††thanks: These authors contributed equally to this work††thanks: These
authors contributed equally to this work

# Dynamical Planckian scaling of charge response at a particle-hole-asymmetric
quantum critical point with Kondo destruction

Ananth Kandala  Department of Physics, University of Florida, Gainesville,
Florida, 32611-8440, USA  Haoyu Hu  Department of Physics & Astronomy, Rice
Center for Quantum Materials, Rice University, Houston, Texas 77005, USA
Qimiao Si<EMAIL_ADDRESS>Department of Physics & Astronomy, Rice Center for
Quantum Materials, Rice University, Houston, Texas 77005, USA  Kevin Ingersent
<EMAIL_ADDRESS>Department of Physics, University of Florida, Gainesville,
Florida, 32611-8440, USA

###### Abstract

Metallic quantum criticality is a central theme in a variety of strongly
correlated systems. Recent experiments have raised the fundamental question of
how the charge response can be singular in cases where the Landau framework of
quantum criticality allows singularity only in the spin channel. Motivated by
this emerging issue, we study the particle-hole-asymmetric regime of a Bose-
Fermi Anderson model with power-law forms for both the bosonic bath spectrum
and the fermionic band density of states. We realize a particle-hole-
asymmetric quantum-critical state where quasiparticles are lost due to a
critical destruction of Kondo screening, and demonstrate a dynamical Planckian
scaling of the charge response. Implications for a new regime of heavy-fermion
quantum criticality and for Mott-Hubbard systems are discussed.

Introduction. Strange metals with a complete absence of quasiparticles are of
extensive current interest in a variety of strongly correlated systems Keimer
and Moore (2017); Paschen and Si (2021); Phillips  _et al._ (2022). Such
states often arise near a quantum critical point (QCP) where one phase
continuously transforms into another at absolute zero. The canonical case of
antiferromagnetic (AF) metallic QCPs is traditionally formulated within the
Landau framework of order-parameter fluctuations Hertz (1976); Millis (1993).
The majority of the Fermi surface does not experience the quantum-critical
fluctuations at the ordering wave vector, so the associated quasiparticles
retain their integrity. It is thus necessary to look beyond the Landau
framework to understand strange metals. An alternative paradigm incorporates
additional quantum-critical degrees of freedom that destroy quasiparticles
over the entire Fermi surface Si  _et al._ (2001); Coleman  _et al._ (2001);
Senthil  _et al._ (2004). This happens when the so-called “large” Fermi
surface loses its quasiparticle weight and jumps to a “small” Fermi surface as
the system crosses the QCP; since the quantum phase transition is continuous,
the quasiparticle weight vanishes everywhere on the Fermi surface. A variety
of experiments, in a growing number of AF heavy-fermion metals Paschen  _et
al._ (2004); Gegenwart  _et al._ (2007); Friedemann  _et al._ (2010); Shishido
_et al._ (2005); Park  _et al._ (2006); Knebel  _et al._ (2008); Custers  _et
al._ (2012); Martelli  _et al._ (2019); Schröder  _et al._ (2000); Kirchner
_et al._ (2020), provide evidence for this class of Kondo-destruction QCP Si
_et al._ (2001); Coleman  _et al._ (2001); Senthil  _et al._ (2004).

A particularly intriguing recent development is the discovery of a singular
critical charge response in YbRh2Si2 with dynamical Planckian
($\hbar\omega/k_{\text{B}}T$) scaling Prochaska  _et al._ (2020). This
behavior occurs near a QCP between AF and paramagnetic metallic phases, so it
is incompatible with the Landau framework Hertz (1976); Millis (1993), in
which only the fluctuations of the order-parameter (here, the spin channel)
should be singular. Since YbRh2Si2 was one of the first heavy fermions to
provide experimental evidence for a sudden jump of the Fermi surface across
the QCP Paschen  _et al._ (2004); Gegenwart  _et al._ (2007); Friedemann  _et
al._ (2010), it is natural to associate the charge-channel singularity with
the beyond-Landau nature of the quantum criticality. Such a charge-channel
singularity is indeed found theoretically near Kondo-destruction transitions
in Kondo-limit models with SU(2) symmetry Cai  _et al._ (2020) or very large
spin degeneracy Zhu  _et al._ (2004); Komijani and Coleman (2019). It has been
argued that the singular charge response is important both for strange-metal
behavior Prochaska  _et al._ (2020) and emergent high-$T_{\rm c}$
superconductivity Hu  _et al._ (2021), making it important to establish the
generality of the phenomenon. In the Kondo limit, which involves local moments
that are necessarily particle-hole-symmetric, the charge-channel singularity
is ultimately traced to the fate of the local moments, and thus develops in
the same energy range as the singular spin response Schröder  _et al._ (2000);
Grempel and Si (2003); Zhu  _et al._ (2003); Glossop and Ingersent (2007); Zhu
_et al._ (2007). What happens beyond away from this symmetric limit is an
intriguing open question.

