The end that knows how the middle was cut

The mode that lives in the mistake found that a perfect repeat forbids a band of frequencies, and that a single mistake in the repeat is permitted to hold exactly one state inside the forbidden band. The state is bound to the mistake and decays away from it on both sides. It was there because the repeat had been broken.

There is a way to put a state in a gap without breaking anything. Take a chain of identical sites joined by links that alternate, strong and weak, strong and weak, all the way along. The chain is perfectly periodic. It has a gap. And whether its ends hold a state in the middle of that gap depends on nothing but which kind of link the chain happens to end on — a choice that the infinite chain, with no ends, cannot even detect.

Two couplings, alternating #

The model is the simplest chain that can do this. Each site holds one state at the same energy, set to zero. Neighbouring sites are coupled, so that a particle can hop between them, and the coupling alternates: $v$ between the two sites of a cell, $w$ from one cell to the next. It is the chain of the frequency a lattice cannot carry with two kinds of spring instead of one, and it describes a molecule as well as a lattice. In polyacetylene, a chain of carbon atoms with alternating single and double bonds, the electrons hop more easily across the short double bonds than across the long single ones, and in 1979 Wu-Pei Su, Robert Schrieffer and Alan Heeger used exactly this model to explain its electrical properties.

With two sites per cell the spectrum splits into two bands, symmetric about zero, with a gap between them of width $2|v - w|$. The figure draws every level of a finite chain of forty sites as the ratio $v/w$ is swept from zero to two. The bands are where the levels crowd, and the gap closes where $v = w$ — where the two couplings are equal, the cell is no longer doubled, and the chain is the plain chain again.

The figure also shows where the two zero-energy states come from. A chain of forty sites always has exactly forty levels, and on the right of the figure all of them sit in the two bands, twenty in each. As $v/w$ falls through one, the gap closes and reopens, and when it reopens one level has left the top of the lower band and one the bottom of the upper band, and both have moved to the middle. The bands each hold nineteen states on the left of the figure. Nothing was created at the ends; two states that belonged to the bulk were handed over to them, and the handover could only happen at the moment the gap was shut.

Everything so far is symmetric about $v = w$. The gap has the same width at $v/w = 0.5$ as at $v/w = 2$, in units of the larger coupling. The two levels in the middle of the gap on the left of the figure, and their absence on the right, are the only thing that is not.

The bulk cannot tell #

For an infinite chain the two choices are not two chains. A chain of alternating strong and weak links is the same object whether one groups each strong link into a cell and calls the weak ones the links between cells, or the other way round; moving every label along by one site turns one description into the other. So the bands must be the same, and the figure confirms it: the energy at each wavenumber is the length of the complex number $v + w\,e^{ik}$, which does not change when $v$ and $w$ are swapped, and the two sets of bands coincide to rounding.

The gap a repeat opens read everything about a periodic structure off its bands: where waves travel, where they are forbidden, how fast they decay. That reading is complete for an infinite structure. What this chain shows is that it is not complete for a finite one. Two structures can share every feature of their bands and still behave differently at their ends, and the difference has to be stored somewhere the bands do not show.

It is stored in the phase of the states rather than their energies. Each band’s states are waves across the chain, and as the wavenumber is carried once across the zone the internal make-up of a state — how its amplitude divides between the two sites of a cell, and with what relative phase — rotates. For one choice of cell that rotation adds up to half a turn, a geometric phase of $\pi$ called the Zak phase; for the other it adds up to nothing. Moving the cell boundary by one site changes the phase by exactly $\pi$, which is why the two descriptions of the same infinite chain differ in it while agreeing in every energy. A quantity that depends on a labelling looks unphysical for an infinite chain, and it becomes physical the moment the chain ends and the labelling is fixed by where the end is.

