# Quantum fluctuations lead to glassy electron dynamics in a good metal

Shashank Kumar Ojha<EMAIL_ADDRESS>Department of Physics, Indian
Institute of Science, Bengaluru 560012, India  Sankalpa Hazra  Contributed
equally  Department of Physics, Indian Institute of Science, Bengaluru 560012,
India  Department of Materials Science and Engineering, The Pennsylvania State
University, University Park, PA 16802, USA  Surajit Bera  Contributed equally
Department of Physics, Indian Institute of Science, Bengaluru 560012, India
Sanat Kumar Gogoi  Department of Physics, Indian Institute of Science,
Bengaluru 560012, India  Department of Physics, Digboi College, Digboi 786171,
India  Prithwijit Mandal  Department of Physics, Indian Institute of Science,
Bengaluru 560012, India  Jyotirmay Maity  Department of Physics, Indian
Institute of Science, Bengaluru 560012, India  A. Gloskovskii  Deutsches
Elektronen-Synchrotron DESY, 22607 Hamburg, Germany  C. Schlueter  Deutsches
Elektronen-Synchrotron DESY, 22607 Hamburg, Germany  Smarajit Karmakar  Tata
Institute of Fundamental Research, 36/P, Gopanpally Village, Serilingampally
Mandal, Ranga Reddy District, Hyderabad, 500107, India  Manish Jain
Department of Physics, Indian Institute of Science, Bengaluru 560012, India
Sumilan Banerjee<EMAIL_ADDRESS>Department of Physics, Indian Institute
of Science, Bengaluru 560012, India  Venkatraman Gopalan  Department of
Materials Science and Engineering, The Pennsylvania State University,
University Park, PA 16802, USA  Srimanta Middey<EMAIL_ADDRESS>Department
of Physics, Indian Institute of Science, Bengaluru 560012, India

###### Abstract

One of the central challenges in condensed matter physics is to comprehend
systems that have strong disorder and strong interactions. In the strongly
localized regime, their subtle competition engenders glassy electrons, which
ceases to exist well before the insulator-to-metal transition is approached as
a function of doping. Here, we report on the discovery of glassy electron
dynamics deep inside the good metal regime of an electron-doped quantum
paraelectric system: KTaO3. We reveal that upon excitation of electrons from
defect states to the conduction band, the excess injected carriers in the
conduction band relax in a stretched exponential manner with a large
relaxation time, and the system evinces simple aging phenomena - a telltale
sign of glass. Most significantly, we observe a critical slowing down of
carrier dynamics below 35 K, concomitant with the onset of quantum
paraelectricity in the undoped KTaO3. Our combined investigation using second
harmonic generation technique, density functional theory and phenomenalogical
modeling demonstrates quantum fluctuation-stabilized soft polar modes as the
impetus for the glassy behavior. This study addresses one of the most
fundamental questions regarding the potential promotion of glassiness by
quantum fluctuations and opens a route for exploring glassy dynamics of
electrons in a well-delocalized regime.

Introduction: The notion of glassy dynamics associated with the electronic
degree of freedom in condensed matter systems was first envisaged by Davies,
Lee and Rice in 1982 [1]. Building heavily on Anderson’s seminal work on the
localization of wave functions in random disordered media [2], it was
predicted that, in a disordered insulator with highly localized electronic
states, the interplay between disorder and long-range Coulomb interaction
(Fig. 1a) should precipitate electronic frustration in real space. Such a
scenario would result in a rugged energy landscape with numerous metastable
states, leading to the emergence of electron glass [3, 4, 5]. This hypothesis
was soon tested on various strongly localized electronic systems, including
granular metals, crystalline and amorphous oxides, and later, it was also
extended to doped semiconductors and two-dimensional electron gases [6, 3, 4,
5, 7]. The typical relaxation time in these systems ranges from a few seconds
to several hours, making them an obvious choice for studying glassy physics in
laboratory timescales. What makes these systems even more intriguing is the
array of perturbations that one can use to effectively drive them away from
equilibrium [4]. Moreover, electron glasses are highly susceptible to quantum
effects, making them an ideal platform for investigating the potential role of
quantum fluctuations in the formation or melting of glasses, which has
remained a pervasive problem in the field of glass [8, 9].

In the antithetical regime of highly delocalized electrons i.e. in a metal,
the screening effect significantly reduces the strength of the electron-
electron and electron-impurity interactions. Consequently, such system
generally possesses a non-degenerate ground state with a well-defined Fermi
surface. As a result, the dynamics of glassy behavior, which involves the
existence of multiple, competing ground states, is incompatible with the
behavior of metals. In fact, the manifestation of glassiness fades away
considerably prior to the transition from insulator to metal, and there is an
absolute lack of any substantiated indication of the presence of glassiness
within a good metal regime [3] where quasi-particle mean free path is larger
than the electron’s wavelength.

In this work, we report on the discovery of glassy dynamics of conduction
electrons in an electron-doped quantum paraelectric system, namely KTaO3, in a
good metal regime. Even more surprising observation is that glassiness is
found to appear in a regime where quantum fluctuations are inherently present
in the system. In pristine KTaO3, the quantum fluctuations associated with the
zero point motion of the atoms prohibits the onset of ferroelectric order
below 35 K (Fig. 1b), and the system’s properties are typically governed by
the presence of associated low energy transverse optical phonon (Fig. 1c)
popularly known as soft polar mode [13]. Our combined transport and optical
second harmonic generation measurements find that properties associated with
the soft polar mode are preserved even deep inside the metallic regime. Most
importantly, such soft modes are found to be directly responsible for emergent
glassy dynamics at low temperatures, which is further corroborated by our
theoretical calculations. Our observation is one of the rarest examples where
quantum fluctuation, which is generally considered as a bottleneck for glass
formation [8, 9], is ultimately accountable for the appearance of glassy
dynamics in a good metallic phase.

Demonstration of good metallic behavior: Several successful attempts have been
made in recent times to introduce free carriers in the bulk as well as at the
surface or interface of KTaO3 [14, 15, 16, 17]. For the current investigation,
metallic samples have been prepared by introducing oxygen vacancies in
pristine single crystalline (001) oriented KTaO3 substrate (for details see
methods section and references [11, 18]). These samples were found to exhibit
quantum oscillations below 10 K, which is signature of a good metal with well
defined Fermi surface. To further testify this, we have also computed
temperature-dependent mean free path of electrons ($l_{e}$) within
Drude–Boltzmann picture. Fig. 1c shows the corresponding plots for degenerate
and non-degenerate case. A dotted vertical line marks the temperature above
which $l_{e}$ becomes shorter than the inverse of Fermi wave-vector ($k_{F}$)
and the sample crosses over from normal metal to a bad metal phase [12]. In
the current work, observation of glassy dynamics is inherently constrained to
temperatures which is much lower than crossover temperature to bad metal phase
and hence for all practical purposes our electron doped KTaO3 system can be
considered as a good metal with well-defined scattering.

Demonstration of glassy dynamics: Oxygen vacancy creation in KTaO3 not only
adds free electrons but also leads to the formation of highly localized defect
states. To determine the exact position of the defect states, valence band
spectrum has been mapped out by using hard X-ray photoelectron spectroscopy
(HAXPES) at P22 beamline of PETRA III, DESY (see methods for more details).
Apart from the well-defined quasi-particle peak, mid-gap states centered at
1.6-1.8 eV is observed (Fig. 2a), which arises due to the clustering of oxygen
vacancies [18].

In this work, we utilize these defect states to perturb the system by
selective excitation of trapped electrons to the conduction band via sub-
bandgap light illumination. Subsequently, the system’s response is studied by
monitoring the temporal evolution of the electrical resistance at a fixed
temperature (see inset of Fig. 2a for transport measurement set-up). For each
measurement, the sample was first cooled down to the desired temperature. Once
the temperature stabilizes, the system was driven out of equilibrium by
shining light for half an hour, thereafter resistance relaxation was observed
for the next 1.5 hours in dark condition. For the next measurement, the sample
was heated to room temperature where the original resistance is recovered.
Fig. 2b shows one set of data recorded with green light ($\lambda$=527 nm) at
several fixed temperatures ranging from 15 K to 150 K.

At first glance, Fig. 2b reveals a striking temperature dependence of the
photo-doping effect (also see Supplementary Information Section 1 and
Supplementary Fig. 1) and the way the system relaxes after turning off the
light is also found to be strongly temperature dependent. Further analysis
reveals that in the off-stage, the resistance relaxes in a stretched
exponential manner [exp(-($t$/$\tau$)β) where $\tau$ is the relaxation time
and $\beta$ (stretching exponent) $<$ 1 (Fig. 2c)], signifying the presence of
multiple relaxation channels for the electron-hole recombination [20]. The
microscopic origin behind the spatial separation of electron-hole pair and
resultant non-exponential relaxation will be discussed later. The temperature
evolution of $\beta$ and $\tau$ obtained from the fitting further reveals a
substantial increase in relaxation time followed by constant $\beta\approx$
0.5 below 35 K (Fig. 2c). This observation is remarkable given the fact that
this crossover temperature precisely coincides with the onset of quantum
fluctuation in undoped KTaO3. Such stretched exponential relaxation with
$\beta$ less than 1 is very often considered as a signature of glassy dynamics
and has been observed in variety of systems [4, 21, 22].

