Higher-order topological insulator

In plain words

A whose protected conducting states sit not on its surfaces but on its edges or corners. An ordinary topological insulator is inside and conducts on every surface. In a higher-order one the surfaces are insulating too, and the protected channels run only along the hinges where two faces meet – one-dimensional wires built into a three-dimensional crystal – or, in a flat material, sit at its corners. Bi₄Br₄, built from chains of bismuth and bromine, is the best-studied layered example.

As the site uses it

The side shows surface states, gapped because the edge states of neighbouring layers couple, with the hinge states inside the gap – the signature of a higher-order topological insulator.

Bi₄Br₄ · Bismuth halide chains Bi₄Br₄

Higher-order topological phases with corner states and the topology of moiré bands are newer questions.

Theory & computation

Going deeper

An ordinary topological insulator conducts on every surface. In a higher-order one the surfaces are gapped as well, and protected channels run only along the hinges where faces meet. In cleaved Bi₄Br₄ the top faces show bulk bands, the side faces a gap, and states inside that gap mark the hinges.

One dimension lower each time

A topological insulator is insulating inside, while its boundary – one dimension lower – must conduct: the surfaces of a three-dimensional crystal, the edges of a two-dimensional sheet. Higher-order topology moves the protected states further down. In a second-order topological insulator in three dimensions the surfaces are gapped as well, and conducting channels survive only on the hinges, the lines where two surfaces meet; in two dimensions the edges are gapped and states sit at the corners (Schindler and colleagues, 2018).

What protects them is time reversal combined with a crystal symmetry, such as a rotation or a mirror. Hinge channels can be , all running one way, or helical, pairs running in opposite directions with opposite ; unlike corner states, hinge states connect the valence and , so they conduct.

Bi₄Br₄

Bi₄Br₄ is built from chains of bismuth and bromine that stack into layers, and its crystals cleave to expose a top (001) surface and side (100) surfaces at steps. focused to below a micrometre, with spin resolution, found only bulk bands on the top surface; on the side surfaces it found with a gap, spin-polarised as calculations predict, and states inside that gap connecting the valence and conduction bands, about 15 meV below the – the signature of hinge states (Zhao and colleagues, 2023).

A of Bi₄Br₄ has been identified as a quantum spin Hall insulator with a gap of about 180 meV, so the hinge states can be read as the of stacked layers, gapped where neighbouring layers couple and surviving at the hinges.

Hard to prove

Hinge states are one-dimensional channels inside a three-dimensional crystal, and most probes average over far more than a hinge. Photoemission sees them only where hinges are dense, at the steps of a cleaved surface; transport and scanning probes must separate them from ordinary edge conduction and from trivial states that can also gather at steps. That is why many proposed higher-order materials are still candidates, and why their spin-polarised one-dimensional channels are discussed for but not yet used.

For specialists

An n-th-order topological insulator has protected gapless modes on boundaries of codimension n: hinge states in three dimensions with gapped surfaces, or corner states in two. Chiral higher-order insulators, protected by time reversal combined with fourfold rotation, carry chiral hinge modes; helical ones, protected by time reversal and , carry Kramers pairs – the class proposed for SnTe and surface-modified Bi₂TeI, BiSe and BiTe. In quasi-one-dimensional Bi₄Br₄, spin-resolved nano-ARPES finds bulk bands on the (001) surface, gapped spin-polarised surface states on (100), and in-gap states connecting them – the signature of hinge states.

Where this comes from

Higher-order topological insulators Schindler et al. · Science Advances 4, eaat0346 (2018)
Topological electronic structure and spin texture of quasi-one-dimensional higher-order topological insulator Bi₄Br₄ Zhao et al. · Nature Communications 14, 8089 (2023) cited by 37

In the news

The newest items in the site’s news feed that use the term, one from each source.

