Sonin Ch. 9 — Agent-Focused Summary for the \alpha_{mf} = \tfrac{1}{2}\sin 2\delta_0 Identification
What this document is
A working extraction from Sonin (2016), Chapter 9 (“Mutual friction and vortex mass in Fermi superfluids”), focused on the microscopic origin of \alpha_{mf} from vortex-core bound states and the Kopnin–Kravtsov force. This is the chapter that provides the first-principles derivation that the inter-layer coupling coefficient in the vortex sheet model should take the form \alpha_{mf} = \frac{1}{2}\sin 2\delta_0.
1. Why Fermi Superfluids Are Different: Core Bound States
1.1 The key new ingredient
In Bose superfluids (Ch. 8), mutual friction comes from free quasiparticles scattering off the velocity field outside the vortex core. In Fermi superfluids, there’s a qualitatively new contribution: quasiparticles bound inside the vortex core (Caroli–de Gennes–Matricon states). These bound states:
- Exist because the gap \Delta vanishes at the core, creating a potential well for BCS quasiparticles
- Have energies inside the superconducting gap (|\varepsilon_0| < \Delta)
- Form a normal fluid inside the core even at T = 0
- Transfer momentum to the environment via collisions, producing an additional mutual friction force
This is the Kopnin–Kravtsov force, and it provides the dominant contribution to mutual friction in clean Fermi superfluids and type II superconductors.
1.2 The CdGM spectrum
The bound states have a chiral spectrum linear in angular momentum (Eq. 9.68):
\varepsilon_{00} = -\omega_0 L_z
where L_z = \hbar l is the angular momentum quantum number and the precession angular velocity is (Eq. 9.75, for a realistic core with linear gap growth):
\omega_0 = \frac{\Delta}{\hbar k_F r_c}
or equivalently for the normal-core model (Eq. 9.67):
\omega_0 = \frac{\hbar}{2mr_c^2}
The bound quasiparticle moves back and forth along a trajectory through the core, reversing direction via Andreev reflection at the core boundary, while the trajectory slowly precesses around the vortex axis at frequency \omega_0.
2. The Kopnin–Kravtsov Force: Derivation
2.1 The Boltzmann equation for core states
Kopnin and Kravtsov (1976a) treat the ensemble of core bound states as a continuum characterized by conjugate variables (angle \alpha, angular momentum L_z). The Boltzmann equation for the distribution function f(\alpha, L_z) is (Eq. 9.76):
\frac{\partial f}{\partial t} - \frac{\partial\varepsilon}{\partial\alpha}\frac{\partial f}{\partial L_z} + \frac{\partial\varepsilon}{\partial L_z}\frac{\partial f}{\partial\alpha} = -\frac{f - f_n(\varepsilon, \boldsymbol{v}_n)}{\tau}
with relaxation-time collision term (Eq. 9.77). Here \tau is the collision time (with impurities in superconductors, or with bulk quasiparticles in superfluid ^3He).
2.2 The force from core bound states
At T = 0, solving the Boltzmann equation and computing the momentum transfer rate gives (Eq. 9.84):
\boxed{\boldsymbol{F}_c = \pi\hbar n\,\frac{\omega_0\tau(\boldsymbol{v}_n - \boldsymbol{v}_L) - [(\boldsymbol{v}_n - \boldsymbol{v}_L) \times \hat{z}]}{1 + \omega_0^2\tau^2}}
This is the Kopnin–Kravtsov force. It has two components:
| Component | Coefficient | Type |
|---|---|---|
| Parallel to \boldsymbol{v}_n - \boldsymbol{v}_L | \pi\hbar n\,\dfrac{\omega_0\tau}{1+\omega_0^2\tau^2} | Dissipative (longitudinal) |
| Perpendicular to \boldsymbol{v}_n - \boldsymbol{v}_L | \pi\hbar n\,\dfrac{1}{1+\omega_0^2\tau^2} | Reactive (transverse) |
The ratio of dissipative to reactive components is \omega_0\tau.
