#  Supplementary Materials for
Observation of a Critical Charge Mode
in a Strange Metal

Hisao Kobayashi,1,2∗ Yui Sakaguchi,1 Hayato Kitagawa,1,2 Momoko Oura,1,2
Shugo Ikeda,1,2 Kentaro Kuga,3 Shintaro Suzuki,3 Satoru Nakatsuji,3,4,5,6∗
Ryo Masuda,2,7 Yasuhiro Kobayashi,2,7 Makoto Seto,2,7
Yoshitaka Yoda,8 Kenji Tamasaku,2
Yashar Komijani,9,10 Premala Chandra,10 Piers Coleman10,11∗

1Graduate School of Material Science, University of Hyogo, 3-2-1 Koto, Hyogo
678-1297, Japan
2RIKEN SPring-8 Center, Hyogo 679-5148, Japan
3Institute for Solid State Physics, University of Tokyo, Kashiwa 277-8581,
Japan
4Department of Physics, University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113-0033,
Japan
5CREST, Japan Science and Technology Agency, Kawaguchi, Saitama 332-0012,
Japan
6Institute for Quantum Matter and Department of Physics and Astronomy,
Johns Hopkins University, Baltimore, Maryland 21218, USA
7Institute for Integrated Radiation and Nuclear Science, Kyoto University,
Osaka 590-0494, Japan
8Japan Synchrotron Radiation Research Institute, Hyogo 679-5198, Japan
9Department of Physics, University of Cincinnati, Cincinnati, Ohio 45221-0011,
USA
10Department of Physics and Astronomy, Rutgers University, Piscataway, New
Jersey 08854, USA
11Hubbard Theory Consortium, Department of Physics,
Royal Holloway, University of London, Egham, Surrey TW20 0EX, UK

∗To whom correspondence should be addressed; E-mail:
<EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Contents

  1. A M$\ddot{\bf{o}}$ssbauer Spectroscopy
    1. A.1 M$\ddot{\text{o}}$ssbauer Spectrum: A Conceptual Discussion
    2. A.2 Importance in Studies of Heavy Electron Materials
    3. A.3 Synchrotron-radiation (SR)-based 174Yb M$\ddot{\text{o}}$ssbauer spectroscopy
    4. A.4 Oscillatory Effects of the Finite-Time Window
  2. B SR-based 174Yb M$\ddot{\text{o}}$ssbauer Spectroscopy of Single Crystal $\beta$-YbAlB4
    1. B.1 Experimental Specifics
    2. B.2 M$\ddot{o}$ssbauer Absorption Line-Shapes
    3. B.3  Spectral Fits and 174Yb Hyperfine Hnteractions in $\beta-$YbAlB4
    4. B.4 Statistical Analysis of the Spectra
    5. B.5 Lamb-M$\ddot{\text{o}}$ssbauer factor on 174Yb nuclei in $\beta$-YbAlB4
  3. C Reproducibility after Thermal Cycling
  4. D Recent Characterization of Samples using Bulk Probes
  5. E Excluded Possibilities for Observed Results
  6. F Planckian Dissipation in $\beta$-YbAlB4
  7. G Supporting Theoretical Discussion
    1. G.1 Kondo Breakdown
    2. G.2 Phonons and Polarons

##  Appendix A M$\ddot{\bf{o}}$ssbauer Spectroscopy

###  A.1 M$\ddot{\text{o}}$ssbauer Spectrum: A Conceptual Discussion

Mössbauer spectroscopy averages the environment of the nucleus over time-
scales of order a nanosecond,  the lifetime of the nuclear excited state. In
our experiments, quantum fluctuations in the charge of the f-electrons couple
capacitively to the 174Yb nucleus, perturbing the nuclear excitation energies.
Provided the time-scale of these charge fluctuations are slower or comparable
to a nanosecond, they modify the line-shape of a nuclear transition.

The nuclear transition $|i\rangle\longrightarrow|f\rangle$ can be treated as a
two-level system with transition energy $E_{f}-E_{i}=\hbar\omega_{0}$. The
classic theory of lineshape broadening was developed by Anderson, Kubo and
Blume (?, ?, ?, ?), before the advent of modern many-body theory. Their theory
can be reformulated in a modern context (?), treating two level nuclear system
as a single fermion with Hamiltonian $H_{0}=\hbar\omega_{0}d^{\dagger}d$. The
energy shift of the nuclear energy level is then described by adding in an
interaction $H_{I}=\alpha n_{f}d^{\dagger}d$, where $\alpha$ is the monopole-
like coupling to the occupancy $\hat{n}_{f}$ of the f-orbitals. Thus a static
valence change of the f-state produces an energy shift $\alpha$ in the
Mössbauer absorption line. With this formulation, the change in the Mössbauer
absorption line-shape (Fig. A.1) can be described in terms of the self-energy
$\Sigma_{d}(\omega)$ of the d-fermion.

Figure A.1: Charge fluctuation in the environment capacitively couple to a
two-level system (e.g. a nuclear transition) and fluctuate its resonate
frequency in time. An absorption spectroscopy on this two-level leads to a
modified lineshape $P(\omega)$ which provide information about the charge
fluctuation spectrum $\chi^{\prime\prime}(\omega)/\omega$.

The absorption cross-section (?) at a Doppler shift energy $\hbar\omega$,
$P(\omega+\omega_{0})$ is then related to the Green’s function of the
d-fermion $G_{d}(\omega)=(\omega-\Sigma_{d}(\omega))^{-1}$ via

$P(\omega+\omega_{0})\propto{\rm
Im}G_{d}(\omega-i\eta)=\frac{\Sigma^{\prime\prime}_{d}}{(\omega-\Sigma^{\prime}_{d})^{2}+\Sigma^{\prime\prime
2}_{d}}.$ (1)

To leading order in perturbation theory,

$\qquad\Sigma^{\prime\prime}_{d}(\omega)=\frac{\alpha^{2}T}{\omega}\chi^{\prime\prime}_{\rho}(\omega),$
(2)

where $\chi^{\prime\prime}_{\rho}(\omega)$ is the dynamical charge
susceptibility of the f-state at frequency $\omega$. In this way, Mössbauer
spectroscopy provides a direct probe of low-frequency valence fluctuations.

The energy shift $\alpha\sim\hbar/\tau$ sets the characteristic averaging
time-scale $\tau\sim$1ns over which Mössbauer probes the surrounding charge
fluctuations. If $\chi^{\prime\prime}(\omega)/\omega$ is described by a
Lorentzian of width $\Lambda_{vf}$, then the resulting power-spectrum develops
a two peaked structure provided the charge fluctuation rate
$\Gamma_{vf}\ll\alpha$ is smaller than $\alpha$, but develops a single peak-
structure when $\Gamma_{vf}\gg\alpha$ is much faster than the isomer shift
$\alpha$.

###  A.2 Importance in Studies of Heavy Electron Materials

In heavy electron materials like $\beta-$YbAlB4, charge fluctuations are known
to lead to valence fluctuations (?, ?). Here strong Coulomb repulsion within
the f orbitals restricts the partially filled f-shell occupancies to two
orbital configurations, which in $\beta-$YbAlB4are the 4f 13(Yb3+) and 4f
14(Yb2+). Hybridization with the conduction sea then causes valence
fluctuations, Yb${}^{2+}\rightleftharpoons$Yb${}^{3+}+e^{-}$. The simultaneous
coexistence of these hybridized configurations was first established by core-
level X-ray spectroscopy; the strength of the separate absorption spectra of
the two instantaneous configurations gives an average Yb valence of about 2.7
in $\beta-$YbAlB4 (?). A series of classic experiments in the late 1970s
searched carefully for the corresponding two valence configurations in
Mössbauer studies (?, ?), but they were never seen. Unlike static mixed
valence known in chemistry (eg in iron oxide), the quantum mechanical valence
fluctuations of a heavy fermion system are homogeneous, corresponding to
hybridization energy scales of $10-1000$K that are too fast to be observed by
Mössbauer spectroscopy; usually the observed isomer shift is a single
absorption feature, centered around the average of the two valences. This is
what makes the new observations so fascinating, for the line-splitting that
develops, without a phase transition, at low temperatures appears to
correspond to a slow-down of dynamic valence fluctuations to the point where
they become visible in the Mössbauer spectroscopy.

###  A.3 Synchrotron-radiation (SR)-based 174Yb M$\ddot{\text{o}}$ssbauer
spectroscopy

The synchrotron-radiation-(SR)-based M$\ddot{\text{o}}$ssbauer spectroscopy is
a new state-of-the-art experimental technique to investigate electronic states
of M$\ddot{\text{o}}$ssbauer atoms in compounds (?, ?). Figure A.2(A) shows
the experimental setup for the SR-based 174Yb ($E_{\gamma}=76.471$keV between
the $I_{g}=0$ and $I_{e}=2$ nuclear states) M$\ddot{\text{o}}$ssbauer
spectroscopy at BL09XU and BL19LXU beamlines on SPring-8 (?). The SR pulse was
monochromatized to approximately 5eV at $E_{\gamma}$ by using a double-crystal
Si (333) monochromator and then encountered a sample including 174Yb nuclei in
a cryostat. After passing through the sample, the monochromatized SR pulse
encountered a single-crystalline YbB12 (not enriched 174Yb) known as a
scatterer. This YbB12 scatterer was kept at 26K and moved in the triangle mode
by a velocity transducer to create a relative Doppler velocity
$\hbar\omega_{\rm D}$ between the sample and the scatterer. Note that very
strong scattering due to electrons in matter occurs promptly after the SR
pulse, but the scattering due to resonant nuclear excitation is delayed by the
finite lifetime of the excited nuclear state. Thus, we independently observe
very weak nuclear resonant scattering in the time-domain measurement. The
scattering intensity of the monochromatized SR pulse passing through the
sample in the forward direction was measured by these delayed resonant
scattering signals from 174Yb nuclei in the YbB12 scatterer after each prompt
SR pulse using a multi-element Si avalanche photodiode (APD) detector in the
spatially incoherent scattering geometry.

