The freely gliding dislocations responsible for the 'giant plasticity' of ultrapure helium-4 crystals — a large, reversible softening of the elastic coefficient c44 below a few hundred millikelvin — have never been shown to move by a specific microscopic mechanism. Haziot, Rojas, Fefferman, Beamish and Balibar, who discovered the effect [Phys. Rev. Lett. 110, 035301 (2013)], posed the question: do these dislocations (or the kinks on them) move by quantum tunneling, or by classical thermal activation over Peierls barriers? They identified the missing input as a microscopic calculation of the kink energy Ek for 4He dislocations and its comparison with kBT in the experimental regime. A decade of follow-up work — dislocation-network spectroscopy, critical-speed, avalanche and creep experiments, classical simulations of rare-gas analogues, path-integral quantum simulations, and authoritative reviews — has accurately mapped the dislocation networks and dynamics, and the classical-versus-quantum interpretation was explicitly contested in a published Comment, but no Ek calculation for 4He and no accepted discrimination exists. The problem asks for the microscopic mechanism of basal dislocation glide in hcp 4He at the lowest temperatures, settled by microscopic calculation, decisive experiment, or a validated mobility theory.

Background

Plasticity is the irreversible deformation of a crystal under large stress, and it is carried by dislocations — line defects in which the lattice is locally sheared by one Burgers vector (the elementary slip step). A dislocation can move by 'glide' along a slip plane, letting the crystal strain at stresses far below the ideal shear strength. In a hexagonal close-packed (hcp) crystal, the easy glide plane is usually the basal plane, and the elastic coefficient governing shear strain in that plane is c44. In ordinary (classical) crystals, glide is resisted by the Peierls barrier: the periodic lattice potential that a straight dislocation must cross as it advances by one Burgers vector between equivalent positions. At low temperature, a dislocation crosses this barrier by nucleating a pair of kinks — short segments connecting the line across adjacent low-energy ('Peierls') valleys — which then spread sideways. Thermal activation over this landscape has a rate set by the kink formation energy Ek and the kink migration barrier, roughly as exp(-E/kBT), so classical glide freezes out at sufficiently low temperature. The quantum alternative, hypothesized since the 1970s in this context, is that kinks or dislocation segments pass through the barrier by tunneling, giving a temperature-independent glide rate; establishing macroscopic quantum tunneling of a mesoscopic defect would be a landmark of mesoscopic quantum mechanics.

Helium-4 is the paradigmatic 'quantum crystal': because of its small mass and weak interatomic forces, zero-point motion is so large that helium does not solidify at all under its own vapor pressure and must be pressurized; its elastic constants are measured in bar rather than gigapascal, hundreds of times softer than conventional solids. The same zero-point motion makes defects unusually mobile at low temperature. It also allows extraordinary chemical purity: crystals can be grown containing only ppb-level 3He isotopic impurities, and these trace impurities bind to and pin dislocations, while above roughly 0.3 K collisions with thermal phonons damp dislocation motion. Between the 3He-pinning and phonon-damping regimes, dislocations in 4He move with almost no dissipation. The field's attention was drawn to dislocations by the discovery (Day and Beamish, 2007) of a large low-temperature stiffening of the shear modulus of solid 4He, first discussed in connection with the 'supersolidity' torsional-oscillator anomalies (supersolidity being the conjecture that solid helium could combine crystalline order with superfluid-like flow) and later understood as 3He pinning of mobile dislocations.

Against this background, Haziot, Rojas, Fefferman, Beamish and Balibar (2013) reported 'giant plasticity': in oriented ultrapure 4He single crystals measured between piezoelectric transducers at kilohertz frequencies, at temperatures from 15 mK to 0.7 K, the resistance to shear nearly vanishes in the softest direction — the coefficient c44 is reduced reversibly by 50–80% — under stresses as small as a microbar, with no measurable dissipation and no dependence on strain amplitude once 3He impurities are expelled into a residual liquid. The effect is strongly anisotropic and shows that dislocations glide freely parallel to the basal planes of the hcp structure; the dislocations move at estimated speeds of order centimeters per second, and their small-strain response is accurately described by the vibrating-string (Granato–Lücke-type) model of a pinned dislocation network, as later established quantitatively by Fefferman et al. (2014).

