Why should a measured response be an integer?
The integer quantum Hall effect presented a theoretical puzzle after its 1980 discovery. A semiconductor contains impurities, edges and a complicated potential, yet its transverse response can approach a simple integer multiple of a fundamental conductance. Explaining a clean electron orbit was insufficient. The harder question was why small changes in microscopic details failed to shift the integer continuously.
In 1982 David Thouless, Mahito Kohmoto, Peter Nightingale and Marcel den Nijs studied electrons in a two-dimensional periodic potential and a magnetic field. Their four-author paper, usually called TKNN, calculated the small current induced by a weak field. Quantization became explicit when all occupied states were separated from empty ones by an energy gap. The integer belonged to the occupied wavefunctions throughout momentum space. This provided a way to understand a stable response even when individual energy bands had a complicated shape.
Sources: [1][2]
A band energy does not contain all the information
A Bloch state is a quantum pattern of motion in a periodic material. Crystal momentum labels its repeating wave, and the wavefunction records amplitude and phase inside one repeating cell. Amplitude gives probability after squaring; phase says where the wave is in its cycle. Two materials can have similarly shaped energy bands but different wavefunctions. A response to a weak electric field depends on how those wavefunctions change, including their relation to other bands.
TKNN used the Kubo formula, a way to calculate the small current induced by a weak electric field. For completely occupied isolated bands at zero temperature, ordinary longitudinal band velocity contributions cancel over momentum space. The transverse response can remain because the states have a geometric structure. Completely occupied bands and a gap are essential here. When a band is partly filled, the response generally depends on the occupied portion and can vary continuously.
Sources: [2]
Geometric phase compares states around a closed path
A wavefunction’s overall phase can be chosen freely at each momentum. That freedom makes a phase written at one isolated point unobservable. Comparisons around a closed loop are different. If an isolated eigenstate follows a sufficiently slow parameter cycle, it can acquire a geometric phase in addition to the phase from elapsed time and energy. Michael Berry’s 1984 paper gave a general account of this effect.
Berry curvature describes the geometric phase per small area of parameter space. For a two-dimensional band the parameters are the two momentum components. Opposite sides of a Brillouin zone, one repeating region of momentum space, identify equivalent crystal momenta, so the full domain has periodic connections. This is the torus used in the mathematics; the physical sample need not be shaped like a doughnut. Locally chosen phases can differ between overlapping patches; their curvature and the full-zone response agree when these patches are joined correctly.
Sources: [3][4]
The Chern number sums curvature over the whole zone
Adding curvature over the complete repeating momentum domain, then dividing by one full turn, defines a dimensionless integer called the Chern number. Each isolated occupied band contributes its own integer. The sum sets the Hall conductance shown in the formula: conductance is measured in siemens, the Chern number has no units, and the other symbols are the positive elementary charge and Planck’s constant. The sign depends on the chosen orientation. Curvature itself has units of squared metres when its two momentum coordinates are wavevectors in inverse metres.
A nonzero integer means that one cannot choose a single smooth periodic phase convention everywhere. Smooth changes of a gapped Hamiltonian change the curvature locally, but its full integral remains the same integer. A band touching permits that integer to change; a touching does not guarantee a change. For a total integer of one, the magnitude of sheet Hall conductance is about 38.74 microsiemens. A global property of wavefunctions thus fixes a voltage-and-current response that survives many microscopic changes.
Sources: [2][4]
Flux insertion offers a second physical explanation
Laughlin’s 1981 argument considered a Hall system wrapped around a cylinder. Slowly changing flux through its hole generates an electric field around the circumference by Faraday’s law. The Hall response transfers charge across the cylinder, from one edge toward the other. One flux quantum returns the Hamiltonian to an equivalent gauge description, even though charge has moved between edges.
Combining the Hall conductance with that flux increment gives an integer number of elementary charges transferred in an ideal integer state. One flux quantum is Planck’s constant divided by the positive elementary charge; multiplying it by the Hall conductance gives the transferred charge. Its direction depends on field and flux orientation. This picture links quantization to gauge invariance and a mobility gap: the interior has no extended states available for ordinary dissipative transport at the relevant energy. It complements the bulk band calculation. The cylinder makes the link between a completed flux cycle and transferred charge especially clear.
