Creasing Changes the Field

What a crease does that a translation cannot

Move a flat sheet across the room and nothing about it changes. Bend it and three distinct things happen at once, and they are worth keeping apart, because they are usually collapsed into the single word strain.

First, the bond lengths change. Carbon–carbon distances on the outside of the fold are stretched and on the inside compressed, which changes the hopping integral between neighbouring orbitals — the number that sets how easily an electron moves from one atom to the next. Second, the orbitals themselves tilt out of the plane, so the clean separation between in-plane σ bonds and out-of-plane π bonds is no longer clean; rehybridisation gives the surface a local dipole and a shifted work function. Third, the bend imposes a strain gradient through the thickness of a sheet one atom thick, which is the geometric content of what gets called quantum orbital flexoelectricity.

Only the first of these is captured by the model in the panel below, and it is the one with the cleanest mathematics.

Strain enters the equation as a magnetic field that is not there

Near its Dirac points, graphene’s low-energy electrons obey a two-dimensional Dirac equation. When the lattice is strained, the modulated hopping integrals do not appear in that equation as a potential. They appear in exactly the place a vector potential appears:

with β ≈ 3 the dimensionless hopping modulation parameter and a = 0.142 nm the bond length. Because it sits in the vector-potential slot, its curl behaves like a magnetic field:

This is the part worth slowing down for. B_s is not a magnetic field. No current makes it, no magnet supplies it, and it does not break time-reversal symmetry — it points the opposite way for electrons at the other Dirac point, which is why the sheet as a whole stays time-reversal symmetric while each valley sees a field. But electrons in one valley respond to it as though it were real: Landau-like levels, deflection, pseudo-Hall response. The geometry is not producing a metaphor for a field. It is producing the term that a field would have produced.

And note what the formula demands. B_s is the derivative of the strain, not the strain. A uniformly stretched sheet — however hard you pull — has no pseudo-field at all. Only a varying strain has one. Which means the field lives where the geometry changes, and a crease is nothing but a place where the geometry changes.

The crease, done as arithmetic

Take a single crease as a Gaussian ridge of height h and half-width w, and let a clamped sheet take up the excess length as in-plane strain:

Everything else follows from that one profile without a further assumption. The apex curvature of a Gaussian ridge is κ = h/w², so the radius of curvature at the top of the fold is R = w²/h — sharpness is set by the square of the width against the height, which is why a slightly tighter crease does far more than a taller one. And because B_s goes as z′z″, it is odd about the crease centre: the two flanks of a single fold carry opposite pseudo-magnetic fields, and the crest between them carries none.

Three regimes, and which one you are in

Push the sliders and the panel changes character three times, which is the real content of the exercise.

Gentle and wide. Small h, large w: strain is a fraction of a percent, the pseudo-field is tens of tesla, and the sheet is still developable — it can be flattened back without stretching. Bond lengths are barely touched. What has changed is the embedding, not the metric.

Sharp. Shrink w and the apex radius collapses as w²/h. Now three things arrive together: hundreds of tesla of pseudo-field on each flank, an electrostatic enhancement at the apex because field lines crowd where curvature is high, and orbital rehybridisation that the elastic model does not know about. This is the regime where a crease is a device: a line along the sheet with different chemistry, different work function, and different transport from the flat material a nanometre away.

Many creases, close together. Add folds and bring them within a few nanometres. The strain fields overlap, the pseudo-fields of adjacent flanks meet, and the pattern — not the individual fold — becomes the object. A periodic array of creases is a superlattice, and superlattices open gaps and fold bands. The sheet has acquired a new spectrum from geometry alone.

What survives flattening, and what does not

The useful discipline here is a single question asked of every effect: does it survive isometric flattening? A crease that can be smoothed out without stretching has zero Gaussian curvature. Everything that dies when you flatten it was a fact about how the sheet sat in space — mean curvature physics, the flexoelectric dipole, the field enhancement at the apex. Everything that requires actual stretching is metric physics, and by Gauss’s theorema egregium a region of nonzero Gaussian curvature cannot be flattened without it. Those are the regions where the pseudo-field is not an artefact of a clamped boundary condition but a consequence the sheet cannot escape.

So the honest version of “changing the formation changes the electromagnetics” is not one claim but a fork. Some of the change is embedding — real, measurable, reversible, and gone the moment the sheet relaxes flat. Some of it is metric — locked in by curvature that no amount of unbending can remove, absent tearing. A great deal of loose talk about topology in two-dimensional materials is the first thing wearing the name of the second.

Why this is the same question

An infinite uniformly charged sheet gives a field that never fades and carries no information: one number at every point. Charge it only in patches and the field acquires a direction, but only at the seams. A flat graphene sheet is the electronic version of the same emptiness — perfect symmetry, nothing happening, one hopping integral everywhere. The crease is the seam.

In both cases the structure is not distributed across the surface and it is not located in the material. It is at the boundary between two regions that differ, and the mathematics says so explicitly: the pseudo-field is a derivative, and a derivative is a statement about difference. Uniformity has no derivative. This is the sense in which a boundary is not where a thing ends but the only place a thing is generated — and here it is not a figure borrowed from somewhere else. It is what the gradient in the formula means.