# Quantum battery based on quantum discord at room temperature

C. Cruz<EMAIL_ADDRESS>Grupo de Informação Quântica, Centro de
Ciências Exatas e das Tecnologias, Universidade Federal do Oeste da Bahia -
Campus Reitor Edgard Santos. Rua Bertioga, 892, Morada Nobre I, 47810-059
Barreiras, Bahia, Brasil.  M. F. Anka<EMAIL_ADDRESS>Instituto de
Física, Universidade Federal Fluminense, Av. Gal. Milton Tavares de Souza s/n,
24210-346 Niterói, Rio de Janeiro, Brasil.  M. S. Reis<EMAIL_ADDRESS>Instituto de Física, Universidade Federal Fluminense, Av. Gal. Milton Tavares
de Souza s/n, 24210-346 Niterói, Rio de Janeiro, Brasil.  R. Bachelard
<EMAIL_ADDRESS>Departamento de Física, Universidade Federal de São
Carlos, Rodovia Washington Luís, km 235 - SP-310, 13565-905 São Carlos, SP,
Brasil  Alan C. Santos<EMAIL_ADDRESS>Departamento de Física,
Universidade Federal de São Carlos, Rodovia Washington Luís, km 235 - SP-310,
13565-905 São Carlos, SP, Brasil

###### Abstract

The study of advanced quantum devices for energy storage has attracted the
attention of the scientific community in the past few years. Although several
theoretical progresses have been achieved recently, experimental proposals of
platforms operating as quantum batteries under ambient conditions are still
lacking. In this context, this work presents a feasible realization of a
quantum battery in a carboxylate-based metal complex, which can store a finite
amount of extractable work under the form of quantum discord at room
temperature, and recharge by thermalization with a reservoir. Moreover, the
stored work can be evaluated through non-destructive measurements of the
compound’s magnetic susceptibility. These results pave the way for the
development of enhanced energy storage platforms through material engineering.

Quantum Battery; Quantum Discord; Ergotropy; Metal Complexes.

Introduction. Batteries are common components in many technological devices,
storing different types of energy and converting it under the form of electric
current Li  _et al._ (2018). Over the past years, the quest for an efficient
use of energy has boosted the development of reliable mechanisms for energy
storage Liu  _et al._ (2019); Manzano-Agugliaro  _et al._ (2013). Recently, a
novel class of batteries has attracted the attention of the scientific
community, namely, quantum batteries (QBs). Chemical batteries convert
chemical energy into electric one through reactions between two species with
different chemical properties Li  _et al._ (2018). Differently, QBs are
constituted of quantum systems, and exploit the superposition principle of
states and its quantum correlations. Based on these, QBs are able to store an
amount of work from the quantum states, called ergotropy, which is extracted
to power quantum devices Ferraro  _et al._ (2018); Alicki and Fannes (2013);
Binder  _et al._ (2015); Santos  _et al._ (2020); Giorgi and Campbell (2015);
Campaioli  _et al._ (2017); Andolina  _et al._ (2019); Çakmak (2020); Santos
_et al._ (2019); Çakmak (2020); Francica  _et al._ (2020); Le  _et al._
(2018)-not necessarily as electric power.

However, there are several challenges to overcome before QBs are put into
practical use, such as storing ergotropy at room temperature (i.e., preserving
quantum correlations or coherence), offering a non-destructive access to the
amount of stored work, and a charge lifetime comparable with that of the
conventional classical batteries. Recently, several devices have been proposed
as QBs, based on different quantum systems, for example, using solid-state and
quantum optical systems, such as superconducting devices Strambini  _et al._
(2020), circuit-QED Andolina  _et al._ (2019); Rossini  _et al._ (2020), and
two-level emitters into waveguides Monsel  _et al._ (2020). However, the
platforms proposed so far for QBs hardly fulfill the above requirements, and
the possibility to place QBs into practical use is still under debate Monsel
_et al._ (2020); Strambini  _et al._ (2020).

