Abstract
Predictions and discoveries of new phases of superfluid 3He in confined geometries, as well as novel topological excitations confined to surfaces and edges of near a bounding surface of 3He, are driving the fields of superfluid 3He infused into porous media, as well as the fabrication of sub-micron to nano-scale devices for controlled studies of quantum fluids. In this report we consider superfluid 3He confined in a periodic geometry, specifically a two-dimensional lattice of square, sub-micron-scale boundaries (“posts”) with translational invariance in the third dimension. The equilibrium phase(s) are inhomogeneous and depend on the microscopic boundary conditions imposed by a periodic array of posts. We present results for the order parameter and phase diagram based on strong pair breaking at the boundaries. The ordered phases are obtained by numerically minimizing the Ginzburg-Landau free energy functional. We report results for the weak-coupling limit, appropriate at ambient pressure, as a function of temperature T, lattice spacing L, and post edge dimension, d. For all d in which a superfluid transition occurs, we find a transition from the normal state to a periodic, inhomogeneous “polar” phase with \(T_{c_{1}} < T_{c}\) for bulk superfluid 3He. For fixed lattice spacing, L, there is a critical post dimension, d c , above which only the periodic polar phase is stable. For d<d c we find a second, low-temperature phase onsetting at \(T_{c_{2}} < T_{c_{1}}\) from the polar phase to a periodic “B-like” phase. The low temperature phase is inhomogeneous, anisotropic and preserves time-reversal symmetry, but unlike the bulk B-phase has only \(\mathtt{D}_{\text{4h}}^{\text{L}+\text{S}}\) point symmetry.
Similar content being viewed by others
Notes
The in-plane chiral phase with period 2L is a periodic version of the texture obtained by Surovtsev and Fomin [21] for a uniform distribution of rod-like impurities embedded in 3He-A.
References
V. Ambegaokar, P. de Gennes, D. Rainer, Phys. Rev. A 9, 2676 (1975)
D. Rainer, M. Vuorio, J. Phys. C, Solid State Phys. 10, 3093 (1977)
P. de Gennes, D. Rainer, Phys. Lett. A 46, 429 (1974)
N.D. Mermin, T.-L. Ho, Phys. Rev. Lett. 36, 594 (1976)
P.M. Walmsley, A.I. Golov, Phys. Rev. Lett. 109, 215301 (2012)
T. Kunimatsu, H. Nema, R. Ishiguro, M. Kubota, T. Takagi, Y. Sasaki, O. Ishikawa, J. Low Temp. Phys. 171, 280 (2013)
L.J. Buchholtz, G. Zwicknagl, Phys. Rev. B 23, 5788 (1981)
Y. Nagato, M. Yamamoto, K. Nagai, J. Low Temp. Phys. 110, 1135 (1998)
A. Vorontsov, J.A. Sauls, Phys. Rev. B 68, 064508 (2003)
L.H. Kjäldman, J. Kurkijärvi, D. Rainer, J. Low Temp. Phys. 33, 577 (1978)
A.B. Vorontsov, J.A. Sauls, Phys. Rev. Lett. 98, 045301 (2007)
M. Freeman, R.S. Germain, E.V. Thuneberg, R.C. Richardson, Phys. Rev. Lett. 60, 596 (1988)
J. Xu, B.C. Crooker, Phys. Rev. Lett. 65, 3005 (1990)
A. Schechter, R.W. Simmonds, R.E. Packard, J.C. Davis, Nature 396, 554 (1998)
L.V. Levitin, R.G. Bennett, A. Casey, B. Cowan, J. Saunders, D. Drung, T. Schurig, J.M. Parpia, Science 340, 6 (2013)
M. Gonzalez, P. Zheng, E. Garcell, Y. Lee, H.B. Chan, Rev. Sci. Instrum. 84, 025003 (2013)
N. Zhelev, R. Bennett, R. Ilic, J. Parpia, L. Levitin, A. Casey, J. Saunders, Bull. Am. Phys. Soc. 58 (2013)
D. Rainer, J.W. Serene, Phys. Rev. B 13, 4745 (1976)
D. Vollhardt, P. Wölfle, The Superfluid Phases of 3He (Taylor & Francis, New York, 1990)
J.A. Sauls, Phys. Rev. B 84, 214509 (2011)
E.V. Surovtsev, I.A. Fomin, J. Low Temp. Phys. 150, 487 (2008)
O.C. Zienkiewicz, R.L. Taylor, J.Z. Zhu, The Finite Element Method: Its Basis and Fundamentals (Butterworth-Heinemann, Stoneham, 2005)
J.R. Shewchuk, in Applied Computational Geometry: Towards Geometric Engineering, ed. by M.C. Lin, D. Manocha. Lecture Notes in Computer Science, vol. 1148 (Springer, Berlin, 1996), pp. 203–222
M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970)
W.W. Hager, H. Zhang, ACM Trans. Math. Softw. 32, 113 (2006)
Acknowledgements
This research is supported by the National Science Foundation (Grant DMR-1106315). We thank David Ferguson for many discussions and critique during the course of this work.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Wiman, J.J., Sauls, J.A. Superfluid Phases of 3He in a Periodic Confined Geometry. J Low Temp Phys 175, 17–30 (2014). https://doi.org/10.1007/s10909-013-0924-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10909-013-0924-4