11institutetext:  University of Connecticut, 196A Auditorium Road, U-3046
Storrs, CT 06269-3046, USA
Nordita, KTH Royal Institute of Technology and Stockholm University,
Roslagstullsbacken 23, SE-106 91 Stockholm, Sweden
Department of Physics, University of Sri Jayewardenepura, Nugegoda, Sri Lanka

# Driven Quantum Matter

Gayanath W. Fernando and R. Matthias Geilhufe 1122 and Adil-Gerai Kussow 11
and W. Wasanthi P. De Silva 33112233

###### Abstract

Single- and many-electron calculations and related dynamics are presented for
a dimer and small Hubbard clusters. Floquet-Bloch picture for a periodic dimer
is discussed with regard to the time dependence of the Peierls gap and the
expectation of the current operator. In driven Fermi-Hubbard clusters, the
time dependence of Charge Gaps and phase separation along with charge pairing
at various cluster sizes indicate the presence and absence of paired electron
states. We examine the effect of electromagnetic time dependent external
perturbations on Hubbard many-electron systems in our search of for precursors
to superconducting states and time crystals. Two principally different kinds
of electromagnetic excitations are analyzed:(1) Recently demonstrated dynamic
modulation of Hubbard parameters due to excitation of certain phonon modes
within the far-infrared domain, and (2) Hubbard Hamiltonian, with fixed
parameters in an electromagnetic field, resonant with transitions between the
ground state and high-energy excited states as possible precursors to
superconductivity, within visible-near-infrared domains.

###### pacs:

73.22.-f

###### pacs:

71.10.Li

###### pacs:

73.22.Gk

##  1 Introduction

During the past decade, our understanding of periodically driven systems has
advanced significantly for example, in cold atoms, electron-phonon and
strongly correlated systems. It is now possible to use fast, time dependent
drivers, such as lasers, to dynamically induce phase/topological changes in
the electronic properties of a given material. Some of the recently developed
ultra-fast techniques have been able to drive condensed matter systems out of
equilibrium into transient novel phases. Such phenomena are related to
symmetry breaking, a well known concept of modern physics. Strongly correlated
materials give rise to a wide variety of interesting and exotic phenomena
leading to complex phase diagrams and dynamical responses. Recent reports of
photo-molecular high temperature superconductivity [1], transient
dehybridization in a $f$-electron system [2] and photo-induced changes in the
second harmonic generation signal from a transition metal oxide [3] are some
examples of correlated electron systems where emergent phases could exist
under certain conditions. Recently, breaking of time translation symmetry has
been observed in Floquet systems, for example, with emergence of a "time
crystal" through persistent oscillations of interacting spins [4].

The unifying themes of this manuscript are stable dynamical phases and their
properties in simple systems, focusing on dynamical tuning of one-and many-
particle gaps. In a periodic dimer where one-particle physics is central, the
ability to open or close gaps has implications in controlling topological and
transport properties while in small Hubbard clusters, tuning many-electron
gaps can lead to non-topological dynamical phase separation. We also examine
the dynamics of paired electron states and their direct and indirect
enhancements, possibly related to unconventional superconductivity. For
example, charge pairing (or electronic phase separation) instabilities could
produce conditions favorable to superconductivity, acting as a precursor. In
addition, several dynamical effects arising from external time dependent
sources (periodically driven and/or photo-induced) could also be favorable in
this regard and are discussed here.

Certain electronic inhomogeneities found in complex oxides [5] are regarded as
resulting from such strong correlations and could be labeled as electronic
phase separation. Here, when doping the half-filled antiferromagnetic
insulator with holes, the energy of the system becomes lower if the holes
cluster to create variations of hole rich and hole poor regions due to
antiferromagnetism. Our group was able to point out various pairing scenarios
using the Hubbard model based on several different bipartite clusters[6, 7]
using exact calculations. These results show similar phase separation at
larger doping levels even around optimal doping at the cluster level, but the
cluster calculations are invariably tied to some uncertainties due to size and
edge effects. However, with the progress of ultra-cold atom traps and the
ability to generate and control small systems, cluster calculations (using
Bose-Hubbard or Fermi-Hubbard-type Hamiltonians) can be expected to provide
useful guidance.

##  2 Dynamical Symmetry Breaking and Floquet Picture

Periodically driven quantum matter can be described in the Floquet-Bloch
formalism, based on a Hamiltonian which is invariant under discrete lattice
translations ${\bf R}_{i}$ and time translations $T$ (the period of the
driver), $H({\bf x}+{\bf R}_{i},t+T)=H({\bf x},t)$ where $T=2\pi/\omega.$ Here
$\omega$ denotes the frequency of the driver. Using the Floquet-Bloch ansatz,
we can express these states as
$\left|\Psi_{\alpha\vec{k}}\right\rangle=e^{\mathrm{i}\left(\vec{k}\cdot\vec{r}-\epsilon_{\alpha,\vec{k}}t\right)}\left|u_{\alpha,\vec{k}}(\vec{r},t)\right>$,
where $\vec{k}$ is the Bloch wave vector, $\epsilon_{\alpha,\vec{k}}$ the
quasi-energy, and $\alpha$ a band index.
$\left|u_{\alpha,\vec{k}}(\vec{r},t)\right>$ are time and lattice periodic
states and tied to Floquet-Bloch creation and annihilation operators
$c^{\dagger}_{\alpha\vec{k}}(t)$ and $c_{\alpha\vec{k}}(t)$, as well as their
Fourier components $c^{\dagger}_{\alpha\vec{k}n}$ and $c_{\alpha\vec{k}n}$,
e.g., $c_{\alpha\vec{k}}(t)=\sum_{n}e^{\mathrm{i}n\omega
t}c_{\alpha\vec{k}n}$. The time-dependent Schrödinger equation can now be
mapped to a time-independent eigenvalue problem

$\sum_{m\beta}(H_{\vec{k}\,\alpha\beta}^{n-m}-m\omega\delta_{mn}\delta_{\alpha\beta})\left|\Phi_{\beta\vec{k}}^{m}\right\rangle=\epsilon_{\alpha}\left|\Phi_{\alpha\vec{k}}^{n}\right\rangle,$
(1)

with
$\left|\Phi_{\alpha\vec{k}}^{n}\right\rangle=c^{\dagger}_{\alpha\vec{k}n}\left|0\right\rangle$.
Note that in practice, only a finite number of Fourier components is taken
into account.

Figure 1:  Floquet band structure for the dimer discussed here. Note that the
zero field bands show the typical Peierls gap at the zone boundaries for the
parameters chosen (see text). Also note the time-dependent floquet bands
(dashed lines) that close the standard Peierls gap in the dimer at the zone
edges at a field strength of $E_{0}/\omega=1.3$. The field strength has a
significant effect on the floquet band dispersions. Figure 2:  Floquet quasi-
energies vs the field strength: Note that at $E_{0}/\omega=1.3$ there are
level crossings that lead to closing the Peierls gap and transitions discussed
in the text. The Fourier modes corresponding to $n,m=-1,0,1$ are included in
the expansion in Eq. 2.

Recently, Gómez-León and Platero [11] have studied a one-dimensional,
spatially periodic dimer under a time-periodic vector potential
$A(t)=A_{0}\sin(\omega t)$ with $A_{0}=qE_{0}/\omega$ and time dependent
hopping $\tau$ (with $E_{0}$ denoting the field strength). The corresponding
time-dependent Hamiltonian at a given k-point may be written as

$H_{\bf k}(t)=\sum_{n,m}\sum_{j,l}\tau(t)_{j,l}c^{\dagger}_{{\bf k},n}c_{{\bf
k},m}\exp(i\kappa(t)\cdot(\rho_{n,j}-\rho_{m,l}))$ (2)

with $\kappa(t)=(-t,{\bf k})$ and $\rho_{n,j}=(n\omega,{\bf R}_{j}).$ Fourier
transforms of the time-dependent hopping $\tau(t)$ are related to field
induced states and this makes it possible to induce various transitions.

Figure 3: Estimated current for different field strengths. Beyond the critical
field strength of $E_{0}/\omega=1.3$, higher harmonics in the current become
dominant. This value coincides with the quasi-energy level crossing shown in
Fig. 2.

Here the time-dependent vector potential ${\bf A}(t)$ is included via the
canonical substitution $\vec{k}\rightarrow\vec{k}+e\vec{A}(t)$. Hence, it is
combined with exponentials $\exp(im\omega t)$ in a Fourier series expansion to
absorb the time dependence into the Hamiltonian. Following time averaging, the
resulting $\omega$-term in the above equation is analogous to an effective
electric field applied along the extra dimension. Some of the emerging quasi-
energies (defined modulo $\omega$) can also be thought of as being tied to
higher topological orders. Non adiabatic processes in such systems increase
the tuning possibilities with the addition of new topological phases.

##  3 Periodic Dimer and Peierls Gap

As a simple example, we focus on the one-dimensional periodic dimer of atoms
and calculate the time dependent effects on the band structure. The expected
ground state in the time independent case corresponds to atoms with equal
spacing. However, this arrangement is unstable and the well-known Peierls
transition [12] results in dimerization or the formation of short and long
bonds (alternate pairs of atoms moving closer). The undriven tight binding
model gives rise to an energy band structure

$E_{\pm}(k)=\pm t^{{}^{\prime\prime}}\sqrt{\lambda^{2}+1+2\lambda\cos(ka)}$
(3)

in the basis of two atoms with $\lambda=t^{{}^{\prime\prime}}/t^{{}^{\prime}}$
being the ratio of hopping parameters corresponding to long and short bonds
and $a$ the lattice translation distance (see Fig. 1).

In this static case, the dimerization leads to an energy gap (Peierls gap) in
the electronic bands at the edges of the Brillouin Zone which lowers the total
energy. This can be affected by a time dependent external field by inducing
metastable states in the gap region. Figs. 1 and 2 show calculated band
structure and quasi-energies for the driven case using parameters
$\lambda=0.7$ and driving frequency $\omega=1.5$ in units of hopping
$t^{{}^{\prime}}$. Note that the Floquet bands at a field strength of
$E_{0}/\omega=1.3$ close the Peierls gap at the zone boundaries due to
crossings seen in the quasi-energies at this field strength (Fig. 2).

