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# absolute permeability formula magnetism

a measure of the permeability of a medium relative to that of a vacuum: The relative permeability of a medium can actually be less than 1. The reciprocal of magnetic permeability is magnetic reluctance. to the flux it produces, permeability takes into account the distribution of the flux within the material. The absolute permeability of free space in SI unit is 2 See answers GauravSaxena01 GauravSaxena01 Hello!! If the current (and therefore H) is increased, the BâH curve OA could be plotted (Fig. and is equal to: The permeability of a medium () is most often quoted in terms of its relative } (the applied magnetic field versus time) of the applied field. Magnetic Field Strength refers to one of two ways that the expression of a magnetic field can take place. as Diamagnetism. "@type": "ListItem", ... Sheet metal â¦ We show magnetic flux with the Greek letter; Ð¤. The relative magnetic permeability is defined as the permeability of a material relative to the permeability of free space (vacuum) which equals roughly 4p x10-7 H/m. often have very high values for . The magnetic permeability of the material is directly proportional to the number of lines passing through it. In SI units, permeability is measured in henries per meter (H/m or â¦ "position": 2, "@type": "ListItem", Furthermore, the formation of a magnetic field takes place when a wire carries an electric current. When this happens, the material actually creates an the permeability relates to the orientation of bound magnetic particles within a material. The permeability of common materials is given in Table 1. },{ The dimensional formula of ... Its absolute permeability in S.I. In applications, relative permeability is often represented as a function of water saturation; however, owing to capillary hysteresis one often resorts to a function â¦ Absolute permeability means the actual permeability. can pass through a medium. The permeability of vacuum is 4Ï×10-7 H/m. We show it with µ. }. This gives units of Î¼ to be T m / A 13.12).If the current is decreased to zero, the curve AC is obtained.OC is called the remanence or residual flux density, that is, the flux â¦ Absolute permeability of material of rod Î¼ = Î¼ + Î¼0 = 1 + ÏmÎ¼0â´ Î¼ = 1 + 499 × 4Ï × 10-7 = 2Ï × 10-4 Hm-1 Absolute permeability of material of rod Î¼ = Î¼ + Î¼0 = 1 + ÏmÎ¼0â´ Î¼ = 1 + 499 × 4Ï × 10-7 = 2Ï × 10-4 Hm-1 ... Frog is leviated in magnetic field produced by the current in vertical solenoid below the frog due to repulsion, â¦ For air and other non-magnetic materials, µr has the value of unity (µr = 1). Copyright maxwells-equations.com, 2012. The permeability is measured in units of Henries/meter [H/m], which has dimensions of Therefore, it may be necessary to multiply the given relative permeability value by 4 p x10 -7 H/m or 1.257 x10 -6 H/m to get the absolute permeability value â¦ Fig. Relative permeability is higher at low values of, Calculate the absolute permeability for a magnetic material whose μ, ${{\mu }_{r}}=\frac{\mu }{{{\mu }_{0}}}$. Home » Magnetism » Absolute and Relative Magnetic Permeability (µ) { 2 Answers 227 Views; What is the formula of speed? units is ... View Answer. Results of the material balance calculations are given below: ... For magnetic â¦ The water viscosity is treated as a constant with a value of 0.6 cp. application to Maxwell's Equations. Electric Field. If the magnetic field exists in a material other than a vacuum, the permeability in the equation is that of the material and not a vacuum. This article deals with magnetic field strength formula. The term was coined in September 1885 by Oliver Heaviside. Magnetic Permeability = Magnetic flux density × [Magnetic field strength]-1. Just as permittivity is associated with the alignment of bound molecules electric dipole moments, inductance per unit length. Relative permeability must be between zero and one. of the magnetic field permittivity of a material relates the For some magnetic materials µ, For any one magnetic material, the relative permeability value can vary considerably, being dependent on the flux density in the material. This is the ratio of the permeability of the material â¦ Here, $$\mu â¦ In these materials, the B and H fields According per Biot-Savartâs law, the magnetic induction at a point P due to a current carrying conductor of length l at a distance r from the point P is given by, ³ B = Î¼ i (l × r) / r ³ where Î¼ is permeability of free space, and B is magnetic field. where is the permeability of the porous medium in single-phase flow, i.e., the absolute permeability. What should I do to remember the formulas of physics? It is certainly different from the magnetic flux density. The dimensional formula of permeability is MLT^-2*A^-2 0 Upvotes 0 Upvotes 0 Downvotes. Similarly, materials can be classified by That is the permeability of free space (a vacuum), which has the abbreviation µo. An absolute permeability of 33 md, porosity of 25%, and an initial water saturation of 30% characterize a saturated oil reservoir that exists at its bubble-point pressure of 1925 psi. Magnetic permeability is defined as the ratio of flux density to the magnetic force which is given as: \(\mu =\frac{B (Tesla)}{H (\frac{A-t}{m})}$$ Absolute Permeability The relative permeability of a medium can actually be less than 1. } ] "@context": "http://schema.org", In this manner, } Am- pereâs work in electromagnetism included the develop- ment of a mathematical formula that relates the inter- action between two current-carrying wires, and in it he statedthatthisinteractionisproportionaltothecurrent and the length of the conductors, but inversely propor- tionaltotheseparationbetweenthewires. "url": "https://electricala2z.com/magnetism/absolute-relative-magnetic-permeability-%C2%B5/", The permeability of a vacuum is denoted by , "@type": "BreadcrumbList", From this the absolute permeability is calculated: ... Driving dependence permeability. Report Abuse; Comment; Related Questions. "@id": "https://electricala2z.com", Relative permeability is the ratio of effective permeability of a particular fluid at a particular saturation to absolute permeability of that fluid at total saturation. If a single fluid is present in a rock, its relative permeability is 1.0. In contrast, Ferromagnetism in materials exhibit hysteresis. The permeability of a material is defined as the ease with which a flux can be created in that material. Î¼o = B/H = Î¼o Î¼r The reference value of absolute permeability in a vacuum is Î¼o = 4Ï x 10-7. For some magnetic materials µr can have a value in the thousands. "name": "Home" A. cosÓ¨ Where Ð¤ is the magnetic flux and unit of Ð¤ is Weber (Wb) B is the magnetic field and unit of B is Tesla A is the area â¦ m= B/ H = Henries/ meter. The magnetic permeability of a material is the degree of magnetization that it receives when it responds linearly to a magnetic field. Magnetic Field, via Equation [1]. However, rather than just being a measure of the ratio of the m.m.f. the permeability is a function of the strenght of the applied fields and the history These notes are useful for quick revision of important concepts before exams. It is possible to apply so much magnetic field force to a ferromagnetic material that no more flux â¦ are linearly related per Equation [1], and the intensity (magnitude) That is the permeability of free space (a vacuum), which has the abbreviation µ, To compare the permeability of any given material with the permeability of free space, it is necessary to use a ratio µ, If the non-magnetic core of a solenoid is replaced with a magnetic material, the flux produced by the same number of ampere-turns may be greatly increased. Relative permeability is higher at low values of flux density.eval(ez_write_tag([[468,60],'electricala2z_com-box-3','ezslot_7',105,'0','0'])); To find the absolute permeability of a material, the permeability of free space is multiplied by the relative permeability of the material: Exampleeval(ez_write_tag([[250,250],'electricala2z_com-medrectangle-3','ezslot_4',106,'0','0']));eval(ez_write_tag([[250,250],'electricala2z_com-medrectangle-3','ezslot_5',106,'0','1'])); Calculate the absolute permeability for a magnetic material whose μr is 800. In this unit we learn magnetic permeability that is the quantity of ability to conduct magnetic flux. Due to polarization, there is permittivity, and due to magnetism, there is permeability. "position": 1, In Greek, the symbol for magnetic permeability is denoted as Î¼, which is measured in units of H/m (Henry per meter) or N/A 2 (newton per ampere squared). ${{\mu }_{r}}=\frac{\mu }{{{\mu }_{o}}}$. 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Home > Latex Error > Latex Error Environment Eqnarray Undefined

# Latex Error Environment Eqnarray Undefined

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## Latex Align Undefined Control Sequence

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## \usepackage{amsmath}

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## Public huts to stay overnight around UK What do you call "intellectual" jobs?

