advanced32 min studyLesson 339 in Physics

Kosterlitz–Thouless vortex unbinding

Explain how compact phase defects produce a two-dimensional transition without ordinary magnetization and derive its characteristic vortex-flow diagnostics.

Before you begin

On this page

The question for today

What you’ll understand

Separate algebraic phase correlations from true spontaneous magnetization.
Derive the logarithmic energy and positional entropy of a vortex.
Interpret dilute-vortex flows, the stiffness jump, and the essential correlation-length singularity.

A two-dimensional planar spin system can change its long-distance behavior without acquiring a nonzero infinite-system magnetization. At low positive temperature, smooth phase fluctuations produce algebraically decaying correlations; at higher temperature, unbound vortices destroy that delicate coherence and correlations become short ranged. The Berezinskii–Kosterlitz–Thouless transition is therefore a warning against identifying every phase transition with a local Landau order parameter that turns on as an ordinary power law.

The phase is compact and can wind

For short-range XY spins, describe the local orientation by an angle theta identified modulo $2\pi$ . Smooth low-energy configurations have a gradient free energy with phase stiffness $\rho_s$ . In two dimensions, their thermal fluctuations already prevent a fixed orientation from having true long-range order at positive temperature under the usual continuous-symmetry and short-range assumptions. The low-temperature phase instead has quasi-long-range order: correlations decay as a power, rather than approaching a nonzero constant.

Compactness allows configurations excluded by a globally smooth single-valued real angle. Around a vortex, the phase changes by an integer multiple of $2\pi$ . Away from its core a unit vortex has gradient magnitude $1/r$ , and integrating squared gradient over annuli gives a logarithmic energy. A core contribution depends on microscopic details. A periodic bulk system has net winding-charge constraints, so the isolated-vortex disk estimate is a useful energy argument, while vortex–antivortex pairs provide the appropriate neutral bulk description.

F_{\mathrm{sw}}=\frac{\rho_s}{2}\int d^2x\,|\nabla\theta|^2,\qquad q=\frac1{2\pi}\oint\nabla\theta\cdot d\boldsymbol\ell\in\mathbb Z,\qquad E_q\simeq\pi\rho_s q^2\ln(L/a)+E_{\mathrm{core}}

rho_s and vortex energies are in joules, theta is dimensionless, q is integer winding, L is system scale and a is core cutoff in metres. The energy estimate assumes a permitted vortex winding in a disk and a nearly uniform stiffness outside the core; neutral bulk pair physics needs both defects.↔ Scroll wide equations horizontally.

Vortex entropy competes with logarithmic energy

The number of possible core positions is proportional to $(L/a)^2$ , giving positional entropy approximately $2k_B\ln(L/a)$ . With dimensionless stiffness $K=\rho_s/(k_BT)$ , a unit-vortex free-energy estimate has logarithmic coefficient $\pi K-2$ . Its change of sign suggests vortex proliferation. The simple argument uses a fixed stiffness and ignores screening by smaller bound pairs, so it does not determine the transition from the bare microscopic stiffness alone.

A dilute gas of vortex pairs supplies the improved picture. Let y denote a dimensionless unit-vortex fugacity, proportional to the Boltzmann factor for its core cost. Coarse graining small pairs reduces the effective stiffness and changes y. In a common normalization, y grows when K is below $2/\pi$ and decreases above it at very small y. The flow of inverse stiffness starts at order y squared because neutral pair contributions are needed.

\frac{dK^{-1}}{d\ell}=4\pi^3y^2+\cdots,\qquad \frac{dy}{d\ell}=(2-\pi K)y+O(y^3),\qquad K=\frac{\rho_s}{k_BT}

K,y and the logarithmic length scale ell are dimensionless; increasing ell means coarser spatial resolution. These conventional leading dilute-vortex flows require y small and suitable short-range two-dimensional XY physics. Numerical fugacity coefficients depend on its normalization; the long-distance threshold concerns renormalized stiffness.↔ Scroll wide equations horizontally.

A fixed line and an essential singularity replace ordinary onset

The low-temperature flows approach a line with y zero and a finite renormalized stiffness $K_R\geq2/\pi$ . The spin-wave correlation exponent is $\eta=1/(2\pi K_R)$ , reaching one quarter at the transition. The limiting stiffness just below the transition obeys $\rho_s^R/(k_BT_{\mathrm{BKT}})=2/\pi$ . This is a universal jump in a renormalized response, not a nonzero magnetization that suddenly appears or a universal value for every microscopic bare coupling.

