Ripplonic Lamb Shift for Electrons on Liquid Helium

Abstract

We study the shift of the energy levels of electrons on helium surface due to the coupling to the quantum field of surface vibrations. As in quantum electrodynamics, the coupling is known, and it is known to lead to an ultraviolet divergence of the level shifts. We show that there are diverging terms of different nature and use the Bethe-type approach to show that they cancel each other, to the leading-order. This resolves the long-standing theoretical controversy and explains the existing experiments. The results allow us to study the temperature dependence of the level shift. The predictions are in good agreement with the experimental data.Electrons above the surface of liquid helium were one of the first observed two-dimensional electron systems (2DESs) [1][2][3][4]. In this system the conceptual simplicity is combined with far from trivial behavior, which allows studying many-body effects in a well-characterized setting. The system displays the highest mobility known for 2DESs, exceeding 2 × 10 8 cm 2 /(V·s) [5] and can be exquisitely well controlled [6]. The electron-electron interaction is typically strong, so that the electrons can form a Wigner solid [7,8] or a strongly correlated liquid with unusual transport properties [9,10]. A number of new many-electron phenomena have been found recently [11][12][13][14][15].An advantageous feature of the system is the simple form of the confining potential. It is formed by the high Pauli barrier at the helium surface and the image potential. One then expects the electron energy spectrum to be well understood. Indeed, already the first experiment on transitions between the subbands of quantized motion normal to the surface showed a good, albeit imperfect agreement with the model [4]. Much work has been done on improving, sometimes empirically, the form of the confining potential, cf. [4,[16][17][18][19]. On the other hand, it has been known that the electrons are also coupled to a bosonic field, the capillary waves on the surface of helium (ripplons), and that this coupling affects the electron energy spectrum [20][21][22][23][24]. The importance of this effect was demonstrated in explaining the Wigner crystallization [8,25] and through cyclotron-resonance measurements [26].In terms of the coupling to a bosonic field, electrons on helium are a close condensed-matter analog of systems studied in quantum electrodynamics. The known form of the coupling [27] and the possibility to control it and to study the interplay of this coupling with the manyelectron effects make the system particularly attractive. In this context, a major problem is that the ripploninduced shift of the electron energy levels of motion normal to the surface contains diverging terms. They come from short-wavelength ripplons. The ultraviolet divergence is strong, as a high power of the wave number. Unless one deals with it carefully, the resulting level shifts become comparable to the electron binding energy for a short-wavelength cutoff approaching twice the interatomic distance. The problem bears ...

2D electrons floating on a suspended atomically thin dielectric

“…The polaron regime of transport appears due to the renormalization of energy spectrum similarly to the case of the 2D electrons on He interacted with the ripplon vibrations. [38] B. Relaxation caused by roughness of ATD Next, we turn to the consideration of the roughnessinduced relaxation which limits the mobility in the case of non-effective relaxation via phonons.…”

“…Fig.4this region is separated by dotted line). The polaron regime of transport appears due to the renormalization of energy spectrum similarly to the case of the 2D electrons on He interacted with the ripplon vibrations [38]. …”

Internal geometric friction in a Kitaev-chain heat engine

“…The method of TDELs was proposed as a fast and convenient way to perform statistical mechanic calculations for an assembly of systems [22,23]. It was applied to semiconductors [33][34][35][36], superfluids [37], optomechanical oscillators [38,39], heat losses in thermoelectric systems [27,28], and thermalization of finite-size systems [24][25][26]. Even before TDELs were proposed, the advantages of temperature-dependent mean potentials and corresponding forces were known in statistical mechanics of fluid mixtures [40].…”

Many-electron system on helium and the color center spectroscopy

“…The spectroscopic observation of transitions between the quantized energy levels [3] was a direct proof of the picture of the electron confinement and the overall nature of the potential. Since then much work has been done on the exact positions and the widths of the spectral lines and their dependence on the temperature and the electron density [4][5][6][7][8][9][10][11][12].…”