To shed light into the gap closing, we calculate the current expectation value

$\left<J(t)\right>=\operatorname{Tr}(\rho(t)J(t)).$ (4)

The current operator is obtained as $J(t)=e\frac{\partial H_{k}(t)}{\partial
k}$, taking into account the Hamiltonian of (2). The density matrix $\rho(t)$
is calculated by composing it into $k$-dependent terms $\rho=\sum_{k}\rho_{k}$
and numerically solving the following Lindblad equation

$\frac{\mathrm{d}\rho_{k}}{\mathrm{d}t}=-\mathrm{i}\left[H_{k}(t),\rho_{k}\right]+\gamma\sum_{k^{\prime}}\left(L_{k,k^{\prime}}\rho_{k^{\prime}}L_{k^{\prime},k}^{\dagger}\right.\\\
\left.-\frac{1}{2}\left\\{L_{k^{\prime},k}L_{k^{\prime},k}^{\dagger},\rho_{k}\right\\}\right).$
(5)

The initial conditions are taken corresponding to the undriven system as
follows:

$\rho_{k}(0)=\left.\rho_{k,0}\right|_{A_{0}=0}=\sum_{n=\pm}f(E_{k\,n},\mu,\beta)\left|kn\right\rangle\left\langle
kn\right|,$ (6)

where $f(E_{k\,n},\mu,\beta)$ denotes the Fermi-Dirac distribution function
for a state with energy $E_{k\,n}$ ($n=\pm$), $\mu$ the chemical potential,
$\beta=(k_{B}T)^{-1}$. Furthermore, $\left|k\pm\right\rangle$ are eigenstates
of the undriven Hamiltonian $\left.H_{k}\right|_{A_{0}=0}$. Here,
$\left|k\,-\right\rangle$ ($\left|k\,+\right\rangle$) denote the occupied
(unoccupied) states at $\mu=0$. The Lindblad jump operators are constructed to
project contributions of unoccupied states (at no driving) to occupied states,
to ensure the system relaxes into the ground state, whenever the driving-field
is turned off,

$L_{k^{\prime}k}=\left|k^{\prime}\,-\right\rangle\left\langle k\,+\right|.$
(7)

After long enough driving, the current develops a regular pattern shown in
Fig. 3 for one driving period. For a critical field strength of
$E_{0}/\omega=1.3$, the current develops a dominant feature related to higher
harmonics. The value of the critical field strength coincides with the level
crossing of the Floquet quasi-energies shown in Fig. 2. Parameters are chosen
similarly as before, with a dissipation of $\gamma=0.05$. A Fourier analysis
shows that the current responds in odd multiples of the driving frequency,
($\omega,3\omega,\dots$). It is very likely that the kinks/distortions around
the extremal points that are indicated by arrows in the steady state current
plot (Fig. 3) at and beyond a critical value of the field strength are due to
quantum interference-related effects due to quasi-energy levels in the
vicinity of a given crossing (as suggested in Ref. [13]).

Figure 4: (a) Doubly occupied sites in the ground state vs $U$ for the
2$\times$4 Hubbard ladder at two fillings: Note that at large negative $U$
values, the curves at given fillings approach their maximum allowed double
occupancies (2 for quarter-filling and 1 for one hole off quarter-filling).
Also note that these double occupancies are fairly low compared to their
allowed maximum values for positive $U$; the inset shows a typical 2$\times$4
ladder with indicated hopping parameters and periodic boundary conditions
along its axis. (b) Charge Gaps for various 2$\times$N ladders at the
indicated fillings, around one hole off quarter-filling. Note how the
stability of the odd electron state vs paired combination varies with N.

##  4  Paired Electron States

Following the study of a one-particle gap closure in the dimer, we now focus
our attention on the driven many-electron gaps and related properties in small
Hubbard clusters. Minimal Hubbard cluster calculations for small ladders near
half-filling have clearly shown a tendency to phase separate [6, 7] leading to
pairing. Here we examine the dynamical behavior of such clusters at various
fillings for similar systems in order to examine the possibility of novel
dynamical behavior. For small clusters, we use exact diagonalization (ED) and
Lanczos [14] method to calculate the eigenvalues and eigenstates of the single
orbital (Fermi) Hubbard Hamiltonian $\mathcal{H}$ with periodic or open
boundary conditions:

$\mathcal{H}=-\sum_{<ij>\sigma}t_{ij}B_{ij\sigma}+\sum_{i}U{n}_{i\uparrow}{n}_{i\downarrow}$
(8)

Here $U$ is the onsite Coulomb repulsion while $t_{ij}$ is the hopping matrix
element from site $i$ to site $j$ and $\sigma$ denotes the spin index with
$B_{ij\sigma}={c}^{\dagger}_{i\sigma}{c}_{j\sigma}+H.c.$ In what follows, we
have considered nearest neighbor hopping (set to $t^{{}^{\prime}}$) and if
indicated, next nearest neighbor hopping (denoted by $t^{{}^{\prime\prime}}$).
Note: We reserve the symbol $t$ to denote time throughout the manuscript. In
addition, we have chosen to study the dynamics in the single orbital model due
to its simplicity and relevance to the cuprates, while acknowledging that a
multi-orbital model would be more appropriate to other classes of high
temperature superconductors.

Before including time dependent parameters in the above Hamiltonian, we focus
first on the static case observables such as Charge Gap and double occupancies
defined using the many-body eigenvalues of $\mathcal{H}$ (note that we have
only included the onsite Coulomb repulsion here but off-site repulsions can be
incorporated easily). The Charge Gap examines the stability of an $n$-electron
state with respect to $n+1$ and $n-1$ many-electron states and is defined as

${\rm Charge~{}Gap}=\Delta_{c}=E(n+1)+E(n-1)-2E(n).$

The energies $E(n)$ in the above equation represent the ground state energies
of the many-electron system at specified electron fillings. Normalized double
occupancies (summed over all the sites) or the average number of doubly
occupied sites in a given state of the cluster is given by

${\rm Double~{}Occupations}=\sum_{i}w_{i}<d_{i}^{\dagger}d_{i}>~{}{\rm
with}~{}d_{i}^{\dagger}=c_{i\uparrow}^{\dagger}c_{i\downarrow}^{\dagger}$

where $i$ represents a site index and $w_{i}$ an associated weight. The
averages shown above can be calculated for the ground state or any higher
lying excited state.

##  5 2$\times$N ladders: Static Case

We illustrate the above defined charge pairing and double occupations using
2$\times$N ladders as an example shown in Fig. 4. Note how the number of
doubly occupied sites (Fig. 4(a)) increases to their expected values at large
negative $U$ values for a 2$\times$4 ladder in the ground state (for example,
at quarter filling where $n=4$, the maximum number of doubly occupied sites =
2). We are searching for ways to induce such enhancements at positive $U$
values (see previous work on ET2X; Ref. [15]).

In Fig. 4(b), where the Charge Gap at one hole off quarter-filling vs $U$ is
shown, the negative gap regions indicate that at those $U$ values, an equally
weighted combination of $n-1$ and $n+1$ (even numbered) electron states is
more stable than the state with exactly $n$ (odd numbered) electrons in this
cluster. We identify this as a phase separation instability. With increasing
$U$, the behavior of the Charge Gap depends on the lattice size, geometry and
many electron interactions. Looking at Figure 4, we observe that for small $U$
values, the negative Charge Gap regions occur in the N = 2$\times$4,
2$\times$8 ladders at the indicated filling. For N=2$\times$8 near quarter
filling, it is energetically favorable for the mixtures of paired $n=8$ and
$n=6$ electron states to coexist instead of the $n=7$ electron state However,
note that for N=2$\times$6 and 2$\times$10 ladders, we observe the opposite
behavior at small $U$ values (i.e., absence of phase separation as defined
above). Even this behavior has its advantages since in this odd electron
state, it is easy to induce a magnetization by applying a small, external
magnetic field. When the paired states (with an even number of electrons) are
stable, these could be classified into states where the number of doubly
occupied sites is relatively high (Paired Electron Crystal (PEC) [15, 16]) or
clusters with antiferromagnetically coupled electrons (with a majority of
singly occupied sites having up or down spins). The conclusion here is that if
there is a possibility of tuning the $U$ parameter in some fashion, it is
possible to induce certain emergent states with desirable magnetic or paired
states, as described in the following sections.

Figure 5: Charge Gap at half-filling for a 5-site cluster for a single orbital
Hubbard model for various values of $t^{"}$ (nearest neighbor hopping
parameter among the outer sites). Energies are measured in units of
$t^{{}^{\prime}}$ (hopping parameter between the center and outer sites) and
open boundary conditions have been imposed here. Note that the negative Charge
Gap regions imply an electronic phase separation where mixtures of many-body
states with electron counts $n=4$ and $6$ can coexist instead of $n=5.$ This
geometry closely resembles the $\kappa$-phase molecular units in CuO4 and in
the organic compound (BEDT-TTF)2I3.

Figure 6:  The time dependence of the Charge Gap (at $t^{"}=1.5$) under a
laser pulse (as described in Equations 10 and 9) in the 5-site cluster. The
driving frequency $\omega=2.5$ in units of hopping. Note that the laser pulse
can drive the phase separation instability at $U$ values as large as 2 with
the help of a small, external magnetic field $h$, as in Eq. 9.

Figure 7:  When a time dependence is introduced to the $t^{"}$ parameter
related to hopping between outer sites of the 5-site cluster, as indicated in
the text (Eq. 11 with A=1.2), some of the negative Charge Gap regions in Fig.
7 are replaced by positive gap regions which may be identified as a
consequence of counter-adiabatic driving. A small magnetic field $h$ (in Eq.
9) can be used to detect magnetic polarizations evident in the positive gap
regions at half-filling.

##  6 Hubbard Clusters: Dynamics

The exact many-body solutions of the (static) 5-site cluster with open
boundary conditions show a high degree of sensitivity of the negative Charge
Gap regions as a function of $U$, for small $U$ values (see Fig. 5). This
provides an ideal platform for a simple enough case study of dynamics. This
cluster closely resembles a CuO4 unit as well as the $\kappa$-phase molecular
unit in the organic compound (BEDT-TTF)-compounds [15]. The Charge Gap at
certain fillings such as half-filling and below, remains negative indicating
possible phase separation and the coexistence of paired states. In order for
this to happen, the onsite interaction parameter $U_{0}$ has to be less than 2
(in units of $t^{{}^{\prime}}$) but it also depends crucially on the hopping
parameter $t^{{}^{\prime\prime}}$ between the nearest neighbor sites on the
boundary (such as between the O atoms in CuO4). Near half-filling, suppose
this cluster is driven by a time dependent pulse at frequency $\omega$ (as
described in the experimental study of Ref. [1]) with time independent
(constant) hopping $t_{ij}=1.5$ between outer sites (in units of
$t^{{}^{\prime}}$ as in Fig. 5).

Using exact diagonalization at every time step, we diagonalize the Hamiltonian

$\mathcal{H}(t)=-\sum_{<ij>\sigma}t_{ij}B_{ij\sigma}+\sum_{i}U(t){n}_{i\uparrow}{n}_{i\downarrow}+h\sum_{i}(n_{i\uparrow}-n_{i\downarrow})$
(9)

with

$U(t)=U_{0}\\{1-\sin^{2}({\omega
t})\exp({-\frac{(t-t_{0})^{2}}{(2t_{0})^{2}}}\\}.$ (10)

By monitoring the Charge Gap as a function of time and $U_{0}$, we see that
the negative regions of the gap can be sustained until the pulse vanishes (see
Fig. 7). In addition, a small, external magnetic field $h$ has been added in
order to keep track of possible magnetic polarization. Near half-filling, in
the negative gap regions, the magnetization remains null while positive gap
regions show a nonzero magnetization (due to an unpaired electron). Also, as
soon as a time variation of the hopping parameter $t^{{}^{\prime\prime}}_{ij}$
on the boundary of the cluster similar to Eq. 10 is introduced, the Charge Gap
shows positive regions for $0.8\leq A\leq 1.4$ where

$t^{{}^{\prime\prime}}_{ij}(t)=A\\{1-\sin^{2}({\omega
t})\exp({-\frac{(t-t_{0})^{2}}{(2t_{0})^{2}}}\\}.$ (11)

This behavior (shown in Fig. 7 for $A=1.2$ in Eq. 11) may be described as
being close to "counter-adiabatic" driving. It is somewhat similar to
introducing some form of non-inertial effects in a classical system, thereby
forcing it into a different path. The corresponding double occupations remain
high at intermittent times indicating the presence of a PEC. For large
$U_{0}$, it is energetically unfavorable to have doubly occupied sites (in the
ground state) due to strong Coulomb repulsion and hence it favors an
Antiferromagnetic (AFM) state. In addition, the relevant moderate $U_{0}$
values below 8 (in units of $t^{{}^{\prime}}$) and strong frustrations (i.e.,
large $t^{"}>0.8$) are also seen to lead to a possible AFM $\rightarrow$ PEC
transition (see also Refs. [15, 16]). By appropriately tuning the parameters
in this problem, the resulting electronic structure and corresponding (paired
vs magnetic) phases can be manipulated easily.