Recent experiments have pointed to a novel regime of quantum-critical heavy-
fermion metals with inherent particle-hole asymmetry. The quasi-kagome
materials CeRhSn and CeIrSn show strange-metal behavior Kim  _et al._ (2003);
Tsuda  _et al._ (2018), including a Grüneisen ratio (of the thermal expansion
to the specific heat) that diverges with an unusual exponent Tokiwa  _et al._
(2015), suggesting an entirely new universality class. Important clues about
this quantum criticality are that CeRhSn is mixed valent Sundermann  _et al._
(2021), implicating the involvement of the charge degrees of freedom, but the
differing responses to in-plane and out-of-plane pressure reveal local-moment
degrees of freedom still to be well-defined Shimura  _et al._ (2021). These
systems motivate study of the interplay between intersite RKKY interactions
and the local Kondo effect in entangled spin and charge channels. Among the
important open questions are whether there is a complete loss of
quasiparticles, whether the charge and spin responses are both singular and,
if so, whether the two singularities can acquire different energy scales, as
indicated experimentally Shimura  _et al._ (2021). Mixed valence is also
expected to be important for other quantum-critical heavy-fermion metals such
as $\beta$-YbAlB4 Matsumoto  _et al._ (2011) and, likely, the $5f$-electron
system PuCoGa5 that superconducts with $T_{c}=18.5$ K (a record high for
$f$-electron systems) Sarrao  _et al._ (2002); Ramshaw  _et al._ (2015).
Theoretically, the role of valence fluctuations in heavy-fermion quantum
criticality has been treated within the Landau framework Miyake and Watanabe
(2014), but it has yet to be systematically studied in the context of beyond-
Landau quantum criticality.

The relevant physics is captured by the Anderson lattice model in the
particle-hole-asymmetric regime, incorporating RKKY interactions between the
local spin degrees of freedom whose importance has been deduced from the
aforementioned anisotropic pressure response of CeRhSn Shimura  _et al._
(2021). A particularly powerful approach to elucidate the competing phases and
quantum criticality of this model is extended dynamical mean-field theory Si
and Smith (1996); Smith and Si (2000); Chitra and Kotliar (2000), in which the
Bose-Fermi Anderson (BFA) model appears as an effective Hamiltonian 111See
Supplemental Material for certain technical details and additional results..
In this work, we study BFA models with a sub-ohmic bosonic-bath spectrum and a
fermionic band that has a power-law pseudogap in its density of states. We
realize an inherently particle-hole-asymmetric quantum-critical state where
quasiparticles are lost due to a critical destruction of Kondo screening, and
demonstrate a dynamical Planckian ($\hbar\omega/k_{\text{B}}T$) scaling of the
charge response.

Model and solution methods. The BFA model is described by the Hamiltonian

$\displaystyle H_{\text{BFA}}$
$\displaystyle=\epsilon_{d}\sum_{\sigma}n_{d\sigma}+Un_{d\uparrow}n_{d\downarrow}$
$\displaystyle\;+\sum_{\mathbf{k},\sigma}\epsilon_{\mathbf{k}}c_{\mathbf{k}\sigma}^{{\dagger}}c_{\mathbf{k}\sigma}+\frac{V}{\sqrt{N_{\mathbf{k}}}}\sum_{\mathbf{k},\sigma}(d_{\sigma}^{{\dagger}}c_{\mathbf{k}\sigma}+\text{H.c.})$
(1)
$\displaystyle\;+\sum_{q,\mu}\omega_{\mathbf{q}}\phi_{\mathbf{q}\mu}^{{\dagger}}\phi_{\mathbf{q}\mu}+\sum_{\mu}\frac{g_{\mu}S_{d}^{\mu}}{\sqrt{N_{\mathbf{q}}}}\sum_{\mathbf{q}}(\phi_{\mathbf{q}\mu}^{{\dagger}}+\phi_{-\mathbf{q}\mu}),$

where $d_{\sigma}^{{\dagger}}$ creates an impurity electron with energy
$\epsilon_{d}$ and spin $z$ component
$\sigma=\pm\frac{1}{2}\equiv\>\uparrow,\downarrow$; $c_{k,\sigma}^{\dagger}$
creates a conduction electron with wave vector $\mathbf{k}$, energy
$\epsilon_{\mathbf{k}}$, and spin $z$ component $\sigma$;
$\phi_{q\mu}^{{\dagger}}$ creates a boson in bath $\mu\in\\{x,y,z\\}$ with
wave vector $\mathbf{q}$ and energy $\omega_{\mathbf{q}}$; and
$S_{d}^{\mu}=\sum_{\sigma,\sigma^{\prime}}d_{\sigma}^{{\dagger}}\frac{1}{2}\tau_{\sigma\sigma^{\prime}}^{\mu}d_{\sigma^{\prime}}$
is Cartesian component $\mu$ of the impurity spin, $\tau^{\mu}$ being a Pauli
matrix. Other quantities entering Eq. (Dynamical Planckian scaling of charge
response at a particle-hole-asymmetric  quantum critical point with Kondo
destruction) are the Coulomb interaction $U$ between electrons in the impurity
level, the local hybridization $V$ between the impurity and conduction
electrons, and the coupling $g_{\mu}$ between component $\mu$ of the
impurity’s spin and the corresponding bosonic bath. $N_{\mathbf{k}}$ and
$N_{\mathbf{q}}$ are the number of $\mathbf{k}$ and $\mathbf{q}$ points,
respectively. We take the electronic and bosonic densities of states to have
the power-law forms

