The domain in reciprocal space

A fundamental domain is a piece of the plane containing one point of every orbit: repeat it by the group and the whole pattern comes back, with nothing drawn twice. This collection has built them for all seventeen plane groups, counted their walls, and found that the share of the plane one occupies is one part in the group’s order and never quite that — because the points on symmetry elements are counted more than once.

The same idea in reciprocal space is the one every calculation over a crystal is built on, and it has a name of its own: the irreducible wedge. A symmetric operator does the same thing at every member of a star, so a sweep over the zone can be replaced by a sweep over one representative of each.

The wedge is a fundamental domain, and not a special construction #

The point group acts on the reciprocal lattice, so it acts on the zone, and the wedge is a fundamental domain for that action. Everything the domains essays established applies unchanged: it contains one point of each orbit, its share is one over the group’s order up to the boundary, and its walls are where the group’s operations act.

That is worth saying because the wedge is usually introduced as a piece of computational folklore — a triangle with corners at Γ, X and M, memorised per lattice type — rather than as the same object the direct-space essays built. It is the same object, and reading it that way makes its properties obvious rather than conventional.

Measured on a grid rather than drawn as a region #

The wedge here is not drawn as a shape. It is computed as a set: take a grid of wavevectors, group them into stars, keep one representative of each.

That produces a count rather than an area, which is the quantity a calculation actually cares about — how many wavevectors it must visit — and it avoids the whole question of which boundary points belong to which wedge, a question with a conventional answer rather than a mathematical one.

The identity being checked is that the star sizes add to the size of the grid. Every wavevector lies in exactly one star; the stars partition the grid; and a modular-arithmetic mistake in the action would break the partition immediately.

Why the share exceeds one over the order #

A count of representatives divided by the size of the grid comes out larger than one over the group’s order, on every group, and the reason is the boundary.

A wavevector fixed by some operation has a short star — the orbit–stabiliser theorem — so it is not divided down as far as a naive count assumes. The wavevectors on the mirror lines and at the special points are exactly those, and they lie on the wedge’s boundary.

For p4m on a grid of sixths, the count is ten of thirty-six, which is 27.8 per cent against an ideal of 12.5. The excess is large because the grid is coarse and its boundary is a big fraction of it; on a finer grid the share falls towards the ideal without ever reaching it.

That behaviour is the same one the direct-space essay measured for fundamental domains: the share is one part in the order plus a boundary term, and the boundary term is where all the interest is.

Why the reduction is legitimate, in one line #

It is worth stating the justification rather than assuming it, because the wedge is used so routinely that the reason is often left out.

Let H(k) be the finite problem at a wavevector, and let g be an operation of the point group. The Bloch operators give a unitary map carrying H(k) to H(g·k), so the two matrices are related by a change of basis: same eigenvalues, same multiplicities, same everything a calculation asks about them.

So the answer at g·k is the answer at k transported, and computing both is computing one thing twice. That is the entire argument, and it holds for any operator commuting with the group rather than for any particular one.

What the argument does not give is the eigenvectors at g·k for free in a useful form — they are the transported ones, and transporting them costs about as much as recomputing them for a small problem. So the saving is real for eigenvalues and integrals over the zone, and thinner for anything that needs the states themselves at every point.

What it buys, in numbers #

The saving is close to the order of the point group, and for a large group that is a large factor.

A p6m-symmetric calculation visits about a seventh of a grid rather than all of it; a p4m one about a quarter; a p1 one visits everything, because nothing is a copy of anything. In three dimensions the factors are correspondingly larger — up to forty-eight for a cubic crystal — which is why the wedge is not an optimisation but a precondition for the calculation being possible at all.

And the saving is exact: the wavevectors not visited are not approximated, skipped or interpolated. Whatever the operator does at one member of a star, it does at every member, so the discarded points carry no information at all.

The grid’s own arithmetic #

The share depends on the grid as well as on the group, and the dependence is arithmetic rather than a matter of resolution.

A grid of N × N wavevectors contains the fractions with denominator dividing N. Which of those are fixed by an operation depends on N’s divisors: a grid of sixths contains the wavevectors of halves and thirds, so it contains the corners and edge centres of both square and hexagonal zones; a grid of fifths contains neither.

