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Magnetic Topological Insulator
Magnetic topological insulators are three dimensional magnetic materials with a non-trivial topological index protected by a symmetry other than time-reversal. In contrast with a non-magnetic topological insulator, a magnetic topological insulator can have naturally gapped surface states as long as the quantizing symmetry is broken at the surface. These gapped surfaces exhibit a topologically protected half-quantized surface anomalous Hall conductivity (e^2/2h) perpendicular to the surface. The sign of the half-quantized surface anomalous Hall conductivity depends on the specific surface termination. Theory Axion coupling The \mathbb_2 classification of a 3D crystalline topological insulator can be understood in terms of the axion coupling \theta. A scalar quantity that is determined from the ground state wavefunction :\theta = -\frac\int_ d^3k \, \epsilon^ \text \Big \mathcal_\alpha \partial_\beta \mathcal_\gamma -i\frac \mathcal_\alpha \mathcal_\beta \mathcal_\gamma \Big ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Magnet
A magnet is a material or object that produces a magnetic field. This magnetic field is invisible but is responsible for the most notable property of a magnet: a force that pulls on other ferromagnetic materials, such as iron, steel, nickel, cobalt, etc. and attracts or repels other magnets. A permanent magnet is an object made from a material that is magnetized and creates its own persistent magnetic field. An everyday example is a refrigerator magnet used to hold notes on a refrigerator door. Materials that can be magnetized, which are also the ones that are strongly attracted to a magnet, are called ferromagnetic (or ferrimagnetic). These include the elements iron, nickel and cobalt and their alloys, some alloys of rare-earth metals, and some naturally occurring minerals such as lodestone. Although ferromagnetic (and ferrimagnetic) materials are the only ones attracted to a magnet strongly enough to be commonly considered magnetic, all other substances respond weakly to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gauge Theory
In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie groups). The term ''gauge'' refers to any specific mathematical formalism to regulate redundant degrees of freedom in the Lagrangian of a physical system. The transformations between possible gauges, called ''gauge transformations'', form a Lie group—referred to as the ''symmetry group'' or the ''gauge group'' of the theory. Associated with any Lie group is the Lie algebra of group generators. For each group generator there necessarily arises a corresponding field (usually a vector field) called the ''gauge field''. Gauge fields are included in the Lagrangian to ensure its invariance under the local group transformations (called ''gauge invariance''). When such a theory is quantized, the quanta of the gauge fields are called '' gauge bosons ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Johns Hopkins University
Johns Hopkins University (Johns Hopkins, Hopkins, or JHU) is a private university, private research university in Baltimore, Maryland. Founded in 1876, Johns Hopkins is the oldest research university in the United States and in the western hemisphere. It consistently ranks among the most prestigious universities in the United States and the world. The university was named for its first benefactor, the American entrepreneur and Quaker philanthropist Johns Hopkins. Hopkins' $7 million bequest to establish the university was the largest Philanthropy, philanthropic gift in U.S. history up to that time. Daniel Coit Gilman, who was inaugurated as :Presidents of Johns Hopkins University, Johns Hopkins's first president on February 22, 1876, led the university to revolutionize higher education in the U.S. by integrating teaching and research. In 1900, Johns Hopkins became a founding member of the American Association of Universities. The university has led all Higher education in the U ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Magnetoelectric Effect
In its most general form, the magnetoelectric effect (ME) denotes any coupling between the magnetic and the electric properties of a material. The first example of such an effect was described by Wilhelm Röntgen in 1888, who found that a dielectric material moving through an electric field would become magnetized. A material where such a coupling is intrinsically present is called a magnetoelectric. Historically, the first and most studied example of this effect is the linear magnetoelectric effect. Mathematically, while the electric susceptibility \chi^e and magnetic susceptibility \chi^v describe the electric and magnetic polarization responses to an electric, resp. a magnetic field, there is also the possibility of a magnetoelectric susceptibility \alpha_ which describes a linear response of the electric polarization to a magnetic field, and vice versa: :P_i= \sum_j \epsilon_0\chi^e_ E_ + \sum_j \alpha_H_j :\mu_0 M_i= \sum_j \mu_0\chi^v_H_ + \sum_j \alpha_E_j, The tensor \alpha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pseudovector
In physics and mathematics, a pseudovector (or axial vector) is a quantity that is defined as a function of some vectors or other geometric shapes, that resembles a vector, and behaves like a vector in many situations, but is changed into its opposite if the orientation of the space is changed, or an improper rigid transformation such as a reflection is applied to the whole figure. Geometrically, the direction of a reflected pseudovector is opposite to its mirror image, but with equal magnitude. In contrast, the reflection of a ''true'' (or polar) vector is exactly the same as its mirror image. In three dimensions, the curl of a polar vector field at a point and the cross product of two polar vectors are pseudovectors. One example of a pseudovector is the normal to an oriented plane. An oriented plane can be defined by two non-parallel vectors, a and b, [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chern Number
In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since found applications in physics, Calabi–Yau manifolds, string theory, Chern–Simons theory, knot theory, Gromov–Witten invariants, topological quantum field theory, the Chern theorem etc. Chern classes were introduced by . Geometric approach Basic idea and motivation Chern classes are characteristic classes. They are topological invariants associated with vector bundles on a smooth manifold. The question of whether two ostensibly different vector bundles are the same can be quite hard to answer. The Chern classes provide a simple test: if the Chern classes of a pair of vector bundles do not agree, then the vector bundles are different. The converse, however, is not true. In topology, differential geometry, and algebraic geometry, it is often important to count how many li ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fermi Level
