Defect-mediated superfluid mass transport in solid $^4$He: which crystalline defects support intrinsic superflow, and how does local defect-core superfluidity map quantitatively to measured global mass flux?

Pollet, Boninsegni, Kuklov, Prokof'ev, Svistunov, and Troyer (Phys. Rev. Lett. 101, 097202 (2008)) argued from first-principles quantum Monte Carlo that ideal hcp solid $^4$He is insulating, but that the local strain concentrated at crystalline defects (dislocation cores, grain boundaries) exceeds the critical strain (~0.1–0.2, i.e. tens of bar) needed to close the vacancy gap and render the defect core superfluid, thereby explaining superfluid-like mass flow through solid $^4$He. The problem is to determine which realistic defect geometries and densities in experimental samples actually support intrinsic superfluid transport, and to establish the quantitative mapping from defect-core superfluidity — including defect structure, network connectivity, temperature dependence, and $^3$He-impurity blocking — to the global mass-flux signals measured in experiments. It remains open: as of 2023–2026, two quantum-simulation groups are in direct published disagreement about whether dislocation cores in hcp $^4$He possess superfluid off-diagonal long-range order at all, the structure-to-flow mapping is indicated experimentally but not derived quantitatively, and the predicted local critical strains have not been verified in real samples.

Background

A supersolid is a hypothetical phase in which a crystal's atoms occupy a regular lattice while a fraction of them flows without friction, i.e. the solid simultaneously breaks translational symmetry and possesses superfluid order. Penrose's question of whether such a phase can exist became urgent when Kim and Chan (Nature 427, 225 (2004)) observed a drop in the period of a torsional oscillator filled with solid helium-4 below ~200 mK, interpreted as non-classical rotational inertia from mass decoupling. In parallel, Sasaki, Ishiguro, Caupin, Maris, and Balibar (Science 313, 1098 (2006)) reported mass flow along grain boundaries — the interfaces between misoriented crystallites — in solid $^4$He at the melting curve. The relevant solid is the hexagonal close-packed (hcp) phase, in which most experiments are done. Two later developments reframed the problem. First, Day and Beamish (Nature 450, 853 (2007)) found that the shear modulus of solid $^4$He stiffens at exactly the same temperatures as the torsional-oscillator period shifts, and the period anomalies are now largely attributed to this classical elastic stiffening rather than bulk supersolidity (see the Colloquium by Boninsegni and Prokof'ev, Rev. Mod. Phys. 84, 759 (2012)). Second, Ray and Hallock (Phys. Rev. Lett. 100, 235301 (2008)) used a 'sandwich' cell — bulk liquid reservoirs connected through a block of solid $^4$He grown off the melting curve — and observed pressure equilibration by mass flow through the solid with critical, superfluid-like dependence on the driving pressure difference, inconsistent with viscous liquid flow through cracks. First-principles simulations are based on Feynman's path-integral formulation, in which quantum exchange appears as permutations of particle world lines in imaginary time; the 'worm algorithm' is a Monte Carlo method that samples these permutations efficiently, and superfluidity is detected through off-diagonal long-range order (ODLRO), i.e. macroscopic long-distance behavior of the one-body density matrix, or through superfluid-density estimators. Ideal hcp $^4$He crystals were shown by such methods to be insulating. Pollet, Boninsegni, Kuklov, Prokof'ev, Svistunov, and Troyer (Phys. Rev. Lett. 101, 097202 (2008)) computed the strain thresholds at which the energy gap for creating a vacancy closes: hydrostatic decompression reaches the gapless regime only inside the metastable liquid density range (with an attractive vacancy–vacancy interaction of binding energy ~1.4 K making the homogeneous supersolid unstable), anisotropic compression requires ~10–12% strain (~50 bar), and pure shear requires a critical strain $u_c \approx 0.15$ (~35 bar) — strains unreachable homogeneously but plausibly exceeded locally at the cores of dislocations (line defects whose Burgers vector measures the lattice mismatch carried around the core; 'screw' dislocations shear parallel to the line, 'edge' dislocations perpendicular to it) and at grain boundaries. Consistent with this, simulations by the same collaboration found that the core of a screw dislocation along the hexagonal axis is superfluid — a quasi-one-dimensional Luttinger liquid (a gapless one-dimensional quantum fluid with algebraic correlations) — and that grain boundaries are generically superfluid with a transition temperature of order 0.5 K at the melting pressure, except for special low-angle orientations. The physics of dislocations in solid $^4$He has since become rich in its own right: 'superclimb' denotes the anomalously enhanced climb (motion of a dislocation line out of its glide plane) of dislocations whose cores are superfluid, and reviews such as Beamish and Balibar (Rev. Mod. Phys. 92, 045002 (2020)) document that dislocations dominate the low-temperature mechanics of solid helium. What is at stake is the microscopic explanation of the only observed frictionless mass transport through a crystal: whether it rides on intrinsic superfluidity of defect cores, and whether defect-scale physics can reproduce the measured flux. Hallock's review (J. Low Temp. Phys. 197, 167 (2019)) summarizes the experimental state: the flux through the solid is a universal function of temperature, is nonlinear in the driving chemical potential 'as might be expected for a Luttinger liquid', and is extinguished by trace $^3$He impurities at a concentration-dependent temperature, with 'strong evidence' that it is carried by dislocation cores.