Figure A.2:  (A) Schematic illustration of the experimental set up for the
synchrotron-radiation-based 174Yb M$\ddot{\text{o}}$ssbauer spectroscopy. The
174Yb nuclear resonance was obtained by synchrotron radiation using a double-
crystal Si (333) monochromator. A sample and the single-crystalline YbB12 were
cooled below 20K and at 26K in each cryostat, respectively. A multielement Si
avalanche photodiode (APD) detector was used to measure delayed incoherent
emission from 174Yb nuclei in the single-crystalline YbB12 in the spatially
incoherent scattering geometry. (B) Typical delay time spectrum of the single-
crystalline YbB12 at 26K. The closed circles with error bar indicate the
observed spectrum. (C) Synchrotron-radiation-based 174Yb
M$\ddot{\text{o}}$ssbauer spectrum of the powder YbB12 sample at 20K. The
delayed scattering signals were accumulated at each Doppler velocity with the
time window from $t_{\rm s}$ to $t_{\rm e}$ as shown in (B). The closed
circles with error bar and the red solid line present the observed and the
analytical spectra, respectively.

The delayed scattering signal $S(\hbar\omega_{\rm D},\tau)$ after each SR
prompt pulse in the nuclear recoil-free absorption process is given by

$S(\hbar\omega_{\rm D},\tau)\propto\int_{0}^{z^{\rm sc}}dz\exp(-\mu_{e}^{\rm
sc}z)\left|\int
d\omega\frac{\Gamma_{0}/2\exp(-i\hbar\omega\tau/\Gamma_{0})}{\hbar(\omega-\omega_{\rm
D})-i\Gamma_{0}/2}A^{\rm sa}(\hbar\omega)A^{\rm
sc}(\hbar\omega,\hbar\omega_{\rm D},z)\right|^{2},$ (3)

where

$A^{\rm sa}(\hbar\omega)=A_{0}^{\rm sa}\exp\left(-i\frac{\mu_{n}^{\rm
sa}z^{\rm sa}\Gamma_{0}/2}{\hbar\omega-i\Gamma_{0}/2}\right),$ $A^{\rm
sc}(\hbar\omega,\hbar\omega_{\rm D},z)=A_{0}^{\rm
sc}\exp\left(-i\frac{\mu_{n}^{\rm
sc}z\Gamma_{0}/2}{\hbar(\omega-\omega_{D})-i\Gamma_{0}/2}\right),$

$\tau=t/\tau_{0}$ is a dimensionless time, $\tau_{0}=\hbar/\Gamma_{0}$ the
lifetime of the excited nuclear state, $A^{\rm sa}(\hbar\omega)$ $(A^{\rm
sc}(\hbar\omega,\hbar\omega_{\rm D},z))$ represents the radiation amplitude in
depth $z^{\rm sa}$ $(z)$ of the sample (scatterer), $z^{\rm sa}$ $(z^{\rm
sc})$ and $\mu_{n}^{\rm sa}$ $(\mu_{n}^{\rm sc})$ denote the thickness and the
nuclear resonant absorption coefficient of the sample (scatterer),
respectively, and $\mu_{e}^{\rm sc}$ represents the electronic absorption
coefficient of the scatterer (?). The delayed scattering signals
$S(\hbar\omega_{\rm D},\tau)$ were accumulated at each $\hbar\omega_{\rm D}$
within the time window from $\tau_{\rm s}=t_{\rm s}/\tau_{0}$ to $\tau_{\rm
e}=t_{\rm e}/\tau_{0}$ to measure a SR-based M$\ddot{\text{o}}$ssbauer
absorption spectrum in the energy-domain measurements. Figure A.2(B) shows a
delay time spectrum $S(\tau)(\propto\int S(\hbar\omega_{\rm
D},\tau)d\omega_{\rm D})$ from 174Yb nuclei in the YbB12 scatterer measured in
our experiment. Since the lifetime of the $I_{e}=2$ excited state of the 174Yb
nucleus is $\tau_{0}=2.58$ns, as seen in Fig. A.2(B), a typical time window in
our experiments was in the range from $\tau_{\rm s}\sim 3.1$ ($t_{\rm s}\sim
8$ns) to $\tau_{\rm e}\sim 6.2$ ($t_{\rm e}\sim 16$ns) to measure SR-based
174Yb M$\ddot{\text{o}}$ssbauer spectra.

###  A.4 Oscillatory Effects of the Finite-Time Window

The intensity of $S(\hbar\omega_{\rm D},\tau)$ around $|\hbar\omega_{\rm
D}|\sim 0$ for the YbB12 sample oscillates as a function of $\tau$ about the
exponential curve as shown in Ref. (?) for 57Fe M$\ddot{\text{o}}$ssbauer
resonance where in general, $|\hbar\omega_{\rm D}|\sim 0$ means energy near
the M$\ddot{\text{o}}$ssbauer transition of 174Yb nuclei in a sample. These
oscillations in $S(\hbar\omega_{\rm D},\tau)$ are dynamical effects of
scattering processes in the sample and the scatterer. Since the frequencies of
the oscillation components vary as $\hbar\omega_{\rm D}$, these oscillation
behaviors in $S(\hbar\omega_{\rm D},\tau)$ are smeared out in $S(\tau)$ as
seen in Fig. A.2(B). Experimentally, the intensities at $\hbar\omega_{\rm D}$
in a spectrum, $S(\hbar\omega_{\rm D})(=\int_{\tau_{\rm s}}^{\tau_{\rm
e}}S(\hbar\omega_{\rm D},\tau)d\tau)$, depend not only on $\hbar\omega_{\rm
D}$ but also on the time window in the accumulation. Thus, a shape of an
absorption component in a spectrum is affected by the time window in the
accumulation and a Lorentz-function line shape with the expected width should
be obtained in a spectrum using a thin-sample with the time window from
$\tau_{s}\sim 0$ to $\tau_{e}\sim\infty$. Figure A.2(C) shows the SR-based
174Yb M$\ddot{\text{o}}$ssbauer spectrum of the powder YbB12 (not enriched
174Yb) sample at 20K. As seen in this figure, the oscillatory components were
observed in background and the full-width at half-maximum of the main
absorption component at 0 mm/s was evaluated to be 1.2 mm/s which is much
narrower than that ($2\Gamma_{0}=2.00$ mm/s) expected from $\tau_{0}$ of the
$I_{e}=2$ excited state of 174Yb nucleus. These phenomena were observed in the
57Fe M$\ddot{\text{o}}$ssbauer spectra measured by the delayed coincidence
method (?) and the SR-based 151Eu M$\ddot{\text{o}}$ssbauer spectroscopy (?),
indicating that these are related to the time-window effect in the
accumulation of $S(\hbar\omega_{\rm D},\tau)$. Thus, this time-window effect
in the SR-based M$\ddot{\text{o}}$ssbauer spectroscopy is important for
M$\ddot{\text{o}}$ssbauer resonances with short $\tau_{0}$s. The observed SR-
based 174Yb M$\ddot{\text{o}}$ssbauer spectrum of YbB12 is well analyzed using
one nuclear transition with this time-window effect and the effective
thicknesses of the YbB12 sample and the scatterer. Since $\tau_{\rm s}\sim
3.1$ is much larger compared to $\tau_{\rm s}\sim 0$, then the higher energy
resolution than the conventional 174Yb M$\ddot{\text{o}}$ssbauer spectroscopy
was achieved in our experiments. The enhanced energy resolution may be
essential for extracting time-dependent hyperfine interaction parameters in
$\beta$-YbAlB4.