$\displaystyle\rho_{f}(\epsilon)$
$\displaystyle=\frac{1}{N_{\mathbf{k}}}\sum_{\mathbf{k}}\delta(\epsilon-\epsilon_{\mathbf{k}})=\rho_{0,c}\Bigl{|}\frac{\epsilon}{D}\Bigr{|}^{r}c\biggl{(}\frac{|\epsilon|}{D}\biggr{)},$
(2) $\displaystyle\rho_{b}(\omega)$
$\displaystyle=\frac{1}{N_{\mathbf{q}}}\sum_{q}\delta(\omega-\omega_{\mathbf{q}})=\rho_{0,b}\Bigl{|}\frac{\omega}{D}\Bigr{|}^{s}\Theta(\omega)\,c\biggl{(}\frac{|\omega|}{D}\biggr{)},$
(3)

where $D$ is an overall energy scale that we henceforth set to 1, $\rho_{0,b}$
and $\rho_{0,c}$ are normalization factors, $\Theta(x)$ is the Heaviside
function, and $c(x)$ is a cutoff function such that $c(x)\to 1$ for $x\ll 1$
and $c(x)\to 0$ for $x\to\infty$. The exponent $r>0$ creates a power-law
pseudogap in the band density of states around the Fermi energy $\epsilon=0$,
while the exponent $s<1$ describes a sub-ohmic dissipative bath. The quantum-
critical behavior of the purely-fermionic power-law Anderson model Withoff and
Fradkin (1990); Bulla  _et al._ (1997); Gonzalez-Buxton and Ingersent (1998);
Logan and Glossop (2000) has been extensively investigated Ingersent and Si
(2002); Fritz and Vojta (2004); Pixley  _et al._ (2012); Chowdhury and
Ingersent (2015), but the previous study of its Bose-Fermi counterpart Pixley
_et al._ (2013) primarily focused on the particle-hole-symmetric limit. Here
we present results obtained at particle-hole asymmetry using the recently-
developed continuous-time quantum Monte Carlo (CTQMC) method for the spin-
isotropic model Cai and Si (2019) (which built on the general method Gull  _et
al._ (2011); Otsuki (2013)) as well as that for the Ising-anisotropic model
Pixley  _et al._ (2011, 2013), and using the numerical renormalization-group
(NRG) method Bulla  _et al._ (2008) for the anisotropic case.

This paper focuses on the illustrative case of powers $r=0.6$ and $s=0.9$ with
$\Gamma_{0}=\pi\rho_{0,c}V^{2}=0.1$, $U=0.03$, and $\epsilon_{d}\neq-U/2$
leading to breaking of particle-hole symmetry. We study (a) an Ising-
anisotropic model with $g_{\mu}=g\delta_{\mu,z}$, describing bosons coupling
to just one component of the impurity spin, and (b) an SU(2)-symmetric model
with $g_{\mu}=g$ involving three equivalent bosonic baths $\mu=x$, $y$, $z$.
Both variants of the model have been solved using CTQMC with
$\epsilon_{d}=-0.05$ and a cutoff function $g(x)=(2\pi)^{-1/2}\exp(-x^{2}/2)$.
We have also solved the Ising version using the NRG with $c(x)=\Theta(1-x)$
and discretization $\Lambda=9$; in these calculations, we generally chose
$\epsilon_{d}\simeq-0.025$ to achieve entry into the quantum-critical regime
at the lowest iteration number (or highest temperature) for the fixed $U$ and
$\Gamma_{0}$ values. The NRG and CTQMC are complementary in that the former
method can reliably compute static quantities arbitrarily close to absolute
temperature $T=0$ and dynamics in the regime $|\omega|\gtrsim T$ 222From this
point on, we adopt units where $D=\hbar=k_{\text{B}}=g\mu_{\text{B}}=1$.,
while the latter provides dynamical properties over all $\omega/T$ and is
practical to apply in the SU(2)-symmetric case more relevant to many heavy-
fermion materials. We show that, despite their differing cutoffs, CTQMC and
the NRG yield a consistent description of universal critical properties.

Figure 1: (a) A low-lying NRG eigenvalue $E_{N}$ vs odd iteration number $N$
for bosonic couplings $g$ close to $g_{c}$ in the Ising BFA model. For
$9\lesssim N\lesssim N^{*}(g)$, $E_{N}$ remains close to its value at the
Kondo-destruction QCP. (b) Binder cumulant $U_{4,s}(\beta,g)$ vs $g$ for the
SU(2) BFA model. (c) Optimized scaling collapse of the data from (b) using
$g_{c}=0.183(1),\nu^{-1}=0.684(15)$, $A=0.56$, and $\phi=0.50$. Legends are
shared between (b), (c).

