# Supplementary information:
Impact of bright-dark exciton thermal population mixing on the brightness of
CsPbBr3 nanocrystals

Mohamed-Raouf Amara  Laboratoire de Physique de l’École Normale Supérieure,
ENS, Université PSL, CNRS, Sorbonne Université, Université Paris-Cité, F-75005
Paris, France  Division of Physics and Applied Physics, School of Physical and
Mathematical Sciences, Nanyang Technological University, 637371, Singapore
Caixia Huo  Division of Physics and Applied Physics, School of Physical and
Mathematical Sciences, Nanyang Technological University, 637371, Singapore
Institute of Materials/School of Materials Science and Engineering, Shanghai
University, Shanghai 200444, China  Shaoxing Institute of Technology, Shanghai
University, Zhejiang 312000, China  Christophe Voisin  Laboratoire de Physique
de l’École Normale Supérieure, ENS, Université PSL, CNRS, Sorbonne Université,
Université Paris-Cité, F-75005 Paris, France  Qihua Xiong  State Key
Laboratory of Low-Dimensional Quantum Physics, Department of Physics, Tsinghua
University, Beijing 100084, China.  Frontier Science Center for Quantum
Information, Beijing 100084, P.R. China  Collaborative Innovation Center of
Quantum Matter, Beijing 100084, P.R. China  Beijing Academy of Quantum
Information Sciences, Beijing 100193, P.R. China  Carole Diederichs
Corresponding author<EMAIL_ADDRESS>Laboratoire de Physique de
l’École Normale Supérieure, ENS, Université PSL, CNRS, Sorbonne Université,
Université Paris-Cité, F-75005 Paris, France  Institut Universitaire de France
(IUF), 75231 Paris, France

##  SI Experimental aspects

###  SI.1 Synthesis and characterisation

Figure S1: Synthesis characterisation. (a) Ensemble room-temperature
photoluminescence and absorption spectra. The excitation energy is shown as
$E_{\text{exc}}$. (b) Size-distribution histogram obtained from transmission
electron microscopy.

Cesium lead bromide nanocrystals (CsPbBr3 NCs) were grown using a modified
previously described method [1]. Hereafter, we highlight only the main
differences. The injection temperature is lowered to
$180\text{\,}\mathrm{\SIUnitSymbolCelsius}$ to yield smaller NCs. After
injection, the reaction was stopped briefly in an ice water bath and then left
at room temperature. The solution is then purified by several rounds of
centrifugation and re-dispersion in hexane until the obtained solution remains
clear and stable. The solution is then filtered using a
$20\text{\,}\mathrm{nm}$ membrane filter to remove any by-products of the
synthesis that might remain after the purification process. Transmission
electron microscopy was performed on the synthesized NCs and revealed the
presence of nearly cubic NCs (average aspect ratio $$0.89$\pm$0.09$$) with a
mean edge length of $7.7\pm 1.2\text{\,}\mathrm{nm}$ and sizes ranging from
$4\text{\,}\mathrm{nm}12\text{\,}\mathrm{nm}$. Detailed information about the
size, morphology and transmission electron microscopy images of the
synthesised NCs can be found in ref. [2] under the label _synthesis A_.

Figure S2: Additional experimental aspects at $4.5\text{\,}\mathrm{K}$. (a)
Representative excitation power density dependence of the integrated emission
intensity of a single NC under continuous wave (CW) and pulsed excitation. (b)
PL intensity of a single NC under CW excitation revealing the stability of the
emission over the course of hours. Time intervals shown in red correspond to
the reoptimisation of the position of the excitation beam and focal, and the
time interval highlighted in light blue shows a continuous small decrease of
the PL intensity due to the slight drift of the sample. (c) Raw coincidence
counts of second-order correlation measurements performed at
$4.5\text{\,}\mathrm{K}$ showing photon antibunching at zero time delay. (d)
PL time trace showing the minimal spectral diffusion observed at our
resolution.

###  SI.2 Sample preparation

For single NC measurements, the solution was diluted $\sim$ $2000$$\times$ in
a 2% solution of polystyrene in toluene. The targeted concentration was
achieved by multiple rounds of dilution for accuracy/reproducibility. The
solution was then spin-coated on either a glass or a distributed Bragg
reflector (DBR) substrate to increase the collection efficiency.

###  SI.3 Optical characterisation

For single particle measurements, samples were mounted on the cold finger of a
continuous liquid-helium flow cryostat cooled down to
$4.5\text{\,}\mathrm{K}$. All optical measurements were performed with a home-
built scanning micro-PL setup. Excitation and detection of the PL emission
were realised with the same 50$\times$ long-working-distance dry microscope
objective with NA=0.65. NCs were excited either by a continuous diode laser at
$457\text{\,}\mathrm{nm}$ or a pulsed $\mathrm{fs}$ frequency-doubled Ti:Sa
laser (repetition rate: $80\text{\,}\mathrm{MHz}$) tunable between
$400\text{\,}\mathrm{nm}$ and $520\text{\,}\mathrm{nm}$. The PL emission was
dispersed with a diffraction grating and detected with a charge-coupled-device
(CCD) camera. For time-resolved measurements, the PL emission was directed via
the second output of the monochromator to two single-photon avalanche
photodiodes (Micro Photon Devices SPADs) with a nominal dark count rate of
less than $50\text{\,}\mathrm{c}\mathrm{t}\mathrm{s}\mathrm{/}\mathrm{s}$. The
overall resolution of the time-resolved setup is further discussed in the
following. This setup layout enabled us to spectrally resolve the emission of
single CsPbBr3 NCs with both the CCD and APDs.

Despite the low excitation power density kept at all times, charged exciton
and biexciton emission is observed on the majority of NCs. Fig. S2a shows a
representative power density dependence of the emission of a single NC
displayed as a function of the average number of excitons generated
respectively per lifetime and per pulse for continuous and pulsed excitation,
assuming an absorption cross section
$\sigma\sim$4\text{\times}{10}^{-14}\text{\,}{\mathrm{cm}}^{2}$$ for an
excitation at $\sim$ $450\text{\,}\mathrm{nm}$ [3, 4]. We verify that the
pulsed excitation experiments are performed in the linear regime where the
number of excitons generated per pulse $N$ is kept below $\langle
N\rangle\sim$ 0.1. The emission intensity is stable over the course of hours
(Fig. S2b) and no blinking is observed even at the smallest time bins (zoom-in
view of Fig. S2b). Furthermore, only minimal spectral diffusion is observed at
our resolution (Fig. S2d). A second-order correlation measurement performed on
the bright exciton emission under pulsed excitation is shown in Fig. S2c and
reveals highly antibunched emission.

##  SII Decay dynamics analysis

Figure S3: Instrument response function (IRF) of the time-correlated single-
photon counting (TCSPC) setup. (a) IRF of the TCSPC setup as a function of
wavelength. (b,c) IRFs at (b) the excitation wavelength
($457\text{\,}\mathrm{nm}$) and (c) at the detection wavelength
($525\text{\,}\mathrm{nm}$) together with the key features: IRF peak width,
tail/peak ratio and IRF tail decay.

###  SII.1 Time-resolved data analysis

As we are concerned with measuring decay times in the order of
$100\text{\,}\mathrm{ps}$ which is close to the nominal time-resolution of the
single-photon detection setup, special attention needs to be given to the
measurement of the instrument response function (IRF). Indeed, the finite
width of the laser pulses as well as the physical and electronic nature of the
single-photon detection process shape the temporal response of decay
measurements. The IRF therefore exhibits (Fig. S3) a characteristic shape with
a sharp and narrow peak and a lower amplitude tail due to the diffusion of
photocarriers generated in neutral regions near the depletion layer [5].

To properly account for this response, we measure the IRF of our setup with a
$5\text{\,}\mathrm{nm}$ step from
$460\text{\,}\mathrm{nm}525\text{\,}\mathrm{nm}$ using the scattering of the
pulsed laser excitation obtained by frequency-doubling a Ti:Sa laser
($80\text{\,}\mathrm{MHz}$ with $\sim$ $100\text{\,}\mathrm{fs}$ pulses). The
laser pulse is highly attenuated using the same filter used during our
experiments and the count rate on the SPADs is kept similar to the single NC
emission count rate (well below 1 photon per pulse). Interestingly, we find
that the shape of the IRF is highly dependent on the detection wavelength even
within this small window. Fig. S3a shows the measured IRFs while in Fig.
S3b,c, we highlight the difference between the IRF measured at
$457\text{\,}\mathrm{nm}$ and $525\text{\,}\mathrm{nm}$ together with the
features of interest: tail/peak ratio, IRF width and characteristic decay time
of the tail. The tail/peak ratio and IRF width show a similar decreasing trend
with increasing wavelength from $0.33$ to $0.11$ and $\sim$
$75\text{\,}\mathrm{ps}$ to $\sim$ $40\text{\,}\mathrm{ps}$, respectively. The
characteristic decay time of the IRF tail is found to be nearly constant
across the entire detection range at $\sim$ $210\text{\,}\mathrm{ps}$. The
characteristic IRF decay time is very similar to the decay times previously
reported in the literature for single NC or ensembles at low temperature [6,
7, 8], it is thus of utmost importance to properly take into account this
response when fitting the time-resolved data.

To analyse the decay curves, we perform a reconvolution fit using a maximum
likelihood estimator to fit the decay curves with multi-exponential decays
convoluted to the IRF as:

$I(t)=\sum_{i=1}^{N}\mathrm{IRF}(t)\otimes
A_{i}\exp\left(-\frac{t-t_{0}}{\tau_{i}}\right)+B$ (S1)

where the IRF is measured within $2.5\text{\,}\mathrm{nm}$ of the decay curve
wavelength, the $A_{i}$ are the amplitudes of the exponential decays, $t_{0}$
is the zero-delay time, $\tau_{i}$ are the exponential decay times and $B$ is
the correction for the decay background. For temperature-dependent decay
measurements, decay curves are acquired as a function of temperature and each
curve is fitted as a convolution of the IRF closest to the emission wavelength
and a multi-exponential decay. This reconvolution technique taking into
account the appropriate IRF thus yields reliable amplitudes for each component
of the decay which cannot be obtained by fitting only the tail of decays
(shown in Fig. S4 in red). This allows us to monitor the temperature-
dependence of the multi-exponential decay of single NCs via both the decay
times and relative component intensity.

Figure S4: Influence of the IRF on the recovered decay parameters. Comparison
of decay times obtained using different fit methods taking into account the
IRF or not. (a,b) Full decay trace showing the raw data and fit without taking
into account the IRF. (a) Decay analysis with the IRF taken at
$485\text{\,}\mathrm{nm}$ together with the corresponding fit. (b) Decay
analysis with the IRF taken at $525\text{\,}\mathrm{nm}$ with the
corresponding fit.

To highlight the importance of the IRF, we show in Fig. S4 a decay curve
acquired at $523.5\text{\,}\mathrm{nm}$ and several fit attempts. The lines in
green, light blue and red show respectively the fit attempts using the IRF
recorded at the closest wavelength of $525\text{\,}\mathrm{nm}$, the IRF
recorded at $485\text{\,}\mathrm{nm}$, and, neglecting the IRF, only fitting
the tail of the decay curve. Each of these fits converges with residuals that
are evenly distributed around 0. As such, they could all seem to be reasonable
but physically they are not. Each yields a different decay time shown in the
figure legend. Interestingly, the IRF tail has a great impact on the decay
time recovered from the fit procedure. When neglecting the IRF in the fit
procedure, the decay time increases by a factor $\sim$2 and becomes close to
the IRF tail decay time $\sim$ $210\text{\,}\mathrm{ps}$. It is thus
necessary, in order to accurately measure such short decay times, to take into
account the IRF close to the emission wavelength particularly in this range as
the detectors response can vary.

