Sonin Ch. 8 — Agent-Focused Summary for the \alpha_{mf} Identification and Inter-Layer Coupling

What this document is

A working extraction from Sonin (2016), Chapter 8 (“Mutual friction”), focused on two problems: (1) deriving the mutual friction coefficients \alpha and \alpha' from microscopic scattering theory, and (2) determining whether \alpha_{mf} correctly describes the coupling between co-rotating vortex sheets separated by a counter-rotating shear layer. Annotated with connections to the E_\text{rep}(h) derivation from the Ch. 3 summary.

1. The Force on a Vortex: Dissipative + Reactive Decomposition

1.1 The friction force structure

When a vortex moves at velocity \boldsymbol{v}_L relative to the normal component at \boldsymbol{v}_{nl} (the local normal velocity near the vortex), the friction force per unit length is (Eq. 8.2):

\boldsymbol{F}_{fr} = D(\boldsymbol{v}_{nl} - \boldsymbol{v}_L) + D'[\hat{z} \times (\boldsymbol{v}_{nl} - \boldsymbol{v}_L)]

Two components: - D (longitudinal/dissipative): Force parallel to relative velocity. This is the “drag” — it dissipates energy. - D' (transverse/reactive): Force perpendicular to relative velocity. This does no work — it’s a Hall-like deflection.

This is the decomposition that maps onto the dual-spin gyroscope model. The reactive component D' rotates the coupling direction; the dissipative component D provides energy transfer. The ratio D'/D is the tangent of the “scattering phase” that becomes the Weinberg angle analog.

1.2 The Magnus force balance

The friction force must be balanced by the Magnus force from the superfluid (Eq. 8.6):

\rho_s[\boldsymbol{\kappa} \times (\boldsymbol{v}_{sl} - \boldsymbol{v}_L)] = \boldsymbol{F}_{fr}

This is the equation of vortex motion — it determines \boldsymbol{v}_L given the local superfluid and normal velocities. It’s the single-vortex version of what enters the HVBK equations macroscopically.

1.3 Connection to macroscopic \alpha and \alpha'

The macroscopic mutual friction parameters \alpha and \alpha' are connected to the microscopic parameters D and D' by (Eq. 8.7):

\alpha - j\alpha' = \frac{1}{\kappa\rho_s}\left[\frac{\ln(r_m/r_l)}{4\pi\rho_n\nu} + \frac{1}{D + jD'} - \frac{1}{j\rho_s\kappa}\right]^{-1}

where j is the complex unit. This is a j-complex impedance formula — the three terms in the denominator represent three channels of momentum transport in series: viscous drag in the normal fluid, microscopic scattering at the vortex, and the Magnus force from the superfluid.

Key for our problem: The inter-layer coupling in the vortex sheet model is analogous to this three-channel structure. The “microscopic scattering” channel is the polar-jet coupling through the vortex cores; the “viscous drag” channel is the turbulent shear layer; and the “Magnus force” channel is the circulation mismatch across the boundary.

2. The Two Regimes and Their Key Results

2.1 Low temperature: ballistic quasiparticle scattering

When the quasiparticle mean free path l_{qp} exceeds the vortex core radius, mutual friction comes from individual quasiparticles scattering off the velocity field around the vortex. The key results for rotons (Eq. 8.40–8.41):

D = \frac{\kappa\rho_n}{\sqrt{2\pi^3}}\frac{\sqrt{\mu T}}{p_0}\left(\ln\frac{p_0}{\sqrt{\mu T}}\right)^3, \qquad D' = -\kappa\rho_n

And for phonons:

D \sim \kappa\rho_n \frac{T}{mc_s^2}, \qquad D' = -\kappa\rho_n

The crucial universal result: D' = -\kappa\rho_n for all quasiparticle types, while D \ll |D'| at low temperatures. This means:

The friction is predominantly reactive (transverse), not dissipative.
The vortex moves approximately with the local centre-of-mass velocity (Eq. 8.42): \boldsymbol{v}_L = \frac{\rho_s}{\rho}\boldsymbol{v}_{sl} + \frac{\rho_n}{\rho}\boldsymbol{v}_{nl}

2.2 Near the critical temperature: Ginzburg-Landau regime

When the core radius exceeds l_{qp}, mutual friction comes from order parameter relaxation. The parameters near T_c are (Eq. 8.116):

d = \frac{1}{4\Lambda\rho_s}\int_0^\infty \frac{r\,dr}{\rho_s(r)}\left[\left(\frac{\partial\rho_s}{\partial r}\right)^2 + \left(\frac{\Lambda\rho_s}{S}\frac{\partial S}{\partial r}\right)^2\right]

d' - 1 = -\frac{\Delta C}{\rho_s S}\frac{T_c - T}{T_c}

where \Lambda \sim (T_c - T)^{-1/3} is the relaxation parameter. The dissipative parameter d diverges as (T_c - T)^{-1/3} approaching the critical point.

