# Bipolar thermoelectric superconducting single-electron transistor

Sebastiano Battisti<EMAIL_ADDRESS>NEST, Istituto Nanoscienze-CNR
and Scuola Normale Superiore, I-56127 Pisa, Italy  Giorgio De Simoni  NEST,
Istituto Nanoscienze-CNR and Scuola Normale Superiore, I-56127 Pisa, Italy
Luca Chirolli  NEST, Istituto Nanoscienze-CNR and Scuola Normale Superiore,
I-56127 Pisa, Italy  Alessandro Braggio<EMAIL_ADDRESS>NEST,
Istituto Nanoscienze-CNR and Scuola Normale Superiore, I-56127 Pisa, Italy
Francesco Giazotto<EMAIL_ADDRESS>NEST, Istituto Nanoscienze-CNR
and Scuola Normale Superiore, I-56127 Pisa, Italy

###### Abstract

Thermoelectric effects in normal metals and superconductors are usually very
small due to the presence of electron-hole symmetry. Here, we show that
superconducting junctions brought out of equilibrium manifest a sizable
bipolar thermoelectric effect that stems from a _strong_ violation of the
detailed balance. To fully control the effect, we consider a thermally biased
SIS’IS junction where the capacitance of the central S’ region is small enough
to establish a Coulomb blockade regime. By exploiting charging effects we are
able to tune the Seebeck voltage, the thermocurrent, and thereby the power
output of this structure, via an external gate. We then analyse the main
figures of merit of bipolar thermoelectricity and we prospect for possible
applications.

††preprint: APS/123-QED

Introduction— Thermal transport and quantum thermodynamics at the nanoscale
have recently attracted a growing interest Benenti  _et al._ (2017); Dubi and
Di Ventra (2011); Kosloff (2013); Seifert (2012); Campisi  _et al._ (2011);
Giazotto  _et al._ (2006); Fornieri and Giazotto (2017); Muhonen  _et al._
(2012); Brunner  _et al._ (2012); Barato and Seifert (2015); Polettini  _et
al._ (2015); Verley  _et al._ (2014); Pietzonka and Seifert (2018); Manikandan
_et al._ (2019); Pekola  _et al._ (2007); Saira  _et al._ (2007); Fornieri
_et al._ (2014); Tirelli  _et al._ (2008); Sothmann  _et al._ (2017), thanks
to the opportunity the thermoelectric effect offers to manipulate heat and
control the energy efficiency of nanodevices Claughton and Lambert (1996);
Ozaeta  _et al._ (2014); Vischi  _et al._ (2019); Giazotto  _et al._ (2015,
2014); Marchegiani  _et al._ (2016); Brandner  _et al._ (2013); Sothmann  _et
al._ (2014a); Esposito  _et al._ (2009); Whitney (2014); Ronetti  _et al._
(2016); Kamp and Sothmann (2019); Hussein  _et al._ (2019). In the linear
regime thermoelectricity requires a broken electron-hole (EH) symmetry, which
also implies a non-reciprocal $IV$ characteristic, i.e., $I(V,\Delta
T)\neq-I(-V,\Delta T)$, where $\Delta T$ is the temperature difference. Indeed
metals, that are almost electron-hole (EH) symmetric, show nearly negligible
Seebeck coefficients Mott and Jones (1958) and present zero thermovoltages in
the superconducting phase. However, in the non-linear regime it has been
demonstrated Marchegiani  _et al._ (2020a, b), that superconducting SIS’
tunnel junctions with a sufficiently suppressed Josephson coupling exhibit a
sizable thermopower due to the spontaneous breaking of EH symmetry, yielding
an effective Seebeck coefficient (${\cal S}$) as large as $\sim 10^{5}$ times
its value in the normal state Germanese  _et al._ (2022, 2023). At the same
time, the EH symmetry determines the full _bipolarity_ of the effect with
reciprocal $IV$ characteristics. The bipolar thermoelectric effect emerges
when a strong temperature difference is suitably applied, and in the presence
of strong asymmetry in the energy gaps of the junction.

In this Letter, we consider an SIS’IS structure where a central
superconducting (SC) island featuring strong Coulomb interaction is sandwiched
between two SC leads via tunnel barriers. In such a system, the origin of the
bipolar thermoelectric properties greatly differs from that of standard
thermoelectricity in quantum dots, and lies in the _strong_ violation of the
detail balance induced by the temperature difference in the junction and the
interacting nature of BCS theory. Furthermore, we exploit the gating
properties of the Coulombic island to control the thermoelectric performances
of the engine. This unique electrical tunability differs from other platforms
Bernazzani  _et al._ (2023), and can be relevant for on-chip energy harvesting
and other energy management purposes in superconducting quantum processors and
radiation sensors Paolucci  _et al._ (2023).

Figure 1:  (a): Scheme of the SIS’IS transistor. The red-coloured parts show
the hot superconductors and the blue-coloured part shows the cold one. $V$ and
$V_{G}$ denote respectively the source-drain and gate voltages. (b):
Sequential tunneling rates versus energy for different values of $T_{hot}$ at
$T_{cold}=0.2~{}T_{C}$. Here we set $\Delta=220~{}\mu$eV (corresponding to Al,
aluminum), $\Delta_{is}=\Delta/2$, which can be obtained using a normal metal-
superconducting bilayer Germanese  _et al._ (2022); Hijano  _et al._ (2023),
and all identical barrier resistances $R=1{\rm M}\Omega$. Figure 2: (a),(b):
Charge current flowing through the SIS’IS structure at thermal equilibrium (a)
and in the non-linear regime (b) versus $V$ and $N_{G}=C_{G}V_{G}/e$. The
colour scale is the same for both graphs. (c): Blow-up of the red-dashed
region shown in panel (b) where the onset of the thermoelectric behaviour is
clearly visible. The calculations parameters are as in Fig. 1.

Model—The SIS’IS structure under investigation is shown in Fig.1(a), and
consists of two superconducting (SC) leads (L,R, red part in Fig.1(a)) with SC
gap $\Delta$ put in tunnel contact with a Coulombic island (central blue part
in Fig.1(a)) with a different SC gap $\Delta_{is}$, via two identical barriers
of resistance $R_{L/R}$. In order to observe bipolar thermoelectricity the
leads are chosen to have a larger gap than the island, $\Delta>\Delta_{is}$,
and they are kept at a temperature $T_{hot}>T_{cold}$ larger than the island
temperature, $T_{is}\equiv T_{cold}=0.2~{}T_{C}$, where $T_{C}$ is the
critical temperature of the superconducting leads. The tunneling barriers are
assumed to be resistive enough to make the Josephson energy negligible with
respect to thermal energy, thus allowing the Josephson coupling to be
neglected Marchegiani  _et al._ (2020c). Yet, in order to observe Coulomb
blockade we assume the charging energy of the central island,
$E_{C}=e^{2}/2C_{tot}$, with $C_{tot}$ the total island capacitance, large
enough that $E_{C}\gg k_{B}T_{l}$ with $l=is,L,R$ and $E_{C}\gtrsim\Delta$.

For sufficiently resistive barriers the full transport properties of the
system can be described through the rates $\Gamma_{j}(\delta U)$, with
$j=R,L$, that describe the tunneling probability through the $j$-th barrier by
the Fermi golden rule Averin and Likharev (1991); Pekola  _et al._ (2008)

$\Gamma_{j}=\frac{1}{e^{2}R_{j}}\int_{-\infty}^{\infty}\\!\\!\\!\\!\\!\\!\\!dE~{}n_{is}(E+\delta
U)[1-f_{is}(E+\delta U)]n_{j}(E)f_{j}(E),$ (1)

where $e$ is the electron charge, $\delta U$ is the electrostatic energy
acquired ($\delta U>0$) or lost ($\delta U<0$) during the tunneling process
Averin and Likharev (1991),
$n_{l}(E)=\left|Re[(E+i\gamma)/\sqrt{(E+i\gamma)^{2}-\Delta^{2}_{l}(T_{l})}]\right|$
is the (smeared by non-zero $\gamma\ll\Delta_{l}$) BCS density of states (DOS)
of the $l=is,L,R$ element Dynes  _et al._ (1984), and $f_{l}(E)$ the Fermi-
Dirac distribution function at equilibrium temperature $T_{l}$. We assumed the
SC island to be fully in equilibrium at its temperature $T_{is}$.

