Field · Emerged 1980 – 2007

Topological Matter

Can the shape of electrons' quantum states, rather than the arrangement of their atoms, define a phase of matter?

5 chapters5 min read7 turning points 1 open problem

Branched from: Solid-State Physics + Phase Transitions
Branched into: Not yet surveyed past here
Figures: Klaus von Klitzing, David Thouless, Robert Laughlin, Daniel Tsui, Horst Störmer, Duncan Haldane, Andre Geim, Konstantin Novoselov, Charles Kane, Eugene Mele, Shou-Cheng Zhang, Laurens Molenkamp

In brief

In 1980 Klaus von Klitzing found that the resistance of a thin layer of electrons in a strong magnetic field comes in steps, and that each step has a value fixed by two constants of nature to high precision, whatever the material, its impurities or its shape. So exact a number from so messy a sample demanded an explanation. It came in 1982 from topology, the branch of mathematics concerned with properties that do not change under smooth deformation, such as the number of holes in a doughnut.

The steps count a topological invariant of the electrons' quantum states, an integer that cannot change without closing the energy gap. From that idea grew a new classification of matter. It predicted, and experiments found, topological insulators, which are insulating inside but conduct on their surfaces. Graphene, a single sheet of carbon atoms, joined the story in 2004. The hunt for exotic particles in such materials has produced some of the field's biggest hopes, and some of its retractions.

Key ideas

Quantum Hall effectEnters 1980: In a thin layer of electrons at low temperature and high magnetic field, the Hall resistance, the voltage across the current divided by the current, is locked to $h/(\nu e^2)$ , where $\nu$ is a whole number.
Topological invariantEnters 1981 – 1982: A whole number computed from the quantum states of all the electrons in a band, which stays fixed under any smooth change. For the quantum Hall effect it is called the Chern number, and it equals $\nu$ .
Fractional chargeEnters 1982 – 1983: In the fractional quantum Hall effect, the electrons act collectively so that the basic excitations carry a fraction of an electron's charge, such as one third.
Edge stateEnters 2005 – 2007: At the boundary of a topological material, where the invariant must change to the value of empty space, conducting states must exist. They carry current along the edge and are protected from being scattered backwards.
Dirac electronsEnters 2004 – 2005: In graphene the electrons behave as if they had no mass, obeying an equation like the one Dirac wrote for relativistic particles, at a speed about 1/300 of the speed of light.

Draws on other domains

↙ Mathematics
Algebraic Topology
Topological phases of matter

Chapter I

An Exact Number from a Dirty Sample

The Hall effect has been known since 1879. Pass a current along a strip in a magnetic field, and the field pushes the charges sideways, building up a voltage across the strip. The ratio of that voltage to the current, the Hall resistance, normally rises smoothly with the field. On the night of 4–5 February 1980, at the high magnetic field laboratory in Grenoble, Klaus von Klitzing measured it for electrons confined to a thin layer at the surface of a silicon transistor, cooled to near absolute zero. The Hall resistance rose in flat steps, and the steps sat at $h/e^2$ divided by a whole number, where $h$ is Planck's constant and $e$ the electron's charge.

That was astonishing. The sample was a commercial device full of impurities and irregular edges, and yet it produced a combination of fundamental constants to high precision. Later measurements pushed the agreement to parts in a billion. Something had to make the answer immune to all the details.

Chapter II

Topology Enters

The explanation came in stages. In 1981 Robert Laughlin gave an argument that the steps must be exact. In 1982 David Thouless and three colleagues showed what the whole number is. It counts how the electrons' quantum states twist as one moves through the band, in the same way that the number of holes counts the shape of a surface in algebraic topology. A smooth change cannot alter a whole number, so impurities and irregular shapes cannot either. Thouless had already brought topology into physics a decade earlier, in the vortex transitions of phase transitions.

Cleaner samples brought another surprise. In 1982 Daniel Tsui and Horst Störmer, using layers grown by Arthur Gossard, found a plateau at one third of the first step. No picture of independent electrons allows it. Laughlin explained it in 1983. At that field the electrons condense into a new quantum liquid, and its excitations carry exactly one third of an electron's charge. The charge of an electron, it seemed, is not the smallest charge that can move through a solid.

