The transition with nothing to order

A transition has an order parameter: a quantity that is zero in the disordered phase and not in the ordered one, whose vanishing defines the transition and whose exponents classify it. That is the standard account, and in two dimensions it cannot apply.

The Mermin–Wagner theorem says a two-dimensional system with a continuous symmetry has no long-range order at any temperature above absolute zero. Long-wavelength fluctuations cost too little energy, there are too many of them, and they destroy the alignment however weak they are — the same counting of modes that decides how many ways there are to vibrate, with the answer going the other way.

So a thin film of superfluid helium is not ordered at any temperature. It is nevertheless superfluid below about a kelvin and not above, and the changeover is sharp.

What is actually different on the two sides #

The thing that changes is not the alignment of anything. It is whether the film’s vortices come in pairs.

A vortex in a superfluid carries one quantum of circulation, and in two dimensions it is a point defect rather than a line. Around it the flow speed falls as one over the distance.

The energy of a single vortex is therefore the integral of the square of that speed over the area, which diverges — logarithmically:

$$E = \pi J \ln\frac{L}{a}$$

for a system of size $L$ and a core of size $a$. In a large film that is a large energy, and a single vortex looks forbidden.

It is not, because the entropy diverges logarithmically too. A vortex can be placed anywhere, and the number of places is $(L/a)^2$, so

$$TS = 2kT\ln\frac{L}{a}.$$

Two logarithms, and the free energy is their difference.

It is worth being explicit about what “topological” is doing in the phrase “topological transition”, since the word is used loosely elsewhere. A vortex cannot be removed by any smooth local change of the field: the phase winds by a whole turn round it, and a whole number cannot be changed a little. So a vortex is a defect that the configuration space itself forbids removing, and the only way to get rid of one is to bring it together with an anti-vortex or to take it out through the boundary. That indestructibility is what makes counting them a sensible thing to do, and it is why the transition is about a number rather than about a magnitude.

A sign, rather than a size #

$$F = (\pi J - 2kT)\ln\frac{L}{a}$$

The bracket changes sign at $kT = \pi J/2$. Below that temperature a single vortex costs an unbounded amount of free energy in a large system and cannot appear on its own; above it a vortex is free, and they appear in whatever numbers they like.

What can appear below the transition is a pair of opposite circulation. Far away their flows cancel, the logarithms cancel with them, and the pair’s energy depends on their separation rather than on the size of the system. So the low-temperature phase is not free of vortices — it is free of unpaired ones.

Kosterlitz and Thouless’s argument, in 1973, is that paragraph. It is a competition between two quantities that both diverge in the same way, and everything about the transition follows from the fact that they diverge at the same rate.

That rate is what makes it two-dimensional. In three dimensions a vortex is a line, its energy grows with the length of the line, and its entropy grows with the same length; the competition still exists but the numbers differ. In one dimension there are no vortices at all. The logarithms are a two-dimensional accident and the transition is one too.

What survives the argument being too crude #

The free-energy estimate is for a single vortex in an otherwise empty system, and it ignores the pairs that are already there — which screen the interaction between any new pair, weakening it, and making unbinding easier than the crude estimate says.

Doing that properly means asking how the stiffness and the vortex density look when the system is examined on larger and larger scales, which is a renormalisation-group calculation of the kind universality rests on and produces two coupled equations. They have a conserved quantity, so their trajectories are hyperbolae, and the picture is unusually clean.

Trajectories on one side of a diagonal flow down to the horizontal axis: the vortices become irrelevant at large scales, and the stiffness settles at whatever value the flow left it. Trajectories on the other side flow up and away: the vortices proliferate, and there is no stiffness.

Two features of that picture are worth naming, because neither occurs in an ordinary transition.

The low-temperature phase is a line of fixed points, not a point. The flow ends anywhere on a segment of the axis, depending on where it started, so every temperature below the transition is a critical point in its own right. The correlations decay as a power law at all of them, with an exponent that varies continuously with temperature.

The separatrix ends at one particular place. A trajectory exactly on the diagonal flows into the origin, and the origin is at a definite value of the stiffness. Whatever the film was made of and whatever its bare stiffness was, if it is at its transition then its renormalised stiffness is that number.

The jump nobody can adjust #

That number is $2/\pi$, in units where the stiffness is measured against the temperature, and the integration gives $0.643$ against a prediction of $0.6366$.

What makes it remarkable is what it does not contain. Not the vortex core energy, not the bare stiffness, not the film’s thickness, not the substrate. Nelson and Kosterlitz’s prediction in 1977 was that every two-dimensional superfluid loses its superfluid density discontinuously, from a value equal to $2/\pi$ times $k$ times its own transition temperature, divided by the appropriate combination of the mass and Planck’s constant.

Films of helium of widely different thicknesses have widely different transition temperatures, and they all sit on that one line. The measurement — a torsional oscillator, whose period changes when part of the helium stops moving with it — is one of the more direct confirmations of a renormalisation-group prediction there is, precisely because there is nothing to fit.