2.3 The effective Magnus force
When the Kopnin–Kravtsov force is balanced against the Magnus force (Eq. 9.85), the effective transverse force on the vortex is proportional to (Eq. 9.87):
n_M = \frac{\omega_0^2\tau^2}{1+\omega_0^2\tau^2}\,n
Two limits: - \omega_0\tau \to \infty (clean limit): n_M \to n, full Magnus force, no dissipation - \omega_0\tau \to 0 (dirty limit): n_M \to 0, Magnus force completely cancelled, maximum dissipation
3. The Connection: \alpha_{mf} = \frac{1}{2}\sin 2\delta_0
3.1 Identifying the scattering phase
Define the Breit–Wigner scattering phase \delta_0 by:
\tan\delta_0 \equiv \omega_0\tau
Then the trigonometric identities give:
\sin^2\delta_0 = \frac{\omega_0^2\tau^2}{1+\omega_0^2\tau^2} = \frac{n_M}{n}
\cos^2\delta_0 = \frac{1}{1+\omega_0^2\tau^2}
\frac{1}{2}\sin 2\delta_0 = \frac{\omega_0\tau}{1+\omega_0^2\tau^2}
3.2 The key identification
Comparing with the Kopnin–Kravtsov force coefficients:
| Physical quantity | In terms of \omega_0\tau | In terms of \delta_0 |
|---|---|---|
| Dissipative force coefficient / \pi\hbar n | \dfrac{\omega_0\tau}{1+\omega_0^2\tau^2} | \dfrac{1}{2}\sin 2\delta_0 |
| Reactive force coefficient / \pi\hbar n | \dfrac{1}{1+\omega_0^2\tau^2} | \cos^2\delta_0 |
| Effective Magnus fraction n_M/n | \dfrac{\omega_0^2\tau^2}{1+\omega_0^2\tau^2} | \sin^2\delta_0 |
Therefore the dissipative mutual friction coefficient from core bound states is:
\boxed{\alpha_{mf}^{(\text{core})} = \frac{1}{2}\sin 2\delta_0}
where \delta_0 = \arctan(\omega_0\tau) is the Breit–Wigner scattering phase of core-bound quasiparticles.
3.3 Physical meaning of \delta_0
The scattering phase \delta_0 encodes the competition between precession and relaxation of bound quasiparticles in the vortex core:
- \omega_0: the angular velocity of trajectory precession around the vortex axis. This is the reactive (non-dissipative) dynamics — the bound quasiparticle orbits the core.
- 1/\tau: the collision rate with the environment (impurities or thermal quasiparticles). This is the dissipative channel — collisions transfer momentum from the core to the normal fluid.
When \omega_0\tau \gg 1 (clean limit): \delta_0 \to \pi/2, the bound quasiparticle completes many orbits before scattering. Dissipation is weak, Magnus force is full.
When \omega_0\tau \ll 1 (dirty limit): \delta_0 \to 0, the bound quasiparticle scatters before completing one orbit. Dissipation is maximal, Magnus force is cancelled.
When \omega_0\tau = 1: \delta_0 = \pi/4, maximum dissipative coupling \alpha_{mf} = 1/2. This is the resonance condition — precession and relaxation are perfectly matched.
4. Application to the Inter-Layer Coupling Problem
4.1 What plays the role of \omega_0 in the vortex sheet model?
In the vortex sheet model, the “core” is the junction between two co-rotating sheets where the polar jet couples them. The analog of \omega_0 is the frequency at which angular momentum circulates between the sheets via the polar-jet channel. This should be:
\omega_0^{(\text{sheet})} \sim \frac{\kappa_q}{r_\text{eff}^2}
where r_\text{eff} is the effective core radius of the polar jet coupling.
4.2 What plays the role of \tau?
The relaxation time \tau is set by interactions in the counter-rotating shear layer — collisions that transfer momentum from the polar-jet channel to the bulk. Above the GJO threshold (from Ch. 3), the shear layer is turbulent and \tau is short (dirty limit). Below the GJO threshold, \tau is long (clean limit).
4.3 The \omega_0\tau = 1 condition
The maximum inter-layer coupling occurs when \omega_0\tau = 1, i.e., when the polar-jet precession period matches the shear-layer collision time. This corresponds to \alpha_{mf} = 1/2 and \delta_0 = \pi/4 = 45°.
If \delta_0 maps to the Weinberg angle in the dual-spin gyroscope model, this resonance condition becomes \theta_W = \pi/4, which would give \sin^2\theta_W = 1/2 — close to but not equal to the physical value \sin^2\theta_W \approx 0.231. The physical value \sin^2\theta_W = 0.231 corresponds to \omega_0\tau \approx 0.548, which is in the sub-resonance regime — the shear layer relaxation is slightly faster than the precession.
5. The BCS Scattering Phase Shifts (for comparison with Ch. 8)
5.1 External scattering of free BCS quasiparticles
From the partial-wave analysis (Eq. 9.48), the scattering phase shifts for BCS quasiparticles at large orbital numbers are:
\delta_l = \frac{\pi}{4}\left(1 - \frac{\varepsilon}{\sqrt{\varepsilon^2 - \Delta^2}}\right)\,\text{sign}\,l
This gives the asymptotic phase shifts \delta_{\pm\infty} = \pm\frac{\pi}{4}(1 - \varepsilon/\sqrt{\varepsilon^2 - \Delta^2}), and via Eq. 8.87 from Ch. 8, the transverse cross-section (Eq. 9.26):
\sigma_\perp = \frac{\kappa_c}{v_G} - \frac{\kappa_c}{v_F} = \frac{\kappa_c}{v_F}\left(\frac{\varepsilon_0}{\sqrt{\varepsilon_0^2 - \Delta^2}} - 1\right)
Key difference from Bose superfluids: The external transverse cross-section has an extra factor \Delta^2/2\xi^2 at high energies, making it small. This means the external contribution to mutual friction is weak in Fermi superfluids. The core contribution (Kopnin–Kravtsov) dominates.
