The BKT Transition: Vortices, Topology and a Nobel Prize

Some phase transitions are not about a symmetry breaking at all. In a thin film of helium or a flat array of tiny magnetic arrows, the change from one phase to another is driven by whirlpools, called vortices, that are bound in pairs at low temperature and break free at high temperature. This is the Berezinskii–Kosterlitz–Thouless (BKT) transition Berezinskii–Kosterlitz–Thouless transition, Wikipedia^[1]. It earned a share of the 2016 Nobel Prize in Physics Nobel Prize in Physics 2016: press release^[2], and it is what our Vortia & Antivortia BKT transition tee is about.

Why should two dimensions have no phase transition?

In 1966 Mermin and Wagner proved rigorously that a one- or two-dimensional isotropic Heisenberg model with short-range interactions cannot be ferromagnetic or antiferromagnetic at any nonzero temperature Mermin and Wagner 1966^[3]. The general lesson is that in low dimensions, long-wavelength fluctuations of a continuous symmetry, the slow twists of the spins, are cheap enough to destroy long-range order Mermin–Wagner theorem, Wikipedia^[4]. The two-dimensional XY model, where each spin is an arrow free to point anywhere in the plane, is the standard example Classical XY model, Wikipedia^[5].

So the textbook picture of a phase transition, where order appears below a critical temperature, cannot apply. Yet experiments on thin helium films found a sharp onset of superfluidity. The paradox was resolved by showing that there is a genuine finite-temperature transition, of a new kind, in which vortices play the central role Nobel Prize in Physics 2016: Scientific Background^[6]. For the more familiar symmetry-breaking kind, see our guide to the Ising model and phase transitions.

How do vortices drive the BKT transition?

A vortex is a point around which the spin direction turns by a full 360°. An antivortex turns the other way. Both are topological defects: no small, smooth change can remove a single one Berezinskii–Kosterlitz–Thouless transition, Wikipedia^[1]. In a system of size L with lattice spacing a, an isolated vortex costs energy E = πJ ln(L/a), where J is the coupling energy between neighbouring spins in joules Nobel Prize in Physics 2016: Scientific Background^[6]. Its centre can sit at about (L/a)² places, so its entropy is S = k_B ln(L²/a²) Nobel Prize in Physics 2016: Scientific Background^[6]. The free energy is

F = E − TS = (πJ − 2k_BT) ln(L/a)

with T in kelvin and k_B = 1.381 × 10⁻²³ J/K the Boltzmann constant Nobel Prize in Physics 2016: Scientific Background^[6]. Below a critical temperature the bracket is positive, so free vortices cost an energy that grows with system size and they cannot exist alone. Above it the bracket is negative and free vortices proliferate. The crossover is at

T_BKT = πJ/(2k_B)

This is Kosterlitz and Thouless’s heuristic energy–entropy argument Nobel Prize in Physics 2016: Scientific Background^[6] Kosterlitz and Thouless 1973^[7]. Below T_BKT, vortices still appear, but only as tightly bound vortex–antivortex pairs. The pairs interact like charges +1 and −1 with a force falling as 1/r, so the defects behave like a two-dimensional neutral Coulomb gas Nobel Prize in Physics 2016: Scientific Background^[6]. The transition is the moment those pairs unbind.

The simple estimate is not exact. For the square-lattice XY model, Monte Carlo simulations give k_BT_c/J = 0.8935(1) Classical XY model, Wikipedia^[5], below the bare value π/2 ≈ 1.571, because bound pairs screen and weaken the interaction. Handling that screening properly needed the renormalization-group equations developed by Kosterlitz Nobel Prize in Physics 2016: Scientific Background^[6].

What is “topological order” in the BKT phase?

Kosterlitz and Thouless proposed a new definition of order for two-dimensional systems with no conventional long-range order. They called it topological order, and noted that the singularities at the transition are very weak Kosterlitz and Thouless 1973^[7]. Below T_BKT the spin correlations decay as a power law rather than staying constant Nobel Prize in Physics 2016: Scientific Background^[6] Berezinskii–Kosterlitz–Thouless transition, Wikipedia^[1]:

C(r) ∼ (a/r)^η, η = k_BT/(2πJ)

Here r is the separation in metres and η is a temperature-dependent exponent. Above T_BKT correlations decay exponentially Berezinskii–Kosterlitz–Thouless transition, Wikipedia^[1]. The phase is called quasi-long-range ordered, and the transition is of infinite order Berezinskii–Kosterlitz–Thouless transition, Wikipedia^[1].

Vadim Berezinskii recognised a year before Kosterlitz and Thouless that vortices could drive a transition in the XY model, though he did not describe its nature correctly, which is why his name leads the acronym Nobel Prize in Physics 2016: Scientific Background^[6].

How was the BKT transition confirmed?