The 2013 authors stated the unresolved question: whether this freely reversible glide at the lowest temperatures is quantum (tunneling of dislocations or of kinks) or classical (thermal activation over Peierls barriers). They noted that the observed dissipation-free, strain-independent, reversible motion suggests tunneling, but also that quantum fluctuations should broaden kinks and thereby reduce Ek; if Ek is below kBT in the experimental range, kinks are irrelevant altogether — as for steps on helium crystal surfaces, whose dynamics is also dissipation-free at low temperature — and the glide is then neither 'tunneling' nor 'activated' but effectively free. They concluded that a microscopic calculation of Ek for 4He dislocations could indicate whether the observed motion is quantum or classical, thanks to a negligible kink energy. The same paper posed two further open directions: the microscopic arrangement and density of the dislocations producing the c44 reduction (a simple Frank network, that is an isotropic tangle of dislocations meeting at nodes, should give less than 5% reduction, so aligned sub-boundaries were suspected), and the nature and dynamics of the hardening defects (probably jogs — short segments by which a dislocation line steps out of its glide plane, where it cannot move freely) produced by large stresses. Later work has substantially advanced those two directions, while the mechanism question remains without an accepted answer.

Problem Statement

Determine the microscopic mechanism by which basal dislocations glide in ultrapure hcp 4He crystals at the lowest temperatures of the giant-plasticity experiments (from roughly 15 mK up to a few hundred mK), as the source paper poses it: is the freely reversible, dissipation-free glide of dislocations parallel to the basal planes — the motion responsible for the 50–80% reversible reduction of c44 — fundamentally quantum (tunneling of dislocation segments or of kinks through the Peierls landscape) or classical (thermally activated motion over Peierls/kink barriers)? The source authors point to one specific theoretical input: a microscopic calculation of the kink energy EkE_k for 4He dislocations that includes quantum zero-point motion, to be compared with kBTk_B T in the experimental regime, which in their words 'could indicate if the dislocation motion is quantum or classical thanks to a negligible kink energy' — if Ek≲kBTE_k \lesssim k_B T, kinks are irrelevant and the glide is effectively free rather than 'tunneling' or 'activated'; if Ek≫kBTE_k \gg k_B T while glide remains undamped and temperature-independent, thermal nucleation of kinks is frozen out and the operative mechanism must then be identified from the remaining mobility inputs — the kink migration barrier, the effective Peierls stress and the dissipation channel — since a line that already carries kinks may still glide classically by kink migration, or by tunneling, or freely; intermediate cases require the kink nucleation and migration barriers to be quantified. Admissible resolutions include: (i) a quantum many-body calculation of EkE_k (and of the relevant migration barriers and effective Peierls stress) for basal dislocations in hcp 4He, with quantified uncertainties, together with the conclusion it licenses about the observed motion; (ii) an experimental discrimination (for example the temperature and strain-rate dependence of glide mobility in the millikelvin regime, or an isotope-based comparison) that excludes all but one of thermal activation, tunneling and effectively free glide, and so identifies the operative mechanism; or (iii) a validated microscopic-to-mesoscopic mobility model whose parameters are fixed ab initio or from independent measurements and which reproduces the measured softening and dissipation of the giant-plasticity experiments with a definite mechanism. Any answer must evaluate the mechanism for the actual experimental system — ultrapure hcp 4He with 3He binding and phonon damping controlled — rather than for a proxy model class, and must distinguish all three mechanism classes (tunneling, thermal activation, effectively free glide) rather than assuming a dichotomy. The classes are separated by the rate-limiting step of the glide motion: thermal activation, in which a barrier large compared with kBTk_B T is surmounted thermally, giving an Arrhenius temperature and strain-rate dependence; tunneling, in which the rate-limiting barrier is not thermally surmountable in the experimental window and is traversed by quantum penetration, giving a temperature-independent but still barrier-controlled rate; and effectively free glide, in which zero-point motion renormalizes the Peierls/kink landscape so that no barrier is rate-limiting (Ek≲kBTE_k \lesssim k_B T, kinks irrelevant, as for steps on helium crystal surfaces). A barrier rendered negligible by zero-point motion is therefore to be reported as effectively free glide rather than as tunneling, and the answer must state which of the three classes the identified rate-limiting step belongs to.