We further verify the absence of additional unresolved decay times longer than
the excitation laser repetition rate. To do so, we check the background level
of photoluminescence decay curves against the background level obtained with
no emitter, only due to the SPADs dark counts, during the same acquisition
time. Both background levels being the same, we can exclude unresolved longer
decay times.

###  SII.2 Model

####  SII.2.1 Rate equations

The evolution of the system, shown in Fig. S5, is governed by rate equations
that can be written in matrix form as

$\mathcal{\dot{P}}=\mathcal{A}\mathcal{P}$

where $\mathcal{P}=\\{P_{i}\\}_{i\leq 4}$ and $\mathcal{A}$ is the transition
matrix given by

$\begin{pmatrix}\dot{p}_{\text{G}}\\\ \dot{p}_{\text{B}}\\\
\dot{p}_{\text{D}}\\\
\dot{p}_{\text{E}}\end{pmatrix}=\begin{pmatrix}0&\Gamma_{\text{B}}&\Gamma_{\text{D}}&0\\\
0&-\left(\Gamma_{\text{B}}+\gamma_{\text{BD}}\right)&\gamma_{\text{DB}}&a\Gamma_{\text{E}}\\\
0&\gamma_{\text{BD}}&-\left(\Gamma_{\text{D}}+\gamma_{\text{DB}}\right)&\left(1-a\right)\Gamma_{\text{E}}\\\
0&0&0&-\Gamma_{\text{E}}\end{pmatrix}\begin{pmatrix}p_{\text{G}}\\\
p_{\text{B}}\\\ p_{\text{D}}\\\ p_{\text{E}}\end{pmatrix}$

At this point, the rate equations are general enough to allow for the
scenarios that we will explore, i.e. one- and two-phonon transitions with two
relative bright-dark orderings: bright over dark and bright under dark. This
versatility is hidden in the transition rates $\gamma_{\text{BD}}$ and
$\gamma_{\text{DB}}$.

The solution of the matrix equation is given by:

$\mathcal{P}=\sum_{i=1}^{N}\alpha_{i}\vec{v}_{i}e^{-\left|\Gamma_{i}\right|t}$

where $\alpha_{i}$ are constants derived from initial conditions, $\Gamma_{i}$
are the eigenvalues of $\mathcal{A}$ and $\vec{v}_{i}$ the corresponding
eigenvectors. At $t=0$, we have $\mathcal{P}=(0,0,0,1)$, such that the
$\alpha_{i}$ are given by the solution of
${\sum_{i}\alpha_{i}\vec{v}_{i}=\mathcal{P}(t=0)=(0,0,0,1)}$.

The total photoluminescence signal is given by

$S(t)=\textstyle\sum_{i}\eta_{i}\Gamma_{i}p_{i}=\sum_{i}A_{i}e^{-\left|\Gamma_{i}\right|t}$

where $\eta_{i}$ is the quantum yield associated with state $i$ and the
amplitudes read
$A_{i}=\textstyle\sum_{j}\eta_{j}\Gamma_{j}\alpha_{i}\vec{v}_{i}^{(j)}$ where
$\vec{v}_{i}^{(j)}$ is the $j$-th component of eigenvector $\vec{v}_{i}$. The
sum $i$ runs over $i=\text{B},\text{D},\text{E}$.

Figure S5: Schematic of the system comprised of a ground state (G), an excited
state (E), a bright exciton state (B) and a dark exciton state (D) together
with the rates involved in the model.

In our experiments, we only acquire the PL of the bright state given by
$S_{\text{B}}(t)=\eta_{\text{B}}\Gamma_{\text{B}}p_{\text{B}}$ which in
general is a tri-exponential decay of the form

$A_{\text{short}}e^{-\Gamma_{\text{short}}t}+A_{\text{long}}e^{-\Gamma_{\text{long}}t}+A_{\text{rise}}e^{-\Gamma_{\text{rise}}t}$

where $A_{\text{short}},A_{\text{long}}\geq 0$ and $A_{\text{rise}}\leq 0$.

In most cases, no rise time is observed (i.e. rise time constant is very short
and/or the amplitude $A_{\text{rise}}$ is negligible) and the decays take the
simpler form

$A_{\text{short}}e^{-\Gamma_{\text{short}}t}+A_{\text{long}}e^{-\Gamma_{\text{long}}t}$

Using the reconvolution technique we can extract the values of the short and
long decay times together with their relative amplitudes. We define the
fractional intensity of the long component $I_{\text{rel},\text{long}}$ that
will be used together with the decay times $\Gamma_{\text{short}}$ and
$\Gamma_{\text{long}}$ for our analysis.

$I_{\text{rel},\text{long}}=\left(\frac{A_{\text{long}}}{\Gamma_{\text{long}}}\right)/\left(\sum_{i}A_{i}/\Gamma_{i}\right)$
(S2)

| One-phonon process | Two-phonon processes
---|---|---
|  |  |  |  |  |
$\gamma_{\text{B}\rightarrow\text{D}}/\gamma_{0}$ | $\left(N+1\right)$ | $N$ | $\left(N_{\text{1}}+1\right)\left(N_{\text{2}}+1\right)$ | $N_{\text{1}}N_{\text{2}}$ | $N_{\text{1}}\left(N_{\text{2}}+1\right)$ | $N_{\text{2}}\left(N_{\text{1}}+1\right)$
$\gamma_{\text{D}\rightarrow\text{B}}/\gamma_{0}$ | $N$ | $\left(N+1\right)$ | $N_{\text{1}}N_{\text{2}}$ | $\left(N_{\text{1}}+1\right)\left(N_{\text{2}}+1\right)$ | $N_{\text{2}}\left(N_{\text{1}}+1\right)$ | $N_{\text{1}}\left(N_{\text{2}}+1\right)$
$\Gamma_{\text{short}}(T\rightarrow 0)$ |  $\Gamma_{\text{B}}$ \+ $\gamma_{0}$ | $\Gamma_{\text{B}}$ |  $\Gamma_{\text{B}}$ \+ $\gamma_{0}$ | $\Gamma_{\text{B}}$ | $\Gamma_{\text{B}}$ | $\Gamma_{\text{B}}$
$\Gamma_{\text{long}}(T\rightarrow 0)$∗ | $\Gamma_{\text{D}}$ |  $\Gamma_{\text{D}}$ \+ $\gamma_{0}$ | $\Gamma_{\text{D}}$ |  $\Gamma_{\text{D}}$ \+ $\gamma_{0}$ | $\Gamma_{\text{D}}$ | $\Gamma_{\text{D}}$
Table S1: Four scenarios arising from two relative orderings and two phonon-
driven mixing schemes. ∗Since the relative intensity of this component is less
than a few % at the lowest temperatures this limit is not used as a
constraint.

####  SII.2.2 Simulations

Transitions between exciton sublevels are modelled as phonon-assisted
mechanisms. Here, we consider both first-order and second-order phonon-driven
transitions with the two possible level orderings: bright state above or below
the dark state. The corresponding bright-dark transfer rates are given in the
main text and summarised in Table S1.

As equations are quite intricated and we have quite a lot of parameters, we
would like to rely on more than just non-linear optimisation as it is often
difficult to assess the relative quality of fits solely based on residuals or
goodness of fit measurements with this high number of parameters. In Table S1,
the low- and high-temperature limits of the short and long decay components
are given and used to constrain the model parameters $\Gamma_{\text{B}}$ and
$\Gamma_{\text{D}}$.

Let us emphasize that $a$ is set to $0.75$ to reflect the exciton fine
structure of our material consisting of a bright triplet and a dark singlet.
In fact, this parameter also controls the maximum decay-component relative
intensity attainable under strong magnetic field, i.e.
$I_{\text{rel,long}}\left(B\gg\Delta_{0}/\mu_{B}\Delta_{\text{BD}}g\right)\rightarrow
1-a$. The set value of $a=0.75$ shows very good agreement with measurements on
ensembles of CsPbBr3 NCs at magnetic fields up to $30\text{\,}\mathrm{T}$
where $I_{\text{rel,long}}($30\text{\,}\mathrm{T}$)\sim 0.25$ [9].

####  SII.2.3 Conversion efficiency

In the case of a dark ground state, we can define the conversion efficiency
from bright to dark, resp. dark to bright, exciton as
$\gamma_{\downarrow}/(\Gamma_{\text{B}}+\gamma_{\downarrow})$, resp.
$\gamma_{\uparrow}/(\Gamma_{\text{D}}+\gamma_{\uparrow})$. In Fig. S6 we show
the conversion efficiencies from bright to dark and dark to bright exciton for
both CdSe-based NCs and LHP NCs. For CdSe-ZnS NCs, the conversion efficiency
from bright to dark exciton is larger than 80% at the lowest temperatures and
increases with temperature. Similarly, for CdSe-CdS NCs, more than 35% of the
bright exciton population is converted to the dark state at the lowest
temperatures and 50% at $\sim$ $26\text{\,}\mathrm{K}$. Conversely, the
conversion efficiency from dark to bright exciton is zero at the lowest
temperatures and reaches 50% at $3\text{\,}\mathrm{K}$ and
$11\text{\,}\mathrm{K}$ respectively. This contrasts with LHPs, where the
conversion efficiency from bright to dark exciton as well as dark to bright
exciton are both zero at the lowest temperatures. The dark to bright
conversion efficiency reaches 50% at 8, 13, 16 and $54\text{\,}\mathrm{K}$,
respectively for FAPbBr3, CsPbI3, FAPbI3 and CsPbBr3 and the bright to dark
conversion efficiency reaches 50% at 33, 37, 73 and $162\text{\,}\mathrm{K}$.
This highlights the weak efficiency of the bright/dark population conversion
in LHP NCs which stems from the second-order nature of the bright/dark
transitions in LHPs and more precisely from the vanishing upward and downward
transition rates at the lowest temperatures coupled with favourable transition
rates.

Figure S6: Conversion efficiencies between bright and dark excitons in several
materials. Dark-to-bright (solid) and bright-to-dark (dashed) exciton
conversion efficiency (a) for CdSe-based NCs with one-phonon bright-dark
population transfer and (b) for LHP NCs with two-phonon bright-dark population
transfer. Transition rates used in this calculation for CsPbBr3 are
$\gamma_{0}=$10\text{\,}\mathrm{n}\mathrm{s}^{-1}$$,
$\Gamma_{\text{B}}=$15\text{\,}\mathrm{n}\mathrm{s}^{-1}$$,
$\Gamma_{\text{D}}=$0.12\text{\,}\mathrm{n}\mathrm{s}^{-1}$$ with phonon
energies $E_{\text{ph},1}=$6\text{\,}\mathrm{meV}$$,
$E_{\text{ph},2}=$19\text{\,}\mathrm{meV}$$ and $a=0.75$ while rates for other
materials are extracted from refs. [10, 11, 12, 13].

