Four signs say where the charge sits

The phase a symmetry turns into a number followed a band’s state once across the Brillouin zone of a chain with an inversion centre and found that it returns with a phase of exactly nought or exactly π, and nothing in between. Those two values turned out to be two positions: the charge of the band is centred on one of the two points of the cell that an inversion centre fixes. The chain’s symmetry, acting on its states, had located its electrons.

A plane crystal whose only symmetry beyond its translations is a centre — the plane group p2 — has four such points in each cell: the corner, the midpoints of the two edges, and the centre. p2’s half-turns sit on all four, and they are four different kinds of point, none carried to another by the group. So the question the chain answered has a sharper form here: on which of the four does a band’s charge sit?

The answer is read off the band at four wavevectors, and it needs one bit from each. At the four points of the zone that inversion maps onto themselves, a band’s state is either even or odd under inversion — a sign. Two comparisons of those signs give the two coordinates of the charge centre, a product of all four says whether there is a centre at all, and none of it requires building the functions that would show where the charge is.

The crystal, and why its centres are empty #

The crystal is built to make the question visible. Its square cell holds four atoms, a quarter of the way along each diagonal from the corner: at $(\tfrac14, \tfrac14)$, $(\tfrac34, \tfrac14)$, $(\tfrac14, \tfrac34)$ and $(\tfrac34, \tfrac34)$. An inversion through the corner carries each atom to another — the first to the fourth, the second to the third — so the arrangement has a centre, and no atom sits on any of the four inversion centres.

Electrons hop between neighbouring atoms. Along the first axis each atom has two neighbours, one across the line through the corner and one across the line through the cell’s middle, and the two hoppings can differ; the same along the second axis. One diagonal hopping, the same on every copy of that diagonal, keeps the centre and removes the mirror lines the square would otherwise have, so the group is p2 and nothing larger. Four versions of the crystal differ only in which bond is the stronger along each axis.

In each version the lowest of the four bands is separated from the others at every wavevector, so its states form a smooth family, and its charge has a well-defined centre — the average position of the Wannier function built from it. Where that centre falls is what the four signs are about.

Four signs at four points #

Inversion sends a wavevector $k$ to $-k$. Four points of the square zone are sent to themselves, up to a lattice vector of the reciprocal lattice: the zone centre Γ, the midpoints of its edges X and Y, and its corner M. At each of those, inversion maps the band’s state to a state at the same wavevector, and since the band is not degenerate there, it maps the state to itself times a sign — even or odd.

The rule is two comparisons. Where the signs at Γ and X agree, the charge centre’s first coordinate is nought; where they differ, it is a half. The signs at Γ and Y decide the second coordinate the same way. For the four crystals the patterns are $(+,+,+,+)$, $(+,-,+,-)$, $(+,+,-,-)$ and $(+,-,-,+)$ at Γ, X, Y, M, and they predict centres at the corner, the middle of one edge, the middle of the other, and the middle of the cell.

The reason is the chain’s reason, applied one line of the zone at a time. Along a line of constant second wavevector through Γ and X, the band restricted to that line is a band of a chain with an inversion centre, and its Zak phase — nought or π — is fixed by whether its two endpoint states have the same parity or opposite parities. That Zak phase, divided by 2π, is the charge centre’s first coordinate. Two lines through Γ, one along each axis, give both coordinates; the fourth point M is the check that the two lines tell a consistent story.

The same centres, from the Berry phase #

A rule that reads a position off four signs deserves a check that shares nothing with it. The charge centre of a band can be computed directly: carry the band’s state once across the zone along the first axis, multiplying the overlaps of neighbouring states, and the phase of the product — the Berry phase of that loop — divided by 2π is the first coordinate of the centre, on that line of the zone.

For all four crystals the Berry phases give the centres the parities predicted, to six decimal places, and they give them on every line of the zone, not only on the lines through Γ: the curves are flat. A flat curve is what a localised charge does — its position does not depend on which line is used to find it — and it is the reason the parities at four points are enough, since they sample two lines and the answer is the same on all of them.

The two computations have nothing in common but the crystal. The parities are four eigenvalues of a symmetry operator at four wavevectors. The Berry phase is a product of sixty overlaps round a loop, and it knows nothing about inversion; it would give a centre for a crystal with no centre at all, where the answer could be anywhere. Here it lands on one of four points every time, and on the point the signs named.

Where the lowest band lives, seen in its bands #

In the crystal whose lowest band is centred at the middle of the cell, the band is even at Γ and M and odd at X and Y. Γ differs from X and from Y, so both coordinates are a half. The band’s charge sits between four atoms, on the point where the strong bonds of both directions meet — which is where a chemist would have put it, a bonding combination centred on the square the strong bonds enclose. The parities have located a bond, not an atom, and they did it without being told where the atoms or the bonds were.

That is the useful content of the rule. A band built from orbitals on atoms can have its charge on those atoms or between them, and which it is has consequences — for the crystal’s polarisation, for where charge collects at a surface, for whether an edge carries states — that no count of electrons per atom reveals. The four signs reveal it, and they are the kind of number a calculation of the band structure produces at almost no cost.

Sixteen patterns, eight centres #

Four signs make sixteen patterns. The rule says what eight of them mean and is silent on the other eight.

The eight whose signs multiply to +1 each name one of the four inversion centres and one of two kinds of orbital sitting there, even or odd under inversion — four places times two parities, which is every way a single band’s charge can be localised on a point the group fixes. The sign at Γ is the orbital’s own parity, and the differences Γ–X and Γ–Y are its position.

