Dynamics of electron bubbles in superfluid ³He-B

Abstract

This work investigates the transport of an electron bubble near the free surface of superfluid ³He-B under applied electric and magnetic fields. Based on a theoretical framework combining the quasiclassical Green’s function and the Lippmann-Schwinger equation, we have calculated the scattering cross section and mobility of the electron bubble, together with their temperature and depth dependences. An electric field shifts the position of the electron bubble and thereby tunes its coupling to the surface bound states. The surface density of states decays with depth, whereas the transport cross section increases with energy and depth; these competing trends compensate, resulting in a nearly depth-independent mobility consistent with the linear dispersion of the surface states. In contrast, an applied magnetic field opens a Zeeman gap in the surface-state spectrum, which breaks the linear dispersion of the bound states. Our results demonstrate that external electric and magnetic fields provide effective control of the spectral structure and scattering properties of the surface bound states.

Keywords

Superfluid ³He / majorana fermion / surface bound state

References

[1] Majorana E. Teoria simmetrica dell’elettrone e del positrone. Il Nuovo Cim. 1937; 14(4):171-184.

[2] Tanaka Y, Tamura S, Cayao J. Theory of Majorana zero modes in unconventional superconductors. Prog Theor Exp Phys. 2024; 2024(8):08C105.

[3] Prada E, San-Jose P, de Moor MWA, Geresdi A, Lee EJH, Klinovaja J, et al. From Andreev to Majorana bound states in hybrid superconductor-semiconductor nanowires. Nat Rev Phys. 2020; 2(10):575-594.

[4] Sato M, Ando Y. Topological superconductors: A review. Rep Prog Phys. 2017; 80(7):076501.

[5] Liang QF, Wang Z, Kawakami T, Hu X. Exploration of Majorana bound states in topological superconductors. Acta Phys Sin. 2020; 69(11):117102.

[6] Li G, Ding H, Wang ZQ, Gao HJ. Majorana zero mode and its lattice construction in iron-based superconductors. Acta Phys Sin. 2024; 73(3):030302.

[7] Chen L, Pan XH, Cao Z, Liu DE, Liu X. Electrostatic environment and Majorana bound states in full-shell topological insulator nanowires. Phys Rev B. 2024; 109(7):075408.

[8] Cao Z, Chen S, Zhang G, Liu DE. Recent progress on Majorana in semiconductor-superconductor heterostructures: Engineering and detection. Sci China Phys Mech Astron. 2023; 66(6):267003.

[9] Lee J, Ikegaya S, Asano Y. Anomalous proximity effect of a spin-singlet superconductor with a spin-orbit interaction. Phys Rev B. 2025; 111(17):174516.

[10] Oshima D, Ikegaya S, Schnyder AP, Tanaka Y. Flat-band Majorana bound states in topological Josephson junctions. Phys Rev Res. 2022; 4(2):L022051.

[11] Löfwander T, Shumeiko VS, Wendin G. Andreev bound states in high-T_c superconducting junctions. Supercond Sci Technol. 2001; 14(5):R53-R77.

[12] Osheroff DD, Richardson RC, Lee DM. Evidence for a new phase of Solid He³. Phys Rev Lett. 1972; 28(14):885-888.

[13] Tsutsumi Y, Kawakami T, Shiozaki K, Sato M, Machida K. Symmetry-protected vortex bound state in superfluid ³He-B phase. Phys Rev B. 2015; 91(14):144504.

[14] Lv BQ, Qian T, Ding H. Experimental perspective on three-dimensional topological semimetals. Rev Mod Phys. 2021; 93(2):025002.

[15] Qi XL, Hughes TL, Raghu S, Zhang SC. Time-reversal-invariant topological superconductors and superfluids in two and three dimensions. Phys Rev Lett. 2009; 102(18):187001.

[16] Autti S, Haley RP, Jennings A, Pickett GR, Poole M, Schanen R, et al. Transport of bound quasiparticle states in a two-dimensional boundary superfluid. Nat Commun. 2023; 14(1):6819.

[17] Murakawa S, Wada Y, Tamura Y, Saitoh M, Aoki Y, Nomura R, et al. Surface condition dependence of transverse acoustic impedance in superfluid ³He. J Phys: Conf Ser. 2009; 150(3):032070.