Figure 1: (a) Sketch of a Cu2(HCOO)4(HCOOH)2(C4H10N2) low-dimensional metal
complex ccd ; ves , with a polyhedron representation of the molecule. The
internal energy levels of the compound composed by a singlet ground-state
$\ket{\beta_{-}}$ and a triplet degenerate subspace ($\ket{\beta_{t}}$). The
arrows stand for the effective spins stemming from the magnetic structure. (b)
Sketch of the discharging and charging processes. (top) The material in
thermal equilibrium with a reservoir at temperature $T$ (with maximum energy
stored achieved in 83 K) is in its singlet ground state, with energy stored
$E_{0}$: (bottom) (I) an external stimulus (red arrow) drives it to the state
$\ket{\downarrow\downarrow}$ (with zero energy stored) by consuming the stored
ergotropy; (II) removing the stimulus charges the QB, by increasing the
population of $\ket{\beta_{-}}$, bringing it back to the singlet ground-state.

In this context, low-dimensional metal complexes (LDMC), such as quantum
antiferromagnets Cruz  _et al._ (2016); Souza  _et al._ (2009); Reis  _et al._
(2012); Čenčariková and Strečka (2020); Cruz and Anka (2020); Souza  _et al._
(2008); He  _et al._ (2017, 2017); Breunig  _et al._ (2017); Čenčariková and
Strečka (2020), appear as promising platforms to implement QBs prototypes.
Indeed, these complexes present a molecular structure that shields them from
environment fluctuations. This includes fluctuations in temperature Reis  _et
al._ (2012); Čenčariková and Strečka (2020), magnetic field Čenčariková and
Strečka (2020); Cruz and Anka (2020); Souza  _et al._ (2008), and pressure
Cruz  _et al._ (2017); Cruz and Anka (2020). LDMCs effectively behave as a
two-qubit system (see Fig.1(a)), which can hold stable quantum correlations
above room temperature Cruz  _et al._ (2016); Reis  _et al._ (2012); Souza
_et al._ (2009).

In this work, we propose metal complexes as room-temperature operating QBs,
where the working substance is a dinuclear copper (II) carboxylate-based
metal-organic complex Cruz  _et al._ (2016). The ergotropy is stored in the
quantum discord between the spins, and the available ergotropy can be measured
non-destructively by monitoring the magnetic susceptibility of the compound.
Finally, the carboxylate structure proves robust against self-discharging
processes Santos (2021).

Metal complexes as effective two-qubit systems. In this work, we focus on a
metal-organic dinuclear cooper (II) compound with chemical formula
Cu2(HCOO)4(HCOOH)2(C4H10N2), whose crystalline structure is obtained through
single-crystal X-ray analysis ccd  (see Fig. 1a). The system is prepared using
dehydrated copper (II) nitrate (Cu(NO3)2), piperazine (C4H10N2), formic acid,
ethanol and distilled water; and refluxed at $\leavevmode\nobreak\ 425$ K for
3 hours. The piperazine organic compound yields a porous environment which
allows for the production of metal-organic frameworks with magnetically
isolated dimers of Cu(II).

The reduced magnetic unit is then formed by two metallic centers of Cu(II),
with electronic configuration $d^{9}$ and $s=1/2$, in a dimeric tetraformate
unit. The syn-syn bond between them, characteristic of carboxylate-based metal
complex, yields a short intermolecular separation of $2.68(2)$ $\AA$ Cruz  _et
al._ (2016); Souza  _et al._ (2009). This provides a nearly-ideal realization
of an isolated two-qubit system Yurishchev (2011); Gaita-Ariño  _et al._
(2019); Moreno-Pineda  _et al._ (2018). It is worth noting that, despite the
specificity of the present LDMC, this behavior is actually encountered in a
broad range of metal complexes, thanks to their large intramolecular
interaction energy, as compared to the intermolecular ones Cruz and Anka
(2020); Yurishchev (2011); Souza  _et al._ (2009, 2008); Čenčariková and
Strečka (2020); Chakraborty and Mitra (2019); Yurishchev (2011); Gaita-Ariño
_et al._ (2019); Moreno-Pineda  _et al._ (2018). The energy of the two coupled
qubit system is provided by the following Hamiltonian:

$\displaystyle
H=E_{0}\left(S_{1}^{(z)}+S_{2}^{(z)}\right)+J\left(\vec{S}_{1}\cdot\vec{S}_{2}\right).$
(1)