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# Tunable spin textures in polar antiferromagnetic hybrid organic–inorganic perovskites by electric and magnetic fields

## Abstract

The hybrid organic–inorganic perovskites (HOIPs) have attracted much attention for their potential applications as novel optoelectronic devices. Remarkably, the Rashba band splitting, together with specific spin orientations in k-space (i.e., spin texture), has been found to be relevant for the optoelectronic performances. In this work, by using first-principles calculations and symmetry analysis, we study the electric polarization, magnetism, and spin texture properties of the antiferromagnetic (AFM) ferroelectric HOIP TMCM-MnCl3 (TMCM = (CH3)3NCH2Cl+, trimethylchloromethyl ammonium). This recently synthesized compound is a prototype of order–disorder and displacement-type ferroelectric with a large piezoelectric response, high ferroelectric transition temperature, and excellent photoluminescence properties as reported by You (Science 357:306, 2017). The most interesting result is that the inversion symmetry breaking coupled to the spin–orbit coupling gives rise to a Rashba-like band splitting and a related robust persistent spin texture (PST) and/or typical spiral spin texture, which can be manipulated by tuning the ferroelectric or, surprisingly, also by the AFM order parameter. The tunability of spin texture upon switching of AFM order parameter is largely unexplored and our findings not only provide a platform to understand the physics of AFM spin texture but also support the AFM HOIP ferroelectrics as a promising class of optoelectronic materials.

## Introduction

The past few years witnessed the extremely rapid development of hybrid organic–inorganic perovskites (HOIPs), which have been shown to be promising optoelectronic materials1,2,3,4,5. HOIP materials have several commom features, including the classical $${\mathrm{ABX}}_3$$ perovskite architecture and the presence of organic cation that occupy the A-site. As for the B-site, it can be occupied not only by main group elements, but also by transition metal atoms such as Mn and Fe, thus introducing magnetic degrees of freedoms into the compound. As for the X-site, it is usually the halogen element. The HOIP materials have some advantages and, in particular, the exceptionally long carrier lifetimes make them very attractive for optoelectronic devices, such as light absorbers and light-emitting diodes6,7,8,9,10,11.

To further enhance the optoelectronic performances of HOIP materials, intense research has been directed to explain the microscopic origin of the long lifetimes9,10,11. Recently, the presence of Rashba band splitting has been suggested to be connected with the carrier lifetimes and to improve their optoelectronic performances12,13,14,15. When lacking spatial inversion symmetry, the spin–orbit coupling (SOC) leads to an effective momentum-dependent magnetic field $${\vec{\mathrm \Omega }}( {\vec k})$$ acting on the spin $$\vec \sigma$$ and the effective SOC Hamiltonian can be written as $$H_{SO} = {\vec{\mathrm \Omega }}( {\vec k})\! \cdot \! \vec \sigma$$16,17. In this case, the SOC will split the spin degeneracy with specific spin orientations (i.e., spin texture) in the momentum k-space, as was first demonstrated by Rashba18 and Dresselhaus et al.19. The spin texture can often be manipulated and even reversed by switching the electric polarization under external electric field, leading to an all-electric and non-volatile control of the spin state20,21,22,23,24. Rashba effects were mainly discussed in non-magnetic lead halide perovskites9,10,11,12,13,14,15,24,25,26,27,28 or non-magnetic ferroelectric (FE) semiconductors20,21,22,23,29,30,31,32,33,34. However, to the best of our knowledge, there are no studies on the spin texture in AFM FE HOIPs. Furthermore, antiferromagnets are very appealing for spintronic applications due to their superior properties, as they produce no stray fields and display intrinsic ultrafast spin dynamics35,36,37. In the last few years, intense theoretical and experimental research have shown that it is possible to realize electric-field control of magnetism in multiferroic materials38,39,40,41. The couplings between polarization, magnetism, and spin textures are still largely unexplored but they could have important applications in magneto-optoelectronic devices. Indeed, some recent reviews have pointed out intriguing spin-optotronic properties in HOIP materials42,43. Therefore, it is interesting to study whether there exists unusual Rashba effects in AFM FE HOIPs and how to manipulate the spin textures.

In this work, we start by considering the compound of TMCM-MnCl3 (TMCM = (CH3)3NCH2Cl+, trimethylchloromethyl ammonium). In 2017, You et al.44 reported that TMCM-MnCl3 is a FE material that shows an excellent piezoelectric response (d33 = 185 pC/N) that is close to that of inorganic piezoelectrics such as BaTiO3 (d33 = 190 pC/N) and a high transition temperature Tc of 406 K. Besides, TMCM-MnCl3 displays excellent photoluminescence properties, with a near-unity photoluminescence emission efficiency44. In our study, we discuss the interplay among FE and magnetic orderings, and spin textures by using density functional theory (DFT). We show that TMCM-MnCl3 is a prototype of order–disorder and displacement-type FE whose polarization can be greatly modified by the halogen atom substitutions. The most important result is that a Rashba-like effect in the band structure leads to robust unidirectional persistent spin texture (PST) and/or spiral spin texture31,45. The spin textures have been predicted to support an extraordinarily long spin lifetime, which is promising for optoelectronic devices46,47. By tuning the FE or, surprisingly, the antiferromagnetic order parameter, we find that the spin texture can be modified significantly. Our results indicate that not only the electric but also the magnetic field can effectively be used to manipulate the spin textures even in AFM but polar HOIP materials such as TMCM-MnCl3. Our results suggest that AFM FE HOIPs is an interesting class of materials that deserves further study.

## Results

At high temperature, TMCM-MnCl3 adopts a paraelectric phase with centrosymmetric space group P63/mmc with disordered organic cations. However, as the temperature decreases, TMCM-MnCl3 undergoes an order–disorder phase transition at around 406 K and crystallizes into the polar Cc phase44. To find the experimental ground-state structure, we rotate the organic cations randomly to consider different configurations and then optimize these structures to calculate their total energies. In our work, we generate about 100 random structures and find that the Cc phase indeed has the lowest total energy, in agreement with the experimental result. It has a monoclinic conventional cell with the distortion angle between a and c axis about 95°. Our optimized lattice constants are a = 9.371, b = 15.548, and c = 6.521Å which are consistent with the experimental report of a = 9.478, b = 15.741, and c = 6.577 Å. As shown in Fig. 1, TMCM-MnCl3 contains four organic cations and four Mn ions in the conventional cell. Their crystal packing is similar to BaNiO3-like perovskite. The Mn ions form the inorganic chains along [001] direction with the ligand ions of Cl, whereas the organic cations are inserted between these inorganic chains. One can see that the freezing of polar organic cations can give rise to the FE polarization along approximately the [10$$\bar 1$$] direction. From crystallographic analysis, it has 6 different polar axes with 12 possible orientations of polarization. This multiaxial characteristic is certainly interesting for fundamental research and practical applications of FE HOIPs48.

To study FE properties, we apply the modern theory of electric polarization49,50. The details of DFT calculations are described in “Methods.” To simulate the antiferroelectric (AFE)–FE transition, we fix two organic cations and rotate the other two cations by introducing an interpolating parameter λ (see Supplementary Note 1 and Supplementary Fig. 1)51,52,53. It is noteworthy that this dimensionless parameter λ is not the usual linear interpolation for atomic positions but it defines the correlated rotation of cations as well as the displacement of the MnCl3 framework. Therefore, it represents the normalized amplitude of the roto-displacive path. As the transition path is artificially assigned to act as a computational tool, the polarization difference between λ = 1 and λ = 0 states has a real physical meaning. Here we define our convention for the coordinates as the x (y) axis being along a (b) axis, respectively. As for the z axis, it is vertical to the xy plane and it has an angle about 5° with the c axis. In TMCM-MnCl3 system, the polarization is evaluated to be 6.12 μC/cm2 approximately along the [10$$\bar 1$$] direction (Px is about 4.18 μC/cm2 and Pz is about −4.48 μC/cm2), which is in rather good agreement with the experimental value of Pz ~ 4.00 μC/cm244. To shed light into the microscopic mechanism of FE polarization, we perform mode decomposition54 with respect to the reference centric phase by considering the different functional units, i.e., organic cations and framework. This approach, called functional mode analysis, has been already used for the analysis of FE polarization in hybrid compounds. Here, functional refers to functional units in the HIOPs, i.e., organic cations and framework. It is useful to disentangle the different contributions to the total polarization by considering the role played by the different functional units. We find that the polarization contains two main contributions, one is from the organic cations about 4.87 μC/cm2 and the other part comes from the distortion of the inorganic framework, which is about 1.15 μC/cm2. The first contribution can be associated to the ordering of the organic cations, whereas the second one can be related to a significant displacement-type contribution. Therefore, TMCM-MnCl3 is a prototype as order–disorder and displacement-type FE.