####  SII.2.4 Integrated intensities

Within the system comprised of a dark and a bright state, we can determine the
total emitted intensity $I_{i}$ of a state $i$ by integrating its population
as $I_{i}=\eta_{i}\int_{0}^{\infty}\Gamma_{i}p_{i}dt$. The stationary solution
can instead be obtained by directly integrating the system of equations, from
which we obtain the following expressions for the integrated intensities of
the bright and dark states for a dark ground state:

$\displaystyle I_{\text{B}}$
$\displaystyle=\eta_{\text{B}}\Gamma_{\text{B}}\frac{p_{\text{B}}(0)\left(\Gamma_{\text{D}}+\gamma_{\uparrow}\right)+p_{\text{D}}(0)\gamma_{\uparrow}}{\left(\Gamma_{\text{B}}+\gamma_{\downarrow}\right)\left(\Gamma_{\text{D}}+\gamma_{\uparrow}\right)-\gamma_{\uparrow}\gamma_{\downarrow}}$
$\displaystyle I_{\text{D}}$
$\displaystyle=\eta_{\text{D}}\Gamma_{\text{D}}\frac{p_{\text{D}}(0)\left(\Gamma_{\text{B}}+\gamma_{\downarrow}\right)+p_{\text{B}}(0)\gamma_{\downarrow}}{\left(\Gamma_{\text{B}}+\gamma_{\downarrow}\right)\left(\Gamma_{\text{D}}+\gamma_{\uparrow}\right)-\gamma_{\uparrow}\gamma_{\downarrow}}$

where $p_{B}(0)=a$ and $p_{D}(0)=1-a$. The ratio of intensity
$I_{\text{B}}/I_{\text{D}}$ therefore reads:

$\frac{I_{\text{B}}}{I_{\text{D}}}=\frac{\eta_{\text{B}}\Gamma_{\text{B}}}{\eta_{\text{D}}\Gamma_{\text{D}}}\frac{p_{\text{B}}(0)\left(\Gamma_{\text{D}}+\gamma_{\uparrow}\right)+p_{\text{D}}(0)\gamma_{\uparrow}}{p_{\text{D}}(0)\left(\Gamma_{\text{B}}+\gamma_{\downarrow}\right)+p_{\text{B}}(0)\gamma_{\downarrow}}$
(S3)

For one-phonon transitions and for the two-phonon sum process, the ratio of
intensities as $T\rightarrow 0$ simplifies to

$\frac{I_{\text{B}}}{I_{\text{D}}}\Bigg{|}_{\text{1ph}}=\frac{I_{\text{B}}}{I_{\text{D}}}\Bigg{|}_{\text{2ph,sum}}=\frac{\eta_{\text{B}}}{\eta_{\text{D}}}\frac{a}{(1-a)+\gamma_{0}/\Gamma_{\text{B}}}$
(S4)

The low temperature intensity ratio $I_{\text{B}}/I_{\text{D}}$ therefore
depends not only on $a$ but also on $\gamma_{0}/\Gamma_{\text{B}}$.

In the case of the two-phonon difference process, the ratio of intensities as
$T\rightarrow 0$ simplifies to

$\frac{I_{\text{B}}}{I_{\text{D}}}\Bigg{|}_{\text{2ph,Raman}}=\frac{\eta_{\text{B}}}{\eta_{\text{D}}}\frac{p_{\text{B}}(0)}{p_{\text{D}}(0)}=\frac{\eta_{\text{B}}}{\eta_{\text{D}}}\frac{a}{1-a}$
(S5)

where $a=0.75$, such that regardless of the actual rates involved in the
system, the ratio of intensity between the bright and dark states is given by
$3\eta_{\text{B}}/\eta_{\text{D}}$. This is in stark contrast with the case of
one-phonon transitions and the two-phonon sum process, where the ratio of
intensities depends on the actual rates involved in the system via
$\gamma_{0}/\Gamma_{\text{B}}$.

In the case of a bright ground state, the relative intensity as $T\rightarrow
0$ reads:

$\frac{I_{\text{B}}}{I_{\text{D}}}\Bigg{|}_{\text{1ph}}=\frac{I_{\text{B}}}{I_{\text{D}}}\Bigg{|}_{\text{2ph,sum}}=\frac{\eta_{\text{B}}}{\eta_{\text{D}}}\frac{a+\gamma_{0}/\Gamma_{\text{D}}}{(1-a)}$
(S6)

The intensity ratio depends not only on $a$ but also on the ratio
$\gamma_{0}/\Gamma_{\text{D}}$.

For the second-order difference process, we find that the low-temperature
relative intensities do not depend on the actual ordering of the exciton
sublevels:

$\frac{I_{\text{B}}}{I_{\text{D}}}\Bigg{|}_{\text{2ph,Raman}}=\frac{\eta_{\text{B}}}{\eta_{\text{D}}}\frac{p_{\text{B}}(0)}{p_{\text{D}}(0)}=\frac{\eta_{\text{B}}}{\eta_{\text{D}}}\frac{a}{1-a}$
(S7)

###  SII.3 Simulation procedure

Here we detail the procedure used to obtain the characteristic temperature
$T_{c}$ in the phase sub-space where all experimental requirements are met.

Figure S7: Phase space mapping of first-order phonon transitions. (a)
Schematic of the system comprised of a ground state (G), an excited state (E),
a bright exciton state (B) and a dark exciton state (D) and the coupling rates
of the model. (b,c) Particular features of interest obtained from simulations
of the decays in the $(\gamma_{0},\,\Delta_{\text{BD}})$ space. (b) Intensity
of the long-decay component at 0 and $100\text{\,}\mathrm{K}$. (c) Long decay
time at $100\text{\,}\mathrm{K}$. (d) Characteristic temperature
$T_{c}^{\text{1ph}}$ in the phase subspace defined by the intersection of the
(b) and (c) subspaces. (e, f) Experimental decay dynamics with (e) the long
and short decay times and (f) the long decay component fractional intensity
together with a fit using parameters in (d).

We calculate the long-decay-time intensity at in the low- and high-temperature
limits, $I_{\text{rel,long}}^{\text{0 K}}$ and $I_{\text{rel,long}}^{\text{100
K}}$, as well as the long decay time in the high-temperature limit
$\tau_{\text{long}}^{100K}$. For the sake of clarity, we restrict the phase
sub-space displayed to the one matching our experimental results, i.e. we only
show the region of phase space where the long-decay-time relative intensity is
lower than 0.1 at $0\text{\,}\mathrm{K}$ and higher than 0.85 at
$100\text{\,}\mathrm{K}$ (Fig. S7b), and where the long decay time is
comprised between $150\text{\,}\mathrm{ps}$ and $350\text{\,}\mathrm{ps}$ at
$100\text{\,}\mathrm{K}$ (Fig. S7c).

The phase subspace in Fig. S7d (same as Fig. LABEL:Mfig:BoDNphmapsb) is then
defined as the intersection of the three subspaces related to the long decay
fractional intensity at low and high temperature (Fig. S7b) and to the long
decay time at high temperature (Fig. S7c). Applying the same procedure for the
opposite fine structure ordering, with a bright exciton below the dark
exciton, yields an empty intersection.

Figure S8: Additional phase space mapping of second-order processes. (a) Sum-
process with $\hbar\omega_{2}=9\hbar\omega_{1}$ such that
$\hbar\omega_{1}+\hbar\omega_{2}=\Delta_{\text{BD}}$. (b) Difference-process
with varying the lower phonon energy $E_{1}=$6\text{\,}\mathrm{meV}$$ and
$E_{1}=$9\text{\,}\mathrm{meV}$$.

For second-order transitions, the same procedure is applied to obtain Fig.
LABEL:Mfig:BoDNphmapsc. This procedure was also applied for the opposite fine
structure ordering, with the bright exciton state below the dark exciton
state, for which we find that only the two-phonon difference model yields a
non-empty intersection.

As discussed in the main text, for both second-order processes the energy of
one of the phonon modes can further be varied while ensuring
$\hbar\omega_{1}+\hbar\omega_{2}=\Delta_{\text{BD}}$ and
$\hbar\omega_{2}-\hbar\omega_{1}=\Delta_{\text{BD}}$. Fig. S8 shows that the
obtained phase spaces are similar to the ones obtained in Fig.
LABEL:Mfig:BoDNphmapsb,c. For the second-order difference process, the
obtained phase space is slightly shifted towards larger $\gamma_{0}$ as the
lowest phonon energy is increased from 3 to 6 and $9\text{\,}\mathrm{meV}$,
matching the optical phonon modes observed as replica in the spectrum.

##  SIII Additional data

###  SIII.1 Bandgap thermal dependence

Figure S9: Temperature-dependence of the emission energy. (a) Bright triplet
emission energy as a function of temperature. In red, we reproduced the
dependence recorded on bulk single crystals in ref. [14]. (b) Emission energy
dependence of the slope of $E(T)$ deduced from (a).

In the investigated temperature range, the emission energy of single CsPbBr3
NCs increases linearly with temperature (Fig. S9a) with an average slope of
$\sim$ $0.3\text{\,}\mathrm{meV}\text{\,}{\mathrm{K}}^{-1}$ (Fig. S9b),
slightly smaller than the low temperature slope measured on single bulk
crystals [14].

###  SIII.2 Decay time statistics

As discussed in the main text, the PL decay of single CsPbBr3 NCs is bi-
exponential at all temperatures with a predominant
sub-$100\text{\,}\mathrm{ps}$ decay channel at the lowest temperatures. In
Fig. S10, we show the distribution of this short decay time at
$4.5\text{\,}\mathrm{K}$ measured across $\sim$ 35 NCs.

Figure S10: Short decay time statistics at $4.5\text{\,}\mathrm{K}$.

###  SIII.3 Temperature-dependent photoluminescence

Fig. S11 shows an example of the full PL temperature-dependence recorded with
the steady-state PL in Fig. S11a and the fitted time-resolved PL in Fig. S11b.

Figure S11: Temperature-dependent photoluminescence. (a) Steady-state PL
spectra as a function of temperature. (b) Time-resolved PL as a function of
temperature with bi-exponential fits.

###  SIII.4 Dark exciton state

We performed magneto-optical spectroscopy on single CsPbBr3 NCs that reveals
the emergence of a dark singlet state below the bright triplet. Based on prior
studies [6, 15, 2, 16] and comparing finite field measurements to zero-field
measurements, we can safely identify the trion and optical phonon replica. For
Fig. S12a, at zero field we can identify the three lowest energy optical
phonon replica. Based on the bright triplet emission energy, the trion
emission is expected around $\sim~{}$30\text{\,}\mathrm{meV}$$ and is not
observed. At finite field, a weak additional peak emerges below the bright
triplet at $\sim$ $14\text{\,}\mathrm{meV}$, close to an optical phonon
replica, and is attributed to the brightened dark singlet. For Fig. S12b, we
observe the weak emergence of two peaks below the bright triplet in addition
to previously identified optical phonon replica. Based on the emission energy
of the bright triplet, the trion emission is around
$\sim~{}$23\text{\,}\mathrm{meV}$$ and we find a peak at
$\sim~{}$21.5\text{\,}\mathrm{meV}$$. It is therefore attributed to the trion
whose emission emerged after prolonged exposure to the excitation laser. The
second peak appears at finite magnetic field at
$\sim~{}$12\text{\,}\mathrm{meV}$$ with a lower energy than the previously
identified low-energy optical phonon replica and is attributed to the dark
singlet.

Figure S12: Magneto-optical spectroscopy of individual CsPbBr3 NCs at
$4.5\text{\,}\mathrm{K}$. (a) NC emitting at $2.48\text{\,}\mathrm{eV}$ and
(b) NC emitting at $2.42\text{\,}\mathrm{eV}$.

