Two ways of being second order

Putting a second band beside a band tests whether the closed form for the first one’s shape survives. It does, exactly, for as long as nothing connects the two — and once something does, the shape departs from it at a rate that is second order in the coupling. That is the answer to the question, and it comes with a complaint attached: a departure that is second order in the ratio of the coupling to the separation is a measurement of the ratio, so knowing it leaves a whole curve of systems consistent with what was seen.

The obvious repair is to measure something else. The obvious objection to the repair is that everything available is second order in the same small quantity, so a second number of the same order looks like the same number again.

This essay is about how badly wrong that objection is, and about the one digit that makes it wrong.

Two numbers, and the exponents they carry #

The system is the standard one: a square net whose every site carries two orbitals, the lower set at $-\Delta/2$ and the upper at $+\Delta/2$, each hopping to its own kind with its own strength, and the two kinds hopping to one another across a bond with strength $t_\perp$. Four parameters, and the whole of the argument is about what a measurement can recover of them.

Two of the numbers a measurement supplies are first order and easy. Each band’s width is its own hopping times the width of the underlying structure’s band, and it is blind to everything else — which is why an angle-resolved measurement reports hoppings and nothing anybody argues about.

The two interesting numbers are the ones the coupling produces, and neither exists at all when $t_\perp = 0$:

• the shape departure — how far the lower band’s fourth-moment ratio has been pushed from the value it takes when the two bands do not talk, which for this structure is exactly $2.25$;
• the excess gap — how much wider the gap between the two bands is than the four band edges alone would put it.

Both vanish quadratically. Neither is a measurement of $t_\perp$ on its own, because neither can be interpreted without knowing $\Delta$.

What separates them is one power, and the way to see it is to stop reasoning about them and differentiate them. The matrix of $\partial \ln(\text{number}) / \partial \ln(\text{parameter})$ is a table of exponents: a cell reading 2 says this number goes as the square of this parameter, and every entry is a central difference taken at two step sizes that have to agree before either is used.

At a coupling of 0.03 and a separation of 12 the two rows come out at

$$\text{shape} \sim t_\perp^{1.989}\,\Delta^{-2.038}, \qquad \text{excess} \sim t_\perp^{2.000}\,\Delta^{-1.036},$$

which is $(2, -2)$ and $(2, -1)$ to the accuracy a finite difference on a diagonalisation supports. The shape is a function of $t_\perp/\Delta$ and of nothing else; the excess gap is a function of $t_\perp^2/\Delta$.

Both are second order in the coupling and they are second order in different variables. That is the sentence the rest of this essay is a consequence of.

Why the difference is there, and not an accident #

Second-order perturbation theory pushes each level of the lower band down by a sum of $|t_\perp|^2$ divided by an energy denominator. The denominator is the separation between that level and the levels it is being pushed by, which is $\Delta$ up to the band widths.

The gap is the difference between two particular levels — the top of the lower band and the bottom of the upper — so it inherits one factor of $t_\perp^2/\Delta$ directly. It is an energy, and it is measured in the same units as $\Delta$.

The shape is a ratio of moments, and a ratio is dimensionless. Whatever the perturbation does to the fourth moment, the answer has to be divided by the square of the second, and the width the second moment carries is set by the hopping rather than by the separation. So the departure arrives as $t_\perp^2/\Delta$ divided by an energy that is itself of order $\Delta$ — the pushing is uneven across the band, and how uneven it is depends on how the denominator varies across the band, which is a fraction of $\Delta$ rather than a fraction of the width.

A dimensionless number and an energy, both built from the same second-order shift. It is not surprising once written down, and the two-band shape calculation has both numbers in front of it without noticing, because both were reported as second order in the coupling and that description hides the difference exactly.

The quotient in which the coupling cancels #

If one number goes as $t_\perp^2/\Delta^2$ and the other as $t_\perp^2/\Delta$, their quotient goes as $\Delta$ — with the coupling cancelling exactly rather than approximately.

That is a strong claim and it is cheap to check. Computing both numbers across a factor of three in the separation, at three different couplings, and dividing:

Each row is one constant. The constant is the same to a part in seventy across the two weakest couplings and has drifted by six per cent by the strongest, which is the next order in $t_\perp$ arriving — exactly where an expansion in the coupling says it should.