##  Appendix B SR-based 174Yb M$\ddot{\text{o}}$ssbauer Spectroscopy of Single
Crystal $\beta$-YbAlB4

###  B.1 Experimental Specifics

High-purity single crystals of $\beta$-YbAlB4 were grown by the Al-flux method
(?). Energy-dispersive X-ray analysis indicated no impurity phases, no
inhomogeneities and a ratio Yb:Al of 1:1 within the detection limits of the
equipment. The SR-based 174Yb M$\ddot{\text{o}}$ssbauer experiments were
carried out at the BL09XU and BL19LXU beamlines on SPring-8 (see Fig. 1 A of
the paper) (?), using SR to excite the $E_{\gamma}=$ 76.471keV
M$\ddot{\text{o}}$ssbauer transition between the $I_{g}=0$ ground and
$I_{e}=2$ excited states of the 174Yb nuclei contained in the compound. The
energy level diagram of the excited 174Yb nuclear state is sensitive to the
surrounding charge configuration and provides real-time information about the
Yb valence (see Fig. 1 C of the paper). Measurements were carried out under
both ambient and external pressure, and the $c$-axis of the single crystalline
$\beta$-YbAlB4 samples was aligned along the propagation vector
$\mbox{\boldmath$k$}_{0}$ of the incident X-ray. Under external pressure, the
single crystalline samples were loaded into a sample cavity of the Inconel
alloy gasket in a clamp-type diamond-anvil cell (DAC) with ruby crystals and
mixtures of methanol-ethanol or Daphne7474 as a pressure-transmitting medium
to ensure hydrostatic conditions at 2K. Pressure was calibrated by measuring
the wavelength shift of the $R_{1}$ luminescence line of the ruby crystals in
the clamp-type DAC. Since the sizes of the single crystalline $\beta$-YbAlB4
samples in the sample cavities were comparable with that of the incident X-ray
beam, the collimators were used to obtain 174Yb M$\ddot{\text{o}}$ssbauer
spectra of satisfactory quality. The high intensity of the incident X-ray made
it possible to carry out these experiments without isotopic purification.

Our measurements were performed using cryostat with liquid Helium bath (SM4000
Oxford Instrument). The vibration of the sample/DAC can be ignored within out
velocity scale for 174Yb Mössbauer spectroscopy. The pressure medium for the
diamond-anvil compression was a methanol-ethanol mixture that works best up to
about 10 GPa. In liquid media and in spectroscopic and diffraction
measurements, this is the preferred medium for diamond-anvil compression. In
resistivity measurements, the wire contacts are connected to the sample using
conductive paste, which unfortunately reacts with alcohol, so methanol-ethanol
mixtures are not used in high-pressure resistivity measurements. Thus there
can be sometimes be apparent discrepances between pressure-dependent transport
and spectroscopy measurements that arise due to the different pressure-
transmitting media involved.

###  B.2 M$\ddot{o}$ssbauer Absorption Line-Shapes

In the 174Yb M$\ddot{\text{o}}$ssbauer effect, the resonance is an electrical
quadrupole $E2$ transition from the $I_{g}=0$ ground to $I_{e}=2$ excited
nuclear states. The five nuclear transitions with $\Delta
I^{z}(=I_{g}^{z}-I_{e}^{z})=0$, $\pm 1$, and $\pm 2$ are allowed between the
ground and excited nuclear states. In no electrical quadrupole and magnetic
hyperfine interaction, these five allowed transitions have the same energy and
then we observe an absorption component in a M$\ddot{\text{o}}$ssbauer
spectrum, as seen in Fig. A.2(C). The pure electrical quadrupole interaction
splits the $I_{e}=2$ excited nuclear state into three sublevels with
$I_{e}^{z}=0$, $\pm 1$, and $\pm 2$ within an axial symmetry by

$\mathcal{H}_{Q}=\frac{1}{4}\frac{eQV_{zz}}{I_{e}(2I_{e}-1)}\left[3(I_{e}^{z})^{2}-I_{e}(I_{e}+1)\right],$
(4)

where the second derivative of the potential $V_{zz}=\partial^{2}V/\partial
z^{2}$ is the $z$ component of the diagonalized electric-field gradient (EFG)
tensor and $Q$ is the nuclear quadrupole moment of the $I_{e}=2$ excited
nuclear state of the 174Yb nucleus. Thus, three allowed nuclear transitions
with $\Delta I^{z}=0$, $\pm 1$, and $\pm 2$ have different energies. In the
nuclear resonant scattering using SR, the polarized SR provides the
possibility of selective excitations in allowed nuclear transitions. Due to
the perfect $\sigma$-polarization of the incident X-ray in our experiments,
the excitation probabilities in these 174Yb nuclear transitions are very
sensitive to the direction of the $z$-axis and the symmetry of the
diagonalized EFG tensor at the Yb site (?).

For $\beta$-YbAlB4 with the orthorhombic $Cmmm$ structure (?), the principal
axes of the diagonalized EFG tensor at one crystallographic Yb site are along
the three crystal axes because the local symmetry at the Yb site is $m2m$.
Since the Yb ions are sandwiched between boron layers and are centered between
seven-member boron rings, the principal $z$-axis of the diagonalized EFG
tensor at the Yb site is parallel to the $c$-axis of $\beta$-YbAlB4, which is
the quantization axis $\mbox{\boldmath$z$}_{J}$ of the hyperfine interactions
at the 174Yb nuclear in $\beta$-YbAlB4. In our experimental conditions where
$\mbox{\boldmath$z$}_{J}$ at the Yb site in $\beta$-YbAlB4 was parallel to
$\mbox{\boldmath$k$}_{0}$, the $I_{g}=0\rightarrow I_{e}^{z}=\pm 1$ nuclear
transitions are only selected in the five $E2$ nuclear transitions of the
174Yb M$\ddot{\text{o}}$ssbauer resonance (?), which have the same energy
within an assumption of the axial symmetric EFG tensor.

The possibilities of selective excitations in allowed nuclear transitions and
the higher energy resolution are crucial to discuss dynamics of valence
fluctuations and the electronic states of the Yb ions in $\beta$-YbAlB4 via
the SR-based 174Yb M$\ddot{\text{o}}$ssbauer spectroscopy.

As seen in Figs. 2 A and B of the paper, the broadenings of two nuclear
transitions are observed in the SR-based 174Yb M$\ddot{\text{o}}$ssbauer
spectra for $\beta$-YbAlB4, which is attributed to relaxation between two
nuclear transitions. In the M$\ddot{\text{o}}$ssbauer resonances, a relaxation
occurs in the same $\Delta I^{z}$ nuclear transitions with different energies
caused by time-dependent hyperfine interactions. The M$\ddot{\text{o}}$ssbauer
absorption line shapes in the presence of the time-dependent hyperfine
interactions have been discussed in the stochastic ways (?, ?, ?, ?). In the
case of slow relaxation, $1/\tau_{f}<2\varepsilon$, where $2\varepsilon$
represents a energy difference between these two nuclear transitions without
the relaxation effects and $\tau_{f}$ denotes the characteristic timescale in
the time-dependent hyperfine interactions, the M$\ddot{\text{o}}$ssbauer
absorption line shape for two nuclear transitions is expressed in the
stochastic model as follow:

$I(\hbar\omega_{\rm
D})\propto\left((1-R)\frac{\Gamma_{0}/2+W+(1/x)(\hbar\omega_{\rm
D}+Wx)}{(\hbar\omega_{\rm
D}+Wx)^{2}+(\Gamma_{0}/2+W)^{2}}+(1+R)\frac{\Gamma_{0}/2+W-(1/x)(\hbar\omega_{\rm
D}-Wx)}{(\hbar\omega_{\rm D}-Wx)^{2}+(\Gamma_{0}/2+W)^{2}}\right),$ (5)

where $2W=1/\tau_{f}$, $x=[(\varepsilon/W)^{2}-1]^{1/2}$, and $R$ represents
small difference in intensity between two nuclear transitions.

In order to combine Eq. 5 with Eq. 3 both amplitude and phase information of
$A^{s}a(\hbar\omega)$ are required. The amplitude
$I(\hbar\omega)\sim|A^{sa}(\hbar\omega)|^{2}$ is given by Eq. 5. To find the
phase, we have fitted Eq. 5 with two Lorentzians which can be easily continued
to the complex plane.

Thus the relaxation brings about a broadening and distortions as well as a
shift of the absorption lines. For $1/\tau_{f}(=2W)\ll 2\varepsilon$,
$I(\hbar\omega_{\rm D})$ is the same as two Lorentz-functions centered at
$\hbar\omega_{\rm D}=\pm Wx\sim\pm\varepsilon$, which is expected in the
absence of any fluctuations in the hyperfine interactions. In the case of fast
relaxation, $1/\tau_{f}>2\varepsilon$, the M$\ddot{\text{o}}$ssbauer
absorption line shape is given by

$I(\hbar\omega_{\rm
D})\propto\left(\frac{(1+1/y)(\Gamma_{0}/2+W(1-y))}{(\hbar\omega_{\rm
D})^{2}+(\Gamma_{0}/2+W(1-y))^{2}}+\frac{(1-1/y)(\Gamma_{0}/2+W(1+y))}{(\hbar\omega_{\rm
D})^{2}+(\Gamma_{0}/2+W(1+y))^{2}}\right),$ (6)

where $y=[1-(\varepsilon/W)^{2}]^{1/2}$. For $1/\tau_{f}(=2W)\gg
2\varepsilon$, $I(\hbar\omega_{\rm D})$ represents a Lorentz-function at
$\hbar\omega_{\rm D}=0$ (the first term in eq. (4)). The fluctuations are so
fast that the nucleus only feel an average hyperfine interactions. The
collapse of a two-peaked structure related to two nuclear transitions occurs
in a M$\ddot{\text{o}}$ssbauer spectrum as $2\varepsilon\cdot\tau_{f}$ is
comparable with and then becomes smaller than unity. Accordingly, the two-
peaked structure observed in the M$\ddot{\text{o}}$ssbauer spectrum manifests
slow relaxations in time-dependent hyperfine interactions.