Loss of quasiparticles. When the Kondo effect occurs, the low-energy physics
is characterized by quasiparticles with small but nonzero weight. The
quasiparticle weight $Z$ is proportional to the energy scale associated with
the development of the composite fermions. We start by searching for a Kondo-
destruction QCP in the Ising-anisotropic BFA model using the NRG method. The
NRG provides a discrete approximation to the many-body spectrum at a sequence
of energy scales $D_{N}\simeq D\Lambda^{-N/2}$, describing the essential
physics at temperatures $T\simeq D_{N}$ for $N=0$, $1$, $2$, $\ldots$. We
locate the Kondo-destruction QCP by determining for trial values of $g$
whether the spectrum approaches, for iterations $N\gtrsim N^{*}(g)$, that of
the particle-hole-asymmetric Kondo fixed point (for $g<g_{c}$) or the bosonic-
localized (Kondo-destroyed) fixed point (for $g>g_{c}$); see Fig. 1(a). A
process of bisection on the $g$ axis increases $N^{*}$ and allows $g_{c}$ to
be determined to one part in $10^{12}$ or better. At $g_{c}$, the energy scale
for the development of the well-defined Kondo singlet has gone to zero. This
means that the quasiparticle weight $Z$ vanishes. The loss of quasiparticles
is also captured in the NRG many-body spectrum at $g_{c}$ Note (1), which no
longer has a Fermi-liquid description.

The Kondo-destruction QCP can be located within the CTQMC approach by
constructing the Binder cumulant $U_{4,s}=\langle M_{z}^{4}\rangle/\langle
M_{z}^{2}\rangle$, where $\langle
M_{z}^{n}\rangle=\langle[\beta^{-1}\int_{0}^{\beta}S_{d}^{z}(\tau)d\tau]^{n}\rangle$
with $\beta=1/k_{B}T$. Near a QCP at $g=g_{c}$, one expects
$U_{4,s}(\beta,g)=f(\beta^{1/\nu}(g-g_{c})/g_{c}+A\beta^{-\phi/\nu})$, where
$f(x)$ is a scaling function and $A\beta^{-\phi/\nu}$ represents the
subleading correction to scaling. The Ising-anisotropic model is addressed in
Fig. S1, while Fig. 1(b) shows raw data for the SU(2)-symmetric model. One can
identify $g_{c}$ as the point where $U_{4,s}$ vs $g$ curves for different
values of $\beta$ cross at a single point. Figure 1(c) presents the result of
optimizing the scaling collapse of the data from Fig. 1(b) with respect to
trial values of $g_{c}$, $\nu$, $A$, and $\phi$. As we will show in the next
section, there is no Pauli form at $g_{c}$ for either the spin or charge
susceptibility even at vanishingly small temperatures. This provides an
alternative way to characterize the absence of Landau quasiparticles at the
Kondo-destruction QCP.

Figure 2:  Temperature dependence of static local susceptibilities at the
Kondo-destruction QCP: (a) spin and (b) charge susceptibilities for the Ising
and SU(2) BFA models, computed using CTQMC. The SU(2) charge data are
multiplied by $1.5$ for clarity. (c) and (d) respectively superimpose Ising
CTQMC spin and charge results on their NRG counterparts. All data for
$\epsilon_{d}=-0.05$.

Singular charge and spin responses. To probe the critical properties, we first
examine the static local spin and charge susceptibilities

$\displaystyle\chi_{s}(T)$ $\displaystyle=-\biggl{.}\frac{\partial\langle
S_{d}^{z}\rangle}{\partial h}\biggr{|}_{h=0}=\int_{0}^{\beta}\langle
S_{d}^{z}(\tau)S_{d}^{z}(0)\rangle\,d\tau,$ (4) $\displaystyle\chi_{c}(T)$
$\displaystyle=-\frac{\partial\langle
n_{d}\rangle}{\partial\epsilon_{d}}=\int_{0}^{\beta}\langle:\\!n_{d}(\tau)\\!:\;:\\!n_{d}(0)\\!:\rangle\,d\tau,$
(5)

where
$:\\!n_{d}\\!:\>=\sum_{\sigma}n_{d,\sigma}-\langle\sum_{\sigma}n_{d,\sigma}\rangle$
and $h$ is a local magnetic field that enters the Hamiltonian through a term
$hS_{d}^{z}$.

As shown in Fig. 2, at the QCP both $\chi_{s}$ and $\chi_{c}$ have divergent
temperature dependences:

$\chi_{\alpha}(T)=C_{\alpha}\,T^{-x_{\alpha}}\quad\text{for }\alpha=s,\,c.$
(6)

CTQMC gives $x_{s}=0.80(1)$ for both the Ising and SU(2) BFA models [Fig.
2(a)], consistent within the numerical uncertainty with the NRG result
$x_{s}=0.7906(1)$ for the Ising BFA model. Figure 2(c), plotted over a much
wider temperature range, shows that CTQMC closely reproduces an NRG
$\chi_{s}(T)$ (calculated for the same values of $U$, $\epsilon_{d}$, and
$\Gamma_{0}$) that is already approaching its asymptotic low-temperature
behavior for $T\lesssim 0.01D$.