In 1988 Duncan Haldane showed that a quantum Hall effect needs no magnetic field. Electrons hopping on a honeycomb lattice with the right pattern of internal fields would do it. The model looked artificial and was put aside.

Chapter III

Topological Insulators and Graphene

A real honeycomb of atoms arrived in 2004. Andre Geim and Konstantin Novoselov in Manchester pulled single sheets of carbon atoms from graphite with adhesive tape and measured their conduction. The electrons in graphene behave like massless particles obeying the relativistic equation of Dirac, and their quantum Hall steps are shifted by a half from those of ordinary electrons, a sign of the Dirac equation at work.

Graphene's honeycomb prompted Charles Kane and Eugene Mele to ask in 2005 whether the coupling between an electron's spin and its motion could replace Haldane's fields. It could, giving a new invariant that takes only two values. Graphene's version is too weak to see, so Shou-Cheng Zhang and his students predicted where to look instead, in thin wells of mercury telluride. In 2007 Laurens Molenkamp's group found the predicted conducting edges on otherwise insulating samples. A topological insulator is insulating inside, but its surface must conduct, because the invariant has to change at the boundary with empty space.

The search then turned to particles. Combining topological materials with superconductors should give Majorana modes, halves of an electron that could store quantum information protected by topology. Signs were reported from 2012 onwards, but a much-publicised 2018 claim was retracted in 2021, and the question is still open.

Chapter IV

A Closer Look: The Ohm Made Exact

The quantum Hall steps sit at $R = R_K/\nu$ , where $\nu$ is a whole number and $R_K$ is the von Klitzing constant. Since 2019 the SI units fix $h = 6.62607015 \times 10^{-34}$ J s and $e = 1.602176634 \times 10^{-19}$ C exactly, so

R_K = \frac{h}{e^2} = 25{,}812.807\,45\ldots \ \Omega .

Filling $\nu$	Hall resistance $R_K/\nu$ (Ω)
1	25,812.807
2	12,906.404
3	8,604.269
4	6,453.202
1/3 (fractional)	77,438.422

Which step appears depends on how many electrons there are per unit of magnetic flux. The flux quantum is $h/e = 4.136 \times 10^{-15}$ weber, and the filling is $\nu = n h / (eB)$ for $n$ electrons per square metre in a field $B$ . A typical layer with $n = 3 \times 10^{15}$ m⁻² reaches $\nu = 1$ at $B = 12.4$ T and $\nu = 2$ near 6.2 T.

Von Klitzing's original paper was titled as a way to measure the fine-structure constant $\alpha$ , the number that sets the strength of electromagnetism. With the magnetic constant fixed, as it then was, at $\mu_0 = 4\pi \times 10^{-7}$ N/A²,

\alpha = \frac{\mu_0 c}{2 R_K}, \qquad \frac{1}{\alpha} = 137.036 .

For resistance standards, laboratories adopted in 1990 a conventional value $R_{K\text{-}90} = 25{,}812.807$ Ω exactly. The 2019 value is larger by about 18 parts in a billion, so every resistance standard in the world shifted by that amount on 20 May 2019. That such a correction mattered at all is a measure of how exact the steps are: devices of gallium arsenide and of graphene give the same value to about a part in ten billion.

Chapter V

A New Kind of Order

Before 1980, phases of matter were classified by their symmetry, as in Landau's theory. Topological matter showed that two phases can have identical symmetry and still differ, in an integer that no local measurement can see. The idea now runs through condensed matter physics, from spin liquids in magnetism to topological superconductivity, and back into mathematics, where physicists' classifications have posed new problems. The biggest prize, a quantum computer protected by topology, is still out of reach.