###  B.3  Spectral Fits and 174Yb Hyperfine Hnteractions in $\beta-$YbAlB4

####  B.3.1 Ambient pressure spectra

The fitting procedures of the SR-based 174Yb M$\ddot{\text{o}}$ssbauer spectra
for $\beta$-YbAlB4 were performed using the equations in Section A.3 with two
selected nuclear transitions, including the effective thicknesses of the
single crystalline $\beta$-YbAlB4 samples and the YbB12 scatterer and the
time-window effect in the time-domain measurements in each experiment (see
Section B.1). Figure B.3(A) shows the refined energies $\varepsilon$ for two
$I_{g}=0\rightarrow I_{e}^{z}=\pm 1$ nuclear transitions in $\beta$-YbAlB4 at
ambient pressure. As shown in this figure, the energy difference between these
nuclear transitions is almost independent of temperature up to 20K.

M$\ddot{\text{o}}$ssbauer spectroscopy has been used to discuss electronic
states of atoms included M$\ddot{\text{o}}$ssbauer isotopes via the nuclear
hyperfine interactions. Although there are three main electrical and magnetic
hyperfine interactions to determine the energies and the eigenstates of the
ground and the excited nuclear states in a M$\ddot{\text{o}}$ssbauer isotope,
the electrical monopole and electrical quadrupole interactions are important
to discuss the electronic state of the atoms in a paramagnetic state. In the
experiments using the single crystalline $\beta$-YbAlB4 samples, the
degenerate $I_{g}=0\rightarrow I_{e}^{z}=\pm 1$ nuclear transitions are only
selected in the five $E2$ nuclear transitions. In a paramagnetic state with no
external magnetic field, the energy of these degenerate nuclear transitions
depends on both the electrical monopole and quadrupole interactions. We do not
evaluate the electrical monopole and quadrupole interactions for two selected
nuclear transitions in $\beta$-YbAlB4 only from the spectrum observed in the
experimental condition with
$\mbox{\boldmath$z$}_{J}\parallel\mbox{\boldmath$k$}_{0}$. Thus, we have
measured the SR-based 174Yb M$\ddot{\text{o}}$ssbauer spectrum at 5K in
another experimental condition to evaluate the electrical monopole and
quadrupole interactions for these two nuclear transitions. In the experiment,
the $c$-axis of the single crystalline $\beta$-YbAlB4 samples was tilted
though 10 deg from the incident X-ray in the horizontal plane, that is, the
angle $\theta$ between $\mbox{\boldmath$z$}_{J}$ and $\mbox{\boldmath$k$}_{0}$
was 10 deg.

As shown in Fig. G.8, the characteristic features in the SR-based 174Yb
M$\ddot{\text{o}}$ssbauer spectrum observed at 5K with $\theta$ = 10 deg are
different from those in the spectrum at 5K with $\theta$ = 0 in Fig. 2 A of
the paper, due to the difference of the excitation probabilities in five
allowed nuclear transitions between these experimental conditions. The
excitation probabilities in these nuclear transitions were evaluated with
$\theta$ = 10 deg. We have analyzed the observed SR-based 174Yb
M$\ddot{\text{o}}$ssbauer spectrum at 5K using the evaluated excitation
probabilities for the five $E2$ nuclear transitions. In this analysis,
furthermore, the $\tau_{f}$ value and the energies for the degenerate
$I_{g}=0\rightarrow I_{e}^{z}=\pm 1$ nuclear transitions were fixed on those
refined from the spectrum observed at 5K with $\theta$ = 0 within the
experimental accuracies. The energy $\varepsilon$ of the $I_{g}=0\rightarrow
I_{e}^{z}=\pm 1$ nuclear transition is given by $\varepsilon=\delta-E_{Q}/2$
using the isomer shift $\delta$ and the electrical quadrupole splitting
$E_{Q}$. As seen in Fig. G.8, the analytical spectrum agrees well with the
observed one. The $\delta$ and $E_{Q}$ values were evaluated to be
$\delta=0.22$ mm/s and $E_{Q}=-2.3$ mm/s and $\delta=-0.76$ mm/s and
$E_{Q}\sim 0$ mm/s for two selected nuclear transitions, respectively, by
using the stochastic model.

As shown in Fig. B.3(B), the intensity ratio $R$ between these two selected
nuclear transitions is independent of temperature up to 20K within our
experimental accuracy. The selected nuclear transitions with $\delta=0.27$
mm/s and $E_{Q}=-2.3$ mm/s and $\delta=-0.76$ mm/s and $E_{Q}\sim 0$ mm/s
correspond to the Yb3+ and Yb2+ ionic states, respectively. Since all Yb sites
are crystallographically equivalent in $\beta$-YbAlB4, we assume the same
recoil-free fraction (Lamb-M$\ddot{\text{o}}$ssbauer factor) of two selected
nuclear transitions. The averaged Yb valence was evaluated to be 2.6(1) from
the refined $R$ values, which is in good agreement with those (Yb+2.75)
obtainede by hard X-ray photoelectron and X-ray absorption near edge structure
spectroscopies (?, ?).

Figure B.3:  (A) Refined energies $\varepsilon$ for two selected
$I_{g}=0\rightarrow I_{e}^{z}=\pm 1$ nuclear transitions and (B) difference in
intensity $R$ between these two nuclear transitions in $\beta$-YbAlB4 as
functions of temperature. The red broken lines are visual guides.

Since the nuclear transition for the Yb2+ state shows a single line feature in
the observed spectra, the contribution of the distribution of the ionic and
electronic charges in the lattice to $E_{Q}$ is negligibly small at the Yb
site in $\beta$-YbAlB4 within our experimental energy resolution. The $E_{Q}$
value extracted for the Yb3+ state thus provides information about the
crystalline-electric-field (CEF) ground state of the Yb3+ $4f$ electrons in
$\beta$-YbAlB4. Since the degenerate $I_{g}=0\rightarrow I_{e}^{z}=\pm 1$
nuclear transitions are selected in the experimental condition with
$\mbox{\boldmath$z$}_{J}\parallel\mbox{\boldmath$k$}_{0}$, the assumption of
the axial symmetric EFG is reasonable within our experimental energy
resolution. The $E_{Q}^{4f}$ due to the interaction between the nuclear
quadrupole moment $Q$ and the quadrupole moment of the CEF ground state of
rare-earth $4f$ electrons is given by

$E_{Q}^{4f}=-\frac{1}{4}e^{2}Q(1-R_{Q})\langle J||\alpha||J\rangle\langle
r^{-3}\rangle_{4f}\langle 3J_{z}^{2}-J(J+1)\rangle,$ (7)

where $\langle r^{-3}\rangle_{4f}$ is the averaged value of $r^{-3}$ for the
$4f$ electrons in a given $J$ state for a rare-earth ion, $\langle
J||\alpha||J\rangle$ represents a reduced matrix element defined by the
operator equivalent and $R_{Q}$ is the shielding factor (?). The states of the
Yb3+ $4f$ electrons are split to four Kramers doublets by the CEF interaction.
In the case where $J_{z}$ is conserved, the expected $E_{Q}^{4f}$ values are
evaluated to be $-14.7$, $-2.1$, $6.3$, and $10.5$ mm/s for the four Kramers
doublets with $J_{z}=\pm 7/2$, $\pm 5/2$, $\pm 3/2$, and $\pm 1/2$,
respectively, using $\langle r^{-3}\rangle_{4f}=13.83$ a.u., and
$R_{Q}=0.21(1)$ with $\langle J||\alpha||J\rangle=2/63$ for ${}^{2}F_{7/2}$
and $Q=2.12$ b (?, ?). As the result of the extracted $E_{Q}$ value for the
Yb3+ state, the ground Kramers doublet for the Yb3+ $4f$ electrons is almost
pure $J_{z}=\pm 5/2$ states in $\beta$-YbAlB4 with a negligible admixture of
the $J_{z}=\pm 1/2$ state.

####  B.3.2 Finite pressure, low temperature spectra.

As seen in Fig. 2 B of the paper, the similar two-peaked structure was
observed in the SR-based 174Yb M$\ddot{\text{o}}$ssbauer spectra at 0.25 and
0.7GPa at 2K. It was confirmed by the SR X-ray diffraction measurements of
$\beta$-YbAlB4 under pressure (?) that the structural symmetry and the
individual atomic coordination parameters do not change up to 3.5GPa at 7K.
Thus, the characteristic change in the features of the observed spectra around
$\sim 1$GPa is not related to a pressure-induced structural transition in
$\beta$-YbAlB4 at 2K. The observed spectra below 1.2GPa at 2K were analyzed by
using the same stochastic model like that at ambient pressure (see Section A.3
and B.2). As seen in Figs. 2 A and B of the paper and Fig. A.2(C), the
absorption component observed in the SR-based 174Yb M$\ddot{\text{o}}$ssbauer
spectrum at 2.3GPa and 2K is much narrower than that in the spectrum at 20 K
and ambient pressure and comparable with that in the spectrum of YbB12. The
observed spectrum at 2.3GPa and 2K was analyzed using a single absorption
component (see Section B.2). Figure G.9(A) shows the refined energies
$\varepsilon$ for two $I_{g}=0\rightarrow I_{e}^{z}=\pm 1$ nuclear transitions
up to 1.2GPa at 2K and the refined absorption-peak position at 2.3GPa and 2K
in $\beta$-YbAlB4. As shown in Fig. G.9(B), the intensity ratio $R$ between
these two selected nuclear transitions is independent of pressure up to 1.2GPa
at 2K within our experimental accuracy. The absorption-peak position at 2.3GPa
and 2K corresponds to the weighted average of two absorption-peak positions
below 1.2GPa at 2K, revealing that the averaged electronic state of the Yb
ions in $\beta$-YbAlB4 is not changed by pressure up to $\sim$2.3GPa at 2K.

