Quantum kink dynamics and the microscopic mobility of dislocations in solid $^4$He

De Gennes (C. R. Physique 7, 561 (2006)) proposed that a dislocation in a quantum crystal such as solid $^4$He remains mobile at zero temperature through quantum tunneling of kink pairs: with kink creation energy $U$ and kink transfer integral $t$, opposite kinks form a bound state controlled by $\varepsilon=(2U-t)/t$, and the glide velocity follows a tunneling-limited law $V\propto\exp(-\sigma^*/\sigma)$ — quantum plasticity, rather than superfluidity, as the origin of anomalous mechanics. He stressed that the theory was schematic (simple-cubic geometry, one kink held fixed, unknown parameters) and listed what a real theory requires: realistic $t$ and $U$ for hexagonal-close-packed (hcp) $^4$He, determination of the realized regime ($\varepsilon\ll1$ versus $t\ll U$), the both-kinks-mobile re-derivation, and the extension to interacting dislocation networks in polycrystals. Experiments (2007-2014) settled the phenomenology: the anomalous mechanics of solid $^4$He is dislocation-mediated (shear-modulus stiffening below ~0.2 K, giant plasticity, $^3$He impurity pinning), not supersolidity. Simulations computed the structure, Peierls stress, effective mass, and position fluctuations of straight dissociated dislocations in hcp $^4$He, and theory proposed quantum-liquid states of kinks and jogs. However, the microscopic quantities de Gennes called for remain uncomputed with controlled errors: the only prior estimates of kink-level quantities for hcp $^4$He (Chui 2010) are phenomenological, resting on fitted, uncontrolled inputs, and fall short of that standard, so the kink energies, tunneling matrices, and the regime parameter remain open. The leading experiment (Haziot et al. 2013) explicitly calls for 'a calculation of $E_k$', the field's own review (Balibar et al. 2016) lists the kink energy and the kink Peierls barrier as open questions, and even the quantum character of the observed mobility is contested (Zhou et al. 2013). The problem is open.

Background

Solid $^4$He is the prototypical quantum crystal: the helium atom is light and only weakly bound by van der Waals attraction, so the crystalline phase exists only under pressure (about 25 bar and above) and the quantum zero-point motion of each atom about its lattice site is a significant fraction of the interatomic spacing. In such a crystal, defect motion need not freeze out classically: energy barriers can be traversed by quantum tunneling, so lattice defects may stay mobile as $T\to0$.

The defects central to mechanics are dislocations — the line defects that carry plastic deformation in essentially all crystals. An edge dislocation is the line terminating an extra half-plane of atoms; a closed circuit around the line fails to close by the lattice vector $\mathbf b$, the Burgers vector. An edge dislocation can translate ('glide') within its glide plane, but the periodic lattice potential — the Peierls barrier, the energy relief resisting translation of the line by one lattice spacing — makes a perfectly straight line immobile: the line must nucleate a pair of kinks, short segments of height $b$ connecting adjacent low-energy Peierls valleys, and pull the kinks apart. In classical crystals kink pairs are created by thermal activation, so plasticity vanishes at low temperature.

The question became urgent when torsional-oscillator experiments (Kim and Chan, 2004) reported that solid $^4$He in a rotating cell appeared to lose part of its rotational inertia below about 200 mK — a 'non-classical rotational inertia' initially interpreted as supersolidity (coexistence of crystalline order and superfluid flow), reviving proposals from the late 1960s based on Bose condensation of quantum vacancies. P.-G. de Gennes (C. R. Physique 7, 561 (2006)) asked instead whether the relevant zero-temperature mechanics of solid $^4$He could be a quantum plasticity of dislocations.

His model: a kink on an edge dislocation has creation energy $U$ and can hop along the line by tunneling with transfer integral $t$ — the tight-binding hopping matrix element between adjacent kink positions one atomic spacing $b$ apart along the line. Holding one kink of a pair fixed for bookkeeping, the pair separation performs a quantum walk with on-site energy $2U$ and hopping $t$. If $2U>t$ the pair forms a bound state; writing $e^p=2U/t\equiv 1+\varepsilon$ with

$$\varepsilon=\frac{2U-t}{t},$$

the bound pair has size $l=b/2p$. In the regime $\varepsilon\ll1$ ($t$ comparable to $U$) the pairs are extended, and the ground-state line is a random walk of persistence length $l$ and hop size $b$, with lateral fluctuation $X\sim(L/l)^{1/2}b$ for a line of length $L$. Under a shear stress $\sigma$ (exerting the Peach–Koehler force $\sigma b$ per unit length on the line), a bound pair must tunnel through a barrier to unbind, giving a strongly nonlinear glide law

$$V=V_0\,\exp(-\sigma^*/\sigma),\qquad \sigma^*=\frac{4}{3}\,\frac{\varepsilon^3 t}{b^3},\qquad V_0\sim tb/\hbar,$$

so that a line density $n$ of such dislocations produces a shear rate $\dot\gamma=nbV$: zero-temperature plastic flow rather than superfluidity, with the mobility controlled entirely by kink tunneling. De Gennes emphasized the schematic character of the construction — simple-cubic geometry (real $^4$He is hexagonal close-packed, hcp), one kink held fixed (in reality both kinks are mobile, changing all numerical coefficients), a contact kink–kink attraction, and no knowledge of the actual values of $t$ and $U$ for $^4$He. He moreover suspected that the mathematically convenient regime $\varepsilon\ll1$ is 'probably not achieved in practice: we rather expect $t\ll U$, giving very tight bound states'.