Turning points7

1980EXPERIMENT
The quantum Hall effect
Working at the high magnetic field laboratory in Grenoble, Klaus von Klitzing measures the Hall resistance of electrons trapped at the surface of a silicon transistor. It forms flat plateaus at values equal to $h/e^2$ divided by a whole number, to high precision, whatever the sample. He received the 1985 Nobel prize.
Klaus von Klitzing
› Sources (1)
- von Klitzing, K., Dorda, G. & Pepper, M. (1980). New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance. Physical Review Letters 45(6): 494–497.
1981 – 1982DISCOVERY
The Hall steps are topological
Robert Laughlin argues in 1981 that the steps follow from the gauge symmetry of electromagnetism. In 1982 David Thouless, Mahito Kohmoto, Peter Nightingale and Marcel den Nijs show that the Hall conductance of a filled band is a topological invariant of its quantum states. Mathematicians soon recognised it as a Chern number, familiar from the topology of curved spaces. Thouless shared the 2016 Nobel prize.
David Thouless, Robert Laughlin
› Sources (2)
- Thouless, D. J., Kohmoto, M., Nightingale, M. P. & den Nijs, M. (1982). Quantized Hall conductance in a two-dimensional periodic potential. Physical Review Letters 49(6): 405–408.
- Laughlin, R. B. (1981). Quantized Hall conductivity in two dimensions. Physical Review B 23(10): 5632–5633.
1982 – 1983DISCOVERY
Fractional charge
Daniel Tsui and Horst Störmer, in very clean samples grown by Arthur Gossard at Bell Labs, find a Hall plateau at one third of the first step, which single electrons cannot explain. Robert Laughlin explains it in 1983 as a new quantum liquid, whose excitations carry a third of an electron's charge. Tsui, Störmer and Laughlin shared the 1998 Nobel prize.
Robert Laughlin, Daniel Tsui, Horst Störmer
› Sources (2)
- Tsui, D. C., Störmer, H. L. & Gossard, A. C. (1982). Two-dimensional magnetotransport in the extreme quantum limit. Physical Review Letters 48(22): 1559–1562.
- Laughlin, R. B. (1983). Anomalous quantum Hall effect: an incompressible quantum fluid with fractionally charged excitations. Physical Review Letters 50(18): 1395–1398.
1988DISCOVERY
A quantum Hall effect without a magnetic field
Duncan Haldane invents a model of electrons hopping on a honeycomb lattice that shows a quantized Hall effect with no overall magnetic field. It showed that the topology belongs to the band structure itself, not to the magnet. The model was long regarded as a curiosity, then became the template for topological insulators. Haldane shared the 2016 Nobel prize.
Duncan Haldane
› Sources (1)
- Haldane, F. D. M. (1988). Model for a quantum Hall effect without Landau levels: condensed-matter realization of the "parity anomaly". Physical Review Letters 61(18): 2015–2018.
2004 – 2005EXPERIMENTContested
Graphene
Andre Geim and Konstantin Novoselov in Manchester peel single layers of carbon atoms from graphite with adhesive tape and measure their electrical properties. In 2005 their group and Philip Kim's at Columbia show that graphene's electrons behave as massless Dirac particles, with an unusual quantum Hall effect. Geim and Novoselov received the 2010 Nobel prize.
What is disputed
Walt de Heer and others criticised the Nobel committee's scientific background for understating earlier work, including de Heer's own graphene grown on silicon carbide and older observations of very thin graphite. The dispute is about the history and its weighting, not about the Manchester measurements.
Andre Geim, Konstantin Novoselov
› Sources (2)
- Novoselov, K. S., Geim, A. K. et al. (2004). Electric field effect in atomically thin carbon films. Science 306(5696): 666–669.
- Novoselov, K. S. et al. (2005). Two-dimensional gas of massless Dirac fermions in graphene. Nature 438: 197–200.
2005 – 2007PARADIGM-SHIFT
Topological insulators
Charles Kane and Eugene Mele show that electron spin can play the role of the magnetic field, giving a new invariant and a quantum spin Hall effect. Andrei Bernevig, Taylor Hughes and Shou-Cheng Zhang predict it in thin layers of mercury telluride, and in 2007 Laurens Molenkamp's group in Würzburg observes the conducting edges. Three dimensional versions were found in bismuth compounds in 2008.
Charles Kane, Eugene Mele, Shou-Cheng Zhang, Laurens Molenkamp
› Sources (3)
- Kane, C. L. & Mele, E. J. (2005). Z2 topological order and the quantum spin Hall effect. Physical Review Letters 95(14): 146802.
- Bernevig, B. A., Hughes, T. L. & Zhang, S.-C. (2006). Quantum spin Hall effect and topological phase transition in HgTe quantum wells. Science 314(5806): 1757–1761.
- König, M. et al. (2007). Quantum spin Hall insulator state in HgTe quantum wells. Science 318(5851): 766–770.
2012 – 2021EXPERIMENTContested
The Majorana chase
Theory predicts that the ends of certain superconducting nanowires should host Majorana modes, halves of an electron that could store quantum information safe from noise. A Delft group reports signs of them in 2012. A 2018 paper in Nature claiming a decisive quantized signal is retracted in 2021, after the full data were found to undermine it, and other claims are withdrawn or corrected. The search continues.
What is disputed
Whether any experiment has yet shown Majorana modes, as opposed to ordinary states that mimic their signatures, is disputed. Critics argue that selective reporting of data played a part in the retracted claims. Supporters point to steady improvements in materials and measurement. In 2025 Microsoft announced a chip it described as topological, but the accompanying Nature paper did not claim proof of Majorana modes, and many physicists remain sceptical.
› Sources (2)
- Mourik, V. et al. (2012). Signatures of Majorana fermions in hybrid superconductor-semiconductor nanowire devices. Science 336(6084): 1003–1007.
- Zhang, H. et al. (2018). Quantized Majorana conductance. Nature 556: 74–79. Retracted 2021.