Above, $S_{n}^{(k)}\\!=\\!(\hbar/2)\sigma_{n}^{k}$, where $\sigma_{n}^{k}$
represent the Pauli matrices and $k\\!\in\\!\\{x,y,z\\}$. The first right-hand
term is the Zeeman Hamiltonian Reis (2013), where
$E_{0}\\!=\\!\mu_{B}g_{z}B_{z}$, $g_{z}$ the isotropic Landé factor; $\mu_{B}$
represents the Bohr magneton; and $B_{z}$ is the external (fixed) magnetic
field intensity. Such term describes the energy levels of each cell of the
battery (qubit):
$H_{0}\ket{\epsilon_{i}}\\!=\\!\epsilon_{i}\ket{\epsilon_{i}}$, with
$\epsilon_{i}\\!\propto\\!E_{0}$ and $\ket{\epsilon_{i}}$ the eigenvalue and
eigenstate, respectively. In particular, the energy scale $E_{0}$ (and, as we
shall see later, the ergotropy of the system - Fig. 1b), is proportional to
the fixed magnetic field $B_{z}$. The Heisenberg Hamiltonian
$H_{\text{int}}=J\left(\vec{S}_{1}\cdot\vec{S}_{2}\right)$ corresponds to the
internal interaction between Cu(II) ions (intra-cell interaction), while $J$
represents the magnetic coupling constant.

The syn-syn metal carboxylate conformation yields a very short metal-to-metal
magnetic interaction, leading to a huge magnetic coupling $J/k_{B}\\!=\\!748$
K between Cu(II) cells. This allows for the existence of stable quantum
correlations above room temperature Reis  _et al._ (2012); Souza  _et al._
(2009). Consequently, as sketched in Fig. 1a, the energy levels of the system
is composed by the singlet state $\ket{\beta_{-}}$ with energy $E_{-}=0$ and
the triply-degenerate subspace
$\\{\ket{\beta_{\text{t}}}\\}\\!=\\!\\{\ket{\beta_{+}},\ket{\downarrow\downarrow},\ket{\uparrow\uparrow}\\}$,
with energy $E_{\text{t}}=J$, where we have defined
$\ket{\beta_{\pm}}\\!=\\!(\ket{\downarrow\uparrow}\pm\ket{\uparrow\downarrow})/\sqrt{2}$.
Thus, the gap $\Delta E>0$ between the ground and first excited ground state
indicates that the system will be in a entangled singlet ground state with
anti-parallel alignment (quantum antiferromagnet).

In particular, for this class of materials, the static magnetic field $B_{z}$
splits the energy levels of the compound, inducing a quantum level crossing
Chakraborty and Mitra (2019); Cruz and Anka (2020); Breunig  _et al._ (2017),
changing its corresponding populations Cruz and Anka (2020) (for details, see
ref. Sup ). However, the strong magnetic coupling, yielded by the syn-syn bond
between the Cu(II) cells, leads to a crossing field of $B_{c}\sim\\!556$ T.
This very large value of the crossing field is due to the large gap
($J_{\text{int}}/k_{B}\\!=\\!748$ K) between the ground (singlet entangled
state) and the first excited states (triplet separable state) yielded by the
syn-syn bond between the Cu(II) cells (see Fig. 1a). Therefore, for any
$B_{z}\ll B_{c}$ ($E_{0}\ll J$), the system behaves as an effective two-level
one, with cycle between the singlet ground state $\ket{\beta_{-}}$ and the
excited triplet $\ket{\beta_{\text{t}}}$ one.

The density matrix for the coupled system in thermal equilibrium can be
written, in the energy basis, as an X-state:

$\displaystyle\rho(T,B_{z})$ $\displaystyle=$
$\displaystyle\frac{e^{{\zeta}}}{2Z}\left[\begin{matrix}2e^{\beta
E_{0}}&0&0&0\\\ 0&{1+e^{-4{\zeta}}}&{1-e^{-4{\zeta}}}&0\\\
0&{1-e^{-4{\zeta}}}&{1+e^{-4{\zeta}}}&0\\\ 0&0&0&2e^{-\beta
E_{0}}\end{matrix}\right]\leavevmode\nobreak\ ,$ (2)

where $Z(T,B_{z})\\!=\\!e^{{\zeta}}+e^{-3{\zeta}}+2e^{{\zeta}}\cosh\left(\beta
E_{0}\right)$ is the partition function, $\beta\\!=\\!1/k_{B}T$ and
${\zeta}\\!=\\!\beta J/4$.