The halogen atoms and H atoms can form a complex hydrogen bonding network with the organic cations, which mainly determine the relative orientations of the organic cations with respect to the framework. Therefore, it may be useful to study how the halogen substitutions may influence the FE polarization. Indeed, the halogen atoms have similar chemical properties, but they differ in electronegativity, which, in turn, will effectively change the electric polarization through hydrogen bond network that is responsible for the complex cations and framework interaction. By changing the halogen atoms in the inorganic framework and/or organic cations, we find that the polarization can be significantly modified (see Supplementary Figs 23).

As for the magnetic ground state, we performed collinear calculations showing that TMCM-MnCl3 has strong AFM interaction within the inorganic MnCl3 chains. This can be understood in terms of Goodenough–Kanamori rule, which predicts a strong AFM super-exchange interaction between two half-filled $$Mn^{2 + }$$ (3d5) ions55,56. However, the interchain interaction between the inorganic MnCl3 chains is weak AFM, as the distance between neighboring chains is large (>9 Å). The energy of different magnetic configurations is shown in Supplementary Fig. 4. The G-type AFM state is the ground state with AFM intrachain and interchain couplings. To accurately evaluate the spin coupling parameters, we adopt a four-state method57,58. The effective spin exchange J for the intrachain Mn–Mn pair is computed to be 12 meV, whereas the interchain interaction is about 0.1 meV. When considering the SOC effect, the non-collinear calculations show that the local spin moments tend to be perpendicular to the MnCl3 chains and the magnetic anisotropy energy is about 0.03 meV/Mn. We note that there is no relevant canting of spins in TMCM-MnCl3 system, i.e., we have a collinear AFM HOIPs compound. Considering that TMCM-MnCl3 has weak interchain interaction, one can apply external fields (e.g., magnetic field) to switch the spins along one direction, i.e., C-type AFM state (intrachain AFM coupling and interchain FM coupling). Therefore, considering the tunable FE and magnetic states, the TMCM-MnCl3 system provides an ideal platform to investigate the interplay between FE ordering, magnetic ordering, and spin textures.

We investigate the electronic properties of valence band maximum (VBM) and conduction band minimum (CBM) by calculating the band structures with/without SOC (see Figs 2a-d). Here, the conventional cell contain four organic cations and four Mn ions (see Fig. 1). When considering SOC, the spin moment is set to be along y axis. The band structures of G-type $${\mathrm{AFM}}_y$$ and C-type $${\mathrm{AFM}}_y$$ states are shown in Fig. 2b, d, respectively. To help understand the spin textures discussed in the following paragraphs, we choose a specific symmetric k-path containing $$k_b$$ and $$k_{ac}$$, which is perpendicular to the polarization (see Fig. 2e, f)23,31,32,59. Here, $$k_b$$ and $$k_{ac}$$ denote the $$k$$ path from $${\mathrm{\Gamma }}$$ (0,0,0) to $${\mathrm{Y}}$$ (0,0.5,0) and $${\mathrm{Q}}$$ (0.5,0,0.5), respectively. To simplify the illustration of Brillouin zone, we simplify the crystal lattice from slightly monoclinic to orthorhombic (see Fig. 2f). For G-type AFM state (see Fig. 2a), our calculations show that the VBM and CBM are located at the $${\mathrm{\Gamma }}$$ point and the partial density of states (DOS) show that the valence band edge contains contributions from Mn-3d and Cl-2p orbitals, whereas the conduction band edge is mainly composed of Mn-3d orbitals (see Supplementary Fig. 5). Due to the symmetry (see below for detailed analysis), all eigenstates are at least twofold degenerate (i.e., spin-up and spin-down states). When taking SOC into account, the Rashba–Dresselhaus effect removes the spin degeneracy into singlets along the symmetry path but it still keeps twofold degeneracy at $${\mathrm{\Gamma }}$$ point (see Fig. 2b). Interestingly, for the C-type $${\mathrm{AFM}}$$ state, the doublet at $${\mathrm{\Gamma }}$$ point splits into two singlets with a sizable spin splitting at VBM about 0.027 eV after inclusion of SOC (see Fig. 2d).

To understand the band degeneracy at $${\mathrm{\Gamma }}$$ point, we perform the symmetry analysis by considering Kramers degeneracy. Considering a Hamiltonian $$\hat H$$ with an eigenvector $$\psi$$ and a real eigenvalue $$\lambda$$ such as $$\hat H\psi = \lambda \psi$$. Let $$\psi \prime = \hat A\psi$$, where $$\hat A$$ commutes with $$\hat H$$. It’s easy to write: $$\hat H\psi {\prime} = \hat H\hat A\psi = \hat A\hat H\psi = \lambda \hat A\psi = \lambda \psi ^{\prime}$$. Hence, both $$\psi$$ and $$\psi^ {\prime}$$ are eigenvectors of $$\hat H$$ with the same eigenvalue $$\lambda$$. One can prove that $$\psi$$ and $$\psi {\prime}$$ are orthogonal to each other if $$\hat A$$ is anti-unitary and $$\hat A^2\psi = - \psi$$, as $$\langle\psi ,\hat A\psi\rangle =\langle \hat A\psi ,\hat A^2\psi \rangle^{\ast}$$ = $$- \langle\hat A\psi ,\psi\rangle^{\ast}= - \langle\psi ,\hat A\psi\rangle$$ and thus $$\langle\psi ,\hat A\psi\rangle = 0$$. Due to the orthogonality, $$\psi$$ is degenerate with $$\psi {\prime} = \hat A\psi$$. Therefore, if $$\hat A$$ commutes with $$\hat H$$ and $$\hat A^2 = - 1$$, the band structure can be double degenerate. In our TMCM-MnCl3 system, the G-type $${\mathrm{AFM}}_y$$ state has the magnetic symmetry of $$\hat M = \{ \left\{ {E{\mathrm{|}}0} \right\},\left\{ {\hat m_{ac}{\mathrm{|}}\frac{{a + b + c}}{2}} \right\},\hat T\left\{ {\hat m_{ac}{\mathrm{|}}\frac{c}{2}} \right\},\hat T\left\{ {E{\mathrm{|}}\frac{{a + b}}{2}} \right\}\}$$, where $$E$$ is identity operator, $$\hat T$$ is time-reversal operator, and $$\hat m_{ac}$$ is the mirror symmetry operator followed by lattice translation. We find that the operator $$\hat A_{\mathrm{b}} = \hat T\left\{ {E{\mathrm{|}}\frac{{a + b}}{2}} \right\}$$ is anti-unitary and commutes with the Hamiltonian at $${\mathrm{\Gamma }}$$ point (here, $$\hat A_{\mathrm{b}}$$ plays a similar role as $$\hat T$$ in the time-reversal invariant case considered by the Kramers degeneracy). The band structure of G-type $${\mathrm{AFM}}_y$$ state is shown in Fig. 2b and we use the subscript of $$\hat A$$ (i.e., b) to index the band structure. Using the properties of half-spin system at the $${\mathrm{\Gamma }}$$ point, we can identify $$\hat A_{\mathrm{b}}^2 = \hat T^2 = - 1$$, leading to twofold degeneracy at the $${\mathrm{\Gamma }}$$ point with SOC effect. As for C-type $${\mathrm{AFM}}_y$$ state, it has the magnetic symmetry of $$\hat M = \{ \left\{ {E{\mathrm{|}}0} \right\},\left\{ {E{\mathrm{|}}\frac{{a + b}}{2}} \right\},\hat T\left\{ {\hat m_{ac}{\mathrm{|}}\frac{{a + b + c}}{2}} \right\},\hat T\left\{ {\hat m_{ac}{\mathrm{|}}\frac{c}{2}} \right\}\}$$. Different from G-type $${\mathrm{AFM}}_y$$ state, we cannot find such an anti-unitary symmetry operator $$\hat A_{\mathrm{d}}$$ to construct Kramers pair, as $$\left( {\hat T\left\{ {m_{ac}{\mathrm{|}}\tau _C} \right\}} \right)^2\psi = \hat T^2m_{ac}^2\psi = - 1\! \cdot \! \left( { - 1} \right)\! \cdot \! \psi = \psi$$, where $$\tau _C = \frac{{a + b + c}}{2}$$ or $$\frac{c}{2}$$. Hence, the energy bands of C-type $${\mathrm{AFM}}_y$$ with SOC are all singlet as shown in Fig. 2d. When turning off SOC, spin is independent from the spatial degrees of freedom and pure spin rotation $$\hat U$$ can be introduced to explain the energy band degeneracy60. $$\hat U$$ can reverse the spin but it is unitary and keeps the momentum invariant. For collinear AFM system without SOC, the wave function $$\phi$$ can be chosen to have a definite Sz value (1/2 for up-spin or -1/2 for down-spin); thus, $$\hat U\phi$$ and $$\phi$$ are orthogonal and form the Kramers pair. Without SOC, G-type AFM state has the symmetry of $$\hat U\left\{ {E{\mathrm{|}}\frac{{a + b}}{2}} \right\}$$. It commutes with the Hamiltonian for all wave vectors and lead to twofold degeneracy in the whole BZ, including path $${{Q}} - {\mathrm{\Gamma }} - {{Y}}$$ as shown in Fig. 2a. As for the C-type AFM state, the twofold degeneracy along $${{Q}} - {\mathrm{\Gamma }}$$ and $${\mathrm{\Gamma }} - {{Y}}$$ (see Fig. 2c) can be ascribed to different symmetry mechanism. The wave vectors in $${{Q}} - {\mathrm{\Gamma }}$$ and $${\mathrm{\Gamma }} - {{Y}}$$ respect the symmetry of $$\hat T\left\{ {m_{ac}{\mathrm{|}}\frac{c}{2}} \right\}$$ and $$\hat U\left\{ {m_{ac}{\mathrm{|}}\frac{c}{2}} \right\}$$, respectively. It is noteworthy that without SOC, one can get $$m_{ac}^2\psi = \psi$$ and hence we can have $$\hat A_{\mathrm{c}} = \hat T\left\{ {m_{ac}{\mathrm{|}}\frac{c}{2}} \right\}$$, $$\hat A_c^2\psi = \hat T^2m_{ac}^2\psi = - 1\! \cdot \! 1\! \cdot \! \psi = - \psi$$. Therefore, both $$\hat T\left\{ {m_{ac}{\mathrm{|}}\frac{c}{2}} \right\}$$ and $$\hat U\left\{ {m_{ac}{\mathrm{|}}\frac{c}{2}} \right\}$$ can form Kramers pair and the corresponding band structure is double degenerate. Apart from these symmetry arguments, we can also apply systematic group theory analysis based on the co-representation of the magnetic point group to understand the spin degeneracy at the $${\mathrm{\Gamma }}$$ point (see Supplementary Note 2). These two methods give the same results. Therefore, to summarize our discussion, the different symmetry operations in G-type $${\mathrm{AFM}}_y$$ state and C-type $${\mathrm{AFM}}_y$$ state can lead to different band degeneracy at the $${\mathrm{\Gamma }}$$ point.