Strong violation of detail balance—The electron tunneling rate from the lead
to the island as a function of the electrostatic energy difference $\delta U$
contains the crucial physics for the bipolar thermoelectric effect, and it is
shown in Fig. 1(b) for different temperatures $T_{hot}$. At equilibrium,
$T_{hot}=T_{cold}$ (dark blue line), we notice a small peak at $\delta
U^{*}=\Delta-\Delta_{is}>0$, which corresponds to the matching of the DOS
divergences of the two different superconductors and it is activated by the
temperature, since at zero temperature it is completely Pauli blocked. At
finite temperature in equilibrium, the rates satisfy the detail balance
$\Gamma(-\delta U)=e^{-\delta U/k_{B}T}\Gamma(\delta U)$, so correspondingly
we expect a small peak [although not visible in Fig. 1(b)] to exist at
negative energies $\delta U=-\delta U^{*}$ as well .

More intriguing physics arises in out-of-equilibrium conditions. By increasing
the lead temperature, $T_{hot}>T_{cold}$, the peak at negative energy emerges
and becomes much stronger than its counterpart at positive energy (brown
line). This is an unusual situation where $\Gamma(-|\delta U|)>\Gamma(|\delta
U|)$, and we identify it as a strong violation of the detail balance. It comes
out as a consequence of a finite temperature difference that populates more
states at higher energy in the hot leads. From Eq. (1) a strong violation is
possible _only if_ for some energy $\epsilon,\delta U>0$ we have

$n_{is}(\epsilon-\delta U)[1-f_{is}(\epsilon-\delta U)]>n_{is}(\epsilon+\delta
U)[1-f_{is}(\epsilon+\delta U)].$ (2)

This inequality constitutes a necessary condition and it depends only on the
island DOS and on its Fermi function. It can be shown 111For more information
refer to the Supplementary Material. that Eq. (S3) is meaningful _only if_
there is a gap asymmetry between the hot and cold side, and the
superconducting DOS features a monotonously decreasing energy dependence for
$|E|>\Delta$ Marchegiani  _et al._ (2020a). Furthermore, in order to be
satisfied, Eq. (S3) _implicitly_ requires that the BCS island DOS shifts with
the electrochemical potential (the electrostatic energy difference $\delta U$
in the rate). Indeed, if the DOS does not depend on $\delta U$ (i.e.,
substituting $n_{is}(\epsilon\pm\delta U)\to n_{is}(\epsilon)$ like in a
semiconductor) Eq. (S3) cannot be satisfied since $n_{is}>0$. Thus, the strong
violation stems essentially from the interacting character captured by the BCS
mean field theory and, as we are going to show,it is the essential precursor
of the bipolar thermoelectric phenomena. The fundamental role of the
interaction clarifies also well why this effect can be associated to a
spontaneous symmetry breaking of the PH symmetry induced by the out-of-
equilibrium condition Marchegiani  _et al._ (2020a); Germanese  _et al._
(2022, 2023).

Current—In the sequential tunneling approximation we get the electric current
through a standard master equation approach Nazarov and Blanter (2009);
Bagrets and Nazarov (2003); Braggio  _et al._ (2006); Flindt  _et al._ (2008),
in which the time-evolution of the population $P_{n}$ of the island charge
state $n$ is determined by
$\dot{P}_{n}=\sum_{n^{\prime}}\mathbf{W}_{nn^{\prime}}P_{n^{\prime}}$. The
kernel $\mathbf{W}_{nn^{\prime}}=\sum_{j=R,L}\Gamma_{j}^{nn^{\prime}}$ with
the diagonal terms $\mathbf{W}_{nn}$ fixed by the conservation of the
probability, $\sum_{n}P_{n}=1$ implying $\sum_{n}\mathbf{W}_{nn^{\prime}}=0$.
The rates $\Gamma_{j}^{n,n\pm 1}=\Gamma_{j}(\delta U_{n,n\pm 1})$ correspond
only to the transitions $n=n^{\prime}\pm 1$ described by the rates of Eq. (1).
The electrostatic energy difference is $\delta U_{n,n\pm 1}=U(n\pm 1)-U(n)\pm
eV/2$ Averin and Likharev (1991); Beenakker (1991); Nazarov and Blanter
(2009), with $V=V_{L}-V_{R}$ the source-drain bias, and
$U(n)=E_{C}(n-q_{is}/e)^{2}/2$ the electrostatic energy, which depends on the
offset charge on the island $q_{is}=C_{g}V_{g}+\sum_{j}C_{j}V_{j}$, with
$C_{g}$ the gate capacitance and $C_{j}$ the $j$-th barrier capacitance (for
which $C_{tot}=C_{g}+C_{L}+C_{R}$). In the stationary limit Beenakker (1991),
the current can be simply computed in the right lead
$I=I_{R}=-I_{L}=e\sum_{n}[\Gamma_{R}^{n,n+1}-\Gamma_{R}^{n,n-1}]P_{n}^{0}$
where $P_{n}^{0}$ is the stationary probability of island charge states. This
general numerical approach can be further simplified in the Coulomb blockade
regime $k_{B}T\ll E_{C}$ by noting that the dominant contribution to the
transport at one resonance is associated only to the tunneling rates involving
neighboring charge states $n-1\rightleftharpoons n$. The current in such case
can be written as

$I_{n}=e\frac{\Gamma_{L}^{f}(n-1)\Gamma_{R}^{f}(n)-\Gamma_{L}^{b}(n)\Gamma_{R}^{b}(n-1)}{\Gamma_{L}^{f}(n-1)+\Gamma_{R}^{f}(n)+\Gamma_{L}^{b}(n)+\Gamma_{R}^{b}(n-1)},$
(3)

where we use a shortened notation for the rates
$\Gamma_{j}^{f/b}(n)=\Gamma_{j}(\pm\delta U_{n,n\pm 1})$. Since we are mainly
interested in the deep subgap regime in the bias range $|eV|<2\Delta$ with
$E_{C}=4\Delta$, we approximate the current by considering the dominant
contribution of two neighboring Coulomb resonances, i.e.,
$I=I_{n}+I_{n+1}+\mathcal{O}(e^{-E_{c}/2k_{B}T_{hot}})$.

Figure 3: Out-of-equilibrium ($T_{hot}=0.7T_{C}$) current $I$ as a function
of: (a) $N_{G}$ for different values of $V$, and (b) $V$ for different values
of $N_{G}$ (respectively horizontal and vertical cuts of Fig. 2(b)). The
dashed lines correspond to the prediction of the simplified model Eq. (3),
showing its validity in the case of $E_{C}\gg eV$. (c): Out-of-equilibrium
zero-bias conductance as a function of $N_{G}$ calculated for the curves in
(b).

Coulomb diamonds— Figure 2(a),(b) show the current $I$ as a function of the
source-drain bias $V$ and the gate-tunable offset charge $N_{G}=C_{g}V_{g}/e$.
Typical Coulomb diamonds appear, which display periodicity in the offset
charge $N_{G}$ in units of the electron charge $e$. The system does not
present any even-odd effect since the average Cooper pair recombination rate
in our system $\Gamma_{r}\simeq 16$kHz Maisi  _et al._ (2013) is much smaller
than the tunneling rates $\Gamma_{j}\sim I/e$ (inverse average electron
dwelling time in the island). Coulomb diamonds at equilibrium are shown in
Fig. 2(a). As a guide for the eye, we show with black dashed lines the
boundaries of the Coulomb diamonds for $\Delta=0$, where the electrostatic
energies vanish, $\delta U_{n,n\pm 1}(N_{G},V)\equiv 0$. As expected, the SC
gap pushes the boundaries of the Coulomb diamonds up in energy, and charge
transport is suppressed in the $(N_{G},V)$ plane domains satisfying
($eV<2\Delta+2\Delta_{is}\approx 3\Delta$).