Extractable work (ergotropy). Unlike classical batteries, a QB is
characterized by a finite amount of work in a quantum system, which can be
extracted via unitary process Allahverdyan  _et al._ (2004). The maximum
amount of available work extractable via unitary processes is called ergotropy
Liu  _et al._ (2019); Alicki and Fannes (2013); Çakmak (2020).

Given the system state $\rho(T,B_{z})$ and the QB internal spectrum of
$H_{0}$, living in an $N$-dimensional Hilbert space, the ergotropy is defined
as the amount of work which can be extracted by unitary operations $V$:

$\mathcal{E}(T,B_{z})\\!=\\!\mbox{Tr}\left[\rho(T,B_{z})H_{0}\right]-\min_{V\in\mathcal{V}}\left\\{\mbox{Tr}\left[V\rho(T,B_{z})V^{\dagger}H_{0}\right]\right\\},$
(3)

where the minimization is taken over the set $\mathcal{V}$ of all unitary
operators acting on the system Allahverdyan  _et al._ (2004). The energy
eigenvalues of the battery self-Hamiltonian $H_{0}$ are
$\epsilon_{1}\\!\leq\\!\epsilon_{2}\\!\leq\\!\cdots\\!\leq\\!\epsilon_{N}$;
and the eigenvalues of $\rho(T,B_{z})$ are
$\varrho_{1}\\!\geq\\!\varrho_{2}\\!\geq\\!\cdots\\!\geq\\!\varrho_{N}$,
associated to eigenvectors $\ket{\varrho_{n}}$ Sup . Then, the ergotropy in
the Eq. (3) can be written as Allahverdyan  _et al._ (2004):

$\displaystyle\mathcal{E}(T,B_{z})=\sum\nolimits_{i,n}^{N,N}\varrho_{n}(T,B_{z})\epsilon_{i}\left(|\langle\varrho_{n}|\epsilon_{i}\rangle|^{2}-\delta_{ni}\right),$
(4)

where we note that this definition is tied to a specific ordering of the
eigenvalues $\varrho_{n}$ of Eq. (2) and $H_{0}$, due to the $\delta_{ni}$
term. This leads to the following expression for the ergtropy per molecule, in
thermal equilibrium:

$\mathcal{E}_{\text{Cu(II)}}=\
E_{0}\left\\{\frac{1-e^{{4\zeta}}\left[\cosh\left(\beta
E_{0}\right)-3\sinh\left(\beta
E_{0}\right)\right]}{e^{{4\zeta}}\left[2\cosh\left(\beta
E_{0}\right)+1\right]+1}\right\\}.$ (5)

On the other hand, the enormous gap between the ground (singlet - entangled)
and the first excited (triplet - separable) state provided by the syn-syn
bound, allows us to address the system in the regime of magnetic
susceptibility at $E_{0}\ll k_{B}T$. From Eq. (5), it is possible to write the
ergotropy $\mathcal{E}_{\text{Cu(II)}}$, in terms of the Bleaney-Bowers
magnetic susceptibility equation Bleaney and Bowers (1952) (for details, see
ref. Sup )

$\mathcal{E}_{E_{0}\ll k_{B}T}(T)=\
E_{0}\frac{k_{B}T\chi(T)}{4N_{A}g^{2}\mu_{B}^{2}}\left[e^{-4\zeta}-1\right]\leavevmode\nobreak\
,$ (6)

where $N_{A}$ is the Avogadro number. As detailed in Sup , this magnetic
susceptibility regime is valid in the limit of small magnetic fields $B_{z}\ll
B_{c}$ compared to the crossing one ($B_{c}\\!\sim\\!556$ T) Chakraborty and
Mitra (2019).