Knowing the spin degeneracy at the $${\mathrm{\Gamma }}$$ point, we can consider the spin texture around this point in the Brillouin zone. Considering that TMCM-MnCl3 displays two long-range ordering, i.e., FE and AFM orderings, it is interesting to see how the spin textures behave under the interplay of these two order parameters. Recently, the electric-field control of spin textures has been shown in non-magnetic FE GeTe thin film20,21,22,23. Here, as we will show below, the spin textures in TMCM-MnCl3 can be tuned by switching not only FE ordering but also by switching the magnetic order parameter in an AFM polar HOIP. This represents a new degree of freedom to play with in the spin-texture tuning, which has been very little studied in the literature. The SOC splits the band structure into two branches, which exhibit similar spin textures but with opposite helicity or orientation. Here we will focus on the inner branches near the $${\mathrm{\Gamma }}$$ point, whereas the spin textures of the outer branches are illustrated in Supplementary Figs 717. To simplify the visualization, we project the spin textures on a specific plane, which is perpendicular to the polarization (see Fig. 2f)23,31,32,59.

In the following, we discuss the spin textures in G-type AFM state. We pay attention to the spin texture at CBM, as the spin value at VBM is small due to the weak band splitting. It is useful to introduce the AFM order parameter defined as $${\mathbf{L}} ={\sum \nolimits_{\boldsymbol{i}}} {\mathbf{S}}^{\boldsymbol{i}} - {\sum \nolimits_{\boldsymbol{j}}} {\mathbf{S}}^{\boldsymbol{j}}$$, where $${\mathbf{S}}^i$$ ($${\mathbf{S}}^j$$) is the spin moment along the positive (negative) axis, respectively. We use the subscript of L to define different AFM state. For example, $${\mathbf{L}}_{\mathbf{G}}$$~y indicates the G-type AFM configuration along the y direction. In addition, we use $${\mathbf{L}}_{\mathbf{G}}$$~ −y to indicate the operation that flip the spin from y to −y direction. The polarization (P) is along the [10$$\bar 1$$] direction, whereas −P is along the [$$\bar 101$$] direction. As we can see in Fig. 3a, it shows a robust PST at CBM. The spin is unidirectional and parallel ($$k_{ac}$$ < 0) or antiparallel ($$k_{ac}$$ > 0) to $$k_b$$ direction (i.e., vertical to the mirror reflection). One can understand the spin texture by considering the magnetic symmetry. The G-type $${\mathrm{AFM}}_y$$ state has the magnetic space symmetry of $$\hat M = \left\{ {\left\{ {E\left| 0 \right.} \right\},\left\{ {\hat m_{ac}\left| {\frac{{a + b + c}}{2}} \right.} \right\},\hat T\left\{ {\hat m_{ac}\left| {\frac{c}{2}} \right.} \right\},\hat T\left\{ {E\left| {\frac{{a + b}}{2}} \right.} \right\}} \right\}$$, which can be labeled with $$\hat M = \{ \hat m,\hat T\hat m,\hat T\}$$. Thus, one can have the following constraints on spin texture: $${\mathbf{S}}\left( k \right) = \hat m{\mathbf{S}}\left( {\hat mk} \right)$$, $${\mathbf{S}}\left( k \right) = \hat T\hat m{\mathbf{S}}\left( {\hat T\hat mk} \right)$$, and $${\mathbf{S}}\left( k \right) = \hat T{\mathbf{S}}\left( {\hat Tk} \right)$$. We note that the PSTs occupy a substantial scale of Brillouin zone. It spans more than 0.04 $${\AA}^{-1}$$ around the $${\mathrm{\Gamma }}$$ point, whereas for comparison the reciprocal wave vector of $$k_b$$ is $${\uppi}/{\mathrm{b}} = 0.20\,{\AA}^{ - 1}$$, which corresponds to the length of symmetry path from (0,0,0) to (0,0.5,0). In this large area, the spin configurations remain nearly unidirectional, which is favorable to support the long spin lifetime of carrier promising for optoelectronic applications31,45,46. Our results suggest that TMCM-MnCl3 is a rare example of Rashba-AFM HOIP FE with robust PST.