In Fig. 2(b) we show the results for the non-equilibrium case,
$T_{hot}=0.7~{}T_{C}>T_{cold}$. Subgap conduction channels become clearly
visible, as thermally excited states promote the stronger emergence of the
matching peak resonances. At integer $N_{G}$ and $|eV|\sim 3\Delta$, where the
transport is dissipative ($IV>0$) even if fully inside the Coulomb blockade
diamond, we clearly see the appearance of yellow (blue) crosses at positive
(negative) $V$. These features stems from the enhancement of the _negative
energy_ peak in the tunneling rate for electrostatic energy $\delta
U\approx-\delta U^{*}$ [see Fig. 1(b)], and are also a direct consequence of
the strong violation of the detailed balance, even if their nature is still
dissipative. More intriguingly, for half-integer $N_{G}$ and $eV\sim\Delta/2$,
the sign of the current becomes opposite to the bias $IV<0$, as shown in the
blow-up of Fig. 2(c). This behavior is a signature of thermoelectricity and it
appears _only_ when a finite temperature difference is applied between the
island and the hot leads. These subgap structures are equivalently present at
positive and negative bias for the _same_ temperature difference, showing the
full _bipolar_ character that is enforced by the EH symmetry of the unbiased
system. The emerging bipolar thermoelectric effect is similar to SIS’ systems
Marchegiani  _et al._ (2020d). Furthermore, our bipolar thermoelectric
superconducting transistor offers the possibility to be manipulated thanks to
Coulomb interaction and associated gating effects. We also notice that, unlike
a conventional quantum-dot thermoelectric effect Staring  _et al._ (1993);
Mahan and Sofo (1996) that is activated by a temperature gradient _between_
the leads, the present effect appears when the leads have the _same_
temperature higher than the Coulomb island Note (1).

Figure 3(a) displays cuts of Fig. 2(b) at different $V$. In a region around
half-integer gate charge $N_{G}$ we clearly see a change of the sign of the
current around the Coulomb resonance opposing to the bias (thermoelectric
effect). Notably, the sign of the thermoelectric current does not change
passing through the resonant value due to the unique bipolar nature, unlike in
the conventional (unipolar) thermoelectric effect in a quantum dot Sothmann
_et al._ (2014b).

Figure 4: a: Seebeck voltage (left y-axis) and nonlinear Seebeck coefficient
$|\mathcal{S}|$ (right y-axis) versus $N_{G}$ for different values of
$T_{hot}$. b: Maximum thermocurrent versus $N_{G}$ for different values of
$T_{hot}$. c: Power output $P_{out}=-\widetilde{I}~{}\widetilde{V}$ versus
$N_{G}$ for different values of the load resistor. $\widetilde{I}$ and
$\widetilde{V}$ are the solutions of the intersection of the $IV$
characteristics with the load line of the resistor. Note that among all
possible solutions only the electrically stable one, i.e., that with
$dI/dV>0$, can be operated by the engine Marchegiani  _et al._ (2020a, c).

In Fig. 3(b) we show cuts of Fig. 2(b) at fixed $N_{G}$. We first focus on the
zero-bias behavior: as we vary $N_{G}$ towards half-integer values the zero-
bias conductance (ZBC) $G_{0}=dI/dV|_{V=0}$ becomes negative at the critical
value $N^{*}_{G}\approx 4.42$ [see Fig. 3(c)]. This behavior blatantly
suggests the spontaneous breaking of the EH symmetry of the system Marchegiani
_et al._ (2020a, d), and highlights the unique capability of the gate control
in our system that, differently from other platforms Bernazzani  _et al._
(2023), allows to continuously tune the emergent bipolar thermoelectric
properties. A negative ZBC, together with the condition $I(0,\Delta T)=0$ as
dictated by EH symmetry, implies the existence of thermoelectricity ($IV<0$)
and the existence of a Seebeck voltage $V_{S}$, since at high-biases $eV\gg
3\Delta$ the system necessarily becomes again dissipative, $IV>0$.
Interestingly, this implies that the Seebeck voltage (i.e., open circuit
voltage defined as $I(V_{S})=0$) is expected to be dependent on the gate
voltage, $V_{S}(N_{G})$, with clear consequences on the gate tunability of the
thermoelectric performance. At finite values of $V$ the current exhibits a
peak changing in sign (thermoelectricity) when $N_{G}$ approaches half-integer
values. The system shows an absolute negative conductance and thereby
thermopower, $\dot{W}=-IV>0$. Furthermore, for some values of $N_{G}$ the $IV$
curve presents more than one resonant peak (see yellow line): this is a
consequence of the Coulomb blockade, since the island electrostatic energy
differences $\delta U$ for different rates depends on the gate $N_{G}$ and
bias $V$, and correspondingly the matching peaks resonance of the dominant
rates appear split for non resonant values $N_{G}\neq 1/2+n$ of the $IV$
curve. Note that a similar double-peak structure can be observed also as a
function of $N_{G}$, as shown in Fig. 3(b).

Thermoelectric figures of merit—The intrinsic nonlinear nature of the above
effect does not allow to describe the thermoelectric figures of merit of our
system via a linear thermoelectric approach Goldsmid (2013). However, we can
still define a Seebeck voltage $V_{S}$ and a nonlinear Seebeck coefficient
${\cal S}=V_{S}/\Delta T$ with $\Delta T=T_{hot}-T_{cold}$. We stress that EH
symmetry implies two Seeback voltages, $\pm V_{S}$, and a _bipolar_ ${\cal S}$
Marchegiani  _et al._ (2020a). Figure 4(a) shows $V_{S}$ (solid line, left
scale) and $|{\cal S}|$ (dashed line, right scale) as a function of $N_{G}$
for different temperatures of the leads. By changing $T_{hot}$, the Seebeck
voltage shows horn-like nonlinear features which are even higher at slightly
lower value of $T_{hot}$. By inspection of Fig. 3(b) we recognize that the
yellow curve has two peaks and for certain values of $\Delta T$ the second
peak can even cross the $I=0$ axis, returning a higher open circuit (Seebeck)
voltage. Analogously, ${\cal S}$ is also similarly affected and its maximal
value is not necessarily associated to the maximal thermovoltage (due to the
nonlinearities it is not even necessarily associated with the highest
temperature difference $\Delta T$). In Fig. 4(b) the maximum thermocurrent
$I_{max}(N_{G})=\operatorname{max}_{0<V<2\Delta/e}[|I(V,N_{G})|]$ as a
function of $N_{G}$ is shown to gradually become zero while lowering
$T_{hot}$, as expected, since the temperature difference is not enough to
trigger the bipolar thermoelectricity Germanese  _et al._ (2022, 2023). The
thermoelectric generator character of the transistor appears when closing the
circuit on a load resistor. Figure 4(c) displays the output power of the
structure as a function of $N_{G}$ for different values of the load resistor,
demonstrating the ability of fine gating control of the output power. The
maximum achievable output power is typically associated with the smallest
possible load resistance, and turns out to depend on several parameters.

Conclusions—We theoretically proposed and analysed a bipolar thermoelectric
superconducting single-electron transistor that enables tuning and control of
the bipolar thermoelectric effect through an applied gate voltage. The
interplay between Coulomb blockade and out-of-equilibrium thermoelectricity
finds its origin in the strong violation of the detail balance, which is
triggered by different SC gaps, a finite temperature difference and,
crucially, by the interacting nature intrinsic to the BCS theory. We
investigated the performance of a fully gate-tunable heat engine that can
provide, with realistic parameters, a nonlinear Seebeck coefficient up to
$\sim 3$ mV/K at subKelvin temperatures. The effect can be implemented in a
device that can produce gate-controlled single-electron thermoelectricity in a
fully superconducting design, thereby fostering interest for on-chip energy
harvesting and management, single-charge electronics, and single-photon
detection Paolucci  _et al._ (2023).

We acknowledge the EU’s Horizon 2020 Research and Innovation Framework
Programme under Grant No. 964398 (SUPERGATE), No. 101057977 (SPECTRUM), and
the PNRR MUR project PE0000023-NQSTI for partial financial support. A.B.
acknowledges the Royal Society through the International Exchanges between the
UK and Italy (Grants No. IEC R2 192166.).

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##  Appendix A Supplementary Informations

##  Appendix B Sufficient conditions for the strong violation of the detailed
balance

It is interesting to discuss which are the conditions to observe the strong
violation of the detailed balance reported in the main text for the golden
rule rate of the Eq. (1). For a Coulomb island, at equilibrium in the
temperatures, the rates usually satisfy the detail balance

$\Gamma(-\delta U)=e^{-\beta\delta U}\Gamma(\delta U),$ (S1)

with the inverse temperature $\beta=1/k_{B}T$. This property guarantees that
the final distribution of the quantum dot states follows the standard
Boltzmann distribution, such that the $n$ charge states of the island has an
occupation probability $p(n)\propto e^{-\beta E_{c}(n)}$, with $E_{c}(n)$ the
energy of the state Nazarov and Blanter (2009). Clearly however, when the
island and the leads have a different temperatures $\beta_{j}\neq\beta_{is}$,
the detail balance is necessarily violated.

