Number-Phase Vortex Effect in Quantum Fluids

• Number-Phase Vortex Effect is the interplay between fixed particle number, global condensate phase winding, and vortex-mediated geometric phases in quantum fluids.
• The effect exposes the breakdown of standard BdG theory near vortex cores, where condensate back-action alters the predicted universal Berry phase.
• Its manifestations in Josephson junctions and vortex electron scattering offer measurable phase shifts and flux offsets, linking topology with experimental observables.

The number–phase vortex effect denotes the coupling between particle-number conservation, condensate phase winding, and vortex-mediated geometric phase in superfluid and superconducting systems. In the formulation developed for a bound quasiparticle transported adiabatically around a unit-winding superfluid vortex, exact $U(1)$ conservation and condensate back action alter the Berry phase from the universal Bogoliubov–de Gennes (BdG) value $\pi$ to a non-universal quantity set by system parameters and trapping geometry (Lin et al., 2017). In a superconducting Josephson geometry, the same underlying vortex phase winding becomes directly visible as a measurable Josephson phase shift, including a reversible $0$–$\pi$ transition and an effective $\Phi_0/2$ flux offset (Golod et al., 2010). In vortex-electron scattering, the phase singularity $e^{i\ell\varphi}$ likewise generates measurable transverse and azimuthal signatures, although that setting concerns the number–phase structure of twisted beams rather than the particle-number-conserving Berry-phase problem of neutral superfluids (Liu et al., 11 Mar 2025).

1. Canonical structure and vortex phase winding

In a neutral fermionic superfluid, the condensate carries a well-defined macroscopic phase $\phi$, while the total particle number $N$ is fixed. The variables $N$ and $\phi$ are canonically conjugate, so any process that winds $\pi$0 by $\pi$1 necessarily shifts $\pi$2, and vice versa. The number–phase vortex effect arises when this conjugacy is combined with the topological phase winding of a vortex and with the adiabatic motion of a localized quasiparticle.

For a unit-winding vortex, the conventional mean-field picture assigns a $\pi$3 winding to the condensate phase. In the superconducting London gauge, the phase outside the vortex core is the polar angle, $\pi$4, and the circulation condition is

$\pi$5

Projected onto a Josephson junction plane, the vortex-induced condensate phase in one electrode can be written as

$\pi$6

This expresses the essential feature of a vortex relevant to the number–phase problem: the magnetic flux or core localization is local, but the condensate phase texture is global.

In the quasiparticle-transport setting, one imagines a single fermionic quasiparticle localized by a weak Zeeman trap and dragged once around such a vortex. The elementary BdG analysis then predicts a Berry phase $\pi$7. The number–phase vortex effect begins precisely where that universal prediction fails.

2. Breakdown of the standard BdG description near the vortex core

The usual BdG treatment replaces the condensate by a $\pi$8-number $\pi$9 and breaks $0$0 particle-number conservation down to $0$1. In that framework, the quasiparticle is mapped to an effective two-level system, and the Berry phase $0$2 follows from a spin-$0$3 texture that sweeps a solid angle $0$4. This is the familiar universal result.

The limitation appears when the bound quasiparticle lies within a distance of order the coherence length, or penetration depth, from the vortex core, where the superfluid velocity is non-zero:

$0$5

In that region, adiabatic transport of the quasiparticle both probes and back-reacts on the moving condensate. Particle-number non-conserving calculations based on BdG equations are then unable to capture the correct physics. The continuity equation,

$0$6

need not hold for individual quasiparticle eigenstates, and the quasiparticle hole component can effectively jump into the condensate without a compensating flow. The same failure can be expressed as a violation of the $0$7-sum rule for the current-current correlator (Lin et al., 2017).

An equivalent diagnostic uses angular momentum. For a number-nonconserving BdG quasiparticle

$0$8

the Cooper-pair part $0$9 carries no well-defined individual current. Depending on the argument used, one is led either to $\pi$0 or to $\pi$1, where $\pi$2 is the flux through the annulus in $\pi$3 units. The data identify this inconsistency as a symptom of BdG breakdown when superflow is present. A common misconception is therefore that the vortex Berry phase remains universally $\pi$4 even arbitrarily close to the core; the number-conserving analysis rejects that conclusion.

3. Number-conserving Berry phase and exact reformulations

With exact particle-number conservation, the many-body ground state for an annulus of circumference $\pi$5 and a trap at angle $\pi$6 is written as

$\pi$7

so that the Berry phase for one revolution of $\pi$8 is

$\pi$9

The phase is therefore $\Phi_0/2$0 times the total canonical angular momentum.

For general flux $\Phi_0/2$1, the derivation combines the Byers–Yang gauge argument at half-integer flux with first-order perturbation in $\Phi_0/2$2. The result is

$\Phi_0/2$3

with

$\Phi_0/2$4

The correction $\Phi_0/2$5 contains condensate contributions through matrix elements of $\Phi_0/2$6 and is bounded by the $\Phi_0/2$7-sum rule, $\Phi_0/2$8. A naive BdG evaluation of $\Phi_0/2$9 diverges, and the data identify that divergence as a sign that continuity can be restored only by twisting the condensate phase globally.

A compact exact form follows from the adiabatic work–energy relation. If $e^{i\ell\varphi}$0 and $e^{i\ell\varphi}$1 are the total energies with and without the bound quasiparticle, then

$e^{i\ell\varphi}$2

Since the quasiparticle energy difference $e^{i\ell\varphi}$3 can be computed with high fidelity, this equation gives the exact Berry phase (Lin et al., 2017).