Figure 8: The calculated double occupancies (averaged doubly occupied sites)
of the ground state vs time $t$ at two fillings of the 2x4 ladder subjected to
an external laser pulse leading to excitations with Hubbard parameter $U$ (as
in Eq. 10 with $U_{0}=3$ and $\omega=3.3$ in units of hopping). Note the
enhancements beyond the ($t=0$) equilibrium values at $t=20$. Such
enhancements in double occupations are likely to favor superconductivity, as
observed by Buzzi et al. [1] for the organic material (see text). The
flattening behavior of the double occupancy at large times is due to the decay
of the laser pulse. Figure 9: The calculated double occupancies (doubly
occupied sites) as a function of the energy eigenvalues (from the ground state
to higher lying excited states) at $U_{0}=2$ and $U_{0}=10$ for a 6-site
(static) Hubbard cluster at half-filling. Note the plateaus near integral
values for $U_{0}=10$ (i.e., large positive Hubbard $U$ parameter). Similar
clusters of eigenvalues exist for $U_{0}=2$ as well in narrow bands of energy.
Note that in general, higher lying eigenstates have higher double occupancies
and using dipole-type excitation ideas discussed in the text, it may be
possible to access those states.

##  7 Optical excitations

Using mid-infrared pulses, modulation of the Hubbard $U$ parameter in an
organic Mott insulator was reported previously in Ref. [17]. Recently,
excitation of local (C=C) molecular vibrations in the CT salt
$\kappa$-ET2Cu[N(CN)2]Br have also been observed to result in a large increase
in carrier mobility and the opening of a superconducting-like optical gap [1].
Apparently, a coherence temperature T∗= 50K, far higher than the equilibrium
superconducting transition temperature TC= 12.5K, has been detected. In order
to model this dynamical behavior, we have monitored the double occupancies in
a 8-site (2$\times$4) Hubbard ladder, 6- and 5-site clusters using the time
dependent Hubbard Hamiltonian given by Eq. 9 at different fillings. The
parameters used in these calculations were obtained from the work of Buzzi et
al. [1] (for laser-induced modulation of the Hubbard $U$ parameter).

A plot of normalized double occupancies (summed over all the sites at
$U_{0})=3$) in the ground state of the 2x4 ladder,
$\sum_{i}w_{i}<d_{i}^{\dagger}d_{i}>$ (as discussed earlier), is also shown in
Fig. 8. Note how the double occupancies show a transient increase from their
equilibrium ($t=0$) value to that at $t=20$. Such enhancements in double
occupations (seen in other clusters as well) are likely to correlate with
increases in mobility and may favor superconductivity, as observed by Buzzi et
al. [1].

##  8 Dipole-type excitations of doubly occupied states of Hubbard clusters

This section is focused on manipulating double occupancies in Hubbard clusters
using methods from Quantum Optics. Based on the many-orbital, time-dependent
Schrödinger equation following Ref. [18], we demonstrate that it is possible
to reach any desirable doubly occupied configuration by a sequence of laser
pulses with a specific frequency $\omega$, and duration $\tau$. The time-
dependent Hamiltonian, $H(t)=H_{0}+H_{1}(t)$ is given by

$\mathcal{H}_{1}(t)=-e\cos(\omega t)\\{\mathbf{\mathcal{\vec{E}}_{el}}\cdot\
\sum_{s=1}^{n}{\vec{r}}_{s}\\}$ (12)

The electric field component here drives the inter-site electronic transitions
between two many-electron states $\left|i\right\rangle$ and
$\left|j\right\rangle$ of the unperturbed Hamiltonian $H_{0}$ as defined
previously. The matrix of the dipole moments $d_{ij}$ or the Rabi frequency
matrix

$\Omega_{R}^{ij}=|d_{ij}|{\mathcal{E}_{el}}/\hbar$

drives the time evolution of the Schrödinger wave function.

Here the calculated well-defined many-body energy levels, $\\{E_{j}\\}$, of
the cluster along with the dipole moment transition matrix, $\\{d_{i,j}\\}$,
are used. The transitions between the ground state, $g$, and the states with
high double occupancies (Fig. 9), $j=doc$, driven by the appropriate dipole
moment, ${d}_{g,doc}$, are excited by the laser pulse having transition
frequency, $\omega_{g,doc}=(E_{doc}-E_{g})/\hbar$. This coherent excitation of
all initially populated ground states into states with high double occupancy
requires a finite-length laser pulse with duration $\tau=\pi/\Omega_{g,doc}$,
where the characteristic frequency
$\Omega_{g,doc}=\sqrt{{\Omega_{R}}^{2}+\omega_{g,doc}^{2}}$. Here the Rabi
frequency, $\Omega_{R}=d_{g,doc}E_{0}/\hbar$, depends on both the amplitude
$E_{0}$ of the electric field of the laser, and the dipole moment,
$d_{g,doc}$, between the ground and the doubly occupied state. Our numerical
estimate of $d_{g,doc}\simeq 5er$ ($r\simeq$ being a nearest neighbor distance
and $e$ = electronic charge) yields a frequency $\Omega_{g,doc}$, with the
required duration of a pulse, $\tau\simeq 2-5$ fs. One can see that all
required parameters of laser excitation favorable to high double occupancy are
within the accessible range. Based on the numerical results from our cluster
studies (see Fig. 9), the energy difference is $\Delta E_{g,doc}\simeq 1.5U$.
Since the typical $U\simeq 2-3eV=2-3t$, the coherent excitation of doubly
occupied states requires a laser within the visible-near-infrared optical
range.

In summary, small condensed matter systems that are driven can give rise to
various emergent phases. Floquet-Bloch effects can lead to quasi-energy states
resulting in gap closures and other unexpected electronic properties. In small
Hubbard clusters, at appropriate $U$ values, it is possible to tune phase
separation, dipolar or magnetic-type instabilities either by variation of
filling, temperature and time dependent perturbations such as femto-second
laser pulses. Some of these calculations can provide insights into the
electronic properties of certain superconductors, including phase separation
and electromagnetically induced emergent phases.

###### Acknowledgements.

We acknowledge the computing resources provided by the Center for Functional
Nanomaterials, which is a U.S. DOE Office of Science Facility, at Brookhaven
National Laboratory under Contract No. DE-SC0012704 and the Swedish national
infrastructure for computing - SNIC provided by the high performance computing
center north (HPC2N). We acknowledge support from a Los Alamos grant through
the U.S. DOE. This research also used funding from the ERC synergy grant
"HERO" (No 810451), VILLUM FONDEN via the Centre of Excellence for Dirac
Materials (Grant No. 11744), the Knut and Alice Wallenberg foundation (
2019.0068) as well as the Vetenskapsrådet (no. 2017.03997). We thank Dr. R. T.
Clay for helpful suggestions and Dr. A. Cavalleri for a preprint of their
recent work. One of us (GWF) gratefully acknowledges hospitality and
discussions with Prof. Hai-Qing Lin and colleagues at the Beijing
Computational Science Research Center.

## References

  * [1] Buzzi et al., M. , Photo-molecular high temperature superconductivity, 2020, arXiv:2001.05389, Phys. Rev. X, 2020, 031028,
  * [2] Leuenberger et al., D., 2018, Phys. Rev. B, 97, 165108,
  * [3] Sheu et al., Y. -M., 2013, Phys. Rev. B, 88, 020101R,
  * [4] Zhang, J. et al., Observation of a discrete time crystal, Nature, 2017, 543, 218,
  * [5] E. Dagotto Complexity in Strongly Correlated Electronic Systems, Science, 2005, 309, 257,
  * [6] Fernando, G. W. and Kocharian, A. N. and Palandage, K. and Wang, Tun and Davenport, J. W., Phys. Rev. B., 2007, 75, 085109, American Physical Society
  * [7] Fang, Kun and Fernando, G. W. and Kocharian, A. N., Physics Letters A, 2012, 376, 538 - 543,
  * [8] Wilczeck, F., Phys. Rev. Lett., 2011, 109, 160401, American Physical Society
  * [9] Bruno, P., Phys. Rev. Lett., 2013, 110, 118901, American Physical Society
  * [10] Watanabe, H. and Oshikawa, M., Phys. Rev. Lett., 2015, 114, 251603, American Physical Society
  * [11] Gómez-León, A. and Platero, G., Phys. Rev. Lett., 2013, 110, 200403, American Physical Society,
  * [12] Ashcroft, N. W. and Mermin, N. D. Solid State Physics, Saunders College, 1976,
  * [13] Russomanno, A. and Silva, A. and Santoro, G. E. Periodic Steady Regime and Interference in a Periodically Driven Quantum System, Phys. Rev. Lett., 2012, 109, 257201
  * [14] Cullum, J.K. and Willoughby, R.A., Lanczos Algorithms for Large Symmetric Eigenvalue Computations: Volume 1, Theory, Classics in Applied Mathematics, 2002, Society for Industrial and Applied Mathematics,
  * [15] De Silva, W. Wasanthi and Gomes, N. and Mazumdar, S. and Clay, R. T., Coulomb enhancement of superconducting pair-pair correlations in a $\frac{3}{4}$-filled model for $\kappa\text{$-$}(\text{BEDT-TTF}){}_{2}X$, Phys. Rev. B, 2016, 93, 205111, American Physical Society,
  * [16] , Clay, R. T. and Mazumdar, S., Physics Reports, 2019, 788, 1-89, Elsevier,
  * [17] Singla et al., R. , THz-frequency modulation of the Hubbard U in an organic Mott insulator, 187401, 2015, 115, Physical Review Letters, American Physical Society,
  * [18] Scully, M. O. and Zubairy, M. S. Quantum Optics, Cambridge, 1997,


#  The Dark Dimension, the Swampland, and the Dark Matter Fraction Composed of
Primordial Near-Extremal Black Holes

Luis A. Anchordoqui Department of Physics and Astronomy,
Lehman College, City University of New York, NY 10468, USA  Department of
Astrophysics, American Museum of Natural History, NY 10024, USA  Ignatios
Antoniadis High Energy Physics Research Unit, Faculty of Science,
Chulalongkorn University, Bangkok 1030, Thailand  Laboratoire de Physique
Théorique et Hautes Énergies \- LPTHE
Sorbonne Université, CNRS, 4 Place Jussieu, 75005 Paris, France  Center for
Cosmology and Particle Physics, Department of Physics,
New York University, 726 Broadway, New York, NY 10003, USA  Dieter Lüst
Max–Planck–Institut für Physik, Werner–Heisenberg–Institut, 80805 München,
Germany  Arnold Sommerfeld Center for Theoretical Physics, Ludwig-Maximilians-
Universität München, 80333 München, Germany

###### Abstract

In a recent publication we studied the decay rate of primordial black holes
perceiving the dark dimension, an innovative five-dimensional (5D) scenario
that has a compact space with characteristic length-scale in the micron range.
We demonstrated that the rate of Hawking radiation of 5D black holes slows
down compared to 4D black holes of the same mass. Armed with our findings we
showed that for a species scale of ${\cal O}(10^{9}~{}{\rm GeV})$, an all-
dark-matter interpretation in terms of primordial black holes should be
feasible for black hole masses in the range $10^{14}\lesssim M/{\rm g}\lesssim
10^{21}$. As a natural outgrowth of our recent study, herein we calculate the
Hawking evaporation of near-extremal 5D black holes. Using generic entropy
arguments we demonstrate that Hawking evaporation of higher-dimensional near-
extremal black holes proceeds at a slower rate than the corresponding
Schwarzschild black holes of the same mass. Assisted by this result we show
that if there were 5D primordial near-extremal black holes in nature, then a
PBH all-dark-matter interpretation would be possible in the mass range
$10^{7}\sqrt{\beta}\lesssim M/{\rm g}\lesssim 10^{21}$, where $\beta$ is a
parameter that controls the difference between mass and charge of the
associated near-extremal black hole.

