Approximate vortex lattices of atomic Fermi superfluid on a spherical surface

The gist: The paper characterizes approximate vortex structures of atomic Fermi superfluids on a spherical surface as an analogue of the planar vortex-lattice problem by two constructions based on Ginzburg-Landau theory.

Theoretical Framework

The study considers an equal-population Fermi superfluid confined to a spherical surface with an effective monopole of strength g at the center of the sphere, scaling the length by radius R and energy by E0 = ħ2/2mF L2/2, where mF is the mass the fermion <ref:2604.05216#pg2>. The system near criticality when the effective magnetic field is strong and the order parameter is relatively small is described by Ginzburg-Landau theory with a free energy functional F2D[ψ] ≃ Z S2 d2r α ψ2 + β2/2 ψ4 + 1/2M ψ∗L2ψ <ref:2604.05216#pg3>. The linearized GL equation associated with the free energy functional is cast as an eigenvalue equation Hangψ = Eψ, where Hang = L2/J squared - (qg)2 <ref:2604.05216#pg4>. The simultaneous eigenstates are the monopole harmonics Yµ,j,m(θ, ϕ) [35]. For a monopole with effective magnetic charge g, the total flux through the surface is 4πg <ref:2604.05216#pg5>. Assigning q = 2e to the effective electric charge carriers of the Cooper pairs, the Dirac quantization condition gives qg = 2eg = N with integer values of N <ref:2604.05216#pg6>. Moreover, the effective flux of a vortex in the Fermi superfluid is ϕv = 2π/q, so the total number of flux quanta on the sphere becomes Nv = 4πgϕv = 2N <ref:2604.05216#pg7>.

Geometric Constructions

The paper identifies two main approaches for constructing vortex lattices using degenerate ground states of the linearized GL equation:

1. Geometric construction: This method employs scaffolds from the random, geodesic-dome, and Fibonacci lattices to construct the order parameter from the degenerate monopole harmonics <ref:2604.05216#pg2>. The zeros of the wavefunction are placed at these scaffold sites <ref:2604.05216#pg2>.

2. Minimization construction: This approach minimizes the free energy by numerically adjusting coefficients to find the solution with the minimal Abrikosov parameter βA <ref:2604.05216#pg2>.

The geometric scaffolds include:

. Random lattice:

This is constructed by drawing random points from a uniform distribution over the surface of a sphere, ensuring a uniform distribution of points since the differential solid angle, domega = sin θ dθ dϕ = -d(cos θ) dϕ, is uniform in cos θ and ϕ <ref:2604.05216#pg5>.

. Geodesic-dome lattice:

This counterpart to the triangular lattice on a sphere may be constructed by projecting a specific class of polyhedra, known as icosadeltahedra, onto the surface of a sphere <ref:2604.05216#pg5>. These polyhedra are characterized by their equilateral triangular faces and icosahedral symmetry <ref:2604.05216#pg5>. Three distinct geodesic-dome classes arise from specific choices of (h, k) <ref:2604.05216#pg5>.

. Fibonacci lattice:

The Fibonacci lattice provides an analytical construction of Nv approximately uniformly spaced points on a sphere, with the spherical coordinates given by θn = arccos(2n − 1)/Nv − 1 and ϕn = 2πnφ−1 (mod 2π), where φ = (1 + √5)/2 is the golden ratio <ref:2604.05216#pg5>. This construction places points along a golden spiral that winds from a pole to its antipole <ref:2604.05216#pg5>.

Numerical Minimization

The minimization construction involves finding the coefficients Cm in Eq. (6) that minimize the Abrikosov parameter βA, which reflects the nonuniformity of the solution <ref:2604.05216#pg2>. This is performed by varying the real and imaginary parts of complex coefficients in Eq. (6), constituting a parameter space of 2(2N + 1) real dimensions <ref:2604.05216#pg5>. The optimization employs two gradient-based algorithms from the SciPy library: The L-BFGS-B method and a trust-region constrained approach <ref:2604.05216#pg5>. Convergence is determined by checking the relative change in the function value and the norm of the gradient <ref:2604.05216#pg5>.

Results Comparison

The comparison between constructions reveals several key findings regarding the Abrikosov parameter βA <ref:2604.05216#pg2>.

. Geometric construction:

For small vortex numbers less than 20, the geodesic-dome lattice yields the smallest βA since perfect lattices are still allowed <ref:2604.05216#pg5>. As the vortex count increases, the geometric construction with the Fibonacci lattice produces comparable βA to the minimization solution, and both extrapolate to a value βA ≈ 1.16 given by the triangular vortex lattice on a 2D plane <ref:2604.05216#pg5>.

