Low-Dimensional Bose Gases and BKT Physics

In two dimensions at nonzero temperature, smooth phase fluctuations destroy true long-range order, but they can leave algebraic correlations and finite stiffness. Vortex–antivortex unbinding then produces the Berezinskii–Kosterlitz–Thouless (BKT) transition, characterized by a universal jump of the renormalized helicity modulus rather than a conventional local order parameter.

Required background. Use the phase–density EFT and the general RG ideas of scaling directions and momentum-shell integration. Helpful background. Free-boson vertex operators provide a complementary normalization check.

Algebraic order from smooth phases

At distances larger than the healing length, the classical free energy is

$$\frac{F}{T}=\frac{K}{2}\int\mathrm d^2x\,(\nabla\theta)^2, \qquad K=\frac{\Upsilon}{T},$$

where $\Upsilon=n_s/m$ is the helicity modulus. Gaussian fluctuations give

$$\big\langle[\theta(\mathbf r)-\theta(0)]^2\big\rangle =\frac{1}{\pi K}\ln\frac{r}{\xi}, \qquad g_1(r)\propto r^{-\eta}, \quad \eta=\frac{1}{2\pi K}.$$

The logarithm rules out ODLRO for any $T>0$, but permits quasi-long-range order. This is the Bose-fluid realization of the infrared result; the general theorem requires short-range interactions and continuous symmetry.

Vortex energetics and the universal jump

A vortex of winding $q$ has $\nabla\theta=q\hat{\boldsymbol\phi}/r$, hence

$$E_v=\pi\Upsilon q^2\ln(L/\xi)+E_{\mathrm{core}}.$$

Its positional entropy is approximately $2\ln(L/\xi)$, so the free-energy balance for $|q|=1$ changes sign near $\pi\Upsilon=2T$. Screening by vortex pairs turns this estimate into the BKT RG flow. With fugacity $y=e^{-E_{\mathrm{core}}/T}$,

$$\frac{\mathrm dK^{-1}}{\mathrm d\ell}=4\pi^3y^2, \qquad \frac{\mathrm dy}{\mathrm d\ell}=(2-\pi K)y.$$

The separatrix gives the universal renormalized jump

$$\Upsilon(T_{\mathrm{BKT}}^-)=\frac{2T_{\mathrm{BKT}}}{\pi}, \qquad \eta(T_{\mathrm{BKT}})=\frac14.$$

The normalization-independent content is the relation between the long-distance stiffness and the correlation exponent. The RG equations and physical interpretation are reviewed in Altland and Simons 2023, § 6.5, pp. 360–372; the jump relation is established in Nelson and Kosterlitz 1977, pp. 1201–1205.

Finite size and traps

At finite $L$, the correlation length cannot exceed the system size, and a fitted power law can mimic the transition over a narrow window. A defensible analysis follows the size drift of the renormalized stiffness or the BKT form $\xi_+\sim\exp[b/\sqrt{T-T_{\mathrm{BKT}}}]$. Trap averaging mixes local stiffnesses; nonequilibrium vortex coarsening is a separate dynamical question.

Exercises

Derive the vortex-energy logarithm for unit winding from the phase free energy.

Solution

Insert $|\nabla\theta|=1/r$: $E_v=(\Upsilon/2)\int_\xi^L2\pi r\,\mathrm dr/r^2=\pi\Upsilon\ln(L/\xi)$. The omitted core energy is nonuniversal and is encoded in the fugacity.

References

• Alexander Altland and Ben Simons, Condensed Matter Field Theory, 3rd ed., Cambridge University Press (2023), § 6.5, pp. 360–372, doi:10.1017/9781108781244.
• David R. Nelson and J. Michael Kosterlitz, “Universal Jump in the Superfluid Density of Two-Dimensional Superfluids,” Physical Review Letters 39 (1977) 1201–1205, doi:10.1103/PhysRevLett.39.1201.