In the charge sector, CTQMC gives $x_{c}=0.44(3)$ for the Ising BFA model and
$x_{c}=0.47(2)$ for its SU(2) counterpart [Fig. (2)(b)], whereas the NRG Ising
BFA result is $x_{c}=0.1164(1)$. Figure 2(d) reveals that the CTQMC and NRG
data coincide closely, but $\chi_{c}(T)$ reaches its asymptotic regime only at
temperatures below those accessible to CTQMC. This behavior can be understood
by considering corrections to Eq. (6), i.e.,
$1/\chi_{\alpha}(T)=C_{\alpha}^{-1}\,T^{x_{\alpha}}+(C^{\prime}_{\alpha})^{-1}\,T^{x^{\prime}_{\alpha}}+\ldots$,
where $x^{\prime}_{\alpha}>x_{\alpha}$. The leading correction is smaller than
$\epsilon$ times the asymptotic term in Eq. (6) for temperatures $T<|\epsilon
C^{\prime}_{\alpha}/C_{\alpha}|^{1/[x^{\prime}_{\alpha}-x_{\alpha}]}$. The NRG
data yield $x^{\prime}_{s}=0.90(4)$, $C^{\prime}_{s}/C_{s}\approx 30$,
$x^{\prime}_{c}=0.24(1)$, and $C^{\prime}_{c}/C_{c}\approx 0.4$. The facts
that $x^{\prime}_{c}-x_{c}\ll x^{\prime}_{s}-x_{s}$ and
$C^{\prime}_{c}/C_{c}\ll C^{\prime}_{s}/C_{s}$ make it necessary to go to much
lower temperatures to access the asymptotic behavior of $\chi_{c}$.

Figure 3: Dynamical local spin susceptibility $\chi_{s}$ at the Kondo-
destruction QCP: (a) Real-frequency Ising NRG data at different temperatures
$T$ collapse when plotted vs $\omega/T$. (b) Imaginary-time SU(2) CTQMC data
at different inverse temperatures $\beta$ collapse with dynamical exponent
$0.2000(1)$ when plotted vs $\pi T/\sin(\pi\tau T)$. Colors in (b) are as
specified in Figs. 1(b) and 1(c).

Dynamical Planckian scaling of critical spin and charge responses. We now
consider the dynamical responses that accompany the loss of quasiparticles at
the Kondo-destruction QCP, beginning with the dynamical spin susceptibility,
which in imaginary time is $\chi_{s}(\tau,T)=\langle
S^{z}(\tau)S^{z}(0)\rangle-\langle S^{z}(0)\rangle^{2}$. Figure 3(a) of the
Ising case shows that real-frequency NRG data over 6 decades of temperature
and more than 8 decades of frequency collapse onto a single curve,
demonstrating a Planckian ($\omega/T$) scaling 333The NRG does not capture the
correct scaling form of the scaling function for $\omega\ll T$ due to
truncation errors; see, e.g., Ref. Ingersent and Si, 2002.. For $\omega\gg T$,
$\chi_{s}(\omega,T)\sim\omega^{-x_{s}}$ up to $\omega\simeq 0.01D$, where
deviations from scaling set in. Dynamical scaling also occurs in the SU(2)
case, as demonstrated by the imaginary-time dependence of the spin
susceptibility from CTQMC. The power-law dependence on $\pi T/\sin(\pi\tau
T)$, shown in Fig. 3(b), describes a scaling collapse of the susceptibility in
terms of $\omega_{n}/T$ Note (1).

Finally, we turn to the principal focus of this work, namely the dynamical
critical charge response probed by
$\chi_{c}(\tau,T)=\langle:\\!n_{d}(\tau)\\!:\;:\\!n_{d}(0)\\!:\rangle$. Figure
4(a) shows the real-frequency dynamics in the Ising case. NRG data spanning 6
decades of temperature and more than 10 decades of frequency show a clear
scaling collapse in terms of $\omega/T$. For $\omega\gg T$,
$\chi_{c}(\omega,T)\sim\omega^{-x_{c}}$, but corrections to scaling set in at
lower frequencies than for $\chi_{s}(\omega,T)$, consistent with the greater
prominence of subleading terms in $\chi_{c}(T)$ [see Fig. 2(d)]. This result
is also corroborated by the CTQMC result for the SU(2) case, as shown in Fig.
4(b). The imaginary-time dependence corresponds to an $\hbar\omega_{n}/k_{B}T$
scaling of the dynamical charge susceptibility. Moreover, the cutoff scale for
the scaling region is considerably different between the charge and spin
susceptibilities, demonstrating that the particle-hole asymmetry has liberated
the scaling of the charge response from its spin counterpart.

Figure 4: Dynamical local charge susceptibility $\chi_{c}$ at the Kondo-
destruction QCP: (a) Real-frequency Ising NRG data at different temperatures
$T$ collapse when plotted vs $\omega/T$. (b) Imaginary-time SU(2) CTQMC data
at different inverse temperatures $\beta$ collapse with dynamical exponent
$0.5728(2)$ when plotted vs $\pi T/\sin(\pi\tau T)$. Colors in (b) are as
specified in Figs. 1(b) and 1(c).

Discussion. Several remarks are in order. First, our results show remarkable
consistency between the quantum-critical behavior in the particle-hole-
asymmetric Bose-Fermi Anderson model in the Ising and SU(2) cases. This is to
be contrasted with the particle-hole-symmetric case, where a recent study
shows that the SU(2) model can have important differences from its Ising
counterpart Cai and Si (2019). The consistency between the two spin symmetries
allows us to reach a very complete understanding of the quantum-critical
behavior, demonstrating the loss of quasiparticles and uncovering dynamical
Planckian scaling. Our results motivate further studies for other combinations
of the key parameters of the model: the pseudogap power $r$ of the fermionic
band and the power-law exponent $s$ of the bosonic bath.