So a grid whose size shares no factor with the interesting denominators has no special wavevectors, every star is full length, and the share is as close to one over the order as it can be. A grid through the special points has short stars and a larger share.

That is the same divisibility statement the box essay makes about which wavevectors exist at all, and the two together are the whole of what arithmetic contributes to choosing a sampling scheme.

The boundary, which is where the conventions live #

Every account of the wedge has an awkward paragraph about its boundary, and this collection can be unusually direct about why.

A wavevector on a mirror line is fixed by that mirror, so it belongs to a star of half the usual size; the wedge must contain it once, and the two halves of the boundary that the mirror exchanges must not both be included. Which half is a convention, exactly as the choice of which faces of a direct-space fundamental domain to include is.

The counting done here sidesteps it: representatives are chosen by a canonical rule over the grid, one per star, and no region is drawn. So the count is convention-free even though a picture of a wedge would not be — which is why the figures show which grid points survive rather than shading a triangle.

The three fundamental domains this collection now has #

It is worth putting the family together, because the word has been used three times here for three different actions and the differences are instructive.

A fundamental domain of a plane group acting on the plane. One piece of the pattern; repeated by the whole group, translations included, it fills the plane. Its area is the cell’s area divided by the point group’s order, and the walls hand back the generators.

An asymmetric unit. The same object as used by the International Tables, chosen as a box for convenience rather than for its walls — one point of each orbit under the space group, with the boundary conventions written out because a box’s faces do not correspond to operations.

The irreducible wedge. One point of each orbit under the point group acting on the zone. The translations play no part, because they act trivially on wavevectors; what remains is a finite group acting on a finite set, which is the simplest of the three.

The third is the easiest and is usually presented as the hardest, because it arrives with a picture of a triangle and a list of corner labels rather than with the sentence “a fundamental domain for a finite group acting on a torus”.

What a wedge cannot save #

Two things the reduction does not help with, and they are worth knowing before a calculation is designed around it.

Anything that is not invariant under the group. A field applied to the crystal, a strain, a defect — each lowers the symmetry, and the wedge of the lowered group is larger. A calculation of a crystal’s response to something that breaks its symmetry has to use the subgroup’s wedge, which for a general perturbation is the whole zone.

Quantities that mix wavevectors. An interaction coupling two wavevectors is not decided by the star of either. The reduction applies to problems that decompose by wavevector, which is exactly the problems where the translations are a symmetry; anything that spoils that decomposition spoils the saving with it.

Both are ordinary and both are the same statement: the wedge is a fundamental domain for a group, so it saves exactly as much as that group is a symmetry of the problem, and not a step more.

Where a calculation puts its points #

The wedge decides how many wavevectors a calculation visits; it does not decide which, and the two questions are usually confused.

A calculation on a finite box has its wavevectors chosen for it: the box of N × N cells holds exactly the fractions with denominator dividing N, and no others exist to be visited. The wedge then reduces that fixed set by the group.

A calculation choosing its own grid — for an integral over the zone, say — chooses a spacing and an offset, and both matter. An offset grid can avoid the special points entirely, which makes every star full length and the reduction as efficient as possible, at the cost of never sampling the points where the symmetry actually forces something. A grid through the special points samples them and pays for it in a larger wedge.

That trade-off is the whole of the practical literature on sampling schemes, and it is decided by exactly the arithmetic in this essay: how many stars a given grid has, and how many of its points are fixed by something.

A count that is not an area #

One more distinction, because it is the difference between the figures here and the triangles in a textbook.

A wedge drawn as a region has an area, and the area is one over the group’s order exactly — no boundary correction, because a boundary is a set of measure zero and contributes nothing to an area.

A wedge computed as a set of grid points has a count, and the count exceeds one over the order, because the boundary points are a positive fraction of a finite grid.

Both are right and they answer different questions. A calculation integrating over the zone with a continuous method wants the area; a calculation visiting grid points wants the count. Quoting the first while doing the second is the commonest way of under-estimating how much work a sampling scheme is, and the discrepancy is largest exactly where it matters — on the coarse grids a first calculation uses.

The same theorem, five times now #

It is worth listing where this collection has now used the orbit–stabiliser theorem, because the wedge is the fifth and the list is a good summary of what the theorem is for.