Here we discuss the interplay between FE ordering, magnetic ordering, and spin texture. In Fig. 3b, we fix the magnetic ordering but reverse the FE polarization from P to −P. One can see that the PST is reversed with the switching of the polarization. The spin transformation rule under space inversion operator $$\hat I$$ is shown in Fig. 3e. When considering the spin configurations at the points $$\mathop{k}\limits^{\rightharpoonup}$$ to -$$\mathop{k}\limits^{\rightharpoonup}$$ related by the space inversion operator $$\hat I$$ (i.e., reversal of polarization), the spin orientations remain unchanged, as $$\hat I{\mathbf{S}}_{\mathbf{P}}\left( k \right) = {\mathbf{S}}_{ - {\mathbf{P}}}\left( { - k} \right) = {\mathbf{S}}_{\mathbf{P}}\left( k \right)$$, where P is the FE order parameter, $${\mathbf{S}}_{ - {\mathbf{P}}}$$ is the new spin after the space inversion. In G-type $${\mathrm{AFM}}_y$$ state, the spin at $$- k$$ point has opposite orientation compared with the spin at $$k$$ point, i.e., $${\mathbf{S}}_{\mathbf{P}}( - k) = - {\mathbf{S}}_{\mathbf{P}}(k)$$, whereas after the space inversion (switching of polarization), one can get $${\mathbf{S}}_{ - {\mathbf{P}}}\left( { - k} \right) = {\mathbf{S}}_{\mathbf{P}}(k)$$. Therefore, we can reverse the PST by switching the FE ordering. As for switching the magnetic ordering, the time-reversal operation $$\hat T$$ will reverse the magnetic ordering parameter L and spin texture changes according to $$\hat T{\mathbf{S}}_{\mathbf{L}}\left( k \right) = {\mathbf{S}}_{ - {\mathbf{L}}}\left( { - k} \right) = - {\mathbf{S}}_{\mathbf{L}}\left( k \right)$$, where $${\mathbf{S}}_{ - {\mathbf{L}}}$$ is the new spin after the time reversal. In Fig. 3c, we fix the FE order and then we flip magnetic ordering from $${\mathbf{L}}_{\mathbf{G}}$$~y to $${\mathbf{L}}_{\mathbf{G}}$$~ −y. We find the PST remains unchanged, as $${\mathbf{S}}_{\mathbf{L}}( - k) = - {\mathbf{S}}_{\mathbf{L}}(k)$$ and $${\mathbf{S}}_{ - {\mathbf{L}}}\left( { - k} \right) = - {\mathbf{S}}_{\mathbf{L}}(k)$$. Apart from the flip of magnetic ordering from L to −L, one can also switch the magnetic ordering to other directions. In Fig. 3d, we rotate the magnetic ordering from $${\mathbf{L}}_{\mathbf{G}}$$~y to $${\mathbf{L}}_{\mathbf{G}}$$~x. Surprisingly, there is PST at not only CBM (see Fig. 3d) but also VBM in comparison with the small spin value at VBM in $${\mathbf{L}}_{\mathbf{G}}$$~y case (see Supplementary Note 3). The G-type $${\mathrm{AFM}}_x$$ state has the magnetic symmetry of $$\hat M = \{ \left\{ {E{\mathrm{|}}0} \right\},\left\{ {\hat m_{ac}{\mathrm{|}}\frac{c}{2}} \right\},\hat T\left\{ {\hat m_{ac}{\mathrm{|}}\frac{{a + b + c}}{2}} \right\},\hat T\left\{ {E{\mathrm{|}}\frac{c}{2}} \right\}\}$$. The rotation part keeps the same with G-type $${\mathrm{AFM}}_y$$ state but the translational part of glide plane changes. We note that the magnetic symmetry is conserved as long as the magnetic ordering lies within the ac plane, i.e., the mirror $$\hat m_{ac}$$. According to the symmetry analysis, the operator $$\hat A = \hat T\left\{ {E{\mathrm{|}}\frac{c}{2}} \right\}$$ commutes with the Hamiltonian accompanied with $$\hat A^2 = \hat T^2 = - 1$$ at the $${\mathrm{\Gamma }}$$ point, leading to twofold degeneracy at $${\mathrm{\Gamma }}$$ point with SOC effect (see Supplementary Fig. 6). Our further calculations demonstrate that when we rotate the magnetic ordering within the ac plane, the PST can be switched along the magnetic ordering (see Supplementary Note 3). It is an interesting result that we can continuously rotate PST by switching the magnetic ordering. Now we can draw the conclusion that the G-type $${\mathrm{AFM}}$$ state shows the robust PST around the $${\mathrm{\Gamma }}$$ point and the PST can be manipulated not only by switching the polarization but also by switching the magnetic ordering. This is certainly a new result, as, so far, the switching of spin-texture chirality has been linked only to the switching of FE polarization while here we point out the active role of the switching of AFM order parameter.

Now we discuss the spin textures in C-type AFM state. TMCM-MnCl3 has weak interchain interaction and one may apply external fields (e.g., magnetic field) to switch the spins along one direction, i.e., C-type AFM state. We find the PST along $$k_{ac}$$ direction, although there is a deviation from the unidirectional spin orientation when moving far away from the $${\mathrm{\Gamma }}$$ point. We note that the PST in G-type $${\mathrm{AFM}}_y$$ state is along the $$k_b$$ direction. This phenomenon shows that one can switch the PST by realizing different magnetic state. As for CBM, it exhibits spiral spin textures with clockwise helicity. This two-dimensional vector field is identical to the characteristic Rashba-like spin texture23,32,59. The C-type $${\mathrm{AFM}}_y$$ state has the magnetic symmetry of $$\hat M = \left\{ {\left\{ {E{\mathrm{|}}0} \right\}} \right.,\left\{ {E{\mathrm{|}}\frac{{a + b}}{2}} \right\},\hat T\left\{ {\hat m_{ac}{\mathrm{|}}\frac{{a + b + c}}{2}} \right\},\hat T\left\{ {\hat m_{ac}{\mathrm{|}}\frac{c}{2}} \right\},$$ which can be labeled with $$\hat M = \{ \hat T\hat m\}$$. Correspondingly, the doublet state at $${\mathrm{\Gamma }}$$ point is lifted into singlets by SOC with a sizable band spin splitting at VBM about 0.027 eV (see Fig. 2d). According to the $$\hat T\hat m$$ symmetry, the PST and spiral spin texture can be understood with $${\mathbf{S}}\left( k \right) = \hat T\hat m{\mathbf{S}}\left( {\hat T\hat mk} \right)$$. At $${\mathrm{\Gamma }}$$ point, there is no spin component along $$k_b$$ direction, as $$\hat T\hat mk$$= k and $${\mathbf{S}}({\mathrm{\Gamma }}) = \hat T\hat m{\mathbf{S}}\left( {\mathrm{\Gamma }} \right)$$. It is interesting that VBM and CBM have same magnetic symmetry but show PST and spiral spin texture, respectively. To the best of our knowledge, it is the first case about the coexistence of PST and spiral spin texture in the same compound.

In Fig. 4c, d, we fix the magnetic ordering but flip the FE polarization from P to −P. The spin textures are switched according to $$\hat I{\mathbf{S}}_{\mathbf{P}}\left( k \right) = {\mathbf{S}}_{ - {\mathbf{P}}}\left( { - k} \right) = {\mathbf{S}}_{\mathbf{P}}\left( k \right)$$. For the VBM of C-type $${\mathrm{AFM}}_y$$ state, the spin at $$- k$$ and $$k$$ point have same orientation, i.e., $${\mathbf{S}}_{\mathbf{P}}( - k) = {\mathbf{S}}_{\mathbf{P}}(k)$$, whereas after the space inversion, one can get $${\mathbf{S}}_{ - {\mathbf{P}}}\left( { - k} \right) = {\mathbf{S}}_{\mathbf{P}}(k)$$. Therefore, when switching the FE polarization, the PST maintains the same spin orientation (see Fig. 4c). However, as for the CBM, the spin at $$- k$$ and $$k$$ point have opposite orientation, i.e., $${\mathbf{S}}_{\mathbf{P}}( - k) = - {\mathbf{S}}_{\mathbf{P}}(k)$$. After the space inversion, it can be $${\mathbf{S}}_{ - {\mathbf{P}}}\left( { - k} \right) = {\mathbf{S}}_{\mathbf{P}}(k)$$. Thus, the helicity of spiral spin texture is reversed (see Fig. 4d). It is interesting about the different tunability of VBM and CBM under same external field. It is also important to note that this compound has been recently synthesized and the switching of polarization has been realized with a well-defined PE loop44; therefore, we expect that the manipulation of spin textures by the external electric field could be easily verified by experiments. In Fig. 4e, f, we fix the FE order but flip the AFM ordering from $${\mathbf{L}}_{\mathbf{C}}$$~y to y to see the variation of spin texture. We find the PST of VBM (see Fig. 4e) is reversed whereas the spiral spin texture of CBM (see Fig. 4f) maintains the same helicity. The variation of spin texture is consistent with the rule of $$\hat T{\mathbf{S}}_{\mathbf{L}}\left( k \right) = {\mathbf{S}}_{ - {\mathbf{L}}}\left( { - k} \right) = - {\mathbf{S}}_{\mathbf{L}}\left( k \right)$$. For VBM, before switching L, $${\mathbf{S}}_{\mathbf{L}}( - k) = {\mathbf{S}}_{\mathbf{L}}(k)$$. After switching L, $${\mathbf{S}}_{ - {\mathbf{L}}}( - k) = - {\mathbf{S}}_{\mathbf{L}}(k)$$. However, for CBM, before switching L, $${\mathbf{S}}_{\mathbf{L}}\left( { - k} \right) = - {\mathbf{S}}_{\mathbf{L}}(k)$$. After switching L, $${\mathbf{S}}_{ - {\mathbf{L}}}( - k) = - {\mathbf{S}}_{\mathbf{L}}(k)$$. The variation is totally different from the spin texture tunability upon the change of FE polarization. Our results demonstrate that one can manipulate the spin textures by switching the AFM order parameter but independently from the electric degrees of freedoms. Furthermore, to the best of our knowledge, we present a unique case in the literature, where there is coexistence of PST and spiral spin texture in the same material.

