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71-420: Skyrmions as topological objects are important in solid-state physics , especially in the emerging technology of spintronics . A two-dimensional magnetic skyrmion , as a topological object, is formed, e.g., from a 3D effective-spin "hedgehog" (in the field of micromagnetics : out of a so-called " Bloch point " singularity of homotopy degree +1) by a stereographic projection , whereby the positive north-pole spin

142-609: A Lax pair for the sine-Gordon equation, in the sense that the zero-curvature equation recovers the PDE rather than them satisfying Lax's equation. The sinh-Gordon equation is given by This is the Euler–Lagrange equation of the Lagrangian Another closely related equation is the elliptic sine-Gordon equation or Euclidean sine-Gordon equation , given by where φ {\displaystyle \varphi }

213-419: A quantum superposition of baryons and resonance states. It could be predicted from some nuclear matter properties. In field theory, skyrmions are homotopically non-trivial classical solutions of a nonlinear sigma model with a non-trivial target manifold topology – hence, they are topological solitons . An example occurs in chiral models of mesons , where the target manifold is a homogeneous space of

284-403: A section of the tangent bundle of the principal fiber bundle of SU(2) over spacetime. This abstract interpretation is characteristic of all non-linear sigma models. The first term, tr ⁡ ( L μ L μ ) {\displaystyle \operatorname {tr} (L_{\mu }L^{\mu })} is just an unusual way of writing the quadratic term of

355-485: A clockwise ( left-handed ) twist of the elastic ribbon to be a kink with topological charge θ K = − 1 {\displaystyle \theta _{\text{K}}=-1} . The alternative counterclockwise ( right-handed ) twist with topological charge θ AK = + 1 {\displaystyle \theta _{\text{AK}}=+1} will be an antikink. Multi- soliton solutions can be obtained through continued application of

426-456: A constant energy density has been added so that the potential is non-negative. With it the first two terms in the Taylor expansion of the potential coincide with the potential of a massive scalar field, as mentioned in the naming section; the higher order terms can be thought of as interactions. The topological charge is conserved if the energy is finite. The topological charge does not determine

497-425: A crystal disrupt periodicity, this use of Bloch's theorem is only an approximation, but it has proven to be a tremendously valuable approximation, without which most solid-state physics analysis would be intractable. Deviations from periodicity are treated by quantum mechanical perturbation theory . Modern research topics in solid-state physics include: Sine%E2%80%93Gordon equation The sine-Gordon equation

568-452: A crystal of sodium chloride (common salt), the crystal is made up of ionic sodium and chlorine , and held together with ionic bonds . In others, the atoms share electrons and form covalent bonds . In metals, electrons are shared amongst the whole crystal in metallic bonding . Finally, the noble gases do not undergo any of these types of bonding. In solid form, the noble gases are held together with van der Waals forces resulting from

639-423: A general theory, is focused on crystals . Primarily, this is because the periodicity of atoms in a crystal — its defining characteristic — facilitates mathematical modeling. Likewise, crystalline materials often have electrical , magnetic , optical , or mechanical properties that can be exploited for engineering purposes. The forces between the atoms in a crystal can take a variety of forms. For example, in

710-457: A good candidate for future data-storage solutions and other spintronics devices. Researchers could read and write skyrmions using scanning tunneling microscopy. The topological charge, representing the existence and non-existence of skyrmions, can represent the bit states "1" and "0". Room-temperature skyrmions were reported. Skyrmions operate at current densities that are several orders of magnitude weaker than conventional magnetic devices. In 2015

781-865: A line of pendula, hanging on a straight line, in constant gravity. Connect the bobs of the pendula together by a string in constant tension. Let the angle of the pendulum at location x {\displaystyle x} be φ {\displaystyle \varphi } , then schematically, the dynamics of the line of pendulum follows Newton's second law: m φ t t ⏟ mass times acceleration = T φ x x ⏟ tension − m g sin ⁡ φ ⏟ gravity {\displaystyle \underbrace {m\varphi _{tt}} _{\text{mass times acceleration}}=\underbrace {T\varphi _{xx}} _{\text{tension}}-\underbrace {mg\sin \varphi } _{\text{gravity}}} and this