The eight whose signs multiply to −1 name nothing. A band with that pattern has an odd Chern number, a quantity measuring how far its states twist across the whole zone, and a band with a non-zero Chern number cannot be gathered into localised functions at all. Time reversal forces the Chern number of every band to vanish, so in a crystal with time-reversal symmetry — any crystal without magnetism or an applied field — only the first eight patterns occur, and the degeneracy time reversal forces is the same symmetry closing off a different set of possibilities.

When the product is minus one #

The last eight are not a formality, and a crystal that breaks time reversal shows them.

Take one site per cell, at the inversion centre, carrying one even orbital and one odd one, and couple them in a way that breaks time reversal but keeps the centre. At one setting of the model the lower band’s parities are $(-,-,-,+)$ — product −1 — and its Berry phase, followed across the zone, does not stay flat: the charge centre moves steadily and arrives one whole cell from where it started. There is no single position to name, because the answer depends on the line of the zone used to find it and changes by a full cell across them. At another setting the parities multiply to +1 and the centre does not move. The four signs said which would happen before the loop was run, and the winding is the Chern number, one.

So the rule has two outputs, not one: a position when the product is +1, and a proof that there is no position when it is −1. Neither needed the band’s functions.

What the rule depends on, and cannot see #

It needs the band to be separate. At each of the four points the band must not be degenerate with another, or its parity is not defined; and across the zone it must be separated by a gap from every other band, or its charge centre is not defined either. The four crystals are built with gaps everywhere, and that is checked at the four points, but a real band structure with crossings needs the rule applied to the whole set of bands below a gap, where it gives the centre of their total charge modulo a cell.

It locates a centre to within the four points and says nothing finer. A charge centred on an inversion centre is all the rule can place, because that is all inversion can pin. A crystal with less symmetry lets the centre wander continuously, as the chain without its centre showed, and the parities then do not exist to read.

The Berry phase is computed on a discrete loop. Sixty wavevectors round each loop, with the overlaps multiplied in order; the phase that comes out is exactly independent of the arbitrary phases of the sixty states, because each state appears once as a bra and once as a ket, and it converges to the continuous value as the points multiply. Six decimal places at sixty points is the agreement quoted.

The crystals are models. Four atoms, a handful of hoppings, numbers chosen to make the four cases clean; nothing here is a material. What carries over is the arithmetic: the rule is a statement about any band of any crystal with the group p2, and the models are where it is checked.

And no figure shows a Wannier function. The rings mark where the charge is centred, which is a single number per coordinate computed from the Berry phase; the shape of the charge around that centre, and how far it spreads, are not computed and are not drawn.

The refusal is chosen for a reason. All four crystals have their lowest band even at Γ, and their centres are all different; a reading of the position from the zone centre’s parity alone would return one answer for four crystals. The position is in the differences, and only the differences.

A symmetry that places electrons #

This is the most concrete use yet of a group acting on functions. The little groups at the four points — the star of each wavevector and the operations that fix it — each contribute a representation, here a single sign; those signs, compared, say where in real space the band’s charge must be. The wavevectors are in reciprocal space and the answer is in the cell, and the bridge between them is the fact that a localised charge on an inversion centre transforms, at each of the four points, with a sign fixed by where it sits.

The same logic runs in reverse, and that is its practical form. Each inversion centre, with an even or odd orbital on it, produces a definite pattern of four signs; the four Wyckoff positions of p2 with their two orbital parities produce eight patterns; and a band whose signs match none of the eight — whose product is −1 — cannot be any combination of localised orbitals on any of them. The census of what localised charges produce, set against what bands actually show, is how a band that is not atomic in origin announces itself.

Where the rule comes from #

The four signs are the smallest case of a computation already made at larger little groups. Where two levels must meet reads the representations of the little group at each special point and finds levels forced together where a representation is two-dimensional; here every little group at the four points is inversion alone, every representation is a sign, and nothing is forced together — but the signs still carry information, because they are fixed by where the band’s charge sits. Which levels join which then follows a band between special points by restricting representations, and it is the same restriction, along a line through Γ and X, that makes the pair of signs into a Zak phase. The crossing at the corner is the case where the representation at a point is two-dimensional and no band can be separated at all, which is exactly when this rule has nothing to read.

The pieces have owners. That the Berry phase of a band is its charge centre is Zak’s, from 1989, and the modern theory of polarisation built on it by King-Smith and Vanderbilt in 1993. That the parities at inversion-invariant points fix the Berry phase along a line, and that their product over all four points gives the parity of the Chern number, was set out for inversion-symmetric insulators by Hughes, Prodan and Bernevig and by Turner, Zhang, Mong and Vishwanath around 2011. The general form — tables of the representations every localised orbital on every Wyckoff position produces, against which a band can be tested — is the theory of band representations, begun by Zak in the 1980s and turned into complete tables for all the space groups in 2017. The two-orbital model with a winding band is Qi, Wu and Zhang’s, from 2006. What is computed here is the plane case checked two ways at once, on crystals built so that the answer is visibly not an atom.

Still open: more than one band, and more than one centre #

The rule was applied here to one band at a time. The electrons of a real crystal fill several, and the quantity that matters is the centre of all their charge together, which is the sum of the individual centres modulo a lattice vector. The parities of a set of filled bands, multiplied point by point, give that sum’s position among the four centres — but they cannot tell two bands centred on the same point from two bands centred on points that add to it. A crystal with two filled bands, one on the corner and one on the middle of the cell, has the same total parities as one with both on the middle of an edge and its twin; telling them apart needs more than inversion.

And the other plane groups have more to say. A group with rotations of order three, four or six has more points of special symmetry in its zone, larger little groups than a sign, and more Wyckoff positions to place charge on. The same census — every localised orbital on every special position, and the pattern of representations it produces at every special wavevector — is the classification of what those groups’ bands can be, and it would say, group by group, which band patterns are atomic and which are not.