A state on one end and one sublattice #

The zero-energy states can be found without a computer, and the calculation shows where the asymmetry lives. Call the two sites of each cell A and B. The chain only ever couples an A site to a B site: within a cell by $v$, between cells by $w$. Look for a state with energy exactly zero that sits on the A sites alone. The condition for zero energy on each B site is that the amplitudes arriving from its two A neighbours cancel:

$$v\,\psi_A(n) + w\,\psi_A(n+1) = 0 \quad\Rightarrow\quad \psi_A(n+1) = -\frac{v}{w}\,\psi_A(n).$$

The chain begins with an A site, so the first amplitude can be anything, and every later amplitude is fixed by the one before. The state is a geometric series with ratio $-v/w$. If $v < w$ it shrinks along the chain and is a perfectly good state, bound to the left end. If $v > w$ it grows without limit and is not a state at all. The same argument from the right end, on the B sublattice, gives the partner state there.

The figure’s exact diagonalisation agrees in every detail. The state has all its weight on the A sites, the probability falls by $(v/w)^2 = 0.25$ per cell, and its energy is zero because it lives on one sublattice only and the chain couples each sublattice only to the other. In a finite chain the two end states overlap very slightly through the middle and split to a tiny energy either side of zero, which is why the figure’s levels are within a millionth rather than exactly at zero; the splitting shrinks by a factor of $v/w$ with every extra cell. For twenty cells at $v/w = 0.5$ that factor has been applied twenty times, $(1/2)^{20} \approx 0.95$ millionths of the coupling, which is the size of the splitting the figure finds.

The decay is the same kind of decay that the wall that is not quite a wall describes for a particle under a barrier. Zero energy is in the gap, no wave can travel there, and a state at that energy can only exist as an amplitude falling away from wherever it is held.

The loop that counts them #

The information the bands lose is in the phase of $v + w\,e^{ik}$ rather than its size. As the wavenumber crosses the zone that complex number traces a circle of radius $w$ centred on $v$. If $w > v$ the circle encloses the origin, and the phase makes one full turn; if $w < v$ it does not, and the phase wobbles and comes back. The number of turns is the winding number, and the figure counts it by adding up the phase’s change around the loop: one for the chain with edge states, zero for the chain without.

A winding number is a whole number, so it cannot change gradually. The only way to take the loop from enclosing the origin to not enclosing it is to drag it through the origin, and a loop through the origin means some wavenumber with zero energy — the gap closing. That is why the two sides of $v = w$ are distinct: no smooth change of the chain that keeps the gap open can turn one into the other. And the winding number equals the number of zero-energy states each end of a finite chain must hold. That statement, that a whole number computed from the infinite bulk counts states at a boundary, is called the bulk–boundary correspondence, and this chain is its simplest case.

The same kind of integer turned up in the two in the flux quantum, where the phase of a superconductor must wind by whole turns round a ring, in the whirlpool that comes in one size, where the phase of a superfluid can only wind by whole turns round a vortex, and in the phase a magnet leaves on a path it never touched, where a phase depends on what a loop encloses rather than on the path. Here the loop is in the space of wavenumbers, and what it encloses is a point where the energy would vanish.

Disorder the edge state ignores #

A state held by a whole number should not care about details, and the figure tests that. In the first set of chains every coupling is changed at random by up to thirty per cent, so the chain is no longer periodic at all. The zero-energy level survives in every one of them. The derivation above explains why: it never used the couplings being equal from cell to cell, only that each B site’s two A neighbours can be made to cancel, and that remains possible whatever the couplings are, as long as the state still shrinks along the chain on average.

In the second set the same chains are also given random energies on the sites themselves. Now an A site’s energy couples the state on it to itself, the argument fails, and the level moves off zero to a typical value of four hundredths of the coupling. The survival has a sharper form than the figure’s thirty per cent. For a chain whose couplings are random but keep the sublattice symmetry, the edge state survives as long as the typical inner coupling is weaker than the typical outer one, where “typical” means the geometric mean — the average of the logarithms. The derivation shows why: the amplitude is multiplied by one $-v/w$ per cell, so after many cells its size is set by the sum of the logarithms of those ratios, and the state is bound to the end exactly when that sum runs to minus infinity. The chain can be made as irregular as desired in every other way. Only the balance of logarithms decides, and a transition happens where it tips.

The protection was never regularity. It was the symmetry that couples A only to B — called chiral or sublattice symmetry — and the level is exactly as robust as that symmetry is.