In the main text, we showed a case where the violation is extreme: when the
”backward” rate (that promotes electron tunneling in energetically
unfavourable state) is favoured in comparison to the ”forward” one, i.e.

$\Gamma(-\delta U)>\Gamma(\delta U),$ (S2)

with $\delta U>0$. By noticing that the functions determining the rate are all
positive, $0\leq f_{j},n_{j}$, and the Fermi distribution satisfies $f_{l}\leq
1$, and by assuming the EH symmetry for the leads, one can easily see that Eq.
(S2) can be satisfied _only if_ for some energy $\delta U>0$ we have

$n_{is}(\epsilon-\delta U)[1-f_{is}(\epsilon-\delta U)]>n_{is}(\epsilon+\delta
U)[1-f_{is}(\epsilon+\delta U)].$ (S3)

Since this is a necessary condition it is instructive to investigate what are
the requirements that the island DOS has to satisfy to promote the strong
violation. We see immediately that for energy independent DOS of the island
$n_{is}(\epsilon)=const$ (non superconducting metallic island) the inequality
cannot be satisfied since the Fermi function is a monotonously decreasing
function of the energy, i.e. $f_{j}(E)\geq f_{j}(E^{\prime})$ for $E\leq
E^{\prime}$. Without any other assumption on the DOS it immediately follows
that an $NIN$ junction cannot ever show any strong violation of the detail
balance. This argument also implies that an $SIN$ junction cannot shown any
strong violation of the detail balance.

In turn, for an $SIS$ junction the inequality Eq. (S2) can be potentially
satisfied due to the fact that, for a standard BCS DOS, it can happen that
$n_{is}(\epsilon-\delta U)>n_{is}(\epsilon+\delta U)$ for
$\epsilon>\Delta_{is}$, compensating the natural opposite behaviour induced by
the $1-f$ term to not violate the inequality. This demonstrates the importance
of the energy dependence of the DOS for, at least, some energies.

However, one needs to exclude the case of two identical superconductors.
Indeed, for a pure $SIS$ case, i.e. $n_{is}(\epsilon)=n_{j}(\epsilon)$, an
exchange in the roles of the DOS between the lead and island is fully
equivalent to a change in sign of the temperature difference, with a
consequent temperature-independent violation of the detailed balance, that
implies the violation of the second principle of thermodynamics. Thus, the
condition of strong violation of the detail balance identifies a physical
mechanism in which for some states in a certain range of energies it is
convenient to go against the electrostatic energy.

It is important to stress again that inequality Eq. (S3) is a _necessary_
condition but _not a sufficient_ one to realize the reported strong violation
(notice that the inequality Eq. (S3) depends only on the island temperature
and obviously at equilibrium the detailed balance must be satisfied, implying
the necessary character of Eq. (S3) but not the sufficient one). As discussed
also in Ref. Marchegiani  _et al._ (2020a, d) the temperature difference has
to be operated in a way that the large gap is kept at a temperature higher
than the small gap. Furthermore, the condition of the monotonously decreasing
density of states for $\epsilon>0$ does not necessarily imply
superconductivity as requirement. This was discussed in the Supplementary
material of Ref. Marchegiani  _et al._ (2020a), for example a Lorentzian
density of state $n_{is}(\epsilon)\propto\gamma/(\gamma^{2}+\epsilon^{2})$, if
always centered in the chemical potential, can also determine the violation of
the inequality, thus enlarging the range of cases where the strong violation
of the detail balance can be indeed similarly reported beyond BCS
superconductivity.

Finally, we wish to importantly note another requirement that needs to be
satisfied in order to make possible the strong violation of the detail
balance. By inspection of Eq. (S3) it follows that the argument of the island
DOS must shift with the energy $\delta U$, i.e. $n_{is}(\epsilon\pm\delta U)$.
Indeed, the DOS shifts with $\delta U$ due to the fact that the SC gap opens
at the Fermi energy as a result of the _intrinsic_ interacting nature of the
Cooper instability. Notably, this complexity is elegantly accounted for by the
BCS mean-field theory, whose simplicity in turn may hide its profound origin.
For example, in a semiconductor featuring a similar band-gap in the spectrum,
the position of the chemical potential is determined by the electron density
and it is not fixed to reside in the gap. It follows that the DOS does not
shift with the electrostatic energy. In turn, in a superconductor when the
electrochemical potential is changed, such as by applying a bias to a
junction, the DOS shifts with it. In the Coulomb island we are considering,
the role of the bias is essentially played by the electrostatic energy
difference $\delta U$.

The importance of the interaction was never particularly stressed before but
the rate analysis discussed here shows it in a crystal clear way and it also
sheds light on the physics of the bipolar thermoelectricity in the whole sense
Marchegiani  _et al._ (2020a, d). Also in that cases we can state that the
mean-field nature of the BCS theory, which properly takes in account the
important part of the interaction, is a crucial ingredient behind the bipolar
thermoelectricity in the tunnelling limit. This conclusion can be concurrently
obtained also by the experimental observation that bipolar thermoelectricity
is connected to a spontaneous breaking of the EH symmetry. In the experimental
demonstration of the bipolar thermoelectric effect we have demonstrated also a
current controlled hysteretic cycle of the junction, which clearly showed the
reality of the spontaneous symmetry breaking of the EH symmetry.

The analysis so far presented assumed a finite $\delta U$, that also implies
nonlinearity in the electrostatic energy difference. This is meaningful, since
the maximum of violation of the detail balance appears at the matching-peak
conditions $\delta U=\pm(\Delta(T)-\Delta_{is}(T))$. However, it is
interesting to look at what happens at very low energies $\delta U\to 0$. By
expanding the rate we find

$\frac{\partial\Gamma_{j}}{\partial\delta U}\Bigl{|}_{\delta
U=0}=\frac{1}{e^{2}R_{j}}\int_{-\infty}^{\infty}d\epsilon\left(\frac{\partial
n_{is}(\epsilon)}{\partial\epsilon}[1-f_{is}(\epsilon)]-n_{is}(\epsilon)\frac{\partial
f_{is}(\epsilon)}{\partial\epsilon}\right)n_{j}(\epsilon)f_{j}(\epsilon).$
(S4)

Clearly the first term is not present if the DOS of the island $n_{is}$ is not
affected by the energy $\delta U$, and we are left with only the second term
that is always positive. So _only if_ the island DOS is monotonously
decreasing, i.e. $\partial_{\epsilon}n_{is}<0$ for $\epsilon>0$ one could even
have a negative $\partial\Gamma_{j}/\partial\delta U<0$ which strongly violate
the detail balance. So the arguments we did before in nonlinear cases ($\delta
U\neq 0$) can be repeated in the linear case. Notably, the presence of a
negative $\partial\Gamma_{j}/\partial\delta U$ in the linear regime of $\delta
U$ is necessarily associated to the interaction effects included by the island
DOS by the BCS theory. A similar phenomenology is also observed at the level
of the current of the SIS’ junction, where at low bias a negative differential
conductance is reported Marchegiani  _et al._ (2020a). This is again connected
to the SSB of the EH symmetry which create an unstable situation, essentially
shown in the strong violation of the detail balance, where the system tries to
push particles against the natural direction dictated by the electrostatic
energy. This intrinsic instability constitutes at the same time an arbitrary
strong violation of the non-equilibrium fluctuation dissipation theorem, which
is nevertheless valid at the temperature equilibrium due to the detail
balance.

Figure S1: (a):Source grain current $I$ as a function of source drain bias $V$
with $T_{cold}=T_{is}=0.2T_{C}$, $T_{l}=0.7T_{C}$ and $T_{r}=1.3T_{l}$, for
different values of $N_{G}$.(b):Source grain current $I$ as a function of
source drain bias $V$ with $\Delta_{is}=\Delta/2$, $\Delta_{l}=\Delta$ and
$\Delta_{r}=0.9\Delta$, for different values of $N_{G}$.

