# Critical slowing down of fermions near a magnetic quantum phase transition

Chia-Jung Yang  Department of Materials, ETH Zurich, 8093 Zurich, Switzerland
Kristin Kliemt  Physikalisches Institut, Goethe-Universität Frankfurt, 60438
Frankfurt, Germany  Cornelius Krellner  Physikalisches Institut, Goethe-
Universität Frankfurt, 60438 Frankfurt, Germany  Johann Kroha  Physikalisches
Institut and Bethe Center for Theoretical Physics, Universität Bonn, 53115
Bonn, Germany  Manfred Fiebig  Department of Materials, ETH Zurich, 8093
Zurich, Switzerland  Shovon Pal  Department of Materials, ETH Zurich, 8093
Zurich, Switzerland  School of Physical Sciences, National Institute of
Science Education and Research, HBNI, Jatni, 752 050 Odisha, India

(September 3, 2024)

###### Abstract

A universal phenomenon in phase transitions is critical slowing down (CSD) —
systems, after an initial perturbation, take an exceptionally long time to
return to equilibrium. It is universally observed in the dynamics of bosonic
excitations, like order-parameter collective modes, but it is not generally
expected to occur for fermionic excitations because of the half-integer nature
of the fermionic spin. Direct observation of CSD in fermionic excitations or
quasiparticles would therefore be of fundamental significance. Here, we
observe fermionic CSD in the heavy-fermion (HF) compound YbRh2Si2 by terahertz
time-domain spectroscopy. HFs are compound objects with a strongly enhanced
effective mass, composed of itinerant and localized electronic states. We see
that near the quantum phase transition in YbRh2Si2 the build-up of spectral
weight of the HFs towards the Kondo temperature $T_{K}\approx 25$ K is
followed by a logarithmic rise of the quasiparticle excitation rate on the
heavy-Fermi-liquid side of the quantum phase transition below $10$ K. A
critical two-band HF liquid theory shows that this is indicative of fermionic
CSD. This CSD is a clear indication that the HF quasiparticles experience a
breakdown near the quantum phase transition, and the critical exponent of this
breakdown introduces a classification of fermionic quantum phase transitions
analogous to thermodynamic phase transitions — solution to a long-standing
problem.

At a continuous phase transition, the ordered and the disordered phases become
equal in energy. As a consequence, the fluctuations between these two states
become infinitely slow. This so-called critical slowing down (CSD) is
universally observed in the dynamics of classical fields which are bosonic in
nature and vanish at the phase transition, like the magnetization, associated
with bosonic magnons, in the case of ferromagnetic orderStatPhys . By
contrast, CSD of fermionic excitations or quasiparticles is generally not
expected to occur since fermions as elementary particles are thought to be
indestructible. However, certain quantum materials known as heavy-fermion (HF)
compounds host composite fermionic quasiparticles. These are quantum
superpositions of itinerant and localized, i.e., heavy, electron states
generated by the Kondo effectKondo1964 ; Hewson1993 , with a low binding
energy parameterized by the Kondo energy scale, the lattice Kondo temperature
$T_{K}$. At a magnetic quantum phase transition (QPT) in such materials (see
Fig. 1a), these brittle, heavy quasiparticles are assumed to disintegrate due
to critical fluctuationsSi2001 ; Coleman2001 ; Loehneysen2007  despite their
fermionic nature, and fermionic CSD would be a unique signature of this
destruction.

Using time-resolved terahertz (THz) spectroscopy, we directly observe such
fermionic CSD as a suppression of the heavy-particle hybridization gap and
flattening of the associated band. This “softening” expands the region in
momentum space where resonant THz absorption is allowed. We observe this as an
increase instead of a Kondo-weight-loss-generated decrease of the Kondo-
related THz signal towards the quantum critical point (QCP) before the heavy
quasiparticle band vanishes altogether below a breakdown temperature $T_{\rm
qp}^{*}$Gegenwart2007 . Moreover, we identify a critical exponent in this
behaviour and thus set the stage for the classification of fermionic quantum
criticality in analogy to the criticality of thermodynamic phase transitions.