A jump of a fixed height is a strange kind of prediction. It is not a critical exponent — it is a value — and it is universal in a stronger sense than an exponent is, because an exponent describes how something approaches zero and this describes what it is when it gets there.

The exponent that runs, and what it means for the low-temperature side #

The most unfamiliar feature of the whole thing is not the jump. It is that below the transition there is no ordinary ordered phase to have been reached.

In an ordinary ordered phase the correlation between two points approaches a non-zero constant as they are separated: that constant is the square of the order parameter, and its existence is what “long-range order” means. Here the correlation decays, as a power law,

$$\langle \psi^*(0)\,\psi(r)\rangle \sim r^{-\eta(T)},$$

with an exponent that is zero at absolute zero and rises continuously to exactly $1/4$ at the transition. So the correlation goes to zero at every temperature, and the order parameter is zero everywhere, and there is still a transition.

The name for this is quasi-long-range order, and it is what the line of fixed points in the flow diagram means physically. An ordinary critical point is one temperature at which correlations are scale-free; here every temperature below the transition is.

The value $1/4$ at the transition is another number nobody can adjust, and it is the same statement as the universal jump seen through a different quantity — the exponent and the stiffness are related, and the stiffness reaching $2/\pi$ is the exponent reaching a quarter. Both are measurable, and X-ray measurements of the correlations in two-dimensional smectic films have found exponents running continuously with temperature exactly as this requires.

A number puts the scale of the effect in view. For a helium film one atomic layer thick the transition sits a little below one kelvin, and thickening the film raises it towards the bulk value of 2.17 K. What the universal jump says is that the superfluid density at the moment of the transition is fixed by that temperature alone: about 0.6 micrograms per square centimetre per kelvin, times the transition temperature, whatever the film is. Thicker films have higher transition temperatures and correspondingly larger jumps, and the ratio is the same for all of them.

Why a jump in a quantity that has no jump anywhere else #

It is worth stopping on how odd the discontinuity is, because nothing in the rest of critical phenomena behaves like it.

At an ordinary continuous transition the order parameter goes to zero smoothly, the way the bulk superfluid fraction does, and the interesting quantity is the power with which it does so. That power is universal and the amplitude in front of it is not — two samples of the same substance share an exponent and differ in prefactor, which is why exponents are the currency of the subject and values are not. Here the currency is inverted: the quantity that is universal is a value, and it is the value the system holds at the last instant before it stops being a superfluid at all.

The reason is that the jump is not a property of the ordered phase running out. It is the condition under which a vortex pair comes apart. A bound pair is held together by an attraction going as the logarithm of the separation, with a coefficient proportional to the superfluid stiffness; the entropy available to a free vortex also goes as the logarithm of the system size, with a coefficient proportional to the temperature. Two logarithms with different coefficients cross at one place, and the crossing does not care what either coefficient was made of. Everything about the material has gone into setting where on the temperature axis the crossing happens, and nothing into what the stiffness is when it does.

That is why the prediction has no free parameter and why a torsional oscillator can test it without knowing anything about the film it is testing.

The same argument in things that are not helium #

Nothing above used superfluidity except as a source of vortices, so the argument runs wherever a two-dimensional system has a continuous symmetry and point defects.

Thin-film superconductors. The same transition, with the charge making the vortex interaction screened beyond the penetration depth rather than logarithmic for ever — which cuts the argument off at a length and rounds the transition. Films thin enough that the screening length exceeds their size show it cleanly.

Two-dimensional crystals. Melting in two dimensions goes by the unbinding of dislocations, and there is a second transition after it, at which disclinations unbind. Between them sits a phase with orientational order but no positional order — the hexatic — which is a state of matter that exists only in two dimensions and was predicted from this argument before it was seen.

Arrays of Josephson junctions. A grid of superconducting islands is the same model built deliberately, with the coupling adjustable, and it was used to test the theory in a way helium does not permit.

And the two-dimensional Coulomb gas. Vortices interacting logarithmically are the same problem as charges in two dimensions, so the transition is also a statement about when a plasma of positive and negative charges ionises — and the Debye screening that decides it is the same screening the renormalisation calculation performs.

What made it hard to believe #

The result took a decade to be accepted, and the reasons are worth setting out because they are about what counts as a transition.

There is no latent heat and no discontinuity in any derivative of the free energy that anybody could find. The free energy and all its derivatives are continuous at the transition, and its singularity is an essential one — every term of a Taylor expansion vanishes. By Ehrenfest’s classification, which sorts transitions by which derivative jumps, this one is of infinite order and therefore not a transition at all.

The specific heat has a peak in the wrong place. It rises to a maximum some way above the transition temperature, because the entropy released by the unbinding is spread over a range rather than concentrated at a point. An experimenter looking for the transition by finding the peak in the specific heat finds it in the wrong place, which is what happened for some years.