One critical test to confirm a glassy phase is the observation of aging
phenomenon wherein the system’s response depends on its age [23]. More
precisely, older systems are found to relax more slowly than younger ones. To
examine this, we prepared the system of desired age by tuning the duration of
light illumination ($t$${}_{\text{ill}}$), and measurements similar to that
shown in Fig. 2b were performed at 15 K (Fig. 3a ). For further analysis, we
only focus on relaxation after turning off the light. In Fig. 3b we plot the
change in resistance in the off-stage ($\Delta$$R$${}_{\text{off}}$)
normalized with a total drop in resistance ($\Delta$$R$${}_{\text{ill}}$) at
the end of illumination. Evidently, with increasing $t$${}_{\text{ill}}$
system’s response becomes more and more sluggish. More precisely, $\tau$
obtained from the fitting is found to scale linearly with $t$${}_{\text{ill}}$
(inset of Fig. 3c). This is the defining criteria for simple or full aging
[23] which is more clear in Fig. 3c where all the curves can be collapsed to a
universal curve by normalizing the abscissa by $t$${}_{\text{ill}}$. A careful
look at Fig. 3c reveals that for larger values of $t$${}_{\text{ill}}$, curves
start to deviate from the universal scaling. This is much clear in Fig. 3d
which contains a similar set of scaling data for another sample with a
different carrier concentration. Such an observation is consistent with the
criteria for aging that $t$${}_{\text{ill}}$ should be much less than the time
required to reach the new equilibrium under perturbation. As evident from Fig.
3a and inset of Fig. 3d, above a critical value of $t$${}_{\text{ill}}$ there
is little change in resistance upon shining the light any further. This
signifies that the system is closer to its new equilibrium and hence aging
ceases to hold at a higher $t$${}_{\text{ill}}$.

Presence of polar nano regions & importance of quantum fluctuations: As
mentioned earlier, the glassy behavior of electrons in conventional electron
glasses results from the competition between disorder and Coulomb interactions
and is only applicable in strongly localized regime [3, 4, 5]. As glassiness
is observed within a good metal regime ($k_{F}$$l_{e}$$>$1) in our oxygen-
deficient KTaO3 samples, we need to find other processes responsible for the
electron-hole separation and complex glassy relaxations in the present case.
In the context of electron doping in another well-studied quantum paraelectric
SrTiO3, a large lattice relaxation (LLR) model involving deep trap levels [24]
has been associated as a dominant cause for prohibiting electron-hole
recombination [25, 26]. However, our analysis of $\tau$ vs. $T$ (Supplementary
Information Section 4 and Supplementary Fig. 4) does not support the
applicability of the LLR model in the present case. In sharp contrast, we will
conclusively demonstrate here that the effective charge separation in such
systems directly correlates with the appearance of polar nano regions (PNRs),
which arise as a direct consequence of the defect dipoles present in a highly
polarizable lattice of quantum paraelectric [27].

In an ordinary dielectric host, an electric dipole can polarize the lattice
only in its immediate vicinity and the correlation length ($r_{c}$) is
generally of the order of unit cell length which further remains independent
of temperature [6]. However the situation is drastically different in highly
polarizable hosts such as KTaO3 where the magnitude of $r_{c}$ is controlled
by the polarizability of the lattice which is inversely proportional to the
soft mode frequency ($\omega$s). Since $\omega$s decreases with decreasing
temperature, $r_{c}$ becomes large at lower temperatures. As a result, PNRs
spanning several unit cells are formed around the defect dipole which is
randomly distributed in the lattice (Fig. 4a).

While PNRs are quite well established in the insulating regime (Supplementary
Information Section 5 and Supplementary Fig. 5), they are expected to vanish
in the metals due to screening effects from free electrons. Surprisingly,
several recent experiments have reported that PNRs can exist even in the
metallic regime [29, 30, 31, 32, 33]. Motivated by these results, we have
carried out temperature-dependent optical second harmonic generation (SHG)
measurement (Fig. 4b) which is a powerful technique to probe PNRs [9]. Fig. 4c
shows the temperature evolution of SHG intensity which is directly
proportional to the volume density of PNRs. As evident, no appreciable SHG
signal is observed at room temperature, however a strong signal enhancement is
observed below 150 K, signifying the appearance of PNR below 150 K in our
metallic sample. Since this onset temperature exactly coincides with the
temperature below which an appreciable photo-doping effect is observed in our
photo conductivity measurements (Supplementary Information Section 6 and
Supplementary Fig. 6), we believe that the internal electric field generated
around such PNRs is the major cause behind driving apart the photo-generated
electron-hole pairs in real space. Another notable observation is that the SHG
intensity is independent of the temperature below 35 K. Since
$r_{c}\propto\omega_{s}^{-1}$ [13], our SHG measurement establishes that
quantum fluctuation stabilized soft polar mode is retained even in metallic
KTaO3 [35].

We have also conducted an investigation into the primary defect dipoles
responsible for the creation of PNRs in our samples. Considering that our
samples were prepared through high-temperature annealing within an evacuated
sealed quartz tube [18], there is a possibility of K vacancies due to its high
volatility. The presence of a certain degree of K vacancy has been indeed
observed in our HAXPES measurements (Supplementary Information Section 7 and
Supplementary Fig. 7). It is widely established that, in $AB$O3 systems, the
off-centering of substitute $B$ atom ($B$ antisite-like defect) in the
presence of $A$ atom vacancy leads to a macroscopic polarization even in a
non-polar matrix [36, 37, 9, 38, 39]. Our density functional theory
calculations (details are in the method section) considering Ta antisite-like
defect has found significant Ta off-centering along [100] and [110] in the
presence of K vacancy (Fig. 4d). Further, we have also computed induced
polarization in the system following the modern theory of polarization where
the change in macroscopic electric polarization is represented by a Berry
phase [12, 15, 42, 13, 44] and the non-zero Berry curvature i.e., Berry phase
per unit area, is taken as a signature of finite polarization in the material.
In Fig.4e, f, we have plotted the Berry curvature in the plane $k_{z}=0$
($\Omega_{xy}(\mathbf{k})$) for the Ta off-centering along [110] and [100],
respectively. From the figure, it is clear that the large contributions to the
Berry curvature are due to the avoided crossings of bands at the Fermi surface
(see inset of Fig. 4f) which are induced by spin-orbit coupling.

Phenomenological model to understand glassy dynamics: We now discuss a
possible mechanism for the emergent glassy dynamics in a metal where
conduction electrons coexist with PNRs. Since the glassiness in the present
case is observed in the dynamical relaxation of excess injected carriers in
the conduction band, it is necessary to have a thorough understanding of the
relaxation processes happening against the backdrop of randomly oriented PNRs.
In indirect-band semiconductors like KTaO3 which have strong electron-lattice
and defect-lattice coupling, the relaxation should be predominantly
nonradiative and manifest itself as the emission of several low-energy phonons
[45]. Further, since inter-band electron-phonon matrix element due to acoustic
phonons are negligible, the electron-hole recombination may primarily involve
soft polar modes at low temperatures, although it is not the lowest energy
phonon [46]. While such a multi-phonon inter-band transition could lead to
multichannel relaxation with large time scales [45], it can never give rise to
collective glassy behavior.

Instead, we suggest the following scenario for the observed glassiness. As was
previously mentioned, there is clear evidence that the internal electric field
around PNRs has a significant impact on electron-hole recombination in our
sample. It has been demonstrated previously [27] that the random interactions
between PNRs (in the limit of dilute defect dipoles) leads to a dipole glass
at low temperatures in KTaO3. Electrons and holes being charged particles
would immediately couple to the complex electric field from the dipole glass
and hence there is a chance that the glassy background of PNRs can induce
glassiness to the free carriers in the system.

In order to study such a possibility, we consider a theoretical model with the
Hamiltonian,

$\displaystyle H$ $\displaystyle=H_{el}+H_{gl}+H_{el-gl},$ (1)

where
$H_{el}=-\sum_{ij}t_{ij}c_{i}^{\dagger}c_{j}-\varepsilon_{0}\sum_{\alpha}f_{\alpha}^{\dagger}f_{\alpha}$
($\varepsilon_{0}>0$) describes the electronic part in terms of creation and
annihilation operators $c_{i}^{\dagger},c_{i}$ ($i=1,\cdots,N_{c}$) and
$f_{\alpha}^{\dagger},f_{\alpha}$ ($\alpha=1,\cdots,N_{f}$) of the $N_{c}$
conduction and $N_{f}$ impurity electronic states, respectively. For our
calculations, we consider various lattices and corresponding hopping
amplitudes $t_{ij}$ to describe several different energy dispersions for the
conduction band, e.g., a band with semicircular DOS
$g(\epsilon)=(1/2\pi)\sqrt{W^{2}-\omega^{2}}\theta(W-|\omega|)$ with band
width $W$ [$\theta(x)$ is heaviside step function] (Fig. 5(a)), and a flat
band with width $W=0$, as discussed in the Supplementary Information Section
9. We take a flat impurity band at energy $-\varepsilon_{0}$, separated by a
gap $\Delta=\varepsilon_{0}-W/2$ from the conduction band minimum. We set the
chemical potential $\mu=0$, at the centre of conduction band (Fig. 5(a)).