Since the 174Yb M$\ddot{\text{o}}$ssbauer spectrum at 2.3GPa and 2K was well
reconstructed by a Lorentz-function, $1/\tau_{f}\gg 2\varepsilon$ in the
stochastic relaxation model. The width of the absorption component of
$\beta$-YbAlB4 sample was refined to be $\Gamma=1.11(1)$mm/s at 2.3GPa and 2K,
which is broader than $\Gamma_{0}=$1.00mm/s expected from $\tau_{0}$ of the
$I_{e}=2$ excited state of the 174Yb nucleus. If we assume that this
broadening comes from the relaxation between two nuclear transitions, $\Gamma$
is given by

$\Gamma=\Gamma_{0}+2W(1-y)\sim\Gamma_{0}+\frac{\varepsilon^{2}}{2}\tau_{f}$

from the first term of eq. (4). The energy difference $2\varepsilon$ between
two $I_{g}=0\rightarrow I_{e}^{z}=\pm 1$ nuclear transitions was estimated to
be $2\varepsilon\sim 2.2(6)$ mm/s from the refined energies $\varepsilon$ for
two $I_{g}=0\rightarrow I_{e}^{z}=\pm 1$ nuclear transitions for the Yb3+ and
Yb2+ ionic states at ambient pressure and below 1.2GPa at 2K. We have
evaluated $\tau_{f}\sim 0.04\tau_{0}~{}(\sim 0.1$ns) at 2.3GPa and 2K. In
M$\ddot{\text{o}}$ssbauer absorption experiments, the broadening from
$\Gamma_{0}$ in the width of the Lorentz-function could occur due to the
effective thickness of a M$\ddot{\text{o}}$ssbauer isotope in a sample. Thus,
the evaluated $\tau_{f}$ value is maximum one for $\beta$-YbAlB4 at 2.3GPa and
2K.

###  B.4 Statistical Analysis of the Spectra

The entire focus of our paper involves the resonant Mössbauer absorption
spectra that lie at low Doppler velocity below $\pm$2mm/s. In this region, the
X-ray scattering off the nucleii is unitary and the only source of error is
the photon count. By contrast, the off-resonance Mössbauer signals at large
Doppler shift are dominated by multiple elastic scattering, leading to
interference effects that give rise to systematic errors in the data that are
absent at low Doppler shift. This leads to modulations in background signal as
a function of Doppler velocity that are related to the finite sample
($\beta$-YbAlB4) and absorber (YbB12) thicknesses, and the finite time window
for photon counts (?, ?, ?, ?).

Now we turn to the analysis of the resonant Mössbauer absorption. There are
two aspects of the data that we wish to highlight that clearly demonstrate a
signal that rises well above the noise level. There are two i

####  B.4.1 Broadening of the Lineshape at low temperature

A plot of FWHM of the spectral dip as a function of temperature and pressure
(Fig. G.10) clearly shows that the width of absorption line broadens at low
temperatures and pressures. This is in marked contrast with the fact that the
spectral peaks usually broaden at high temperatures. Furthermore, the
evolution of the FWHM vs. pressure correlates with transition between a
strange metal to a Fermi liquid in previous transport experiments.

####  B.4.2 Statistical significance of double-peak structure

We have done a thorough statistical analysis of the data. The quality of the
fit is dependent on the regression and total sum of squares, given by

$SSR=\frac{1}{N}\sum_{i=1}^{N}(y_{i}-f_{i})^{2},\qquad
SST=\frac{1}{N}\sum_{i=1}^{N}(y_{i}-\bar{y})^{2}$ (8)

where $y_{i}$ is the data, $f_{i}$ is the fit to data,
$\bar{y}=\frac{1}{N}\sum_{i}y_{i}$ is the average of the data set and $N$ is
the number of data points. These variables are used to compute the standard
error via $\sigma_{tot}=\sqrt{SSR}$. However, it is more systematic to define
two new parameters $R^{2}$ and adjusted $\bar{R}^{2}$ parameters:

$R^{2}=1-\frac{SSR}{SST},\qquad\bar{R}^{2}=1-\frac{N-1}{N-P-1}(1-R^{2})$ (9)

where the latter compensates for the number of fitting parameters, $P$.

Fig. G.11(A) shows the attempt to fit the experimental lineshape using a
single Gaussian or Lorentzian peak. Such single-peak fits do not capture the
data in either the central region or the tails, as quantified by the low
values of $R^{2}$ and $\bar{R}^{2}$. If, following Referee 3. we disregard the
key central region as noise, we can fit the tails using a Gaussian or
Lorentzian lineshape, providing a noise background against which we can assess
the probability that the splitting is a statistical anomaly. This fit is shown
in blue in Fig. G.11(B).

The blue baseline, based on the assertion that there is no splitting allows us
to now assess whether the observed deviation from a single peak is
statistically significant. The error bars in the measurement have been
reliably established from the standard deviation in photon counts. Taking the
pessimistic assumption that each deviating point is the result of Gaussian
noise about the single peak background, we then have two points with a
standard deviation of 2$\sigma$ a central point with a standard deviation of
3$\sigma$. The probability of this deviation is then given by

${\cal P}={\cal E}(2\sigma)^{2}{\cal E}(3\sigma)\equiv{\cal E}(4.84\sigma)$
(10)

where

${\cal
E}(x)=\frac{1}{\sqrt{2\pi\sigma}}\int_{x}^{\infty}e^{-\frac{x^{2}}{2\sigma^{2}}}.$
(11)

In other words, this data represents a 4.84$\sigma$ event, an event that would
occur less than one in a million experiments. This analysis, combined with the
other data that shows a systematic broadening at low temperatures and low
pressures establishes that our results meet the gold standard (5$\sigma$) for
statistically significant experimental observations.

By contrast, the double-peak fit to the data, a fit grounded in the Anderson-
Kubo formula for dynamical fluctuation, is significantly more successful, as
shown in Fig. G.11(C). It is these fits that were used to extract the valence
fluctuation rate $\tau_{f}^{-1}$. Comparing the $R^{2}$ and $\bar{R}^{2}$ of
these fits, we conclude that first the analytical Kubo-Anderson fit and next a
Gaussian fit are the best fits to the data.

###  B.5 Lamb-M$\ddot{\text{o}}$ssbauer factor on 174Yb nuclei in
$\beta$-YbAlB4

The Lamb-M$\ddot{\text{o}}$ssbauer factor $f_{\rm LM}$ describing the recoil-
free fraction of the nuclear absorption, is related to atomic motions for
M$\ddot{\text{o}}$ssbauer isotopes in a matter. The $f_{\rm LM}$ is generally
described by

$f_{\rm
LM}=\langle\exp[i(\mbox{\boldmath$k$}_{0}-\mbox{\boldmath$k$}_{f})\cdot\mbox{\boldmath$r$}]\rangle=\exp[-\langle[\mbox{\boldmath$k$}_{0}\cdot\mbox{\boldmath$r$}]^{2}\rangle]=\exp[-k_{0}^{2}\langle\Delta
z^{2}\rangle],$ (12)

where $\Delta z$ is an atomic displacement from a regular position in a matter
along the direction of $\mbox{\boldmath$k$}_{0}$. In scattering experiments,
this is the effects of the zero point and temperature motions of the atom
bound in a matter. The temperature dependence of $f_{\rm LM}$ is

$f_{\rm LM}(T)=\exp\left(-E_{\rm
R}\int_{0}^{\infty}\frac{F(E)}{E}\coth\frac{\beta E}{2}dE\right)$ (13)

using the local (partial) phonon density of states $F(E)$ for a
M$\ddot{\text{o}}$ssbauer atom, where $\beta=\frac{1}{k_{\rm B}T}$, $k_{\rm
B}$ is Boltzmann constant and $E_{\rm R}(=\frac{E_{\gamma}^{2}}{2mc^{2}})$
recoil energy. In the Debye approximation for $\nu(E)$, $f_{\rm LM}(T)$
becomes

$f_{\rm LM}(T)=\exp\left(-\frac{6E_{\rm R}}{k_{\rm B}\Theta_{\rm
D}}\left[\frac{1}{4}+\left(\frac{T}{\Theta_{\rm
D}}\right)^{2}\int_{0}^{\Theta_{\rm D}/T}\frac{x}{e^{x}-1}dx\right]\right).$
(14)

In the case of $T\ll\Theta_{\rm D}$, $f_{\rm LM}(T)$ has a formula of

$f_{\rm LM}(T)=\exp\left(-\frac{E_{\rm R}}{k_{\rm B}\Theta_{\rm
D}}\left[\frac{3}{2}+\left(\frac{\pi}{\Theta_{\rm
D}}\right)^{2}T^{2}\right]\right).$ (15)

In SR-based M$\ddot{\text{o}}$ssbauer spectroscopy, the relative intensity $I$
integrated in the energy region of the spectrum with absorption components for
a sample is proportional to the number and $f_{\rm LM}$ of
M$\ddot{\text{o}}$ssbauer resonant nuclei in the sample. The same single
crystalline $\beta$-YbAlB4 sample was used in our SR-based 174Yb
M$\ddot{\text{o}}$ssbauer experiments at ambient pressure. The size of this
sample was much larger than that of the incident X-ray beam, indicating that
the number of 174Yb nuclei in the beam was same in our experiments. We have
evaluated $I$ proportional to $f_{\rm LM}(T)$ from the observed SR-based 174Yb
M$\ddot{\text{o}}$ssbauer spectra at ambient pressure with correction of
scattering conditions including the time-window effects.