††preprint: MPP-2024-5††preprint: LMU-ASC 01/24

##  I Introduction

The Swampland program aims at understanding which are the “good” low-energy
efective field theories (EFTs) that can couple to gravity consistently (e.g.
the landscape of superstring theory vacua) and distinguish them from the “bad”
ones that cannot Vafa:2005ui . In theory space, the boundary setting apart the
good theories from those downgraded to the swampland is characterized by a set
of conjectures classifying the indispensable properties of an EFT to enable a
consistent completion into quantum gravity. These conjectures provide a
catwalk from quantum gravity to astrophysics, cosmology, and particle physics
Palti:2019pca ; vanBeest:2021lhn ; Agmon:2022thq .

For instance, the distance conjecture (DC) predicts the appearance of infinite
towers of states that become exponentially light and trigger the collapse of
the EFT at infinite distance limits in moduli space Ooguri:2006in . Associated
to the DC is the anti-de Sitter (AdS) distance conjecture, which correlates
the dark energy density to the mass scale $m$ characterizing the infinite
tower of states, $m\sim|\Lambda|^{\alpha}$, as the negative AdS vacuum energy
$\Lambda\to 0$, with $\alpha$ a positive constant of ${\cal O}(1)$
Lust:2019zwm . In addition, under the premise that this scaling behavior holds
in de Sitter (dS) – or quasi dS – space, an unbounded number of massless modes
also materialize in the limit $\Lambda\to 0$.

As demonstrated in Montero:2022prj , applying the AdS-DC to dS space could
help elucidate the radiative stability of the cosmological hierarchy
$\Lambda/M_{p}^{4}\sim 10^{-120}$, because it connects the size of the compact
space $R_{\perp}$ to the dark energy scale $\Lambda^{-1/4}$ via
$R_{\perp}\sim\lambda\ \Lambda^{-1/4}$, where the proportionality factor is
estimated to be within the range $10^{-1}<\lambda<10^{-4}$. Actually, the
previous relation between $R_{\perp}$ and $\Lambda$ derives from constraints
by theory and experiment. On the one hand, since the associated Kaluza-Klein
(KK) tower contains massive spin-2 bosons, the Higuchi bound Higuchi:1986py
provides an absolute upper limit to $\alpha$, whereas explicit string
calculations of the vacuum energy (see e.g. Itoyama:1986ei ; Itoyama:1987rc ;
Antoniadis:1991kh ; Bonnefoy:2018tcp ) yield a lower bound on $\alpha$. All in
all, the theoretical constraints lead to $1/4\leq\alpha\leq 1/2$. On the other
hand, experimental arguments (e.g. constraints on deviations from Newton’s
gravitational inverse-square law Lee:2020zjt  and neutron star heating
Hannestad:2003yd ) lead to the conclusion encapsulated in
$R_{\perp}\sim\lambda\ \Lambda^{-1/4}$; namely, that there is one extra
dimension of radius $R_{\perp}$ in the micron range, and that the lower bound
for $\alpha=1/4$ is basically saturated Montero:2022prj . A theoretical
amendment on the connection between the cosmological and KK mass scales
confirms $\alpha=1/4$ Anchordoqui:2023laz . Assembling all this together, we
can conclude that the KK tower of the new (dark) dimension opens up at the
mass scale $m_{\rm KK}\sim 1/R_{\perp}$. For the dark dimensions scenario, the
five-dimensional (5D) Planck scale (or species scale where gravity becomes
strong Dvali:2007hz ; Dvali:2007wp ; Cribiori:2022nke ; vandeHeisteeg:2023dlw
) is given by

$M_{*}\sim m_{\rm KK}^{1/3}M_{p}^{2/3}\,,$ (1)

where $M_{p}$ is the reduced Planck mass. Thus, since the size of dark
dimension in the micron scale, $m_{\rm KK}\sim 1~{}{\rm eV}$ and so $M_{*}\sim
10^{9}~{}{\rm GeV}$.

In previous work Anchordoqui:2022txe ; Anchordoqui:2022tgp , we first
calculated the decay rate of primordial black holes (PBH) perceiving the dark
dimension and demonstrated that the rate of Hawking radiation slows down
compared to 4D black holes of the same mass. Then, we used this result to show
that the mass range supporting a 5D PBH all-dark-matter interpretation is
extended compared to that in the 4D theory by 3 orders of magnitude in the low
mass region. As a natural outgrowth of this work, herein we study the Hawking
evaporation of near-extremal 5D black holes. More concretely, we generalize
the 4D results obtained in deFreitasPacheco:2023hpb  to $d$-dimensions. We
then discuss the impact of our findings in assessing the dark matter fraction
that could be composed of PBHs. Since Hawking evaporation of near-extremal 5D
black holes proceeds at a slower rate than the corresponding Schwarzschild
black holes of the same mass, we show herein that near extremality could
further relax the lower mass bound range of a PBH all-dark-matter
interpretation.

The outline of the paper is as follows. In Sec. II we summarized the results
of our previous work. In Sec. III we provide an overview of near-extremal
black holes and discuss the various charges that can potentially bring
together the inner and outer horizons. In Sec. IV we lay out a proof of
principle for primordial near-extremal black holes investigating a model in
which the charge leading to extremality is carried by dark electrons living in
the bulk. In Sec. V, we first adopt generic entropy arguments to derive the
scaling behavior of the decay rate of higher-dimensional near-extremal black
holes. After that, armed with our findings we investigate how near-extremal
black holes perceiving the dark dimension could modify the constraints on a
PBH all-dark-matter interpretation. The paper wraps up in Sec. VI with some
conclusions.

##  II Primordial black hole dark matter interpretation

It has long been speculated that black holes could be produced from the
collapse of large amplitude fluctuations in the early universe
Zeldovich:1967lct ; Hawking:1971ei ; Carr:1974nx ; Carr:1975qj . For an order
of magnitude estimate of the black hole mass $M$, we first note that the
cosmological energy density scales with time $t$ as $\rho\sim 1/(Gt^{2})$ and
the density needed for a region of mass $M$ to collapse within its
Schwarzschild radius is $\rho\sim c^{6}/(G^{3}M^{2})$, so that primordial
black holes (PBHs) would initially have around the cosmological horizon mass
Carr:2020xqk

$M\sim\frac{c^{3}t}{G}\sim 10^{15}\left(\frac{t}{10^{-23}~{}{\rm
s}}\right)~{}{\rm g}\,,$ (2)

with $M_{p}=1/\sqrt{8\pi G}$. This means that a black hole would have the
Planck mass ($M_{p}\sim 10^{-5}~{}{\rm g}$) if they formed at the Planck time
($10^{-43}~{}{\rm s}$), $1~{}M_{\odot}$ if they formed at the QCD epoch
($10^{-5}~{}{\rm s}$), and $10^{5}M_{\odot}$ if they formed at $t\sim 1~{}{\rm
s}$, comparable to the mass of the holes thought to reside in galactic nuclei.
This back-of-the-envelope calculation suggests that PBHs could span an
enormous mass range. Despite the fact that the mass spectrum of these PBHs is
yet to be shaped, on cosmological scales they would behave like a typical cold
dark matter particle.

Nevertheless, an all-dark-matter interpretation in terms of PBHs is severely
constrained by observations Carr:2020xqk ; Green:2020jor ; Villanueva-
Domingo:2021spv . Of relevance to our investigation, the extragalactic
$\gamma$-ray background Carr:2009jm  and on the CMB spectrum Clark:2016nst
constrain PBH evaporation of black holes with masses $\lesssim 10^{17}~{}{\rm
g}$, whereas the non-observation of microlensing events from the MACHO
Macho:2000nvd , EROS EROS-2:2006ryy , Kepler Griest:2013aaa , Icarus
Oguri:2017ock , OGLE Niikura:2019kqi  and Subaru-HSC Croon:2020ouk
collaborations constrain black holes with masses $\gtrsim 10^{21}~{}{\rm g}$.

Microscopic black holes of Schwarzschild radii smaller than the size of the
dark dimension are: bigger, colder, and longer-lived than a usual 4D black
hole of the same mass Argyres:1998qn . Indeed, Schwarzschild black holes
radiate all particle species lighter than or comparable to their temperature,
which in four dimensions is related to the mass of the black hole by

$T_{s}=\frac{M_{p}^{2}}{8\pi M}\sim\,\bigg{(}\frac{M}{10^{16}\,{\rm
g}}\bigg{)}^{-1}~{}{\rm MeV}\,,$ (3)

whereas for five dimensional black holes the temperature mass relation is
found to be Anchordoqui:2022txe

$T_{s}\sim\frac{1}{2\pi r_{s}}\sim\sqrt{\frac{3}{64}}\ \frac{1}{\pi}\
\frac{M_{p}\ \Lambda^{1/8}}{\lambda^{1/2}\
M^{1/2}}\sim\left(\frac{M}{10^{10}~{}{\rm g}}\right)^{-1/2}~{}{\rm MeV}\,,$
(4)

where

$r_{s}(M)\sim\frac{1}{M_{*}}\left[\frac{16}{3}\ \frac{M}{M_{*}}\
\zeta(4)\right]^{1/2}$ (5)

is the 5D Schwarzschild radius Myers:1986un  and $\zeta(4)=\pi^{2}/90$. The
second relation in (4) applies to the dark dimension scenario. For the
numerical estimate of (4), we have taken $\lambda\sim 10^{-3}$ as suggested by
astrophysical observations Anchordoqui:2022ejw ; Noble:2023mfw . It is evident
that 5D black holes are colder than 4D black holes of the same mass. The
Hawking radiation causes a 4D black hole to lose mass at the following rate
Keith:2021guq