[18] Jiang WG, Barquist CS, Gunther K, Lee Y, Chan HB. Superfluid ³He-B surface states in a confined geometry probed by a microelectromechanical oscillator. Phys Rev B. 2023; 108(2):024508.

[19] Okuda Y, Nomura R. Surface Andreev bound states of superfluid ³He and Majorana fermions. J Phys Condens Matter. 2012; 24(34):343201.

[20] Bunkov YM, Gazizulin RR. Direct observation of the specific heat of Majorana quasiparticles in superfluid³He-B. Sci Rep. 2020; 10(1):20120.

[21] Murakawa S, Tamura Y, Wada Y, Wasai M, Saitoh M, Aoki Y, et al. New anomaly in the transverse acoustic impedance of superfluid ³He-B with a wall coated by several layers of ⁴He. Phys Rev Lett. 2009; 103(15):155301.

[22] Ikegami H, Kono K. Texture of superfluid ³He probed by a Wigner solid. Phys Rev Lett. 2006; 97(16):165303.

[23] Nagato Y, Higashitani S, Nagai K. Subgap in the surface bound states spectrum of superfluid ³He-B with rough surface. J Low Temp Phys. 2018; 190(5):277-301.

[24] de M Carvalho AM, Marques AS, Silva GT, Garcia GQ, Furtado C. Scattering of massless quasi-particles in the ³He-A superfluid. Int J Geom Methods Mod Phys. 2025; 22(13):2550115.

[25] Ikegami H, Bum Chung S, Kono K. Mobility of ions trapped below a free surface of superfluid ³He. J Phys Soc Jpn. 2013; 82(12):124607.

[26] Shindou R, Furusaki A, Nagaosa N. Quantum impurity spin in Majorana edge fermions. Phys Rev B. 2010; 82(18):180505.

[27] Leiderer P. Ions at helium interfaces: A review. Entropy. 2026; 28(1):109.

[28] Matos JCA, Giannakeas P, Ciardi M, Pohl T, Rost JM. Transitional supersolidity in ion doped helium droplets. Phys Rev Lett. 2025; 134(18):186001.

[29] Skrbek L. Ion pools underneath the surface of superfluid ⁴He: A playground of classical two-dimensional physics. J Low Temp Phys. 2023; 212(5):232-250.

[30] Salomaa M. The mobility of negative ions in superfluid ³He. Contemp Phys. 1982; 23(2):169-188.

[31] Autti S, Haley RP, Jennings A, Pickett GR, Surovtsev EV, Tsepelin V, et al. Drag on cylinders moving in superfluid ³He-B as the dimension spans the coherence length. J Low Temp Phys. 2024; 217:264-278.

[32] Poitrenaud J, Williams FIB. Precise measurement of effective mass of positive and negative charge carriers in liquid helium II. Phys Rev Lett. 1972; 29(18):1230-1232.

[33] Rudd MJ, Yapa PS, Shook AJ, Maciejko J, Davis JP. Strong-coupling corrections to hard domain walls in superfluid ³He-B. Phys Rev B. 2021; 104(9):094520.

[34] Tsutsumi Y. Scattering theory on surface Majorana fermions by an impurity in ³He-B. Phys Rev Lett. 2017; 118(14):145301.

[35] Regan RC. Quantized vortices in superfluid ³He [dissertation]. Evanston: Northwestern University; 2020. Available from: https://www.proquest.com/openview

[36] Arrayás M, Trueba JL, Uriarte C, Zmeev DE. Design of a system for controlling a levitating sphere in superfluid ³He at extremely low temperatures. Sci Rep. 2021; 11(1):20069.

[37] Scott JW, Nguyen MD, Park D, Halperin WP. Magnetic susceptibility of andreev bound states in superfluid ³He-B. Phys Rev Lett. 2023; 131(4):046001.

[38] Shevtsov O, Sauls JA. Electron bubbles and Weyl fermions in chiral superfluid ³He-A. Phys Rev B. 2016; 94(6):064511.

[39] Baym G, Pethick CJ, Salomaa M. Mobility of negative ions in superfluid ³He-B. J Low Temp Phys. 1979; 36(3):431-466.