The subsequent experimental context within which the question now sits: a shear-modulus stiffening of solid $^4$He below about 0.2 K with the same dependence on $^3$He impurity content, strain amplitude, and annealing as the torsional-oscillator signals (Day and Beamish, 2007); 'giant plasticity' — nearly lossless glide of dislocation networks along the basal planes (the close-packed planes of the hcp structure) in the window roughly 0.1–0.3 K, bounded below by $^3$He binding to dislocations and above by damping from thermal phonons (Haziot et al. 2013); dislocation networks organized as two-dimensional sub-boundary arrays of very long ($`\sim`$100–230 $\mu$m) parallel lines (Fefferman et al. 2014); and the consensus that the torsional-oscillator period shifts are an elastic-coupling effect rather than bulk superfluidity (Boninsegni and Prokof'ev, 2012). What this phenomenology has not settled is the microscopic mechanism and its parameters — whether dislocation glide at $T\to0$ is quantum kink tunneling, reflects an effectively vanishing kink energy, or is residual classical activated motion — which is precisely the unresolved content of de Gennes' question.

Problem Statement

Turn de Gennes' schematic kink-tunneling theory of a single dislocation into a quantitative, testable microscopic theory of dislocation mobility in real solid $^4$He. Here a kink is a short step, of height one Burgers vector $b$ (the lattice translation by which a closed circuit around the dislocation line fails to close), connecting adjacent low-energy valleys of the dislocation line in the periodic lattice potential; kinks are the elementary carriers of glide and, in a quantum crystal, can move along the line by quantum tunneling. The problem comprises four jointly linked parts, as posed in the source paper's concluding remarks; they serve one common objective — a predictive microscopic account of zero-temperature dislocation motion in the prototypical quantum crystal.

1. Kink energetics and tunneling parameters in the correct lattice. Compute, from realistic He–He interactions and the true hexagonal-close-packed (hcp) crystal structure of $^4$He — including the dissociation of basal dislocations into Shockley partial dislocations (partials with reduced Burgers vectors) separated by a ribbon of stacking fault (a local fault in the stacking sequence of the close-packed planes) — the kink shapes, the kink creation energy $U$, and the kink tunneling transfer integral $t$ (or the equivalent kink effective mass and Peierls-relief barrier for kink motion along the line), at the pressures and molar volumes of the mechanical experiments; and thereby determine the regime parameter $\varepsilon=(2U-t)/t$, i.e. whether $^4$He lies in the extended-bound-pair regime $\varepsilon\ll1$ or in the tight-binding regime $t\ll U$ that de Gennes considered more likely.

2. The corrected single-line mobility law. Re-derive the zero-temperature glide dynamics with both kinks of a pair mobile (numerical coefficients, not only scaling) for the dissociated hcp core, and determine the resulting mobility law and its characteristic stress scale — the analogue of the glide law $V=V_0\exp(-\sigma^*/\sigma)$, where $V$ is the glide velocity under shear stress $\sigma$ and $V_0\sim tb/\hbar$ is a velocity scale, with characteristic stress $\sigma^*=(4/3)\varepsilon^3 t/b^3$ (here $b$ is the Burgers-vector magnitude and $t$, $U$ are as in part 1).

3. Mechanism discrimination against experiment. Decide, quantitatively, whether the measured dislocation mobility of ultrapure $^4$He crystals in the giant-plasticity window (nearly lossless basal glide, bounded below by $^3$He pinning and above by phonon damping) is governed by quantum kink tunneling, by an effectively vanishing kink energy and kink Peierls barrier, or by residual classical thermal activation — the alternatives the later literature explicitly left open and could not distinguish.

4. From single line to network. Carry the microscopic kink parameters through to the interacting dislocation networks of polycrystalline $^4$He — ladders of dislocations forming grain boundaries and sub-boundary arrays with long-range elastic interactions — and determine whether quantum-mobile networks account quantitatively for the anomalous macroscopic mechanical response (the shear-modulus anomaly and its dependence on $^3$He concentration, strain amplitude, frequency, and crystal orientation), and what role, if any, mass flow along dislocation cores plays alongside glide.

References

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