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852-411: A practical way to create and access magnetic skyrmions under ambient room-temperature conditions was announced. The device used arrays of magnetized cobalt disks as artificial Bloch skyrmion lattices atop a thin film of cobalt and palladium . Asymmetric magnetic nanodots were patterned with controlled circularity on an underlayer with perpendicular magnetic anisotropy (PMA). Polarity is controlled by

923-456: A pseudosphere uniquely up to rigid transformations . There is a theorem, sometimes called the fundamental theorem of surfaces , that if a pair of matrix-valued bilinear forms satisfy the Gauss–Codazzi equations, then they are the first and second fundamental forms of an embedded surface in 3-dimensional space. Solutions to the sine-Gordon equation can be used to construct such matrices by using

994-994: A skyrmion can be approximated by a soliton of the Sine–Gordon equation ; after quantisation by the Bethe ansatz or otherwise, it turns into a fermion interacting according to the massive Thirring model . The Lagrangian for the skyrmion, as written for the original chiral SU(2) effective Lagrangian of the nucleon-nucleon interaction (in (3 + 1)-dimensional spacetime), can be written as where L μ = U † ∂ μ U {\displaystyle L_{\mu }=U^{\dagger }\partial _{\mu }U} , U = exp ⁡ i τ → ⋅ θ → {\displaystyle U=\exp i{\vec {\tau }}\cdot {\vec {\theta }}} , τ → {\displaystyle {\vec {\tau }}} are

1065-482: A solution φ {\displaystyle \varphi } is E = ∫ R d x ( 1 2 ( φ t 2 + φ x 2 ) + m 2 ( 1 − cos ⁡ φ ) ) {\displaystyle E=\int _{\mathbb {R} }dx\left({\frac {1}{2}}(\varphi _{t}^{2}+\varphi _{x}^{2})+m^{2}(1-\cos \varphi )\right)} where

1136-465: A solution of the sine-Gordon equation, although with some caveats: if the surface is complete, it is necessarily singular due to the Hilbert embedding theorem . In the simplest case, the pseudosphere , also known as the tractroid, corresponds to a static one-soliton, but the tractroid has a singular cusp at its equator. Conversely, one can start with a solution to the sine-Gordon equation to obtain

1207-488: A tailored magnetic-field sequence and demonstrated in magnetometry measurements. The vortex structure is imprinted into the underlayer's interfacial region by suppressing the PMA by a critical ion-irradiation step. The lattices are identified with polarized neutron reflectometry and have been confirmed by magnetoresistance measurements. A recent (2019) study demonstrated a way to move skyrmions, purely using electric field (in

1278-406: A traveling kink and a standing breather or a traveling antikink and a standing breather results in a phase shift of the standing breather. In the process of collision between a moving kink and a standing breather, the shift of the breather Δ B {\displaystyle \Delta _{\text{B}}} is given by where v K {\displaystyle v_{\text{K}}}

1349-450: A tunable multilayer system in which two different types of skyrmions – the future bits for "0" and "1" – can exist at room temperature. Solid-state physics Solid-state physics is the study of rigid matter , or solids , through methods such as solid-state chemistry , quantum mechanics , crystallography , electromagnetism , and metallurgy . It is the largest branch of condensed matter physics . Solid-state physics studies how

1420-457: Is a second-order nonlinear partial differential equation for a function φ {\displaystyle \varphi } dependent on two variables typically denoted x {\displaystyle x} and t {\displaystyle t} , involving the wave operator and the sine of φ {\displaystyle \varphi } . It was originally introduced by Edmond Bour  ( 1862 ) in

1491-431: Is a solution of the sine-Gordon equation Then the system where a is an arbitrary parameter, is solvable for a function ψ {\displaystyle \psi } which will also satisfy the sine-Gordon equation. This is an example of an auto-Bäcklund transform, as both φ {\displaystyle \varphi } and ψ {\displaystyle \psi } are solutions to