Preprintnot yet peer reviewed 14 Oct 2026 arXiv

Non-Hermitian Two-Kink Control of Corner States in Twisted Chern Bilayers

Corner states are bound by boundary-mass defects, but this geometric anchoring restricts their spatial and internal control. Focusing on assembled from with opposite Chern numbers, we show that a mass-phase kink carried by a non-Hermitian interlayer mass and an independent…

Theory

Preprintnot yet peer reviewed 13 Oct 2026 arXiv

Pseudo-relativistic fermions and topological states in zinc-blende semiconductors and their heterostructures (Habilitation thesis)

Tracing my scientific path from many-body physics in nanostructures to quantum matter, this Habilitation à diriger des recherches (HDR) memoir presents only part of my research since defending two PhD theses in Toulouse and Nizhny Novgorod in 2011. Focusing on pseudo-relativistic fermions and…

TheoryGraphene

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Metal, semimetal and insulator

The three basic ways a solid treats electricity, set by how its electrons fill the energies open to them. In a metal they fill a band of energies only partly, so the smallest push sets them moving, and it conducts at any temperature. In an insulator they fill a band completely and a wide gap separates it from the next, so nothing can move. A semimetal sits in between: two bands just touch or overlap a little, leaving a small number of mobile electrons and holes. NbSe₂ is a metal, graphene a semimetal, hBN an insulator.

A classification by where the Fermi level falls in the band structure. In a metal it crosses a partly filled band, leaving a Fermi surface, a carrier density of order one per atom (about 10²² cm⁻³) and a resistivity that falls on cooling. In a band insulator it lies in a gap of several electronvolts; a semiconductor is an insulator whose gap is small enough to be doped or gated. A semimetal has a slight overlap between conduction and valence bands (graphite, bismuth, bulk WTe₂) or bands that touch at isolated points (graphene), giving low and nearly equal densities of electrons and holes. Interactions can open gaps that band theory does not predict: Mott insulators from half-filled bands, excitonic insulators from semimetals.

Full explanation

Chirality

The property of an object that cannot be turned into its own mirror image, like a left and a right hand or a corkscrew that turns one way. Molecules can be chiral, and so can crystals: in tellurium the atoms form screw-shaped chains that wind either to the left or to the right. Electrons, spins and vibrations can take on a handedness too, and chiral materials can treat left- and right-turning light, or spins pointing one way or the other, differently.

The absence of improper symmetry operations (mirror planes, inversion, rotoreflections), so that an object and its mirror image are distinct enantiomorphs. Chiral crystals belong to the 65 Sohncke space groups, such as trigonal tellurium (P3₁21 or P3₂21), whose helical chains give natural optical activity, current-induced magnetisation and radial spin textures through spin–orbit coupling. Chirality also describes Weyl fermions, whose chirality sets the chiral anomaly and the chiral magnetic effect; edge modes that propagate one way; spin spirals and skyrmions with a fixed sense of rotation from the Dzyaloshinskii–Moriya interaction; phonons carrying angular momentum in the valleys of TMDCs; and proposed chiral orders such as loop currents in kagome metals and Mn₃Si₂Te₆. Barron’s distinction between true and false chirality decides which physical effects a given handedness allows.

Full explanation

Spin

A built-in property of every electron that makes it a tiny magnet. Measure it along any direction and you only ever find one of two answers, ‘up’ or ‘down’. Despite the name, nothing is actually spinning – the word stuck from an early picture – but the magnetism is real: countless electron spins lined up are what make a fridge magnet stick. Spintronics tries to carry information in spin rather than in charge.

The intrinsic angular momentum of the electron, ħ/2, with a magnetic moment of almost exactly one Bohr magneton; a measurement along any axis yields one of two values. In solids spin couples to orbital motion through spin–orbit coupling and to other spins through exchange; in 2D materials it sets magnetic order, spin–valley locking and how long a spin survives as a carrier of information.

Full explanation

Surface state

Electron states that exist only at the surface of a crystal, because the crystal stops there. Atoms at a surface have lost their neighbours on one side, so electrons can take energies there that the inside of the crystal does not allow. Usually such states are a nuisance that traps charge. In a topological insulator they are the point: the inside insulates while the surface is guaranteed to conduct, with each electron’s spin tied to its direction of motion.