And the correlation length diverges too fast to measure. Its exponential form means it goes from small to enormous over a narrow range, and outside that range it looks like nothing is happening.

What settled it was the universal jump, because it is a prediction of a number rather than of a shape. A theory that predicts a curve can usually be fitted to data by adjusting something; a theory that predicts $2/\pi$ and nothing else cannot.

Watching the vortices instead of inferring them #

Everything above is inferred from a stiffness, an exponent or a specific heat — quantities averaged over a whole sample, from which the presence of free vortices is deduced. It is possible to look at the vortices.

Take a gas of cold atoms and squeeze it into two parallel planes, each two-dimensional in the sense that motion across the plane is frozen out. Release them and let the two clouds expand into one another. Where their phases agree, the matter waves interfere constructively; the result is a pattern of straight fringes, and the contrast of those fringes measures how well the phase in one plane matches the phase in the other across the field of view.

Below the transition the fringes are straight, and their contrast falls off with the length over which it is measured — as a power law, which is the quasi-long-range order of the section above, made directly visible as the fading of a fringe rather than deduced from a response function.

Above it the fringes acquire dislocations: places where one fringe ends and the pattern shifts by half a period. A dislocation in the interference pattern is what a free vortex in one of the clouds looks like, because going round a vortex the phase winds by a whole turn and the fringe has to accommodate it. So the photograph shows the objects themselves appearing as the temperature is raised through the transition, one at a time, in the numbers the theory predicts.

That experiment, done in 2006, also read the exponent off the contrast’s decay and found it running with temperature and reaching about a quarter at the transition — the second of the two universal numbers on this page, measured in a gas of a few thousand atoms rather than in a helium film. Kosterlitz and Thouless shared the Nobel Prize a decade later.

What Mermin and Wagner did not say #

The theorem quoted at the top is the most over-read result in this part of the subject, and the transition described here is the reason it is worth reading precisely.

Mermin and Wagner forbid long-range order: the correlation between two points cannot approach a non-zero constant as they are separated. It is regularly restated as “two-dimensional crystals cannot exist”, and that does not follow — as the low-temperature phase of this essay demonstrates, since it has no long-range order and is perfectly well a superfluid.

For a two-dimensional crystal the theorem says that an atom’s mean square displacement from its lattice site grows with the size of the sample, and it grows logarithmically. A logarithm is the gentlest divergence available: taking a sample from a thousand atoms across to a hundred million across, a factor of a hundred thousand, doubles the wandering. So a flake large enough to hold in tweezers is, for every practical purpose, a crystal, and its diffraction pattern shows peaks that are power-law singularities rather than true delta functions — sharper than any instrument can tell from the real thing.

Graphene exists, and it also does the other thing the theorem’s premises exclude: it ripples out of the plane. A sheet embedded in three dimensions has a direction available for its long-wavelength fluctuations that a strictly two-dimensional one does not, and the rippling stabilises it further.

The distinction to carry away is between three behaviours rather than two. Correlations that decay exponentially are a disordered phase; correlations that approach a constant are an ordered one; and correlations that decay as a power law are neither, are what two dimensions permit, and are what the whole of this transition is about.

Where the model runs out #

Everything is for an infinite film and every film is finite. The logarithms are cut off at the sample size, so the transition is rounded over a width that shrinks only as one over the logarithm of the size — which is a very slow approach and means finite-size effects are visible in samples that would be enormous by any other standard.

The vortices are treated as point objects with a core energy and no structure. A real vortex core is a region where the order parameter is suppressed, of a size the coherence length sets, and near the transition that length grows. Where it becomes comparable with the vortex separation the description stops.

The approach to the transition is not a power law. The correlation length diverges as an exponential of one over the square root of the temperature difference, not as a power, so every exponent that describes it is infinite. That is what makes the transition so hard to identify numerically: quantities that would be sharp at an ordinary transition here approach their limits with corrections that fall off logarithmically, and simulations of quite large systems can be fitted convincingly with the wrong theory.

And there is no transition at all in three dimensions by this mechanism. The bulk superfluid transition is an ordinary one with a genuine order parameter and finite exponents, and its universality class is a different one. A film thick enough is a three-dimensional system, so somewhere between the two there is a crossover, and where it sits depends on how the thickness compares with the coherence length rather than on any fixed number of atomic layers.

The ladder from here #

Later rungs on this anchor: the hexatic phase and the two-stage melting of a two-dimensional crystal; the roughening transition of a crystal surface, which is the same argument applied to steps rather than to vortices; the transition in a Josephson array and what makes it a controllable version; and the finite-size scaling that has to be done before any numerical study of any of these means anything.

The neighbouring ladders are the point at which the two become one, which is a transition of the ordinary kind with an order parameter and exponents, the whirlpool that comes in one size, where the defect this transition is about is derived, and the knot the field cannot untie, where a topological quantity again decides what a system is allowed to do.