Where the chain has been seen #

In polyacetylene the zero-energy state appears not at the end of a molecule but in the middle, at a domain wall where the pattern of strong and weak bonds switches from one assignment to the other. Such a wall is an end for each of the two halves, and it binds a state at zero energy. Su, Schrieffer and Heeger found that these walls carry charge without spin, or spin without charge — an electron’s two properties coming apart — and the effect was measured in doped polyacetylene, whose conductivity rises by many orders of magnitude.

The domain wall shows what the zero-energy state means for charge. A wall between the two dimerisations holds one state at zero energy that is shared between the band below and the band above: counting the states, half of it belongs to each. If it is empty the region around the wall carries half an electron’s charge fewer than the bands would, and if it is full, half more. In polyacetylene each state holds two electrons of opposite spin, and the halves combine into the stranger combinations Su, Schrieffer and Heeger found: a wall with charge and no spin, or a wall with spin and no charge. A particle’s quantum numbers are not guaranteed to come in the bundles they come in for a free electron, and a whole number counted from the bulk is enough to split them.

The chain has since been built deliberately. Atoms in an optical lattice with alternating wells have had the winding measured directly, as a geometric phase acquired across the zone, in 2013. Arrays of coupled optical waveguides and microwave resonators show the edge states as light or microwaves trapped at the end of the array. And mechanical chains of rotors and springs have been designed with floppy modes stuck to one end, the same mathematics applied to a structure’s vibrations. The 2016 Nobel Prize, to David Thouless, Duncan Haldane and Michael Kosterlitz, recognised the wider idea that whole numbers of this kind classify phases of matter — the same Kosterlitz and Thouless whose transition is the transition with nothing to order.

Where the model stops #

Only neighbours are coupled. Hopping to the next-nearest site couples A to A and breaks the sublattice symmetry, moving the edge states off zero exactly as the random site energies did. Real polyacetylene has such couplings, weakly, and its zero modes are protected only as far as they are small.

The particles do not interact. Everything here is a single particle on a chain. Electrons repel each other, and in interacting systems the counting of protected edge states can change; for some symmetries two protected states can pair up and cancel, which the non-interacting winding number does not know.

The chain is long. The exact zero energy belongs to a semi-infinite chain. A finite chain splits its two end states by an amount that falls exponentially with length, which is tiny at twenty cells for $v/w = 0.5$ but not at three cells, or near $v = w$ where the decay is slow.

And one dimension is special. In two and three dimensions the analogous whole numbers are defined differently and protect states that travel along edges and surfaces rather than sitting at points; the chain is the simplest member of a family whose larger members are more subtle.

What the pictures cannot show #

The edge-state figure draws probabilities, and the probabilities cannot show the sign that the derivation depends on. The amplitude on successive A sites alternates in sign, $-v/w$ each time, and the cancellation on the B sites between is what makes the energy zero; the bars show only its square.

Nor can any figure localise the winding number. It is not a property of any site or any wavenumber; it is a property of the whole loop, and a change at one wavenumber can only alter it by closing the gap. A drawing of the loop shows it, but only because the drawing shows the loop all at once.

Still open: whether a protected zero mode can hold a qubit #

The chain’s zero-energy end states have a close cousin in superconducting wires, where a similar whole number protects zero modes of a stranger kind, called Majorana modes, that are their own antiparticles. A pair of them at the two ends of a wire would store information non-locally, in a way ordinary noise acting on one end cannot disturb — sidestepping the fragility that the state that cannot be copied makes unavoidable for ordinary quantum bits, where no backup is allowed — and that is the basis of proposals for topologically protected quantum computation.

Whether such modes have been made and seen is not settled. Signatures reported in semiconductor nanowires from 2012 onwards have been argued to have mundane explanations, and at least one prominent claim was retracted; newer devices and protocols designed to rule out the alternatives are being tested. The mathematics of protection is clear, as the simple chain shows. Whether a real material keeps the protecting symmetry well enough, cold enough and long enough for the protection to be useful is the open question.

The habit worth keeping is the one the loop teaches. When two systems share every local property, look for a whole number that neither can change smoothly. The bands of the two chains are identical at every wavenumber; the number of times a loop goes round a point is not, and it is the number that the ends obey.