates, based on a refined shear deformation shell theory and free of the well-known shear locking effect. Exact constitutive equations, including higher order moments-strains relations, are also deduced for the adequate analysis of thick shells. To circumvent the shear locking problems, the mixed interpolation of the tensorial components (MITC) of the linear strain tensor is used. An exhaustive study of different surfaces is performed, especially in doubly curved shells, and interesting conclusions of the higher order modes of vibration and the strain energy of the element are derived. Other desirable features like a low computational effort, a straightforward extension to nonlinear formulation and applications for composite shells are found in this novel and general formulation. Very good results in the proposed practical cases have been found.

Fluctuation Theory in Chemical Kinetics

In this research, we study the stability properties of chemical reactions of arbitrary orders. In a given chemical experiment, we focus on the formation of a chemical equilibrium by optimizing the reaction rate. Under infinitesimal simultaneous variations of the concentrations of reacting species, the binary component equilibrium is achieved when either one of the orders or concentrations of reactants vanishes. The chemical concentration capacities of the components are calculated to describe the local stability of the equilibrium. The correlation between the components is obtained as the mixed second-order derivative of the rate with respect to concentrations. The global stability analysis is performed by introducing a symmetric matrix with its diagonal components as the chemical capacities and off-diagonal components as the local correlation. We find that the local chemical stability requires the orders of the reactants to be either negative or larger than unity. The corresponding global stability requires the positivity of a cubic factor over the orders of the reactants. In short, our consideration illustrates how a chemical reaction takes place by attaining its activation state and asymptotically approaches the equilibrium when two components are mixed with arbitrary orders. Qualitative discussions are provided to support our analysis towards the formation of an optimized equilibrium. Finally, along with future directions, we discuss verification of our model towards the formation of carbon-based reactions, formation of organic/inorganic chemical equilibria and catalytic oxidation of C O − H 2 mixtures in presence of Pt.

Multiparticle [formula omitted] mechanics with F(4) superconformal symmetry

We present a new multiparticle model of N=8 mechanics with superconformal F(4) symmetry. The system is constructed in terms of two matrix N=4 multiplets. One of them is a bosonic matrix (1,4,3) multiplet and another is a fermionic (0,4,4) one. Off-diagonal bosonic components of the (1,4,3) multiplet are chosen to take values in the flag manifold U(n)/[U(1)]n and they carry additional gauge symmetries. The explicit form of the F(4) supersymmetry generators is found. We demonstrate that the F(4) superalgebra constructed contains as subalgebras two different D(2,1;α=−1/3) superalgebras intersecting over the common sl(2,R)⊕su(2) subalgebra.

3-D studies of MT data in the Central Polish Basin: Influence of inversion parameters, model space and transfer function selection

3-D inversion of geophysical data has become a practical, commonly used tool in the interpretation of magnetotelluric (MT) data. However, 3-D inversion is a non-linear and non-unique task and requires adopting additional assumptions. In this paper, we analyze the influence of selection of inversion parameters in the case of previously published, real model for Pomerania region (Poland). This zone is a regional scale 2-D structure, which locally becomes 3-D. The analyzed model was accomplished by inversion of full impedance and tippers from 31 MT stations with the period range from 10 s to 10,000 s. The inversion parameters were chosen to provide the best possible agreement of the model with well-known geophysical and geophysical data.The purpose of the presented work is to determine how the change of some inversion parameters influences previously obtained model, and therefore we check which structure features remain in spite of the change of the implied parameters. A number of tests were performed employing an implementation of the freely available ModEM inversion package. These included the choice of an appropriate initial model and a model covariances evaluation as a means to control smoothness between adjoining model areas. Additionally, we compared the models calculated from all transfer functions (impedances and tippers) and from subsets (only tippers and off-diagonal components of the impedance tensor and tippers). Moreover, the tests with the model coordinate system rotation were performed.Our analysis has shown that the values of covariance matrix coefficients have an evident effect on the results. However, the multiple use of the covariance matrix with small coefficients gives a similar result as its single use with higher coefficients. The initial model and the selection of the transfer function for inversion can significantly change the model. However, changing the orientation of the coordinate system from geographic to the one that is aligned with a regional strike direction does not cause noticeable changes in the solution obtained. It can be only noted that for the rotated coordinate system the obtained model becomes closer to the model of a 2-D regional structure.

Hamiltonian theory for quantum Hall systems in a tilted magnetic field: Composite-fermion geometry and robustness of activation gaps

We use the Hamiltonian theory developed by Shankar and Murthy to study a quantum Hall system in a tilted magnetic field. With a finite width of the system in the $z$ direction, the parallel component of the magnetic field introduces anisotropy into the effective two-dimensional interactions. The effects of such anisotropy can be effectively captured by the recently proposed generalized pseudo-potentials. We find that the off-diagonal components of the pseudo-potentials lead to mixing of composite fermions Landau levels, which is a perturbation to the picture of $p$ filled Landau levels in composite-fermion theory. By changing the internal geometry of the composite fermions, such a perturbation can be minimized and one can find the corresponding activation gaps for different tilting angles, and we calculate the associated optimal metric. Our results show that the activation gap is remarkably robust against the in-plane magnetic field in the lowest Landau level.