In our experiments under pressure at 2K, as seen in Fig. 2 B of the paper,
174Yb M$\ddot{\text{o}}$ssbauer spectra of satisfactory quality were obtained
by using the collimators to absorb incident X-ray photons without passing
though the single crystalline $\beta$-YbAlB4 sample in the clamp-type DAC,
which were made by Re and Pb metals with 250 and 800$\mu$m thickness,
respectively. Although the thicknesses of these samples were same within our
experimental accuracies, the transmission intensities of the incident X-ray
photons with $E_{\gamma}$ were slightly different though these collimators. We
have corrected these absorption effects to evaluate $I$ proportional to
$f_{\rm LM}(p)$ from the observed SR-based 174Yb M$\ddot{\text{o}}$ssbauer
spectra.

##  Appendix C Reproducibility after Thermal Cycling

Figure C.4: Mössbauer spectra of the same $\beta$-YbAlB4 sample studied in the
paper after a thermal cycling at 2K under ambient pressure to confirm the
reproducibility of the splitting. For comparison, the analytical curve in red
is the same fit as that used in the 2K ambient pressure data in the paper (Fig
2A in paper).

Fig. C.4 shows the result of Mössbauer spectroscopy of the the same
$\beta$-YbAlB4 sample studied in the paper in a different thermal cycle at 2K
and under ambient pressure. Although the details have changed the
characteristic double-peak feature persists.

##  Appendix D Recent Characterization of Samples using Bulk Probes

In this section, we briefly present new thermodynamical data taken from
recently grown $\beta-$YbAlB4 crystals, similar to the ones used for the
Mössbauer spectroscopy. As seen in Fig. D.5, the absence of any thermodynamic
transition in $M/H$ and $C/T$ support the interpretation of the Mössbauer data
as discussed in the next section. Fig. D.6 indicates the sharpness of
superconducting transition for the same samples.

Figure D.5: (A) Magnetic susceptibility vs. temperature for field parallel to
ab-plane and c-axis. (B) Heat capacity of $\beta-$YbAlB4 and $\alpha$-LuAlB4.
The latter is subtracted from the former to get the electronic contribution to
the $C/T$ shown in red. Figure D.6: Resistivity of $\beta$-YbAlB4 as a
function of temperature for two different residual resistivity ratios (RRRs).
Note the sharpness of the superconducting transition for the selected single
crystals with RRR$>$100 used in our Mössbauer experiments.

##  Appendix E Excluded Possibilities for Observed Results

In the following we discuss a series of cross-checks to confirm the overall
consistency of the results and that support of the charge fluctuation
interpretation of the data.

  * •

Nuclear Origins. The quantization axis $\mbox{\boldmath$z$}_{J}$ of the
diagonalized electric-field gradient (EFG) tensor at the Yb site with the
$m2m$ local symmetry is parallel to the $c$-axis of $\beta$-YbAlB4 with an
orthorhombic $Cmmm$ structure, indicating
$\mbox{\boldmath$z$}_{J}\parallel\mbox{\boldmath$k$}_{0}$ in our experiments.
As a result of the angular momentum selection rule, only a single degenerate
nuclear transition $I_{g}=0\rightarrow I_{e}^{z}=\pm 1$ is probed out of the
five $E2$ nuclear transitions ($\Delta I^{z}=0,\pm 1,$ and $\pm 2$) of the
174Yb M$\ddot{\text{o}}$ssbauer resonance (see Section B.2) (?).

  * •

Magnetic Origins The absence of long-range magnetic order in $\beta$-YbAlB4
(?, ?) rules out magnetic hyperfine interactions as the origin of this
splitting. Generally, the electrical monopole interaction in the hyperfine
interaction is related to a valence state of a rare-earth ion.

  * •

Disorder. All Yb sites are crystallographically equivalent in $\beta$-YbAlB4
and it is known, from the SR X-ray diffraction measurements (?), that the
structural $Cmmm$ symmetry and the individual atomic coordination parameters
do not change up to 3.5GPa at 7K. Moreover, the residual resitivity ratio RRR
more than 100 reveals that there is no quenched disorder in this material at
low temperature.

  * •

Antiferroquadrupolar (AFQ) Order. From the Mössbauer spectra alone, it is not
possible to exclude the scenario of an AFQ configuration with staggered
electric field gradients. However such a state of long-range order can be
directly ruled out from other observations. Such quadrupolar transitions are
well known in other heavy fermion systems (e.g. UPd3 (?) and CeB6 (?)) where
they lead to a singularity in the specific heat and a cusp in the resistivity.
Neither features are observed. Finally, there is no signature of any
quadrupolar structural change in the high resolution X-ray diffraction
patterns.

  * •

Charge Density Wave Scenario There are two good reasons why a charge density
wave (CDW) scenario can be ruled out as an explanation for the observed
effects we present here:

First, there is no evidence in $\beta$-YbAlB4 for any phase transition into a
partially gapped phase in either transport or thermodynamic measurements
(Figs. D.5,D.6), which all display smooth evolution as a function of
decreasing temperature with no observable discontinuities or other features.

More generally, we can rule out any ordered frozen configurations, which would
have produced a thermodynamic signal. The alternative, disordered frozen
configurations would play the role of quenched disorder, a possibility that is
ruled out by the tiny residual resistivity of less than 1$\mu\Omega$cm in the
$\beta$-YbAlB4 samples used in the Mössbauer experiments.

Second, the observed isomer shift in the Mössbauer spectra is inconsistent
with a CDW. Isomer shifts result from small modifications of nuclear
excitation spectra induced by changes in the occupancies of orbitals with
significant atomic penetration. 3d orbitals of transition metals and
4f-orbitals of rare earth atoms that penetrate deep into an atom,
“communicating” their valence to the nucleus via their Coulomb-coupling to the
innermost s-orbitals. The observed isomer shifts for 174Yb Mössbauer isotopes,
Yb2+ and Yb3+ ions are separated by about $1$mm/s [see W. Henning, $et$ $al$.
Z. Phys. 241 138 (1971)], and the splitting observed to develop between the
two peaks in the spectrum of $\beta$-YbAlB4 at 2 K is of the same order. A CDW
would typically occur in a conduction band orbital that does not penetrate far
inside the atom and thus does not influence the nuclear spectrum; moreover, a
CDW typically involves a fractional modulation in charge, far smaller than the
integral charge required to produce known isomer shifts.

  * •

Relaxational Broadened Static Double Peak Finally, one may envision a
relaxational broadening mechanism to account for the discrepancy of the static
fit to the lineshape. An electronic relaxation broadening mechanism, produced
for example by a phonon-induced change in the Yb environment would have to be
slow on a Mössbauer time scale to broaden the peaks. Mössbauer spectroscopy
averages the environment of the nucleus over time-scales of order a nano-
second. Anything faster than this scale simply leads to a single, motionally
narrowed absorption line. To avoid this fate, a relaxational excitation would
have to operate at at the micro-volt scale, a scale that is orders of
magnitude smaller that the characteristic scale of collective electronic or
lattice excitations.

  * •

Mixture of Nuclear Transitions. By varying the incident angle of the X-ray
w.r.t. to the c-axis to 10 deg, it is possible to excite other nuclear
transitions which have different energies due to different coupling to the
electronic degrees of freedom and to allow us to extract the the isomer
(electrical monopole) shift $\delta$ and the electrical quadrupole splitting
$E_{Q}$ shifts (see Section B.2 and B.3). For the two different charge states
of Yb ions in $\beta$-YbAlB4, these values were estimated to be $\delta=0.27$
mm/s and $E_{Q}=-2.3$ mm/s and $\delta=-0.72$ mm/s and $E_{Q}=0$ mm/s for two
selected 174Yb nuclear transitions, respectively. Since the total angular
momentum $J$ is a good quantum number, the electrical quadrupole interaction
derives from the nonspherical $4f$ electron distribution (?). The second of
these transitions with $E_{Q}=0$ corresponds to the nuclear transitions of the
isotropic $(J=0)$ Yb2+ ion with ${}^{1}S_{0}$. The difference in $\delta$
between these nuclear transitions is comparable with that between the Yb3+ and
Yb2+ ionic state reported by the conventional 174Yb M$\ddot{\text{o}}$ssbauer
spectroscopy (?). Furthermore, these extracted $\delta$ values are compatible
with the electron densities at the Yb nuclei for the Yb2+ and Yb3+ ionic
states in $\beta$-YbAlB4 calculated by using the full-potential Korringa-Kohn-
Rostoker method (?, ?). Thus, the selected nuclear transition with
$\delta=0.27$ mm/s and $E_{Q}=-2.3$ mm/s is identified as the nuclear
transition of the Yb3+ configuration with ${}^{2}F_{7/2}$.