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1562-410: Is an example of an integrable PDE . Among well-known integrable PDEs, the sine-Gordon equation is the only relativistic system due to its Lorentz invariance . This is the first derivation of the equation, by Bour (1862). There are two equivalent forms of the sine-Gordon equation. In the ( real ) space-time coordinates , denoted ( x , t ) {\displaystyle (x,t)} ,

1633-442: Is broadly considered to be the subfield of condensed matter physics, often referred to as hard condensed matter, that focuses on the properties of solids with regular crystal lattices. Many properties of materials are affected by their crystal structure . This structure can be investigated using a range of crystallographic techniques, including X-ray crystallography , neutron diffraction and electron diffraction . The sizes of

1704-600: Is called a kink and represents a twist in the variable φ {\displaystyle \varphi } which takes the system from one constant solution φ = 0 {\displaystyle \varphi =0} to an adjacent constant solution φ = 2 π {\displaystyle \varphi =2\pi } . The states φ ≅ 2 π n {\displaystyle \varphi \cong 2\pi n} are known as vacuum states, as they are constant solutions of zero energy. The 1-soliton solution in which we take

1775-531: Is called an elastic collision . The kink-kink solution is given by φ K / K ( x , t ) = 4 arctan ⁡ ( v sinh ⁡ x 1 − v 2 cosh ⁡ v t 1 − v 2 ) {\displaystyle \varphi _{K/K}(x,t)=4\arctan \left({\frac {v\sinh {\frac {x}{\sqrt {1-v^{2}}}}}{\cosh {\frac {vt}{\sqrt {1-v^{2}}}}}}\right)} while

1846-466: Is conserved due to topological reasons and it is always an integer. For this reason, it is associated with the baryon number of the nucleus. As a conserved charge, it is time-independent: d B / d t = 0 {\displaystyle dB/dt=0} , the physical interpretation of which is that protons do not decay . In the chiral bag model , one cuts a hole out of the center and fills it with quarks. Despite this obvious "hackery",

1917-773: Is equivalent to the sine-Gordon equation φ u v = sin ⁡ φ {\displaystyle \varphi _{uv}=\sin \varphi } . The zero-curvature equation is so named as it corresponds to the curvature being equal to zero if it is defined F μ ν = [ ∂ μ − A μ , ∂ ν − A ν ] {\displaystyle F_{\mu \nu }=[\partial _{\mu }-A_{\mu },\partial _{\nu }-A_{\nu }]} . The pair of matrices A u {\displaystyle A_{u}} and A v {\displaystyle A_{v}} are also known as

1988-499: Is mapped onto a far-off edge circle of a 2D-disk, while the negative south-pole spin is mapped onto the center of the disk. In a spinor field such as for example photonic or polariton fluids the skyrmion topology corresponds to a full Poincaré beam (a spin vortex comprising all the states of polarization mapped by a stereographic projection of the Poincaré sphere to the real plane). A dynamical pseudospin skyrmion results from

2059-723: Is the one-soliton solution with a {\displaystyle a} related to the boost applied to the soliton. The topological charge or winding number of a solution φ {\displaystyle \varphi } is N = 1 2 π ∫ R d φ = 1 2 π [ φ ( x = ∞ , t ) − φ ( x = − ∞ , t ) ] . {\displaystyle N={\frac {1}{2\pi }}\int _{\mathbb {R} }d\varphi ={\frac {1}{2\pi }}\left[\varphi (x=\infty ,t)-\varphi (x=-\infty ,t)\right].} The energy of

2130-558: Is the sine-Gordon equation, after scaling time and distance appropriately. Note that this is not exactly correct, since the net force on a pendulum due to the tension is not precisely T φ x x {\displaystyle T\varphi _{xx}} , but more accurately T φ x x ( 1 + φ x 2 ) − 3 / 2 {\displaystyle T\varphi _{xx}(1+\varphi _{x}^{2})^{-3/2}} . However this does give an intuitive picture for