To model the dynamics of the glassy background, which may result from either a
single PNR or randomly-distributed coupled assembly of PNRs, we consider a
system with $N_{g}$ degrees of freedom $\\{x_{\mu}\\}$ (Supplementary
Information Section 9). These position-like variables, related to the electric
dipoles inside the PNRs, can be thought of as a multi-coordinate
generalization of the usual _single configuration coordinate_ [24, 45, 26] for
a defect or impurity in semiconductors. Such single coordinate defect model,
though can give rise to very slow relaxation of photoresistivity [24, 45, 26],
is unlikely to lead to collective aging phenomena seen in our experiment. For
our model, we thus assume a collective glassy background that gives rise to a
two-step relaxation, $C_{gl}(t)=\langle
x_{\mu}(t)x_{\mu}(0)\rangle=A\exp(-t/\tau_{s})+B\exp[-(t/\tau_{\alpha})^{\beta}]$
as a function of time $t$, for the dynamical correlation of $x_{\mu}$ at
temperature $T$. The relaxation consists of a short-time exponential decay
with the time scale $\tau_{s}$ and a long-time stretched exponential
$\alpha$-relaxation with time scale $\tau_{\alpha}(T)$ and stretching exponent
$\beta$ [47, 48]. We assume that $\tau_{s}$ is temperature independent,
whereas $\tau_{\alpha}$ increases with decreasing temperature. We also vary
the coefficients $A(T)$ and $B(T)$ with $T$ such that the relative strength of
the stretched exponential part increases at lower temperatures (Supplementary
Information Section 9).

The glassy PNRs lead to a transition between the conduction band and impurity
state via a coupling $H_{el-
gl}=\sum_{i\alpha\mu}(V_{i\alpha\mu}c_{i}^{\dagger}f_{\alpha}+\mathrm{h.c})x_{\mu}$,
where we take $V_{i\alpha\mu}$ as Gaussian complex random numbers to keep the
model solvable. The coupling can arise either due to direct coupling of the
electric field of the PNR to the electrons, or indirectly via coupling between
PNR and electrons mediated by phonons, e.g., the soft polar optical phonon
mode.

To gain an analytical understanding of the electron-hole recombination
dynamics we consider a limit where there are no _backactions_ of the
conduction electrons on the impurity electrons and the glass (Supplementary
Information Section 9).

The description of the relaxation of the photo-excited electrons requires the
consideration of the out-of-equilibrium quantum dynamics, which is beyond the
scope of this paper. Instead, we consider an equilibrium dynamical
correlation, namely the (connected) density-density correlation function
$C_{el}(t)=\langle n_{i}(t)n_{i}(0)\rangle-\langle n_{i}(0)\rangle^{2}$ as a
function of time $t$ for the density $n_{i}=c_{i}^{\dagger}c_{i}$ of the
conduction electron. The correlation function $C_{el}(t)$ can be computed
exactly at a temperature $T$ in the toy model (Eq. (1)). In the limit of no
backaction of the conduction electrons, the electronic correlation function
can be written as a convolution of the spectral function of the glass
(Supplementary Information Section 9). As a result, the glass spectral
function, which contains the information of the multiple time-scales and their
non-trivial temperature dependence, may directly induce glassiness in the
electronic relaxation.

To verify the above scenario, we numerically compute $C_{el}(t)$ for two-step
glass correlation functions $C_{gl}(t)$, shown in Fig. 5(b) (upper panel) for
three temperatures $T=0.5,0.3,0.1\ll\Delta$, where the temperature dependence
is parameterized by $B(T)$ ($A=1-B$) and $\tau_{\alpha}(T)$ (Fig. 5(b) (lower
panel)). We take a temperature independent exponent $\beta=0.5$ and
$\tau_{\alpha}(T)\sim T^{-2.8}$, consistent with our experimental results
(Fig. 2(c)). Here $\tau_{s}^{-1}$ ($\tau_{s}$) is set as the unit of energy
(time) ($\hbar=1$). The calculated $C_{el}(t)$ is plotted in Fig. 5(c) for a
flat conduction band ($W=0$) (upper panel) and a semicircular conduction band
DOS with $W=0.01$, and electron-glass coupling $V=0.3$. As shown in Fig. 5(c),
we find that for a local glassy bath, whose bandwidth is comparable or larger
than the electronic energy scales $W$ and $\Delta$, the complex glassy
correlation, namely the two-step relaxation with long and non-trivial
temperature-dependent time scale, is also manifested in the electronic
relaxation.

Outlook: While we do observe complex relaxations in such a toy model, these
are still weak in contrast to the actual experimental results. We expect that
a more realistic theory considering the direct coupling with a real space
distribution of PNRs embedded in the sea of conduction electrons would yield a
strong glass. This would further demand intricate knowledge about the nature
of interactions in the presence of PNRs which is still a subject of debate.
Interestingly, these questions form the basis for understanding the nature of
conduction in an interesting class of materials known as polar metals [49, 31,
32], and hence we believe that our finding of glassy relaxations in presence
of PNRs will be crucial in building the theory of conduction in quantum
critical polar metals. Further, the observation of glassy dynamics deep inside
the good metallic regime is in sharp contrast with the conventional
semiconductors where glassy relaxation ceases to exist just before the
insulator- metal transition (IMT) is approached from the insulating side. This
raises the question about the envisaged role of glassy freezing of electrons
as a precursor to IMT apart from the Anderson and Mott localization [50, 3].

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## Methods

Sample preparation: Oxygen deficient KTaO3 single crystals were prepared by
heating as received 001 oriented pristine KTaO3 substrate (from Princeton
Scientific Corp.) in a vacuum sealed quartz tube in presence of titanium wire.
For more details we refer to our previous work [11, 18].

Dielectric measurement: Temperature-dependent dielectric measurement was
performed in a close cycle cryostat using an impedance analyzer from Keysight
Technology Instruments (model No. E49908).

Transport measurement: All the transport measurements were carried out in an
ARS close cycle cryostat in van der Pauw geometry using a dc delta mode with a
Keithley 6221 current source and a Keithley 2182A nanovoltmeter and also using
standard low-frequency lock-in technique. Ohmic contacts were realized by
ultrasonically bonding aluminum wire or by attaching gold wire with silver
paint.

Light set-up: Light of the desired wavelength was passed through the optical
window of the cryostat. A home-built setup consisting of a diffuser and lens
was used to make light fall homogeneously over the sample (see inset of Fig.
2a ). Commercially available light-emitting diodes from Thor Labs were used as
a light source. Incident power on the sample was measured with a laser check
handheld power meter from coherent (Model No: 54-018).

HAXPES measurement: Near Fermi level and K 2p core level spectra were
collected at Hard X-ray Photoemission Spectroscopy (HAXPES) beamline (P22) of
PETRA III, DESY, Hamburg, Germany using a high-resolution Phoibos electron
analyzer [51]. Au Fermi level and Au 4f core level spectra collected on a gold
foil (mounted on the same sample holder) were used as a reference for making
the correction to the measured kinetic energy. The chamber pressure during the
measurement was $\sim$ 10-10 Torr. An open cycle Helium flow cryostat was used
to control the sample temperature.

Second harmonic generation measurement: SHG measurements were performed under
reflection off the sample at a 45-degree incidence angle. A $p$-polarised 800
nm beam from a Spectra-Physics Spirit-NOPA laser was used as the fundamental
beam (pulse width: 300 fs, repetition rate: 1 MHz), and was focused on the
sample surface. The $p$-polarised SHG intensity generated by the sample, was
measured by a photo multiplier tube. The sample temperature was controlled by
a helium-cooled Janis 300 cryostat installed with a heating element.

Density functional theory: The noncolinear density functional theory
calculations were carried out using the QUANTUM ESPRESSO package[52]. In this
calculation optimized norm-conserving pseudopotentials [53, 54, 55] were used
and for the exchange-correlation functional [56] we have incorporated Perdew,
Burke and Ernzerhof generalized gradient approximation (PBE-GGA). For the unit
cell, the Brillouin zone was sampled with $8\crossproduct 8\crossproduct 8$
$k$-points. The wave functions were expanded in plane waves with an energy up
to 90 Ry. Since the effect of SOC in KTaO3 is quite remarkable, we have
employed full-relativistic pseudopotential for the Ta atom. The structural
relaxations were performed until the force on each atom was reduced to 0.07
eV/$\mathrm{\SIUnitSymbolAngstrom}$. The Berry phase calculations are carried
out as implemented in the Wannier90 code [16, 58]. We have used a $41\times
41$ 2D k-mesh for the Berry curvature calculations. No significant change in
the result is observed on increasing the k-mesh up to $101\times 101$ (see
Supplementary Information Section 8 for further details).

## Data availability

The data that support the findings of this work are available from the
corresponding authors upon reasonable request.