Signatures of Kondo quasiparticle destruction were suspected from the
dynamical scaling of the magnetic susceptibilitySchroeder2000 , specific heat
measurementsCusters2003 , Hall effect measurements of the carrier
densityPaschen2004 , and optical conductivity measurementsProchaska2020 . The
conjectures entering the interpretation of these measurements have been
challenged, howeverVojta2008 ; Woelfle2011 . Time-resolved THz spectroscopy is
a unique tool to probe the heavy quasiparticle dynamics and to resolve these
questions. Specifically, HF materials respond to an incident ultrashort THz
pulse by the emission of a time-delayed reflexWetli2018 . This “echo” is a
response from the reconstructing Kondo ground state after its destruction by
the incident pulse. Hence, time acts as a filter separating the Kondo-
sensitive delayed pulse from the Kondo-insensitive main pulse so that the
former is a background-free response of the HF state. Specifically, the
amplitude and the delay time of the echo pulse are proportional to the HF
spectral weight and the Kondo coherence time
$\tau_{K}=2\pi\hslash/k_{B}T_{K}$, respectively (with $\hslash$ the Planck and
$k_{B}$ the Boltzmann constant). As a method, this technique was introduced
through experiments on CeCu6-xAuxWetli2018 ; Pal2019 ; Yang2020 .

Figure 1: Exploring fermionic quantum criticality by time-resolved THz
reflectivity. a, Schematic phase diagram of YbRh2Si2 with characteristic
energy scales. b, Reflected THz signal from the YbRh2Si2 sample (red) and from
a Pt mirror reference (black). The time traces are normalized by the maximum
field amplitude at $t=0$ ps. The delayed, purely Kondo-related “echo” pulse is
visible in the interval between 1.3 and 2.6 ps, where it distinctly differs
from the Pt reference signal. c, Signal of the delayed pulse in YbRh2Si2 at
$B_{\perp}=0$ for various temperatures.

In YbRh2Si2 we now apply this new method to directly measure the fermionic
CSD. YbRh2Si2 is a prototypical HF compound. In zero magnetic field it is
antiferromagnetic below the Neél temperature $T_{N}=72$ mK. It undergoes a QPT
to a Kondo HF liquid at a critical magnetic field of $B_{\perp}\approx 66$ mT
perpendicular to the $c$ axisTrovarelli2000 ; Gegenwart2002  (see Fig. 1a))
induced by the Ruderman-Kittel-Kasuya-Yosida (RKKY) magnetic interaction
between the Yb momentsRuderman1954 ; Kasuya1956 ; Yosida1957 ; Nejati2017 .
Alternatively, a 6% substitution of Rh by Ir creates a QCP at zero
fieldKrellner2012 ; Friedemann2009 . YbRh2Si2 has a Kondo temperature of
$T_{K}\approx 25$ K, high enough to enable a wide quantum-critical region and
to permit us to search for signs of CSD in the range $T^{*}_{\rm qp}<T<T_{K}$,
in contrast to the case of CeCu6-xAux. In our temperature-dependent, time-
resolved THz reflection spectroscopy measurements we crossed the QCP by
varying the magnetic field or the Ir concentration. The 1.5-cycle THz pulses
of $\sim 2$ ps duration were incident onto the $c$-cut Yb(Rh1-xIrx)2Si2
samples ($x=$0, 0.06). The echo pulses were analysed as described
elsewhereWetli2018 . With $T_{K}=25$ K we obtain a delay time $\tau_{K}\approx
1.9$ ps for these in agreement with the data in Figs. 1b and 1c. We therefore
chose the time window for the analysis from $1.3$ to $2.6$ ps.