$\displaystyle\left.\frac{dM}{dt}\right|_{\rm evap}$ $\displaystyle=$
$\displaystyle-\frac{M_{p}^{2}}{30720\ \pi\ M^{2}}\ \sum_{i}c_{i}(T_{s})\
\tilde{f}\ \Gamma_{s}$ (6) $\displaystyle\sim$ $\displaystyle-7.5\times
10^{-8}\ \left(\frac{M}{10^{16}~{}{\rm g}}\right)^{-2}\ \sum_{i}c_{i}(T_{s})\
\tilde{f}\ \Gamma_{s}~{}{\rm g/s}\,,$

whereas a 5D black hole has an evaporation rate of Anchordoqui:2022txe

$\displaystyle\left.\frac{dM}{dt}\right|_{\rm evap}$ $\displaystyle\sim$
$\displaystyle-9\ \pi^{5/4}\zeta(4)T_{s}^{2}\ \sum_{i}c_{i}(T_{s})\ \tilde{f}\
\Gamma_{s}\sim-\frac{27\ \Lambda^{1/4}\ M_{p}^{2}}{64\ \pi^{3/4}\ \lambda\ M}\
\sum_{i}c_{i}(T_{s})\ \tilde{f}\ \Gamma_{s}$ (7) $\displaystyle\sim$
$\displaystyle-1.3\times 10^{-12}\ \left(\frac{M}{10^{16}~{}{\rm
g}}\right)^{-1}\ \sum_{i}c_{i}(T_{s})\ \tilde{f}\ \Gamma_{s}~{}{\rm g/s}\,,$

where $c_{i}(T_{s})$ counts the number of internal degrees of freedom of
particle species $i$ of mass $m_{i}$ satisfying $m_{i}\ll T_{s}$,
$\tilde{f}=1$ $(\tilde{f}=7/8)$ for bosons (fermions), and where
$\Gamma_{s=1/2}\approx 2/3$ and $\Gamma_{s=1}\approx 1/4$ are the (spin-
weighted) dimensionless greybody factors normalized to the black hole surface
area Anchordoqui:2002cp . In the spirit of Emparan:2000rs , we neglect KK
graviton emission because the KK modes are excitations in the full transverse
space, and so their overlap with the small (higher-dimensional) black holes is
suppressed by the geometric factor $(r_{s}/R_{\perp})$ relative to the brane
fields. Thus, the geometric suppression precisely compensates for the enormous
number of modes, and the total contribution of all KK modes is only the same
order as that from a single brane field. On top of that, the 5D graviton has 5
helicities, but the spin-1 helicities do not have zero modes, because we
assume the compactification has $S^{1}/\mathbb{Z}_{2}$ symmetry and so the
$\pm 1$ helicities are projected out. The greybody factors of spin-2 particles
strongly suppress massless graviton emission on the brane
$\Gamma_{s=2}/\Gamma_{s=1/2}\lesssim 10^{-3}$, and the emission of $\pm 1$
helicities in the bulk is also suppressed; see e.g., Fig. 2 of Ref.
Ireland:2023zrd . Contribution from the spin-0 depends on the radion mass.
Since the addition of one scalar does not modify the order of magnitude
calculations of this work, throughout we neglect the graviton emission. At the
end of the concluding section we comment on the feasibility of detecting
graviton emission on the brane. Now, comparing (6) and (7) it is easily seen
that 5D black holes live longer than 4D black holes of the same mass.

Integrating (7) we can parametrized the 5D black hole lifetime as a function
of its mass and temperature,

$\tau_{s}\sim 14\ \Bigg{(}\frac{M}{10^{11.33}~{}{\rm g}}\Bigg{)}^{2}\
\left(\frac{4}{\sum_{i}c_{i}(T_{s})\ \tilde{f}\ \Gamma_{s}}\right)^{-1}~{}{\rm
Gyr}\,,$ (8)

where we have used (4) to estimate that $T_{s}\sim$ and therefore
$c_{i}(T_{s})$ receives a contribution of 6 from neutrinos and 2 from photons,
yielding $\sum_{i}c_{i}(T_{s})\ \tilde{f}\ \Gamma_{s}=4$. Armed with (8) it is
easily seen that for a species scale of ${\cal O}(10^{9}~{}{\rm GeV})$, an
all-dark-matter interpretation in terms of 5D black holes must be feasible for
masses in the range

$10^{14}\lesssim M/{\rm g}\lesssim 10^{21}\,.$ (9)

As we advanced in the Introduction, this range is extended compared to that in
the 4D theory by 3 orders of magnitude in the low mass region.

At this stage, it is worthwhile to point out that a stunning coincidence is
that the size of the dark dimension $R_{\perp}\sim$ wavelength of visible
light. This means that the Schwarzschild radius of 5D black holes is well
below the wavelength of light. For point-like lenses, this is the critical
length where geometric optics breaks down and the effects of wave optics
suppress the magnification, obstructing the sensitivity to 5D PBH microlensing
signals Croon:2020ouk .

##  III Near-extremal black holes

Asymptotically flat, static, and spherically symmetric charged (or rotating)
black holes can be categorized as generalizations of the popular Schwarzschild
metric. Such charged black holes carry additional quantum numbers, which make
their properties change drastically and unique new phenomena arise. A far
reaching hallmark of rotating black holes or those which are electrically
(and/or magnetically) charged is their thermodynamical property dubbed
extremality (i.e. zero temperature). Extremal black holes are in essence
stable gravitational objects with finite entropy but vanishing temperature,
and so the contribution to the gravitational energy completely originates in
the electromagnetic charges and/or rotational angular
momentum/spin.111Actually, when the temperature reduces to zero, the entropy
reduces to the logarithm of the number of degenerate ground states, which is
zero if the ground state is not degenerate Hawking:1994ii ; Page:2000dk .
Extremality also implies that the inner (Cauchy) and outer (event) horizons do
coincide, leading to a vanishing surface gravity. The Reissner-Nordström (RN)
metric describes the simplest extremal black hole, which has its mass equal to
its charge in appropriate units.

It has long been suspected that any electromagnetic charge or spin would be
lost very quickly by any 4D black hole population of primordial origin. On the
one hand, the electromagnetic charge of a Reissner-Nordstrom (RN) black hole
is spoiled by the Schwinger effect Schwinger:1951nm , which allows pair-
production of electron-positron pairs in the strong electric field outside the
black hole, leading to the discharge of the black hole and subsequent
evaporation Gibbons:1975kk ; Hiscock:1990ex . On the other hand, a rapidly
rotating Kerr black hole Kerr:1963ud  spins down to a nearly non-rotating
state before most of its mass has been given up, and therefore it does not
approach to extremal when it evaporates Page:1976ki . All in all, near-
extremal primordial RN black holes or Kerr black holes are not expected to
prevail in the universe we live in.

Adding to the story, it was pointed out in Alonso-Monsalve:2023brx  that
primordial black holes could grow by absorbing unconfined quarks and gluons.
Given Debye screening, the quark-gluon plasma must be color neutral on long
length scales $l\gg\lambda_{D}$, but could have a nontrivial distribution of
color charge across shorter length scales $l\sim\lambda_{D}$. In particular,
there could exist regions with net color charge, whose spatial extent is set
by $\lambda_{D}(T)$ Manuel:2003zr ; Manuel:2004gk . If this were the case,
then black holes would acquire a net color charge Alonso-Monsalve:2023brx .
However, after the QCD confinement transition, the medium would cease to
screen the primordial black hole enclosed charge ($\lambda_{D}\to\infty$), and
therefore it would become energetically (very) costly for any primordial black
hole to maintain its color charge.

An alternative interesting possibility is to envision a scenario where the
black hole is charged under a generic unbroken $U(1)$ symmetry (dark photon),
whose carriers (dark electrons with a mass $m^{\prime}_{e}$ and a gauge
coupling $e^{\prime}$) are always much heavier than the temperature of the
black hole Bai:2019zcd . This implies that the charge $Q$ does not get
evaporated away from the black hole and remains therefore constant. Strictly
speaking, the pair production rate per unit volume from the Schwinger effect
can be slowed down by arbitrarily decreasing $e^{\prime}$, whereas the weak
gravity conjecture (WGC) imposes a constraint on the charge per unit mass;
namely, for each conserved gauge charge there must be a sufficiently light
charge carrier such that

$e^{\prime}q/m_{e^{\prime}}\geq\sqrt{4\pi}\sqrt{(d-3)/(d-2)}M_{p}^{-(d-2)/2}\,,$
(10)

where $q$ is the integer-quantized electric charge of the particle and $M_{p}$
is the reduced Planck mass Arkani-Hamed:2006emk . Setting
$e=e^{\prime}=\sqrt{4\pi\alpha}$ the (4D) Schwinger effect together with the
WGC lead to a bound on the minimum black hole mass of near extremal black
holes with evaporation time longer than the age of Universe, $M_{ne}\gtrsim
5\times 10^{15}~{}{\rm g}(m_{e^{\prime}}/10^{9}~{}{\rm GeV})^{-2}$ Bai:2019zcd
.

The latest chapter in the story is courtesy of dS backgrounds. If a black hole
is embedded in a dS background, there is an additional bound on
$m_{e^{\prime}}$ from the festina lente (FL) conjecture Montero:2019ekk . This
is because the RN-dS line element comprises two horizons accessible to an
observer outside the black hole: (i) the familiar event horizon of the charged
black hole and (ii) the cosmological horizon. Usually, the black hole and the
cosmological horizons would have different temperatures, and so they cannot be
in thermal equilibrium. Considering large black holes whose size is comparable
to the dS radius and demanding their evaporation avoids superextremality leads
to the festina lente bound: for every charged state in the theory,

$m_{e^{\prime}}^{4}\gtrsim(e^{\prime}q)^{2}\frac{(d-1)(d-2)}{2\ M_{p}^{2-d}\
\ell_{d}^{2}}\,,$ (11)

where $\ell_{d}$ is the dS radius. This bound is satisfactorily satisfied in
our universe for the electron.

##  IV Higher-dimensional Schwinger pair production

The metric of a $d$-dimensional RN-like dS black hole has the form

$ds^{2}=U(r)\ dt^{2}-U^{-1}(r)\ dr^{2}-d\Omega_{d-2}^{2}\,,$ (12)

where $d\Omega_{d-2}$ is the line element of a flat space of $d-2$ dimensions
in spherical coordinates and

$U(r)=1-\frac{2M}{M_{*}^{d-2}\ r^{d-3}}+\frac{(e^{\prime}Q)^{2}}{4\pi\
M_{*}^{d-2}\ r^{2d-6}}-\frac{r^{2}}{\ell_{d}^{2}}\,,$ (13)

where $M_{*}$ is the $d$-dimensional Planck scale and the coupling
$e^{\prime}$ is taken as a parameter. A well motivated scenario emerges if the
SM has charges under the $U(1)$ field, such that $e^{\prime}$ becomes of the
order of SM gauge couplings divided by the square root of the volume of the
internal space.

Before proceeding, we pause to note that the FL inequality (11) remains the
same in any number of dimensions, since the gauge coupling has units of
Energy2-d/2 Montero:2021otb . However, it is important to stress that the FL
bound only applies to black holes of size comparable to the cosmological
horizon and therefore it is not of direct interest for scales smaller than
$R_{\perp}$. The dS-WGC is relevant to black holes with a horizon radius
smaller than $R_{\perp}$ Antoniadis:2020xso . However, for large $\ell_{4}$
values, dS-WGC constraints on the particle spectrum can also be safely
neglected. Hereafter, we proceed under the assumption of a (nearly) flat 5D
Minkowski background and neglect the last term in (13). We further assume that
(10) is satisfied. For details, see the Appendix.