##  Appendix C Differences with the conventional thermoelectricity in quantum
dots

In this section we want to discuss the crucial differences between the bipolar
thermoelectric effects of the system studied in the main text and the unipolar
thermoelectricity that usually appears in conventional quantum dots Staring
_et al._ (1993); Mahan and Sofo (1996). Quantum dots can be tuned via an
external gating to be _non_ EH symmetric Averin and Likharev (1991), thus
generating a thermoelectric effect. It is indeed well know that the quantum
dots can show a gate dependent linear thermoelectric effect when the thermal
gradient is applied throughout the system, i.e. when the temperatures of the
source and drain leads are different. Notably, the sign of the Seebeck
coefficient can be changed by appropriately operating with a gate. The gate is
indeed capable to move the system from electron-like to hole-like characted.
However, we term this effect unipolar since once the gating condition is
fixed, a change in the thermal gradient causes a sign change in the
thermoelectrical response. Furthermore, the presence of a thermal gradient
between the leads makes, in general, the structure also non-symmetric by
inversion, so in case of an unipolar thermoelectric effect a non-reciprocal IV
characteristic $I(V,\Delta T)\neq-I(-V,\Delta T)$ also appears. This situation
strongly differs from the _bipolar_ thermoelectric effect reported in main
text, where the temperature difference between isalnd and leads does not
determines the sign of the thermoelectricity that it is only determined by the
spontaneous symmetry breaking. This is a real crucial difference between the
two effects. Indeed, in the system we discuss in the main text, it is crucial
that we apply a thermal gradient between the central island an the two
external leads (kept at the same temperature). This configuration is not
expected to generate any net thermoelectric effect classifiable as a standard
_unipolar_ thermoelectrical contribution. Indeed, the thermal gradients at the
two junctions have opposite signs with respect to the current flow trough the
system and, for an identical left/right tunnel barriers we consider, no net
unipolar thermoelectrical contribution is expected to be generated. Notice
that this thermal configuration is well known to not generate a net (unipolar)
thermoelectric effect. As a comparison, thermoelectric cells usually combine
$p$\- and $n$-like material with opposite signs of the thermoelectrical
coefficient (thermo-couple configuration). In conclusion, the temperature
configuration considered in the system uniquely proves the bipolar nature of
the effect. This has been discussed also elsewhere Marchegiani  _et al._
(2020d) but, in the main text, we demonstrated that the same arguments
substantially apply also in the presence of Coulombic effects.

At the same time, one can speculate that if the right/left barrier symmetry is
broken a net linear thermoelectrical effect may eventually emerge. General
considerations coming from EH symmetry of the system Benenti  _et al._ (2017),
at the linear level, suggest that this configuration would not generate any
linear thermoelectric effect. However, the bipolar effect is a nonlinear
effect, requiring a finite temperature difference and sometimes also finite
bias, so those general linear arguments do not applied. It is interesting to
explore how a different set of conditions for the system (and the system
parameters) affects the reported evidences. We tested a different set of cases
changing the structure in Fig. 1(a) keeping the same configuration of
temperatures (cold island and with the hot leads at the same temperatures). In
particular we tested all the following configurations:

  * •

Full metallic structure (no superconductors) with equal tunnel barriers
($R_{l}=R_{r}$),

  * •

Full metallic structure with different tunnel barriers ($R_{l}\neq R_{r}$),

  * •

Metallic dot with superconductive leads with equal tunnel barriers
($R_{l}=R_{r}$),

  * •

Metallic dot with superconductive leads with different tunnel barriers
($R_{l}\neq R_{r}$),

  * •

Superconductive dot with metallic leads with equal tunnel barriers
($R_{l}=R_{r}$),

  * •

Superconductive dot with metallic leads with different tunnel barriers
($R_{l}\neq R_{r}$).

_None_ of these configurations showed any kind of thermoelectricity. This
result can be seen as the consequence of the fact that thermoelectricity in
our system is strongly correlated with the nature of the BCS DOS of island and
leads, as pointed out in the previous discussion on the necessary conditions
for the bipolar thermoelectricity.

##  Appendix D Robustness of the effect

However, clearly, if the temperature of the external right/left
superconducting leads are different one can expect a contribution coming for
the standard unipolar thermoelectrical effect. So even in the case of not
perfect temperature profile an experiment can address the signature of the
reported effect. Similarly one could speculate what happen in the case that
the gap of the superconducting leads are not exactly identical, as we simply
assumed in the main text. In other words, hereafter we want to discuss the
robustness of the effect with respect to a change in the temperatures or gaps
profiles assumed, for simplicity in the main text. We will consider separately
the two most interesting cases: i) the temperature of the leads is not
perfectly equal and ii) the SC gap of the leads is not perfectly equal.

Taking into analysis the first case, in Fig. S1(a) it can be observed the IV
characteristic when the temperature of the right lead is higher than the one
of the left lead. Here we have $T_{is}=0.2T_{C}$, $T_{l}=0.7T_{C}$,
$T_{r}=0.8T_{C}$ where $T_{is}$ is the temperature of the island and $T_{l}$,
$T_{r}$ are the temperatures of the left and right lead respectively. These
curves are still reminiscent of the bipolar thermoelectric effect discussed in
the main text. However, on top of that we can recognize the contribution of
the _unipolar_ thermoelectric effect that manly shifts the intercept of the
curves al zero bias. Indeed these IV curves loose the reciprocity, $I(V,\Delta
T)=-I(-V,\Delta T)$, as a direct consequence of the emerging unipolar
thermoelectricity that not symmetric temperature configuration of the external
leads will determine being associated to the _conventional_ thermoelectrical
behavior of the quantum dots. However, if the quantum dot is gated exactly at
the Coulomb resonance, such as $N_{G}=4.5$, the EH symmetry of the symmetry is
again restored: no linear unipolar thermoelectric effects and the reciprocity
of IV curves is fully restored. However, also in that case, intriguing
differences can be seen with respect to the result reported in the main text.
In particular the difference between the IV curvers where two thermoelectric
peaks appear at the Coulomb resonance where in the main text we have only a
one-peak matching peak for that gate value. Indeed the temperature difference
of the leads determine also a different value for the superconducting gaps in
the leads, i.e. $\Delta_{r}(T_{r})\neq\Delta_{l}(T_{l})$, as predicted by BCS
tinkham2004. So, correspondely, there are two different matching peaks values
depending on which barriers the tunnelling happens. This result in the double
peak structure also at resonance. However, even if some of the peculiarity of
the bipolar thermoelectric effect are lost, at least comparing Fig. S1(a) with
the main text figures, we can still conclude the existence of bipolar
thermoelectric effect component. Many of the figure of merits discussed in the
paper are roughly still valid and are not substantially ruined by a moderate
temperature inequality in the leads.

Finally we wish to show the possible consequence of the fact that two external
lead have a slightly different zero temperature gaps. However, for simplicity,
the temperature configuration is taken exaclty the same of the main text. In
this case it can be seen from Fig. S1(b) that the situation remind aspects of
the previous case. Anyway, here also the right/left symmetry is broken. In
this case too, we can conclude that the effect described in the main text is
sufficiently robust and do not need fine tuning of the parameters to be
observable.


# Strong cosmic censorship for a black hole in loop quantum gravity

Cai-Ying Shao<EMAIL_ADDRESS>MOE Key Laboratory of Fundamental Physical
Quantities Measurement, Hubei Key Laboratory of Gravitation and Quantum
Physics, PGMF, and School of Physics, Huazhong University of Science and
Technology, Wuhan 430074, Hubei, China  Cong Zhang<EMAIL_ADDRESS>Department Physik, Institut für Quantengravitation, Theoretische Physik III,
Friedrich-Alexander-Universität Erlangen-Nürnberg, Staudtstraße 7/B2, 91058
Erlangen, Germany  Wei Zhang<EMAIL_ADDRESS>Department of Physics,
Beijing Normal University, Beijing 100875, China  Cheng-Gang Shao
<EMAIL_ADDRESS>MOE Key Laboratory of Fundamental Physical Quantities
Measurement, Hubei Key Laboratory of Gravitation and Quantum Physics, PGMF,
and School of Physics, Huazhong University of Science and Technology, Wuhan
430074, Hubei, China

###### Abstract

A fine gravitational theory is essentially expected to deal with the problem
of spacetime singularities. Loop quantum gravity as one branch of quantum
gravity is potential to explore the nature of black holes. Recently, according
to the quantum Oppenheimer-Snyder model in loop quantum cosmology, a novel
loop quantum corrected black hole in de Sitter spacetime has been discovered.
Here, we focus on examining the strong cosmic censorship(SCC) based on such a
quantum modified black hole by considering a massless neutral scalar field
perturbation. As a result, we find that the SCC is destroyed as the black hole
approaches to the near-extremal limit. Notably, the critical value of the
black hole mass ratio for such a violation increases with the increase of the
cosmological constant. It was implied the cosmological constant plays an
important role in moderating the violation of the SCC.

