It is the Landau branch near its minimum: one density-wave quantum with energy Δ at wavevector k₀.
The smoke-ring picture is historical. In 2-D films a lower branch appears: Δ/k_B ≈ 5 K, k₀ ≈ 1.8 Å⁻¹, m* = 0.20 m₄ (source · typo).
The normal density is the momentum carried by thermal quasiparticles per unit drift velocity. In two dimensions ρₙ²ᴰ = −(ħ²/2)∫ d²k/(2π)² k² ∂n_B/∂ε. Near the roton minimum n_B ≈ e^{−ε/k_BT} (Δ ≫ k_BT), k² ≈ k₀², and the integral over the ring of radius k₀ gives the width of the Gaussian in q = k − k₀:
which is the paper's eq (12). The 3-D roton formula has the same structure with k₀⁴ and T^{−1/2} replaced by the 3-D phase-space factors. Varga 2022 (eq 6, local) converts the areal density to a suppression of the 3-D density in a channel of height D: Δρₙ = (2/D) ρₙ²ᴰ, i.e. one such layer at each wall.
Why “quasiparticle”: ε(k) is a branch of the dynamic structure factor S(k,ω); the roton is the quantum of that branch at k₀. It carries momentum and energy, not circulation, and it has no conserved winding number (see §4).
The channel is — roton wavelengths thick. “2-D roton” does not mean the film is a roton wavelength thick; it means what the sources say: a dense wall layer counted per area and a strictly 2-D spectrum with a lower gap.
Rotational spiral? No source supports one.
Point vortices in a film are a 2-D Coulomb gas. Below T_KT they are bound in pairs; at T_KT the universal jump fixes ρₛ²ᴰ/T_KT.
Live: the Kosterlitz flow (K, y) and T_KT versus channel height, with and without the roton term ρₛ²ᴰ = ρ_sb D_c − Δρₙ²ᴰ.
Linearise the Coulomb gas about the dilute limit. A pair at separation r = a₀e^l screens the interaction of larger pairs; integrating out the shell (l, l+dl) renormalises the stiffness and the fugacity (Nelson 1995 eq 2.48): dK⁻¹/dl = 4π³y² + O(y⁴), dy/dl = (2 − πK)y + O(y³). The fixed line y = 0 is attractive for K > 2/π (pairs stay bound, y → 0) and repulsive for K < 2/π (y grows, free vortices). The last trajectory that still reaches y = 0 ends exactly at K = 2/π; translating K(∞) = 2/π back to ρₛ²ᴰ gives eq (1).
In the paper's static estimate (Table 1, columns 2–3) the recursion is not run: eq (1) is applied directly to the bare ρₛ²ᴰ, i.e. ε ≈ 1 below T_KT (p. 5). The dynamic fit (§6) does run the recursion, truncated at l_max = ln(r_D/a₀), and defines T_KT^fit by K(l_max) = 2/π.
The paper's eq (17) carries a third, velocity term ½ z I₁(z)/I₀(z), z = 2πK₀e^l V_s/V₀, following ref [10]; the fit treats V_s/V₀ as a parameter. It is set to zero in this static picture and live in §6.
Rotons: none (no winding number). ⁴He vortices: a density-set Berry phase, not statistics. Laughlin quasiholes: abelian, θ = π/m. Ising and Fibonacci anyons: non-abelian, need a degenerate fusion space.
Drag the probe around the anchor and read the phase under each rule.
Vortices ↔ charges of a 2-D Coulomb gas (§3). A vortex taken round a loop picks up φ = 2π n A: huge, area-dependent, and continuous in the density. Exchanging two identical ⁴He vortices therefore gives no quantised angle — no anyon class.
In the composite-boson Chern–Simons picture a Laughlin quasihole is a vortex, and its loop phase is 2π × enclosed charge/e = 2π/m per quasihole enclosed. ⁴He lacks flux attachment, so its vortex phase counts atoms, not a fixed fraction — θ is not defined.
Need a degenerate fusion space: σ × σ = 1 + ψ (Ising), τ × τ = 1 + τ (Fibonacci). Candidate hosts for Ising σ: half-quantum vortices of p+ip pairing — not ⁴He. Fibonacci τ: Z₃ Read–Rezayi at ν = 12/5 (checked), no superfluid realisation.
| object | winding number | loop phase | fusion space | class |
|---|---|---|---|---|
| roton | none (density wave) | dynamical only, ∫ε dt/ħ | — | not topological |
| ⁴He vortex | ±1 in the phase | 2π n A (density-set) | 1-dim | none: no quantised θ |
| Laughlin quasihole | ±1 (composite-boson vortex) | 2π w/m | 1-dim | abelian, θ = π/m |
| Ising σ | half-quantum vortex + Majorana | diag(e^{−iπ/4}, e^{3iπ/4}) per loop | 2^{n−1} | non-abelian (not universal) |
| Fibonacci τ | parafermion quasihole | diag(e^{−8πi/5}, e^{6πi/5}) per loop | Fib(3n−2) | non-abelian, universal |
Rule A (⁴He vortex). The vortex's position (x₀, y₀) is a slow parameter of the many-body state; the Berry connection is the momentum density integrated over the fluid, and the curvature is 2π × local atom density. Around a loop of area A: γ = 2π n A (the Magnus/Kutta–Joukowski phase). It also follows from the composite picture of (b) with the “flux” set to one quantum per atom. Sources unsourced (Haldane–Wu 1985, Ao–Thouless 1993).
