Vortices and Topological Defects in Paired Matter

A vortex is a configuration whose order-parameter phase winds nontrivially around a core where the amplitude must be suppressed or the order parameter must leave its low-energy manifold. In a neutral two-dimensional superfluid its energy grows logarithmically with system size; in a charged superconductor magnetic screening cuts off that growth at the penetration depth. Winding topology constrains the defect, but it does not make every vortex-core eigenstate topologically protected.

Required background. Superfluid phase stiffness supplies the phase-only energy and normalization.

Helpful background. BdG theory supplies core spectra and their finite-size tests.

Winding and the core

For a one-component complex pair field $\Delta=\lvert\Delta\rvert e^{i\theta}$, a loop $C$ outside all cores obeys

$$n=\frac{1}{2\pi}\oint_C\boldsymbol\nabla\theta\mathbin{\cdot}\mathrm d\boldsymbol\ell\in\mathbb Z.$$

The integer follows from single-valuedness of $\Delta$. It classifies maps from the loop into the vacuum manifold $S^1$, since $\pi_1(S^1)=\mathbb Z$. In a multicomponent order parameter the vacuum manifold and identifications can differ, permitting fractional vortices, domain walls attached to vortices, or defects unstable to intercomponent locking. The homotopy calculation must use the actual physical manifold rather than the symbol “complex gap.”

For a neutral fluid with phase free energy $F=(\rho_s/2)\int\mathrm d^2x\,(\nabla\theta)^2$, the far-field solution is $\theta=n\varphi$. Cutting off the radial integral at core scale $\xi$ and system size $R$ gives

$$E_n=\pi\rho_s n^2\log\frac{R}{\xi}+E_{\mathrm{core}}.$$

The core scale is of order the coherence length in weak-coupling BCS theory, but its precise energy is not fixed by phase-only EFT. BdG or microscopic theory is needed inside $r\lesssim\xi$.

The corresponding circulation for constituent mass $m$ and pair phase is

$$\kappa=\oint\mathbf v_s\mathbin{\cdot}\mathrm d\boldsymbol\ell =\frac{2\pi n}{2m}$$

in natural units. Restoring $\hbar$ gives $h n/(2m)$ for a paired fermion superfluid.

Neutral vortices, Abrikosov vortices, and BKT proliferation

In a charged condensate the gradient becomes $\nabla\theta-2e\mathbf A$. Supercurrents are screened beyond $\lambda_L$, and a type-II vortex carries quantized flux while spreading its magnetic field over $\lambda_L$. The hierarchy $\kappa_{\mathrm{GL}}=\lambda_L/\xi>1/\sqrt2$ favors separated flux tubes between lower and upper critical fields; the inequality is a Ginzburg–Landau result near its regime of validity, not a universal microscopic classification far from $T_c$.

In two-dimensional neutral matter, entropy competes with the logarithmic energy. A single vortex has positional entropy $S\simeq2\log(R/\xi)$, suggesting the balance $F\simeq(\pi\rho_s-2T)\log(R/\xi)$. Renormalization of vortex pairs makes this heuristic precise as the BKT stiffness jump. In a finite sample the proliferation is a crossover and depends on boundaries and inhomogeneity. The defect mechanism and renormalized transition are developed in Berezinskii 1971, pp. 493–500 and Kosterlitz and Thouless 1973, §§2–4.

Pinning changes vortex motion and transport but not the basic circulation quantization. Disorder can also create low-energy core structure and broaden transitions; conclusions about defect statistics or protected modes require tests beyond imaging flux.

Core states are a separate question

An ordinary $s$-wave vortex hosts Caroli–de Gennes–Matricon states with characteristic spacing $\omega_0\sim\Delta^2/E_F$ Caroli, de Gennes, and Matricon 1964, pp. 307–309. Particle–hole symmetry makes the spectrum approximately symmetric, and finite resolution can merge several levels into an apparent zero-bias feature. A protected Majorana zero mode requires a nontrivial bulk BdG invariant, the correct defect class, an open bulk gap, and stability to allowed perturbations; those conditions are developed on the topological BdG page.

The chapter diagram links stiffness to defect energy while keeping core spectroscopy and topological classification distinct.

Winding fixes circulation and long-distance energetics. Gauge screening, core regularization, dimensionality, and the bulk topological class determine the stronger conclusions. Original schematic, not to scale.

The paired-matter claim test matrix states the corresponding failure tests.

Exercise

Compare one double vortex with two single vortices. Ignore core overlap and calculate the leading logarithmic energies.

Solution

A charge-$n$ vortex has $E_n\simeq\pi\rho_sn^2\log(R/\xi)$. One $n=2$ vortex therefore costs $4\pi\rho_s\log(R/\xi)$, while two well-separated $n=1$ vortices cost approximately $2\pi\rho_s\log(R/\xi)$ plus their interaction and two core energies. The quadratic winding cost favors splitting unless confinement, geometry, or multicomponent structure changes the balance.

References

• Berezinskii, V. L. (1971). “Destruction of long-range order in one-dimensional and two-dimensional systems having a continuous symmetry group I.” Soviet Physics JETP 32, 493–500. JETP archive.
• Caroli, C., de Gennes, P. G., and Matricon, J. (1964). “Bound fermion states on a vortex line in a type II superconductor.” Physics Letters 9, 307–309. doi:10.1016/0031-9163(64)90375-0.
• Kosterlitz, J. M., and Thouless, D. J. (1973). “Ordering, metastability and phase transitions in two-dimensional systems.” Journal of Physics C 6, 1181–1203. doi:10.1088/0022-3719/6/7/010.