## Acknowledgements

SM acknowledges SERB, India for a core research grant CRG/2022/001906.
Portions of this research were carried out at the light source PETRA III DESY,
a member of the Helmholtz Association (HGF). We would like to thank Dr.
Anuradha Bhogra and Dr. Thiago Peixoto for their assistance at beamline P22.
Financial support by the Department of Science & Technology (Government of
India) provided within the framework of the India@DESY collaboration is
gratefully acknowledged. S.H. and V.G. acknowledge support from the US
Department of Energy under grant no. DE-SC-0012375 for temperature-dependent
second-harmonic generation measurements. SB acknowledges support from SERB
(CRG/2022/001062), DST, India. SKG and MJ gratefully acknowledge Supercomputer
Education and Research Centre, IISc for providing computational facilities
SAHASRAT and PARAM-PRAVEGA. SKG acknowledges DST-Inspire fellowship
(IF170557). SKO acknowledges the wire bonding facility at the Department of
Physics, IISc Bangalore. SKO and SM thank Professor D. D. Sarma for giving
access to quartz tube sealing setup and dielectric measurement setup in his
lab. SKO also thanks Shivam Nigam for experimental assistance.

## Competing interests

The authors declare no competing interests.

Figure 1: Electron glass, quantum paraelectricity and temperature dependence
of electron’s mean free path in KTaO3-δ a. A schematic to describe the concept
of electron glass in a disordered insulator with highly localized electronic
states. Here the solid colored cones represent the random disorder potential
and yellow-filled circles represent electrons trapped in them. Disorder tries
to make the distribution of doped carriers random, however long-range Coulomb
interactions ($U_{1}$, $U_{2}$) try to make the distribution homogeneous
leading to electronic frustration. b. Temperature dependent dielectric
constant ($\epsilon$) of pristine (001) oriented KTaO3 single crystal recorded
at an AC frequency of 10 MHz. Sitting on the verge of quantum critical point,
KTaO3 is long known for its peculiar dielectric properties wherein the onset
of quantum fluctuation leads to a marked departure from classical paraelectric
behavior below 35 K leading to saturation of $\epsilon$ at low temperature.
This crossover entails an interesting non-classical $T^{2}$ dependence of the
inverse of $\epsilon$ below 35 K (shown in the inset) which can be taken as a
sign of the onset of quantum paraelectricity [10]. c. Unit cell of pristine
KTaO3 along with the transverse optical soft mode. d. In the Drude-Boltzman
picture, conductivity ($\sigma$) of metal is given by
$\sigma$=$n$$e$2$\tau$/$m$∗=$e$2($k_{F}$)2$l_{e}$/3$\pi$2$\hbar$ where $n$ is
the carrier density, $\tau$ is scattering time constant, $k_{F}$ is Fermi wave
vector, $m$∗ is the effective mass of electrons and $l_{e}$ is the electron’s
mean free path. $l_{e}$ calculated using this formula for a sample with
$n$=5.7$\crossproduct$1017 cm-3 has been shown by a blue curve. At such a
dilute limit, the Fermi surface has little anisotropy (see our previous work
[11]), keeping this in mind $k_{F}$ has been calculated by effectively mapping
the ellipsoidal Fermi surface onto a spherical Fermi surface. As evident from
the plot, above $\sim$140 K, $l_{e}$ becomes shorter than the inverse of
$k_{F}$ and the system enters into a bad metal phase. Further, at such dilute
density and with $m$∗=0.56 $m_{e}$ the Fermion degeneracy temperature is found
to be 68 K. Above this temperature, electrons become non-degenerate and the
modified $l_{e}$ is given by $l^{\prime}_{e}$$\sim$ $l_{e}$ $n$-1/3/$\Lambda$
[12] where $\Lambda$ is the de Broglie thermal wavelength given by
$\Lambda$=$h$/$\sqrt{2\pi m^{*}k_{B}T}$. The orange curve shows the calculated
$l_{e}$ for the non-degenerate case. As evident, this exercise only shifts the
curve to a little higher temperature, however, the conclusions remain the
same.

Figure 2: HAXPES and transport measurements under light a. Near Fermi level
electronic states measured at 40 K using hard X-ray photoemission spectroscopy
on the KTaO3-δ sample. Binding energy was corrected by taking Au Fermi level
as a reference. In the figure, Au Fermi level intensity has been reduced by
thousand times and shifted rightward for comparison. Inset shows a home-built
setup for transport measurement under light illumination (see methods for more
details). Green arrow shows the incoming photon which excites trapped
electrons to the conduction band (shown with slate-gray color). b. Temporal
evolution of resistance under green light illumination ($\lambda$ = 527 nm,
power = 145 $\mu$ watt) for 30 minutes measured at several fixed temperatures.
After 30 minutes, resistance relaxation was observed in dark for the next 1.5
hours. For comparative analysis, change in resistance has been converted into
relative percentage change ($\Delta$$R$/$R_{0}$)$\crossproduct$100\.
Additional measurements with red light have been shown in Supplementary
Information Section 2 and Supplementary Fig. 2. c. Temperature dependence of
stretching exponent ($\beta$) and relaxation time ($\tau$) obtained from
fitting of resistance relaxation in light off stage with a stretched
exponential function. We further emphasize that, while the value $\beta$ has
been obtained by fitting the data for 1.5 hours, the same value is obtained
even when the data is fitted for a longer period of up to 24 hours
(Supplementary Information Section 3 and Supplementary Fig. 3).

Figure 3: Aging phenomena a. Relative percentage change in resistance
($\Delta$$R$/$R_{0}$)$\crossproduct$100 measured at 15 K for different
$t$${}_{\text{ill}}$ ranging from 0.5 min to 30 min. Thereafter, resistance
relaxation was observed for the next 1.5 hours in every case. b. Temporal
evolution of resistance relaxation in off-stage for different values of
$t$${}_{\text{ill}}$. Since the amount of doping depends on the duration of
light illumination, the change in resistance in the off-stage
($\Delta$$R$${}_{\text{off}}$) has been normalized with a total drop in
resistance ($\Delta$$R$${}_{\text{ill}}$) at the end of illumination [19]. c.
Upon re-scaling the time axis with $t$${}_{\text{ill}}$, all the curves in
off-stage (except $t$${}_{\text{ill}}$ = 30 min) are found to fall on a
universal curve. Inset shows the linear relationship between $\tau$ and
$t$${}_{\text{ill}}$ which is the defining criteria for simple/full aging. d.
Simple aging observed for another sample at 15 K. Inset shows one
representative resistance relaxation data at 15 K.

Figure 4: Polar nano regions, second harmonic generation measurement and
density functional theory a. A schematic depicting real space random
distribution of polar nano regions in a highly polarizable host lattice. b. A
schematic of the experimental setup for measuring optical second harmonic
generation signal in reflection geometry (for more details see methods
section). c. Temperature-dependent SHG intensity measured on KTaO3-δ sample.
d. A portion of the relaxed structure of supercell of size
2$\crossproduct$2$\crossproduct$2 of KTaO3 with Ta off-centering along the
(100) and (110) direction around K vacancy. For more details, see the methods
section and Supplementary Information Section 8 and Supplementary Fig. 8. e.
Plot of Berry curvature $\Omega_{xy}(\mathbf{k})$ (shown in color-map) for
$k_{z}=0$ and bands (solid contours) intersecting the Fermi surface for the
system having an antisite-like Ta defect along (110) direction. f. Same as e.
for the system with an antisite-like Ta defect along (100) direction. The
inset shows the spin-orbit induced avoided crossing of bands yielding large
contributions to the Berry curvature. We get average polarization of 5.8 $\mu
C/cm^{2}$ and 11.1 $\mu C/cm^{2}$ for Ta2+ antisite-like defect along (110)
and (100) respectively for the supercell of size $2\crossproduct
2\crossproduct 2$. We also note that, we do not observe any polarization in
pristine KTaO3.

Figure 5: Theoretical calculations a. A schematic of band diagram considered
for computing inter band transitions when the conduction electron is coupled
to glassy background from randomly oriented polar nano regions. b. (Upper
panel) The glass correlation function $C_{gl}(t)$ vs. $t$ for three
temperatures for $\tau_{s}(T)=1$, $b=0.5$. (Lower panel) Plot of
$\tau_{\alpha}(T)\sim T^{-2.8}$ vs. $T$ (left y axis, in units of $\tau_{s}$)
and $B(T)$ (right y axis) vs. $T$ ($A=1-B$) for three temperatures. c. (upper
panel) The density-density correlation function of conduction electron
$C_{el}(t)$ vs. $t$ for three temperatures for flat ($W=0$) conduction band.
(lower panel) $C_{el}(t)$ vs. $t$ for three temperatures for semicircular
conduction band with $W=0.01$

.

Supplementary Information

List of contents

Section 1. Sheet resistance vs temperature plot.

Section 2. Experiment with red light.

Section 3. Long-time relaxation of persistence photo-resistance.

Section 4. Deviation of persistence photo-resistance relaxation time ($\tau$)
from activated behavior at low temperature.

Section 5. Dielectric loss in pristine KTaO3.

Section 6. Correspondence between the appearance of polar nano regions and
photo-doping effect.

Section 7. Presence of potassium vacancy.

Section 8. Berry phase calculations.

Section 9. Toy model of complex inter-band electronic relaxation mediated by a
glassy bath.

Supplementary Section 1: Sheet resistance vs temperature plot.