Therefore, Eq. (6) represents one of the main results of this letter, in which
the evaluation of the amount of stored ergotropy in metal complexes are
experimentally accessible, by measuring a macroscopic property of the system:
the magnetic susceptibility.

Ergotropy measurement and quantumness of metal-carboxylate QBs. Up to date,
proposals of QBs have been done through an ergotropy destructive readout,
through state tomography Liu  _et al._ (2019); Ferraro  _et al._ (2018); Le
_et al._ (2018); Strambini  _et al._ (2020); Monsel  _et al._ (2020). The QB
proposed here provides a different approach to read the stored ergotropy, by
measuring the available work of a carboxylate-based QB accessible without
altering the stored energy, this is a strong request for designing realistic
QBs. Fig. 2a presents the theoretical (black line) and experimental (open
circles) behavior of the ergotropy as a function of the temperature, where the
experimental data was obtained from the measurement of the magnetic
susceptibility of the substance. The theoretical curve was plotted by Eq. (5),
considering the Earth’s magnetic field of $B_{z}\\!=\\!10^{-4}$ T, and the
experimental parameters $g=2$ ($d^{9}$ ions) and $J/k_{B}\\!=\\!748$ K, in a
good agreement with the laboratory environment in which batteries should
operate. It is worth highlighting that by decreasing the temperature we get
the maximum amount of ergotropy
$\mathcal{E}^{\text{max}}_{\text{Cu(II)}}\\!\approx\\!1.12$ mJ/mol, where
values larger than $99.75\%$ of $\mathcal{E}^{\text{max}}_{\text{Cu(II)}}$ can
be reached for temperatures $T_{\text{max}}\\!\leq\\!100$ K reaching the
$100\%$ below $83$ K. Moreover, as can be seen, at room temperature ($\sim
293$ K), the amount of stored energy is $75\%$ of the maximum one. This
suggests that the carboxylate-based QB could operate under daily life
conditions.

To explore the metal-carboxylate as a QB, we now present experimental results
that give access to the ergotropy, which here corresponds to the quantum
discord. The characterization of the quantumness of the battery is done by
computing entanglement of formation ($\mathcal{F}_{E}$) Hill and Wootters
(1997); Wootters (1998) and the quantum discord based on Schatten 1-norm
$\mathcal{D}(T)$ Ciccarello  _et al._ (2014), in terms of the magnetic
susceptibility Sup . First, we find that the ergotropy as function of the
temperature is given by the Schatten 1-norm quantum discord as Sup

$\mathcal{E}_{E_{0}\ll k_{B}T}(T)=2E_{0}\mathcal{D}(T),\leavevmode\nobreak\
\leavevmode\nobreak\ \forall\leavevmode\nobreak\ \leavevmode\nobreak\ J>0.$
(7)

From above equation we observe that, for any dinuclear metal complex of
spin-1/2 with an antiparallel alignment ($J\\!>\\!0$), the maximum amount of
extracted work is related to the existence of genuinely quantum correlations
beyond entanglement between the Cu(II) ions, which is quantified by quantum
discord. The inset in Fig. 2a show experimental result for the discord,
showing that the decay of the ergotropy with temperature corresponds to the
decay of Schatten 1-norm quantum discord.

(a)

(b)

Figure 2: (a) Experimental measurement of ergotropy, in mili Joule per mole,
as function of the environment temperature $T$ (in Kelvin) for the metal-
carboxylate QB proposed. The inset shows the experimental quantum discord as
function of the temperature. Black solid line denote theoretical results and
open circles show the experimental result, computed for the Earth’s magnetic
field $B_{z}\\!=\\!10^{-4}$ T. Vertical dashed line indicates room temperature
$T\\!=\\!293$ K. (b) Theoretical (solid line curve) and experimental result
(open circles) for entanglement of formation, compared with the theoretical
result for normalized ergotropy (dot-dashed line).

