The phase a symmetry turns into a number

Where two levels must meet reads a band structure’s degeneracies off the symmetry of a wavevector: what the group at a point allows decides which levels touch there. That is a statement about energies. This essay asks the same kind of question about the states, and the answer is a different kind of object.

What the loop leaves behind #

Fix a band and follow its Bloch state as the wavevector crosses the Brillouin zone from one edge to the other. Because the zone is a closed loop — its two edges are the same wavevector — the state comes back to itself. It does not come back with the same phase.

The phase it accumulates is the Zak phase, the one-dimensional case of Berry’s phase, and it is not a property of any single state. Each |u(k)⟩ is defined only up to a phase of its own; what survives is what the loop does, because the arbitrary phases cancel round a circuit.

That cancellation is also how the phase is computed. Sample the zone at N points, take the overlap ⟨uⱼ | uⱼ₊₁⟩ between neighbours, multiply them all together and take the argument of the product. Every state appears once as a bra and once as a ket, so a phase attached to it cancels against itself, and the product’s argument is unchanged by any choice at all.

It is worth being precise about which loop is being followed, because “across the zone” hides a choice. The zone is a circle: its two ends are the same wavevector, since they differ by a reciprocal lattice vector, and a Bloch state at one end is the state at the other multiplied by a fixed factor that depends on where the basis functions were placed. Fixing that factor is called choosing a periodic gauge, and it is the one part of the setup that is not free — a different choice moves the phase by a whole multiple of 2π, which is why the phase is defined modulo a turn and not absolutely. Everything else about the states may be chosen at will.

The second row of that figure is the more surprising one. A discretisation of an integral is normally an approximation that improves as the mesh is refined; this one is exact at four points and stays exact. That is the signature of a quantity forced to a value by a symmetry: there is nothing to converge to, because the answer was never anywhere else.

One consequence of that exactness is practical and is the reason this quantity is computed the way it is. A Berry phase written as an integral of a connection needs the states to be differentiable in the wavevector, which means choosing their phases smoothly across the zone — a real difficulty, and one that gets worse in more dimensions. The product of overlaps needs nothing of the kind: the states may be handed over in any order with any phases, as a numerical eigensolver produces them, and the answer is right. The discretisation is not an approximation to the integral, it is a better-behaved object that has the same value.

Why two values and not a continuum #

The chain has an inversion centre, sitting midway between its two sites. Inversion sends k to −k, so it maps the zone loop to itself traversed backwards, and a phase accumulated round a loop changes sign when the loop is reversed. So the phase equals its own negative — modulo a full turn, since a phase is only defined that far — and γ = −γ (mod 2π) has exactly two solutions: 0 and π.

That is the whole argument, and it is worth noticing how little it uses. Nothing about the strength of the hoppings, nothing about the shape of the bands, nothing about how the states were computed. A symmetry constrains a quantity to a set with two elements, and the model’s job is only to say which.

The test of a claim like that is whether it can be lost, and it can. Raise one site and lower the other by a staggered energy and the inversion centre is gone; the phase then takes values that are neither zero nor π, moving smoothly with the distortion, and returning to exactly π when the distortion is switched off. A quantity that stays quantised however the model is deformed would be a quantity quantised by the model rather than by the symmetry.

It is worth asking what a different symmetry would allow, because that is what turns one result into a rule. A mirror through a site does the same thing as inversion here, since in one dimension they coincide; a chain with no symmetry at all constrains nothing and the phase is free. In two and three dimensions the same argument runs with whatever operations fix the direction being traversed, and the allowed values are the solutions of γ = −γ when the operation reverses the loop and of γ = γ when it does not. So the set of allowed phases is read off the little group of the direction, exactly as the star of a wavevector reads the allowed degeneracies off the little group of a point.

Where the number means nothing #

One row of the sweep needs separating from the rest, and it is the one where the argument fails rather than where it succeeds.

A Zak phase belongs to a band. At equal hoppings the two bands of this chain meet at the zone boundary, and a band that touches another one is not a band over the whole zone. The product of overlaps is still computable — a product of complex numbers always has an argument — and what comes out is neither zero nor π.

So the quantity to test is the gap, not the phase. A phase off a quantum means either that the symmetry has been broken or that there was no band to begin with, and only the gap tells those apart. That is why the computation reports the gap beside every answer, and why the gapless row is in the figure rather than quietly excluded from the sweep.

There is a second reason the gapless point matters: it is where the two values change places. The phase cannot move continuously from zero to π while staying quantised, so a chain cannot pass from one value to the other without its gap closing somewhere. The two regimes are separated by a closing, and the closing is not an accident of this model — it is what having two allowed values and a continuous parameter forces.

There is a family resemblance to the degeneracies this ladder is otherwise about, and the difference is instructive. A glide sticks two levels together forces two energies to coincide at a zone boundary, and the forcing is visible in the band structure as a contact. Here nothing is visible in the band structure at all: the two regimes have gaps of the same size and dispersions of the same shape, and what distinguishes them is a property of the states that no plot of energies shows. That is the general lesson of the subject — a symmetry constrains more than the spectrum, and some of what it constrains has to be looked for on purpose.

The phase is a position #

The reading that makes all of this crystallography rather than topology is the one that turns the number back into a length.

γ/2π is the Wannier centre of the occupied band: the position within the cell that the band’s localised orbital is centred on, measured in units of the lattice constant and defined modulo one. Computing it for the two regimes gives 0 and 1/2.