As depicted in Fig. 2b of the main text, shining light above 150 K does not
have any noticeable effect. This observation is further supported by the plot
shown in Supplementary Fig. 1, where we compare the $R_{S}$ vs $T$ curve
obtained in the dark taken in the cooling run with the measurement conducted
during the heating run after exposing the sample to green light (power= 145
$\mu$ Watt) for 30 minutes at 10 K. It is evident from the plot that the two
curves converge around 150 K, indicating a lack of any significant photo-
doping above 150 K.

Supplementary Figure 1: Temperature dependent sheet resistance
($R$${}_{\text{S}}$) of oxygen-deficient KTaO3 sample
($n$=5.7$\crossproduct$1017 cm-3) taken in a cooling run in dark (sky blue
curve). The slate-gray color curve shows data taken in a heating run after
shining light for 30 minutes at 10 K.

Supplementary Section 2: Experiment with red light.

Supplementary Figure 2: a. Temporal evolution of resistance under red light
illumination ($\lambda$ = 650 nm, power = 60 $\mu$ watt) for 30 minutes
measured at several fixed temperatures. After 30 minutes, resistance
relaxation was observed in dark for the next one hour. This measurement was
repeated for 3 consecutive cycles. For comparative analysis, change in
resistance has been converted into relative percentage change
($\Delta$$R$/$R_{0}$)$\crossproduct$100\. b. Temperature dependence of
stretching exponent ($\beta$) obtained from fitting of resistance relaxation
in light off stage with a stretched exponential function for all three cycles.
c. Temperature dependence of corresponding relaxation time ($\tau$) for all
three cycles.

In the main text, we have presented all the measurements conducted using the
green light. In this section, we provide an additional set of data obtained in
three consecutive cycles (see Supplementary Fig. 2a) using a red light of
wavelength $\lambda$ = 650 nm, power = 60 $\mu$ Watt). As evident from
Supplementary Fig. 2b and 2c, similar to the data obtained with green light,
we observe a constant value of stretching exponent ($\beta$) followed by a
significant enhancement in relaxation time ($\tau$) below 35 K. This
observation aligns with the findings from the measurements using green light
and suggests that the behavior observed is not specific to a particular
wavelength of light.

Supplementary Section 3. Long-time relaxation of persistence photo-resistance.

In Fig. 2c of the main text and Supplementary Fig. 2b, the value of the
stretching exponent was determined by fitting the resistance relaxation during
the off-stage, which lasted for 1.5 hours. To ensure that the resistance
relaxation was substantial enough to yield a reliable fitting, we conducted an
additional measurement where the resistance relaxation was observed for an
entire day (see Supplementary Fig. 3). It is important to emphasize that even
when the data is fitted for an extended period of up to 24 hours, the same
value of $\beta$ = 0.5 is obtained.

Supplementary Figure 3: Temporal evolution of resistance under green light
illumination ($\lambda$ = 527 nm, power = 145 $\mu$ Watt) for 30 minutes
measured at 15 K. After 30 minutes, resistance relaxation was observed in dark
for the next 24 hours. For comparative analysis, change in resistance has been
converted into relative percentage change
($\Delta$$R$/$R_{0}$)$\crossproduct$100\.

Supplementary Section 4. Deviation of persistence photo-resistance relaxation
time ($\tau$) from activated behavior at low temperature.

In the large lattice relaxation (LLR) model, the recombination of electron-
hole pairs occurs through the thermal excitation of electrons over an energy
barrier. This process is an activated process of the Arrhenius type. To
validate this model, we have plotted ln $\tau$ against 1000/$T$ in
Supplementary Fig. 4. It is evident from the plot that the data does not
exhibit the expected linear behavior at low temperatures signifying that the
LLR model can not account for our experimental observations.

Supplementary Figure 4:  Arrhenius plot of photo-resistance relaxation time
($\tau$). As evident, a clear deviation from activation behavior is observed
below 35 K.

Supplementary Section 5. Dielectric loss in pristine KTaO3.

In its ideal form, KTaO3 is perfectly centrosymmetric and hence, should not
possess any electric dipole moment. However, the presence of impurities and
disorder can disrupt the crystal’s inversion symmetry, leading to the
development of permanent electric dipoles [1, 2, 3, 4, 5]. In highly
polarizable host such as KTaO3, these dipoles polarize the surrounding lattice
leading to formation of polar nano regions (PNRs) [6]. Under an applied AC
electric field, these PNRs act as a source of dielectric losses, which can be
observed as a peak in the loss tangent (tan $\delta$ =
$\epsilon$${}^{{}^{\prime\prime}}$/$\epsilon$${}^{{}^{\prime}}$, where
$\epsilon$${}^{{}^{\prime}}$ and $\epsilon$${}^{{}^{\prime\prime}}$ are the
real and imaginary parts of the complex dielectric function, respectively).

Our temperature-dependent measurement of dielectric function indeed reveals
the presence of PNRs even in our pristine KTaO3 single crystal (see inset of
Supplementary Fig. 5). Further, frequency-dependent measurements reveal that
the dielectric loss is an activated process emphasizing that the PNRs in
pristine crystals are very dilute and independent and hence their relaxation
can be approximated by a single Debye-like process.

Supplementary Figure 5:  Arrhenius plot of dielectric relaxation frequency
($f$) for pristine KTaO3. Inset shows the dielectric loss tangent (tan
$\delta$) around 54 K at $f$ = 50 kHz.

Supplementary Section 6. Correspondence between the appearance of polar nano
regions and photo-doping effect.

In order to study the correlation between PNRs and photo-doping effect, we
have compared the temperature-dependent total SHG intensity with the fraction
of resistance (in terms of relative percentage change
($\Delta$$R$/$R_{0}$)$\crossproduct$100) which has not been recovered at the
end of 1.5 hours after turning off the light. We call this quantity the
persistence photo resistance denoted with the symbol $\delta$. See the inset
of the bottom panel of Supplementary Fig. 6 for the definition of $\delta$. As
evident, the appearance of finite signal in SHG exactly coincides with the
$\delta$, strongly signifying the direct role of PNRs behind the effective
electron-hole separation of our samples.

Supplementary Figure 6:  (upper panel) Temperature-dependent SHG intensity
measured on oxygen deficient KTaO3 sample. (lower panel) Temperature
dependence of persistence photo resistance ($\delta$) in terms of relative
percentage change at the end of 1.5 hours after turning off the light (see the
inset of the lower panel for the definition of $\delta$).

Supplementary Section 7. Presence of potassium vacancy.

To investigate the presence of potassium vacancy in our oxygen-deficient KTaO3
sample, K 2p core levels were collected at the Hard X-ray Photoemission
Spectroscopy (HAXPES) beamline (P22) at PETRA III, DESY. Supplementary Fig. 7a
shows one representative experimental data (recorded with an incident photon
energy of 5800 eV at room temperature) along with its fitting with the
convolution of Lorentzian and Gaussian functions. It is evident from the
figure that, in addition to the K 2p 3/2 peak at 292.65 eV arising from the
lattice, there is an extra peak appearing at 293.1 eV. It has been previously
shown that the presence of an additional peak at a higher binding energy is a
characteristic feature of a potassium vacancy in the system [7]. In order to
further understand the potassium vacancy profile in our sample, we have
carried out measurements with varying photon energy from 3500 eV to 5800 eV
(see Table I for the values of mean free path (MFP) and mean escape depth
(MED)). As evident from Supplementary Fig. 7b, there are hardly any
significant alterations in the line shape of the spectra as the photon energy
was increased. This observation indicates that the potassium vacancy profile
is homogeneous throughout the bulk of the sample.

Supplementary Figure 7: a Deconvolution of K2p core level spectrum (for oxygen-deficient KTaO3 sample) by using the convolution of Lorentzian and Gaussian function. This data was recorded at room temperature with photon energy $h$$\nu$ = 5800 eV. Yellow-filled circles denote the experimental data and a solid black line denotes the simulated spectra. Curves filled with gray and sky blue color correspond to the lattice potassium and potassium vacancy respectively. The dashed black line denotes the Shirley background and the red-filled curve corresponds to the difference between the experimental data and simulated spectra. b K 2p core level spectra recorded with varying incident photon energy. All the spectra were recorded at room temperature. Photon energy (eV) | MFP (nm) | MED (nm)
---|---|---
3400 | 5.1 | 15.3
4600 | 6.4 | 19.2
5800 | 7.7 | 23.1
Table 1: Table of mean free path (MFP) and mean escape depth (MED) with
increasing photon energy. MFP was calculated from the formula $m$(K.E.)γ where
K.E. is the kinetic energy of the ejected electrons and the values of $m$ and
$\gamma$ were taken to be 0.12 and 0.75 respectively [8]. Further, MED was
roughly assumed to be 3 times MFP which would roughly account for 95 percent
of the total collected intensity.

Supplementary Section 8. Berry phase calculations.