Sources: [5]
A slow one-dimensional cycle can pump an integer amount
Thouless found a related result in 1983 for a one-dimensional periodic potential that changes slowly through a cycle. Momentum supplies one periodic parameter and the stage of the cycle supplies another. Together they form a two-dimensional parameter space with a Chern number. For an isolated fully occupied band in the ideal adiabatic limit, the number transported per cycle is that integer.
The corresponding Wannier centre, the position associated with a localized band state, shifts by an integer number of unit cells. The distance is the unit-cell length, which can span two neighbouring wells in a doubled lattice. If it is five hundred nanometres and the integer is one, ten cycles shift the centre by five micrometres. Position can move by a noninteger amount during a cycle. Quantization refers to the completed cycle; transitions into another band, unsuitable occupations or finite-system effects can spoil the ideal result.
Sources: [6][7]
Optical lattices made the cycle and its controls visible
Cold-atom experiments published in the 2016 volumes of Nature Physics tested Thouless pumping with optical superlattices. Nakajima and colleagues loaded ultracold ytterbium fermions into a potential made from two standing-wave patterns. Changing their relative phase and depths changed the sequence of wells and barriers. Absorption images measured the atomic cloud’s centre after successive cycles. They compared paths that did or did not surround the gap-closing point and tested dependence on cycle speed and temperature.
Lohse and colleagues independently used bosonic atoms in a controlled superlattice and observed quantized cloud displacement. Preparing atoms in an excited band could reverse the direction relative to the lowest band. Such comparisons distinguish band geometry from merely dragging particles with a moving well. These experiments record displacement and density in neutral atoms. The same mathematical response becomes visible through motion of a cloud in a different apparatus.
Sources: [7][8]
Light on a chip can carry the same kind of integer
Nothing in the Chern number requires electrons. In 2008 Duncan Haldane and Srinivas Raghu showed that light in a periodic structure has bands too, and that those bands can carry a Chern number when magnetized material makes light running one way behave differently from light running the other way. They predicted that at frequencies inside a gap between bands, light would travel along the edge of such a structure in one direction only. In 2009 Zheng Wang, Yidong Chong, John Joannopoulos and Marin Soljačić built one from magnetized ferrite rods and sent microwaves along its edge. A metal obstacle in the path did not send the waves back: with no backward edge state at that frequency, they went around it.
At the near-infrared wavelengths used in optical fibre, magnetic effects in materials are at least a hundred times weaker than at microwave frequencies, so Mohammad Hafezi and colleagues used geometry instead. On a silicon chip they laid out a 10 by 10 grid of ring-shaped waveguides, each tens of micrometres across, joined by small linking rings. Shifting each linking ring slightly gives light that hops around any square of four rings the same extra phase, the role a magnetic field plays for an electron. Light circling one way in the rings feels the opposite field from light circling the other way, so the chip has two one-way edges running in opposite directions. They stay apart as long as little light reverses inside a ring.
In 2013 the group imaged the faint light scattered from each ring with an infrared camera. At a frequency in an edge band, light entered at one corner and travelled along the edge in a strip one or two rings wide. On a chip with one edge ring left out, it went tightly around the gap and reached the output without spreading into the middle.
Sources: [10][11][12][13]
Robustness has physical conditions
Disorder destroys simple crystal-momentum labels, which calls for a formulation beyond perfectly repeating bands. Niu, Thouless and Wu developed a many-body boundary-condition approach in 1985 that also addresses disorder and interactions. It preserves the importance of a gap or mobility gap. Fractional Hall liquids require a many-body treatment because their response belongs to correlated electrons, including the resulting fractionally charged excitations.
The shared idea is a whole-cycle or whole-domain response. Local curvature may move around as a potential changes, and an atom cloud may take a different path during a pump cycle. As long as the relevant gap and occupation conditions persist, the complete response keeps its integer. Temperature and overly rapid driving can excite other states, which is why the experimental controls remain part of observing that stability.
Sources: [9][2]