Applications

Where it is used

Metrology
The ohm from the quantum Hall effect
Since 1990 national laboratories have realised the ohm with quantum Hall devices, which agree with each other to parts in a billion. When the SI units were redefined in 2019, Planck's constant and the electron's charge were fixed exactly, which made $h/e^2$ an exact number of ohms. Every calibrated resistance meter is traced to it.
› Sources (1)
- BIPM (2019). The International System of Units (SI), 9th edition. Bureau International des Poids et Mesures.
Topology↗ Mathematics · Algebraic Topology
A periodic table of topological matter
In 2009 Alexei Kitaev classified all possible topological insulators and superconductors, using K-theory and Bott periodicity from algebraic topology. The result is a table in which the possible invariants repeat as the dimension changes. Questions raised by the physics, such as how symmetry and interactions change the classification, have become active research in topology itself.
› Sources (2)
- Kitaev, A. (2009). Periodic table for topological insulators and superconductors. AIP Conference Proceedings 1134: 22–30.
- Freed, D. S. & Moore, G. W. (2013). Twisted equivariant matter. Annales Henri Poincaré 14(8): 1927–2023.

Open problems

Where the map runs out

Open

Non-Abelian anyons and topological quantum computing

Open as of 2026. Simpler, Abelian anyons were observed in 2020, and non-Abelian anyons have been simulated on quantum processors since 2023, but interference experiments in real materials have not yet shown the non-Abelian kind beyond doubt.

In two dimensions, particles need not be ordinary bosons or fermions. Some predicted excitations, called non-Abelian anyons, remember the order in which they have been moved around each other. Information stored in that braiding would be immune to most noise. Do such particles exist in any real material, and can they be controlled?

Why it is hard: The candidates, such as the fractional quantum Hall state at filling 5/2 and Majorana modes in nanowires, need extremely clean samples at millikelvin temperatures, and their signatures can be imitated by ordinary effects. A convincing test means braiding them and seeing the predicted change of state.
What resolving it unlocks: A quantum computer whose qubits are protected by topology rather than by constant error correction, and a new kind of particle statistics seen in nature.

› Sources (1)

Nayak, C., Simon, S. H., Stern, A., Freedman, M. & Das Sarma, S. (2008). Non-Abelian anyons and topological quantum computation. Reviews of Modern Physics 80(3): 1083–1159.