So the separation can be read off a pair of measurements with no fitting anywhere. Divide the excess gap by the shape departure, divide by 4.23, and the answer is $\Delta$. The coupling then follows from either number alone, because the thing that was blocking it — not knowing the separation — has been removed.

This is worth stating in the negative, because that is the form in which it is useful. There is no experiment on this system that measures $\Delta$ directly at second order; there is a pair of measurements neither of which contains it and whose quotient is proportional to it.

What two one-per-cent measurements leave #

A quotient is an answer to can it be done. The useful question is how well, and that is the covariance of the inversion rather than the inversion itself.

Take both numbers known to one per cent — an honest precision for a band shape read off a photoemission spectrum, and a generous one. Propagating that through the exponent matrix gives the region of parameter space still consistent with what was seen:

• the coupling to 1.13 per cent;
• the separation to 1.40 per cent.

The amplification is a factor of about 1.4, which is what a condition number of 6.38 buys once it is divided among two well-separated directions. Two per cent measurements of two dimensionless-and-not numbers pin a coupling that never appears alone in either of them to a per cent and a bit.

The pair that fixes nothing, and gets worse as the model gets better #

Now the control, and it is the part of this essay worth carrying away.

The upper band has a shape too, and it departs from $2.25$ in the same way for the same reason. Measuring it is no harder than measuring the lower band’s — arguably easier, since it is the empty one. It is a second measurement, of a second quantity, of the same system.

It is worth almost nothing. Its exponents are $(2.017, -2.038)$ against the lower band’s $(1.989, -2.038)$: the same function of the same ratio. Its level curve in the parameter plane lies along the first one rather than across it, and a pair of measurements whose curves are parallel fixes a direction and not a point.

With the same one per cent precision on both:

The confidence region is thirty-six times larger in area, from two measurements of the same precision on the same system, differing only in which two numbers were read off it.

And the direction it is spread along says what has gone wrong, which a size alone would not. It runs along $t_\perp$ and $\Delta$ rising together — the direction that leaves their ratio unchanged, which is precisely the direction both observables are blind to. That is a degeneracy rather than an imprecision, and the distinction matters: no amount of averaging removes it.

The last column of that figure is the one that is genuinely strange on first reading. The same-kind pair’s condition number is 12.5 at a coupling of 0.15, 27.3 at 0.1, 42.2 at 0.08, 106.2 at 0.05 and 292.6 at 0.03.

It gets worse as the coupling gets weaker. Every intuition about measurement says the opposite: a small perturbation is a clean one, second-order perturbation theory is more nearly exact, and the model being fitted is more nearly true.

All of that is correct and it is the reason. In the limit where second-order perturbation theory is exact, the two band shapes are the same function of $t_\perp/\Delta$ — not similar functions, the same one. The only thing distinguishing them is the fourth-order remainder, which is what a stronger coupling supplies. So the pair is separable to exactly the extent that the expansion the experiment is being interpreted in has started to fail, and it becomes exactly singular in the limit where that expansion is perfect.

A pair of measurements that gets less informative the better the model holds is the signature of a degeneracy, and it is worth naming because the diagnostic runs the other way from every rule of thumb about noise. A fit whose parameter errors shrink when the data are taken further into a regime where the model is doubtful has not found a better measurement. It has found a place where two of its observables have stopped being copies of one another, and it is buying its precision with the model’s own error.

All four, from four numbers #

The two easy numbers were set aside at the start and they are what makes the whole inversion practical rather than a curiosity.

Each band’s width reads its own hopping with an exponent of $0.999$ and reads every other parameter at $10^{-3}$ or below. So the two widths fix the two hoppings on their own, and the two second-order numbers are left carrying the two parameters that are hard. The four-by-four inversion comes out at a condition number of 12.6, and one per cent measurements of all four numbers fix

$$t_\perp \text{ to } 1.54\%, \quad \Delta \text{ to } 2.56\%, \quad \text{and each hopping to } 1.00\%.$$

A condition number of twelve on a four-parameter inversion is not a triumph of numerical analysis; it is what happens when a set of observables has been chosen so that two of them are first order in different things and the other two are second order in different things. The choice of what to measure is doing more work here than any amount of care in the fitting, which is the same conclusion a parameter that never finds a value reached from the other end — there, a third parameter had no interior optimum because nothing in the data responded to it.