We also investigate the spin textures with other magnetic configurations (see Supplementary Figs 717). By manipulating the magnetic order parameter with different orientation and different magnetic state, the corresponding spin texture will change accordingly and it is the origin of magneto-crystalline anisotropy61. This property is dual of the spin-texture electric-anisotropy first discussed in the HOIP material (NH2CHNH2)SnI362 where it has been shown that the spin-texture topology is modified significantly upon variations of the direction of the electric polarization. The sensitivity of the topology of spin texture to variation/switching of the magnetic order parameter could have far reaching consequences in AFM spintronics, as this property could be exploited in AFM memory elements: the change in spin-texture topology of relevant electronic bands should be detectable in terms of magneto-optical Kerr effect, as already shown in the metal-organic framework material [C(NH2)3]Cr[(HCOO)3]63. Further study is in progress to verify these properties. Our results clearly suggest that one could manipulate the spin texture via tuning the magnetic ordering at different levels: by fixing the magnetic configurations but changing the L orientation in space, or by changing the different realization of L. It has been shown that AFM materials can be manipulated by applying magnetic fields36,64,65. The magnetic moments can be appreciably rotated in a quasi-static manner within the Stoner-Wohlfarth model66. In this picture, the ordered magnetic state is preserved when the magnetization is reversed and a spin-flop field can rotate the magnetic moments by 90° 67. Besides, the AFM state could be reoriented by optical excitation68,69, exchange bias70,71, strain72,73, and other different approaches36,65. We note that the manipulation of FE polarization and magnetic configuration was realized in the classical multiferroic material $${\mathrm{TbMnO}}_3$$39 and $${\mathrm{BiFeO}}_3$$40. Therefore, the FE and magnetic orderings in a polar AFM HOIP system could be tuned and the spin texture can be manipulated at the same time, thus leading to interesting magneto-optoelectronic applications.

Our study shows the possibility of tuning spin textures by electric and magnetic fields in AFM HOIP FEs and enhancing its optoelectronic performance, although there remain some challenges such as the wide bandgap and low magnetic ordering temperatures. In our TMCM-MnCl3 system, the bandgap is calculated to be 3.75 eV which is larger than the ideal bandgap suitable for optoelectronic applications. It is reported that TMCM-MnCl3 displays excellent photoluminescence properties with a near-unity photoluminescence emission efficiency44, thus our DFT calculations may overestimate the bandgap. Xiong and colleagues74,75 proposed that TMCM-MnCl3 can be further engineered through element substitution and molecular design so as to optimize for a desired physical properties, as shown by bandgap engineering. Taking the characteristic HOIP material MAPbI3 as an example, the band gap can be easily tuned from 1.2 to 3.0 eV by engineering chemical composition75. The magnetic ordering temperature can be improved as well. In our TMCM-MnCl3 system, the Néel temperature is low due to the weak interchain interaction, which can be ascribed to the large organic cation. The Néel temperature could be improved with smaller organic cation. Furthermore, the substitution on B-site magnetic ions can enhance the magnetic ordering temperature. For example, it is reported that the HOIP material ((CH3)4P)FeBr4 exhibits coupled dielectric and magnetic phase transitions above room temperature76. It is important to note that our study puts forward the concept that one can manipulate the spin texture by applying magnetic fields. The external magnetic field can stabilize the AFM ordering and raise the Néel temperature. Therefore, we hope to stimulate the search of high temperature AFM HOIP FEs in the future.

## Disussion

In this work, we propose the manipulations of spin textures in the AFM HOIP FE TMCM-MnCl3. By using first-principles calculations, we identify a Rashba-like splitting in the band structure. The symmetry analyses based on magnetic space group are used to explain the band degeneracy. We find robust PST in G-type AFM state and it can be effectively manipulated by switching not only polarization but also magnetic ordering. We also find the coexistence of PST and typical spiral spin texture, depending of the relevant band electronic states, in C-type AFM state. To the best of our knowledge, this is the first case of coexistence of PST and spiral spin texture in the same compound. By manipulating the FE, and, interestingly, the magnetic order parameter, the spin texture can be modified significantly. Our work introduces new directions in the field of spin-texture manipulation by external fields, which goes beyond the usual electric-field control of Rashba effect in non-magnetic materials. Considering that TMCM-MnCl3 belongs to the important class of HOIPs, which is relevant to optoelectronic research, we expect that, this study could suggest new magneto-optoelectronic properties in HOIPs. As the switching of polarization in TMCM-MnCl3 has been experimentally demonstrated44, we hope to stimulate new experiments to verify manipulations of spin textures in TMCM-MnCl3 by electric and/or magnetic fields. We expect that AFM HOIP FEs have the potential to improve the optoelectronic performance and give a new strategy to design new multifunctional materials.

## Methods

### DFT calculations

Our first-principles calculations are performed within DFT. The interactions of valence electrons and ions is treated with the projector augmented wave method77 as implemented in the Vienna ab-initio simulation package (VASP)78. The exchange-correlation potential is described by the Perdew–Burke–Ernzerhof functional79. The plane wave cutoff energy is fixed to be 550 eV, and all atomic positions are optimized until each component of the atomic force is smaller than 0.01 eV/Å. The $$4 \times 2 \times 4$$ k-point mesh is used for the Brillouin integration. The electric polarization is calculated by using the Berry phase method50,80. In this approach, we first define a centrosymmetric reference phase which shows an AFE alignment of dipoles in the unit cell and then we continuously rotate and translate the organic cations to reach the FE phase by defining a roto-displacive path in the configuration space. In our work, we take the Van der Waals interactions into account by DFT-D3 correction method81,82 as implemented in the VASP software. The correlated nature of Mn 3d state is included by Hubbard-like corrections with repulsion energy U=3 eV and Hund coupling energy J = 1 eV. Our calculations show that small variation of U and J does not change the main results of our study. To calculate the spin textures, the mean values of the sigma matrices are evaluated at the relevant electronic states with different k-points around a reference point in the Brillouin zone.

## Data availability

The data that support the findings of this study are available from the corresponding authors upon reasonable request.

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## Acknowledgements

This work is supported by NSFC 11825403, the Program for Professor of Special Appointment (Eastern Scholar), the Qing Nian Ba Jian Program, and the Fok Ying Tung Education Foundation. F. L. thanks Dr. K. Liu and J. Li for useful discussion. T.G. thanks the kind hospitality of CNR-SPIN c/o Department of Chemical and Physical Science of University of L’Aquila where this project was initiated during the visiting period from 8 November 2017 to 14 January 2018. A.S. thanks the warm hospitality of W.Li, X.-H. Bu (Nankai University), H. Wu (Fudan University), and W. Ren (Shanghai University), where this work was partially finalized.

## Author information

Authors

### Contributions

H.X. and A.S. proposed and supervised the project. F.L. and T.G. performed the first-principles calculations with the help from J.J. and J.F. F.L., T.G., and J.J. prepared the initial draft of the paper. All authors contributed to the writing and revision of the paper. F.L. and T.G. contribute equally to this work.

### Corresponding authors

Correspondence to Hongjun Xiang or Alessandro Stroppa.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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Reprints and Permissions

Lou, F., Gu, T., Ji, J. et al. Tunable spin textures in polar antiferromagnetic hybrid organic–inorganic perovskites by electric and magnetic fields. npj Comput Mater 6, 114 (2020). https://doi.org/10.1038/s41524-020-00374-8

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• ### Discovery Principles and Materials for Symmetry-Protected Persistent Spin Textures with Long Spin Lifetimes

• Xue-Zeng Lu
•  & James M. Rondinelli

Matter (2020)


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• 1.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
Bounds on Hilbert Functions2013Licentiate thesis, comprehensive summary (Other academic)

This thesis is constituted of two articles, both related to Hilbert functions and h-vectors. In the first paper, we deal with h-vectorsof reduced zero-dimensional schemes in the projective plane, and, in particular, with the problem of finding the possible h-vectors for the union of two sets of points of given h-vectors. In the second paper, we generalize the Green’s Hyperplane Restriction Theorem to the case of modules over the polynomial ring.

• 2.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
Bounds on Hilbert Functions and Betti Numbers of Veronese Modules2014Doctoral thesis, comprehensive summary (Other academic)

The thesis is a collection of four papers dealing with Hilbert functions and Betti numbers.In the first paper, we study the h-vectors of reduced zero-dimensional schemes in $\mathbb{P}^{2}$ . In particular we deal with the problem of findingthe possible h-vectors for the union of two sets of points of given h-vectors. To answer to this problem, we give two kinds of bounds on theh-vectors and we provide an algorithm that calculates many possibleh-vectors.In the second paper, we prove a generalization of Green’s Hyper-plane Restriction Theorem to the case of finitely generated modulesover the polynomial ring, providing an upper bound for the Hilbertfunction of the general linear restriction of a module M in a degree dby the corresponding Hilbert function of a lexicographic module.In the third paper, we study the minimal free resolution of theVeronese modules, $S_{n,d,k}=\oplus_{i\geq 0} S_{k+id}$, where  $S = \mathbb{K}[x_1 , . . . , x_n ]$ by giving a formula for the Betti numbers in terms of the reduced homology of the squarefree divisor complex. We prove that $S_{n,d,k}$ is Cohen-Macaulay if and only if k < d, and that its minimal resolutionis linear when k > d(n − 1) − n. We prove combinatorially that the resolution of $S_{2,d,k}$ is pure. We provide a formula for the Hilbert seriesof the Veronese modules. As an application, we calculate the completeBetti diagrams of the Veronese rings $S_{3,4,0} , S_{3,5,0} and S_{4,3,0}$ .In the fourth paper, we apply the same combinatorial techniques inthe study of the properties of pinched Veronese rings, in particular weshow when this ring is Cohen-Macaulay. We also study the canonicalmodule of the Veronese modules.