In both cases, an additional peak appears at finite magnetic field at a lower
energy than the previously identified low-energy optical phonon replica, with
an energy splitting well above the bulk bright-dark splitting of
$\sim~{}$3.5\text{\,}\mathrm{meV}$$ [16]. Using the recently determined
scaling law, we can compare our results to those obtained on weakly confined
CsPbI3 NCs [16]. Using ${E_{g}=$2.342\text{\,}\mathrm{eV}$}$ and
${R_{y}=$33\text{\,}\mathrm{meV}$}$ [17], we find normalised energies
${(E_{X}-E_{g})/R_{y}=2.36\ \text{and}\ 4.2}$ with corresponding bright-dark
splittings ${\Delta_{\text{BD}}/\left(R_{y}^{2}/E_{g}\right)=27\ \text{and}\
30}$ which fall within the range of values obtained for CsPbI3 NCs in a
similar confinement regime and therefore provides further support for these
identifications.

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# Hawking Flux from a Black Hole with Nonlinear Supertranslation Hair

Feng-Li Lin  linfl(at)gapps.ntnu.edu.tw Department of Physics, National Taiwan
Normal University, Taipei 11677, Taiwan  Shingo Takeuchi
shingo.portable(at)gmail.com Phenikaa Institute for Advanced Study and Faculty
of Basic Science, Phenikaa University, Hanoi 100000, Vietnam

###### Abstract

We study the Hawking flux from a black hole with soft hair by the anomaly
cancellation method proposed by Robinson and Wilczek. Unlike the earlier
studies considering the black hole with linear supertranslation hair, our
study takes into account the supertranslation hair to the quadratic order,
which then yields the angular dependent horizon. As a result, highly
nontrivial kinetic-mixings appear among the spherical Kaluza-Klein modes of
the (1+1)d near-horizon reduced theory, which obscures the traditional
derivation of the Hawking flux. However, after a series of field re-
definitions, we can disentangle the mode-mixings into canonical normal modes,
but the reduced metrics for these normal modes are mode-dependent. Despite of
this, the resultant Hawking flux turns out to be mode-independent and remains
the same as the Schwarzschild’s one. Thus, one cannot tell the black holes
with nonlinear supertranslation hairs from the Schwarzschild’s one by
examining the Hawking flux, so that the nonlinear soft hairs can be thought as
the microstates.

##  I Introduction

Black hole information paradox Mathur:2009hf  invoked by the discovery of
Hawking radiation Hawking:1974rv ; Hawking:1974sw , indicates our insufficient
understanding of quantum gravity and the nature of microstates accounting for
the Bekenstein-Hawking entropy Bekenstein:1973ur . The microstates by
definition should share the common macroscopic observables, i.e., the so-
called hard hairs, which in black hole physics is the mass, charge and angular
momentum, and equivalently the Hawking temperature and Hawking flux in the
thermodynamic sense. Therefore, the additional degrees of freedom accounting
for the black hole microstates can only be the so-called soft hairs, which
will not change the hard hairs but may change the near- horizon properties
such as it shape and local geometry. Accordingly, the soft hairs proposed in
Hawking:2016msc ; Strominger:2014pwa ; Strominger:2017zoo  can be microstates
if they also keep the thermal properties of black hole intact. The soft hairs
are the conserved charges of the infinite-dimensional symmetries, i.e., the
BMS symmetries Bondi:1962px ; Sachs:1962wk ; Sachs:1962zza  of the asymptotic
flat spacetime, and can be observed through the gravitational memory effect
Braginsky:1986ia ; Braginsky&Thorne ; Blanchet:1992br . Moreover, the soft
hairs being treated as microstates are also proposed in Hawking:2016msc  as a
resolution to information paradox. The key idea of Hawking:2016msc  is to view
the formation and evaporation of a black hole as the scattering process, the
infinite relations imposed by BMS symmetries are argued in Hawking:2016msc  to
be used to map the soft hairs of the collapsing matters into the outgoing
Hawking flux, and thus help to resolve the information paradox. However, the
further analysis Mirbabayi:2016axw ; Bousso:2017dny  showed that the soft
modes are decoupled from the scattering of the hard modes after appropriately
dressing the in and out states. This will preserve the Hawking flux and thus
the microstate nature of the linear soft hairs, as shown in
Javadinazhed:2018mle .

As far as we know, only two examples of exact soft-hairy black hole solutions
have been constructed. One is considered by Hawking, Perry and Strominger
(HPS) Hawking:2016sgy  by throwing the soft-hair shockwave into the
Schwarzschild black hole, and the other is explicitly constructed by Compere
and Long (CL) Compere:2016hzt  through nontrivial coordinate transformations.
Their main difference is that the HPS black hole carries linear
supertranslation hair but the CL’s one carries nonlinear one. We will show in
this study, the nonlinear effect will deform the horizon into non-spherical
shape and this non-spherical horizon will obscure the derivation of the
Hawking flux.

As shown in the next section, the surface gravity of the supertranslated black
hole remains the same with the Schwarzschild’s one. This implies the same
Hawking temperature and possibly the same Hawking flux. However, the non-
spherical horizon makes the task of evaluating Hawking flux nontrivial even if
the final result remains the same as Schwarzschild case. Therefore, we in this
paper will try to resolve the issue of evaluating the Hawking flux for the
non-spherical horizon, and clarify the role of nonlinear soft hairs as
microstates.

There are various ways to derive the Hawking radiation from conventional black
holes by considering the quantum field theory (QFT) in curved space. Besides
Hawking’s original derivation Hawking:1974sw , Wilczek and his collaborators
have developed two alternative methods of deriving the Hawking radiation.
First, a tunneling method proposed by Parikh and Wilczek Parikh:1999mf  is to
calculate the WKB amplitude of classical forbidden path to evaluate the
radiation spectrum. Later, motivated by the earlier trace-anomaly proposal
Christensen:1977jc  for 2d gravity, Wilczek and his collaborators
Robinson:2005pd ; Iso:2006wa ; Iso:2006ut  utilize the chiral nature of the
QFT in the reduced (1+1)d near horizon region due to the absence of ingoing
modes, which results in the (consistent) gravitational anomaly
AlvarezGaume:1983ig ; Bertlmann:2000da

$\nabla_{\mu}T^{\mu}{}_{\nu}=\frac{1}{\sqrt{-g}}\partial_{\mu}N^{\mu}_{\nu}$
(1)

where for the Schwarzschild-like metric $ds^{2}=-f(r)dt^{2}+dr^{2}/f(r)$, the
non-vanishing components of $N^{\mu}_{\nu}$ is

$N^{r}_{t}={1\over 192}(f^{\prime 2}+f^{\prime\prime}f),\quad
N^{t}_{r}={-1\over 192f^{2}}(f^{\prime 2}-f^{\prime\prime}f).$ (2)

Thus, an extra flux, i.e., the Hawking flux, is induced to cancel the
gravitational anomaly so that the underlying covariance is preserved.

The Hawking flux for the HPS black hole and its Vaidya type generalization has
been investigated in Chu:2018tzu  by adopting the tunneling method of
Parikh:1999mf , and in Compere:2019rof  (see also Javadinazhed:2018mle ) by
using Hawking’s original method. Both conclude that the Hawking flux from
HPS’s remains the same as the Schwarzschild’s one. In this Letter we will
consider this issue for the CL’s one by anomaly cancellation method of
Robinson:2005pd . As mentioned, the non-spherical horizon deformed by the
nonlinear soft hair poses a challenge to examine the thermal nature of its
Hawking flux. Indeed, the complication arises when performing the dimensional
reduction since the non-spherical horizon shape induces nontrivial mode-
mixings in the (1+1)d reduced theory, which prevent identifying the normal
modes and obtaining their associated Hawking flux. Despite of this, we will
show that the mode-mixings can be removed by nontrivial field-redefinitions so
that the resultant Hawking flux is still the same as the Schwarzschild’s one.

The remaining of the paper is organized as follows. In the next section, we
expand the metric for the static soft-hairy black hole Compere:2019rof  up to
the second order of the soft hair function, then transform it into the
Schwarzschild coordinate for a chosen soft hair function and check its surface
gravity to be the same as the one without soft har. In section III we perform
the dimensional reduction to obtain the near-horizon reduced theory and spell
out the structure of mode-mixings. In section IV we remove the infinite mode-
mixings by a series of nontrivial field re-definitions, and then arrive the
diagonal reduced theory, and based on which, in section V for each “normal
mode” we obtain the Hawking temperature from the reduced metric, and the
Hawking flux by the method of anomaly cancellation. We find the results are
the same as the ones for Schwarzschild black hole. In section VI we conclude
our paper and discuss why our results should hold for general quadratic soft
hair. Besides, in Appendix A we collect the details of the metric of
supertranslated black hole considered in this paper, and in Appendix B we
collect the full details of the near-horizon reduced theory, which is the
starting point up to some field re-definitions for the method of anomaly
cancellation to derive the Hawking flux.

##  II Black holes with supertranslation soft-hairs

Supertranslations are the direction-dependent time-like Killing
transformations on the null-like infinities of the asymptotically flat
spacetimes, the metric of which in the Bondi coordinates specifically for the
past null infinity ${\cal I}^{-}$ takes the following form

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle-
dv^{2}+2dvdr+r^{2}\gamma_{AB}dz^{A}dz^{B}$ (3) $\displaystyle+$
$\displaystyle{2M\over
r}dv^{2}+rC_{AB}dz^{A}dz^{B}-D^{B}C_{AB}dvdz^{A}+\cdots\qquad$

where $\gamma_{AB}$ is the metric of unit 2-sphere, $D_{A}$ is the covariant
derivative with respect to $\gamma_{AB}$. The Bondi mass aspect $M$ and the
traceless tensor $C_{AB}$ depend on $(v,z^{A})$, and characterize the high
degeneracy of classical vacua of general relativity. The associated Killing
field preserving the above asymptotic metric in the Bondi gauge is as follows:

$\xi_{C}=C\partial_{v}-{1\over r}D^{A}C\partial_{A}+{1\over
2}D^{2}C\partial_{r}$ (4)

where $C=C(z^{A})$. We can also obtain the Bondi metric for the the future
null infinity ${\cal I}^{+}$ in a similar way in terms of the outgoing
Eddington-Finkelstein coordinate $(u,r,z^{A})$.

What we will be interested is the CL type supertranslated black hole which is
the vacuum solution of Einstein equation and carries the supertranslation
charges of the above asymptotic BMS symmetries. Its metric in the isotropic
coordinate takes the following form Compere:2019rof :

$\displaystyle ds^{2}$
$\displaystyle=-\frac{\left(1-\frac{M}{2\rho_{s}}\right)^{2}}{\left(1+\frac{M}{2\rho_{s}}\right)^{2}}dt^{2}+\left(1+\frac{M}{2\rho_{s}}\right)^{4}\times$
(5)
$\displaystyle\left(d\rho+[((\rho-E)^{2}+U)\gamma_{AB}+(\rho-E)C_{AB}]dz^{A}dz^{B}\right)\qquad$

where the auxiliary radial coordinate $\rho_{s}$ is related to the radial
coordinate $\rho$ by $\rho_{s}=\sqrt{(\rho-C-C_{(0,0)})^{2}+D_{A}CD^{A}C}$,
here $C_{(0,0)}$ denotes the constant mode of $C=C(z^{A})$ in spherical
harmonic expansion. Besides,

$\displaystyle C_{AB}:=-(2D_{A}D_{B}-\gamma_{AB}D^{2})C,$ $\displaystyle
U:={1\over 8}C_{AB}C^{AB},\qquad E:={1\over 2}D^{2}C+C-C_{(0,0)}.$ (6)

This metric reduces to Schwarzschild black hole of mass $M$ in the isotropic
coordinate when setting $C=0$.