For $d$ dimensions, the Schwinger probability (per unit volume and unit time)
of pair creation in a constant electric field is found to be

$\Gamma_{d}=\frac{(2J+1)}{(2\pi)^{d-1}}\
\sum_{n=1}^{\infty}(-1)^{(2J+1)(n+1)}\
\left(\frac{e^{\prime}E^{\prime}}{n}\right)^{d/2}\ \exp\left(-\frac{\pi
nm_{e^{\prime}}^{2}}{e^{\prime}E^{\prime}}\right)\,,$ (14)

where $E^{\prime}$ is the dark electric field,
$e^{\prime}Q=\sqrt{4\pi}M/M_{*}^{(d-2)/2}$, and $J$ is the spin of the
produced particles Bachas:1992bh . A point worth noting at this juncture is
that for $d>6$ the RN-dS solution is gravitational unstable Konoplya:2008au ;
Konoplya:2013sba  and so we focus our calculation on the interesting case of
$d=5$ that characterizes the dark dimension scenario Montero:2022prj . For
$d=5$, the spin $J$ is half-integer and so (14) can be rewritten as

$\Gamma_{5}=\frac{1}{8\pi^{4}}\sum_{n=1}^{\infty}\
\left(\frac{e^{\prime}E^{\prime}}{n}\right)^{5/2}\ \exp\left(-\frac{\pi
nm_{e^{\prime}}^{2}}{e^{\prime}E^{\prime}}\right)\,.$ (15)

It is of interest to make a comparison between the outer horizon radius of the
5D black hole,

$r_{+,5d}=\left[\frac{M+\sqrt{M^{2}-(e^{\prime}Q)^{2}\
M_{*}^{3}/(4\pi)}}{M_{*}^{3}}\right]^{1/2}=\sqrt{\frac{M}{M_{*}^{3}}}\,,$ (16)

and that of a 4D black hole $r_{+,4d}=M/M_{p}^{2}$ with the same $M$ and
$e^{\prime}Q$. It follows that $r_{+,5d}>r_{+,4d}\Leftrightarrow
M<M_{p}^{4}/M_{*}^{3}$, which if we take $M_{*}\sim 10^{9}~{}{\rm GeV}$
implies that $M<10^{45}~{}{\rm GeV}$ and $r_{+,5d}<1~{}\mu{\rm m}$. This in
turn entails that for the length scale of interest, the outer horizon of a 5D
RN black hole is larger than the corresponding 4D black hole. If this were the
case, then the electric field strength in the outer horizon would be smaller
and it would easier to suppress Schwinger production pairs in five than in
four dimensions.

##  V Higher-dimensional near-extremal black hole decay rate

The suppression of the near-extremal black hole decay rate with respect to
that of Schwarzschild black holes of the same mass advertised in the
Introduction is evident in the order of magnitude calculation that follows.

For a $d$-dimensional spacetime, the relation between the black hole entropy
$S$ and its mass $M$ is Anchordoqui:2001cg

$S=4\pi\ M\ r_{s}/(d-2)\sim(M/M_{*})^{(d-2)/(d-3)}\,.$ (17)

For a Schwarzschild black hole, the temperature scales with entropy as

$T_{s}\sim M_{*}\ S^{-1/(d-2)}$ (18)

and the black hole decay rate scales as

$\Gamma_{s}\sim T_{s}\,.$ (19)

For near-extremal black holes, however, the temperature scales as

$T_{ne}\sim\frac{c}{S}$ (20)

where $c=\sqrt{M^{2}-(e^{\prime}Q)^{2}M_{*}^{d-2}/(4\pi)}$ Cribiori:2022cho .
For $(e^{\prime}Q)M_{*}^{(d-2)/2}/(2M\sqrt{\pi})\ll 1$, it follows that $c\sim
M$, which leads to the non-extremal relation between $S$ and $c$, i.e. $c\sim
M_{*}\ S^{(d-3)/(d-2)}$. However, for the near-extremal case with
$M\sim(e^{\prime}Q)M_{*}^{(d-2)/2}/\sqrt{4\pi}$, the scaling of the $c$ and of
the temperature in terms of $S$ is considered in Basile:2024dqq , and one has
to expand the square root to see that the leading term cancels and the sub-
leading term provides

$c\sim M_{*}\ \sqrt{\beta}\ S^{(d/2-2)/(d-2)}\,,$ (21)

which in turn leads to

$T_{ne}\sim M_{*}\ \sqrt{\beta}\ S^{-d/(2d-4)}=M_{*}S^{-1/(d-2)}\
\sqrt{\beta/S}\,,$ (22)

with $M=M_{*}S^{(d-3)/(d-2)}(1+2S^{-1})$ and
$e^{\prime}QM_{*}^{(d-2)/2}/\sqrt{4\pi}=M_{*}S^{(d-3)/(d-2)}[1+(2+\beta)S^{-1}]$,
and where $\beta$ is an order-one parameter that controls the differences
between the masses and charges of particle species and hence also the
difference between mass and charge of the associated near-extremal black hole.
Therefore,

$T_{ne}=T_{s}\sqrt{\beta/S}$ (23)

and so it follows that

$\Gamma_{ne}\sim T_{ne}=\sqrt{\beta/S}\ \Gamma_{s}\,.$ (24)

Altogether, the evaporation rate of near-extremal black holes would be
suppressed by a factor of $\sqrt{\beta/S}$ with respect to that of
Schwarzschild black holes of the same mass.

Next, in line with our stated plan, we investigate how near-extremal black
holes could modify the PBH range given in Eq. (9). To do so, we consider a
black hole with $M\sim 10^{5}~{}{\rm g}$. From (4) we see that such a black
hole has a temperature $T_{s}\sim 707~{}{\rm MeV}$. This means that
$c_{i}(T_{s})$ receives a contribution of 2 from photons, 6 from neutrinos, 4
from electrons, 4 from muons, 12 from quarks (up, down, and strange), and 8
from gluons, yielding $\sum_{i}c_{i}(T_{s})\ \tilde{f}\ \Gamma_{s}=53/3$.
Substituting these figures into (8) we find that the lifetime of a
$10^{5}~{}{\rm g}$ Schwarzschild black hole is $\tau_{s}\sim 4\times
10^{-5}~{}{\rm yr}$. For a near extremal black hole of the same mass, the
temperature would be $T_{ne}\sim 10^{-6}\sqrt{\beta}~{}{\rm eV}$, where we
have used (23). Bearing this in mind we find that the lifetime of the near-
extremal black hole would be $\tau_{ne}\sim 20/\sqrt{\beta}~{}{\rm Gyr}$. Now,
a rough order of magnitude estimate suggests that if there were 5D primordial
near-extremal black holes in nature, then a PBH all-dark-matter interpretation
would be possible in the mass range

$10^{7}\sqrt{\beta}\lesssim M/{\rm g}\lesssim 10^{21}\,.$ (25)

Note that by tuning the $\beta$ parameter we can have a PBH all-dark-matter
interpretation with very light 5D black holes. Note also that

$\hat{c}=c/M\sim\sqrt{\beta/S}\,,$ (26)

which quantifies the near-extremality, is very small because of the large
entropy.

##  VI Conclusions

We have studied the decay rate of near-extremal black holes within the context
of the dark dimension. Using generic entropy arguments we have demonstrated
that Hawking evaporation of higher-dimensional near-extremal black holes
proceeds at a slower rate than the corresponding Schwarzschild black holes of
the same mass. Armed with our findings we have shown that if there were 5D
primordial near-extremal black holes in nature, then a PBH all-dark-matter
interpretation would be possible in the mass range $10^{7}\sqrt{\beta}\lesssim
M/{\rm g}\lesssim 10^{21}$, where $\beta$ is a parameter that controls the
difference between mass and charge of the associated near-extremal black hole.

The possible existence of near-extremal PBHs evaporating today remains an open
question. We have discussed herein an interesting possibility in which the
black hole is charged under a generic unbroken $U(1)$ symmetry of the dark
dimension, whose carriers are always much heavier than the temperature of the
black hole, and so the charge does not get evaporated away from the black hole
and remains therefore constant. Alternatively, it has been speculated in
Arbey:2019jmj  that PBHs may have been formed with a spin above the Thorne’s
limit $a_{*}<0.998$ of astrophysical objects Thorne:1974ve , and actually near
the Kerr extremal value $a_{*}<1$ set by the third law of thermodynamics
Bardeen:1973gs , where $a_{*}=a/M$, with $a\equiv J_{k}/M$ the spin parameter
and $J_{k}$ the black hole angular momentum. If this were the case, then PBHs
may be still spinning today. Further investigation along these lines is
obviously important to be done.

We end with an observation. The spectrum of graviton emission from black hole
evaporation peaks at a frequency which is an order one factor times the
temperature of a Schwarzschild black hole, $\omega_{\rm peak}\sim T_{s}$
Ireland:2023avg . For ultra-light black holes $M\sim 10M_{*}$, the spectrum
peaks at $\omega_{\rm peak}\sim M_{*}(M_{*}/M)^{1/2}\sim 10^{8}~{}{\rm GeV}$.
It was recently speculated that in scenarios with large-extra dimensions
graviton emission from ultra-light PBHs may be observed by future
gravitational wave detectors Ireland:2023zrd . Here we generalized the
estimate of Ireland:2023zrd  to the dark dimension scenario. Firstly, we note
that after accounting for the redshift in energy density and frequency due to
the cosmological expansion between evaporation and today the gravitational
wave spectrum of a $10M_{*}$ PBH would have a peak at a frequency of
$10^{12}\lesssim f/{\rm Hz}\lesssim 10^{14}$; see Fig. 4 of Ref.
Ireland:2023zrd . This frequency is in the range of JURA Beacham:2019nyx  and
OSQAR II OSQAR:2015qdv  experiments. Secondly, the gravitational wave energy
density can be estimated from Fig. 5 of Ref. Ireland:2023zrd  and is given by
$10^{-8}<\Omega_{\rm GW}h^{2}<10^{-6}$. Finally, we note that such a
gravitational wave energy density is orders of magnitude below the current
sensitivity of JURA and OSQAR II Ireland:2023avg .

## Acknowledgements

The work of L.A.A. is supported by the U.S. National Science Foundation (NSF
Grant PHY-2112527). The work of D.L. is supported by the Origins Excellence
Cluster and by the German-Israel-Project (DIP) on Holography and the
Swampland.

## Appendix

For completness, in this Appendix we provide of concise summary of the salient
characteristics of 4D RN-dS black holes. For $d=4$, the blackening function
(13) is given by

$U(r)=1-\frac{2GM}{r}+\frac{(e^{\prime}Q)^{2}G}{4\pi\ \
r^{2}}-\frac{r^{2}}{\ell_{4}^{2}}\,.$ (27)

With the change of variables ${\cal M}=GM$ and ${\cal
Q}=\sqrt{G}Qe^{\prime}/\sqrt{4\pi}$ we can rewrite (27) in the compact form

$U(r)=1-\frac{2{\cal M}}{r}+\frac{{\cal
Q}^{2}}{r^{2}}-\frac{r^{2}}{\ell_{4}^{2}}\,.$ (28)

4D RN-dS configurations generally admit three horizons, which are located at
$r=r_{h}$ where (28) vanishes, i.e. $\left.U(r)\right|_{r=r_{h}}=0$, yielding
a quartic polynomial. The number of real roots is dictated by the sign of the
discriminant locus $D$ of the quartic polynomial

$\frac{D}{16}=\frac{{\cal M}^{2}}{\ell_{4}^{2}}-\frac{{\cal
Q}^{2}}{\ell_{4}^{2}}-\frac{27{\cal M}^{4}}{\ell_{4}^{4}}+\frac{36{\cal
M}^{2}{\cal Q}^{2}}{\ell_{4}^{4}}-\frac{8{\cal
Q}^{4}}{\ell_{4}^{4}}-\frac{16Q^{6}}{\ell_{4}^{6}}\,.$ (29)

For $D\geq 0$, the quartic polynimial has four real-valued roots. However, one
of them is always negative and therefore unphysical. Then, the spacetime can
have a maximum of three causal horizons, which are dubbed: the Cauchy (a.k.a.
inner) horizon $r_{-}$, the event (a.k.a. outer) horizon $r_{+}$, and the
cosmological horizon $r_{c}$. Note that $r_{+}$ and $r_{c}$ are the horizons
which are accessible to an observer outside of the black hole.