A form is an orbit of a face; its size is how many faces appear together. A Wyckoff position is an orbit of a point; its size is how many atoms one independent atom accounts for. A reflection’s multiplicity is an orbit in reciprocal space; its size is how many measurements one number gets. A star is an orbit of a wavevector; its size is how much of a sweep is a copy. And a fundamental domain’s share is one over the order, corrected by the points with stabilisers.

Five counts, five fields, one theorem. That is the sort of accounting this collection exists to make legible, and each of the five was arrived at independently in its own literature before anybody noticed the others.

One number a reader can carry #

If a single fact from this rung is worth remembering, it is the ratio rather than the construction.

A calculation over a crystal with a point group of order n does about 1/n of the work an unsymmetric one would — a factor of eight for a square lattice’s full group, twelve for a hexagonal one, forty-eight for a cubic crystal. That factor is why symmetry is not a decoration on computational crystallography but the thing that makes it affordable, and it is bought entirely by the observation that a symmetric operator does the same thing at every member of a star.

The correction to that ratio — the boundary, which makes the real share larger — matters at the coarse grids a first calculation uses and vanishes as the grid refines. Quoting 1/n while working on a grid of sixths under-states the work by a factor of two, and quoting the measured share is the honest version of a familiar claim.

Where this goes #

The unfinished part is three dimensions, where the wedges are the ones every electronic-structure calculation uses and where their boundaries are surfaces rather than lines. The arithmetic is identical and the bookkeeping is heavier.

The nearer neighbour is the box, which supplies the set of wavevectors the wedge then reduces — and between them they answer the two practical questions a calculation on a crystal has to settle before it begins: which wavevectors exist, and which of them have to be visited.

Every representative carries a weight #

The share measured here is a count of representatives, and the count is only half of what a calculation needs. The other half is already computed and is easy to drop.

A representative stands for its whole star. If a wavevector’s star has eight members, the answer computed there is the answer at eight points of the grid, and any quantity averaged over the zone must count it eight times.

So the average is weighted, not plain. Add the values at the representatives with each multiplied by its star size, and divide by the size of the grid. Add them plainly and the result is wrong — and wrong in a specific direction, because the representatives with short stars are exactly the high-symmetry ones, so an unweighted average over-counts the zone’s most special points.

The size of the error is the size of the discrepancy this essay measures. For p4m on a grid of sixths the wedge holds ten of thirty-six wavevectors, so an unweighted average weights each of the ten equally at a tenth, while the correct weights range from one part in thirty-six for a point fixed by the whole group up to eight parts in thirty-six for a general one. The two averages are not close.

The star sizes are therefore not a by-product of the reduction. They are half of its output. A wedge without weights is a list of points with no way to use it, which is why the count and the star sizes are computed together above rather than the count alone.

The operation the point group does not contain #

There is one more symmetry acting on the wedge, it is not in the crystal’s point group, and leaving it out is the commonest way a calculation visits twice as many wavevectors as it needs.

Reversing time reverses a wavevector. A state travelling one way becomes a state travelling the other, with the same energy, so the levels at $\mathbf{k}$ and at $-\mathbf{k}$ agree — in any crystal without magnetic order, whatever its point group.

So the effective group for the band energies contains the inversion, whether or not the crystal does. For a centrosymmetric crystal this changes nothing, since the inversion was already there. For a crystal without a centre it halves the work again, and the wedge that a purely spatial argument computes is twice as large as the one the calculation needs.

The extra symmetry is not a symmetry of everything. It relates energies, and it relates them because energies are real numbers unchanged by conjugation. Quantities that are odd under time reversal — a current, a magnetisation — are not equal at $\mathbf{k}$ and $-\mathbf{k}$, and a calculation that reduces its sampling by this factor and then averages such a quantity has folded together two contributions that should have cancelled.

And it interacts with how the grid is chosen. Sampling grids are often offset from the zone centre to improve convergence, and an offset that is not itself carried onto the grid by the point group destroys the star structure this essay counts — leaving a calculation that thinks it has a wedge and has instead a wedge of the smaller group its shifted grid actually possesses. The offsets that are safe are the ones the group permits, which is the same compatibility question the wedge itself is an answer to.