Fig. 2 shows the time-integrated intensity of the THz echo pulse for
Yb(Rh1-xIrx)2Si2 for $x=0$ and $B_{\perp}=214$ mT and for $x=0.06$ and
$B_{\perp}=0$. All plots exhibit a similar maximum value of the spectral Kondo
weight near $25$ K in good agreement with $T_{K}$. Aside from a shift towards
higher temperature right at the QCP, the maximum shows no systematic field
dependence. Below the peak temperature, the signal initially decreases with
temperature for all magnetic fields. This can be attributed to the reduced
thermal broadening of THz-induced interband transitions and is reproduced by
the theory introduced below.

On the antiferromagnetic side of the QCP (Figs. 2a and 2b), the signal
continues to decrease but remains finite down to the lowest experimentally
achieved temperature of 2 K. We note that this temperature and field range
(2.0 K $\leq T\leq$ 20 K, $B_{\perp}\lesssim 66$ mT) is within the white area
of the phase diagram in Fig. 1a, the so-called quantum-critical fanCusters2003
; Friedemann2009 , where thermodynamic and transport properties are dominated
by quantum-critical fluctuationsTrovarelli2000 ; Gegenwart2002 ; Prochaska2020
. However, our THz time-delay spectroscopy is not directly sensitive to these
fluctuations, but exclusively to the HF quasiparticle spectral weightWetli2018
; Pal2019 . Therefore, the behavior in Figs. 2a and 2b indicates that the
Kondo effect remains partially intact in this temperature range. This is
reasonable since we are still a factor of $\sim 25$ above $T_{N}$ so that the
heavy quasiparticles are not entirely destroyed by the impending
antiferromagnetic order..

At quantum criticality (Figs. 2d and 2h), the temperature dependence changes
drastically. The initial signal decrease with temperature is now followed by a
logarithmic increase of the THz echo signal that persists down to the lowest
observed temperature. Note qualitative similarity between the field-tuned
(Fig. 2d) and chemically tuned (Fig. 2h) quantum critical systems, however
with different logarithmic slopes. On the HF liquid side of the QCP the signal
increase towards the lowest temperatures is still present and gradually fades
away with distance from the QCP (Figs. 2e-g). Its onset may also be
conjectured at $50$ mT on the antiferromagnetic side. (Fig. 2c).

To understand the striking logarithmic increase towards low temperature, we
analyze the THz echo signal theoretically. In a HF system, the strongly
correlated, flat band produced by the Kondo effect hybridizes with the light
conduction band to generate a structure with a lower $(n=1)$ and an upper
$(n=2)$ band with avoided crossingPaschen2016 ; Feng2017 ; Pal2019 , as shown
in Figs. 3a to 3c. While low-temperature thermodynamic and transport
experiments probe the lower, occupied band only, resonant THz spectroscopy
involves transitions between both bands, which calls for a two-band theory.
Our according, critical HF liquid theory shows (see Methods) that the $n=1,2$
bands with dispersions $\varepsilon_{n\vec{p}}$ (where $\vec{p}$ is the
crystal-electron momentum) have distinct momentum- and temperature-dependent
spectral weights $z_{n\vec{p}}$. These cross over from $z_{n\vec{p}}\approx 1$
in the strongly dispersive region to $z_{n\vec{p}}=a\,z_{0}\ll 1$ in the flat
region of these bands, see Figs. 3d to 3f and insets. Here, $a\ll 1$ is the
spectral weight of the local, single-ion Kondo resonance, which builds up
logarithmically from above $T_{K}$ and then saturates towards a constant value
for $T<T_{K}$. Further, $z_{0}=(T/T_{0})^{\alpha}$ is a suppression factor
with a critical exponent $\alpha$. It describes the destruction of
quasiparticle spectral weight as the QCP is approached by lowering the
temperature $T$ below the onset temperature for quantum criticality, $T_{0}$.
For the latter, experiments revealed $T_{0}\approx T_{K}$ in
YbRh2Si2Gegenwart2006 .

