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Write ${ϵ}_k={\hslash }^2{k}^2/(2m)$, $E_k=\hslash \omega$, and take $\mu >0$. The Bogoliubov spectrum is gapless and positive for $k\ne 0$. Its small- and large-momentum limits areThe low-energy excitations are collective sound modes; high-energy excitations approach particles with a mean-field energy offset. With healing length $\xi =\hslash /\sqrt{2m\mu }$ and $q=|k|\xi$,The crossover occurs at $q$ of order one, not at an additional energy gap.

The critical velocity of a homogeneous condensate follows because the Landau criterion compares an excitation energy with the energy that flow can release by transferring its momentum. Creating a mode is energetically possible when $E_k-\hslash \mathbf{k}\cdot \mathbf{v}\le 0$. For this isotropic spectrum,The infimum is the long-wavelength speed of sound. This tests energetic stability against the stated excitations; it is not a calculation of a dissipation rate or a universal vortex-nucleation threshold.

For an external potential varying slowly on the healing scale, the Thomas–Fermi approximation for a condensate neglects density-gradient kinetic energy in the stationary Gross–Pitaevskii equation. In occupied regions it givesHere $\rho$ is the appropriate local number density; in the ring it is a linear density and $g$ is the one-dimensional coupling. This is the local-density sound speed of a condensate. The formula applies where $\mu >V$, not in an empty region where its square root would be imaginary.

Normalize the condensate order parameter to the atom number:up to an arbitrary constant phase. A wavefunction normalized to one instead has amplitude $1/\sqrt{2\pi R}$, with $N|\psi {|}^2$ then providing density. Acting with $L_z=-i\hslash {\partial }_{\theta }$ gives angular momentum $n\hslash$ per atom and $Nn\hslash$ in total for the uniform ring. The superfluid velocity isThe interaction chemical potential is ${\mu }_0=gN/(2\pi R)$; a flowing uniform state also has the constant kinetic contribution ${\hslash }^2{n}^2/(2m{R}^2)$ to its total chemical potential. Its speed of sound is set by the interaction term. Applying $|v|<\sqrt{{\mu }_0/m}$ givesFor a uniform ring condensate, this is the continuum phonon criterion used here. Quantization of the available excitation wavelengths in a very small ring can modify a finite-size threshold.

For the nonrotating equilibrium with a raised left half, let ${\rho }_L$ and ${\rho }_R$ be the bulk linear densities. Within the same Thomas–Fermi approximation for a condensate,When both halves are occupied, normalization fixesand thereforeThese expressions require $V_0<2{\mu }_0$. Beyond that threshold, extending them would give a negative left density and is unphysical. Instead the left bulk empties, all $N$ atoms occupy a segment of length $\pi R$, andAt equality the left bulk just reaches zero density. kinetic energy rounds the interface and produces tails beyond the Thomas–Fermi approximation for a condensate, so an exactly empty half is the idealized bulk result.

A nonconstant $V(\theta )$ breaks rotational symmetry:Consequently angular momentum is no longer a good quantum number, and an eigenfunction need not be $e^{in\theta }$ times a constant amplitude. Nevertheless, while the order parameter is nonzero around the whole ring, single-valuedness requiresThe integer is a phase winding of a condensate, not an assertion that the inhomogeneous state remains an angular-momentum eigenstate. A superfluid phase slip can change it if the amplitude vanishes locally.

Conservation of particles gives ${\partial }_t\rho +{\partial }_s(\rho v)=0$. For steady density the one-dimensional particle current $j=\rho v$ is constant. Since the two halves have equal lengths,The current bottleneck in an inhomogeneous condensate occurs because a lower-density region has faster flow at the same current; it also has a smaller speed of sound. This makes the raised half the limiting region.

For $V_0=2{\mu }_0/3$ the equilibrium approximation givesThe winding condition and equal lengths requireThus the left-hand sound-speed test isThe right-hand condition is only $n^2<3m{R}^2{\mu }_0/{\hslash }^2$, so it is less restrictive.

This local sound-speed estimate for ring-flow stability uses the step-potential equilibrium in a ring condensate densities together with the local sound-speed test, as suggested by the preceding approximations. A quantitatively self-consistent finite-current solution also includes $m{v}^2/2$ and the quantum pressure of a condensate in the stationary Gross–Pitaevskii equation, allowing flow to change the density. Smooth steps suppress reflection but do not alone remove those terms. The $3/8$ result is the stated local-density estimate; it should not be promoted to an exact stability theorem for every finite interacting ring.

For a one-component superfluid with nonzero order-parameter magnitude on a closed loop, the phase defines a map from the loop into the circle. Its winding number isThe fundamental group ${\pi }_1(S^1)=\mathbb{Z}$ means that smooth deformations preserving nonzero amplitude cannot change this integer. With superfluid velocity ${\mathbf{v}}_s=\hslash \nabla \theta /m_B$, it gives quantized circulation $\oint {\mathbf{v}}_s\cdot d\mathbf{l}=nh/m_B$. A vortex core permits the order parameter to vanish and carries the winding; a superfluid phase slip changes a loop's winding when a core crosses it. For paired neutral fermions, the relevant condensed-object mass is the pair mass.

For charged pairs, gauge-covariant phase rigidity gives fluxoid and flux quantization in a superconductor:The magnitude of the pair flux quantum is $h/(2e)$. Magnetic flux alone approaches an integer multiple of this value when the circulating-velocity term on a bulk contour is negligible. The topological condition is not a claim that an arbitrary finite loop has exactly quantized magnetic flux independent of its screening current.

There are also momentum-space invariants, distinct from real-space vortex winding. A fully gapped two-dimensional Bogoliubov--de Gennes Hamiltonian can have a Chern number obtained from its occupied quasiparticle bands' Berry curvature. In a chiral paired model it can protect edge modes while the bulk gap and relevant symmetry class are maintained. Such an invariant can change only when those assumptions fail, for example through a bulk gap closing. A nodal d-wave state requires a different analysis; it is not automatically a fully gapped Chern superconductor.