Second, the particle-hole-asymmetric Bose-Fermi Anderson model represents,
through the EDMFT framework, an effective description of the Anderson lattice
model in the mixed-valent regime. Our finding of the dynamical Planckian
scaling of the charge responses at the Kondo-destruction QCP implicates a
singular charge response in a new class of heavy-fermion QCPs. Optical
conductivity measurements in the quantum-critical regime of CeRhIn and CeIrIn
offer an important avenue for progress. Another way to elucidate this new
regime of heavy-fermion quantum criticality will be to ascertain the degree to
which the temperature and energy windows for dynamical scaling are different
between the charge and spin channels.

In the heavy-fermion compound $\beta$-YbBAl4, evidence has been advanced for a
critical charge mode in the quantum-critical regime Kobayashi  _et al._
(2022). Our work motivates the examination of the associated charge responses
as a function of both frequency and temperature.

Third, the Bose-Fermi Anderson model also characterizes the one-band Hubbard
model with explicit intersite exchange interactions, i.e., the $t$-$J$-$U$
Hubbard model Smith and Si (2000). Away from half-filling, the model is
particle-hole-asymmetric. A similar effective model also arises when the $J$
interactions are randomly distributed Chowdhury  _et al._ (2021); Joshi  _et
al._ (2020). In this way, our determination of the dynamical Planckian scaling
of the charge response sheds light on both the singular density fluctuations,
as observed in the electron energy-loss spectrum (EELS) in Bi-2212 Mitrano
_et al._ (2018), and the related singularities as revealed by the
$\hbar\omega/k_{\text{B}}T$ scaling of the optical conductivity van der Marel
_et al._ (2003).

Summary. We have shown that the Bose-Fermi Anderson model serves as an
exemplary setting for particle-hole-asymmetric quantum criticality, in which
the quasiparticles are lost due to critical Kondo destruction. We have
uncovered a dynamical Planckian scaling arising from the quasiparticle loss.
Our results deepen understanding of the emerging physics of singular charge
response in beyond-Landau metallic quantum critical points, and identify a new
regime of heavy-fermion quantum criticality in which to explore this physics.
Furthermore, our work provides new insights into the emerging charge-channel
singularities near the optimal carrier doping of Mott-Hubbard systems.

We thank Silke Paschen and D. T. Adroja for useful discussions. Work at Rice
University was primarily supported by the NSF Grant No. DMR-1920740 and by the
Robert A. Welch Foundation Grant No. C-1411 (H.H. and Q.S.). Computational
work has used the Shared University Grid at Rice funded by NSF under Grant
EIA-0216467, a partnership between Rice University, Sun Microsystems, and
Sigma Solutions, Inc., the Big-Data Private-Cloud Research Cyberinfrastructure
MRI-award funded by NSF under Grant No. CNS-1338099 and by Rice University,
and the Extreme Science and Engineering Discovery Environment (XSEDE) by NSF
under Grant No. DMR170109. One of us (Q.S.) acknowledges the hospitality of
the Aspen Center for Physics, which is supported by NSF grant No. PHY-1607611.