We have investigated the probable mechanism for the realization of PNRs
theoretically using noncollinear density functional (DFT) theory. Antisite
defect in perovskite materials like SrTiO3 (STO) and complex perovskite oxides
(Ca, Sr)3Mn2O7 is known for causing macroscopic polarization in the system [9,
10]. Due to the similarities in the electronic properties of STO and KTO, we
expect KTO to develop polarization due to antsite defects. We have created Ta
antisite defect in a supercell of size $2\crossproduct 2\crossproduct 2$
conventional unit cell. In our calculations, the Ta off-centering is
considered along (100) and (110) directions. The relaxed structures are shown
in Supplementary Fig. 8a. In Supplementary Fig. 8b we have shown the band
structure of the system with Ta antisite defect along (110). The red solid
lines are the bands calculated using DFT within PBE-GGA[11]. In order to
calculate the macroscopic polarization in the system, we use the modern theory
of polarization. The change in electronic contribution to the polarization
$\Delta P$ is defined as [12, 13]

$\Delta P=-\dfrac{e}{2\pi}\phi$ (2)

where $\phi$ is the Berry phase, which is the integral of the Berry curvature
over a surface $S$ bounded by a closed path in $k-$space, i.e., [14]

$\phi=\int_{S}\Omega(\mathbf{k})d\mathbf{k}$ (3)

One can calculate Berry curvature using Bloch states $u_{n}(\mathbf{k})$ as

$\Omega_{\alpha\beta}(\mathbf{k})=\sum_{n}f_{n}(\mathbf{k})\Omega_{n,\alpha\beta}(\mathbf{k})=\sum_{n}-2f_{n}(\mathbf{k})\text{Im}\innerproduct{\partialderivative{u_{n}(\mathbf{k})}{k_{\alpha}}}{\partialderivative{u_{n}(\mathbf{k})}{k_{\beta}}}$
(4)

where $\alpha,\beta$ are the cartesian indices and $f_{n}(\mathbf{k})$ is the
occupation number of the state $n$. Ta antisite defects in KTO have a
partially occupied band. As it well known, taking derivatives of
$u_{n}(\mathbf{k})$ in a finite-difference scheme in the presence of band
crossing and avoided crossing becomes difficult. We followed the procedure
described by X. Wang et. al. [15] for the calculation of Berry curvature using
Wannier functions as implemented in the Wannier90 code [16]. In Supplementary
Fig. 8b, we have shown the Wannier interpolated bands in black dotted lines.
From the figure, it is clear that both the set of bands calculated using DFT
and Wannier interpolation match very well near the Fermi level. In
Supplementary Fig. 8c, we have shown the total Berry curvature for the
antisite Ta defect along (110) in the plane $k_{z}=0$ calculated using 2D
k-mesh of size $41\crossproduct 41$ (top panel) and $101\crossproduct 101$
(bottom panel). While there are small quantitative differences, the
qualitative features in both the panels remain the same.

Supplementary Figure 8:  a. The relaxed structure of KTaO3 (KTO) with Ta off-
centering along the (100) and (110) direction around K vacancy (antisitelike
Ta defect) for a supercell of size 2$\crossproduct$2$\crossproduct$2\. b.
Bandstructure plot for (110) direction antisitelike Ta defect calculated using
density functional theory within PBE-GGA (shown in red solid lines) and
Wannier interpolated bands (shown in black dotted lines). The blue solid line
is the Fermi energy. c. Plot of total Berry curvature in the plane $k_{z}=0$
(in log scale) for the (110) antisitelike Ta defect structure calculated on a
2D k-mesh of sample-size $41\crossproduct 41$ (top panel) and
$101\crossproduct 101$ (bottom panel).

Supplementary Section 9. Toy model of complex inter-band electronic relaxation
mediated by a glassy bath.

As discussed in the main text, we consider the following model of coupled
electron (el)-glass (gl) system.

$\displaystyle H$ $\displaystyle=H_{el}+H_{gl}+H_{el-gl}$ (5a) $\displaystyle
H_{el}$
$\displaystyle=-\sum_{i,j=1}^{N_{c}}t_{ij}c_{i}^{\dagger}c_{j}-\varepsilon_{0}\sum_{\alpha=1}^{N_{f}}f_{\alpha}^{\dagger}f_{\alpha},~{}~{}~{}~{}(\varepsilon_{0}>0)$
(5b) $\displaystyle H_{gl}$
$\displaystyle=\sum_{\mu=1}^{N_{g}}\frac{p_{\mu}^{2}}{2m}+U(\\{x_{\mu}\\})$
(5c) $\displaystyle H_{el-gl}$
$\displaystyle=\sum_{i\alpha\mu}(V_{i\alpha\mu}c_{i}^{\dagger}f_{\alpha}+\mathrm{h.c.})x_{\mu}$
(5d)

The electronic part consists of a conduction band and a flat impurity band at
energy $-\varepsilon_{0}$. We consider three different energy dispersions for
the conduction band, corresponding to different lattices and hopping
amplitudes $t_{ij}$, namely – (1) a flat band or $\delta$ function density of
states (DOS) with bandwidth $W=0$, (2) a semicircular DOS,
$g(\epsilon)=(1/2\pi)\sqrt{W^{2}-\omega^{2}}\theta(W-|\omega|)$ with band
width $W$ [$\theta(x)$ is heaviside step function], e.g., corresponding to a
Bethe lattice, and (3) DOS corresponding to three-dimensional (3d) simple
cubic lattice with nearest-neighbour hopping $W/12$. The band gap between the
conduction band minimum and the impurity band is $\Delta=\varepsilon_{0}-W/2$.
We set the chemical potential at the center of the conduction band, i.e.,
$\mu=0$, as appropriate for a metallic system (see Fig. 5(a) of main text).

The Hamiltonian $H_{gl}$ for a set of particles with positions
$\\{x_{\mu}\\}$, and their canonically conjugate momenta $p_{\mu}$
($[x_{\mu},p_{\nu}]=\imath\delta_{\mu\nu}$ with $\hbar=1$) models the dynamics
of a local glassy background. The exact form of the inter-particle interaction
$U(\\{x_{\mu}\\})$ is not crucial for our calculations. However, as an
example, we take
$U(\\{x_{\mu}\\})=\sum_{\mu<\nu<\gamma}J_{\mu\nu\gamma}x_{\mu}x_{\nu}x_{\gamma}$
with the spherical constraint $\sum_{\mu}x_{\mu}^{2}=N_{g}$, corresponding a
well-known solvable model for glasses, namely the infinite-range spherical
$p$-spin glass model with $p=3$-spin coupling [17]. Here, $J_{\mu\nu\gamma}$
is real Gaussian random number with zero mean and variance
$\overline{J^{2}_{\mu\nu\gamma}}=3!J^{2}/2N_{g}^{2}$, where the overline
denotes averaging over different realizations of $J_{\mu\nu\gamma}$. The
particular scaling with $N_{g}$ ensures extensive free energy in the
thermodynamic limit $N_{g}\to\infty$. The $p$-spin glass model, with both
quantum [17] and classical dissipative [18] dynamics, undergoes a glass
transition at temperature $T_{g}\sim J$. For temperature $T\gtrsim T_{g}$, the
model gives rise to a supercooled liquid regime, like standard structural
glasses [19], with complex two-step relaxation for the dynamical correlation
function

$\displaystyle C_{gl}(t)$ $\displaystyle=\overline{\langle
x_{\mu}(t)x_{\mu}(0)\rangle}=Ae^{-|t|/\tau_{s}}+Be^{-(|t|/\tau_{\alpha})^{\beta}}$
(6)

, having a short-time exponential decay, and long-time stretched exponential
decay.

To obtain a solvable model for the coupled el-gl system, we also take the
coupling $V_{i\alpha\mu}$ between the glass, and the conduction and impurity
electronic states, infinite range. Here $V_{i\alpha\mu}$ is a complex Gaussian
random number with zero mean and variance
$\overline{|V_{i\alpha\mu}|^{2}}=V/(N_{c}N_{f}N_{g})^{1/3}$. The particular
scaling ensures a well-defined thermodynamic limit
$N_{c},N_{f},N_{g}\to\infty$ with finite ratios $p_{f}=N_{f}/N_{c}$ and
$p_{g}=N_{g}/N_{c}$. The ratios allow us tune the backaction of one part of
the system on the other. For example, for simplicity, and to gain an
analytical understanding, as discussed below, we take $p_{f},p_{g}\gg 1$, such
that the back actions of the conduction electrons on the impurity electrons
and the glass are negligible. We expect our main conclusions to be valid for
$p_{f},p_{g}\approx 1$, i.e., when the backaction is substantial.

For the above-disordered model (Supplementary Eq. (5)), using the standard
replica method [20, 17], we calculate the disorder-averaged connected
equilibrium dynamical density-density correlation function for the conduction
electrons, $C_{el}(t)=\overline{\langle n_{i}(t)n_{i}(0)\rangle}-\langle
n_{i}(0)\rangle^{2}$, at temperature $T\ll\Delta$. Here
$n_{i}=c_{i}^{\dagger}c_{i}$ is the conduction electron density. The
correlation function $C_{el}(t)$ captures the relaxation of the thermally
excited electrons in the conduction band at a finite temperature.