2201-461: Is the velocity of the kink, and ω {\displaystyle \omega } is the breather's frequency. If the old position of the standing breather is x 0 {\displaystyle x_{0}} , after the collision the new position will be x 0 + Δ B {\displaystyle x_{0}+\Delta _{\text{B}}} . Suppose that φ {\displaystyle \varphi }

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2272-441: The σ i {\displaystyle \sigma _{i}} are the Pauli matrices . Then the zero-curvature equation ∂ v A u − ∂ u A v + [ A u , A v ] = 0 {\displaystyle \partial _{v}A_{u}-\partial _{u}A_{v}+[A_{u},A_{v}]=0}

2343-730: The American Physical Society . The DSSP catered to industrial physicists, and solid-state physics became associated with the technological applications made possible by research on solids. By the early 1960s, the DSSP was the largest division of the American Physical Society. Large communities of solid state physicists also emerged in Europe after World War II , in particular in England , Germany , and

2414-524: The Bäcklund transform to the 1-soliton solution, as prescribed by a Bianchi lattice relating the transformed results. The 2-soliton solutions of the sine-Gordon equation show some of the characteristic features of the solitons. The traveling sine-Gordon kinks and/or antikinks pass through each other as if perfectly permeable, and the only observed effect is a phase shift . Since the colliding solitons recover their velocity and shape , such an interaction

2485-470: The Hodge star , in this context). As a physical quantity, this can be interpreted as the baryon current; it is conserved: ∂ μ B μ = 0 {\displaystyle \partial _{\mu }{\mathcal {B}}^{\mu }=0} , and the conservation follows as a Noether current for the chiral symmetry. The corresponding charge is the baryon number: Which

2556-495: The Klein–Gordon Lagrangian plus higher-order terms: An interesting feature of the sine-Gordon equation is the existence of soliton and multisoliton solutions. The sine-Gordon equation has the following 1- soliton solutions: where and the slightly more general form of the equation is assumed: The 1-soliton solution for which we have chosen the positive root for γ {\displaystyle \gamma }

2627-678: The Soviet Union . In the United States and Europe, solid state became a prominent field through its investigations into semiconductors , superconductivity , nuclear magnetic resonance , and diverse other phenomena. During the early Cold War, research in solid state physics was often not restricted to solids, which led some physicists in the 1970s and 1980s to found the field of condensed matter physics , which organized around common techniques used to investigate solids, liquids, plasmas, and other complex matter. Today, solid-state physics

2698-618: The isospin Pauli matrices , [ ⋅ , ⋅ ] {\displaystyle [\cdot ,\cdot ]} is the Lie bracket commutator, and tr is the matrix trace. The meson field ( pion field, up to a dimensional factor) at spacetime coordinate x {\displaystyle x} is given by θ → = θ → ( x ) {\displaystyle {\vec {\theta }}={\vec {\theta }}(x)} . A broad review of

2769-428: The pion decay constant . (In 1 + 1 dimensions, this constant is not dimensional and can thus be absorbed into the field definition.) The second term establishes the characteristic size of the lowest-energy soliton solution; it determines the effective radius of the soliton. As a model of the nucleon, it is normally adjusted so as to give the correct radius for the proton; once this is done, other low-energy properties of

2840-426: The rho meson (the nuclear vector meson ) and the pion; the skyrmion relates the value of this constant to the baryon radius. The local winding number density (or topological charge density) is given by where ϵ μ ν α β {\displaystyle \epsilon ^{\mu \nu \alpha \beta }} is the totally antisymmetric Levi-Civita symbol (equivalently,

2911-442: The structure group where SU( N ) L and SU( N ) R are the left and right chiral symmetries, and SU( N ) diag is the diagonal subgroup . In nuclear physics , for N = 2, the chiral symmetries are understood to be the isospin symmetry of the nucleon. For N = 3, the isoflavor symmetry between the up, down and strange quarks is more broken, and the skyrmion models are less successful or accurate. If spacetime has

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2982-486: The absence of electric current). The authors used Co/Ni multilayers with a thickness slope and Dzyaloshinskii–Moriya interaction and demonstrated skyrmions. They showed that the displacement and velocity depended directly on the applied voltage. In 2020, a team of researchers from the Swiss Federal Laboratories for Materials Science and Technology (Empa) has succeeded for the first time in producing

