First-principles microscopic determination of the quasiparticle interaction and scattering amplitude of normal liquid helium-3

The 1958 review by Abrikosov and Khalatnikov consolidated Landau's Fermi-liquid theory of liquid 3He and recorded six acknowledged limitations: imprecisely known low-temperature heat capacity and effective mass; an unknown microscopic quasiparticle collision probability $w(\theta,\varphi)$; the apparent experimental inaccessibility of zeroth sound and of the predicted light-scattering satellites; the Cooper-pairing instability signaled by a logarithmic singularity of the interaction function at backscattering; an unvalidated simplified collision integral in the $\omega\tau\sim1$ crossover; and the absence of generalizations to anisotropic systems. Later literature closed most of these: precision calorimetry fixed $m^*(P)$, zeroth sound was observed acoustically in 1966 and by inelastic x-ray scattering in 2007, superfluid 3He was discovered in 1972 and explained as spin-triplet p-wave pairing, and Fermi-liquid theory of anisotropic systems became standard. The surviving open core is the collision-law item: a quantitative first-principles determination of the quasiparticle interaction function and scattering probability of real liquid 3He from the interatomic potential, sufficient to predict the measured effective mass, Landau parameters, and transport prefactors without phenomenological input. The standard practical determinations still take measured Landau parameters as input, and the recent fully microscopic attempts remain quantitatively discrepant with experiment.

Background

Landau's theory of Fermi liquids describes degenerate systems of interacting particles with half-integer spin — liquid helium-3, conduction electrons in metals — at low temperature. Its elementary excitations are quasiparticles: particle-like states near the Fermi surface (the surface in momentum space separating filled from empty single-particle states at $T=0$, of radius $p_0$), each carrying the spin and, in a Galilean-invariant system, the momentum of a bare particle. The quasiparticle energy is linearized near the Fermi surface through an effective mass $m^*$, and is shifted by the surrounding quasiparticles through an interaction function $f(\chi)$, where $\chi$ is the angle between the two momenta on the Fermi surface. The angular harmonics of $f$ are the Landau parameters (spin-symmetric $F_\ell^s$ and spin-antisymmetric $F_\ell^a$); they fix the compressibility, the spin susceptibility, the effective mass ($m^*/m=1+F_1^s/3$), and the collective modes. Two consequences are central here. First, the kinetic equation for the quasiparticle distribution contains a collision integral governed by the probability $w(\theta,\varphi)$ for the collision of two quasiparticles on the Fermi surface, where $\theta$ is the angle between the incoming momenta and $\varphi$ is the angle between the plane spanned by the incoming momenta and the plane spanned by the outgoing momenta; the collision time grows as $\tau\propto T^{-2}$ at low temperature, giving the Fermi-liquid transport laws — viscosity $\eta\propto T^{-2}$ and thermal conductivity $\kappa\propto 1/T$ — with numerical prefactors set by angular averages of $w$. Second, in the collisionless regime $\omega\tau\gg1$ the kinetic equation supports zeroth sound, a propagating density oscillation with no classical analogue, while at $\omega\tau\ll1$ ordinary hydrodynamic (first) sound holds; the crossover region $\omega\tau\sim1$ is a region of strong absorption whose description requires the full collision integral.

The 1958 review by Abrikosov and Khalatnikov presented this theory (with Landau's microscopic, field-theoretic derivation at $T=0$ appended, relating $f$ to the quasiparticle forward scattering amplitude) and applied it to liquid 3He: heat capacity and spin susceptibility, viscosity and thermal conductivity, sound dispersion and absorption across the hydrodynamic-to-collisionless crossover, and the spectrum of light scattered by quasiparticle-density fluctuations, including sharp sidebands (satellites) in the Rayleigh scattering — light scattered by thermal density fluctuations — shifted by the zero-sound frequency. The review was explicit about its limitations, recording six: (i) the experimental heat-capacity curve reached only the onset of the linear-in-$T$ regime, so the coefficient $\gamma$ of the linear specific heat and hence $m^*$ were only approximate ($m^*\approx1.43\,m_3$, where $m_3$ is the mass of a bare helium-3 atom); (ii) the collision probability $w(\theta,\varphi)$ was not known microscopically for the real liquid, so the transport prefactors were only order-of-magnitude estimates obtained from $w\sim(2\pi/\hbar)f^2$; (iii) direct observation of zeroth sound needed frequencies above roughly $10^8$ Hz, and the light-scattering satellites required $T\lesssim0.05$ K in the visible with an intensity the authors themselves estimated to be at or beyond the limit of measurability; (iv) the perturbative $f(\chi)$ is logarithmically singular at backscattering $\chi\to\pi$, the same singularity that underlies Cooper pairing — the binding of quasiparticle pairs of opposite momentum, the mechanism of superconductivity — so the conditions for a superfluid transition of 3He were open; (v) the crossover analysis used a simplified, relaxation-time-type model collision integral whose adequacy for realistic interactions needed validation; (vi) anisotropic systems, such as conduction electrons in metals, lay outside the developed framework.

Five of these six limitations have since been closed by direct experimental and theoretical work. What survives is item (ii): the quasiparticle interaction function and collision probability of real liquid 3He are still not known from first principles. The standard practical determinations reconstruct them from measured Landau parameters; a calculation starting from the interatomic potential that reproduces the measured effective mass, Landau parameters, and transport prefactors has not been achieved.

Problem Statement

The problem is the surviving open core of the limitations recorded by Abrikosov and Khalatnikov (1958), namely their item (ii), stated essentially as they stated it: the angular- and azimuth-dependent quasiparticle collision probability $w(\theta,\varphi)$, together with the Landau interaction function $f(\chi)$ and the quasiparticle scattering amplitude on the Fermi surface to which they are related, is not known microscopically for liquid helium-3. The task: starting from the microscopic Hamiltonian of normal liquid 3He with a realistic helium-helium interatomic potential, derive with controlled and quantified approximations the quasiparticle interaction function $f(\chi)$ over the full angular range in $\chi$, including the backscattering region $\chi\to\pi$ (with $\chi$ the angle between the two quasiparticle momenta on the Fermi surface) that controls pairing, and the collision probability $w(\theta,\varphi)$ over its full angular range in $\theta$ and $\varphi$, and demonstrate the result by quantitatively predicting — without taking any measured Fermi-liquid quantity as input — the effective mass $m^*$ and the leading spin-symmetric and spin-antisymmetric Landau parameters $F_0^s$, $F_0^a$, $F_1^s$ and $F_1^a$ (the leading coefficients of the angular-harmonic expansion of the interaction function), and the numerical prefactors of the low-temperature viscosity and thermal conductivity, with the comparison made at the reference state of saturated vapor pressure in the zero-temperature limit, the accuracy target being agreement with the corresponding measured values to within their experimental uncertainty. Determinations that reconstruct $w(\theta,\varphi)$ from measured Landau parameters — induced-interaction or scattering-amplitude constructions — lie outside the first-principles target of this problem.

References

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