First-principles microscopic derivation of the full excitation spectrum of superfluid liquid 4He (Landau, 1949)

Landau's 1949 letter 'On the Theory of Superfluidity' stated the excitation spectrum of superfluid liquid helium empirically - phonons at low momentum and a 'roton' minimum at intermediate momentum - and declared its derivation from first principles for the real, strongly interacting liquid an open problem. The empirical side is now complete: inelastic neutron scattering has measured the full dispersion up to its termination at the 'Pitaevskii plateau', and microscopic many-body methods (Feynman-Cohen backflow, shadow wave functions, dynamic many-body theory, quantum Monte Carlo with analytic continuation) reproduce the main quasiparticle branch to a few percent. What still does not exist is a parameter-free derivation from the atomic Hamiltonian - analytic or numerical - whose approximation scheme is controlled and whose error on $\epsilon(k)$ is certified, predicting the spectrum of real 4He: sound velocity, maxon energy, roton gap and effective mass, and the behavior of the quasiparticle branch up to the Pitaevskii plateau, with quantified accuracy. The second-sound dispute with Tisza, the empirical determination of the spectrum, the phonon-roton kinetic theory of viscosity, and the He3 impurity problem are settled and lie outside the scope of this task, which concerns only the first-principles derivation of the spectrum.

Background

Liquid helium-4 (4He, the bosonic isotope) does not solidify at saturated vapor pressure even at zero temperature, and below 2.17 K it becomes superfluid (this phase is called He II): it flows without viscosity and transports heat anomalously. Landau's two-fluid model (Landau, 1941; reprint in Khalatnikov's book on superfluidity) treats He II as the superposition of an inert 'superfluid' component carrying no entropy and a 'normal' component made of thermal excitations, each excitation having a definite energy-momentum relation $\epsilon(p)$ , now usually written as a function of wave number $k$ , $\epsilon(k)$ . Landau's spectrum is a single curve: at small $k$ , sound quanta ('phonons') with $\epsilon(k)=\hbar ck$ , where $c$ is the sound velocity; at intermediate $k$ a local minimum, the 'roton' minimum, fitted by $\epsilon(k)=\Delta+\frac{(\hbar k-\hbar k_0)^2}{2\mu}$ with roton gap $\Delta$ , roton momentum $\hbar k_0$ , and effective mass $\mu$ ; the local maximum between the phonon and roton regions is called the 'maxon'. Superfluidity itself was argued to follow: a body moving slower than the minimum of $\epsilon(k)/\hbar k$ cannot dissipate energy by creating excitations. 'Second sound' is a second wave mode of the two-fluid model in which the normal and superfluid components oscillate out of phase as a temperature wave; its velocity $u_2$ tends to $c/\sqrt{3}$ at low temperature if the normal component consists of phonons with a linear spectrum (Landau's prediction), and to zero if the normal component is a classical gas of atoms (Tisza's competing prediction). The 1949 letter is a short reply to Tisza's adjacent comment in the same issue of Physical Review. In it Landau credits Tisza with priority for the macroscopic two-fluid description, rejects Tisza's atomistic interpretation of the normal component, states the single phonon-roton curve as the result of an unpublished investigation, and lists what remains open: the exact form of $\epsilon(k)$ at intermediate and high momenta; experimental verification of the predicted second-sound velocity minimum near 1 K; a kinetic theory of viscosity built from phonon-roton collisions; the theory of He3 (the fermionic helium isotope) impurities in He II; and - the item that survives as the core of this problem - a first-principles derivation of the spectrum of the real, strongly interacting liquid, as opposed to the weakly interacting Bose gas treated by Bogoliubov theory (the perturbative small-density theory of Bose gases, which fails qualitatively for helium; the fraction of atoms in the zero-momentum quantum state, the 'condensate fraction', is only about 7 percent at saturated vapor pressure in the most recent estimates). 'First principles' here means: start from the nonrelativistic quantum-mechanical Hamiltonian of $N$ helium atoms interacting through a realistic He-He pair potential (of the general type parametrized by Aziz and coworkers from spectroscopy and beam data), and predict $\epsilon(k)$ with no parameter fitted to helium excitation or thermodynamic data. The obstacle is that liquid helium is a strongly correlated Bose liquid with no small parameter. The first microscopic relation, due to Feynman (1954), expresses the excitation energy through the static structure factor $S(k)$ - the Fourier transform of the liquid's pair correlation function, measurable by diffraction - as $\epsilon(k)=\hbar^2 k^2/(2mS(k))$ , with $m$ the atomic mass. This gives the correct phonon slope at small $k$ but overestimates the roton energy by roughly a factor of two; the 'backflow' correction of Feynman and Cohen (1956), in which the moving excitation drags a current of surrounding atoms, repairs the roton qualitatively but is not a controlled expansion. Later microscopic work falls into three families. (i) Correlated variational theories: the shadow wave function, an ansatz in which auxiliary 'shadow' variables are integrated over to generate many-particle (not merely pairwise) correlations, which reproduces the measured dispersion to high accuracy but with a variational bias of unknown size. (ii) Dynamic many-body theory (DMBT), which solves coupled equations for one- and two-particle Green's functions (propagators encoding single- and pair-excitation dynamics) with self-consistent renormalization. (iii) Quantum Monte Carlo (QMC), stochastic sampling that solves the many-body Schrödinger equation with controlled statistical errors for ground-state static properties, but whose extension to dynamical excitations requires analytically continuing correlations from imaginary time to real frequency - a mathematically ill-posed inversion regularized, for example, by GIFT (Genetic Inversion via Falsification of Theories, a maximum-entropy-type method). The measured spectrum itself is known from inelastic neutron scattering (neutrons exchange energy and momentum with the liquid, mapping $\epsilon(k)$ directly) up to the 'Pitaevskii plateau': the region where the single-quasiparticle branch flattens near twice the roton gap, above which an excitation can decay into a roton pair and no longer remains sharp. Phenomenological density functionals such as the Orsay-Trento functional (Dalfovo et al., 1995) describe helium quantitatively but fit their parameters to helium data, so they do not count as first-principles derivations.

Problem Statement

Derive, from first principles, the low-temperature single-quasiparticle excitation spectrum of superfluid liquid 4He. Concretely: starting from the nonrelativistic many-body Hamiltonian of $N$ helium-4 atoms with a fixed, realistic He-He pair interaction (and, if required, ab initio three-body forces), with no parameter adjusted to helium excitation or thermodynamic data, predict the dispersion $\epsilon(k)$ of the liquid at a specified density (the density being stated by the derivation, e.g. saturated vapor pressure): the phonon slope $c$ , the maxon energy, the roton gap $\Delta$ , roton wave number $k_0$ , and effective mass $\mu$ , and the behavior of the branch up to the region where the single-excitation dispersion remains well defined as it approaches the Pitaevskii plateau, with the accuracy and uncertainty of the predictions reported. A derivation qualifies only if it sets out the approximation scheme by which $\epsilon(k)$ is obtained and bounds that scheme's residual error by an independently computable quantity, quoted as a numerical error budget for $\epsilon(k)$ ; for a systematic expansion this means evidence that the successive contributions decrease together with an explicit bound on the neglected remainder. A result whose error bars the scheme cannot bound - an unknown variational bias, or an ill-posed analytic continuation from imaginary time - and a curve whose accuracy is inferred only from agreement with the measured branch, do not meet this test.

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