To see how the ergotropy is stored as quantum correlations beyond
entanglement, we also consider the entanglement of formation $\mathcal{F}_{E}$
as shown in Fig. 2b. To this end, we normalized the ergotropy by its maximum
value
$\mathcal{E}^{\text{norm}}_{\text{Cu(II)}}\\!=\\!\mathcal{E}_{\text{Cu(II)}}/\mathcal{E}^{\text{max}}_{\text{Cu(II)}}$,
and thus both quantities $\mathcal{F}_{E}$ and
$\mathcal{E}^{\text{norm}}_{\text{Cu(II)}}$ are defined in the same interval
(i.e., $[0,1]$). Fig. 2b suggests that the entanglement is not adequate to
explain the qualitative behavior of the ergotropy as function of the
temperature. Thus, as expected from Eq. (7), there is ergotropy even without
entanglement, but not without quantum discord. In addition, as theoretically
detailed in Sup , by increasing the range of temperature, one finds that above
the temperature of entanglement $T_{\text{e}}=J/(\ln(3)k_{B})$ in which
$\mathcal{F}_{E}(T\\!\geq\\!T_{\text{e}})\\!=\\!0$, the ergotropy is is
$16.6\%$ of the maximum one
($\mathcal{E}^{\text{norm}}_{\text{Cu(II)}}(T\\!\geq\\!T_{\text{e}})\\!>\\!0$)
and satisfies the Eq. (7). Therefore, these results lead to the conclusion
that other quantum correlations beyond entanglement needs to taken into
account to describe the amount of ergotropy stored in metal complexes.

Charging and discharging of metal-carboxylate QBs. Due to the high magnetic
interaction obtained from the syn-syn bond in the metal-carboxylate compounds,
the huge magnetic coupling ($J\\!=\\!748$ K) supports the existence of stable
quantum correlations at room temperature ($T\approx 293$ K) Cruz  _et al._
(2016). More specifically, it allows the studied carboxylate-based metal
complex to store a finite amount of ergotropy as quantum discord up to a
threshold temperature $513$ K (obtained through thermogravimetric analysis
reported in ref. Cruz  _et al._ (2016)) under which this material starts to
degrade. Moreover, the strong magnetic coupling makes the system practically
immune to the magnetic field variations in the experimentally feasible region
of small magnetic fields $B_{z}\\!\ll\\!B_{c}$. Hence, in this regime the role
of magnetic field is to define the energy scale ($E_{0}$) of the battery, as
it can be seen in Eq. (6). Therefore, this system does not need a time-
dependent external field for the charging process, differently from the
several kinds of QBs already proposed in literature Campaioli  _et al._
(2017); Andolina  _et al._ (2019); Santos  _et al._ (2020); Le  _et al._
(2018); Kwon  _et al._ (2018); Monsel  _et al._ (2020); Kamin  _et al._
(2020).

Since there is no additional decoherence channel, the dimeric cells are
magnetically isolated, and the material does not degrade below the threshold
temperature. Thus, the system remains in the state given by Eq. (2) and it is
able to store a finite amount of energy quantum discord stable at room
temperature. Thus, the charge lifetime is defined by the existence of non-zero
genuine quantum correlations between the cells of the battery. Since the
ergrotopy increases as we decrease the temperature, one can charge the QB the
submitting the battery to thermal contact with a cold reservoir, under
presence of a static reference magnetic field $B_{z}\ll B_{c}$. Therefore, the
charging process of this carboxylate-based QB is given by a mechanism known in
the literature as charging assisted by thermalization Hovhannisyan  _et al._
(2020).

In this regard, a charging and discharging steps of a major cycle can be
presented for this QB, as sketched in Fig. 1b: (I) an external stimulus (e.g.,
electromagnetic field pulses Moreno-Pineda  _et al._ (2018); Gaita-Ariño  _et
al._ (2019), or pressure Cruz  _et al._ (2017); Cruz and Anka (2020)), lowers
the degree of quantum discord of the system, drives it to the triplet state
subspace (more specifically the state $\ket{\downarrow\downarrow}$),
discharging the battery by consuming the stored work Francica  _et al._
(2020); Monsel  _et al._ (2020); Hovhannisyan  _et al._ (2020); Çakmak (2020);
(II) removing the stimulus, the material, in thermal equilibrium with a
reservoir, returns to the singlet ground-state by increasing the population of
$\ket{\beta_{-}}$, charging the battery (see top figure of the charging-
discharging process). It is worth noting that these two steps belong to a
major cycle and further discussions will be presented elsewhere.

