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ACKNOWLEDGMENTS
We thank the staff at the Institute Laue Langevin, Heinz Maier-Leibnitz
Zentrum, ISIS Neutron and Muon Source, Paul Scherrer Institut, the
spallation neutron source at Oak Ridge National Laboratory, E. Villard,
P. Chevalier, and J. Frank for support; J. Locatelli for the instrument
control system; the ILL cluster computing facility for IT support; and
M. Kugler for initial implementation of the model and support in early
experiments.Funding:Supported by the DFG in the framework of
TRR80 (project E1, project-id 107745057), SPP 2137 (Skyrmionics)
under grant PF393/19 (project-id 403191981), ERC Advanced Grants
291079 (TOPFIT) and 788031 (ExQuiSid), and Germany’s excellence
strategy EXC-2111 390814868 (A.B. and C.P.); DFG grant GE 971/5-1
(R.G.); the DFG in the framework of SFB 1143 (project A07; project-id
247310070), as well as grants GA 1072/5-1 (project-id 270344603)
and GA 1072/6-1 (project-id 324327023) (M.G.); and the LANL Directed
Research and Development program via the Directed Research project
“A New Approach to Mesoscale Functionality: Emergent Tunable
Superlattices (20150082DR)”(D.M.F. and M.J.). Early work by M.G.
and J.W. on the spin wave theory was supported by the Institute for
Materials Science at Los Alamos (RR2015). This research used
resources at the Spallation Neutron Source, a DOE Office of Science
User Facility operated by the Oak Ridge National Laboratory. The
datasets collected at LET have the DOIs 10.5286/ISIS.E.RB1620412 and
10.5286/ISIS.E.RB1720033. The datasets collected at the ILL have
the DOIs 10.5291/ILL-DATA.INTER-413, 10.5291/ILL-DATA.INTER-436,
10.5291/ILL-DATA.INTER-477, 10.5291/ILL-DATA.4-01-1597, and
10.5291/ILL-DATA.4-01-1621. Data recorded at RESEDA were collected
under proposal P00745-01. Additional datasets (not shown) were
collected at MIRA (proposals 13511 and 15633) MLZ, Garching,
Germany; TASP (proposals 20181324 and 20151888) at PSI, Villigen,
Switzerland; and CNCS at SNS Oak Ridge, USA.Author contributions:
T.W., D.M.F., and M.J. proposed and designed the ToF and TAS
experiments. T.W. proposed and designed the polarized TAS
experiments. C.P. proposed the MIEZE experiment. A.B. grew the single
crystal denoted sample 1. T.W., D.M.F., L.B., C.F., H.G., and M.J.
performed the experiments. P.S., G.S.T., M.B., R.B., D.V., C.F., M.S.,
and R.G. supported the experiments at neutron spectrometers under
their responsibility. T.W., C.P., P.B., M.J., and M.G. conceived the
interpretation. J.W. and M.G. performed the theoretical calculations.