3053-520: The asymptotic lines. The first fundamental form of the surface is and the second fundamental form is L = N = 0 , M = sin ⁡ φ {\displaystyle L=N=0,M=\sin \varphi } and the Gauss–Codazzi equation is φ u v = sin ⁡ φ . {\displaystyle \varphi _{uv}=\sin \varphi .} Thus, any pseudospherical surface gives rise to

3124-412: The baryon number is conserved; i.e. that the proton does not decay. The Skyrme Lagrangian is essentially a one-parameter model of the nucleon. Fixing the parameter fixes the proton radius, and also fixes all other low-energy properties, which appear to be correct to about 30%, a significant level of predictive power. Hollowed-out skyrmions form the basis for the chiral bag model (Cheshire Cat model) of

3195-1605: The connection components A μ {\displaystyle A_{\mu }} are given by A u = ( i λ i 2 φ u i 2 φ u − i λ ) = 1 2 φ u i σ 1 + λ i σ 3 , {\displaystyle A_{u}={\begin{pmatrix}i\lambda &{\frac {i}{2}}\varphi _{u}\\{\frac {i}{2}}\varphi _{u}&-i\lambda \end{pmatrix}}={\frac {1}{2}}\varphi _{u}i\sigma _{1}+\lambda i\sigma _{3},} A v = ( − i 4 λ cos ⁡ φ − 1 4 λ sin ⁡ φ 1 4 λ sin ⁡ φ i 4 λ cos ⁡ φ ) = − 1 4 λ i sin ⁡ φ σ 2 − 1 4 λ i cos ⁡ φ σ 3 , {\displaystyle A_{v}={\begin{pmatrix}-{\frac {i}{4\lambda }}\cos \varphi &-{\frac {1}{4\lambda }}\sin \varphi \\{\frac {1}{4\lambda }}\sin \varphi &{\frac {i}{4\lambda }}\cos \varphi \end{pmatrix}}=-{\frac {1}{4\lambda }}i\sin \varphi \sigma _{2}-{\frac {1}{4\lambda }}i\cos \varphi \sigma _{3},} where

3266-498: The course of study of surfaces of constant negative curvature as the Gauss–Codazzi equation for surfaces of constant Gaussian curvature −1 in 3-dimensional space . The equation was rediscovered by Frenkel and Kontorova ( 1939 ) in their study of crystal dislocations known as the Frenkel–Kontorova model . This equation attracted a lot of attention in the 1970s due to the presence of soliton solutions, and

3337-412: The electrons are modelled as a Fermi gas , a gas of particles which obey the quantum mechanical Fermi–Dirac statistics . The free electron model gave improved predictions for the heat capacity of metals, however, it was unable to explain the existence of insulators . The nearly free electron model is a modification of the free electron model which includes a weak periodic perturbation meant to model

3408-515: The equation reads: where partial derivatives are denoted by subscripts. Passing to the light-cone coordinates ( u ,  v ), akin to asymptotic coordinates where the equation takes the form This is the original form of the sine-Gordon equation, as it was considered in the 19th century in the course of investigation of surfaces of constant Gaussian curvature K  = −1, also called pseudospherical surfaces . Consider an arbitrary pseudospherical surface. Across every point on

3479-535: The forms obtained above. The study of this equation and of the associated transformations of pseudospherical surfaces in the 19th century by Bianchi and Bäcklund led to the discovery of Bäcklund transformations . Another transformation of pseudospherical surfaces is the Lie transform introduced by Sophus Lie in 1879, which corresponds to Lorentz boosts for solutions of the sine-Gordon equation. There are also some more straightforward ways to construct new solutions but which do not give new surfaces. Since

3550-462: The geometric interpretation of L μ {\displaystyle L_{\mu }} is presented in the article on sigma models . When written this way, the U {\displaystyle U} is clearly an element of the Lie group SU(2), and θ → {\displaystyle {\vec {\theta }}} an element of the Lie algebra su(2). The pion field can be understood abstractly to be