For a square-well Zeeman trap, the bound-state spectrum is

$e^{i\ell\varphi}$4

with

$e^{i\ell\varphi}$5

and hence

$e^{i\ell\varphi}$6

At $e^{i\ell\varphi}$7, the Berry phase is exactly $e^{i\ell\varphi}$8. Away from that point, it deviates by a non-universal amount controlled by $e^{i\ell\varphi}$9, $\phi$0, and $\phi$1.

4. Condensate deformation, odd-particle states, and topological implications

The number-conserving resolution requires explicit condensate deformation. The odd-fermion ground state is not a simple BdG single-quasiparticle excitation of the even-particle ground state. Lin and Leggett propose the ansatz

$\phi$2

with

$\phi$3

Its physical content is explicit: the extra fermion current inside the trap is compensated by a condensate counterflow outside, so that

$\phi$4

everywhere.

This compensation produces entanglement between quasiparticle and condensate degrees of freedom. The many-body ground state of an odd number of fermions therefore involves superfluid condensate deformation due to the presence of the bound quasiparticle. That effect is beyond the standard BdG description. The data further state that the condensate affects the part of the Berry phase not accounted for in the usual BdG framework, and that the corrected Berry phase becomes non-universal, depending on condensate density, gap $\phi$5, and trap geometry (Lin et al., 2017).

The implications extend to Majorana physics. In $\phi$6 superfluids or proximitized topological-insulator platforms, BdG predicts that exchanging two vortices carrying Majorana zero modes accumulates a non-Abelian Berry phase of $\phi$7, or more generally a unitary in the $\phi$8 ground-state manifold. The number-conserving analysis shows that when vortex cores come within a few coherence lengths of one another, superflow between them cannot be ignored and particle-number–phase entanglement modifies braiding Berry phases by non-universal amounts of order $\phi$9. The paper presents this as a potential limitation on topological protection if coherence lengths cannot be made arbitrarily small or if charging-energy effects lock the condensate phase over finite regions.

5. Josephson detection of vortex phase and the static phase-bias realization

A closely related superconducting manifestation concerns direct phase readout from a single Abrikosov vortex. Such a vortex carries a flux quantum,

$N$0

localized at its center, but induces a global $N$1 phase rotation in the superconducting condensate. The long-range gauge field outside the region pierced by magnetic field is attributed to the Aharonov–Bohm effect, and Josephson junctions provide phase-sensitive detectors of that rotation.

For a junction of length $N$2, the supercurrent obeys

$N$3

while the second London equation gives

$N$4

Integration across the barrier yields

$N$5

In zero external field, the phase difference is dominated by the vortex contribution,

$N$6

so that

$N$7

The observed Josephson phase shift is equal to the polar-angle difference spanned by the vortex across the junction (Golod et al., 2010).

Experimentally, the critical current in an in-plane field follows the Fraunhofer form

$N$8

with $N$9. In the presence of the vortex, the total phase is $N$0, and the entire Fraunhofer pattern is shifted horizontally by an effective flux $N$1 satisfying

$N$2

When the vortex approaches the junction, with $N$3 and $N$4, the phase profile becomes approximately a step of height $N$5 and

$N$6

The junction then enters the $N$7–$N$8 state. The reported manifestations are a minimum of $N$9 at $\phi$0, a half-period shift of the Fraunhofer lobes, and doubling of the periodicity on one side. Since

$\phi$1

the case $\phi$2 implies

$\phi$3

In long junctions, this offset locks at $\phi$4. The same summary interprets the vortex as a static and local “phase battery,” because varying $\phi$5 continuously tunes the static phase difference from $\phi$6 up to $\phi$7 and beyond in principle, with corresponding consequences for Cooper-pair number–phase complementarity.

6. Related vortex-beam number–phase correlations

A distinct but related usage of number–phase language appears in vortex-particle scattering. A twisted electron is an eigenstate of $\phi$8 with eigenvalue $\phi$9, and its idealized monochromatic Bessel-beam wave function is

$\pi$00

The phase singularity $\pi$01 and its associated azimuthal momentum are the key ingredients.

In the elastic $\pi$02 setup analyzed in the superkick framework, the vortex electron is modeled as a Laguerre–Gaussian wave packet and the probe as a tightly focused Gaussian. After the transverse integrations, the overlap integral contains the phase factor

$\pi$03

with $\pi$04 determined by the impact parameter and the total final transverse momentum. Because $\pi$05 depends on $\pi$06, the two-particle probability distribution is shifted away from $\pi$07. In the semiclassical limit,

$\pi$08

and equivalently

$\pi$09

The same analysis yields a single-particle azimuthal modulation,

$\pi$10

whose contrast is an order of magnitude larger for vortex–Gaussian collisions than for Gaussian–Gaussian collisions and peaks around $\pi$11 (Liu et al., 11 Mar 2025).

Neither the modulation nor the net shift appears for ordinary non-vortex beams. The paper therefore presents the superkick effect as a direct, kinematic signature of the number–phase structure of twisted electrons and as access to the topological charge $\pi$12. This is not the same problem as the particle-number-conserving Berry phase of a bound quasiparticle in a superfluid vortex, but it shows that vortex phase singularities can generate experimentally resolvable number–phase correlations across very different physical regimes.