Due to the nontrivial relation between $\rho$ and and $\rho_{s}$, the metric
(5) is to all orders of $C$ in $C$-expansion, thus the supertranslation hair
here is nonlinear in contrast to the HPS’s one. Up to the first order of $C$,
$\rho=\rho_{s}+C+C_{(0,0)}+{\cal O}(C^{2})$ so that
$d\rho=d\rho_{s}+\partial_{A}Cdz^{A}$, then we can transform the metric (5) up
to the first order of $C$ to the Schwarzschild coordinate $(t,r,z^{A})$ by the
relation $r=\rho_{s}(1+{M\over 2\rho_{s}})^{2}$. The resultant metric has the
same $tt$\- and $rr$-components as the Schwarzschild metric so that the
position of the horizon and surface gravity are the same as the ones of
Schwarzschild black hole. One expects the Hawking flux is thermal as usual.
This result is the same as the one obtained in Chu:2018tzu  for the HPS black
which also carries just the linear supertranslation hair.

We now expand the metric to the second order of $C$, and examine the same
issue. For concreteness and simplicity, we consider the following soft
hair111We chose this mode because we expect it will dominate when forming a
soft-hairy black hole from the coalescence of binary black holes due to the
evidences from the empirical studies of the emitted gravitational wave in the
merger and ringdown phases, see e.g. Berti:2005ys :

$C=\varepsilon MY_{2}^{0}(\theta,\phi)=\varepsilon
M\sqrt{\frac{5}{16\pi}}\left(3\cos^{2}\theta-1\right)$ (7)

so that $C_{(0,0)}=0$. Here, $\varepsilon$ is a small parameter to keep track
of the order in the $C$ expansion of the metric. The factor of $M$ in (7) is
just our convention. Moreover, to avoid lengthy formulas, we will just write
down the metric in the near-horizon region relevant for the anomaly
cancellation method of deriving Hawking flux. The full metric is given in the
Appendix A. The resultant near-horizon metric in the Schwarzschild (not the
isotropic) coordinate up to ${\cal O}(r-r_{h})$ and ${\cal
O}(\varepsilon^{2})$ is

$ds^{2}=g_{tt}dt^{2}+g_{rr}dr^{2}+2g_{r\theta}drd\theta+g_{\theta\theta}d\theta^{2}+g_{\phi\phi}d\phi^{2}$

with

$\displaystyle g_{tt}$ $\displaystyle=$ $\displaystyle-
g_{rr}^{-1}=-{r-r_{h}\over 2M}(1-{45\varepsilon^{2}\sin^{2}2\theta\over
8\pi}),$ (8) $\displaystyle g_{\theta\theta}$ $\displaystyle=$ $\displaystyle
r^{2}\Big{(}1+6\sqrt{5\over\pi}\varepsilon\cos 2\theta+{5\varepsilon^{2}\over
16\pi}(75+69\cos 4\theta)\Big{)},$ (9) $\displaystyle g_{r\theta}$
$\displaystyle=$
$\displaystyle-3\varepsilon\sqrt{10M^{3}\over\pi(r-r_{h})}\sin
2\theta+{15\varepsilon^{2}M^{2}\sin 4\theta\over 2\pi(r-r_{h})},$ (10)
$\displaystyle g_{\phi\phi}$ $\displaystyle=$ $\displaystyle
r^{2}\sin^{2}\theta\Big{(}1+3\varepsilon\cos^{2}\theta\Big{\\{}2\sqrt{5\over\pi}+{5\over
4\pi}\varepsilon$ (11) $\displaystyle-{35\over
8\pi}(r-r_{h})\varepsilon+{55\over 4\pi}\varepsilon\cos
2\theta\Big{\\}}\Big{)}$

and

$r_{h}=2M(1-{15\varepsilon^{2}\sin^{2}2\theta\over 16\pi}).$ (12)

Note that $g_{tt}=0$ at $r=r_{h}$ so that the horizon position is shifted from
$2M$ to $r_{h}$, and the more important it is now angular dependent. To make
sure $r=r_{h}$ is indeed a null hypersurface $\cal N$, we evaluate the norm of
the normal vector $\ell$ to a family of surfaces $S=r-r_{h}$ Townsend:1997ku ,
i.e.,

$\ell^{2}\propto g^{\mu\nu}\partial_{\mu}S\partial_{\nu}S$ (13)

and find that $\ell^{2}$ indeed vanishes at $r=r_{h}+{\cal
O}(\varepsilon^{3})$. We then evaluate the surface gravity $\kappa$ for the
time-like Killing vector $\xi=\partial_{t}$ (for static solutions)
Townsend:1997ku  and obtain

$\kappa=\sqrt{-{1\over 2}(D^{\mu}\xi^{\nu})(D_{\mu}\xi_{\nu})|_{\cal
N}}={1\over 4M}+{\cal O}(\varepsilon^{3}).$ (14)

We see that the surface gravity equals to the one of Schwarzschild black hole
even though the horizon is no longer spherical symmetric. Below we will
calculate the associated Hawking flux to examine if the quadratic
supertranslation hairs can be the microstates.

##  III Near-horizon reduced theory

We will calculate the Hawking flux à la the anomaly cancellation method
Robinson:2005pd , by first finding the reduced action of a free scalar theory
in the near-horizon region of the background (II)

$\displaystyle S$ $\displaystyle=$ $\displaystyle\int
d^{4}x\sqrt{-g}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi^{*},$ (16)
$\displaystyle=$ $\displaystyle\int
d^{4}x\sqrt{-g}[g^{tt}|\partial_{t}\phi|^{2}+g^{rr}|(\partial_{r}+A_{\theta}\partial_{\theta})\phi|^{2}$
$\displaystyle\qquad\qquad-{(g^{r\theta})^{2}\over
g^{rr}}|\partial_{\theta}\phi|^{2}]$

where the last term involving ${(g^{r\theta})^{2}\over g^{rr}}$ is sub-leading
in the near horizon limit and can be neglected, and the gauge field
$A_{\theta}\equiv{g^{r\theta}\over g^{rr}}$ is the $\theta$-component not the
$t$-component so that it is irrelevant when considering the Hawking flux, thus
we will omit it in what follows.

To proceed further, we decompose the scalar field in the spherical harmonic
expansion, i.e., $\phi=\sum_{\ell,m}\phi_{lm}Y_{l}^{m}$. Therefore, the
relevant near-horizon action up to ${\cal O}(\varepsilon^{2})$ is explicitly
given by

$S_{n.h.}=\sum_{k,n,l,m}\int dtdr\int
d\Omega\;\Lambda\;Y_{k}^{n*}Y_{l}^{m}(\bar{g}^{tt}|\partial_{t}\phi|^{2}+\bar{g}^{rr}|\partial_{r}\phi|^{2})$
(17)

where

$\displaystyle\bar{g}^{tt}$ $\displaystyle=$
$\displaystyle-(\bar{g}^{rr})^{-1}=-{2M\over
r-2M}(1-{15\varepsilon^{2}M\sin^{2}2\theta\over 8\pi(r-2M)})$ (18)
$\displaystyle\Lambda$ $\displaystyle=$ $\displaystyle
4M^{2}\sin\theta\Big{(}1+12\varepsilon Y_{2}^{0}+\varepsilon^{2}({120\over
7\sqrt{\pi}}Y_{4}^{0}+{54\over 7}\sqrt{5\over\pi}Y_{2}^{0})\Big{)}.$

After performing the spherical integral, the near-horizon action involves the
mode-mixings and takes the following form:

$\displaystyle S_{n.h.}$ $\displaystyle=$ $\displaystyle\sum_{k,n,l,m}\int
d^{2}x\;4M^{2}\Lambda_{kn,lm}((g_{\rm
eff})^{tt}_{kn,lm}\partial_{t}\phi^{*}_{kn}\partial_{t}\phi_{lm}$ (19)
$\displaystyle+(t\rightarrow r))$

where

$\displaystyle\Lambda_{kn,lm}$ $\displaystyle=$ $\displaystyle\int
d\Omega\;\Lambda\;Y_{k}^{n*}Y_{l}^{m}$ (20) $\displaystyle\equiv$
$\displaystyle\Lambda_{lm}^{(0)}\delta_{kl}\delta_{mn}+\Lambda_{lm}^{(2)}\delta_{k-2,l}\delta_{mn}+\Lambda_{lm}^{(4)}\delta_{k-4,l}\delta_{mn}\qquad$

with $\Lambda^{(K)}_{kn,lm}$ is a function of $k,n,l,m$ and ${\cal
O}(\varepsilon^{K/2})$, and its explicit form is given in Appendix B. The
kinetic mixing matrices are

$(g_{\rm eff})^{tt}_{kn,lm}=-{1\over(g_{\rm
eff})^{rr}_{kn,lm}}=-\frac{2(M_{\rm eff})_{kn,\,lm}}{r-2(M_{\rm
eff})_{kn,\,lm}}+O\left(\varepsilon^{3}\right),$ (21)

with

$(M_{\rm eff})_{kn,\,lm}=M+\frac{15}{16\pi}\frac{{\cal
I}_{kn,\,lm}}{\Lambda_{kn,\,lm}}\varepsilon^{2}$ (22)

where ${\cal I}_{kn,\,lm}=\int d\Omega\;\sin^{2}2\theta\;Y_{k}^{n*}Y_{l}^{m}$.
Note that on arriving the RHS of (22), we have used the fact that $r\simeq
2M+{\cal O}(\varepsilon^{2})$ in the near horizon region222See Appendix B for
more details though it will not affect the main conclusion..

From the above we can see that the reduced near-horizon action (19) has high
level of kinetic-mixings among spherical harmonic modes. Thus, it is
impossible to interpret the kinetic mixing matrices as the reduced effective
metric and determine the Hawking temperature of the scalar modes unless we can
remove the mode-mixings. This is very different from the case of Schwarzschild
and Kerr black holes for which the reduced action has no mode-mixing and all
the modes see the same reduced metric and Hawking temperature.

As a naive try, we can truncate the kinetic mixings at some level and
diagonalize the subspace of the truncated modes. For example, a two-by-two
kinetic mixing matrix (e.g., $(g_{\rm eff})^{tt,rr}$ modulo an overall factor)
has the following structure:

$\left(\begin{array}[]{cc}1+\alpha_{0}\varepsilon+\varepsilon^{2}\mu_{0}\mp\frac{2\varepsilon^{2}\zeta_{0}M}{r-2M}&\beta_{0}\varepsilon+\varepsilon^{2}\nu_{0}\mp\frac{2\varepsilon^{2}\eta_{0}M}{r-2M}\\\
\beta_{0}\varepsilon+\varepsilon^{2}\nu_{0}\mp\frac{2\varepsilon^{2}\eta_{0}M}{r-2M}&1+\alpha_{2}\varepsilon+\varepsilon^{2}\mu_{2}\mp\frac{2\varepsilon^{2}\zeta_{2}M}{r-2M}\\\
\end{array}\right)$

where $\mp$ correspond to $tt$\- and $rr$-components, respectively. Note that
these two kinetic mixing matrices can be diagonalized simultaneously. After
diagonalization and modulo the normalization factor, we get the reduced
metrics up to ${\cal O}(\varepsilon^{2})$ seen by the normal modes labelled by
$\pm$ in the following form

$g_{\rm eff,\pm}^{tt}=-(g_{\rm eff,\pm}^{rr})^{-1}=-{2M\over r-2M}(1\pm
c_{0}\varepsilon^{2}{2M\over 2M-r})$ (23)

where $c_{0}$ is an $M$-independent constant. Using these reduced metrics we
can find the Hawking temperature via surface gravity by using (1+1)d version
of (14) and the Hawking flux by using the method of anomaly cancellation. It
is easy to see that both quantities are the same as the ones of Schwarzschild
black hole of the same mass although the two modes see different metrics.
However, this is far from the full story as we have done severe level
truncation from an infinite mode-mixings. Below we remove the mode-mixings by
field re-definitions.