Figure 2: Temperature dependence of the resonant THz absorption across the QCP
in YbRh2Si2. a–g, Evolution of the THz absorption by resonant Kondo
quasiparticle excitations at various external magnetic fields $B_{\perp}$ as a
function of temperature. The weights are derived from the integrated intensity
of the echo pulses emitted in the time window of $1.3$ – $2.6$ ps time window
(see Figs. 1b, c). h, Temperature dependence of the resonant THz absorption in
quantum critical Yb(Rh1.94Ir0.06)2Si2. Fits of Eq. 1 are plotted as blue lines
in d and h.

As mentioned, the THz echo pulse at $\tau_{K}=2\pi\hslash/k_{B}T_{K}$ is
solely sensitive to the breakup-and-recovery dynamics of the HFs and not to
other THz absorption channelsWetli2018 ; Pal2019 ; Yang2020 . Therefore, its
intensity exclusively depends on the quasiparticle weight and the phase space
available for THz-induced excitations. Specifically, the echo-pulse intensity
is proportional to the probability

$P(T)=A\int d^{3}p\ z_{1\vec{p}}\ z_{2\vec{p}}\
f(\varepsilon_{1\vec{p}})\,[1-f(\varepsilon_{2\vec{p}})]\
W(\Delta\varepsilon_{\vec{p}})$ (1)

for the resonant excitation of electrons from the lower to the upper band at
an energy difference
$\Delta\varepsilon_{\vec{p}}=\varepsilon_{2\vec{p}}-\varepsilon_{1\vec{p}}$.
Here, $f(\varepsilon_{n\,\vec{p}})$ is the Fermi-Dirac distribution function.
In Eq. (1), it describes the probability that the $n=1$ band is occupied and
the $n=2$ band is empty before the THz absorption process. $W(\hslash\omega)$
is the spectrum of the incident THz pulse, which is a Gaussian distribution of
width $\Gamma$ centered around the central frequency $\Omega_{\rm THz}$. With
$\Delta\varepsilon_{\vec{p}}\approx\hslash\Omega_{\rm THz}$, the THz-induced
interband transition becomes resonantly allowed. The integral runs over all
electron momenta, and the factor $A$ is a temperature-independent constant,
proportional to the intensity of the incident THz pulse and to the modulus
square of the electric-dipole transition-matrix element between the two bands.

Figure 3: Band structure and Kondo weight calculations towards the QCP. a–c,
Band structure of the conduction band (steep slope) and the Kondo state (flat
band) resulting in a hybridized lower (brown) and upper (orange) branch. d–f,
Momentum- and temperature-dependent quasiparticle weights $z_{1\vec{p}}$,
$z_{2\vec{p}}$ in the lower (brown) and upper (orange) bands as well as the
product $z_{1\vec{p}}\cdot z_{2\vec{p}}$, all calculated from the two-band
critical Fermi liquid theory. g, h, Resonant THz absorption strength at (red
and dark yellow) and away from (blue) the QCP as calculated for the system
parameters of Yb(Rh1-xIrx)2Si2, ($x=0,\,0.06$) and CeCu6-xAux, respectively.

When the probability for HF formation, $a\,z_{0}\propto(T/T_{K})^{\alpha}$,
tends to zero at the QCP, the heavy bands flatten according to the two-band HF
liquid theory. Also, the hybridization gap vanishes, and with that the
quasiparticle energy in the heavy regions of both bands ($n=1,2$) approaches
the Fermi energy $E_{F}$, see Figs. 3d to 3f. This means that the oscillation
frequency of fermionic quasiparticles,
$\omega_{n\vec{p}}=(\varepsilon_{n\vec{p}}-E_{F})/\hslash$, vanishes, which is
indicative of a fermionic CSD. In turn, it implies an expansion of the region
in the momentum space where resonant THz transitions are allowed, seen as
broadening of the shaded areas in Figs. 3a to 3c. The interplay of these two
counteracting effects, quasiparticle destruction and phase-space expansion,
leads to a non-monotonic temperature dependence of the THz absorption strength
$P(T)$. An expansion of Eq. (1)for small $z_{0}(T)$ predicts a logarithmic
increase of $P(T)\propto\ln[1/z_{0}(T)]$ towards low temperatures down to the
region of $T_{\rm qp}^{*}$. The full numerical evaluation of Eq. 1 leads to
the behavior shown in Fig. 3g. It reproduces the Kondo maximum near
$T_{K}\approx 25$ K and at quantum criticality indeed to a logarithmic
increase across an intermediate temperature window $T_{\rm qp}^{*}<T<T_{K}$
according to