Recent experimental results and theoretical predictions find a strongly
anisotropic hybridization between the Yb $4f$ electrons and conduction
electrons in $\beta$-YbAlB4, suggsting its pure $J_{z}=\pm 5/2$ ground doublet
states (?, ?, ?), attributed to the crystalline-electric-field (CEF) ground
state of the Yb $4f$ electrons. Interestingly, the extracted $E_{Q}$ value
indicates that the ground Kramers doublet is almost pure $J_{z}=\pm 5/2$
states (see Section IV-a). The CEF ground state of the Yb3+ $4f$ electrons
infered from our M$\ddot{\text{o}}$ssbauer studies is thus fully consistent
with both the theoretical material modeling, the single-ion crystal field
analysis (?, ?) and recent Core-level spectroscopy measurements (?).

##  Appendix F Planckian Dissipation in $\beta$-YbAlB4

Figure F.7: The resistivity of $\beta$-YbAlB4 showing $\rho\sim T$ for over
the range $0.5$-$25$K (as labeled by arrows) and a slightly different slope
over the range $25$K to $150$K, from (?).

We now establish that the strange metal regime of $\beta$-YbAlB4 lies in the
Planckian dissipation regime. The resistivity in the temperature range between
0.25K and 25K reported in (?), [Fig. ] has a linear temperature dependence.
From these data, we extract

$\rho=\rho_{0}+A_{\rho}T,\qquad\text{where}\qquad A_{\rho}=0.4\mu\Omega
cm/K\qquad 0.5K<T<25K.$ (16)

We use the Drude formula

$\sigma=\frac{ne^{2}\tau_{tr}}{m}\qquad\to\qquad\frac{1}{\tau_{tr}}=\frac{ne^{2}\rho}{m}$
(17)

to find the scattering rate. The effective mass can be eliminated using Fermi
temperature:

$\Big{(}\frac{1}{\tau_{tr}}\Big{)}_{\rm
3D}=\frac{k_{B}T}{\hbar}\times\zeta,\qquad\zeta=\frac{A_{\rho}^{\square}T_{F}}{h/2e^{2}},$
(18)

In $\beta$-YbAlB4, the lattice parameters are $a=0.73nm$, $b=0.93nm$ and
$c=0.35nm$. Therefore, the system is relatively three-dimensional.
Nevertheless, we have cast the scattering rate in terms of an effective “sheet
density” $A_{\rho}^{\square}\equiv A_{\rho}(3\pi^{2}n)^{1/3}$ so that the
results can be compared to those of Cuprate (?). We assume that each unit cell
contributes roughly one electron to the charge density, i.e. $n\sim 6\times
10^{27}m^{-3}$. This is consistent with our measurements the Hall effect (?).
The Fermi temperature is obtained directly from the heat capacity $C=\gamma T$
with $\gamma={\pi^{2}}k_{B}^{2}\nu(\epsilon_{F})/3$ which can be written as

$\gamma_{\rm 3D}=\frac{\pi^{2}}{2}k_{B}\frac{nV}{T_{F}}$ (19)

Considering $\gamma=0.1$ J/K2/mole (?), we have to set
$k_{B}nV=k_{B}N_{A}=8.3$, finding $T_{F}=273$K in the 3D model, small but
plausible values for a heavy-fermion metal. Finally,

$\Big{(}\frac{1}{\tau_{\rm tr}}\Big{)}_{\rm
3D}=\frac{k_{B}T}{\hbar}\times\zeta,\qquad\zeta=0.4$ (20)

While the precise value of $\zeta$ depends on the assumptions of the model,
the important point is that it is a constant of $\zeta\sim O(1)$. Above 25K
the increased linear resistivity coefficient corresponds to
$\tau^{-1}_{tr}=0.63\tau^{-1}_{Pl}$ (25K$<T<$150K) in terms of the Planckian
scattering rate $\tau^{-1}_{Pl}=k_{B}T/\hbar$. As discussed in the paper, the
change in the slope of linear resistivity is correlated with the change in the
dynamics of phonons. The appearance of a transport scattering rate of order
$k_{B}T/\hbar$ is the hallmark of a Planckian metal.

##  Appendix G Supporting Theoretical Discussion

###  G.1 Kondo Breakdown

The Kondo breakdown scenario can be captured by a number of approaches,
including the slave-boson hybridization (?), the numerical renormalization
group studies of the pseudogap single-impurity Anderson model (?) or Schwinger
boson large-N study of the Kondo lattice (?). Fig. G.12(A) shows the
divergence of the charge susceptibility at a Kondo breakdown transition
studied with the latter method. Within a simple Anderson model

$H_{\rm
Anderson}=H_{c}+U_{0}(n_{d})^{2}+\sum_{\sigma=\uparrow,\downarrow}V_{0}(c^{\dagger}_{k\sigma}d_{\sigma}+h.c.)+\mu_{d}n_{d},\qquad
n_{d}=\sum_{\sigma=\uparrow\downarrow}d^{\dagger}_{\sigma}d_{\sigma},$ (21)

the hybridization and the interaction are renormalized $V_{0}\to V=ZV_{0}$ and
$U_{0}\to 0$ due to the Kondo effect, where the wavefunction renormalization
$Z$ is given by

$Z=[1-\partial_{\omega}\Sigma_{f}(\omega)]^{-1}.$ (22)

At the Kondo-breakdown transition, the localization of the $d$ electrons means
that their effective mass diverges and the $Z\to 0$, further renormalizing the
hybrdization to zero $V\to 0$ (?, ?, ?). The fluctuation time-scale is
associated with the hybrdization $V=ZV_{0}$

$\tau^{-1}_{f}\sim\pi\rho V^{2}$ (23)

where $\rho$ is the density of states of the conduction electrons at the Fermi
energy. Therefore, the renormalization $V\to 0$ at the Kondo break-down leads
to slow charge fluctuations.

Fig. G.12(A) shows the static charge susceptibility vs. temperature across a
Kondo breakdown transition between a FL phase and a spin-liquid, computed
using Schwinger boson approach (?). The $\chi_{\rho}\sim 1/T$ at the
transition indicates to the emergence of a sharp zero-frequency peak in the
dynamical charge susceptibility
$-\beta\chi^{\prime\prime}_{\rho}(\omega)/\omega$, as shown in the inset of
Fig. G.12(B). Fig. G.12(B) also shows the result of this critical charge
susceptibility on the M$\ddot{\text{o}}$ssbauer line shape where a single peak
at high temperature is splitted into two peaks proportional to $\alpha^{2}$.

In spite of qualitative agreement, the splitting observed in the paper is too
large compared to the temperature and cannot be entirely accounted for by the
mechanism discussed here.

The resolution of this paradox involves the role of electron-phonon coupling:
The phonons respond to the charge accumulation by dressing the charges into
polarons, thereby additionally renormalizing the hybridization and thus the
width of the dynamical charge susceptibility.

###  G.2 Phonons and Polarons

The physical essence can be captured by a very simplified model

$H=H_{\rm
Anderson}+\sum_{q}[\omega_{q}a_{q}^{\dagger}a_{q}+g_{q}(a_{q}^{\dagger}+a_{q})n_{d}].$
(24)

The normal phononic modes are related to the crystal displacement and momentum
as

$\Delta
z=\sum_{q}\sqrt{\frac{\hbar}{2m\omega_{q}}}(a_{q}+a^{\dagger}_{q}),\qquad
p_{z}=\sum_{q}i\sqrt{\frac{\hbar m\omega_{q}}{2}}(a^{\dagger}_{q}-a_{q})$ (25)

The phonon mode is approximated by the Debye model with a characteristic
energy scale $k_{\rm B}\Theta_{\rm D}$ and $g_{q}$ is the coupling of the
phonons $\omega_{q}$ to the occupation of the $f$-electrons. The effect of
polarons is non-perturbative in $g_{q}$ and can be best captured by the so-
called Lang-Firsov unitary transformation (?)