T.W., D.M.F., L.B., H.G., and C.F. performed the data analysis. T.W. and
M.G. implemented the theoretical skyrmion model for use with TAS
resolution convolution. T.W., C.P., P.B., and M.G. wrote the manuscript
with input from all authors. All authors discussed the results and
reviewed the manuscript.Competing interests:The authors declare
that they have no competing interests.Data and materials availability:
The computer code for the spin wave simulations used in this paper
can be found at the repository given in ( 39 ). All data needed to evaluate
the conclusions in the paper are present in the paper and/or the
supplementary materials and available at the repository given in ( 43 ).SUPPLEMENTARY MATERIALS
science.org/doi/10.1126/science.abe4441
Materials and Methods
Supplementary TextFigs. S1 to S22
Tables S1 to S4
References ( 44 – 80 )21 August 2020; resubmitted 30 April 2021
Accepted 31 January 2022
10.1126/science.abe4441QUANTUM OPTICSBreakdown of topological protection by cavity
vacuum fields in the integer quantum Hall effect
Felice Appugliese^1 *, Josefine Enkner^1 , Gian Lorenzo Paravicini-Bagliani^1 , Mattias Beck^1 ,
Christian Reichl^2 , Werner Wegscheider^2 , Giacomo Scalari^1 , Cristiano Ciuti^3 , Jérôme Faist^1 *The prospect of controlling the electronic properties of materials via the vacuum fields of cavity
electromagnetic resonators is emerging as one of the frontiers of condensed matter physics. We found
that the enhancement of vacuum field fluctuations in subwavelength split-ring resonators strongly
affects one of the most paradigmatic quantum protectorates, the quantum Hall electron transport
in high-mobility two-dimensional electron gases. The observed breakdown of the topological protection
of the integer quantum Hall effect is interpreted in terms of a long-range cavity-mediated electron
hopping where the anti-resonant terms of the light-matter coupling Hamiltonian develop into a finite
resistivity induced by the vacuum fluctuations. Our experimental platform can be used for any
two-dimensional material and provides a route to manipulate electron phases in matter by means of
vacuum-field engineering.A
n intriguing aspect of quantum field
theories is the description of empty
space as permeated by electromagnetic
vacuum fluctuations. Energy conserva-
tion forbids any process that would lead
to net energy extraction from such states, thus
implying that such vacuum fields cannot be
detected by direct absorption. Nonetheless,
there is much experimental evidence of their
existence: Spontaneous emission, the Lamb
shift, and the Casimir effect can only be ex-
plained by invoking the role of vacuum fields
( 1 ). More recently, technological developments
of both laser sources and optical nanocavities
in the strong light-matter coupling offer a new
perspective about vacuum fields, one in which
those fluctuations can be sensed directly ( 2 , 3 )
and used to engineer new properties of matter
( 4 – 7 ) without illumination. Polaritons, optical
excitations in the strong light-matter coupling
regime, can also be used as sensitive probes of
many-body states such as fractional quantum
Hall states ( 8 ) and Wigner crystals ( 9 ). Confin-
ing the light on a strongly subwavelength scale
is a key element that enabled the achievement
of record high interaction strengths ( 10 – 13 ) inwhich the anti-resonant terms of light-matter
coupling play an important role ( 14 ). Cavity-
controlled superconductivity ( 15 , 16 ), long-range
ferroelectric order ( 17 , 18 ), and cavity-mediated
superradiance ( 4 ), all exploiting the cou-
pling of electrons or dipoles to vacuum fields,
have recently been investigated. Experimen-
tally, however, unambiguous identification of
modified equilibrium properties of solid-state
quantum phases of matter ( 19 ) through vac-
uum fields remains an open challenge.
We recently demonstrated the role of Landau
polaritons ( 20 ) in controlling the DC bulk mag-
netotransport ( 21 ) in a semiconductor elec-
tronic gas, even in the absence of external
illumination. The amplitude of the Shubnikov–
de Haas oscillations was modified by the pres-
ence of a terahertz resonator embedding the
Hall bar. However, because a change in the
shape of the oscillations in DC magnetoresis-
tivity is sensitively linked to variations of
scattering processes, the phenomenology is
not universal and every sample is unique. Here,
we investigated transport in the integer quan-
tum Hall regime, where the topological protec-
tion of the edge currents results in a quantized
resistance and where the effect of vacuum fields
can be unambiguously revealed.
At low magnetic fields, it is possible to de-
scribe transport in the presence of impurity
scattering only as a broadening of the Landau
levels’density of states. In the high–magnetic
field regime, the random impurity potential
instead causes the localization of electrons.1030 4 MARCH 2022•VOL 375 ISSUE 6584 science.orgSCIENCE
(^1) Institute of Quantum Electronics, ETH Zürich, Zürich 8093,
Switzerland.^2 Laboratory for Solid State Physics, ETH Zürich,
Zürich 8093, Switzerland.^3 CNRS, Laboratoire Matériaux et
Phénomènes Quantiques, Université de Paris, F-75013 Paris,
France.
*Corresponding author. Email: [email protected] (F.A.);
[email protected] (J.F.)
†Present address: CNRS, ISIS, University of Strasbourg, 67000
Strasbourg, France.
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