$P(T)=\alpha\,A\,\ln(T_{K}/T)\qquad,$ (2)

observing this behavior in Figs. 2d and 2h is thus a unique experimental
signature of fermionic quasiparticle CSD in the Yb(Rh1-xIrx)2Si2 system.

Upon further decreasing the temperature to $T<T^{*}_{\rm qp}$, $P(T)$
approaches zero as the HF weight disappears altogether, see inset of Fig. 3g.
The low-temperature scale $T_{\rm qp}^{*}$ is thus defined as the position of
the signal maximum between the logarithmic increase and its ultimate collapse
towards $T\to 0$. As seen in Fig. 3g (inset), $T_{\rm qp}^{*}$ depends on the
critical exponent $\alpha$ and is several orders of magnitude lower than the
Kondo scale $\sim T_{K}$, possibly undetectably small. A low-temperature scale
$T^{*}$ has also been observed as a maximum in the magnetic
susceptibilityFriedemann2009  whose microscopic origin has, however, remained
unclear. Since in Yb(Rh1-xIrx)2Si2, $T^{*}$ and our theoretically predicted
$T_{\rm qp}^{*}$ are in the same temperature range, we conjecture that both
may be of the same physical origin - the competition of fermionic CSD and
quasiparticle breakdown at the QCP. As a crossover temperature, $T_{\rm
qp}^{*}$ remains non-zero, but can be exceedingly small, depending on
$\alpha$, see inset of Fig. 3g. Note that away from criticality, we have
$\alpha\to 0$ so that the logarithmic low-temperature behavior does not occur
in agreement with Figs. 2e to 2g. The blue curves in Figs. 2d and 2h represent
the evaluation of Eq. (1) for the spectrum $W(\hslash\omega)$ of the THz
pulses used in our experiment. Considering that $\alpha$ is the only
adjustable parameter apart from the overall signal amplitude and that we use
the same value $T_{K}\approx 25$ K for both curves, the agreement between
theory and data is excellent.

We can now extract the critical exponent $\alpha$ by comparing the logarithmic
slope $s_{\mathrm{low}}$ associated with the the CSD at low temperatures
($T^{*}_{\rm qp}<T<T_{K}$) from Eq. (2) with the slope $s_{\mathrm{high}}$ of
the standard logarithmic behaviour of the Kondo weight at high temperature
($T>T_{K}$) according to $P(T)=A\,\ln(T_{K}/T)$. This directly leads to
$\alpha=s_{\mathrm{low}}/s_{\mathrm{high}}$. From the experimental data we
find $\alpha=0.10$ for Yb(Rh1-xIrx)2Si2 at $x=0$, $B_{\perp}=66$ mT in Fig. 2d
and $\alpha=0.28$ at $x=0.06$, $B_{\perp}=0$ in Fig. 2h. Note that different
critical behavior for QPTs by magnetic fields and chemical pressure have been
observed beforeLoehneysen2001 . It is a signature of the different types of
quantum-critical fluctuations in the two cases.

Our theory also explains why the logarithmic low-temperature increase of THz
absorption indicating CSD cannot be observed in the CeCu6-xAux system. For
this material, the ratio $T_{K}/T^{*}$ is substantially smaller than in the
Yb(Rh1-xIrx)2Si2 system so that the effects of the buildup of Kondo weight and
of the CSD overlap to such an extent that the latter is obscured, as seen in
Fig. 3h and in agreement with experimentWetli2018 .