Those are not arbitrary numbers. A one-dimensional cell has exactly two Wyckoff positions of site symmetry −1: the site at the origin and the bond centre at a half. A Wannier function of a band with inversion symmetry must be centred on one of them, because it must be symmetric about its own centre and there is nowhere else to be symmetric about. So the quantisation of the Zak phase, stated in the language of the band, is the statement that a symmetric orbital sits at a symmetric position, stated in the language of the cell.

That equivalence is worth holding onto, because it makes the result feel inevitable rather than exotic. Nothing here needs the word topology. It needs a cell, its site symmetries, and the observation that an orbital respecting a symmetry has to sit where the symmetry lets it.

A note about the localised orbital, since it has been introduced without being constructed. A Wannier function is the Fourier transform of the Bloch states over the zone, and it depends on the gauge chosen for those states — different smooth gauges give differently shaped orbitals. What does not depend on the gauge is where the orbital’s centre of charge sits, and that centre is the Zak phase. So “the electron sits here” is a statement with more content than the picture suggests: the shape of the thing sitting there is a choice, and its position is not.

What a convention moves #

The centre is a position in a cell, and where the cell is drawn is a choice.

Declaring that the second site belongs to the cell next door changes no distance and no energy. It changes the bookkeeping, and the centre computed from that bookkeeping moves — by half a lattice constant per cell of relabelling, which is the displaced site’s share of the band rather than a whole quantum.

So the value of the centre is a statement about a chosen cell, and the invariant statement is that it sits at a position an inversion centre fixes. This is not a defect. It is the same ambiguity that makes a crystal’s electric polarisation defined only modulo a quantum, and the reason a polarisation is reported as a change between two states of a material rather than as an absolute number.

One more consequence of the position reading is worth drawing out, because it says what happens at a surface. A chain in the topological regime has its electrons centred on bonds; cut the chain between two sites and the cut passes through the middle of an orbital, leaving a half-filled state at the end. A chain in the trivial regime has its electrons on sites and a cut passes between orbitals, leaving nothing. So the two values of the phase predict the presence or absence of an end state, and the prediction is about a boundary made from a quantity computed entirely in the bulk. That correspondence is the reason the subject is studied, and it follows here from the pictures of where the orbital sits rather than from anything harder.

What this says about a crystal #

Three things, and the second is the one used constantly without being stated.

The first is that the Zak phase is the electronic contribution to the polarisation. A crystal’s dipole moment per cell is not the dipole moment of a cell, because that depends on where the cell is drawn — the classic difficulty with defining polarisation at all — and the modern account replaces it by exactly this phase, with exactly this modulo. The quantum is one electron charge times a lattice vector divided by the cell volume, and it appears here as the 2π a phase is defined within.

The second is that a centrosymmetric crystal has zero polarisation, and the argument above is why. If the crystal has an inversion centre then every band’s phase is zero or π, so the total polarisation sits on a quantum — which is to say it is zero, since zero and a whole number of quanta are the same value. What a group forbids to happen makes the same kind of statement about transitions; this is it made about a ground-state property, and it is why ferroelectricity requires a polar class before anything about a material is known.

The third is about what a distortion does. A ferroelectric transition takes a centrosymmetric structure to a polar one, and the figure of the broken symmetry is the picture of what happens: the phase leaves its quantum continuously as the distortion grows, so the polarisation is proportional to the distortion at small amplitudes and is a genuine order parameter. The descent of symmetry is where that relation between a lost operation and a newly allowed quantity is set out generally.

The third of those is worth its own sentence. A quantity that is always zero is trivially quantised and proves nothing, so the sweep is required to produce both values — five chains at zero and five at π — before the quantisation counts as a finding. A model that only ever gave one of them would be consistent with the symmetry argument and would be no evidence for it.

There is one more crystallographic reading, and it is about which distortion a structure chooses. An order parameter is a representation makes the general point that a symmetry-breaking distortion transforms as an irreducible representation of the parent group. The staggered energy used above is exactly such a distortion — it is odd under the inversion it destroys — and the phase’s departure from a quantum is a quantity that is zero in the parent phase and non-zero below the transition, which is the definition of an order parameter rather than an analogy to one. So the polarisation is not merely permitted by the symmetry descent; it is one of the coordinates the descent is measured in. The crossing at the corner is where the same parent group’s constraints are read at a different point of the zone.

Where this stops #

The model is a two-site chain with nearest-neighbour hoppings, and it is a model. What it demonstrates is the structure of the argument — a symmetry, a quantity it constrains, two allowed values, and what happens when the symmetry goes — and nothing about a real material’s numbers follows from it. A real calculation replaces the two-by-two matrix by a self-consistent one and the sum over one band by a sum over all the occupied ones, and neither changes the argument.

Two dimensions and three are where the subject becomes large, and none of it is here. The Zak phase generalises to a Berry phase along each direction, the quantisation by inversion becomes a quantisation of each component, and the Wannier centres become a set of positions in the cell rather than one — which is the point at which the subject turns into the symmetry-based classification of band structures, and the point at which the site symmetries of the cell start doing the whole of the work.

A word about what the model leaves out that a reader may reasonably expect. Nothing here is quantum-mechanically self-consistent: the hoppings are given rather than computed from a charge density, so the chain does not decide for itself which regime it is in. A real material does, and the deciding is an energy comparison the symmetry argument has nothing to say about — the symmetry says which values are available, and the energetics say which one a material takes. Both halves are needed and only the first is here, which is the usual division of labour between a symmetry argument and a calculation.

And the phase computed here is for one band in isolation. Where bands cross or touch, the individual phases are not separately defined and only the total over a set of bands is, which is the multi-band version of the gapless row above. That is the usual situation in a real crystal and it is the reason the quantity people compute is a sum over the occupied bands rather than a phase for each.