Rule B (Laughlin). Arovas–Schrieffer–Wilczek (checked): taking a quasihole around a loop gives γ = −2π⟨n⟩_R; a second quasihole inside the loop changes ⟨n⟩_R by −ν = −1/m, so the relative phase for one full loop is Δγ = 2πν = 2π/m, i.e. exchange θ = π/m. The number is set by ν, not by the loop's area, because the state is incompressible.
Rule C (Ising, Fibonacci). With a degenerate space, a loop acts as a matrix: a full loop of one σ around another is R² in the fusion basis {1, ψ}: Kitaev 2006 Table 1 (local, ν = 1, κ = 1) gives R₁^{σσ} = e^{−iπ/8}, R_ψ^{σσ} = e^{3iπ/8}, so R² = diag(e^{−iπ/4}, e^{3iπ/4}). Fibonacci: R₁^{ττ} = e^{−4πi/5}, R_τ^{ττ} = e^{3πi/5} (standard values; not in the local extracts — unsourced), giving R² = diag(e^{−8πi/5}, e^{6πi/5}). The two eigenphases differ, so the loop is not a phase but a rotation in the fusion space; braids with a third anyon do not commute with it. Read–Rezayi 1999 (checked): for Z₃ parafermions and 3n quasiholes the space has Fibonacci dimension F_{3n−2}.
Rotons. There is no slow parameter with a winding: a roton is an excitation with definite momentum, so moving it around a loop produces only the dynamical phase, and two rotons have ordinary Bose statistics.
In a channel the normal fluid is clamped, so only ρₛ moves: c₄² ≈ (ρₛ/ρ)c₁². The device is a Helmholtz resonator (eqs 8–9): the symmetric mode f₀ reads the bulk-like ρ_sb; the antisymmetric mode f₁ pushes superfluid through the nanochannel and reads ρ_sc/ε.
Each filling channel is a slug of superfluid (mass 2ρ_sb a l per basin) driven by the basin pressure: 2ρ_sb a l ÿ₁ = −2(ρ_sb/ρ) a δp₁ (eq 2). The basin pressure is set by its compliance: δp₁ = k_p(2aρ_sb y₁ − a_c ρ⁰_sc y_c)/(ρ(2A² + V_Bχk_p)) (eq 6), where k_p is the membrane stiffness, A and V_B the basin area and volume, χ the compressibility. Try y ∝ e^{iωt}.
Symmetric mode (y₁ = y₂, y_c = 0): ω₀² = (ρ_sb/ρ²) 2ak_p/(l(2A² + V_Bχk_p)) — eq (8). Antisymmetric mode (y₁ = −y₂): the nanochannel slug moves with ε ρ⁰_sc a_c l_c ÿ_c = (ρ⁰_sc/ρ) a_c (δp₁ − δp₂), giving ω₁² = ω₀²[1 + ρ⁰_sc a_c l/(ε ρ_sb a l_c)] — eq (9). Note: eq (5) as printed has no factor a_c on the pressure term; with that factor the dimensions balance and eq (9) follows (page observation).
Then ρ_sb/ρ = f₀²(T)/f₀²(0) (eq 10) and ρ_sc/ρ = (f₁² − f₀²)/(f₁²(0) − f₀²(0)) (eq 11), assuming ρ_sb = ρ⁰_sc = ρ and ε ≈ 1 at T → 0. f(0) must be extrapolated from above 1.2 K (SI B, eqs 21–22): the choice shifts the extracted roton gap. With nominal geometry (A = π(2 mm)², H = 460 nm, a = 0.1 mm × H, l = 2 mm, k_p = 10⁷ N/m, χ = 1/(ρc₁²) with c₁ ≈ 238 m/s — UNSOURCED, the paper takes χ from ref [24]) the model gives f₀(0) ≈ 0.5 kHz against measured 0.75–1.14 kHz; the SI absorbs this in fitted α, β.
At frequency ω the Coulomb gas is probed only out to r_D = √(14D/ω): ε(ω) = 1 + ε_b + ε_f. The dissipation is Q⁻¹ ∝ Im ε/|ε|² (eq 14) after isolating the channel (eq 15).