##  IV Removal of mode-mixing by field re-definitions

We now perform a series of field re-definitions to disentangle the mode-
mixings and find the reduced metric for each unmixed mode. We first introduce
the shorten notation: $(A)_{lm,lm}\rightarrow(A)_{lm}$ for some quantity $A$
such as $g^{tt,rr}_{\rm eff}$ or $\Lambda^{(K)}$, then we can rewrite (19)
into the following:

$\displaystyle S_{n.h.}$ $\displaystyle=$ $\displaystyle 4M^{2}\int
dtdr\sum_{l,m}\sum_{K=0,2,4}L^{(K)}_{lm}$ (24) $\displaystyle\equiv$
$\displaystyle 4M^{2}\int dtdr\sum_{l,m}L_{lm}$ (25)

with

$L^{(K)}_{lm}=(g^{tt}_{\rm
eff})_{l+K\,m}\Lambda^{(K)}_{lm}\partial_{t}\phi^{*}_{l+K\,m}\partial_{t}\phi_{lm}+(t\rightarrow
r).$ (26)

Using the fact

$\frac{(g_{\rm eff})^{tt,rr}_{l+K\,m}}{(g_{\rm eff})^{tt,rr}_{lm}}\sim
1+\varepsilon^{2},\quad\Lambda_{lm}^{(K)}\sim\varepsilon^{K/2},$ (27)

and performing the rescaling:
$\phi_{lm}\rightarrow\phi_{lm}/\sqrt{\Lambda^{(0)}_{lm}}$, then

$\displaystyle L_{lm}$ $\displaystyle=$ $\displaystyle(g^{tt}_{\rm
eff})_{lm}\partial_{t}(\phi^{*}_{lm}+2\Gamma^{*}_{lm})\partial_{t}\phi_{lm}+(t\rightarrow
r)$ (28) $\displaystyle=$ $\displaystyle(g^{tt}_{\rm
eff})_{lm}\Big{(}|\partial_{t}\varphi_{lm}|^{2}-(\overline{\Lambda}^{(2)}_{lm})^{2}|\partial_{t}\phi_{l+2\,m}|^{2}\Big{)}$
$\displaystyle+(t\rightarrow r)$

with

$\displaystyle\varphi_{lm}$ $\displaystyle\equiv$
$\displaystyle\phi_{lm}+\Gamma_{lm},$ (29) $\displaystyle\Gamma_{lm}$
$\displaystyle\equiv$
$\displaystyle\sum_{K=2,4}\overline{\Lambda}^{(K)}_{lm}\phi_{l+K\,m},$ (30)
$\displaystyle\overline{\Lambda}^{(K)}_{lm}$ $\displaystyle=$
$\displaystyle{1\over
2}\frac{\Lambda^{(K)}_{lm}}{\sqrt{\Lambda^{(0)}_{lm}\Lambda^{(0)}_{l+K\,m}}}\sim{\cal
O}(\varepsilon^{K/2}).$ (31)

If there were no second term on the RHS of (28), we would have achieved the
goal of removing the mode-mixing and the $\varphi_{lm}$’s will be the normal
modes. However, we can “slide” the second term in the summation of
$\sum_{lm}L_{lm}$. To proceed, we first notice that

$(g^{tt}_{\rm eff})_{lm}(\overline{\Lambda}^{(2)}_{lm})^{2}=(g^{tt}_{\rm
eff})_{l+2\,m}(\overline{\Lambda}^{(2)}_{lm})^{2}+O\left(\varepsilon^{3}\right)$
(32)

by using (27) and (31). Next, we re-organize $\sum_{lm}L_{lm}$ as follows:

$\displaystyle\sum_{l,m}L_{lm}$ $\displaystyle=$
$\displaystyle(\sum_{l=0}^{1}\sum_{|m|=0}^{l}+\sum_{l=2}^{\infty}\sum_{|m|=l-1}^{l})(g^{tt}_{\rm
eff})_{lm}|\partial_{t}\varphi_{lm}|^{2}$ (33) $\displaystyle+$
$\displaystyle\sum_{l=2}^{\infty}\sum_{|m|=0}^{l-2}(g^{tt}_{\rm
eff})_{lm}\Big{(}|\partial_{t}\varphi_{lm}|^{2}-(\overline{\Lambda}^{(2)}_{l-2\,m})^{2}|\partial_{t}\phi_{lm}|^{2}\Big{)}$
$\displaystyle+(t\rightarrow r).$

The modes appearing in the RHS of the first line are all unmixed and the
normal modes are just $\varphi_{lm}$ for $l=0,1$ with $|m|=l$ and for
$l=2,\cdots,\infty$ with $|m|=l-1,l$. We thus only need to bring the RHS of
the second line into the canonical form by the following field rescaling:
$\phi_{lm}\rightarrow\phi_{lm}/\sqrt{1-(\overline{\Lambda}^{(2)}_{l-2\,m})^{2}}$
for $l=2,\cdots,\infty$ with $|m|=0,\cdots,l-2$. Besides the scaling, we
should also use the fact that under this scaling,
$\Gamma_{lm}^{(K)}\rightarrow\Gamma_{lm}^{(K)}+O\left(\varepsilon^{3}\right)$
and
$\partial_{t}\phi\,\Gamma_{lm}^{(K)*}\rightarrow\partial_{t}\phi\,\Gamma_{lm}^{(K)*}+O\left(\varepsilon^{3}\right)$
for all $l$ and $m$, where $K=2,4$, then we can achieve the “sliding” and also
ensure the already canonical modes, i.e., the $\varphi_{lm}$ in the first line
of (33), are not affected. Without the above fact, the sliding does not work
simply from the naive argument of completing the square. This indicates the
subtlety of our procedure.

Note that the above procedure of field-redefinition goes almost the same for
the $rr$-part. Therefore, after all the field re-definitions and the
“slidings”, we can bring the reduced action (19) into the following unmixed
canonical form

$S_{n.h.}=4M^{2}\int dtdr\sum_{l,m}\Big{(}(g^{tt}_{\rm
eff})_{lm}|\partial_{t}\varphi_{lm}|^{2}+(g^{rr}_{\rm
eff})_{lm}|\partial_{r}\varphi_{lm}|^{2}\Big{)}$ (34)

with $(g^{tt,rr}_{\rm eff})_{lm}$ being $(g^{tt,rr}_{\rm eff})_{lm,lm}$ given
in (21) and (22).

##  V Hawking flux à la anomaly cancellation

Based on the canonical reduced action (34), we see that each normal mode sees
different metric which depend on $l,m$. Especially, the position of the
horizon seen by the normal mode $\varphi_{lm}$ is at

$r=2(M_{\rm eff})_{lm,lm}\equiv 2(M_{\rm eff})_{lm}.$ (35)

Despite that, we can calculate the “Hawking temperature for each normal mode”
$(T_{H})_{lm}$ based on the corresponding effective reduced metric, which
remarkably turns out to be mode-independent, i.e., $(T_{H})_{lm}=T_{H}$, and
is the same as the Schwarzschild’s one up to ${\cal O}(\varepsilon^{2})$ as

$T_{H}=\Big{(}{1\over 2\pi}|\partial_{r}(g^{tt}_{\rm
eff,\pm})^{-1}|\Big{)}|_{r=2(M_{\rm eff})_{lm}}={1\over 8\pi M}+{\cal
O}(\varepsilon^{3}).$ (36)

This result is in consistent with the surface gravity obtained in (14).

For each normal mode we consider the associated Hawking flux by the method of
anomaly cancellation as for the Schwarzschild case. The only difference is now
the reduced metrics are mode-dependent. For the reduced theory (34) we can
write down the conservation law for both the consistent and covariant stress
tensors with gravitational anomaly for each normal mode, respectively,

$\displaystyle\nabla_{\mu}T^{\mu}{}_{\nu,\,lm}$
$\displaystyle=\frac{1}{96\pi\sqrt{-(g_{\rm
eff})_{lm}}}\epsilon^{\beta\delta}\partial_{\delta}\partial_{\alpha}\Gamma^{\alpha}_{\nu\beta,\,lm},$
(37) $\displaystyle\nabla_{\mu}\widetilde{T}^{\mu}{}_{\nu,\,lm}$
$\displaystyle=-\frac{1}{96\pi\sqrt{-(g_{\rm
eff})_{lm}}}\epsilon_{\mu\nu}\partial^{\mu}R_{lm}.$ (38)

By adopting the method of anomaly cancellation Robinson:2005pd ; Iso:2006wa ;
Iso:2006ut , despite of the above anomalous Ward identity for horizon modes,
we shall require the invariance of total effective action and the boundary
condition by requiring the covariant stress tensor vanishes on the horizon,
i.e.,

$(\widetilde{T})_{\mu\nu,\,lm}\big{|}_{r=(r_{h\,(eff)})_{lm}}=0,$ (39)

such that there is no out-going mode from the horizon. From the above, one can
deduce that the Hawking flux $a_{o}$ is related to the gravitational anomaly
$N^{r}_{t}$ in (2) via

$a_{o}=N^{r}_{t}|_{r=2(M_{\rm eff})_{lm}}={\pi\over 12}T_{H}^{2}.$ (40)

We can find that the Hawking flux is the same as the Schwarzschild’s one.

##  VI Discussion and Conclusion

By exploiting the anomaly cancellation method, we in this paper have shown the
Hawking flux for the non-rotating black hole with quadratic supertranslation
hair is the same as the Schwarzschild’s one even though the non-trivial mode-
mixings arise in the near-horizon reduced field theories due to the non-
spherical shape of the horizon. For concreteness we have chosen a particular
soft hair $C$ function (i.e., $Y_{2}^{0}(\theta,\phi)$) to proceed our
calculations, however our conclusion should hold for the more generic
quadratic supertranslation hairs333We indeed have tried the quadratic soft
hair with the simpler function $C=\epsilon MY^{0}_{1}(\theta,\phi)$ and find
that the Hawking flux is the same as the ones for Schwarzschild black hole,
though we do not give the details here.. This is easy to be appreciated by the
following facts: (1) as long as the soft hair $C$ function is
$\phi$-independent but still $\theta$-dependent, the near-horizon metric
should still takes the form of (8)-(11) but with different sub-leading
coefficients of $\epsilon$ expansions. (2) These changes will only affect the
mode coefficients ${\cal I}^{A_{i},B_{i},C_{i},D_{i}}_{lm}$ given in Appendix
B, which will then affect the mode-dependent part of the reduced metric (21).
(3) However, since the Hawking flux (40) does not depend on the details of the
mode-dependent part of the reduced metric, we will expect to get the same
result. (4) Thus, our result should be universal although it needs further
study if the $C$ function is also $\phi$-dependent.