• 3.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
Cohen-Macaulay Property of pinched Veronese Rings and Canonical Modules of Veronese  ModulesManuscript (preprint) (Other academic)

In this paper, we study the Betti numbers of pinched Veronese rings, by means of the reduced homology of the squarefree divisor complex. In particular, we study the Cohen-Macaulay property of these rings. Moreover, in the last section we compute the canonical modules of the Veronese modules.

• 4.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
Green’s Hyperplane Restriction Theorem: an extension to modules2015In: Journal of Pure and Applied Algebra, ISSN 0022-4049, E-ISSN 1873-1376, Vol. 219, no 8, p. 3506-3517Article in journal (Refereed)

In this paper, we prove a generalization of Green's Hyperplane RestrictionTheorem to the case of modules over the polynomial ring, providing in particularan upper bound for the Hilbert function of the general linear restrictionof a module M in a degree d by the corresponding Hilbert function of alexicographic module.

• 5.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
Syzygies of the Veronese Modules2016In: Communications in Algebra, ISSN 0092-7872, E-ISSN 1532-4125, Vol. 44, no 9, p. 3890-3906Article in journal (Refereed)

We study the minimal free resolution of the Veronese modules, S-n,S- d,S- k = circle plus S-i >= 0(k+id), where S = K[x(1), ... , x(n)], by giving a formula for the Betti numbers in terms of the reduced homology of some skeleton of a simplicial complex. We prove that S-n,S- d,S- k is Cohen-Macaulay if and only if k < d, and that its minimal resolution is pure and has some linearity features when k > d (n - 1) - n. We prove combinatorially that the resolution of S-2,S- d,S- k is pure. We show that HS(S-n,S- d,S- k; z) = 1/(n-1)d(n-1)/dz(n-1) [z(k+n-1)/1-z(d)]. As an application, we calculate the complete Betti diagrams of the Veronese rings K[x, y, z]((d)), for d = 4, 5, and K[x, y, z, u]((3)).

• 6.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
Syzygies of Veronese modulesManuscript (preprint) (Other academic)
• 7.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
The h-vector of the union of two sets of points in the projective plane2012In: Le Matematiche, ISSN 2037-5298, E-ISSN 0373-3505, Vol. 67, no 1Article in journal (Refereed)

Given two h-vectors, handh0, we study which are the possible h-vectors for the union of two disjoint sets of points in P2, respectively associated to h and h0and how they can be constructed. We will give some bounds for the resulting h-vector and we will show how to construct the minimal h-vector of the union among all possible ones.

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
Packing equal disks in a unit square: An immunological optimization approach2014In: International Workshop on Artificial Immune Systems, AIS 2015/ICSI3 2015 - Systems Immunology, Immunoinformatics and Immune-computation: Immunology without Borders, Proceedings, Institute of Electrical and Electronics Engineers (IEEE), 2014, article id 7469239Conference paper (Refereed)

Packing equal disks in a unit square is a classical geometrical problem which arises in many industrial and scientific fields. Finding optimal solutions has been proved to be NPhard, therefore, only local optimal solutions can be identified. We tackle this problem by means of the optimization Immune Algorithm (optIA), which has been proved to be among the best derivativefree optimization algorithms. In particular, OPTIA is used to pack up to 150 disks in a unit square. Experimental results show that the immune algorithm is able to locate the putative global optimum for all the instances. Moreover, a comparison with the Covariance Matrix Adaptation Evolution Strategy (CMA-ES) shows that OPTIA is more robust.

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# A certain Acyclic Partition of a digraph

Has the following object been defined in the literature? What is it called? And what literature studies it? Are there other characterizations of this? What properties are known?

Let $G$ be a directed loopless connected graph. The Acyclic Partition (AP) of $G$ is a partition of vertices $S = \{S_1, S_2, \ldots, S_k\}$ of $G$ with the following properties:

1) Each simple cycle $(v_{i_1}, \ldots, v_{i_n})$ of $G$ is contained in exactly one $S_i \in S$.

2) If $u,v \in S_i \in S$, and there is a path between $u$ and $v$, then all paths between $u$ and $v$ are completely contained in $S_i$.

If $S_i, S_j \in S$ and $(S \setminus \{S_i, S_j\}) \cup \{S_i \cup S_j\}$ is an AP, then I call $S$ reducible.

For a given graph $G$, I am interested in its irreducible AP of maximum cardinality.

Update:

I'd like to clarify between an irreducible AP and a Strong Components decomposition. In the example graph shown in the image below, a Strong Components algorithm would return three components $\{\{1,2\}, \{3\}, \{4,5\}\}$ , assuming a single vertex is defined to be connected to itself.

The Strong Component decomposition described above would be an AP. But since $\{\{1,2, 3\}, \{4,5\}\}$ is also an AP and is irreducible, it would be the preferred AP. (Irreducible APs are not necessarily unique.) This illustrates how APs are different from Strong Component decompositions.

Thank you.

• Looks like the partition into strong components to me. – Brendan McKay Aug 25 '14 at 7:10
• For a two-vertex unidirectional path, Strong Components would give two components, whereas its irreducible AP would be a single set containing both vertices. (The Strong Components Decomposition would be a reducible AP, since the union of the two sets is also an AP.) In general, it appears that a Strong Components Decomposition is a subset of an AP. But thanks for pointing out the very close connection to Strong Components that I had overlooked. – Swapnil Bhatia Aug 25 '14 at 15:34
• Can you give an example of an irreducible AP of maximum cardinality having more than one cell? – Brendan McKay Aug 26 '14 at 3:11
• In your example, $\lbrace\lbrace 1,2,3\rbrace\lbrace 4,5\rbrace\rbrace$ is not irreducible, since $\lbrace\lbrace 1,2,3,4,5\rbrace\rbrace$ is also an AP. For this reason, any irreducible AP with more than one cell must have at least three cells. – Brendan McKay Aug 27 '14 at 2:36
• Oh, yes! Thanks. I need to put some conditions on when $S_i, S_j$ in the definition of reducible, e.g. that they must be acyclic. But it does look like Strong Components are very close to what I am looking for. (The context is that I would like to define vertices that are guaranteed to indicate progress, say if the graph represented a process or a computation. – Swapnil Bhatia Sep 6 '14 at 4:04


text:
## freely rotating chain

in polymers
https://doi.org/10.1351/goldbook.F02518
A hypothetical @L03546@ molecule, free from short-range and long-range interactions, consisting of infinitely thin rectilinear segments (bonds) of fixed length, jointed at fixed bond angles; the torsion angles of the bonds can assume all values with equal @P04855@.
Source:
Purple Book, 1st ed., p. 50 [Terms] [Book]


text:
# List coloring of $k$th powers (2013)

Originator(s):    Xuding Zhu (presented by D. West - REGS 2013)

Definitions: The $k$th power of a graph $G$, written $G^k$, is the graph with vertex set $V(G)$ in which distinct vertices are adjacent if the distance between them in $G$ is at most $d$. The list chromatic number of $G$, written $\chi_\ell(G)$, is the least $k$ such that whenever the vertices are given lists of size $k$, a proper coloring can be chosen in which each vertex receives a color from its list. Always $\chi_\ell(G)\ge\chi(G)$; a graph $G$ is chromatic-choosable if equality holds.

Background: The famous List Coloring Conjecture asserts that every line graph is chromatic-choosable. Equivalently, proper edge-colorings can be chosen from lists of size $\chi'(G)$ on the edges. Published first by Bollob\'as and Harris [BH], the conjecture was independently formulated earlier by Albertson and Collins in 1981 and by Vizing as early as 1975 (both unpublished).

Motivated by this conjecture, Kostochka and Woodall [KW] asked whether the square of every graph is chromatic-choosable. Kim and Park [KP] answered the question in the negative, using complete families of mutually orthogonal Latin squares to construct graphs whose squares are large complete multipartite graphs and hence are not chromatic-choosable.