The sharpest test is a prediction by Nelson and Kosterlitz: at the transition, the superfluid density of a two-dimensional film should drop from a universal value to zero Nelson and Kosterlitz 1977^[8]. In SI form, with m the helium-4 atomic mass in kg and ρ_s the areal superfluid density in kg/m²:

ρ_s(T_c) = 2m²k_BT_c/(πħ²)

For helium-4 they quoted a ratio ρ_s/T_c = 3.52 × 10⁻⁹ g cm⁻² K⁻¹, about 3.5 × 10⁻⁸ kg m⁻² K⁻¹ Nelson and Kosterlitz 1977^[8]. It does not depend on film thickness or on any microscopic detail. Bishop and Reppy measured the superfluid mass of helium films on an oscillating Mylar substrate and found good agreement with this jump Bishop and Reppy 1978^[9] Nobel Prize in Physics 2016: Scientific Background^[6]. Later, a BKT crossover was observed in a trapped two-dimensional atomic gas Hadzibabic et al. 2006^[10]. BKT physics is also found in Josephson junction arrays and thin superconducting films Berezinskii–Kosterlitz–Thouless transition, Wikipedia^[1].

Superfluid helium itself is explained in what a superfluid is, and its frictionless flow is on our Superfluid tee.

Why did the BKT transition win a Nobel Prize?

The 2016 prize went half to David Thouless and half to Duncan Haldane and Michael Kosterlitz, “for theoretical discoveries of topological phase transitions and topological phases of matter” Nobel Prize in Physics 2016: press release^[2]. The BKT transition, in which topology rather than symmetry breaking decides the phase, is one of the topological phase transitions that citation refers to Nobel Prize in Physics 2016: Scientific Background^[6].

Quick FAQ

Is a BKT phase really ordered?

Not in the usual sense. There is no net magnetisation or condensate phase across the whole sample, but correlations decay slowly, as a power law Berezinskii–Kosterlitz–Thouless transition, Wikipedia^[1].

Why is it called a topological transition?

Because the defects that drive it are labelled by a whole number, how many times the spins wind around the core, and no small smooth change can alter that number Berezinskii–Kosterlitz–Thouless transition, Wikipedia^[1] Nobel Prize in Physics 2016: Scientific Background^[6].

What is the critical temperature exactly?

It depends on the system; only the jump ratio is universal Nelson and Kosterlitz 1977^[8]. For more cold-matter physics, browse our Thermodynamics collection.

References

1. Wikipedia, “Berezinskii–Kosterlitz–Thouless transition”. https://en.wikipedia.org/wiki/Berezinskii%E2%80%93Kosterlitz%E2%80%93Thouless_transition
2. NobelPrize.org, “The Nobel Prize in Physics 2016” press release (Thouless, Haldane and Kosterlitz). https://www.nobelprize.org/prizes/physics/2016/press-release/
3. N. D. Mermin and H. Wagner (1966), “Absence of Ferromagnetism or Antiferromagnetism in One- or Two-Dimensional Isotropic Heisenberg Models,” Physical Review Letters 17, 1133–1136 (DOI 10.1103/PhysRevLett.17.1133; metadata verified via Crossref). https://doi.org/10.1103/PhysRevLett.17.1133
4. Wikipedia, “Mermin–Wagner theorem”. https://en.wikipedia.org/wiki/Mermin%E2%80%93Wagner_theorem
5. Wikipedia, “Classical XY model”. https://en.wikipedia.org/wiki/Classical_XY_model
6. Nobel Committee for Physics, “Topological Phase Transitions and Topological Phases of Matter”, Scientific Background on the Nobel Prize in Physics 2016 (PDF). https://www.nobelprize.org/uploads/2018/06/advanced-physicsprize2016-1.pdf
7. J M Kosterlitz and D J Thouless (1973), “Ordering, metastability and phase transitions in two-dimensional systems,” Journal of Physics C: Solid State Physics 6, 1181–1203 (DOI 10.1088/0022-3719/6/7/010; metadata verified via Crossref). https://doi.org/10.1088/0022-3719/6/7/010
8. David R. Nelson and J. M. Kosterlitz (1977), “Universal Jump in the Superfluid Density of Two-Dimensional Superfluids,” Physical Review Letters 39, 1201–1205 (DOI 10.1103/PhysRevLett.39.1201; metadata verified via Crossref). https://doi.org/10.1103/PhysRevLett.39.1201
9. D. J. Bishop and J. D. Reppy (1978), “Study of the Superfluid Transition in Two-Dimensional ⁴He Films,” Physical Review Letters 40, 1727–1730 (DOI 10.1103/PhysRevLett.40.1727; metadata verified via Crossref). https://doi.org/10.1103/PhysRevLett.40.1727
10. Zoran Hadzibabic et al. (2006), “Berezinskii–Kosterlitz–Thouless crossover in a trapped atomic gas,” Nature 441, 1118–1121 (DOI 10.1038/nature04851; metadata verified via Crossref). https://doi.org/10.1038/nature04851

Written by Pixelated Physics. Every factual claim is linked to the source we checked; points that are still debated are labelled open.