3621-453: The ideal arrangements, and it is these defects that critically determine many of the electrical and mechanical properties of real materials. Properties of materials such as electrical conduction and heat capacity are investigated by solid state physics. An early model of electrical conduction was the Drude model , which applied kinetic theory to the electrons in a solid. By assuming that

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3692-427: The individual crystals in a crystalline solid material vary depending on the material involved and the conditions when it was formed. Most crystalline materials encountered in everyday life are polycrystalline , with the individual crystals being microscopic in scale, but macroscopic single crystals can be produced either naturally (e.g. diamonds ) or artificially. Real crystals feature defects or irregularities in

3763-552: The interaction between the conduction electrons and the ions in a crystalline solid. By introducing the idea of electronic bands , the theory explains the existence of conductors , semiconductors and insulators . The nearly free electron model rewrites the Schrödinger equation for the case of a periodic potential . The solutions in this case are known as Bloch states . Since Bloch's theorem applies only to periodic potentials, and since unceasing random movements of atoms in

3834-550: The kink-antikink solution is given by φ K / A K ( x , t ) = 4 arctan ⁡ ( v cosh ⁡ x 1 − v 2 sinh ⁡ v t 1 − v 2 ) {\displaystyle \varphi _{K/AK}(x,t)=4\arctan \left({\frac {v\cosh {\frac {x}{\sqrt {1-v^{2}}}}}{\sinh {\frac {vt}{\sqrt {1-v^{2}}}}}}\right)} Another interesting 2-soliton solutions arise from

3905-400: The large-scale properties of solid materials result from their atomic -scale properties. Thus, solid-state physics forms a theoretical basis of materials science . Along with solid-state chemistry , it also has direct applications in the technology of transistors and semiconductors . Solid materials are formed from densely packed atoms, which interact intensely. These interactions produce

3976-526: The material contains immobile positive ions and an "electron gas" of classical, non-interacting electrons, the Drude model was able to explain electrical and thermal conductivity and the Hall effect in metals, although it greatly overestimated the electronic heat capacity. Arnold Sommerfeld combined the classical Drude model with quantum mechanics in the free electron model (or Drude-Sommerfeld model). Here,

4047-437: The mechanical (e.g. hardness and elasticity ), thermal , electrical , magnetic and optical properties of solids. Depending on the material involved and the conditions in which it was formed, the atoms may be arranged in a regular, geometric pattern ( crystalline solids , which include metals and ordinary water ice ) or irregularly (an amorphous solid such as common window glass ). The bulk of solid-state physics, as

4118-468: The negative root for γ {\displaystyle \gamma } is called an antikink . The form of the 1-soliton solutions can be obtained through application of a Bäcklund transform to the trivial (vacuum) solution and the integration of the resulting first-order differentials: for all time. The 1-soliton solutions can be visualized with the use of the elastic ribbon sine-Gordon model introduced by Julio Rubinstein in 1970. Here we take

4189-451: The non-linear sigma model; it reduces to − tr ⁡ ( ∂ μ U † ∂ μ U ) {\displaystyle -\operatorname {tr} (\partial _{\mu }U^{\dagger }\partial ^{\mu }U)} . When used as a model of the nucleon, one writes with the dimensional factor of f π {\displaystyle f_{\pi }} being

4260-462: The nucleon are automatically fixed, to within about 30% accuracy. It is this result, of tying together what would otherwise be independent parameters, and doing so fairly accurately, that makes the Skyrme model of the nucleon so appealing and interesting. Thus, for example, constant g {\displaystyle g} in the quartic term is interpreted as the vector-pion coupling ρ–π–π between

4331-429: The nucleon. The exact results for the duality between the fermion spectrum and the topological winding number of the non-linear sigma model have been obtained by Dan Freed . This can be interpreted as a foundation for the duality between a quantum chromodynamics (QCD) description of the nucleon (but consisting only of quarks, and without gluons) and the Skyrme model for the nucleon. The skyrmion can be quantized to form