Figure 4: Phase diagram of field-tuned YbRh2Si2 as measured by THz time-delay
spectroscopy, using the discrete magnetic-field values shown in Fig. 2. The
color code represents the resonant THz absorption, and the dashed line marks
the critical field of 66 mT.

Summarizing in Fig. 4 the measured THz absorption data in a $T$-versus-$B$
phase diagram reveals that on the HF liquid side of the QCP ($B_{\perp}>66$
mT) $P(T)$ is enhanced for $T<10$ K due to the CSD effect and as explained by
our two-band HF liquid theory. By contrast, on the antiferromagnetic side
($B_{\perp}\lesssim 50$ mT) we observe a reduction of the Kondo-weight-related
absorption but no fermionic CSD. This suggests that in this region the RKKY
interaction strongly affects the quasiparticle dynamics such that the the two-
band HF liquid theory is not valid hereRuderman1954 ; Kasuya1956 ; Yosida1957
.

To conclude, we observed a logarithmic low-temperature increase in the
resonant quasiparticle excitation probability $P(T)$ near a magnetic quantum
phase transition in heavy-fermion materials. We identified this logarithmic
increase as a unique signature of fermionic quasiparticle CSD, that is, a
vanishing quasiparticle frequency, near a quantum phase transition with
fermionic breakdown. Since, in contrast to thermodynamic and transport
properties, our time-resolved THz spectroscopy is exclusively sensitive to the
HF quasiparticle dynamics as opposed to thermal fluctuations, we could further
extract the fermionic critical exponent $\alpha$ of the vanishing
quasiparticle weight. The critical behaviour of $\alpha$ suggests to define
the heavy quasiparticle weight as an order parameter for quantum phase
transitions with fermionic breakdown. It also sets the stage for a
classification of fermionic quantum phase transitions in terms of their
critical exponent, analogous to thermodynamic phase transitions.

Methods
Experimental
Single-crystalline, $c$-oriented Yb(Rh1-xIrx)2Si2 platelets ($x=0,\,0.06$)
with dimensions of $2\times 3\times 0.07$ mm3 were grown from indium flux as
described in the literatureKrellner2012 . The sample surface is freshly
polished before the THz measurements. Samples are mounted onto a Teflon
holder, where two permanent magnets placed above and below the sample generate
a magnetic field of up to 214 mT in the easy magnetic plane perpendicular to
the tetragonal c-axis. We use a temperature-controlled Janis SVT-400 helium-
reservoir cryostat operable in the range from 1.9 to 325 K. The THz
experiments are performed in a $90^{\circ}$ reflection geometry with THz light
in a spectral range from 0.1 to 3 THz polarized perpendicular to the
crystallographic c-axis.

We generate single-cycle THz pulses by optical rectification in a 0.5-mm-thick
(110)-cut ZnTe single crystal, using 90% of an amplified Ti:Sapphire laser
output (wavelength 800 nm, pulse duration 50 fs, pulse repetition rate 1 kHz,
2.5 mJ pulse energy). The energy of the THz pulse is in the range of a few nJ.
The residual 10% of the 800-nm beam is then used for free-space electro-optic
sampling of the reflected THz light from the sample. Both, the THz and the
800-nm beams, are collinearly focused onto a 0.5-mm-thick (110)-cut ZnTe
detection crystal. To increase the accessible time delay between the THz and
the 800-nm pulses, Fabry-Pérot resonances from the faces of the detection
crystal are suppressed by a 2-mm-thick THz-inactive (100)-cut ZnTe crystal.
The THz-inactive crystal is optically bonded to the back of the detection
crystal. The THz-induced ellipticity of the 800-nm beam is measured using a
quarter-wave plate, a Wollaston prism and a balanced photodiode.