Lastly, our results obtained in this study are non-trivial and against
expectations. The reason for “non-trivial” is that our result can be obtained
through the highly complicated calculation as in section IV, field re-
definitions and disentanglement of mode-mixings. The reason for “against
expectations” is that supertranslation and superrotation map one spacetime
solution of Einstein equation to another solution, and they differ by the soft
hairs which are the associated charges of supertranslation and superrotation.
The values of the soft hairs are determined from the process of the black hole
formation, thus the soft hairs are considered to correspond to the microscopic
degrees of freedom encoded in the spacetime of a black hole Compere:2016hzt ;
Hawking:2016sgy . Naively one may expect that the Hawking temperature and
Hawking flux will be modified by the addition of the degrees of freedom
associated with soft hairs.

Instead, our results show that the Hawking temperature and flux are invariant
under supertranslations, which means the Hawking temperature and flux are
independent of the supertranslation hairs, at least up to the quadratic order,
which may be related to the fact that the black hole mass (measured as the
Bondi mass) is invariant under supertranslations. This may also be the case
for the superrotations. Therefore, our results imply that the nonlinear soft
hairs can be thought as the degrees of freedom associated with the black hole
microstates, which may change the shape of the horizon but not the macroscopic
thermodynamical quantities such as the Hawking temperature and flux.

### Acknowledgements

FLL is supported by Taiwan Ministry of Science and Technology (MoST) through
Grant No. 106-2112-M-003-004-MY3, and he also thank NCTS for partial financial
support.

##  Appendix A Metric of supertranslated black hole

In this Appendix we give some details about the metric we adopt for our study.

By adopting the soft hair function $C=\varepsilon
MY_{2}^{0}(\theta,\phi)=\varepsilon
M\sqrt{\frac{5}{16\pi}}\left(3\cos^{2}\theta-1\right)$ (see also (7) in the
main text), the metric with quadratic supertranslation hair we will study take
the following form in the isotropic coordinate

$ds^{2}=-\frac{\left(1-\frac{M}{2\rho_{s}}\right)^{2}}{\left(1+\frac{M}{2\rho_{s}}\right)^{2}}dt^{2}+\left(1+\frac{M}{2\rho_{s}}\right)^{4}(d\rho^{2}+\rho^{2}d\tilde{\Omega}_{2})$
(41)

where the radial coordinate $\rho$ is related to $\rho_{s}$ by

$\displaystyle\rho_{s}^{2}=\frac{45\varepsilon^{2}M^{2}\sin^{2}(2\theta)}{16\pi}+\left(\rho-\frac{1}{8}\sqrt{\frac{5}{\pi}}\varepsilon
M\left(3\cos 2\theta+1\right)\right)^{2},$

and the metric of the deformed 2-sphere is given by

$d\tilde{\Omega}_{2}=(1+\epsilon_{\theta\theta})d\theta^{2}+\sin^{2}\theta(1+\epsilon_{\phi\phi})d\phi^{2}$
(42)

with

$\displaystyle\epsilon_{\theta\theta}$ $\displaystyle=$
$\displaystyle\frac{1}{4\rho}\sqrt{\frac{5}{\pi}}\varepsilon
M(9\cos(2\theta)-1)+\frac{5\varepsilon^{2}M^{2}(1-9\cos(2\theta))^{2}}{64\pi\rho^{2}}$
$\displaystyle+O\left(\varepsilon^{3}\right),$
$\displaystyle\epsilon_{\phi\phi}$ $\displaystyle=$ $\displaystyle{1\over
4\rho}\sqrt{5\over\pi}\varepsilon M(3\cos(2\theta)+5)+{5\over
64\pi}\varepsilon^{2}M^{2}(3\cos(2\theta)+5)^{2}$
$\displaystyle+O\left(\varepsilon^{3}\right).$

Next, we like to rewrite the metric (41) into the form in the Schwarzschild
coordinate by requiring

$\displaystyle\left(1-\frac{2\mu(\rho)}{r}\right)=\frac{\left(1-\frac{M}{2\rho_{s}}\right)^{2}}{\left(1+\frac{M}{2\rho_{s}}\right)^{2}},$
(43)
$\displaystyle\frac{1}{1-\frac{2\mu(\rho)}{r}}\left(\frac{dr}{d\rho}\right)^{2}=\left(1+\frac{M}{2\rho_{s}}\right)^{4}$
(44)

so that the $tt$ and $rr$-part of the resultant metric will take the following
form

$-\left(1-\frac{2\mu(\rho)}{r}\right)dt^{2}+\frac{1}{1-\frac{2\mu(\rho)}{r}}dr^{2}+\cdots.$

Solving the above conditions for the Schwarzschild radial coordinate $r$ and
$\mu(\rho)$, and use the solutions to perform the coordinate transformations
to Schwarzschild coordinate. After lengthy calculations, we arrive the
following metric in Schwarzschild coordinate

$ds^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}$ (45)

with nonzero components given as follows:

$\displaystyle g_{tt}=$ $\displaystyle-\left(1-\frac{2M}{r}\right)$
$\displaystyle+\frac{15\varepsilon^{2}\sin^{2}2\theta\left(M^{5}-3M^{3}\left(\sqrt{r(r-2M)}-M+r\right)^{2}\right)}{4\pi
r\left(r+\sqrt{r(r-2M)}\right)^{2}\left(\sqrt{r(r-2M)}-M+r\right)^{2}}$
$\displaystyle+O\left(\varepsilon^{3}\right),$

$\displaystyle g_{rr}=$
$\displaystyle\frac{1}{1-\frac{2M}{r}}+\frac{15\varepsilon^{2}\sin^{2}2\theta}{4\pi
M(r-2M)^{2}}\bigg{\\{}M^{3}+3M^{2}r-6Mr^{2}$
$\displaystyle-2\left(\sqrt{r^{5}(r-2M)}-r^{3}\right)+4Mr\sqrt{r(r-2M)}\bigg{\\}}+O\left(\varepsilon^{3}\right),$

$\displaystyle g_{\theta\theta}=$ $\displaystyle
r^{2}+\frac{3\sqrt{\frac{5}{\pi}}\varepsilon M\cos
2\theta\left(\sqrt{r(r-2M)}+r\right)^{4}}{2\left(\sqrt{r(r-2M)}-M+r\right)^{3}}+{N_{\theta}\over
D_{\theta}}+O\left(\varepsilon^{3}\right),$

with

$\displaystyle D_{\theta}=$ $\displaystyle 8\pi
r(r-2M)^{2}\Big{[}M^{4}-32Mr^{3}-24M\sqrt{r^{5}(r-2M)}$
$\displaystyle+20M^{2}r\left(2r+\sqrt{r(r-2M)}\right)-4M^{3}\left(4r+\sqrt{r(r-2M)}\right)$
$\displaystyle+8\left(r^{4}+\sqrt{r^{7}(r-2M)}\right)\Big{]}$

and

$\displaystyle N_{\theta}=$ $\displaystyle
15\varepsilon^{2}M^{2}\Big{[}-4M^{5}r^{2}+8M^{4}\left(2\sqrt{r^{5}(r-2M)}+13r^{3}\right)$
$\displaystyle-5M^{3}\left(32\sqrt{r^{7}(r-2M)}+73r^{4}\right)$
$\displaystyle+M^{2}\left(268\sqrt{r^{9}(r-2M)}+409r^{5}\right)$
$\displaystyle-6M\Big{(}26\sqrt{r^{11}(r-2M)}+31r^{6}\Big{)}$
$\displaystyle+30\Big{(}\sqrt{r^{13}(r-2M)}+r^{7}\Big{)}$ $\displaystyle+\cos
4\theta\Big{\\{}4M^{5}r^{2}-8M^{4}\left(2\sqrt{r^{5}(r-2M)}+r^{3}\right)$
$\displaystyle+M^{3}\left(-32\sqrt{r^{7}(r-2M)}-115r^{4}\right)$
$\displaystyle+M^{2}\left(116\sqrt{r^{9}(r-2M)}+191r^{5}\right)-102Mr^{6}$
$\displaystyle-84M\sqrt{r^{11}(r-2M)}+18\left(\sqrt{r^{13}(r-2M)}+r^{7}\right)\Big{\\}}\Big{]},$

and

$\displaystyle g_{r\theta}=$
$\displaystyle-\frac{3\sqrt{\frac{5}{\pi}}\varepsilon M\sin
2\theta\left(r+\sqrt{r(r-2M)}\right)^{4}}{8\sqrt{r(r-2M)}\left(\sqrt{r(r-2M)}-M+r\right)^{3}}$
$\displaystyle+\frac{15\varepsilon^{2}M^{4}r\sin
4\theta}{4\pi(r-2M)\left(\sqrt{r(r-2M)}-M+r\right)^{3}}+O\left(\varepsilon^{3}\right),$

$\displaystyle g_{\phi\phi}=$ $\displaystyle
r^{2}\sin^{2}\theta+\frac{3\sqrt{\frac{5}{\pi}}\varepsilon
M\sin^{2}2\theta\left(\sqrt{r(r-2M)}+r\right)^{4}}{8\left(\sqrt{r(r-2M)}-M+r\right)^{3}}$
$\displaystyle+\frac{15\varepsilon^{2}M^{2}\sin^{2}2\theta}{8\pi
r^{2}(r-2M)^{2}\left(\sqrt{r(r-2M)}-M+r\right)^{4}}H_{\phi}$
$\displaystyle+O\left(\varepsilon^{3}\right),$

with

$\displaystyle H_{\phi}=$
$\displaystyle-4M^{5}r^{3}+8M^{4}\left(2\sqrt{r^{7}(r-2M)}+7r^{4}\right)$
$\displaystyle-M^{3}\left(64\sqrt{r^{9}(r-2M)}+125r^{5}\right)$
$\displaystyle+M^{2}\left(76\sqrt{r^{11}(r-2M)}+109r^{6}\right)$
$\displaystyle-6M\left(6\sqrt{r^{13}(r-2M)}+7r^{7}\right)$
$\displaystyle+6\left(\sqrt{r^{15}(r-2M)}+r^{8}\right)$
$\displaystyle+\Big{\\{}4M^{5}r^{3}-16M^{4}\left(\sqrt{r^{7}(r-2M)}+2r^{4}\right)$
$\displaystyle+M^{3}\left(16\sqrt{r^{9}(r-2M)}+5r^{5}\right)$
$\displaystyle+M^{2}\left(20\sqrt{r^{11}(r-2M)}+41r^{6}\right)$
$\displaystyle-6M\left(4\sqrt{r^{13}(r-2M)}+5r^{7}\right)$
$\displaystyle+6\left(\sqrt{r^{15}(r-2M)}+r^{8}\right)\Big{\\}}\cos 2\theta.$

From the above, we can solve the position of the horizon $r_{h}$ from the
condition $g_{tt}(r_{h})=0$, and the result is

$\displaystyle
r_{h}=2M-\frac{15\varepsilon^{2}M\sin^{2}(2\theta)}{8\pi}+O\left(\varepsilon^{3}\right).$
(46)

We see that $r_{h}$ receive an angular-dependent correction to $r=2M$ at
$O(\varepsilon^{2})$. This angular correction of the horizon position results
in the nontrivial mode-mixing when performing the dimensional reduction.