††preprint: APS/123-QED

##  I Introduction

Spacetime singularities, characterized by infinite curvature or density, have
been a subject of great interest and curiosity in the fields of gravitation
theory and relativistic astrophysics. According to the singularity theorems
proved by Hawking and Penrose, the existence of singularities is unavoidable
in generic gravitational collapses. The presence of singularities poses
profound challenges to our understanding of the universe within the context of
classical general relativity. One specific concern is the existence of naked
singularities, which are singularities that are not hidden within a black hole
event horizon and thus could be observed by outside observers, breaking down
the predictive power of classical general relativity.

In order to alleviate such a loss of predictability, Penrose proposed the
cosmic censorship conjectures [1]. One is called weak cosmic censorship
conjecture (WCC), which asserts that any spacetime singularity formed in a
generic gravitational collapse should be covered by a black hole horizon. It
is obvious that WCC guarantees the predictive power of classical general
relativity only in the spacetime region outside of the black hole. While the
predictability of classical general relativity inside of the black hole is
further restored by the other conjecture, named the strong cosmic censorship
conjecture (SCC), which claims in a colloquial style that the timelike
singularities are not allowed, or can be formulated equivalently as a more
rigorous mathematical statement that the Cauchy horizon inside of the black
hole is unstable for the generic perturbations and thus inextendible. This is
the case for Kerr and Reissner-Nordstrom black holes, where the would-be
timelike singularity does not lead to the violation of the SCC because the
Cauchy horizon becomes singular and inextendible due to the exponential
blueshift effect of the perturbations along it. Actually, in asymptotically
flat spacetimes, the SCC is always valid except for the accelerating black
holes[2, 3]. However, the validity of SCC will become more complicated in the
asymptotically de Sitter spacetimes. A positive cosmological constant leads to
an exponential decay of the external perturbations, which can compete with the
aforementioned blueshift effect along the Cauchy horizon [4, 5]. Thus the
validity of the SCC depends on which one will win in the competition. To be
more specific, the SCC has recently been found violated in the nearly extremal
charged Reissner-Nordstrom de Sitter (RNdS) black hole by the scalar field [6,
7, 8, 9, 10], the fermionic field [11, 12, 13], and the gravito-
electromagnetic field [14]. In addition, as to the rotating Kerr de Sitter
black hole, the SCC can be respected by the bosonic field perturbations [15,
16], but violated by the fermionic field perturbation [17]. While for the
Kerr-Newman de Sitter black hole, the SCC is still violated by both the scalar
and fermionic fields [18]. Last but not least, it is noteworthy that other
factors, such as the smoothness of initial data, nonlinear effect, dark matter
and dark energy, space-time dimensions, and quantum effect of the perturbation
fields, could also impact the validity of the SCC [19, 20, 21, 22, 23, 24].

On the other hand, the presence of singularities in classical general
relativity highlights the necessity for a theory of quantum gravity (QG) that
combines the principles of quantum mechanics and general relativity. Among the
various approaches to QG, loop quantum gravity (LQG) has shown great promise
with significant advancements made (see, e.g., [25, 26, 27, 28, 29, 30, 31]
and the references therein). By applying the procedure of loop quantization to
spherically symmetric black holes, one has gained many insights into the
quantum nature of black holes [32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42],
where the singularity of the Schwarzschild black hole is believed to be
resolved through the effects of LQG as it should be the case, although the
specific detail of how this resolution occurs is scheme dependent. In
particular, with the quantum Oppenheimer-Snyder model in loop quantum
cosmology, a new quantum black hole model has been derived most recently [39],
where the Schwarzschild singularity is resolved by a transition region that
contains an inner horizon. As a result, the global structure of such a quantum
black hole model resembles that of the charged Reissner-Nordstrom black hole.
In this sense, the SCC is still plagued potentially by the emergence of the
inner Cauchy horizon if one immerses this quantum modified black hole in de
Sitter space.

The purpose of this paper is to examine whether the SCC holds for such a
quantum modified black hole in de Sitter space. To this end, we first follow
the same procedure developed in [39] to derive the modified metric of the loop
quantum black hole in de Sitter space in the next section. Then we present the
dynamics of a neutral massless scalar perturbation and derive Christodoulou’s
formulation of the SCC in terms of quasinormal modes in Sec. III. With the
above preparation, we use different numerical methods to calculate the
quasinormal modes and explore the validity of the SCC in Sec. IV. Finally, the
concluding remarks are presented in the last section.

##  II The loop quantum gravity corrected geometry of the black hole in de
Sitter space

Let us follow the precedure introduced in [39] to get the quantum modified
spacetime by considering the quantum Oppenheimer-Snyder model. In this model,
the entire spacetime is divided into two regions. One region comprises a
pressureless dust ball with a constant density, and the other region is a
vacuum outside the dust ball.

In the region with dust, we introduce a coordinate
$(\tau,\tilde{r},\theta,\phi)$ with $0<\tilde{r}<\tilde{r}_{0}$ which adapts
the symmetry of the dust ball. Then, the metric of the ball takes the form

$ds^{2}_{\rm in}=-d\tau^{2}+a(\tau)^{2}(d\tilde{r}^{2}+d\Omega^{2}),$ (1)

where $d\Omega^{2}=d\theta^{2}+\sin^{2}\theta d\phi^{2}$. The dynamics of the
scale factor $a(\tau)$ is governed by the LQC modified Friedmann equation

$H^{2}=\left(\frac{\dot{a}}{a}\right)^{2}=\frac{8\pi
G}{3}\rho(1-\frac{\rho}{\rho_{c}})+\frac{\Lambda}{3},\quad\rho=\frac{M}{\frac{4}{3}\pi\tilde{r}_{0}^{3}a^{3}},$
(2)

where the deformation parameter $\rho_{c}$ denotes the critical density
defined as $\rho_{c}=\sqrt{3}/(32\pi^{2}\gamma^{3}G^{2}\hbar)$ with the
Barbero-Immirzi parameter $\gamma$. $M$ is the mass of the ball with radius
$a(\tau)\tilde{r}_{0}$. It should be noted that the current work adds a
cosmological constant term to the modified Friedmann equation, different from
the initial model considered in [39]. Eq.(2) reverts to the usual Friedmann
equation in the classical regime where $\rho\ll\rho_{c}$. However, in the
quantum regime where $\rho$ is comparable with $\rho_{c}$ so that the
spacetime curvature becomes Planckian, the deformation term will prevent the
matter density $\rho(\tau)$ from reaching infinity which thus prevents the
formation of the singularity. Indeed, according to Eq.(2), at the moment
$\tau_{b}$ with $\rho(\tau_{b})=\rho_{c}\left[1+\sqrt{1+\Lambda/(2\pi
G\rho_{c})}\,\right]/2$, one has $H=0$, which signifies a change of the
dynamics of the ball from the collapsing phase to the expanding phase at
$\tau_{b}$.

In the outside region of the dust ball, we assume the spacetime to be
spherically symmetric and static, as done in [39]. We can use the coordinates
$(t,r,\theta,\phi)$ to describe this region, which is adapted to the symmetry
of the spacetime. In this coordinate, the metric of the outside region reads

$ds^{2}_{\rm out}=-f(r)dt^{2}+g(r)^{-1}dr^{2}+r^{2}d\Omega^{2},$ (3)

where $f(r)$ and $g(r)$ are two unknown functions to be determined. In order
to determine the unknown functions $f(r)$ and $g(r)$, we need to find the
inner most boundary of the outside region which is glued with the dust ball
surface. The junction condition for the gluing requires that the reduced
3-metrics and the extrinsic curvatures along the gluing surfaces obtained from
the 4-metrics $ds^{2}_{\rm in}$ and $ds^{2}_{\rm out}$ respectively are
continuous. It should be noted that the worldlines
$\tau\mapsto(\tau,\tilde{r}_{0},\theta,\phi)$ of each particle on the surface
of the dust ball is a timelike geodesic without rotation. This implies that
the inner most surface of the outside region is also composed of the
congruence of freely falling timelike geodesics associated to the metric
$ds^{2}_{\rm out}$. Moreover, let $\tau\to(t(\tau),r(\tau),\theta,\phi)$ be a
geodesic in the innermost surface of the outside region, with $\tau$ being the
length of the geodesic. Then, the surfaces are glued by the identification
$(\tau,\tilde{r}_{0},\theta,\phi)\sim(t(\tau),r(\tau),\theta,\phi)$. The
calculation could be simplified by such a junction condition.