Problem Statement

Establish, microscopically and quantitatively, whether crystalline defects in hcp solid $^4$He — dislocation cores and grain boundaries — support intrinsic superfluid mass transport, and derive the mapping from local defect-core superfluidity to the global mass-flow signals measured in experiments. Concretely, an answer must determine: (1) for realistic defect geometries (edge and screw dislocations with their actual core structures, including possible core splitting into partial dislocations, and grain boundaries of various misorientations), whether and at what transition temperature the core supports superfluidity/off-diagonal long-range order, settling the current dispute between quantum-simulation methodologies for dislocation cores; (2) how defect density and network connectivity convert the local superfluid response into the macroscopic mass flux, its universal temperature dependence, its nonlinear dependence on driving chemical potential, and its extinction by $^3$He impurities at concentration-dependent temperatures, as observed in the UMass sandwich-type mass-flow experiments; and (3) whether the predicted local critical strains/stresses (of order 0.1–0.2 in strain, tens of bar in stress, per Pollet et al. (2008)) actually occur at defect cores in experimental samples. Completion means a validated account — consistent first-principles simulation plus quantitative agreement with measured flux magnitudes and their temperature, chemical-potential, and impurity dependences, and a determination of whether the predicted local critical strains/stresses actually occur at defect cores in experimental samples — or a demonstration that defect-core superfluidity cannot account for the observed transport, with identification of what does.

References

1. L. Pollet, M. Boninsegni, A. B. Kuklov, N. V. Prokof'ev, B. V. Svistunov, and M. Troyer, 'Local Stress and Superfluid Properties of Solid $^4$He', Phys. Rev. Lett. 101, 097202 (2008). DOI: 10.1103/PhysRevLett.101.097202, https://arxiv.org/abs/0805.3713
2. E. Kim and M. H. W. Chan, 'Probable observation of a supersolid helium phase', Nature 427, 225–227 (2004). DOI: 10.1038/nature02220, https://doi.org/10.1038/nature02220
3. S. Sasaki, R. Ishiguro, F. Caupin, H. J. Maris, and S. Balibar, 'Superfluidity of Grain Boundaries and Supersolid Behavior', Science 313, 1098–1100 (2006). DOI: 10.1126/science.1130879, https://doi.org/10.1126/science.1130879
4. M. W. Ray and R. B. Hallock, 'Observation of Unusual Mass Transport in Solid hcp $^4$He', Phys. Rev. Lett. 100, 235301 (2008). DOI: 10.1103/PhysRevLett.100.235301, https://arxiv.org/abs/0805.0009
5. M. Boninsegni, A. B. Kuklov, L. Pollet, N. V. Prokof'ev, B. V. Svistunov, and M. Troyer, 'Luttinger Liquid in the Core of a Screw Dislocation in Helium-4', Phys. Rev. Lett. 99, 035301 (2007). DOI: 10.1103/PhysRevLett.99.035301, https://arxiv.org/abs/0705.2967
6. L. Pollet, M. Boninsegni, A. B. Kuklov, N. V. Prokof'ev, B. V. Svistunov, and M. Troyer, 'Superfluidity of Grain Boundaries in Solid $^4$He', Phys. Rev. Lett. 98, 135301 (2007). DOI: 10.1103/PhysRevLett.98.135301, https://arxiv.org/abs/cond-mat/0702159
7. J. Day and 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
8. M. Boninsegni and N. V. Prokof'ev, 'Colloquium: Supersolids: What and where are they?', Rev. Mod. Phys. 84, 759–776 (2012). DOI: 10.1103/RevModPhys.84.759, https://arxiv.org/abs/1201.2227
9. R. B. Hallock, 'Mass Flux Experiments in Solid $^4$He: Some History, Recent Work and the Current Status', J. Low Temp. Phys. 197, 167–186 (2019). DOI: 10.1007/s10909-019-02214-6, https://doi.org/10.1007/s10909-019-02214-6
10. J. Beamish and S. Balibar, 'Mechanical behavior of solid helium: Elasticity, plasticity, and defects', Rev. Mod. Phys. 92, 045002 (2020). DOI: 10.1103/RevModPhys.92.045002, https://doi.org/10.1103/RevModPhys.92.045002
11. A. B. Kuklov, N. V. Prokof'ev, and B. V. Svistunov, 'Disorder-induced quantum properties of solid 4He', Low Temp. Phys. 46, 459–464 (2020). DOI: 10.1063/10.0001048, https://doi.org/10.1063/10.0001048
12. L. Liu and A. B. Kuklov, 'Superclimbing dislocation with a Coulomb-type interaction between jogs', Phys. Rev. B 97, 104510 (2018). DOI: 10.1103/PhysRevB.97.104510, https://doi.org/10.1103/PhysRevB.97.104510
13. M. de Koning and 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://www.nature.com/articles/s41535-022-00533-8
14. M. de Koning, W. Cai, C. Cazorla, and J. Boronat, 'Absence of Off-Diagonal Long-Range Order in hcp $^4$He Dislocation Cores', Phys. Rev. Lett. 130, 016001 (2023). DOI: 10.1103/PhysRevLett.130.016001, https://arxiv.org/abs/2301.02854
15. M. Boninsegni, A. B. Kuklov, L. Pollet, N. Prokof'ev, and B. V. Svistunov, 'Comment on “Absence of Off-Diagonal Long-Range Order in hcp $^4$He Dislocation Cores”', Phys. Rev. Lett. 131, 189601 (2023). DOI: 10.1103/PhysRevLett.131.189601, https://arxiv.org/abs/2303.12174
16. M. de Koning, W. Cai, C. Cazorla, and J. Boronat, 'Reply to “Comment on Absence of off-diagonal long-range order in hcp $^4$He dislocation cores”', Phys. Rev. Lett. 131, 189602 (2023). DOI: 10.1103/PhysRevLett.131.189602, https://arxiv.org/abs/2312.04580