$U=\exp[\sum_{q}u_{q}(a_{q}^{\dagger}-a_{q})n_{d}],$ (26)

where $n_{d}=\sum_{\sigma}d^{\dagger}_{\sigma}d_{\sigma}$. As a result of this
transformation, we have

$V\to Ve^{\sum_{q}u_{q}(a_{q}^{\dagger}-a_{q})},\qquad g_{q}\to
g_{q}-u_{q}\omega_{q},\qquad U\to U-\sum_{q}\omega_{q}u_{q}^{2}.$ (27)

Therefore, choosing $u=g_{q}/\omega_{q}$ eliminates the electron-phonon
coupling. But the hybridization is renormalized:

$V\to V\langle
e^{\sum_{q}u_{q}(a_{q}^{\dagger}-a_{q})}\rangle=Ve^{-\sum_{q}u_{q}^{2}(n_{q}+1/2)}.$
(28)

Here, $n_{q}$ is the number of bosons $\omega_{q}$ and it is given by the
Bose-Einstein distribution in a thermal state $n_{q}=n_{B}(\omega_{q})$ where
$n_{B}(\omega)=[1+e^{\omega/T}]^{-1}$ at temperature $T$. At low-temperature
thermal fluctuations are suppressed $n_{B}(\omega_{p})=0$, and only the
quantum fluctuations contribute to the suppression of the renormalized matrix
element of the hybridization $V$. Note that as a result of the unitary
transformation

$a_{q}^{\dagger}a_{q}\to
U^{\dagger}a_{q}^{\dagger}a_{q}U=(a_{q}^{\dagger}-u_{q}n_{d})(a_{q}-u_{q}n_{d})=a_{q}^{\dagger}a_{q}-u_{q}n_{d}(a_{q}^{\dagger}+a_{q})+u^{2}_{q}n_{d}^{2}.$
(29)

Therefore, the phonon occupancy departs from the thermal equilibrium to
$n_{q}=\langle
a^{\dagger}_{q}a_{q}\rangle=n_{B}(\omega_{q})+u_{q}^{2}n_{d}^{2}$. For
simplicity, we limit the discussion to the transition between empty
($n_{d}=0$) and singly occupied ($n_{d}=1$) infinite-U Anderson model.
Therefore, the quantum fluctuation contribution $N_{p}\equiv\sum_{q}u^{2}_{q}$
in the exponent of Eq. (28) can be interpreted as the total number of phonons
excited by the change in the charge $n_{d}=0\to n_{d}=1$ and involved in the
polaron. This leads to the additional slowing-down of the charge fluctuations

$\tau^{-1}_{f}\sim\pi\rho V^{2}e^{-2N_{p}}.$ (30)

This discussion and the interpretation of $N_{p}$ as the number of phonons
excited by the charge goes back to (?, ?).

Figure G.8:  Synchrotron-radiation-based 174Yb M$\ddot{\text{o}}$ssbauer
spectrum of $\beta$-YbAlB4 at 5K. The $c$-axis of the single crystalline
$\beta$-YbAlB4 samples was tilted though 10 deg from the incident X-ray. The
closed circles with error bar indicate the observed spectrum and the red solid
line presents the analytical spectrum. The extracted subspectra for two Yb
states are shown by the bar-diagrams. The relative intensities of the bar-
diagrams represent the evaluated excitation probabilities for the allowed
174Yb nuclear transitions in the experimental conditions.

Figure G.9:  (A) Refined and evaluated energies $\varepsilon$ for the nuclear
transitions and (B) difference in intensity $R$ between these two nuclear
transitions in $\beta$-YbAlB4 at 2K as functions of pressure. In (A), the
solid circles and squares with error bar presents the refined $\varepsilon$
for two selected $I_{g}=0\rightarrow I_{e}^{z}=\pm 1$ nuclear transitions,
respectively, below 1.2GPa. The crosses represent the weighted average
$\varepsilon$ values of two nuclear transitions below 1.2GPa. The solid
triangle with error bar indicates the refined $\varepsilon$ for the single
absorption component in the SR-based 174Yb M$\ddot{\text{o}}$ssbauer spectrum
of $\beta$-YbAlB4 at 2.3GPa and 2K. The red and blue broken lines are visual
guides.  Figure G.10:  Evaluated FWHM of the dip in the observed spectra.
Figure G.11: Various fits to the experimental lineshape at 2K and ambient
pressure. Figure G.12:  (A) The static charge susceptibility as a function of
$T$ for various values of $T_{K}/J_{H}$ shows the divergence of the charge
susceptibility at the Kondo breakdown QCP between a FL (blue) and a spin-
liquid (green). (B) Inset: The dynamical charge susceptibility vs. frequency
at the QCP, shows a sharp feature developing as the temperature is lowered
from red to blue. Main panel: Consequent splitting of the Mössbauer lineshape
at low-temperature as a result of the critical charge fluctuation.

## References and Notes

  * 1. P. W. Anderson, J. Phys. Soc. Japan 9, 316 (1954).
  * 2. R. Kubo, J. Phys. Soc. Japan 9, 935 (1954).
  * 3. M. Blume, J. A. Tjon, Phys. Rev. 165, 446 (1968).
  * 4. M. Blume, Phys. Rev. 174, 351 (1968).
  * 5. Y. Komijani, P. Coleman, Phys. Rev. B 94, 085113 (2016).
  * 6. C. M. Varma, Rev. Mod. Phys. 48, 219 (1976).
  * 7. E. V. Sampathkumaran, Hyperfine Interactions 27, 183 (1986).
  * 8. M. Okawa, et al., Phys. Rev. Lett. 104, 247201 (2010).
  * 9. R. L. Cohen, M. Eibschütz, K. W. West, Phys. Rev. Lett. 24, 383 (1970).
  * 10. I. Nowik, Hyperfine interactions 13, 89 (1983).
  * 11. L. Ruby, S., J. Phys. Colloques 35, C6 (1974).
  * 12. M. Seto, et al., Phys. Rev. Lett. 102, 217602 (2009).
  * 13. R. Masuda, et al., Appl. Phys. Lett. 104, 082411 (2014).
  * 14. G. V. Smirnov, et al., Phys. Rev. A 76, 043811 (2007).
  * 15. F. J. Lynch, R. E. Holland, M. Hamermesh, Phys. Rev. 120, 513 (1960).
  * 16. D. W. Hamill, G. R. Hoy, Phys. Rev. Lett. 21, 724 (1968).
  * 17. M. Seto, et al., Journal of Physics: Conference Series 217, 012002 (2010).
  * 18. R. T. Macaluso, et al., Chem. Mater. 19, 1918 (2007).
  * 19. J. P. Hannon, G. T. Trammell, M. Blume, D. Gibbs, Phys. Rev. Lett. 61, 1245 (1988).
  * 20. Y. H. Matsuda, et al., Journal of the Korean Physical Society 62, 1778 (2013).
  * 21. S. Ofer, I. Nowik, S. G. Cohen, Chemical Applications of M$\ddot{\text{o}}$ssbauer Spectroscopy (Academic Press, New York, p. 427, 1968).
  * 22. A. J. Freeman, R. E. Watson, Phys. Rev. 127, 2058 (1962).
  * 23. Meyer, C., Gros, Y., Hartmann-Boutron, F., Capponi, J.J., J. Phys. France 40, 403 (1979).
  * 24. Y. Sakaguchi, et al., J. Phys. Soc. Jpn. 85, 023602 (2016).
  * 25. W. Sturhahn, Journal of Physics: Condensed Matter 16, S497 (2004).
  * 26. Y. Matsumoto, et al., Science 331, 316 (2011).
  * 27. T. Tomita, K. Kuga, Y. Uwatoko, P. Coleman, S. Nakatsuji, Science 349, 506 (2015).
  * 28. H. C. Walker, et al., Phys. Rev. Lett. 97, 137203 (2006).
  * 29. K. Hanzawa, T. Kasuya, Journal of the Physical Society of Japan 53, 1809 (1984).
  * 30. H. Winkelmann, et al., Phys. Rev. Lett. 81, 4947 (1998).
  * 31. W. Henning, G. Bahre, P. Kienle, Zeitschrift fur Physik A Hadrons and nuclei 241, 138 (1971).
  * 32. M. Ogura, H. Akai, Journal of Physics: Condensed Matter 17, 5741 (2005).
  * 33. H. Akai. Private communication.
  * 34. A. H. Nevidomskyy, P. Coleman, Phys. Rev. Lett. 102, 077202 (2009).
  * 35. Y. Matsumoto, et al., J. Phys. Soc. Jpn. 84, 024710 (2015).
  * 36. A. Ramires, P. Coleman, A. H. Nevidomskyy, A. M. Tsvelik, Phys. Rev. Lett. 109, 176404 (2012).
  * 37. K. Kuga, et al., Phys. Rev. Lett. 123, 036404 (2019).
  * 38. S. Nakatsuji, et al., Nature Physics 4, 603 (2008).
  * 39. A. Legros, et al., Nature Physics 15, 142 (2019).
  * 40. E. C. T. O’Farrell, Y. Matsumoto, S. Nakatsuji, Physical Review Letters 109, 176405 (2012).
  * 41. C. Pépin, Phys. Rev. Lett. 98, 206401 (2007).
  * 42. J. H. Pixley, S. Kirchner, K. Ingersent, Q. Si, Phys. Rev. Lett. 109, 086403 (2012).
  * 43. Y. Komijani, P. Coleman, Phys. Rev. Lett. 122, 217001 (2019).
  * 44. I. G. Lang, Y. A. Firsov, JETP 16, 1301 (1962).
  * 45. D. Sherrington, P. Riseborough, J. Phys. Colloques 37, C4 (1976).
  * 46. A. C. Hewson, D. M. Newns, Journal of Physics C: Solid State Physics 12, 1665 (1979).