Key insight for our model: The dissipative parameter d depends on gradients of \rho_s and S inside the vortex core, while the reactive parameter d' - 1 depends only on the global entropy difference between the normal and superfluid states. The reactive part is “topological-ish” (determined by bulk thermodynamics), while the dissipative part requires microscopic core structure. This matches the structure of the Weinberg angle derivation, where the reactive/dissipative decomposition of \alpha_{mf} maps to \cos\theta_W / \sin\theta_W.

3. The Transverse Force: Aharonov–Bohm Effect and Scattering Phase Shifts

3.1 Why the transverse force matters for \alpha_{mf}

The transverse cross-section (which determines D') has a universal form (Eq. 8.26):

\sigma_\perp = \frac{\kappa}{v_G}

This is independent of the details of the quasiparticle spectrum — it depends only on the circulation quantum and the group velocity. Its origin is the Aharonov–Bohm effect: the sound wave acquires different phases on the two sides of the vortex, producing interference in a narrow forward cone.

3.2 Partial-wave analysis: the scattering phase shifts \delta_l

In the partial-wave expansion (§8.5), the transverse cross-section is given by the Cleary formula (Eq. 8.81):

\sigma_\perp = \frac{1}{k}\sum_l \sin(2\delta_l - 2\delta_{l+1})

For the Aharonov–Bohm problem, the phase shifts are (Eq. 8.78):

\delta_l = (|l| - |l - \gamma|)\frac{\pi}{2}

where \gamma = -\kappa k / 2\pi c_s is the AB parameter (ratio of circulation to wavelength). This gives:

\sigma_\perp = -\frac{1}{k}\sin 2\pi\gamma

For small \gamma: \sigma_\perp \approx -2\pi\gamma/k = \kappa/c_s, recovering the universal result.

3.3 The key structural formula

For general scattering (not just AB), when the phase shifts \delta_l are small, the transverse cross-section depends only on the asymptotic phase shifts at l \to \pm\infty (Eq. 8.87):

\boxed{\sigma_\perp = \frac{2(\delta_{-\infty} - \delta_\infty)}{k}}

This is the equation that connects to \alpha_{mf} = \frac{1}{2}\sin 2\delta_0 in the Kopnin framework. In the Kopnin theory (Ch. 9), \delta_0 is the Breit–Wigner scattering phase of quasiparticles off vortex-core bound states, and the mutual friction coefficient comes from the same partial-wave structure. The difference is that in Ch. 8 the scattering is off the velocity field outside the core, while in Ch. 9 it’s off bound states inside the core. Both produce forces with the same dissipative/reactive decomposition.

4. The Berry Phase Connection

4.1 Berry phase and the transverse force (§8.6)

The Berry phase accumulated when a vortex is transported adiabatically around a closed loop is (Eq. 8.99):

\Delta S_B = -A\frac{\kappa}{2\pi}\oint(d\boldsymbol{l} \cdot \boldsymbol{j})

where A is the loop area and \boldsymbol{j} is the total mass current. The key debate (Ao–Thouless vs. Sonin/Iordanskii): does the current circulation include normal-fluid circulation?

Resolution (Thouless et al. 2001; Sonin 2002): Yes. The total current circulation is \oint(d\boldsymbol{l}\cdot\boldsymbol{j}) = \rho_s\kappa + \rho_n\kappa_n, where the normal circulation \kappa_n is given by (Eq. 8.95):

\kappa_n = -\frac{\mathcal{D}'}{\rho_n}

Relevance to \alpha_{mf}: The Berry phase is not purely topological — it depends on the microscopic scattering that determines D' and hence \kappa_n. Sonin states this explicitly: “Topology is not sufficient to determine the transverse force on the vortex. The magnitude of the transverse force must be determined not from the Berry phase, but vice versa.” This means the \alpha_{mf} identification in the vortex sheet model cannot be fixed by topology alone — it requires the microscopic calculation from Ch. 9.

4.2 The normal circulation at the inter-layer boundary

For our problem, the normal circulation \kappa_n at the boundary between co-rotating sheets has a direct physical interpretation: it’s the circulation of the normal component in the counter-rotating shear layer. Using (Eq. 8.96):

\kappa_n = \frac{\kappa}{1 + [\kappa\ln(r_m/r_l)/4\pi\nu]^2}

At low temperatures where \nu is small (the mean free path is large), \kappa_n \to 0 — the normal circulation is suppressed by viscous effects. At high temperatures near T_c, \kappa_n \to \kappa — the normal circulation equals the superfluid circulation.