States localised at a crystal surface and decaying into the bulk, at energies forbidden in the bulk for that in-plane momentum. Ordinary Tamm and Shockley states come from the broken periodicity, dangling bonds or reconstruction, and can be removed or shifted by passivation. Topological surface states are required by a non-trivial bulk invariant: a single spin–momentum-locked Dirac cone in Bi₂Se₃-family topological insulators, protected against backscattering by time-reversal symmetry, and Fermi arcs joining the projections of Weyl points in Weyl semimetals. In thin films the top and bottom surface states hybridise and open a gap below a critical thickness, about six quintuple layers for Bi₂Se₃. In a 2D material the boundary of a topological phase is its edge, and the counterpart of the surface state is the edge state.

Full explanation

Fermi level

Roughly, the energy up to which a material’s electron states are filled – like the water line in a partly filled glass. Where it sits relative to the band gap decides how many charges can move and whether they are electrons or holes, the empty places electrons leave behind. A gate voltage raises and lowers it, like pouring water in or out.

The electrochemical potential of electrons: the energy at which a state has 50 % occupation in thermal equilibrium. Its position relative to the band edges sets carrier density and type, and where electron and hole densities balance – at the Dirac point of graphene, near mid-gap in an undoped semiconductor – the material is at charge neutrality, where a gated 2D device conducts least; at metal–semiconductor contacts, interface states can pin it and fix the Schottky barrier.

Full explanation

Edge state

Electrons that can only travel along the border of a sheet while its interior stays insulating. In a topological material these border lanes are guaranteed to exist, and in the cleanest cases they carry current without losing energy to scattering.

States localised at a sample boundary and dispersing inside the bulk gap. In quantum Hall and Chern insulators they are chiral, one-way channels; in a quantum spin Hall insulator they are helical, counter-propagating with opposite spins and protected by time reversal, giving e²/h per edge in the ideal case – monolayer 1T′-WTe₂ keeps that edge conduction up to about 100 K. Trivial edges carry states too, from dangling bonds, reconstructions and graphene’s zigzag edge, so the evidence for topology is quantised conductance and its magnetic-field dependence, not conduction at the edge alone.

Full explanation

Mirror symmetry

A crystal has a mirror symmetry if reflecting it in a plane leaves it unchanged. For a 2D sheet the question that matters most is whether it looks the same from above and from below. The answer decides how its electrons’ spins behave, which band crossings are protected, and even how a calculation of it has to be set up.

Invariance under reflection through a plane. For a layer the horizontal mirror in the plane of the sheet matters most: it forbids an out-of-plane dipole, so a slab without it needs a dipole correction in a periodic calculation, and in a 2H monolayer that also lacks inversion it keeps spin–orbit-split spins pointing out of plane, the origin of Ising pairing. Vertical mirrors can protect band crossings – mirror-protected nodal lines in PbTaSe₂, mirror Chern numbers in the SnTe class of topological crystalline insulators – and losing a mirror at a structural transition is how some chain compounds become ferroelectric.

Full explanation

Topological phase

A state of matter told apart not by how its atoms are arranged but by a whole number that describes how its electrons’ waves twist across the crystal. It is the way a doughnut differs from a ball: by its one hole. A whole number cannot change a little, so what it guarantees, such as current running along an edge without loss, survives defects and dirt until the band gap itself closes.

A gapped phase characterised by a topological invariant of its occupied bands – a Chern number, a Z₂ index, a winding number – that cannot change under deformations that keep the gap open and any protecting symmetry intact. Where regions with different invariants meet, the gap must close, which forces boundary states: chiral edge channels in Chern insulators, helical ones in quantum spin Hall insulators, Fermi arcs in Weyl semimetals, Majorana modes in topological superconductors. Topological order in the strict sense – the long-range entanglement of fractional quantum Hall states and spin liquids – is a distinct, stronger notion.

Full explanation

In plain words

As the site uses it

Going deeper

One dimension lower each time

Bi4Br4

Hard to prove

For specialists

Where this comes from

In the news

Non-Hermitian Two-Kink Control of Corner States in Twisted Chern Bilayers

Pseudo-relativistic fermions and topological states in zinc-blende semiconductors and their heterostructures (Habilitation thesis)

Bi₄Br₄