Question 1: (Zhu) Does there exist a constant $k$ such that the $k$th power of every graph is chromatic-choosable? (SOLVED)

Comments: At the workshop in Jinhua where Kim announced the result of [KP], it was noticed that the construction can be modified also to create graphs $G$ such that $\chi_\ell(G^3)\gt\chi(G^3)$, so $k=3$ is not sufficient. Jon Noel reported that the question has been answered in the negative for all odd powers. Xuding Zhu reported that a group in Korea answered in the negative for all powers. A group of students at REGS 2013 did so also. Noel suggested a modified question, also posted here:

Question 2: (Noel) Does there exist a subquadratic function $f$ such that always $\chi_\ell(G^2)\le f(\chi(G^2))$?

Comment: Noel noted that it is trivial to obtained a quadratic upper bound.

References:
[BH] B. Bollob\'as and A.J. Harris, List-colourings of graphs, Graphs Combin. 1 (1985), 115-127.
[KP] S.-J. Kim and B. Park, Counterexamples to the list square coloring conjecture, 2013 preprint, available at http://arxiv.org/abs/1305.2566.
[KW] A.V. Kostochka and D.R. Woodall, Choosability conjectures and multicircuits, Discrete Math. 240 (2001), no. 1-3, 123-143.


text:
## Basic College Mathematics (9th Edition)

Acute Angle: Angles that are less than 90 $^{\circ}$ Obtuse Angle: Angles that are greater than 90$^{\circ}$ Straight Angle: Angles equal to 180$^{\circ}$ Right Angle: Angles equal to exactly 90$^{\circ}$
Acute Angle: Angles that are less than 90 $^{\circ}$ Obtuse Angle: Angles that are greater than 90$^{\circ}$ Straight Angle: Angles equal to 180$^{\circ}$ Right Angle: Angles equal to exactly 90$^{\circ}$


text:
# ALPS 2019 An Alpine LHC Physics Summit

ALPS2019 - (other alps conferences)
22 - 27 April, 2019
Obergurgl, Austria
published February 08, 2021

Continuing the tradition of 2018, 2017 and 2016, we will meet in the Austrian Alps to discuss recent results and developments in experimental and theoretical particle physics. The dates of the meeting are April 22 - 27, 2019 and the following topics are at the focus of the meeting:

Beyond the Standard Model physics

Higgs and Standard Model physics

Neutrino physics

Dark matter

Future colliders

The days are divided into morning sessions with keynote talks, and  late afternoon sessions that will comprise shorter presentations on more specialized topics and the Young Scientist Forum. The meeting will take place at the Obergurgl University Centre, in the ski-resort town of Obergurgl, located in the upper Oetztal Valley in Tyrol, Austria.

### Editorial Board

• Alberto Escalante del valle
HEPHY - Institut fur Hochenergiephysik
• Martin Flechl
HEPHY Vienna
• Suchita Kulkarni
University of Graz
Sessions
Future facilities
Flavour physics
Dark matter
Beyond the Standard Model physics
The Higgs and Standard Model
 Future facilities The European Strategy for Particle Physics and its Update in 2020 PoS(ALPS2019)002 M. Jeitler Belle II status and Physics prospects PoS(ALPS2019)006 C. La Licata News on the CLIC physics potential PoS(ALPS2019)008 A.F. Zarnecki and  On behalf of the CLICdp collaboration Combined Explanations of $(g-2)_\mu$, $(g-2)_e$ and Implications for a Large Muon EDM∗ PoS(ALPS2019)009 A. Crivellin and M. Hoferichter Inert Doublet Model Signatures at Future e+e- Colliders PoS(ALPS2019)010 A.F. Zarnecki, J. Kalinowski, J. Klamka, P. Sopicki, W. Kotlarski, T.N. Robens and D. Sokolowska Flavour physics Angular analysis of the $e^{+}e^{-} \to D^{(*)}D^{*}$ process near the open-charm threshold using initial-state radiation at Belle PoS(ALPS2019)013 V. Zhukova Neutrinos at colliders and lepton number violation PoS(ALPS2019)015 M. Hirsch B-physics anomalies, fluctuations and patterns: a status report PoS(ALPS2019)016 S. Descotes-genon Displaced Heavy Neutrinos at the LHC and Beyond PoS(ALPS2019)020 W. Liu (Semi-)leptonic decays of $D_{(s)}$ Mesons at BESIII PoS(ALPS2019)021 Y. Yang and  On behalf of the BESIII collaboration Dark matter Dark Matter at the Intensity Frontier: the new MESA electron accelerator facility PoS(ALPS2019)022 L. Doria, P. Achenbach, M. Christmann, A. Denig and H. Merkel Status of non-supersymmetric models with GAMBIT PoS(ALPS2019)027 P. Scott and  on behalf of the GAMBIT collaboration Above-ground direct searches for WIMPs PoS(ALPS2019)029 J. Gascon Dark matter/ new physics searches at BESIII PoS(ALPS2019)030 V. Prasad Beyond the Standard Model physics Towards implications of asymptotically safe gravity for particle physics PoS(ALPS2019)034 A. Eichhorn and A. Held Latest results from the NA62 experiment at CERN PoS(ALPS2019)037 N. Lurkin Towards the spectrum of the SU(2) adjoint Higgs model PoS(ALPS2019)038 V. Afferrante, A. Maas and P. Toerek LHC limits on gluinos and squarks in the minimal Dirac gaugino model PoS(ALPS2019)041 H. Reyes-González The Higgs and Standard Model Vacuum Stability Constraints in Models with Extended Scalar Sectors PoS(ALPS2019)043 J. Wittbrodt Higgs physics beyond the Standard Model PoS(ALPS2019)044 T. Stefaniak Exotic bottomonium-like hadrons PoS(ALPS2019)045 V. Baru, E. Epelbaum, A. Filin, C. Hanhart, A. Nefediev and Q. Wang Bottom and strange Yukawa couplings at the NNLO in the MSSM PoS(ALPS2019)046 M. Ghezzi Top Pair Spin Correlation Studies in CMS PoS(ALPS2019)047 B. Isildak Flavour anomaly updates from ATLAS and CMS PoS(ALPS2019)048 A. Bragagnolo Precision EW results from CMS PoS(ALPS2019)050 M. Cipriani and  on behalf of the CMS Collaboration Measurement of top polarization and top pair spin correlations at CMS at $\sqrt{s} =$ 13 TeV PoS(ALPS2019)051 A. Aizuddin Bin Anuar and  on behalf of the CMS Collaboration Electroweak bosons experimental status PoS(ALPS2019)055 S. Bhattacharya


text:
Table 7. Total Economic Impact of Physician's Clinic and Hospital Revenue with Multipliers
Physician*ClinicMultiplierPhysician TotalHospitalMultiplierHospital TotalTotal Impact
1$310,7621.15$357,376$396,5601.28$507,597$864,973 2$202,1101.15$232,427$376,0161.28$481,300$714,685
3$356,9031.15$410,438$656,3901.28$840,179$1,250,617 4$182,2061.15$209,537$325,0151.28$416,019$625,556
5$171,6551.15$197,403$314,0081.28$401,930$599,333 6$175,4061.15$201717$192,7301.28$253,094$454,411
7$197,0761.15$226,375$393,0421.28$503,094$729,731 8$122,5401.15$140,921$197,8851.28$253,293$394,214
9$196,8481.15$226,375$393,0421.28$503,094$729,469 10$169,8321.15$195,307$260,3961.28$333,307$528,613
Total:$6,885,602 Mean:$688,560
• * The physicians are in no particular order.


text:
(48G, 50g) Define a new symbolic constant
09-14-2014, 09:24 PM (This post was last modified: 09-14-2014 09:25 PM by ettlz.)
Post: #1
 ettlz Junior Member Posts: 13 Joined: Aug 2014
(48G, 50g) Define a new symbolic constant
This is a trick I've found to define a new symbolic constant (typically transcendental, or with no proof to the contrary) for use in algebraics; that is, the symbol is unchanged on EVAL but changes to its approximate value with →NUM. In this example, we define the Euler--Mascheroni constant gammaE = 0.577...

First, define the program SYMBCONST:

Code:
« → n x   «     IF x '\pi' SAME     THEN n     ELSE       CASE n 'gammaE' SAME         THEN 0.577215664902         END       END     END   » »

Now define gammaE:

Code:
« 'SYMBCONST(QUOTE(gammaE),\pi)' EVAL »

To see this at work:

Code:
'SIN(gammaE)' EVAL  →  'SIN(gammaE)' 'SIN(gammaE)' →NUM  →  0.5456928233204

This trick relies on →NUM changing the \pi to 3.14159265359 in the second argument to SYMBCONST; it holds no other special meaning. (A more elegant solution would be to find some way to hook →NUM.) More constants can be added with additional clauses in SYMBCONST's CASE.
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