The paper fits ε_b alone and finds D ≈ 10⁻¹¹ cm² s⁻¹, seven orders below the usual ~10⁻⁴; it reports that adding ε_f “was found to produce negligible differences”. Below: both models fitted to the L1 data, numbers only.
Bound pairs. The static dielectric function at scale l is ε̃(l) = K₀/K(l). At frequency ω only pairs with r < r_D = √(14D/ω) respond (the pair diffuses a distance r_D in a cycle), so Re ε(ω) = ε̃(l_max). The lag of pairs near r_D gives Im ε(ω) = (π/4) dε̃/dl at l_max; with dK/dl = −4π³K²y² this is (π/4)·4π³K₀y² = π⁴K₀y²(l_max) — eq (19). (Leggett L12 eqs 10–11 give the same structure “up to numerical constants of order unity”.)
Dissipation. The antisymmetric mode has ω₁² ∝ ρ_sb/ρ² + ρ⁰_sc a_c l/(ε ρ² a l_c); with ε complex, Q⁻¹ = Re ω₁/Im ω₁ (eq 13) and to leading order in ε⁻¹ and a_c/a: Q⁻¹ ≈ (Im ε/|ε|²)(ρ⁰_sc a_c l)/(ρ_sb a l_c) — eq (14), i.e. Im(−1/ε) times a geometric prefactor, the quantity torsional oscillators measure with prefactor ρₛ²ᴰA/M. The nanochannel part is isolated as Q⁻¹ = Q⁻¹_1c − Q⁻¹_0d f_0d/f_1c (eq 15), the f₀ mode measuring the inlet viscous background.
Free vortices. Above T_KT unbound vortices of areal density n_f move with mobility μ = D/k_BT across the superflow and dissipate linearly (Leggett L12 eq 7: friction ∝ n_f ∝ ξ₊⁻²). In the Coulomb-gas language they are free charges: a conductivity σ_f = n_f q² μ adds ε_f = i·4πσ_f/ω to the dielectric function. With q² ∝ K₀k_BT the page writes ε_f = i·4π²K₀n_f D/ω; the exact O(1) prefactor in AHNS [21] (paywalled) is UNSOURCED here and is absorbed in the fitted amplitude. Bishop & Reppy's original fit carried an “overall constant multiplying the free-vortex contribution” F (Leggett L12 eq 13), so including such a term is the historical practice; the paper omitted it in the final fit.
Where n_f comes from. The page's main free model takes ξ₊ from the same recursion that gives ε_b: l* is the scale at which y(l) reaches 1 (pairs are unbound beyond ξ₊ = a₀e^{l*}), n_f = F/ξ₊², so ε_f = i(4π²/14)K₀F·e^{2(l_max − l*)} — one new parameter, nested (F → 0 is the paper's model exactly), and dependent only on D/a₀² like the rest. A second variant was also fitted: n_f = n_f0 exp(−2b/√t) with t = (T − T_KT^fit)/T_KT^fit, the textbook asymptotic form (Leggett eq 5). In these channels the dissipation peak spans ~10–20 mK, i.e. t ≲ 0.01, where exp(−2b/√t) ≈ e^{−20b}: with b of order 1 the term is dead on arrival, and the fits left n_f0 and b essentially at their starting values with no change in RSS (kept in the table for the record). The point is structural: a reduced temperature measured against T_KT is the wrong coordinate when ρₛ(T) itself collapses 10 mK above T_KT; the recursion-derived ξ₊ carries that collapse automatically.
Two fitting facts the table depends on. (i) The bound-only RSS surface in (D, y₀, V_s/V₀) has more than one minimum; for 15 nm the fit started from the paper's parameters stopped at a higher one, and the lower one was only found by starting from the free-vortex solution with F = 0. The table therefore uses, for each model, the best of several starts (paper's parameters, the other model's solution, the other cutoff's solution with D rescaled by (ξ/a₀)², the earlier runs' solutions), and whenever the free model ends lower the bound model is restarted from it with F = 0, so the nested comparison is between true minima by construction. (ii) With V_s/V₀ held near the paper's 0.1 the model cannot be evaluated at large D at all: z = 2πK₀e^l·V_s/V₀ grows with the flow and saturates y for every T once l_max ≳ 5, so the D profile is followed by continuation with V_s/V₀ ∝ e^(−Δl_max). At the literature value D ~ 10⁻⁴ the model therefore needs a correspondingly smaller V_s/V₀ (the refitted value is quoted in the measured statement below) — a statement about the model's velocity term, not about helium.
Two caveats the page inherits from the paper. D is taken constant although it is expected to grow near T_KT (p. 8); and the model's T_KT is defined by K(l_max) = 2/π at the fitted D, so changing D moves T_KT^fit as well as the width.