Since we are only interested in the near-horizon region when considering the
Hawking flux by the anomaly cancellation method, we will further take the near
horizon limit of the above metric, namely, taking $r\simeq r_{h}$ and keep up
to the leading order of ${\cal O}(r-r_{h})$. The resultant near-horizon metric
is

$ds^{2}=g^{(n.h.)}_{\mu\nu}dx^{\mu}dx^{\nu}$

with the nonzero components up to $O(\varepsilon^{2})$ as follows:

$\displaystyle g^{(n.h.)}_{tt}$ $\displaystyle=$
$\displaystyle-(g^{(n.h.)}_{rr})^{-1}=-{r-r_{h}\over
2M}\left(1-{45\varepsilon^{2}\sin^{2}2\theta\over 8\pi}\right),$
$\displaystyle g^{(n.h.)}_{\theta\theta}$ $\displaystyle=$ $\displaystyle
r^{2}\Big{(}1+6\sqrt{5\over\pi}\varepsilon\cos 2\theta+{5\varepsilon^{2}\over
16\pi}(75+69\cos 4\theta)\Big{)},$ $\displaystyle g^{(n.h.)}_{r\theta}$
$\displaystyle=$
$\displaystyle-3\varepsilon\sqrt{10M^{3}\over\pi(r-r_{h})}\sin
2\theta+{15\varepsilon^{2}M^{2}\sin 4\theta\over 2\pi(r-r_{h})},$
$\displaystyle g^{(n.h.)}_{\phi\phi}$ $\displaystyle=$ $\displaystyle
r^{2}\sin^{2}\theta\Big{(}1+3\varepsilon\cos^{2}\theta\Big{\\{}2\sqrt{5\over\pi}+{5\over
4\pi}\varepsilon$ $\displaystyle-{35\over 8\pi}(r-r_{h})\varepsilon+{55\over
4\pi}\varepsilon\cos 2\theta\Big{\\}}\Big{)}.$

##  Appendix B Reduced action

In this Appendix we give some more details about the dimensionally reduced
action.

The more detailed form of the reduced action of (19) in the main text is

$\displaystyle\sum_{kn}\sum_{lm}\int
d^{2}x\,\Bigg{\\{}\phi_{kn}^{*}\Big{(}-\frac{2M}{r-2M}\Lambda_{lm,\,kn}$
$\displaystyle-\frac{15M^{2}\varepsilon^{2}}{4\pi(r-2M)^{2}}{\cal
I}^{C}_{kn,\,lm}\Big{)}\partial_{t}\partial_{t}\phi_{lm}$
$\displaystyle+\,\phi_{kn}^{*}\partial_{r}\left(\frac{r-2M}{2M}\Lambda_{lm,\,kn}-\frac{15M^{2}\varepsilon^{2}}{16\pi}{\cal
I}^{C}_{kn,\,lm}\right)\partial_{r}\phi_{lm}\Bigg{\\}}$ $\displaystyle\equiv$
$\displaystyle\sum_{kn}\sum_{lm}\int d^{2}x\,\Lambda_{kn,\,lm}\Big{(}g_{\rm
eff})^{tt}_{kn,\,lm}\partial_{t}\phi^{*}_{kn}\partial_{t}\phi_{lm}$
$\displaystyle+(g_{\rm
eff})^{rr}_{kn,\,lm}\partial_{r}\phi^{*}_{kn}\partial_{r}\phi_{lm}\Big{)}$

with

$(g_{\rm eff})^{tt}_{kn,lm}=\left(-(g_{\rm
eff})^{rr}_{kn,lm}\right)^{-1}\equiv-\frac{2M}{r-2M-\frac{{\cal
I}^{C}_{kn,\,lm}}{\Lambda_{kn,\,lm}}\frac{15M\varepsilon^{2}}{8\pi r}}$ (47)

and

$\displaystyle\Lambda_{kn,\,lm}=$
$\displaystyle\,4M^{2}\Bigg{\\{}\Bigg{(}1+\frac{3}{2}\sqrt{\frac{5}{\pi}}\left(1+3{\cal
I}^{A_{0}}_{lm}\right)\varepsilon$ $\displaystyle+\frac{45}{2\pi}\left(-{\cal
I}^{B_{0}}_{lm}+3{\cal
I}^{D_{0}}_{lm}\right)\varepsilon^{2}\Bigg{)}\delta_{kl}\delta_{mn}$
$\displaystyle+\frac{9}{2}\left(\sqrt{\frac{5}{\pi}}{\cal
I}^{A_{2}}_{lm}\varepsilon+\frac{5}{\pi}\left(-{\cal I}^{B_{2}}_{lm}+3{\cal
I}^{D_{2}}_{lm}\right)\varepsilon^{2}\right)\delta_{k-2,l}\delta_{mn}$
$\displaystyle+\frac{135}{2\pi}{\cal
I}^{D_{4}}_{lm}\varepsilon^{2}\delta_{k-4,l}\delta_{mn}\Bigg{\\}}$
$\displaystyle\equiv$
$\displaystyle\,\Lambda_{lm}^{(0)}\delta_{kl}\delta_{mn}+\Lambda_{lm}^{(2)}\delta_{k-2,l}\delta_{mn}+\Lambda_{lm}^{(4)}\delta_{k-4,l}\delta_{mn}.$

where

$\displaystyle\int d\Omega\cos 2\theta\,(Y_{k}^{n})^{*}Y_{l}^{m}$
$\displaystyle=$
$\displaystyle-\frac{4m^{2}-1}{4l^{2}+4l-3}\delta_{kl}\delta_{nm}$
$\displaystyle+\frac{2(-1)^{2m}\sqrt{\frac{((l+1)^{2}-m^{2})((l+2)^{2}-m^{2})}{(2l+1)(2l+5)}}}{2l+3}\delta_{k-2,l}\delta_{nm}$
$\displaystyle\equiv$ $\displaystyle\,{\cal
I}^{A_{0}}_{lm}\,\delta_{kl}\delta_{nm}+{\cal
I}^{A_{2}}_{lm}\,\delta_{k-2,l}\delta_{nm}\equiv{\cal I}^{A}_{lm,\,kn},$
$\displaystyle\int d\Omega\cos^{2}\theta\,(Y_{k}^{n})^{*}Y_{l}^{m}$
$\displaystyle=$
$\displaystyle\frac{2l^{2}+2l-2m^{2}-1}{4(l+1)^{2}-4(l+1)-3}\delta_{kl}\delta_{nm}$
$\displaystyle+\frac{(-1)^{2m}\sqrt{\frac{((l+1)^{2}-m^{2})((l+2)^{2}-m^{2})}{(2l+1)(2l+5)}}}{2l+3}\delta_{k-2,l}\delta_{nm}$
$\displaystyle\equiv$ $\displaystyle\,{\cal
I}^{B_{0}}_{lm}\,\delta_{kl}\delta_{nm}+{\cal
I}^{B_{2}}_{lm}\,\delta_{k-2,l}\delta_{nm}\equiv{\cal I}^{B}_{lm,\,kn},$

$\displaystyle\int d\Omega\sin^{2}(2\theta)\,(Y_{k}^{n})^{*}Y_{l}^{m}$
$\displaystyle=$ $\displaystyle 8(-1)^{2m}\delta_{kl}\delta_{nm}\times$
$\displaystyle\frac{\left(l(l+1)\left(l^{2}+l-5\right)+2l(l+1)m^{2}-3m^{4}+3\right)}{(2l-3)(2l-1)(2l+3)(2l+5)}$
$\displaystyle+4(-1)^{2m-2l}\delta_{k-2,l}\delta_{nm}\sqrt{\frac{((l+1)^{2}-m^{2})((l+2)^{2}-m^{2})}{4l^{2}+12l+5}}$
$\displaystyle\times\frac{(-1)^{2l}(2l-1)(2l+7)-4(-1)^{2m}\left(l(l+3)-7m^{2}\right)}{7(2l-1)(2l+3)(2l+7)}$
$\displaystyle-4(-1)^{2m}\sqrt{\frac{((l+1)^{2}-m^{2})((l+2)^{2}-m^{2})}{(2l+1)(2l+3)^{2}(2l+9)}}$
$\displaystyle\times\sqrt{\frac{((l+3)^{2}-m^{2})((l+4)^{2}-m^{2})}{(2l+5)^{2}(2l+7)^{2}}}\delta_{k-4,l}\delta_{nm}$
$\displaystyle\equiv$ $\displaystyle\,{\cal
I}^{C_{0}}_{lm}\,\delta_{kl}\delta_{nm}+{\cal
I}^{C_{2}}_{lm}\,\delta_{nm}\delta_{k-2,l}+{\cal
I}^{C_{4}}_{lm}\,\delta_{nm}\delta_{k-4,l}\equiv{\cal I}^{C}_{lm,\,kn},$

$\displaystyle\int d\Omega\cos
2\theta\cos^{2}\theta\,(Y_{k}^{n})^{*}Y_{l}^{m}$ $\displaystyle=$
$\displaystyle\frac{(-1)^{2m}\delta_{kl}\delta_{nm}}{(2l-3)(2l-1)(2l+3)(2l+5)}\Big{(}12m^{4}+30m^{2}+3$
$\displaystyle+2\left(-8l(l+1)m^{2}+l(l+1)(2l(l+1)-7)\right)\Big{)}$
$\displaystyle+(-1)^{2m-2l}\delta_{k-2,l}\delta_{nm}\sqrt{\frac{((l+1)^{2}-m^{2})((l+2)^{2}-m^{2})}{4l^{2}+12l+5}}$
$\displaystyle\times\frac{8(-1)^{2m}\left(l(l+3)-7m^{2}\right)+5(-1)^{2l}(2l-1)(2l+7)}{7(2l-1)(2l+3)(2l+7)}$
$\displaystyle+2(-1)^{2m}\delta_{k-4,l}\delta_{nm}\sqrt{\frac{((l+1)^{2}-m^{2})((l+2)^{2}-m^{2})}{(2l+1)(2l+3)^{2}(2l+9)}}$
$\displaystyle\times\sqrt{\frac{((l+3)^{2}-m^{2})((l+4)^{2}-m^{2})}{(2l+5)^{2}(2l+7)^{2}}}$
$\displaystyle\equiv$ $\displaystyle\,{\cal
I}^{D_{0}}_{lm}\,\delta_{kl}\delta_{nm}+{\cal
I}^{D_{2}}_{lm}\,\delta_{nm}\delta_{k-2,l}+{\cal
I}^{D_{4}}_{lm}\,\delta_{nm}\delta_{k-4,l}\equiv{\cal I}^{D}_{lm,\,kn}.$

Note that we have used the contraction rule of spherical harmonics to relate
the product of two spherical harmonics to the Clebsch-Gordan coefficients,
i.e.,

$\displaystyle\int
d\Omega~{}(Y_{l_{1}}^{m_{1}})^{*}(Y_{l_{2}}^{m_{2}})^{*}Y_{L}^{M}=\sqrt{\frac{(2l_{1}+1)(2l_{2}+1)}{4\pi(2L+1)}}$
$\displaystyle\times\langle l_{1}0~{}l_{2}0|L0\rangle\langle
l_{1}m_{1}~{}l_{2}m_{2}|LM\rangle,$ (48)

and then evaluate the above integrals explicitly. However, as shown in the
main text, the final answer for the Hawking flux will not depend on the
details of ${\cal I}^{A_{i},B_{i},C_{i},D_{i}}_{lm}$ so that our result should
be quite universal for the general quadratic supertranslation hair.

We comment that we can further simplify the reduced metric given in (47) by
approximating $r=2M+{\cal O}(\varepsilon^{2})$, and then arrive the form given
in (21) and (22) of the main text. However, the final result for the Hawking
flux will be the same for either using this approximation or not.

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