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it with a point at infinity, one also gets a 3-sphere. Solutions to Skyrme's equations in real three-dimensional space map a point in "real" (physical; Euclidean) space to a point on the 3-manifold SU(2). Topologically distinct solutions "wrap" the one sphere around the other, such that one solution,
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no matter how it is deformed, cannot be "unwrapped" without creating a discontinuity in the solution. In physics, such discontinuities are associated with infinite energy, and are thus not allowed.
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modeling a physical system, as the solitons themselves owe their stability to topological considerations. The specific "topological considerations" are usually due to the appearance of the
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are specified, has a non-trivial homotopy group that is preserved by the differential equations. The topological quantum number of a solution is sometimes called the
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221:, could appear or disappear depending on the phase of a field, linking phase transitions to shifts in topological quantum numbers. In the 1980s,
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through its bijective association, so the isomorphism is in the category of topological groups. By taking real three-dimensional space, and
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Vanhala, Tuomas I.; Siro, Topi; Liang, Long; Troyer, Matthias; Harju, Ari; TörmÀ, PÀivi (2016-06-02).
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In the above example, the topological statement is that the 3rd homotopy group of the three sphere is
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theoretical model demonstrated that materials can possess topological quantum numbers like the
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Physical quantities that take discrete values because of topological quantum physical effects
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shows that a fractional number of fermions repelled over the ultraviolet cutoff. So the
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763:"Topological Phase Transitions in the Repulsively Interacting Haldane-Hubbard Model"
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Hao, Ningning; Zhang, Ping; Wang, Zhigang; Zhang, Wei; Wang, Yupeng (2008-08-26).
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in the description of the problem, quite often because the boundary, on which the
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The concept of topological quantum numbers being created or destroyed during
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is a topological quantum number. The origin comes from the fact that the
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722:"Topological edge states and quantum Hall effect in the Haldane model"
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656:"Topological Phase Transitions and Topological Phases of Matter"
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monopole model demonstrated how topological structures, such as
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considerations. Most commonly, topological quantum numbers are
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694:, Cambridge: Cambridge University Press, pp. 256â266,
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Scientific
Background on the Nobel Prize in Physics 2016.
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and so the baryon number can only take on integer values.
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Topological
Quantum Numbers in Nonrelativistic Physics
688:"The 't HooftâPolyakov Monopole Solution and Topology"
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gets multiplied by a fractional number depending on
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394:A generalization of these ideas is found in the
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381:{\displaystyle \pi _{3}(S^{3})=\mathbb {Z} }
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617:Topological string theory
801:Thouless, D. J. (1998).
396:WessâZuminoâWitten model
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170:or a higher-dimensional
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