So far, we have built our model and sketched the calculation to get the metric
of the outside region by the junction condition. Then, just following the
procedure shown in [39], we get

$f(r)=g(r)=1-\left(\frac{2GM}{r}+\frac{\Lambda r^{2}}{3}-\frac{\alpha
G^{2}M^{2}}{r^{4}}\left(1+\frac{\Lambda r^{3}}{6GM}\right)^{2}\right)$ (4)

where $\alpha=16\sqrt{3}\pi\gamma^{3}G\hbar$, proportional to the Planck area,
is the quantum deformation parameter. It should be noted that the metric (4)
is valid only for $r>r_{b}$ with $r_{b}$ denoting the minimal radial of the
dust ball at which the bounce occurs [39]. For convenience, we set $G=\hbar=1$
and $\gamma=0.2375$ in the remainder of this paper. In Fig. 1, we plot the
values of $f(r)$ depending on $r$ for $\Lambda=0.1$, in which $M$ can take
different values. As shown in the figure, for $M$ bigger than some extreme
value $M_{\rm Ext}$, the metric function $f(r)$ has three roots, corresponding
to the three horizons of the black hole. They are respectively the Cauchy
horizon ${r_{i}}$, the event horizon ${r_{h}}$ and the cosmological horizon
${r_{c}}$, with $r_{i}<r_{h}<r_{c}$. If one decreases the mass of the black
hole for the given cosmological constant, the Cauchy and the event horizons
gradually approach. When the Cauchy horizon coincides with the event horizon,
the mass reaches an extreme value, which is denoted as $M_{\text{ext}}$.For
$M<M_{\rm ext}$, the event horizon disappears resulting in a naked
singularity. This case is thus prohibited by the WCC. For research purposes,
our focus here is only on a black hole with three horizons.

Figure 1: The metric function $f(r)$ versus $r$ obtained for given
$\Lambda=0.1$.

##  III Quasinormal modes and strong cosmic censorship

Now, we consider a massless neutral scalar field perturbation in the above
background. The equation of motion in such a curved spacetime is governed by
the following Klein-Gordon equation:

$\square\Phi=0.$ (5)

According to spherical symmetry of the spacetime, the scalar field can be
expanded as

$\Phi=\frac{\phi(r)}{r}Y_{lm}(\theta,\varphi)e^{-i\omega t},$ (6)

where ${Y_{lm}}(\theta,\varphi)$ is the spherical harmonics function. By
plugging it into the Klein-Gordon equation, the master equation in the radial
part reads

$\left({\frac{{{d^{2}}}}{{d{r_{*}^{2}}}}+{\omega^{2}}-{V_{eff}}(r)}\right)\phi(r)=0,$
(7)

where the effective potential is given by

${V_{eff}}(r)=f(r)\left[{\frac{{l(l+1)}}{{{r^{2}}}}+\frac{{f^{\prime}(r)}}{r}}\right].$
(8)

$dr_{*}$ is tortoise coordinate, which is defined as $dr_{*}=\frac{dr}{f(r)}$.
Physically, there only exist purely ingoing waves near the event horizon and
purely outgoing waves near the cosmological horizon [43]. Thus the boundary
conditions are imposed as

$\phi(r)\approx{e^{-i\omega
r_{*}}}\left({r\to{r_{h}}}\right),\quad\phi(r)\approx{e^{i\omega
r_{*}}}\left({r\to{r_{c}}}\right).$ (9)

Then, the discrete quasinormal frequencies can be derived by solving the
equation of motion with the above boundary conditions.

On the other hand, if one imposes purely ingoing wave near the event horizon,
the solution from the equation of motion has both the outgoing and ingoing
waves near the Cauchy horizon, which can be expressed as

${\phi_{ingoing}}\approx{e^{-i\omega
u}}{\left({r-{r_{i}}}\right)^{\frac{{i\omega}}{{{\kappa_{i}}}}}},\quad\phi_{outgoing}\approx
e^{-i\omega u},$ (10)

where $u$ is outgoing coordinate defined as $u=t-r_{*}$ and ${\kappa_{i}}$ is
the surface gravity of Cauchy horizon defined as
${\kappa_{i}}=\left|{\frac{1}{2}{f^{\prime}}\left({{r_{i}}}\right)}\right|$.
Obviously, the ingoing wave has non-smooth radial dependence, which results in
the potential non-smoothness behavior in the energy momentum tensor of the
scalar field. Commonly, the violation of the SCC implies the weak solution can
be extended beyond the Cauchy horizon. In other words, the energy-momentum
tensor consisting of the square of its first derivative for the scalar field
can be integrable at the Cauchy horizon, which requires [6]

$\beta=-\frac{{{\mathop{\rm
Im}\nolimits}\omega}}{{{\kappa_{i}}}}>\frac{1}{2}.$ (11)

for all the quasinormal modes. On the contrary, as long as one finds the
lowest lying quasinormal modes with the criterion $\beta\leq\frac{1}{2}$, the
SCC is preserved. Hence, in order to check the validity of the SCC, we
exclusively focus on the lowest-lying quasinormal modes in the remainder of
this paper.

##  IV Numerical methods and relevant results

In this section, we will use two numerical methods to accurately calculate the
lowest-lying quasinormal modes and present some relevant results. Presently,
many numerical computations of quasinormal modes have been developed with high
precision [44, 45, 46]. Here, we introduce the finite difference method [47]
to obtain the numerical evolution of the scalar field and then extract the
quasinormal spectrum from the data samples with Prony method [48]. In order to
check the correctness of our results, we also employ the matrix method [49].
Besides that, WKB approximation [50, 51, 52] is also performed for the
quasinormal modes with large $l$ s. Note that there are three distinct
families to classify the relevant quasinormal modes, namely, the near-extremal
modes with $l=0$, the de Sitter modes with $l=1$, and the photon sphere modes
with large $l$ s. In what follows, we are going to explore the neutral
massless scalar field with these three modes.

First, it is necessary to perform a coordinate transformation to derive the
double null coordinates, which is defined as $u=t-{r_{*}}$ and $v=t+{r_{*}}$.
Accordingly, the Klein-Gordon equation can be expressed as

$-4\frac{{{\partial^{2}}\phi}}{{\partial u\partial
v}}={V_{{\rm{eff}}}}(r(u,v))\phi.$ (12)

According to finite difference scheme, the data at $N$ can be obtained from
$W$, $E$, and $S$, such that the above equation of motion gives rise to

${\phi_{N}}={\phi_{W}}+{\phi_{E}}-{\phi_{S}}-\Delta u\Delta
v{V_{{\rm{eff}}}}(r(u,v))\frac{{{\phi_{W}}+{\phi_{E}}}}{8},$ (13)

where the indices $N,W,E,S$ denote grid-points, respectively corresponding to
the points $N\equiv(u+\Delta,v+\Delta)$, $W\equiv(u,v+\Delta)$,
$E\equiv(u+\Delta,v)$, and $S\equiv(u,v)$ with $\Delta$ the step width of
$(u,v)$. The time-domain profile will appear soon, once one provides the
specific initial conditions

$\phi(u,0)=0,\quad\phi(0,v)={e^{-\frac{{{{\left({v-{v_{c}}}\right)}^{2}}}}{{2{\sigma^{2}}}}}},$
(14)

where ${v_{c}}$ and $\sigma$ correspond to the center and width of the
Gaussian wave packet. The resulting temporal evolution $\phi(t,{r_{*}})$ can
be obtained from equally elapsed late-time data.