Conclusions. In summary, this Letter shows that carboxylate-based magnetic
systems are promising platforms for QBs. The available work is stored as
quantum discord, i.e., genuine quantum correlations, at room temperature. Due
to the syn-syn bound a strong magnetic interaction emerges,leading to a large
gap between the singlet ground state and the first excited one (see Fig. 1),
which makes this material suitable for engineering room temperature QBs. In
this sense, an external (fixed) magnetic field $B_{z}$ can be used to tune the
total available work for the battery. In addition, the ergotropy measurement
of the QB proposed here is done by non-destructive experimental techniques
associated with the measurement of the compounds magnetic susceptibility, and
no energy is lost during the readout of the available work. The strong intra-
cell interactions also make this system largely immune to the self-discharging
process, which allows it to store energy as long as there is quantum discord
in the system at room temperature and beyond, up to threshold temperature (513
K) in which the material degrades. Moreover, at room temperature, the
efficiency of this complex is approximately $75\%$ of the maximum one
(obtained at $T\\!<\\!83$ K), in which quantum correlations appear, once
again, as a harvestable resource of great interest for quantum technologies
Sapienza  _et al._ (2019).

Since a realistic implementation of a quantum battery has not yet been settled
in an LDMC, the role of its chemical aspects when operating it as a QB remains
to be studied. Nevertheless, the results presented in this letter open a broad
avenue for research of metal complexes as candidate platforms for QBs and the
development of enhanced energy storage platforms through material engineering.
For example, quantum correlations in metal complexes can be handled by
controlling external parameters, as structural factors and thermodynamic
properties Wasielewski  _et al._ (2020); Čenčariková and Strečka (2020); Cruz
_et al._ (2017); Cruz and Anka (2020). Thus, different kinds of control of the
gap between the ground and excited state in dinuclear spin-1/2 metal complexes
QBs can be done through material engineering techniques Gaita-Ariño  _et al._
(2019); Moreno-Pineda  _et al._ (2018); Wasielewski  _et al._ (2020). In
addition, quantum properties of solid-state systems are drastically affected
by thermal decohering effects, hindering the development of feasible quantum
batteries that operate at room temperature. This letter shows that the study
on carboxylate-based materials can change this view and paves the way for the
enhancement and stability of the charging and energy storage processes in QBs.

###### Acknowledgements.

The authors would like to thank P. Brandão and A. M. dos Santos for the
compound’s data. A. C. Santos is supported by São Paulo Research Foundation
(FAPESP) (Grant No 2019/22685-1). R. B. benefited from Grants from São Paulo
Research Foundation (FAPESP, Grants No. 2018/15554-5 and 2019/22685-1). R. B.
benefited from grants from the National Council for Scientific and
Technological Development (CNPq, Grant Nos. 313886/2020-2 and 409946/2018-4).
M.S. Reis thanks FAPERJ for financial support. M.S. Reis belongs to the INCT
of Refrigeração e Termofísica funding by the National Counsel of Technological
and Scientific Development (CNPq), Grant No. 404023/2019-3. This study was
financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível
Superior - Brasil (CAPES) - finance code 001.

Supplemental Material for:
Quantum battery based on quantum discord at room temperature C. Cruz,1,∗ M. F.
Anka,2,† M. S. Reis,2,‡ R. Bachelard,3,§ and Alan C. Santos3,¶ 1Grupo de
Informação Quântica, Centro de Ciências Exatas e das Tecnologias, Universidade
Federal do Oeste da Bahia - Campus Reitor Edgard Santos. Rua Bertioga, 892,
Morada Nobre I, 47810-059 Barreiras, Bahia, Brasil. 2Instituto de Física,
Universidade Federal Fluminense, Av. Gal. Milton Tavares de Souza s/n,
24210-346 Niterói, Rio de Janeiro, Brasil. 3Departamento de Física,
Universidade Federal de São Carlos, Rodovia Washington Luís, km 235 - SP-310,
13565-905 São Carlos, SP, Brasil<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS>
##  Appendix A Quantum level-crossing in a metal complex