What is quoted, and what is computed #

Nothing here is quoted. There is no measurement here at all: every number is a property of the map from parameters to observables, and the one per cent is a stated hypothesis about an experiment rather than a report of one.

That is the honest shape for this question and it is worth saying why. An uncertainty computed from a Jacobian is knowable before any data exist — it depends on the model and on the point, not on what came back — which is exactly when it is worth knowing, because it is the stage at which the choice of what to measure is still open.

The exponents are central differences taken at two step sizes a factor of two apart, and a derivative whose two estimates disagree by more than five parts in a thousand is refused rather than returned. That check is not decoration: the first version of the inversion used a single step of $10^{-6}$, which is far too small for a quantity obtained by diagonalising a matrix, and returned a determinant of the wrong sign.

The condition numbers are the ratio of the largest singular value of the exponent matrix to the smallest, and the singular values come from the eigenvalues of its Gram matrix — computed by the Jacobi method used for every spectrum here.

What this cannot say #

A condition number is not an error bar. It says how a stated measurement precision is amplified; it says nothing about whether the measurement is unbiased, whether the model is the right one, or whether the band a spectrum shows is the band this Hamiltonian has. A model that is wrong can be beautifully conditioned.

The precision is linearised. Everything here is the local behaviour of the map at one point, so a one per cent region is honest and a fifty per cent one — the same-kind pair’s — is a statement that the region is large rather than a description of its shape. At fifty per cent the linearisation has stopped being a description, which strengthens the conclusion rather than weakening it: the pair is worse than the number says.

The two orbitals are of the same parity. They have to be, for the inter-orbital hop to be a real symmetric matrix element that this site’s checked eigensolver can take, which means the model is not a σ set and a π set — between those the coupling is exactly zero and there is nothing to measure. It is a model of two bands that mix, and not of any particular pair.

And the excess gap needs a reference. It is measured against the four uncoupled band edges, which are written down rather than computed a second time with the coupling switched off. That is deliberate — a stabilisation is measured from somewhere is this collection’s standing warning about references — but it does mean the excess is only as good as the claim that the widths are the uncoupled widths, and at a strong coupling they are not.

What the inversion requires #

The two second-order numbers come out at $(2, -2)$ and $(2, -1)$ to within six hundredths of an exponent, at a coupling weak enough for the expansion to hold.

Their quotient is one constant times the separation, across a factor of three in it, to a fifth of a per cent at the weakest coupling tested.

The mixed pair inverts and the same-kind pair does not: one per cent measurements fix the separation to 1.40 per cent one way and to 51 per cent the other.

The same-kind pair’s conditioning is monotone in the coupling, and monotone the wrong way — checked as a sequence rather than at a point, because a determinant evaluated once would have shown a non-zero number and said nothing.

And a model built to be degenerate is refused. Two observables constructed as $p_1p_2$ and $(p_1p_2)^3$ return nothing determined rather than a large uncertainty, together with the direction they cannot see — which comes back as $(1, -1)/\sqrt{2}$ rather than as merely small. A procedure that reports $10^{17}$ where it should report that nothing is determined is how an underdetermined fit gets published.

Still open: the overlap threshold, and two different nets #

The obvious open question is the third parameter set aside here by symmetry. The two hoppings were treated as independently measurable because each band’s width reports its own, and that is true only while the bands are far enough apart for its band to be identifiable — the width of a band that overlaps another one is not a measurable quantity at all, and whether two bands overlap is a question about four numbers has an answer. Running the same exponent matrix across the overlap threshold would say where the easy half of the inversion stops being easy, and the answer is a boundary in the parameter plane rather than a number.

The nearer question is what happens to the quotient when the underlying structure is not the same for both bands. Everything here has one net carrying two orbitals, so both bands are the same shape scaled by their own hopping and the uncoupled fourth-moment ratio is one number for both. A real pair of bands need not be — a σ set on a chain-like path and a π set on a different one have different coordinations, hence different second moments and different shapes before anything mixes. The constant 4.23 would then be two constants, and whether the quotient still cancels the coupling is a question with a definite answer that the same calculation reaches by changing which graph the upper set is built on.