We obtain the replicated partition function,
$Z^{n}=\int\mathcal{D}(\bar{c},c)\mathcal{D}(\bar{f},f)\mathcal{D}x\exp(-S[\bar{c},c,\bar{f},f,x])$,
as coherent-state imaginary-time path integral over the fermionic Grassmann
fields $\\{\bar{c}_{ia},c_{ia},\bar{f}_{\alpha a},f_{\alpha a}\\}$ and
position variables $\\{x_{ia}\\}$, with replica index $a=1,\cdots,n$ and the
action,

$\displaystyle S_{n}$
$\displaystyle=\int_{0}^{1/T}d\tau\sum_{ij,a}\bar{c}_{ia}(\tau)[(\partial_{\tau}-\mu)\delta_{ij}-t_{ij}]c_{ja}(\tau)+\int_{0}^{1/T}d\tau\sum_{\alpha,a}\bar{f}_{\alpha
a}(\tau)(\partial_{\tau}-\mu-\varepsilon_{0})f_{\alpha a}(\tau)$
$\displaystyle+\int_{0}^{1/T}d\tau\bigg{[}\frac{1}{2}\sum_{\mu,a}[m({\partial_{\tau}x_{\mu
a}})^{2}+zx_{\mu a}^{2}]+\sum_{\mu\nu\gamma,a}J_{\mu\nu\gamma}x_{\mu
a}(\tau)x_{\nu a}(\tau)x_{\gamma a}(\tau)\bigg{]}$
$\displaystyle+\int_{0}^{1/T}d\tau\sum_{i\alpha\mu,a}[V_{i\alpha\mu}\bar{c}_{ia}(\tau)f_{\alpha
a}(\tau)x_{\mu a}(\tau)+\mathrm{h.c.}],$ (7)

where the Lagrange’s multiplier $z(\tau)$ imposes the spherical constraint on
$x_{\mu a}(\tau)$. After averaging over the distributions of
$J_{\mu\nu\gamma}$ and $V_{i\alpha\mu}$, we obtain $\overline{Z^{n}}$ and the
corresponding action as

$\displaystyle\tilde{S}_{n}$
$\displaystyle=\int_{0}^{1/T}d\tau\sum_{ij,a}\bar{c}_{ia}(\tau)[(\partial_{\tau}-\mu)\delta_{ij}-t_{ij}]c_{ja}(\tau)+\int_{0}^{1/T}d\tau\sum_{\alpha,a}\bar{f}_{\alpha
a}(\tau)(\partial_{\tau}-\mu-\varepsilon_{0})f_{\alpha a}(\tau)$
$\displaystyle+\frac{1}{2}\int^{1/T}_{0}d\tau\sum_{\mu a}x_{\mu
a}(\tau)(-m\partial^{2}_{\tau}+z)x_{\mu a}(\tau)+N_{c}\int d\tau
d\tau^{\prime}\sum_{ab}\bigg{[}V^{2}(p_{f}p_{g})^{1/3}G_{ab}(\tau,\tau^{\prime})\mathcal{G}_{ba}(\tau^{\prime},\tau)Q_{ab}(\tau,\tau^{\prime})$
$\displaystyle-\frac{J^{2}p_{g}}{4}Q^{3}_{ab}(\tau,\tau^{\prime})\bigg{]},$
(8)

where we have introduced the large $N$-fields,

$\displaystyle G_{ab}(\tau,\tau^{\prime})$
$\displaystyle=-\frac{1}{N_{c}}\sum_{i}c_{ia}(\tau)\bar{c}_{ib}(\tau^{\prime})$
(9a) $\displaystyle\mathcal{G}_{ab}(\tau,\tau^{\prime})$
$\displaystyle=-\frac{1}{N_{f}}\sum_{\alpha}f_{\alpha a}(\tau)\bar{f}_{\alpha
b}(\tau^{\prime})$ (9b) $\displaystyle Q_{ab}(\tau,\tau^{\prime})$
$\displaystyle=\frac{1}{N_{g}}\sum_{\mu}x_{\mu a}(\tau)x_{\mu
b}(\tau^{\prime})$ (9c)

To promote the above as fluctuating dynamical fields, conjugate fields
$\Sigma_{ba}(\tau^{\prime},\tau)$, $\sigma_{ba}(\tau^{\prime},\tau)$ and
$\Pi_{ab}(\tau,\tau^{\prime})$ are introduced for $G,~{}\mathcal{G}$ and $Q$,
respectively, e.g., by using the relation

$\displaystyle\int\mathcal{D}G\prod_{a\tau,b\tau^{\prime}}\delta(N_{c}G_{ab}(\tau,\tau^{\prime})+\sum_{i}c_{ia}(\tau)\bar{c}_{ib}(\tau^{\prime}))=\int\mathcal{D}G\mathcal{D}\Sigma
e^{-\int d\tau
d\tau^{\prime}\Sigma_{ba}(\tau^{\prime},\tau)[N_{c}G_{ab}(\tau,\tau^{\prime})+\sum_{i}c_{ia}(\tau)\bar{c}_{ib}(\tau^{\prime})]}=1$
(10)

As a result, we can now integrate out the fields $(\bar{c},c)$, $(\bar{f},f)$,
and $x$. Assuming replica diagonal ansatz, e.g.,
$G_{ab}(\tau,\tau^{\prime})=\delta_{ab}G(\tau,\tau^{\prime})$, we obtain
$\overline{Z^{n}}=\int\mathcal{D}(G,\mathcal{G},Q,\Sigma,\sigma,\Pi)e^{-nS_{eff}}$
and the effective action

$\displaystyle S_{\rm eff}$ $\displaystyle=-N_{c}\int d\epsilon
g(\epsilon)\Tr\ln\big(-\partial_{\tau}+\mu-\epsilon-\Sigma\big{missing})-N_{f}\Tr\ln\big(-\partial_{\tau}+\mu+\varepsilon_{0}-\sigma\big{missing})+$
$\displaystyle\frac{N_{g}}{2}\Tr\ln\big(-m\partial^{2}_{\tau}+z-\Pi\big{missing})-N_{c}\int
d\tau
d\tau^{\prime}\big{[}\Sigma(\tau,\tau^{\prime})G(\tau^{\prime},\tau)+p_{f}\sigma(\tau,\tau^{\prime})\mathcal{G}(\tau^{\prime},\tau)+\frac{p_{g}J^{2}}{4}Q(\tau,\tau^{\prime})^{3}$
$\displaystyle-\frac{p_{g}}{2}\Pi(\tau,\tau^{\prime})Q(\tau^{\prime},\tau)-V^{2}(p_{f}p_{g})^{1/3}G(\tau,\tau^{\prime})\mathcal{G}(\tau^{\prime},\tau)Q(\tau,\tau^{\prime})\big{]}$
(11)

In the large $N_{c}$ limit, the saddle point solution is obtained by varying
the above action with respect to $G,\Sigma,\mathcal{G},\sigma,Q,\Pi$ and
setting the variations to zero. Due to time-translation invariance at
equilibrium, e.g., $G(\tau,\tau^{\prime})=G(\tau-\tau^{\prime})$ and
$z(\tau)=z$. As a result, we can write the saddle point equations in the
following form after performing a Matsubara Fourier transform first, e.g.,
$G(\tau-\tau^{\prime})\to G(\imath\omega_{n})$ with fermionic Matsubara
frequency $\omega_{n}$, and then doing an analytical continuation,
$G(\imath\omega_{n})\to G_{R}(\omega+\imath 0^{+}$), to real frequencies,
where $G_{R}$ is the retarded Green’s function.

$\displaystyle G_{R}(\omega)$ $\displaystyle=\int d\epsilon
g(\epsilon)\frac{1}{\omega+\mu-\epsilon-\Sigma_{R}(\omega)}$ (12a)
$\displaystyle\mathcal{G}_{R}^{-1}(\omega)$
$\displaystyle=\omega+\mu+\varepsilon_{0}-\sigma_{R}(\omega)$ (12b)
$\displaystyle Q_{R}^{-1}(\omega)$
$\displaystyle=-m\omega^{2}+z-\Pi_{R}(\omega)$ (12c)
$\displaystyle\Sigma(\tau)$
$\displaystyle=V^{2}(p_{f}p_{g})^{1/3}\mathcal{G}(\tau)Q(\tau)$ (12d)
$\displaystyle\sigma(\tau)$
$\displaystyle=V^{2}p_{g}^{1/3}p_{f}^{-2/3}G(\tau)Q(\tau)$ (12e)
$\displaystyle\Pi(\tau)$
$\displaystyle=\frac{3J^{2}}{2}Q^{3}(\tau)+V^{2}p_{f}^{1/3}p_{g}^{-2/3}G(\tau)\mathcal{G}(-\tau)$
(12f)

In the limit $p_{f},p_{g}\gg 1$, we can approximate $\sigma(\tau)\approx 0$
and $\Pi(\tau)\approx(3J^{2}/2)Q^{3}(\tau)$, and neglect the back action of
the conduction electrons on the impurity states and the glass.