## References

  * Keimer and Moore (2017) B. Keimer and J. E. Moore, Nat. Phys. 13, 1045 (2017).
  * Paschen and Si (2021) S. Paschen and Q. Si, Nat. Rev. Phys. 3, 9 (2021).
  * Phillips  _et al._ (2022) P. W. Phillips, N. E. Hussey, and P. Abbamonte, arXiv preprint arXiv:2205.12979 (2022).
  * Hertz (1976) J. Hertz, Phys. Rev. B 14, 1165 (1976).
  * Millis (1993) A. J. Millis, Phys. Rev. B 48, 7183 (1993).
  * Si  _et al._ (2001) Q. Si, S. Rabello, K. Ingersent, and J. Smith, Nature 413, 804 (2001).
  * Coleman  _et al._ (2001) P. Coleman, C. Pépin, Q. Si, and R. Ramazashvili, J. Phys. Cond. Matt. 13, R723 (2001).
  * Senthil  _et al._ (2004) T. Senthil, M. Vojta, and S. Sachdev, Phys. Rev. B 69, 035111 (2004).
  * Paschen  _et al._ (2004) S. Paschen, T. Lühmann, S. Wirth, P. Gegenwart, O. Trovarelli, C. Geibel, F. Steglich, P. Coleman, and Q. Si, Nature 432, 881 (2004).
  * Gegenwart  _et al._ (2007) P. Gegenwart, T. Westerkamp, C. Krellner, Y. Tokiwa, S. Paschen, C. Geibel, F. Steglich, E. Abrahams, and Q. Si, Science 315, 1049 (2007).
  * Friedemann  _et al._ (2010) S. Friedemann, N. Oeschler, S. Wirth, C. Krellner, C. Geibel, F. Steglich, S. Paschen, S. Kirchner, and Q. Si, Proceedings of the National Academy of Sciences 107, 14547 (2010).
  * Shishido  _et al._ (2005) H. Shishido, R. Settai, H. Harima, and Y. Ōnuki, J. Phys. Soc. Jpn. 74, 1103 (2005).
  * Park  _et al._ (2006) T. Park, F. Ronning, H. Q. Yuan, M. B. Salamon, R. Movshovich, J. L. Sarrao, and J. D. Thompson, Nature 440, 65 (2006).
  * Knebel  _et al._ (2008) G. Knebel, D. Aoki, J.-P. Brison, and J. Flouquet, J. Phys. Soc. Jpn. 77, 114704 (2008).
  * Custers  _et al._ (2012) J. Custers, K. A. Lorenzer, M. Müller, A. Prokofiev, A. Sidorenko, H. Winkler, A. M. Strydom, Y. Shimura, T. Sakakibara, R. Yu, Q. Si, and S. Paschen, Nat. Mater. 11, 189 (2012).
  * Martelli  _et al._ (2019) V. Martelli, A. Cai, E. M. Nica, M. Taupin, A. Prokofiev, C.-C. Liu, H.-H. Lai, R. Yu, K. Ingersent, R. Küchler, A. M. Strydom, D. Geiger, J. Haenel, J. Larrea, Q. Si, and S. Paschen, Proc. Natl. Acad. Sci. U.S.A. 116, 17701 (2019).
  * Schröder  _et al._ (2000) A. Schröder, G. Aeppli, R. Coldea, M. Adams, O. Stockert, H. v. Löhneysen, E. Bucher, R. Ramazashvili, and P. Coleman, Nature 407, 351 (2000).
  * Kirchner  _et al._ (2020) S. Kirchner, S. Paschen, Q. Chen, S. Wirth, D. Feng, J. D. Thompson, and Q. Si, Rev. Mod. Phys. 92, 011002 (2020).
  * Prochaska  _et al._ (2020) L. Prochaska, X. Li, D. C. MacFarland, A. M. Andrews, M. Bonta, E. F. Bianco, S. Yazdi, W. Schrenk, H. Detz, A. Limbeck, Q. Si, E. Ringe, G. Strasser, J. Kono, and S. Paschen, Science 367, 285 (2020).
  * Cai  _et al._ (2020) A. Cai, Z. Yu, H. Hu, S. Kirchner, and Q. Si, Phys. Rev. Lett. 124, 027205 (2020).
  * Zhu  _et al._ (2004) L. Zhu, S. Kirchner, Q. Si, and A. Georges, Phys. Rev. Lett. 93, 267201 (2004).
  * Komijani and Coleman (2019) Y. Komijani and P. Coleman, Phys. Rev. Lett. 122, 217001 (2019).
  * Hu  _et al._ (2021) H. Hu, A. Cai, L. Chen, L. Deng, J. H. Pixley, K. Ingersent, and Q. Si, arXiv preprint arXiv:2109.13224 (2021).
  * Grempel and Si (2003) D. Grempel and Q. Si, Phys. Rev. Lett. 91, 026401 (2003).
  * Zhu  _et al._ (2003) J. Zhu, D. Grempel, and Q. Si, Phys. Rev. Lett. 91, 156404 (2003).
  * Glossop and Ingersent (2007) M. Glossop and K. Ingersent, Phys. Rev. Lett. 99, 227203 (2007).
  * Zhu  _et al._ (2007) J.-X. Zhu, S. Kirchner, R. Bulla, and Q. Si, Phys. Rev. Lett. 99, 227204 (2007).
  * Kim  _et al._ (2003) M. S. Kim, Y. Echizen, K. Umeo, S. Kobayashi, M. Sera, P. S. Salamakha, O. L. Sologub, T. Takabatake, X. Chen, T. Tayama, T. Sakakibara, M. H. Jung, and M. B. Maple, Phys. Rev. B 68, 054416 (2003).
  * Tsuda  _et al._ (2018) S. Tsuda, C. L. Yang, Y. Shimura, K. Umeo, H. Fukuoka, Y. Yamane, T. Onimaru, T. Takabatake, N. Kikugawa, T. Terashima, H. T. Hirose, S. Uji, S. Kittaka, and T. Sakakibara, Phys. Rev. B 98, 155147 (2018).
  * Tokiwa  _et al._ (2015) Y. Tokiwa, C. Stingl, M.-S. Kim, T. Takabatake, and P. Gegenwart, Sci. Adv. 1, e1500001 (2015).