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4402-496: The polarisation of the electronic charge cloud on each atom. The differences between the types of solid result from the differences between their bonding. The physical properties of solids have been common subjects of scientific inquiry for centuries, but a separate field going by the name of solid-state physics did not emerge until the 1940s , in particular with the establishment of the Division of Solid State Physics (DSSP) within

4473-777: The possibility of coupled kink-antikink behaviour known as a breather . There are known three types of breathers: standing breather , traveling large-amplitude breather , and traveling small-amplitude breather . The standing breather solution is given by φ ( x , t ) = 4 arctan ⁡ ( 1 − ω 2 cos ⁡ ( ω t ) ω cosh ⁡ ( 1 − ω 2 x ) ) . {\displaystyle \varphi (x,t)=4\arctan \left({\frac {{\sqrt {1-\omega ^{2}}}\;\cos(\omega t)}{\omega \;\cosh({\sqrt {1-\omega ^{2}}}\;x)}}\right).} 3-soliton collisions between

4544-422: The same equation, that is, the sine-Gordon equation. By using a matrix system, it is also possible to find a linear Bäcklund transform for solutions of sine-Gordon equation. For example, if φ {\displaystyle \varphi } is the trivial solution φ ≡ 0 {\displaystyle \varphi \equiv 0} , then ψ {\displaystyle \psi }

4615-525: The sine-Gordon equation is odd, the negative of any solution is another solution. However this does not give a new surface, as the sign-change comes down to a choice of direction for the normal to the surface. New solutions can be found by translating the solution: if φ {\displaystyle \varphi } is a solution, then so is φ + 2 n π {\displaystyle \varphi +2n\pi } for n {\displaystyle n} an integer. Consider

4686-544: The sine-gordon equation. One can produce exact mechanical realizations of the sine-gordon equation by more complex methods. The name "sine-Gordon equation" is a pun on the well-known Klein–Gordon equation in physics: The sine-Gordon equation is the Euler–Lagrange equation of the field whose Lagrangian density is given by Using the Taylor series expansion of the cosine in the Lagrangian, it can be rewritten as

4757-644: The solution, even up to Lorentz boosts. Both the trivial solution and the soliton-antisoliton pair solution have N = 0 {\displaystyle N=0} . The sine-Gordon equation is equivalent to the curvature of a particular s u ( 2 ) {\displaystyle {\mathfrak {su}}(2)} - connection on R 2 {\displaystyle \mathbb {R} ^{2}} being equal to zero. Explicitly, with coordinates ( u , v ) {\displaystyle (u,v)} on R 2 {\displaystyle \mathbb {R} ^{2}} ,

4828-470: The stereographic projection of a rotating polariton Bloch sphere in the case of dynamical full Bloch beams. Skyrmions have been reported, but not conclusively proven, to appear in Bose–Einstein condensates , thin magnetic films, and chiral nematic liquid crystals , as well as in free-space optics. As a model of the nucleon , the topological stability of the skyrmion can be interpreted as a statement that

4899-403: The surface there are two asymptotic curves . This allows us to construct a distinguished coordinate system for such a surface, in which u  = constant, v  = constant are the asymptotic lines, and the coordinates are incremented by the arc length on the surface. At every point on the surface, let φ {\displaystyle \varphi } be the angle between

4970-444: The topology S× R , then classical configurations can be classified by an integral winding number because the third homotopy group is equivalent to the ring of integers, with the congruence sign referring to homeomorphism . A topological term can be added to the chiral Lagrangian, whose integral depends only upon the homotopy class ; this results in superselection sectors in the quantised model. In (1 + 1)-dimensional spacetime,

5041-655: The total baryon number is conserved: the missing charge from the hole is exactly compensated by the spectral asymmetry of the vacuum fermions inside the bag. One particular form of skyrmions is magnetic skyrmions , found in magnetic materials that exhibit spiral magnetism due to the Dzyaloshinskii–Moriya interaction , double-exchange mechanism or competing Heisenberg exchange interactions . They form "domains" as small as 1 nm (e.g. in Fe on Ir(111)). The small size and low energy consumption of magnetic skyrmions make them

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