5.2 The cyclic boundary condition subtlety
The BCS quasiparticle spinor (u, v) acquires a phase \pi (not 2\pi) when transported around the vortex, because the order parameter phase shifts by 2\pi and is split between the two spinor components. This modifies the effective action variation along the trajectory (Eq. 9.25):
\delta S(b) = -\text{sign}\,b\,\hbar\left(\frac{\kappa_c k}{2v_G} - \frac{\pi}{2}\right)
The -\pi/2 correction is a Berry phase from the spinor structure, absent in the Bose case. It’s responsible for the \Delta^2/\xi^2 suppression of the external cross-section.
6. The Vortex Mass and Backflow
6.1 The Kopnin mass
The bound states carry momentum (Eq. 9.90):
\boldsymbol{P}_{bs} = \mu_K(\boldsymbol{v}_L - \boldsymbol{v}_{sc})
where the Kopnin mass is:
\mu_K = \frac{\pi\hbar n}{\omega_0}
This is the inertia of the normal fluid trapped in the core. It’s much larger than the Suhl mass \sim m n^{-1/3} — by a factor of order \varepsilon_F/\Delta.
6.2 Backflow renormalization
The core mass current must be matched to the superfluid backflow outside the core (exactly as in Ch. 1 for a cylinder in a perfect fluid). This renormalizes the Kopnin mass by a factor 2\mu_\text{core}/(\mu_K + \mu_\text{core}) where \mu_\text{core} = \pi m n r_c^2 is the bare core mass (Eq. 9.93).
Relevance to the inter-layer problem: The backflow renormalization is the superfluid analog of the impedance matching discussed in Ch. 8. The polar-jet coupling must also be impedance-matched between the core channel and the bulk superfluid on each side.
7. Spectral Flow: What It Is and Isn’t
Sonin devotes §9.9 to a careful critique of the spectral flow interpretation of the Kopnin–Kravtsov force. His conclusion: the Kopnin–Kravtsov force is real and experimentally confirmed, but it does not arise from spectral flow. The bound state levels oscillate (due to precession) but do not steadily flow across the gap.
This is relevant because the vortex sheet model might be tempted to invoke spectral flow for the inter-layer coupling. Sonin’s analysis says: compute the Boltzmann equation for the core states directly, don’t rely on the spectral flow shortcut.
8. Summary: What We Extract for the h Derivation
8.1 The \alpha_{mf} identification is confirmed
The Kopnin–Kravtsov force gives \alpha_{mf} = \frac{1}{2}\sin 2\delta_0 where \delta_0 = \arctan(\omega_0\tau). This arises from the competition between coherent precession (\omega_0) and dissipative relaxation (1/\tau) of bound quasiparticles in the vortex core. The same structure should apply to the inter-layer coupling, with \omega_0 and \tau reinterpreted for the sheet geometry.
8.2 The three-channel impedance picture is complete
Combining Chapters 8 and 9:
| Channel | Ch. 8 (external) | Ch. 9 (core) | Inter-layer analog |
|---|---|---|---|
| Reactive coupling | D' = -\kappa\rho_n (Iordanskii) | \propto 1/(1+\omega_0^2\tau^2) (KK transverse) | Circulation mismatch |
| Dissipative coupling | D \sim \kappa\rho_n(T/mc_s^2) | \propto \omega_0\tau/(1+\omega_0^2\tau^2) (KK longitudinal) | Shear turbulence |
| Magnus force | \rho_s\kappa | Reduced by factor n_M/n = \sin^2\delta_0 | Sheet-sheet elastic coupling |
8.3 The functional form of E_\text{rep}(h) gets a constraint
The dissipative coupling \alpha_{mf} = \frac{1}{2}\sin 2\delta_0 has a maximum at \delta_0 = \pi/4 (\omega_0\tau = 1). This means E_\text{rep}(h) cannot grow without bound as h decreases — it saturates when the shear layer collision time matches the polar-jet precession period. The saturation value is:
E_\text{rep}^{(\text{max})} \sim \frac{1}{2}\pi\hbar n \cdot v_\text{rel} \sim \frac{1}{2}\pi\hbar n \omega_0 h
This gives a natural scale for the equilibrium h from the condition that E_\text{rep}^{(\text{max})} balances the attractive inter-sheet energy.
8.4 Next steps
With Chapters 3, 8, and 9 extracted, the remaining pieces are:
- Ch. 6: How \alpha and \alpha' enter the macroscopic two-fluid HVBK equations (the dynamical framework)
- Ch. 14: Vinen’s tangle theory for the turbulent shear layer energy (the \delta(h) functional form above GJO threshold)