  * Sundermann  _et al._ (2021) M. Sundermann, A. Marino, A. Gloskovskii, C. Yang, Y. Shimura, T. Takabatake, and A. Severing, Phys. Rev. B 104, 235150 (2021).
  * Shimura  _et al._ (2021) Y. Shimura, A. Wörl, M. Sundermann, S. Tsuda, D. T. Adroja, A. Bhattacharyya, A. M. Strydom, A. D. Hillier, F. L. Pratt, A. Gloskovskii, A. Severing, T. Onimaru, P. Gegenwart, and T. Takabatake, Phys. Rev. Lett. 126, 217202 (2021).
  * Matsumoto  _et al._ (2011) Y. Matsumoto, S. Nakatsuji, K. Kuga, Y. Karaki, N. Horie, Y. Shimura, T. Sakakibara, A. H. Nevidomskyy, and P. Coleman, Science 331, 316 (2011).
  * Sarrao  _et al._ (2002) J. L. Sarrao, L. A. Morales, J. D. Thompson, B. L. Scott, G. R. Stewart, F. Wastin, J. Rebizant, P. Boulet, E. Colineau, and G. H. Lander, Nature 420, 297 (2002).
  * Ramshaw  _et al._ (2015) B. J. Ramshaw, A. Shekhter, R. D. McDonald, J. B. Betts, J. N. Mitchell, P. H. Tobash, C. H. Mielke, E. D. Bauer, and A. Migliori, Proceedings of the National Academy of Sciences 112, 3285 (2015).
  * Miyake and Watanabe (2014) K. Miyake and S. Watanabe, J. Phys. Soc. Japan 83, 061006 (2014).
  * Si and Smith (1996) Q. Si and J. L. Smith, Phys. Rev. Lett. 77, 3391 (1996).
  * Smith and Si (2000) J. L. Smith and Q. Si, Phys. Rev. B 61, 5184 (2000).
  * Chitra and Kotliar (2000) R. Chitra and G. Kotliar, Phys. Rev. Lett. 84, 3678 (2000).
  * Note (1) See Supplemental Material for certain technical details and additional results.
  * Withoff and Fradkin (1990) D. Withoff and E. Fradkin, Phys. Rev. Lett. 64, 1835 (1990).
  * Bulla  _et al._ (1997) R. Bulla, T. Pruschke, and A. C. Hewson, Journal of Physics: Condensed Matter 9, 10463 (1997).
  * Gonzalez-Buxton and Ingersent (1998) C. Gonzalez-Buxton and K. Ingersent, Phys. Rev. B 57, 14254 (1998).
  * Logan and Glossop (2000) D. E. Logan and M. T. Glossop, Journal of Physics: Condensed Matter 12, 985 (2000).
  * Ingersent and Si (2002) K. Ingersent and Q. Si, Phys. Rev. Lett. 89, 076403 (2002).
  * Fritz and Vojta (2004) L. Fritz and M. Vojta, Phys. Rev. B 70, 214427 (2004).
  * Pixley  _et al._ (2012) J. Pixley, S. Kirchner, K. Ingersent, and Q. Si, Physical Review Letters 109, 086403 (2012).
  * Chowdhury and Ingersent (2015) T. Chowdhury and K. Ingersent, Phys. Rev. B 91, 035118 (2015).
  * Pixley  _et al._ (2013) J. Pixley, S. Kirchner, K. Ingersent, and Q. Si, Physical Review B 88, 245111 (2013).
  * Cai and Si (2019) A. Cai and Q. Si, Phys. Rev. B 100, 014439 (2019).
  * Gull  _et al._ (2011) E. Gull, A. J. Millis, A. I. Lichtenstein, A. N. Rubtsov, M. Troyer, and P. Werner, Rev. Mod. Phys. 83, 349 (2011).
  * Otsuki (2013) J. Otsuki, Phys. Rev. B 87, 125102 (2013).
  * Pixley  _et al._ (2011) J. Pixley, S. Kirchner, M. Glossop, and Q. Si, in  _Journal of Physics: Conference Series_ , Vol. 273 (IOP Publishing, 2011) p. 012050.
  * Bulla  _et al._ (2008) R. Bulla, T. A. Costi, and T. Pruschke, Rev. Mod. Phys. 80, 395 (2008).
  * Note (2) From this point on, we adopt units where $D=\hbar=k_{\text{B}}=g\mu_{\text{B}}=1$.
  * Note (3) The NRG does not capture the correct scaling form of the scaling function for $\omega\ll T$ due to truncation errors; see, e.g., Ref. Ingersent2002.
  * Kobayashi  _et al._ (2022) H. Kobayashi, Y. Sakaguchi, H. Kitagawa, M. O. S. Ikeda, K. Kuga, S. Suzuki, S. Nakatsuji, R. Masuda, Y. Kobayashi, M. Seto, _et al._ , arXiv preprint arXiv:2202.12462 (2022).
  * Chowdhury  _et al._ (2021) D. Chowdhury, A. Georges, O. Parcollet, and S. Sachdev, arXiv preprint arXiv:2109.05037 (2021).
  * Joshi  _et al._ (2020) D. G. Joshi, C. Li, G. Tarnopolsky, A. Georges, and S. Sachdev, Phys. Rev. X 10, 021033 (2020).
  * Mitrano  _et al._ (2018) M. Mitrano, A. A. Husain, S. Vig, A. Kogar, M. S. Rak, S. I. Rubeck, J. Schmalian, B. Uchoa, J. Schneeloch, R. Zhong, G. D. Gu, and P. Abbamonte, Proceedings of the National Academy of Sciences 115, 5392 (2018).
  * van der Marel  _et al._ (2003) D. van der Marel, H. J. A. Molegraaf, J. Zaanen, Z. Nussinov, F. Carbone, A. Damascelli, H. Eisaki, M. Greven, P. H. Kes, and M. Li, Nature 425, 271 (2003).