Following Romans:1991nq , we define the phase space of 4D RN-dS black holes as
the 3D parameter space spanned by the mass ${\cal M}$, charge ${\cal Q}$ and
de Sitter radius $\ell_{4}$. To respect the Cosmic Censorship Conjecture
Penrose:1969pc , we require ${\cal M}\geq 0$ and $D\geq 0$, which ensures that
all three horizons are real and satisfy $r_{-}\leq r_{+}\leq r_{c}$. We also
require $\ell_{4}\geq 0$ to exclude AdS. We refer to the region that respects
these conditions as physical phase space $D$.

The confluence of two or the three horizons defines an extremal limit at the
boundary of the physical phase space, that we denote by $\partial D$ and is
characterized by $D=0$. There are three extremal limits dubbed cold
($r_{-}=r_{+}$), Narai ($r_{+}=r_{c}$) and ultracold ($r_{-}=r_{+}=r_{c}$).
The near horizon geometry for each of the extremal limits are AdS${}_{2}\times
S^{2}$, dS${}_{2}\times S^{2}$, and Mink${}_{2}\times S^{2}$, see e.g.
Castro:2022cuo . In Fig. 1 we show the space of 4D RN-dS solutions. The shaded
area is usually referred to as “shark fin” due to its shape.

Figure 1: The family of 4D RN-dS black holes. The gray shaded region
represents the physical phase space of sub-extremal solutions. The boundary of
this allow region has two branches: the left (or cold) branch corresponds to
RN-dS extremal black holes and the right branch corresponds to charged Nariai
black holes, for which the event and cosmological horizons coincide. The blue
star where the two branches intersect, stands for the ultracold solution. The
dashed line indicates the lukewarm solutions with ${\cal M}=|{\cal Q}|$, where
the Cauchy and event horizons have the same temperature. The ${\cal Q}=0$ axis
of neutral black holes indicates a big crunch singularity. Adapted from
Montero:2019ekk .

The ultracold near-extremal limit of the shark fin diagram is a moduli space
point that represents Minkowski spacetime and lies at an infinite distance of
any other spacetime independently of the geodesic path used to reach it
Luben:2020wix . The distance to the ultracold geometry is then consistent with
the AdS-DC Lust:2019zwm . On the other hand, the geometric distance of any
spacetime in the 4D RN-dS family to the origin is finite Luben:2020wix . This
implies that black holes will evaporate back to empty de Sitter space if the
FL bound is satisfied.

On the other hand, in the small curvature limit the dS-WGC implies that there
is at least one state with mass $m$ and charge $q$ satisfying

$m^{2}<\frac{(e^{\prime}q)^{2}}{4\pi
G}-\frac{(e^{\prime}q)^{4}}{12\pi^{2}\ell_{4}^{2}}-\frac{G(e^{\prime}q)^{6}}{32\pi^{3}\ell_{4}^{4}}+{\cal
O}\left(\frac{1}{\ell_{4}^{6}}\right)\,,$ (30)

which reproduces the known WGC in flat space

$e^{\prime}q>\sqrt{4\pi G}\
m\left(1+\frac{G^{2}m^{2}}{2\ell_{4}^{2}}+\cdots\right)\,,$ (31)

for $\ell_{4}\to\infty$ Antoniadis:2020xso .