Next, to extract the quasinormal mode from the temporal evolution data, Prony
method is a very useful tool, which as an extension of the Fourier
decomposition, is of great significance for signal processing and data
analysis. The late-time signal at a certain ${r_{*}}$ is composed of a set of
quasinormal modes, which can be expanded as

$\phi(t)=\sum\limits_{j=1}^{p}{{C_{j}}}{e^{-i{\omega_{j}}t}}.$ (15)

The time interval of the time-domain profile is between ${t_{0}}$ and
$t={t_{0}}+qh$, where $h$ is the time interval of each point. $q$ as the
number of sample signals is an integer and satisfies $q=2p$. For convenience,
every sample is labeled by an integer $n$. According to the above formula, the
time-domain data at any time can be expressed as

${x_{n}}=\sum\limits_{j=1}^{p}{{{\tilde{C}}_{j}}}z_{j}^{n},$ (16)

where
${x_{n}}=\phi\left({{t_{0}}+nh}\right),z_{j}={e^{-i{\omega_{j}}h}},{{\tilde{C}}_{j}}={C_{j}}{e^{-i\omega{t_{0}}}}$.
In order to find ${z_{j}}$, it is necessary to introduce a polynomial function

$A(z)=\prod\limits_{j=1}^{p}{\left({z-{z_{j}}}\right)}=\sum\limits_{i=0}^{p}{{\alpha_{i}}}{z^{p-i}},$
(17)

with ${\alpha_{0}}=1$. Obviously, for any integer $j$ from 1 to $p$,
$A({z_{j}})=0$. Thus, it’s easy to obtain the sum

${\sum\limits_{i=0}^{p}{{\alpha_{i}}}{x_{j-i}}=\sum\limits_{i=0}^{p}{{\alpha_{i}}}\sum\limits_{k=1}^{p}{{C_{k}}}z_{k}^{j-i}=\sum\limits_{k=1}^{p}{{C_{k}}}z_{k}^{j-p}A\left({{z_{k}}}\right)=0.}$
(18)

Considering ${\alpha_{0}}=1$, the above equation can be rewritten as

$\sum\limits_{i=1}^{p}{{\alpha_{i}}}{x_{j-i}}=-{x_{j}}.$ (19)

Thus, we can get $p$ equations after taking $j$ from $p+1$ to $q$ such that
${\alpha_{i}}$ can be solved. After substituting ${\alpha_{i}}$ into Eq. (17),
${z_{j}}$ can be derived easily. Then the quasinormal modes are obtained with
the relation ${\omega_{j}}=\frac{i}{h}\ln\left({{z_{j}}}\right)$. The
coefficients ${C_{i}}$ can also be found according to Eq. (16).

Table 1: The lowest-lying quasinormal modes $\frac{{-{\mathop{\rm Im}\nolimits}(\omega)}}{{{\kappa_{i}}}}$ with different modes, where the results are obtained by different numerical methods for ${M_{E{\rm{xt}}}}/M=0.95,\Lambda=0.01$.  Methods |  l=0 |  l=1 |  l=10 |  l=20
---|---|---|---|---
Prony method |  0.364164 |  0.344098 |  0.337448 |  0.335744
Matrix method |  0.375631 |  0.343185 |  0.337483 |  0.334483
WKB approximation |  |  |  0.338027 |  0.337964
Table 2: The lowest-lying quasinormal modes $\frac{{-{\mathop{\rm Im}\nolimits}(\omega)}}{{{\kappa_{i}}}}$ with different modes, where the results are obtained by different numerical methods for ${M_{E{\rm{xt}}}}/M=0.98,\Lambda=0.01$.  Methods |  l=0 |  l=1 |  l=10 |  l=20
---|---|---|---|---
Prony method |  0.673372 |  0.629653 |  0.617419 |  0.614077
Matrix method |  0.684612 |  0.633103 |  0.617494 |  0.612216
WKB approximation |  |  |  0.618559 |  0.618447

As a comparison, we further resort to the matrix method to ensure the accuracy
of numerical results. By introducing reasonable wave function with a new
variable $Y(y)$ and changing the equation of motion into a regular form in the
interval $[0,1]$, the Eq.(7) can be transformed into a matrix equation in the
form of $\Gamma(\omega)\mathcal{Y}=0$ with the boundary condition
$Y(0)=Y(1)=0$, where $\Gamma(\omega)$ is a matrix and $\mathcal{Y}$ a vector
given by ${\mathcal{Y}_{i}}=Y({y_{i}})$. The quasinormal modes can be
determined by solving the nonlinear algebraic equation
$\det(\Gamma(\omega))=0$.

In Tab.1 and 2, we present low-lying quasinormal modes for the massless
neutral scalar field obtained from both the Prony method and the matrix
method. As shown in Tab.1 and 2, the numerical results derived by both methods
are consistent with each other, and their accuracy error is controlled within
5 percent, which demonstrates the reliability of our numerical calculations.
Moreover, we also employ the WKB approximation to calculate low-lying
quasinormal modes with large $l$ s and find the correlation results converge
with those of other methods. It is noted that as the mass of the black hole is
close to the extremal limit, $\frac{{-{\mathop{\rm
Im}\nolimits}(\omega)}}{{{\kappa_{i}}}}$ becomes larger and larger. This
indicates that the SCC might be violated in the near extremal regime.

Figure 2:  The lowest-lying quasinormal modes with the frequency
$\beta=\frac{{-{\mathop{\rm Im}\nolimits}(\omega)}}{{{\kappa_{i}}}}$ as a
function of the black hole mass ratio ${M_{E{\rm{xt}}}}/M$ , where the dotted
magenta horizontal line represents the threshold value $\beta=\frac{1}{2}$ and
the dotted cyan vertical line denotes the critical value of the mass ratio for
the violation of the SCC.

Figure 3:  The lowest-lying quasinormal modes with the frequency
$\beta=\frac{{-{\mathop{\rm Im}\nolimits}(\omega)}}{{{\kappa_{i}}}}$ as a
function of the cosmological constant $\Lambda$, where the dotted magenta
horizontal line represents the threshold value $\beta=\frac{1}{2}$ and the
dotted cyan vertical line denotes the critical value of $\Lambda$ for the
restoration of the SCC.

As a demonstration, Fig.2 presents the variation of $\beta$ with the black
hole mass ratio ${M_{E{\rm{xt}}}}/M$ for different cosmological constants for
the given $l$. As expected, when the cosmological constant is fixed, the SCC
will only be violated as the mass ratio exceeds a certain critical value. In
addition, the critical value of the mass ratio for the violation of the SCC
increases with the cosmological constant. To test this further, we also plot
the variation of $\beta$ with the cosmological constant $\Lambda$ for
different black hole mass ratios ${M_{E{\rm{xt}}}}/M$ in Fig.3. It is noted
that the larger the cosmological constant is, the harder the SCC is violated.
It seems that the cosmological constant play an important role in recovering
the SCC. Furthermore, the critical value for $\Lambda$ to rescue the SCC
becomes larger with the increase of the mass ratio.

Figure 4:  The density plots of $\beta$ in the $\frac{M_{Ext}}{M}-\Lambda$
plane, where the region above the red line indicates the violation of the SCC.

Finally, to display the behavior of $\beta$ more intuitively, we present the
density plots of $\beta$ in the $\frac{M_{Ext}}{M}-\Lambda$ plane in Fig.4.
The critical threshold $\beta=1/2$ is marked as a solid red line. Only in the
area above that can the SCC be violated. As one can see, the dashed line in
black $\frac{M_{Ext}}{M}=0.99$ illustrates that the SCC is violated in the
case with a smaller cosmological constant but respected when the cosmological
constant is large enough. But anyhow, the SCC will always be violated if only
the black hole approaches a highly near-extremal limit. The critical value of
the black hole mass ratio for the violation is increased with the increase of
the cosmological constant. To a certain degree, the cosmological constant can
moderate the violation of the SCC.

##  V Concluding remarks

In this paper, we consider the perturbation of the massless neutral scalar
field on the top of a loop quantum gravity corrected black hole in de Sitter
spacetime. We obtain the low-lying quasinormal modes by employing the Prony
method and the matrix method. Based on these, we further explore the validity
of the SCC under such a perturbation. As a result, the SCC is always violated
when the black hole approaches the extremal one. It is found that the larger
the cosmological constant is, the harder the SCC is violated, which implies
the cosmological constant plays an important role in alleviating such a
violation.

We conclude our paper by pointing out a potential tension between the SCC and
LQG. If the SCC is valid, it means that the Cauchy horizon becomes singular
and inextendible. But the effect from LQG is supposed to resolve any potential
singularity. If this is the case, the Cauchy horizon is supposed to keep
smooth and extendible in LQG even in the presence of our scalar field,
invalidating the SCC even in the validity regime of the SCC we have found in
this paper. Such a tension may be solved by shifting our perspectives. For
instance, although the emergent Cauchy horizon in the full LQG is smooth and
extendible, making it the predictive power lost beyond the Cauchy horizon,
such a loss might simply be the classical manifestation of quantum
uncertainty. In this sense, the SCC should be discarded. To have a deep
understanding of this issue, it is better to explore what the emergent
classical geometry really looks like in loop quantum gravity coupled to the
quantum scalar field. But this is utterly beyond the scope of this paper and
expected to be reported somewhere else.

## Acknowledgements

This work is supported by the National Key ${\rm{R\&D}}$ Program of China
under Grant No.2021YFC2203001 and Grant No.2022YFC2204602, the Natural Science
Foundation of China Grant No.11925503, Grant No.12075026, and Grant No.
12275022.

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