. Numerical minimization:

The minimization method produces lowβA solutions at some of the geometric-dome lattices <ref:2604.05216#pg5>. For Nv = 72, the minimization leads to a lattice structure resembling the geodesic-dome lattice, and the value of βA from minimization is typically lower than that from the geometric construction using the geodesic-dome lattice as a scaffold <ref:2604.05216#pg2>.

. Large Nv limit:

As Nv increases, both structures extrapolate to converge to 1.16 (the cross symbol) <ref:2604.05216#pg3>. The extrapolation confirms the numerical accuracy of our geometric and minimization constructions <ref:2604.05216#pg3>. The Fibonacci lattice provides a manageable approximation to the fully numerical minimization on the spherical surface <ref:2604.05216#pg3>.

Implications

The study suggests that approximate lattices manage to find an agreeable pattern in the large Nv limit, despite the lack of a genuine regular spherical lattice <ref:2604.05216#pg3>. The various possibilities of vortex configurations on a spherical surface illustrate interesting physics interfacing geometry, many-body systems, and collective excitations <ref:2604.05216#pg2>. Vortex cores are typically much smaller than the size of the atomic cloud, set by the healing length ξ <ref:2604.05216#pg2>. While bosonic superfluids show vortices as holes in the density profile, BCS fermionic superfluids have order parameter vanishing inside but particle density not necessarily vanishing completely due to unpaired fermions <ref:2604.05216#pg2>. The study concludes that spherical vortex systems of Fermi superfluids may be quasi-crystalline in nature without an obvious network of 5–7 dislocation pairs <ref:2604.05216#pg2>.

Conclusion

Given the constraint that perfect lattice structures on a spherical surface with more than 20 vertices are impossible, the investigation characterizes approximate vortex-lattice structures of Fermi superfluid on a spherical surface given by different constructions according to the Ginzburg-Landau theory in the presence of a monopole field <ref:2604.05216#pg2>. The geometric constructions provide intuitive structures based on degenerate monopole harmonics, while minimization generates configurations with lower nonlinear energies <ref:2604.05216#pg2>. As the number of vortices increases, the Fibonacci lattice converges towards the minimization solutions and provides an asymptotic approximation <ref:2604.05216#pg2>. The various possibilities of vortex configurations on a spherical surface thus illustrate interesting physics interfacing geometry, many-body systems, and collective excitations <ref:2604.05216#pg2>.

Acknowledgments

K. S. was supported by the NSF (Grant No. PHY2310656) <ref:2604.05216#pg2>. C. C. C. was supported by the DOE (Grant No. DE-SC0025809) <ref:2604.05216#pg2>. Part of the research was conducted using Pinnacles (NSF MRI, No. 2019144) at the Cyberinfrastructure and Research Technologies (CIRT) at University of California, Merced <ref:2604.05216#pg2>.

1. textbfTheoretically informed lattice construction for curved space simulations: The improved AI can now construct approximate vortex lattices on spherical surfaces by utilizing scaffolds like random, geodesic-dome, and Fibonacci lattices to construct the order parameter from the degenerate monopole harmonics. This allows for more realistic modeling of Fermi superfluids in thin spherical shell geometries where perfect planar lattices are impossible (While planar Fermi superfluids form Abrikosov vortex lattices under magnetic or effective gauge fields, spherical geometry forbids perfect lattices above 20 vortices).

2. textbfAdaptive energy minimization for complex topological systems: The improved AI can employ numerical optimization to minimize the free energy by numerically adjusting the coefficients to find the solution with the minimal Abrikosov parameter. This capability allows it to generate configurations that are energetically favorable, potentially leading to a more physically accurate description of vortex distribution than purely geometric constructions.

3. textbfLarge-N limit extrapolation for asymptotic accuracy: The improved AI can perform extrapolations of both geometric and minimization solutions as the number of vortices increases, noting that the Fibonacci lattice and minimization solution converge to the same value βA ≈ 1.16. This enables the system to provide reliable approximations in the large Nv limit, suggesting a method for simulating highly dense vortex configurations with known asymptotic behavior.

4. textbfTopology verification via current circulation analysis: The improved AI can verify that constructed vortex solutions are topological excitations by checking the gauge-invariant current circulating around each vortex using the formula provided in Eq. (7). This ensures that the simulated structures possess the correct topological character, confirming they are genuine vortices rather than just wave function zeros.

5. textbfModel selection based on physical constraints: The improved AI can use the results to inform experimental design by identifying which lattice structure is most physically relevant for a given vortex number, such as noting that the geodesic-dome prescription yields the smallest βA for certain vortex numbers, guiding experimentalists toward structures that minimize nonlinear energy.