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Topological magnons
Topological magnons are magnons in magnetically ordered systems whose energy bands carry nontrivial topology, such as nonzero Chern numbers and protected chiral edge states, first observed by neutron scattering in a kagome ferromagnet in 2015.
Topological materials
Topological materials are crystalline solids whose electronic band structures have nontrivial topology, including topological insulators with protected conducting surface states and topological semimetals; over half of known stoichiometric, nonmagnetic materials are topological.
Topological measurement-based quantum computation
Topological measurement-based quantum computation is a quantum computing model in which single-qubit measurements on a 3D cluster state provide fault tolerance with a 0.75% error threshold.
Topological mechanical metamaterial
A topological mechanical metamaterial is an artificially structured material whose vibrational properties are governed by topological invariants, enabling protected wave propagation along edges and defects.
Topological mechanical metamaterials with Maxwell lattices
Topological mechanical metamaterials with Maxwell lattices are engineered lattices on the verge of mechanical instability whose phonon band topology directs soft deformation and fracture to specific protected edges.
Topological mechanics of gyroscopic metamaterials
Topological mechanics of gyroscopic metamaterials studies lattices of coupled spinning gyroscopes that break time-reversal symmetry and carry disorder-resistant one-way edge modes, first demonstrated experimentally at the University of Chicago in 2015.
Topological metamaterial
A topological metamaterial is an artificially engineered periodic structure whose wave behavior is governed by topological invariants, producing edge waves that travel around defects without backscattering.
Topological order
Topological order is a type of order in gapped quantum phases of matter, introduced by Xiao-Gang Wen in 1989, arising from long-range quantum entanglement rather than symmetry.
Topological origami and kirigami metamaterials
Topological origami and kirigami metamaterials are mechanical metamaterials made from folded or cut sheets whose behavior is governed by topological invariants, a field begun in 2016.
Topological phase transition
A topological phase transition is a change between phases of matter distinguished by a topological invariant, such as a Chern number, requiring the bulk energy gap to close at the transition point.
Topological phononic crystals
Topological phononic crystals are periodic structures whose phonon bands carry a nontrivial topological invariant, forcing defect-immune edge, surface, or corner modes inside band gaps that resist backscattering.
Topological phononics
Topological phononics is the field applying topological band theory to phonons and sound or elastic waves in metamaterials, producing robust, defect-immune wave states with uses in waveguides and sensing.
Topological photonics
Topological photonics is a research field that applies topological ideas from condensed matter physics to control the flow of light in robust, backscattering-resistant edge states.
Topological photonics in waveguide arrays
Topological photonics in waveguide arrays is the experimental study of topological states of light in coupled optical waveguides, including Floquet and higher-order topological insulators, with applications in lasers and integrated photonic chips.
Topological plasmonic lattices
Topological plasmonic lattices are periodic arrays of metallic nanoparticles or waveguides whose plasmon bands have nontrivial topology, producing edge, interface, and corner modes protected against backscattering.
Topological plasmonics
Topological plasmonics is the branch of nanophotonics applying topological band theory to plasmonic systems, aiming to combine subwavelength light confinement with defect-immune, unidirectional propagation of surface waves.
Topological polaritons
Topological polaritons, first proposed in 2014, are hybrid light-matter quasiparticles whose nontrivial band structure supports protected one-way edge states that resist backscattering from disorder and defects.
Topological protection
Topological protection is the robustness of certain physical properties, typically gapless boundary states, against disorder, defects, and smooth changes, guaranteed by a topological invariant of the bulk.
Topological pumping
Topological pumping is a quantum transport phenomenon in which slow cyclic modulation of system parameters moves particles by an integer number of unit cells per cycle, with the charge set by a topological invariant.
Topological quantum computation
Topological quantum computation is a fault-tolerant approach to quantum computing that encodes information in non-Abelian anyons and computes by braiding them; Microsoft pursues it with its Majorana chips.
Topological quantum field theory
A topological quantum field theory (TQFT) is a quantum field theory whose amplitudes depend only on the topology of the underlying manifolds, introduced by Edward Witten in 1988.
Topological quasicrystalline metamaterials
Topological quasicrystalline metamaterials are artificially engineered structures arranged with quasiperiodic, non-periodic order that host topologically protected edge states, first demonstrated photonic in 2016 using a Penrose lattice of helical waveguides.
Topological quasicrystals
Topological quasicrystals are quasiperiodic structures, natural or artificial, that host topological phases of waves or electrons with protected edge, corner, or point states, first shown in photonic Penrose lattices.
Topological qubit
A topological qubit is a proposed quantum bit storing information non-locally in quasiparticles called Majorana zero modes, proposed by Alexei Kitaev to gain fault tolerance from hardware itself.
Topological rainbow trapping
Topological rainbow trapping (TRT) is a wave-control phenomenon in which waves of different frequencies are spatially separated and trapped at distinct locations along a structure, with localization protected by topological states.
Topological Rayleigh waves
Topological Rayleigh waves are Rayleigh surface waves whose propagation is governed by topological properties of engineered metamaterials, confining energy to surfaces or interfaces and resisting disorder, defects, and backscattering.
Topological semimetal
A topological semimetal (TSM) is a gapless material whose conduction and valence bands cross at points, lines, or surfaces stabilized by topology and crystal symmetry.
Topological soliton
A topological soliton is a stable, particle-like solution of nonlinear field equations that cannot be continuously deformed into the vacuum; examples include monopoles, Skyrmions, and superconducting vortices.
Topological sorting
Topological sorting, also called toposort, is the ordering of a directed graph's vertices so that every edge points from earlier to later; it exists only for directed acyclic graphs.
Topological space
A topological space, also called a topological structure, is a set with a collection of open sets capturing nearness and continuity without distance, the basic object of point-set topology.