The Fermi level of a solid-state body is the thermodynamic work required to add one electron to the body. It is a thermodynamic quantity usually denoted by ''µ'' or ''E''F for brevity. The Fermi level does not include the work required to remove the electron from wherever it came from. A precise understanding of the Fermi level—how it relates to electronic band structure in determining electronic properties, how it relates to the voltage and flow of charge in an electronic circuit—is essential to an understanding of solid-state physics. In band structure theory, used in solid state physics to analyze the energy levels in a solid, the Fermi level can be considered to be a hypothetical energy level of an electron, such that at thermodynamic equilibrium this energy level would have a ''50% probability of being occupied at any given time''. The position of the Fermi level in relation to the band energy levels is a crucial factor in determining electrical properties. The Fermi le ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Berry Phase
In classical and quantum mechanics, geometric phase is a phase difference acquired over the course of a cycle, when a system is subjected to cyclic adiabatic processes, which results from the geometrical properties of the parameter space of the Hamiltonian. The phenomenon was independently discovered by S. Pancharatnam (1956), in classical optics and by H. C. Longuet-Higgins (1958)See page 12 in molecular physics; it was generalized by Sir Michael Berry in (1984). It is also known as the Pancharatnam–Berry phase, Pancharatnam phase, or Berry phase. It can be seen in the conical intersection of potential energy surfaces and in the Aharonov–Bohm effect. Geometric phase around the conical intersection involving the ground electronic state of the C6H3F3+ molecular ion is discussed on pages 385–386 of the textbook by Bunker and Jensen. In the case of the Aharonov–Bohm effect, the adiabatic parameter is the magnetic field enclosed by two interference paths, and it ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unitary Transformation
In mathematics, a unitary transformation is a transformation that preserves the inner product: the inner product of two vectors before the transformation is equal to their inner product after the transformation. Formal definition More precisely, a unitary transformation is an isomorphism between two inner product spaces (such as Hilbert spaces). In other words, a ''unitary transformation'' is a bijective function U : H \to H_2\, between two inner product spaces, H and H_2, such that \langle Ux, Uy \rangle_ = \langle x, y \rangle_ \quad \text x, y \in H. Properties A unitary transformation is an isometry, as one can see by setting x=y in this formula. Unitary operator In the case when H_1 and H_2 are the same space, a unitary transformation is an automorphism of that Hilbert space, and then it is also called a unitary operator. Antiunitary transformation A closely related notion is that of antiunitary transformation, which is a bijective function :U:H_1\to H_2\, between two co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bloch's Theorem
In condensed matter physics, Bloch's theorem states that solutions to the Schrödinger equation in a periodic potential take the form of a plane wave modulated by a periodic function. The theorem is named after the physicist Felix Bloch, who discovered the theorem in 1929. Mathematically, they are written where \mathbf is position, \psi is the wave function, u is a periodic function with the same periodicity as the crystal, the wave vector \mathbf is the crystal momentum vector, e is Euler's number, and i is the imaginary unit. Functions of this form are known as Bloch functions or Bloch states, and serve as a suitable basis for the wave functions or states of electrons in crystalline solids. Named after Swiss physicist Felix Bloch, the description of electrons in terms of Bloch functions, termed Bloch electrons (or less often ''Bloch Waves''), underlies the concept of electronic band structures. These eigenstates are written with subscripts as \psi_, where n is a discr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Index
In the fields of chemical graph theory, molecular topology, and mathematical chemistry, a topological index, also known as a connectivity index, is a type of a molecular descriptor that is calculated based on the molecular graph of a chemical compound. Topological indices are numerical parameters of a graph which characterize its topology and are usually graph invariant. Topological indices are used for example in the development of quantitative structure-activity relationships (QSARs) in which the biological activity or other properties of molecules are correlated with their chemical structure. Calculation Topological descriptors are derived from hydrogen-suppressed molecular graphs, in which the atoms are represented by vertices and the bonds by edges. The connections between the atoms can be described by various types of topological matrices (e.g., distance or adjacency matrices), which can be mathematically manipulated so as to derive a single number, usually known as grap ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Berry Connection And Curvature
In physics, Berry connection and Berry curvature are related concepts which can be viewed, respectively, as a local gauge potential and gauge field associated with the Berry phase or geometric phase. The concept was first introduced by S. Pancharatnam as geometric phase and later elaborately explained and popularized by Michael Berry in a paper published in 1984 emphasizing how geometric phases provide a powerful unifying concept in several branches of classical and quantum physics. Berry phase and cyclic adiabatic evolution In quantum mechanics, the Berry phase arises in a cyclic adiabatic evolution. The quantum adiabatic theorem applies to a system whose Hamiltonian H(\mathbf R) depends on a (vector) parameter \mathbf R that varies with time t. If the n'th eigenvalue \varepsilon_n(\mathbf R) remains non-degenerate everywhere along the path and the variation with time ''t'' is sufficiently slow, then a system initially in the normalized eigenstate , n(\mathbf R(0))\rangle ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