On the disordered side, the correlation length grows as an exponential of a constant divided by the square root of reduced temperature. That essential singularity is not described by a finite ordinary power-law nu. Finite systems can make it especially difficult to distinguish the asymptotic behavior from a large effective exponent. Long-range forces, discrete anisotropy, interlayer coupling, or electromagnetic screening can change the appropriate model, so the short-range neutral XY assumptions must accompany the claim of BKT scaling.

Work it through

An initial dilute-vortex flow step

Use the displayed leading flow equations with $K=0.50$ , $y=0.020$ , and $\Delta\ell=0.10$ . Make one forward-Euler estimate of y and inverse stiffness, using pi=3.1416. Decide whether the initial change promotes or suppresses vortex proliferation.

The fugacity multiplier in its derivative is $2-\pi K=2-1.5708=0.4292$ . Thus $dy/d\ell=0.4292(0.020)=0.008584$ , positive for these initial parameters.
Over the short scale step, $\Delta y\simeq0.0008584$ , so $y_{\mathrm{new}}\simeq0.0208584$ . This is a local estimate using initial derivatives, not a full solution across the transition.
The inverse-stiffness change is $4\pi^3(0.020)^2(0.10)\simeq0.004961$ . Starting from $K^{-1}=2$ , this gives approximately 2.004961, hence K decreases to about 0.4988. Reduced stiffness and growing fugacity initially reinforce vortex proliferation.

The result

The first estimated step gives y≈0.02086 and K≈0.4988; it heads toward greater vortex activity, within a local dilute-gas approximation.

The living notebook

Explore the model

Interactive lab

Interactive lab · waves

A compact phase identifies its endpoints

For each independent run, Reset and set BOTH Phase difference and Amplitude ratio=1. Compare −180° and +180°, predicting the same complete cancellation, then Step. The fixed-frequency copropagating-wave model shows phase periodicity only; it has no two-dimensional orientation field, winding loop, vortex core, stiffness, or fugacity.

SumFirst componentSecond component

Sum at x = 0: 0.866
Resultant amplitude: 1.732
Phase difference: 60°

Phase difference60°Amplitude ratio1

t = 0 s

Two waves share the same frequency and wave number. Playback advances t in sin(2x − t) + a sin(2x − t + phase). Parameter changes pause playback and recompute the model at the current elapsed time. Reset restores both parameters and returns time to zero. Playback starts paused. Adjust either parameter, then choose Play or Step.

Look forAngles separated by a full turn describe the same relative phase. Compact phase is necessary background for vortices, but this two-wave lab cannot create or unbind them.

Pause & reason

Check your understanding

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Read the checkpoint explanation

The BKT low-temperature phase has quasi-long-range order with algebraic correlations and vortices bound in pairs. Under the usual short-range continuous-symmetry assumptions, it does not possess conventional positive-temperature spontaneous magnetization.

Your turn

Try it yourself

For the isolated-vortex disk estimate, take $K=0.50$ , $L/a=100$ , and $E_{\mathrm{core}}=k_BT$ . Estimate the unit-vortex free energy divided by thermal energy, including positional entropy. Explain why this number is not an exact transition criterion.

Take a moment to make a prediction or work it out on paper. Open a hint when you need one.

Give me a hint

Combine the energy and entropy logarithms before evaluating them: the coefficient is pi K−2. Use ln100≈4.6052.

Compare with the solution

The estimate is $\Delta F/(k_BT)=1+(\pi K-2)\ln100=1+(1.5708-2)(4.6052)\simeq-0.9766$ . The positional entropy wins over the specified core plus logarithmic energy within this illustrative estimate. It does not establish an exact finite-system phase transition or its temperature: bound pairs renormalize stiffness, the core cost is nonuniversal, and bulk periodic boundary conditions require net-charge constraints. The universal threshold concerns long-distance renormalized stiffness at the transition, obtained from the collective vortex problem rather than this single-defect arithmetic.

Keep the idea

Take it with you

Compact phase supports winding defects that a globally smooth angle misses.
BKT criticality separates algebraic correlations from a vortex-disordered regime without ordinary magnetization.
Universal stiffness and essential length scaling refer to the renormalized long-distance theory.

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