References

  1. A. Haziot, X. Rojas, A. D. Fefferman, J. R. Beamish, S. Balibar, "Giant Plasticity of a Quantum Crystal", Physical Review Letters 110, 035301 (2013). DOI: 10.1103/PhysRevLett.110.035301, https://doi.org/10.1103/PhysRevLett.110.035301; preprint arXiv:1211.2976, https://arxiv.org/abs/1211.2976
  2. J. Day, J. Beamish, "Low-temperature shear modulus changes in solid 4He and connection to supersolidity", Nature 450, 853–856 (2007). DOI: 10.1038/nature06383, https://doi.org/10.1038/nature06383
  3. C. Zhou, C. Reichhardt, M. J. Graf, J.-J. Su, A. V. Balatsky, I. J. Beyerlein, "Comment on 'Giant Plasticity of a Quantum Crystal'", Physical Review Letters 111, 119601 (2013). DOI: 10.1103/PhysRevLett.111.119601, https://doi.org/10.1103/PhysRevLett.111.119601; preprint arXiv:1303.2679, https://arxiv.org/abs/1303.2679
  4. A. Haziot, X. Rojas, A. D. Fefferman, J. R. Beamish, S. Balibar, "Haziot et al. Reply:" (reply to the Comment on "Giant Plasticity of a Quantum Crystal"), Physical Review Letters 111, 119602 (2013). DOI: 10.1103/PhysRevLett.111.119602, https://doi.org/10.1103/PhysRevLett.111.119602
  5. A. Haziot, A. D. Fefferman, F. Souris, J. R. Beamish, H. J. Maris, S. Balibar, "Critical dislocation speed in helium-4 crystals", Physical Review B 88, 014106 (2013). DOI: 10.1103/PhysRevB.88.014106, https://doi.org/10.1103/PhysRevB.88.014106; preprint arXiv:1307.7686, https://arxiv.org/abs/1307.7686
  6. A. D. Fefferman, F. Souris, A. Haziot, J. R. Beamish, S. Balibar, "Dislocation networks in helium-4 crystals", Physical Review B 89, 014105 (2014). DOI: 10.1103/PhysRevB.89.014105, https://doi.org/10.1103/PhysRevB.89.014105; preprint arXiv:1310.3400, https://arxiv.org/abs/1310.3400
  7. F. Souris, A. D. Fefferman, A. Haziot, N. Garroum, J. R. Beamish, S. Balibar, "Search for Dislocation Free Helium 4 Crystals", Journal of Low Temperature Physics 178, 149–161 (2015). DOI: 10.1007/s10909-014-1251-0, https://doi.org/10.1007/s10909-014-1251-0
  8. A. Lisunov, V. Maidanov, V. Rubanskyi, S. Rubets, E. Rudavskii, S. Smirnov, V. Zhuchkov, "Thermally activated and quantum plasticity of solid 3He at temperatures below 0.5 K", Physical Review B 92, 140505 (2015). DOI: 10.1103/PhysRevB.92.140505, https://doi.org/10.1103/PhysRevB.92.140505
  9. A. N. Malmi-Kakkada, O. T. Valls, C. Dasgupta, "Dislocation Mobility and Anomalous Shear Modulus Effect in 4He Crystals", Journal of Low Temperature Physics 186, 259–274 (2017). DOI: 10.1007/s10909-016-1689-3, https://doi.org/10.1007/s10909-016-1689-3; preprint arXiv:1610.08354, https://arxiv.org/abs/1610.08354
  10. S. Sempere, A. Serra, J. Boronat, C. Cazorla, "Dislocation Structure and Mobility in Hcp Rare-Gas Solids: Quantum versus Classical", Crystals 8, 64 (2018). DOI: 10.3390/cryst8020064, https://doi.org/10.3390/cryst8020064
  11. Z. G. Cheng, J. Beamish, "Plastic Deformation in a Quantum Solid: Dislocation Avalanches and Creep in Helium", Physical Review Letters 121, 055301 (2018). DOI: 10.1103/PhysRevLett.121.055301, https://doi.org/10.1103/PhysRevLett.121.055301
  12. J. Beamish, "Plastic Deformation and Creep in Solid Helium", Journal of Low Temperature Physics 197, 187–207 (2019). DOI: 10.1007/s10909-019-02231-5, https://doi.org/10.1007/s10909-019-02231-5
  13. J. Beamish, S. Balibar, "Mechanical behavior of solid helium: Elasticity, plasticity, and defects", Reviews of Modern Physics 92, 045002 (2020). DOI: 10.1103/RevModPhys.92.045002, https://doi.org/10.1103/RevModPhys.92.045002
  14. M. de Koning, W. Cai, "Dislocation-position fluctuations in solid 4He as collective variables in a quantum crystal", npj Quantum Materials 7, 119 (2022). DOI: 10.1038/s41535-022-00533-8, https://doi.org/10.1038/s41535-022-00533-8