5. Multi-Scale Momentum Transport (Fig. 8.5)

Sonin identifies three spatial regimes for momentum transport around a vortex:

Scale	Region	Physics
r < \lambda (wavelength) or r < r_c (core)	Scattering region	Quasiparticle-vortex interaction; determines D and D'
r_c < r < l_{qp}	Ballistic region	Free quasiparticle streaming; no collisions
l_{qp} < r < r_O (Oseen length)	Viscous subregion	Two-fluid hydrodynamics; viscous momentum transport; determines \boldsymbol{v}_{nl} \to \boldsymbol{v}_n correction
r > r_O	Inertial subregion	Non-linear hydrodynamics; normal circulation \kappa_n required to transport transverse force

Application to the inter-layer problem: The shear layer between co-rotating sheets occupies the “inertial subregion” scale — it’s where the transverse momentum from the microscopic vortex-core coupling must be transported across the boundary. The normal circulation \kappa_n is the mechanism. The GJO instability (from Ch. 3) sets the threshold for when this transport becomes turbulent.

6. What Chapter 8 Gives Us for the h Derivation

6.1 The reactive/dissipative decomposition is real

The D'/D ratio (or equivalently \alpha'/\alpha) is a measurable quantity with clear physical meaning. The reactive component dominates at low temperatures. This validates the structural assumption in the Weinberg angle derivation that the coupling has both reactive and dissipative channels.

6.2 The coupling is mediated by scattering phases

The transverse cross-section \sigma_\perp = 2(\delta_{-\infty} - \delta_\infty)/k depends on phase shifts at large angular momentum. The Kopnin framework (Ch. 9) gives the core phase shift \delta_0 from Breit–Wigner resonance, which provides the complementary piece. The full mutual friction combines both contributions.

6.3 The impedance formula (Eq. 8.7) is the template for inter-layer coupling

The three-channel impedance structure — viscous drag + microscopic scattering + Magnus force — should generalize to the inter-layer problem as: shear layer turbulence + polar-jet core coupling + circulation mismatch. The j-complex structure automatically gives the correct reactive/dissipative decomposition.

6.4 Normal circulation is required

The transverse force cannot propagate without normal-fluid circulation (§8.6). In the inter-layer problem, this means the counter-rotating shear layer must support a net normal circulation to transmit the reactive component of the coupling force. If the shear layer is turbulent (above GJO threshold), this circulation may be partially or fully destroyed, changing the effective coupling from reactive-dominated to dissipative-dominated.

7. What We Still Need from Ch. 9

Chapter 8 treats mutual friction from scattering off the velocity field outside the core (phonons, rotons). For the \alpha_{mf} = \frac{1}{2}\sin 2\delta_0 identification, we need:

§9.5–9.6: Andreev bound states in the vortex core — the CdGM spectrum that produces the Breit–Wigner resonance
§9.7: The Kopnin–Kravtsov force — mutual friction from vortex-core bound state scattering, which gives the \sin 2\delta_0 formula
§9.9: Spectral flow — the connection between bound-state dynamics and macroscopic transport

The distinction matters: Ch. 8 gives D' = -\kappa\rho_n from external scattering (universal, topological-ish), while Ch. 9 gives additional core contributions from internal bound states (non-universal, depends on gap structure). For the inter-layer coupling, the polar jet travels through the core, so it’s the Ch. 9 physics that controls \alpha_{mf}.

8. Extraction Sequence for the Full Coupling Derivation

Start with the impedance formula (Eq. 8.7) as the structural template: three channels in series.
Identify the channels for the inter-layer problem:
- Channel 1: Turbulent shear layer (analog of viscous drag, \sim \ln(r_m/r_l)/4\pi\rho_n\nu)
- Channel 2: Polar-jet core coupling (analog of 1/(D+jD'), computed from Ch. 9)
- Channel 3: Circulation mismatch / Magnus force (analog of 1/j\rho_s\kappa)
Compute Channel 2 from Ch. 9: The Kopnin–Kravtsov force gives the core contribution with \alpha_{mf} = \frac{1}{2}\sin 2\delta_0.
Compute Channel 1 from Ch. 3 + Ch. 14: The GJO instability threshold sets the transition from laminar to turbulent shear; Vinen’s tangle theory (Ch. 14) gives the energy content.
Assemble E_\text{rep}(h) from the total impedance: the energy stored in the coupling is the work done by the friction force against the relative velocity, integrated over the shear layer thickness.