## References

  * (1) C. Vafa,  The string landscape and the swampland, [arXiv:hep-th/0509212 [hep-th]].
  * (2) M. van Beest, J. Calderón-Infante, D. Mirfendereski and I. Valenzuela,  Lectures on the Swampland Program in string compactifications, Phys. Rept. 989, 1-50 (2022) doi:10.1016/j.physrep.2022.09.002 [arXiv:2102.01111 [hep-th]].
  * (3) E. Palti,  The swampland: introduction and review, Fortsch. Phys. 67, no.6, 1900037 (2019) doi:10.1002/prop.201900037 [arXiv:1903.06239 [hep-th]].
  * (4) N. B. Agmon, A. Bedroya, M. J. Kang and C. Vafa,  Lectures on the string landscape and the swampland, [arXiv:2212.06187 [hep-th]].
  * (5) H. Ooguri and C. Vafa,  On the Geometry of the String Landscape and the Swampland, Nucl. Phys. B 766, 21-33 (2007) doi:10.1016/j.nuclphysb.2006.10.033 [arXiv:hep-th/0605264 [hep-th]].
  * (6) D. Lüst, E. Palti and C. Vafa,  AdS and the Swampland, Phys. Lett. B 797, 134867 (2019) doi:10.1016/j.physletb.2019.134867 [arXiv:1906.05225 [hep-th]].
  * (7) M. Montero, C. Vafa and I. Valenzuela,  The dark dimension and the Swampland, JHEP 02, 022 (2023) doi:10.1007/JHEP02(2023)022 [arXiv:2205.12293 [hep-th]].
  * (8) A. Higuchi,  Forbidden mass range for spin-2 field theory in de Sitter space-time, Nucl. Phys. B 282 (1987), 397-436 doi:10.1016/0550-3213(87)90691-2
  * (9) H. Itoyama and T. R. Taylor,  Supersymmetry restoration in the compactified $O(16)\times O(16)$-prime heterotic string theory, Phys. Lett. B 186 (1987), 129-133 doi:10.1016/0370-2693(87)90267-X
  * (10) H. Itoyama and T. R. Taylor,  Small cosmological constant in string models, FERMILAB-CONF-87-129-T.
  * (11) I. Antoniadis and C. Kounnas,  Superstring phase transition at high temperature, Phys. Lett. B 261 (1991), 369-378 doi:10.1016/0370-2693(91)90442-S
  * (12) Q. Bonnefoy, E. Dudas and S. Lüst,  On the weak gravity conjecture in string theory with broken supersymmetry, Nucl. Phys. B 947 (2019), 114738 doi:10.1016/j.nuclphysb.2019.114738 [arXiv:1811.11199 [hep-th]].
  * (13) J. G. Lee, E. G. Adelberger, T. S. Cook, S. M. Fleischer and B. R. Heckel,  New test of the gravitational $1/r^{2}$ law at separations down to 52 $\mu$m, Phys. Rev. Lett. 124, no.10, 101101 (2020) doi:10.1103/PhysRevLett.124.101101 [arXiv:2002.11761 [hep-ex]].
  * (14) S. Hannestad and G. G. Raffelt,  Supernova and neutron star limits on large extra dimensions reexamined, Phys. Rev. D 67, 125008 (2003) [erratum: Phys. Rev. D 69, 029901 (2004)] doi:10.1103/PhysRevD.69.029901 [arXiv:hep-ph/0304029 [hep-ph]].
  * (15) L. A. Anchordoqui, I. Antoniadis, D. Lüst and S. Lüst,  On the cosmological constant, the KK mass scale, and the cut-off dependence in the dark dimension scenario, Eur. Phys. J. C 83, no.11, 1016 (2023) doi:10.1140/epjc/s10052-023-12206-2 [arXiv:2309.09330 [hep-th]].
  * (16) G. Dvali,  Black holes and large $N$ species solution to the hierarchy problem, Fortsch. Phys. 58, 528-536 (2010) doi:10.1002/prop.201000009 [arXiv:0706.2050 [hep-th]].
  * (17) G. Dvali and M. Redi,  Black hole bound on the number of species and quantum gravity at LHC, Phys. Rev. D 77, 045027 (2008) doi:10.1103/PhysRevD.77.045027 [arXiv:0710.4344 [hep-th]].
  * (18) N. Cribiori, D. Lüst and G. Staudt,  Black hole entropy and moduli-dependent species scale, Phys. Lett. B 844, 138113 (2023) doi:10.1016/j.physletb.2023.138113 [arXiv:2212.10286 [hep-th]].
  * (19) D. van de Heisteeg, C. Vafa, M. Wiesner and D. H. Wu,  Species scale in diverse dimensions, [arXiv:2310.07213 [hep-th]].
  * (20) L. A. Anchordoqui, I. Antoniadis and D. Lüst,  Dark dimension, the swampland, and the dark matter fraction composed of primordial black holes, Phys. Rev. D 106, no.8, 086001 (2022) doi:10.1103/PhysRevD.106.086001 [arXiv:2206.07071 [hep-th]].
  * (21) L. A. Anchordoqui, I. Antoniadis and D. Lüst,  The dark universe: Primordial black hole $\leftrightharpoons$ dark graviton gas connection, Phys. Lett. B 840, 137844 (2023) doi:10.1016/j.physletb.2023.137844 [arXiv:2210.02475 [hep-th]].
  * (22) J. A. de Freitas Pacheco, E. Kiritsis, M. Lucca and J. Silk,  Quasiextremal primordial black holes are a viable dark matter candidate, Phys. Rev. D 107, no.12, 123525 (2023) doi:10.1103/PhysRevD.107.123525 [arXiv:2301.13215 [astro-ph.CO]].
  * (23) Y. B. Zel’dovich and I. D. Novikov,  The hypothesis of cores retarded during expansion and the hot cosmological model, Soviet Astron. AJ (Engl. Transl. ), 10, 602 (1967)
  * (24) S. Hawking,  Gravitationally collapsed objects of very low mass, Mon. Not. Roy. Astron. Soc. 152, 75 (1971)
  * (25) B. J. Carr and S. W. Hawking,  Black holes in the early Universe, Mon. Not. Roy. Astron. Soc. 168, 399-415 (1974)
  * (26) B. J. Carr,  The primordial black hole mass spectrum, Astrophys. J. 201, 1-19 (1975) doi:10.1086/153853
  * (27) B. Carr and F. Kuhnel,  Primordial black holes as dark matter: Recent developments, Ann. Rev. Nucl. Part. Sci. 70 (2020), 355-394 doi:10.1146/annurev-nucl-050520-125911 [arXiv:2006.02838 [astro-ph.CO]].
  * (28) A. M. Green and B. J. Kavanagh,  Primordial Black Holes as a dark matter candidate, J. Phys. G 48 (2021) no.4, 043001 doi:10.1088/1361-6471/abc534 [arXiv:2007.10722 [astro-ph.CO]].
  * (29) P. Villanueva-Domingo, O. Mena and S. Palomares-Ruiz,  A brief review on primordial black holes as dark matter, Front. Astron. Space Sci. 8, 87 (2021) doi:10.3389/fspas.2021.681084 [arXiv:2103.12087 [astro-ph.CO]].
  * (30) B. J. Carr, K. Kohri, Y. Sendouda and J. Yokoyama,  New cosmological constraints on primordial black holes, Phys. Rev. D 81, 104019 (2010) doi:10.1103/PhysRevD.81.104019 [arXiv:0912.5297 [astro-ph.CO]].
  * (31) S. Clark, B. Dutta, Y. Gao, L. E. Strigari and S. Watson,  Planck constraint on relic primordial black holes, Phys. Rev. D 95, no.8, 083006 (2017) doi:10.1103/PhysRevD.95.083006 [arXiv:1612.07738 [astro-ph.CO]].
  * (32) R. A. Allsman et al. [MACHO],  MACHO project limits on black hole dark matter in the 1-30 solar mass range, Astrophys. J. Lett. 550, L169 (2001) doi:10.1086/319636 [arXiv:astro-ph/0011506 [astro-ph]].
  * (33) P. Tisserand et al. [EROS-2],  Limits on the MACHO content of the Galactic halo from the EROS-2 survey of the Magellanic clouds, Astron. Astrophys. 469, 387-404 (2007) doi:10.1051/0004-6361:20066017 [arXiv:astro-ph/0607207 [astro-ph]].
  * (34) K. Griest, A. M. Cieplak and M. J. Lehner,  Experimental limits on primordial black hole dark matter from the first 2 yr of Kepler data, Astrophys. J. 786, no.2, 158 (2014) doi:10.1088/0004-637X/786/2/158 [arXiv:1307.5798 [astro-ph.CO]].
  * (35) M. Oguri, J. M. Diego, N. Kaiser, P. L. Kelly and T. Broadhurst,  Understanding caustic crossings in giant arcs: characteristic scales, event rates, and constraints on compact dark matter, Phys. Rev. D 97, no.2, 023518 (2018) doi:10.1103/PhysRevD.97.023518 [arXiv:1710.00148 [astro-ph.CO]].
  * (36) H. Niikura, M. Takada, S. Yokoyama, T. Sumi and S. Masaki,  Constraints on Earth-mass primordial black holes from OGLE 5-year microlensing events, Phys. Rev. D 99, no.8, 083503 (2019) doi:10.1103/PhysRevD.99.083503 [arXiv:1901.07120 [astro-ph.CO]].
  * (37) D. Croon, D. McKeen, N. Raj and Z. Wang,  Subaru-HSC through a different lens: Microlensing by extended dark matter structures, Phys. Rev. D 102, no.8, 083021 (2020) doi:10.1103/PhysRevD.102.083021 [arXiv:2007.12697 [astro-ph.CO]].
  * (38) P. C. Argyres, S. Dimopoulos and J. March-Russell,  Black holes and submillimeter dimensions, Phys. Lett. B 441 (1998), 96-104 doi:10.1016/S0370-2693(98)01184-8 [arXiv:hep-th/9808138 [hep-th]].
  * (39) R. C. Myers and M. J. Perry,  Black holes in higher dimensional space-times, Annals Phys. 172, 304 (1986) doi:10.1016/0003-4916(86)90186-7
  * (40) L. A. Anchordoqui,  Dark dimension, the swampland, and the origin of cosmic rays beyond the Greisen-Zatsepin-Kuzmin barrier, Phys. Rev. D 106 (2022) no.11, 116022 doi:10.1103/PhysRevD.106.116022 [arXiv:2205.13931 [hep-ph]].
  * (41) N. T. Noble, J. F. Soriano and L. A. Anchordoqui,  Probing the Dark Dimension with Auger data, Phys. Dark Univ. 42 (2023), 101278 doi:10.1016/j.dark.2023.101278 [arXiv:2306.03666 [hep-ph]].
  * (42) C. Keith and D. Hooper,  511 keV excess and primordial black holes, Phys. Rev. D 104 (2021) no.6, 063033 doi:10.1103/PhysRevD.104.063033 [arXiv:2103.08611 [astro-ph.CO]].
  * (43) L. Anchordoqui and H. Goldberg,  Black hole chromosphere at the CERN LHC, Phys. Rev. D 67 (2003), 064010 doi:10.1103/PhysRevD.67.064010 [arXiv:hep-ph/0209337 [hep-ph]].
  * (44) R. Emparan, G. T. Horowitz and R. C. Myers,  Black holes radiate mainly on the brane, Phys. Rev. Lett. 85, 499-502 (2000) doi:10.1103/PhysRevLett.85.499 [arXiv:hep-th/0003118 [hep-th]].
  * (45) A. Ireland, S. Profumo and J. Scharnhorst,  Gravitational waves from primordial black hole evaporation with large extra dimensions, [arXiv:2312.08508 [gr-qc]].
  * (46) S. W. Hawking, G. T. Horowitz and S. F. Ross,  Entropy, Area, and black hole pairs, Phys. Rev. D 51 (1995), 4302-4314 doi:10.1103/PhysRevD.51.4302 [arXiv:gr-qc/9409013 [gr-qc]].
  * (47) D. N. Page,  Thermodynamics of near extreme black holes, [arXiv:hep-th/0012020 [hep-th]].
  * (48) J. S. Schwinger,  On gauge invariance and vacuum polarization, Phys. Rev. 82, 664-679 (1951) doi:10.1103/PhysRev.82.664
  * (49) G. W. Gibbons,  Vacuum Polarization and the Spontaneous Loss of Charge by Black Holes, Commun. Math. Phys. 44, 245-264 (1975) doi:10.1007/BF01609829
  * (50) W. A. Hiscock and L. D. Weems,  Evolution of Charged Evaporating Black Holes, Phys. Rev. D 41, 1142 (1990) doi:10.1103/PhysRevD.41.1142
  * (51) R. P. Kerr,  Gravitational field of a spinning mass as an example of algebraically special metrics, Phys. Rev. Lett. 11 (1963), 237-238 doi:10.1103/PhysRevLett.11.237
  * (52) D. N. Page,  Particle Emission Rates from a Black Hole II: Massless Particles from a Rotating Hole, Phys. Rev. D 14, 3260-3273 (1976) doi:10.1103/PhysRevD.14.3260
  * (53) E. Alonso-Monsalve and D. I. Kaiser,  Primordial Black Holes with QCD Color Charge, [arXiv:2310.16877 [hep-ph]].
  * (54) C. Manuel and S. Mrowczynski,  Local equilibrium of the quark gluon plasma, Phys. Rev. D 68, 094010 (2003) doi:10.1103/PhysRevD.68.094010 [arXiv:hep-ph/0306209 [hep-ph]].
  * (55) C. Manuel and S. Mrowczynski,  Whitening of the quark gluon plasma, Phys. Rev. D 70, 094019 (2004) doi:10.1103/PhysRevD.70.094019 [arXiv:hep-ph/0403024 [hep-ph]].
  * (56) Y. Bai and N. Orlofsky,  Primordial Extremal Black Holes as Dark Matter, Phys. Rev. D 101, no.5, 055006 (2020) doi:10.1103/PhysRevD.101.055006 [arXiv:1906.04858 [hep-ph]].
  * (57) N. Arkani-Hamed, L. Motl, A. Nicolis and C. Vafa,  The String landscape, black holes and gravity as the weakest force, JHEP 06, 060 (2007) doi:10.1088/1126-6708/2007/06/060 [arXiv:hep-th/0601001 [hep-th]].
  * (58) M. Montero, T. Van Riet and G. Venken,  Festina Lente: EFT Constraints from Charged Black Hole Evaporation in de Sitter, JHEP 01, 039 (2020) doi:10.1007/JHEP01(2020)039 [arXiv:1910.01648 [hep-th]].
  * (59) M. Montero, C. Vafa, T. Van Riet and G. Venken,  The FL bound and its phenomenological implications, JHEP 10, 009 (2021) doi:10.1007/JHEP10(2021)009 [arXiv:2106.07650 [hep-th]].
  * (60) I. Antoniadis and K. Benakli,  Weak gravity conjecture in de Sitter space-time, Fortsch. Phys. 68 (2020) no.9, 2000054 doi:10.1002/prop.202000054 [arXiv:2006.12512 [hep-th]].
  * (61) C. Bachas and M. Porrati,  Pair creation of open strings in an electric field, Phys. Lett. B 296, 77-84 (1992) doi:10.1016/0370-2693(92)90806-F [arXiv:hep-th/9209032 [hep-th]].
  * (62) R. A. Konoplya and A. Zhidenko,  Instability of higher dimensional charged black holes in the de-Sitter world, Phys. Rev. Lett. 103, 161101 (2009) doi:10.1103/PhysRevLett.103.161101 [arXiv:0809.2822 [hep-th]].
  * (63) R. A. Konoplya and A. Zhidenko,  Instability of $D$-dimensional extremally charged Reissner-Nordstrom(-de Sitter) black holes: Extrapolation to arbitrary $D$, Phys. Rev. D 89, no.2, 024011 (2014) doi:10.1103/PhysRevD.89.024011 [arXiv:1309.7667 [hep-th]].
  * (64) L. A. Anchordoqui, J. L. Feng, H. Goldberg and A. D. Shapere,  Black holes from cosmic rays: Probes of extra dimensions and new limits on TeV scale gravity, Phys. Rev. D 65 (2002), 124027 doi:10.1103/PhysRevD.65.124027 [arXiv:hep-ph/0112247 [hep-ph]].
  * (65) N. Cribiori, M. Dierigl, A. Gnecchi, D. Lüst and M. Scalisi,  Large and small non-extremal black holes, thermodynamic dualities, and the Swampland, JHEP 10 (2022), 093 doi:10.1007/JHEP10(2022)093 [arXiv:2202.04657 [hep-th]].
  * (66) I. Basile, N. Cribiori, D. Lüst and C. Montella,  Minimal black holes and species thermodynamics, [arXiv:2401.06851 [hep-th]].
  * (67) A. Arbey, J. Auffinger and J. Silk,  Evolution of primordial black hole spin due to Hawking radiation, Mon. Not. Roy. Astron. Soc. 494 (2020) no.1, 1257-1262 doi:10.1093/mnras/staa765 [arXiv:1906.04196 [astro-ph.CO]].
  * (68) K. S. Thorne,  Disk accretion onto a black hole II: Evolution of the hole, Astrophys. J. 191 (1974), 507-520 doi:10.1086/152991
  * (69) J. M. Bardeen, B. Carter and S. W. Hawking,  The Four laws of black hole mechanics, Commun. Math. Phys. 31 (1973), 161-170 doi:10.1007/BF01645742
  * (70) A. Ireland, S. Profumo and J. Scharnhorst,  Primordial gravitational waves from black hole evaporation in standard and nonstandard cosmologies, Phys. Rev. D 107 (2023) no.10, 104021 doi:10.1103/PhysRevD.107.104021 [arXiv:2302.10188 [gr-qc]].
  * (71) J. Beacham, C. Burrage, D. Curtin, A. De Roeck, J. Evans, J. L. Feng, C. Gatto, S. Gninenko, A. Hartin and I. Irastorza, et al. Physics Beyond Colliders at CERN: Beyond the Standard Model Working Group Report, J. Phys. G 47 (2020) no.1, 010501 doi:10.1088/1361-6471/ab4cd2 [arXiv:1901.09966 [hep-ex]].
  * (72) R. Ballou et al. [OSQAR],  New exclusion limits on scalar and pseudoscalar axionlike particles from light shining through a wall, Phys. Rev. D 92 (2015) no.9, 092002 doi:10.1103/PhysRevD.92.092002 [arXiv:1506.08082 [hep-ex]].
  * (73) L. J. Romans,  Supersymmetric, cold and lukewarm black holes in cosmological Einstein-Maxwell theory, Nucl. Phys. B 383 (1992), 395-415 doi:10.1016/0550-3213(92)90684-4 [arXiv:hep-th/9203018 [hep-th]].
  * (74) R. Penrose,  Gravitational collapse: The role of general relativity, Riv. Nuovo Cim. 1 (1969), 252-276 doi:10.1023/A:1016578408204
  * (75) A. Castro, F. Mariani and C. Toldo,  Near-extremal limits of de Sitter black holes, JHEP 07 (2023), 131 doi:10.1007/JHEP07(2023)131 [arXiv:2212.14356 [hep-th]].
  * (76) M. Lüben, D. Lüst and A. R. Metidieri,  The black hole entropy distance conjecture and black hole evaporation, Fortsch. Phys. 69 (2021) no.3, 2000130 doi:10.1002/prop.202000130 [arXiv:2011.12331 [hep-th]].