In this limit, following the numerical procedure similar to that in reference
[21, 22], we can numerically solve the above self-consistency equations for
various conduction electron DOS $g(\epsilon)$ to obtain the retarded functions
$G_{R}(\omega)$ and $Q_{R}(\omega)$, whereas the retarded Green’s function of
the impurity band is given by $\mathcal{G}_{R}(\omega)=(\omega+\imath
0^{+}+\mu+\varepsilon_{0})^{-1}$. However, instead of self-consistently
solving for $Q_{R}(\omega)$, we use the Supplementary Eq. (6) to obtain the
spectral function of the glass from the fluctuation-dissipation relation

$\displaystyle\rho_{gl}(\omega)$
$\displaystyle=-\frac{1}{\pi}\mathrm{Im}Q_{R}(\omega)=-\frac{1}{\pi}\tanh\left(\frac{\omega}{2T}\right)C_{gl}(\omega),$
(13)

where $C_{gl}(\omega)=\int_{-\infty}^{\infty}dte^{\imath\omega t}C_{gl}(t)$.
As a result, the conduction electron self-energy can be obtained as

$\displaystyle\Sigma_{R}(\omega)$ $\displaystyle=V^{2}\int
d\omega_{1}d\omega_{2}\rho_{f}(\omega_{1})\rho_{gl}(\omega_{2})\frac{n_{F}(\omega_{1})n_{B}(\omega_{2})-n_{F}(-\omega_{1})n_{B}(-\omega_{2})}{\omega-\omega_{1}-\omega_{2}+\imath
0^{+}},$ (14)

where $n_{F}(\omega)$ and $n_{B}(\omega)$ are Fermi and Bose functions,
respectively, and $\rho_{f}(\omega)=\delta(\omega+\varepsilon_{0})$ is the
spectral function of the impurity electrons. Here we have redefined
$V^{2}(p_{f}p_{g})^{1/3}$ as $V^{2}$. Thus, using $\Sigma_{R}(\omega)$ in
Supplementary Eq. (12a), we can obtain the conduction electron Green’s
function $G_{R}(\omega)$.

###  .1 Density density correlation function

To capture the manifestation of the glassy relaxation in the electron-hole
recombination process we look into the connected density-density correlator
$C_{el}(t)=\langle n_{i}(t)n_{i}(0)\rangle-\langle n_{i}(0)\rangle^{2}$, which
captures the relaxation of the thermally excited carriers at temperature $T$.
$C_{el}(t)$ can be obtained from the imaginary-time correlation function
$C_{el}(\tau)=\langle n(\tau)n(0)\rangle-\langle n(0)\rangle^{2}$, where
$n(\tau)=(1/N_{c})\sum_{i}c^{\dagger}_{i}(\tau)c(\tau)$. In the large-$N_{c}$
limit, the connected correlator is given by the bubble diagram and can be
expressed as $C_{el}(\tau)=G(\tau)G(-\tau)$. Performing Matsubara Fourier
transformation and then analytically continuing $i\Omega_{n}\to\omega+i0^{+}$,
where $\Omega_{m}$ is bosonic Matsubara frequency, we can obtain the retarded
correlator

$\displaystyle C_{el,R}(\omega)=\int
d\omega_{1}d\omega_{2}\rho_{c}(\omega_{1})\rho_{c}(\omega_{2})\bigg{(}\frac{n_{F}(\omega_{1})n_{F}(-\omega_{2})-n_{F}(-\omega_{1})n_{F}(\omega_{2})}{\omega_{1}-\omega_{2}-\omega_{n}-i0^{+}}\bigg{)},$
(15)

where $\rho_{c}(\omega)=-(1/\pi)\mathrm{Im}G_{R}(\omega)$ is the conduction
electron spectral function. Using the fluctuation-dissipation theorem, we can
relate $C_{el,R}(\omega)$ to the Fourier transform of the real-time density-
density correlation function $C_{el}(t)$ as
$C_{el}(\omega)=\coth(\omega/2T)\text{Im}C_{el,R}(\omega)$. Finally,
performing the inverse Fourier transform we obtain the real-time density-
density correlation function
$C_{el}(t)=\int^{\infty}_{-\infty}(d\omega/2\pi)e^{-i\omega t}C_{el}(\omega)$.
We show the results for the numerically computed $C_{el}(t)$ in Fig. 5(c) of
the main text for (1) a flat conduction band ($W=0$), and (2) a semicircular
conduction band DOS, using $C_{gl}(t)$ from Supplementary Eq. (6) (Fig. 5(b),
main text). For the latter, we take a temperature-independent stretching
exponent $\beta=0.5$ and the $\alpha$-relaxation time $\tau_{\alpha}(T)$,
which varies as $\sim T^{-2.8}$, consistent with our experimental results
(Fig. 2(c), main text). We also vary the coefficient $B(T)\leq 1$ such that it
increases with decreasing temperature, while $A(T)=1-B(T)$. This leads to a
more dominant long-time stretched exponential part in $C_{gl}(t)$ at lower
temperatures, compared to the short-time exponential decay in Supplementary
Eq. (6). In Supplementary Fig. 9, we show the results for the conduction band
energy dispersion corresponding to a nearest-neighbour tight binding model on
a simple cubic lattice. We can see that in all the cases the glassy relaxation
of the bath influences the relaxation of the conduction electrons leading
complex and temperature dependent relaxation profile for $C_{el}(t)$.

We can obtain a simple analytical understanding of the above results as
follows. From Supplementary Eq. (14), we can obtain for $\mu=0$

Supplementary Figure 9:  The density-density correlation function of
conduction electron $C_{el}(t)$ vs. $t$ for three temperatures corresponding
to a nearest-neighbour tight binding model on a simple cubic lattice

$\displaystyle\gamma(\omega)={\rm Im}\Sigma_{R}(\omega)$ $\displaystyle=-\pi
V^{2}\rho_{gl}(\omega+\varepsilon_{0})\big{(}n_{F}(-\varepsilon_{0})+n_{B}(-\varepsilon_{0}-\omega)\big{)}$
(16)

Thus, by defining $k(\omega)=\mathrm{Re}\Sigma_{R}(\omega)$, we obtain from
Eq.(12a)

$\displaystyle{\rm Im}G_{R}(\omega)$ $\displaystyle=\int d\epsilon
g(\epsilon)\frac{\gamma(\omega)}{(\omega-\epsilon-k(\omega))^{2}+\gamma(\omega)^{2}}$
(17)

As a result, from Supplementary Eq. (15) we get

$\displaystyle{\rm Im}C_{el,R}(\omega)$ $\displaystyle=\int
d\omega_{1}\rho_{c}(\omega_{1})\rho_{c}(\omega_{1}-\omega)\big{(}n_{F}(\omega_{1})-n_{F}(\omega_{1}-\omega)\big{)}$
(18)

The above can be expressed as

$\displaystyle{\rm Im}C_{R}(\omega)$ $\displaystyle=\frac{1}{\pi^{2}}\int
d\epsilon_{1}d\epsilon_{2}g(\epsilon_{1})g(\epsilon_{2})\int
d\omega_{1}\bigg{[}\frac{\gamma(\omega_{1})}{(\omega_{1}-\epsilon_{1}-k(\omega_{1}))^{2}+\gamma(\omega_{1})^{2}}$
$\displaystyle\frac{\gamma(\omega_{1}-\omega)}{(\omega_{1}-\omega-\epsilon_{2}-k(\omega_{1}-\omega))^{2}+\gamma(\omega_{1}-\omega)^{2}}\bigg{]}\big{(}n_{F}(\omega_{1})-n_{F}(\omega_{1}-\omega)\big{)}$
(19)

Thus,

$\displaystyle C_{el}(t)=\int_{-\infty}^{\infty}d\omega_{1}d\omega
e^{-\imath\omega
t}\rho_{gl}(\omega_{1}+\varepsilon_{0})\rho_{gl}(\omega_{1}+\varepsilon_{0}-\omega)F(\omega,\omega_{1})$
(20)

can be written as a convolution over the glass spectral function
$\rho_{gl}(\omega)$, where

$\displaystyle F(\omega,\omega_{1})$ $\displaystyle=\frac{V^{4}}{2\pi}\int
d\epsilon_{1}d\epsilon_{2}\frac{g(\epsilon_{1})g(\epsilon_{2})\coth\left(\omega/2T\right)[n_{F}(\omega_{1})-n_{F}(\omega_{1}-\omega)]}{[(\omega_{1}-\epsilon_{1}-k(\omega_{1}))^{2}+\gamma(\omega_{1})^{2}][(\omega_{1}-\omega-\epsilon_{2}-k(\omega_{1}-\omega))^{2}+\gamma(\omega_{1}-\omega)^{2}]}$
$\displaystyle\big{(}n_{F}(-\varepsilon_{0})+n_{B}(-\varepsilon_{0}-\omega_{1})\big{)}\big{(}n_{F}(-\varepsilon_{0})+n_{B}(-\varepsilon_{0}-\omega_{1}+\omega)\big{)}$
(21)

The spectral function of the glass contains the information about the multiple
time scales and the non-trivial temperature dependence of the glassy
relaxation. Thus if appropriate conditions on the electronic energy scales $W$
and $\varepsilon_{0}$ ($\Delta$) are made relative to the energy scales of the
glass, e.g., $\tau_{s}^{-1}$ and $\tau_{\alpha}^{-1}$, then the complex
relaxation of the conduction electrons can be obtained. We numerically find
that these conditions are met if the glassy bath is broad, i.e., the bandwidth
of the glass is comparable or larger than the electronic energy scales.

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