Theoretical
We construct a phenomenological, critical two-band Fermi liquid theory to
describe the THz-induced resonant transitions from the heavy conduction to the
light valence band. Electrons in the light band have the dispersion
$\varepsilon_{p}^{(0)}$ measured relative to the Fermi energy $E_{F}$, have
spectral weight unity, and are assumed to be non-interacting. The parameters
of the heavy electron states, generated by the Kondo effect, are controlled by
the Kondo scale $T_{K}$. Their energy lies close to the Fermi level $E_{F}$,
shifted by $\Delta\approx\pm k_{B}T_{K}$ above (particle-like HF systems) or
below (hole-like HF systems) $E_{F}$Hewson1993 ; Feng2017 . They have a
strongly reduced spectral weight $a(T)$ which reaches $a(0)=T_{K}/\gamma\ll 1$
at $T=0$ and decreases logarithmically for temperatures $T>T_{K}$Hewson1993 .
Here, $\gamma$ is the effective hybridization of the rare-earth $4f$ orbitals
with the conduction electron states. We also take a residual interaction of
the quasiparticles within the heavy band into account. It implies an
additional reduction of both, the quasiparticle weight and of the heavy band
shift, by the local quasiparticle weight factor $z_{0}(T)$ (see main text),
$a\to z_{0}\,a$, $\Delta\to z_{0}\Delta$. Taking now the hybridization $V$
between the light, uncorrelated band and the heavy band into account and
calculating the hybridized band structure, the band energies are obtained as

$\varepsilon_{1,2\,\vec{p}}=\frac{1}{2}\left[\varepsilon_{\vec{p}}^{(0)}+z_{0}\Delta\pm\sqrt{(\varepsilon_{\vec{p}}^{(0)}-z_{0}\Delta)^{2}+4z_{0}a|V|^{2}}\right],$
(3)

and are shown, for different temperatures, in Figs. 2d to 2f. Due to the
hybridization, the quasiparticle weights in the lower (1) and upper (2) bands
become momentum dependent and read

$z_{1,2\,\vec{p}}=\frac{(1+z_{0}a)(\varepsilon_{1,2\,\vec{p}}-z_{0}\Delta)-z_{0}a(\varepsilon_{\vec{p}}^{(0)}-z_{0}\Delta)}{2(\varepsilon_{1,2\,\vec{p}}-z_{0}\Delta)-(\varepsilon_{\vec{p}}^{(0)}-z_{0}\Delta)}\
,$ (4)

as shown in Figs. 2a to 2c. Inserting these expressions into Eq. (1) and
adjusting the two parameters $T_{\rm K}/D$ and $\alpha$, where $D$ is the free
conduction bandwidth, leads to the curves shown in Figs. 2d, h and 3g, h.
Symmetry implies that this result for the THz absorption is the same for
particle-like and hole-like HF systems.

Acknowledgement.
This work was financially supported by the Swiss National Science Foundation
(SNSF) via project No. 200021_178825 (M.F., S.P., C.J.Y.) and by the Deutsche
Forschungsgemeinschaft (DFG) via SFB/TR 185 (277625399) OSCAR and the Cluster
of Excellence ML4Q (90534769, project C4) (J.K.) as well as via SFB/TRR 288
(422213477, project A03) and grant no. KR3831/4-1 (K.K, C.K.). S.P. further
acknowledges the support by ETH Career Seed Grant SEED-17 18-1.

Author contributions.
All authors contributed to the discussion and interpretation of the experiment
and to the completion of the manuscript. S.P. and C.J.Y. performed the
experiment and the data analysis. K.K. and C.K. provided the YbRh2Si2 and
Yb(Rh0.94Ir0.06)2Si2 samples. J.K. performed the theoretical analysis. J.K.
and M.F. conceived the work, and S.P. supervised the experiments. S.P., J.K.
and M.F. drafted the manuscript.

Competing interests.
The authors declare that they have no competing financial interests.

Correspondence.
Correspondence and requests for materials should be addressed to, M.F. (email:
manfred.fiebig@mat.ethz.ch), J.K. (email<EMAIL_ADDRESS>or S.P.
(email<EMAIL_ADDRESS>
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