Effect of Torsion Stress on the Offset and Sensitivity of Diagonal and Off-Diagonal GMI in Amorphous Wires.

In this paper, the torsional stress effect on Giant Magneto-Impedance (GMI) was studied in Co-rich amorphous wires. The study, which was conducted in the context of the development of a current clamp based on GMI, considered torsion as a parameter of the influence of this sensor. Both diagonal, Z11, and off-diagonal, Z21, components of the impedance tensor were investigated. The samples were Co-rich wires with a 100 µ diameter. The wires were twisted positive and negative angles with respect to a reference position. For each component of the impedance, the intrinsic sensitivity and offset were measured as a function of the rotation angle. The results showed that the sensitivity of the diagonal component at a given working point slightly increased for angles between −90° to +90°, whereas the sensitivity was almost constant for the off-diagonal component at zero-field. The intrinsic offset in the diagonal configuration was almost unchanged for the rotation angles considered, whereas this offset increased in the off-diagonal configuration. Furthermore, the GMI ratio of Z11 was also measured as a function of the rotation angle for comparison purposes with known data. The maximum of this ratio was obtained for a rotation angle of about 50°.

Quadratic and Cubic Optical Nonlinearities of Y-Shaped and Distorted-H-Shaped Arylalkynylruthenium Complexes.

Straightforward syntheses of bis[bis{1,2-bis(diphenylphosphino)ethane}ruthenium]-functionalized 1,3,5-triethynylbenzene-cored complexes via a methodology employing "steric control" permit facile formation of Y-shaped Sonogashira coupling products and distorted-H-shaped homo-coupled quadrupolar products. Cyclic voltammetric data from these products reveal two reversible metal alkynyl-localized oxidation processes for all complexes. The wavelengths of the linear optical absorption maxima are dominated by the nature of the peripheral alkynyl ligand rather than the substituent at the unique arm of the "Y" or at the quadrupolar complex "core". The quadratic optical nonlinearities of the Y-shaped complexes were assessed by the hyper-Rayleigh scattering technique at 800 nm and employing 100 fs light pulses; introduction of donor NEt2 and/or acceptor NO2 to the wedge periphery resulted in non-zero nonlinearities, with the largest βHRS,800 values being observed for the complexes containing the 4-nitrophenylalkynyl ligands. Depolarization ratios are consistent with substantial off-diagonal first hyperpolarizability tensor components and 2D nonlinear character. Computational studies employing time-dependent density functional theory have been employed to assign the key low-energy transitions in the linear optical spectra and to compute the quadratic nonlinear optical tensorial components. Cubic optical nonlinearities of the quadrupolar complexes were assessed by the Z-scan technique over the range 500-1600 nm and employing 130 fs light pulses; two-photon absorption cross-sections for these distorted-H-shaped complexes are moderate to large in value (up to 5500 GM at 880 nm), while one example displays significant three-photon absorption (1300×10-80 cm6 s2 at 1200 nm).

Scalar Casimir effect in a linearly expanding universe

We investigate quantum vacuum effects for a massive scalar field, induced by two planar boundaries in background of a linearly expanding spatially flat Friedmann–Robertson–Walker spacetime for an arbitrary number of spatial dimensions. For the Robin boundary conditions and for general curvature coupling parameter, a complete set of mode functions is presented and the related Hadamard function is evaluated. The results are specified for the most important special cases of the adiabatic and conformal vacuum states. The vacuum expectation values of the field squared and of the energy–momentum tensor are investigated for a massive conformally coupled field. The vacuum energy–momentum tensor, in addition to the diagonal components, has nonzero off-diagonal component describing energy flux along the direction perpendicular to the plates. The influence of the gravitational field on the local characteristics of the vacuum state is essential at distances from the boundaries larger than the curvature radius of the background spacetime. In contrast to the Minkowskian bulk, at large distances the boundary-induced expectation values follow as power law for both massless and massive fields. Another difference is that the Casimir forces acting on the separate plates do not coincide if the corresponding Robin coefficients are different. At large separations between the plates the decay of the forces is power law. We show that during the cosmological expansion the forces may change the sign.

Unusual one-way edge state in acoustic gyroscopic continuum

Unusual one-way edge states have been observed in composite structures composed of periodic lattices loaded with gyroscopes. Here, we provide a continuum-mechanics understanding to the one-way edge state by formulating surface state equations of acoustic gyroscopic mediums with Hermite mass density tensor. We discover that the unidirectional edge effect arises from nontrivial off-diagonal components of Hermite densities, which causes the symmetric breaking of surface wave propagation towards forward and backward directions. Theoretical predictions on the velocity and decay length of surface waves coincide excellently with numerical simulations. The unidirectional edge state in a two-interface gyroscopic medium is also analyzed. Due to the rotational symmetry in geometry, the unidirectional edge state on one interface is able to prevent itself from the coupling to surface waves on the other interface regardless of the slab thickness. With these anomalous effects, surface waves residing on gyroscopic mediums can flow around the edge defects without back-scatterings, or can be split into two beams of equal energy magnitudes. Our findings may make a bridge that would help to reach the design of non-reciprocal composite materials via an effective medium approach.