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An experimental atomic physicist who coauthored a test of the Berezinskii–Kosterlitz–Thouless transition in trapped ultracold gases. Interference measurements examined how coherence changes with temperature, giving a laboratory probe of ordering that differs from conventional three-dimensional Bose-Einstein condensation.^[ref]

For the physical free energythe Boltzmann factor is $e^{-F_{\theta }/(k_{B}T)}$. At $T>0$, the Gaussian thermal phase fluctuations therefore obeyThe temperature factor is necessary when ${\rho }_s$ is the energy stiffness defined by the parent. Defining a dimensionless stiffness $K={\rho }_s/(k_{B}T)$ instead replaces $k_{B}T/{\rho }_s$ by $1/K$.

The difference cancels the spatially uniform phase mode. At any fixed separation, $1-\cos (\mathbf{k}\cdot \mathbf{x})$ removes the small-$k$ singularity. What grows in low dimensions is the variance as the separation tends to infinity.

In one dimension, the integral equals $|x|/2$, so the variance is $(k_{B}T/{\rho }_s)|x|$. In two dimensions, with microscopic cutoff $a$ and $r\gg a$, it grows asFor the Gaussian field, the order parameter correlation isIt decays exponentially in one dimension and algebraically in two. Thus nonzero-temperature Gaussian fluctuations do not leave conventional long-range order in these dimensions. This is the long-wavelength mechanism related to the Mermin-Wagner theorem; vortices and interactions require the fuller theory of the Berezinskii–Kosterlitz–Thouless transition in the two-dimensional case.

The phase stiffness is the coefficient of $\frac{1}{2}\int d^{d}x(\nabla \theta {)}^2$ in the long-distance free energy. Its renormalized value determines the thermal phase fluctuations and algebraic correlation function of a two-dimensional O(N) model with $N=2$.

Operationally impose a small phase difference $\varphi$ between opposite sides of a square sample of side $L$. The smooth minimum has $\theta =\varphi x/L$, so the two-dimensional excess free energy isThe derivative concerns a prescribed twist at the stated observation scale; in $d$ dimensions its conventional size factor is $L^{d-2}$. This response definition includes thermal fluctuations and vortices after equilibration, whereas a microscopic gradient coefficient need not. A finite phase stiffness requires no nonzero spatially uniform order parameter; the Berezinskii–Kosterlitz–Thouless transition is the instructive example.

Topological quantum matter has robust low-energy properties controlled by global topology rather than local order parameters.

The central question is which quantum states can be joined by a smooth change of a local Hamiltonian while keeping a nonzero bulk spectral gap. A Chern insulator makes this question concrete: two band insulators can have different integer Chern numbers, forcing a gap closure between them. The Haldane phase supplies a one-dimensional example in which a protecting symmetry matters. These are distinct from intrinsic topological order, with its anyons and topology-dependent ground-state degeneracy. The Berezinskii–Kosterlitz–Thouless transition concerns a separate mechanism involving vortices; topology organizes more than one kind of physical phenomenon.

The renormalized phase stiffness has this discontinuity at the Berezinskii–Kosterlitz–Thouless transition. The algebraic correlation function has exponent $1/4$ at the transition from the ordered side.

The stiffness here is an energy in two dimensions, not an areal mass density. For a neutral superfluid of particles of mass $m$, let $n_s$ be the superfluid number density per unit area. Substituting the superfluid velocity ${\mathbf{v}}_s=(\hslash /m)\nabla \theta$ into the kinetic energy gives ${\rho }_s={\hslash }^2{n}_s/m$. Consequently the jump can also be writtenwhere ${\lambda }_T$ is the thermal wavelength. It is the large-distance superfluid number density, rather than the total density or a locally inferred condensate density, that enters this relation.

Within the vortex-pair renormalization flow, $y$ changes from irrelevant to relevant at $\pi K_R=2$, with $K_R$ the large-distance dimensionless phase stiffness. Combining this with the spin wave result $\eta =1/(2\pi K_R)$ gives $\eta =1/4$. Above the transition the homogeneous equilibrium phase stiffness vanishes at infinite observation scale. A discontinuous response coefficient is compatible with a diverging correlation length; it does not imply a latent heat.^[ref]

A theoretical physicist who developed early ideas about phase transitions in two-dimensional systems with continuous symmetry. His work helped establish how strong fluctuations can remove ordinary long-range order while topological defects still control a sharp transition. John Michael Kosterlitz explicitly credits Berezinskii's earlier related ideas in the history of the Berezinskii–Kosterlitz–Thouless transition.^[ref]

An experimental atomic physicist who coauthored a trapped-gas study of the Berezinskii–Kosterlitz–Thouless transition. Its interference measurements connected the loss of long-distance coherence with vortex physics, showing how a carefully confined gas can test ideas about two-dimensional ordering.^[ref]