Quantum antiferromagnets typically present a quantum level-crossing in the
presence of an external magnetic field $B_{z}$, splitting its energy levels
and changing the corresponding populations Chakraborty and Mitra (2019); Cruz
and Anka (2020); Breunig  _et al._ (2017). In particular for the carboxylate
based metal complex described in the main text, the magnetic field splits the
degeneracy of the triply-degenerate subspace
$\\{\ket{\beta_{\text{t}}}\\}\\!=\\!\\{\ket{\beta_{+}},\ket{\downarrow\downarrow},\ket{\uparrow\uparrow}\\}$,
inducing a level-crossing between its singlet-ground state and the first
excited one when the field reaches a critical value Chakraborty and Mitra
(2019); Breunig  _et al._ (2017). This critical value can be calculated
through the evaluation of the populations. From the Hamiltonian:

$\displaystyle
H=E_{0}\left(S_{1}^{(z)}+S_{2}^{(z)}\right)+J\left(\vec{S}_{1}\cdot\vec{S}_{2}\right),$
(S1)

and the spectral decomposition of the state $\rho(T,B_{z})=e^{-\beta H}/Z$,
where $Z=\mbox{Tr}{\left[e^{-\beta{H}}\right]}$, the battery state eigenvalues
$\varrho_{i}$ (population) and its corresponding eigenvectors can be written
as:

$\displaystyle\varrho_{1}$
$\displaystyle=\frac{1}{1+e^{\zeta}(1+2\cosh\left(\beta
E_{0}\right))}\rightarrow\ket{\beta_{-}},$ (S2) $\displaystyle\varrho_{2}$
$\displaystyle=\frac{e^{\zeta+\beta E_{0}}}{1+e^{\zeta}(1+2\cosh\left(\beta
E_{0}\right))}\rightarrow\ket{\uparrow\uparrow},$ (S3)
$\displaystyle\varrho_{3}$
$\displaystyle=\frac{e^{\zeta}}{1+e^{\zeta}(1+2\cosh\left(\beta
E_{0}\right))}\rightarrow\ket{\beta_{+}},$ (S4) $\displaystyle\varrho_{4}$
$\displaystyle=\frac{e^{\zeta-\beta E_{0}}}{1+e^{\zeta}(1+2\cosh\left(\beta
E_{0}\right))}\rightarrow\ket{\downarrow\downarrow},$ (S5)

where
$\ket{\beta_{\pm}}\\!=\\!(\ket{\downarrow\uparrow}\pm\ket{\uparrow\downarrow})/\sqrt{2}$;
$\beta=1/k_{B}T$; ${\zeta}\\!=\\!-\beta J$; and $E_{0}\\!=\\!\mu_{B}gB_{z}$,
with $g$ being the isotropic Landé factor and $\mu_{B}$ the Bohr magneton.
Since the quantum level crossing is temperature independent, Fig. S1 shows the
populations $\varrho_{n}$ as function of the magnetic field at room
temperature $T=293$ K, obtained from the experimental parameters $g=2$
($d^{9}$ ions) and $J/k_{B}\\!=\\!748$ K Cruz  _et al._ (2016). The inset
shows the magnetic field dependence of the energy levels

$\displaystyle E_{1}$ $\displaystyle=0\rightarrow\ket{\beta_{-}},$ (S6)
$\displaystyle E_{2}$
$\displaystyle=J-E_{0},\rightarrow\ket{\uparrow\uparrow},$ (S7) $\displaystyle
E_{3}$ $\displaystyle=J\rightarrow\ket{\beta_{+}},$ (S8) $\displaystyle E_{4}$
$\displaystyle=J+E_{0}\rightarrow\ket{\downarrow\downarrow},$ (S9)

obtained from the spectral decomposition of the system Hamiltonian, Eq. (S1).
As can be seem, the magnetic field over the populations is driven our system
from a maximally entangled ground state
$\ket{\beta_{-}}\\!=\\!\left[\ket{\downarrow\uparrow}-\ket{\uparrow\downarrow}\right]/\sqrt{2}$,
to a separable ground state $\ket{\uparrow\uparrow}$, when the magnetic field
surpass the critical value $B_{c}\approx 556$ T ($E_{0}\\!=\\!J$). This huge
value of crossing field is a consequence of the strong magnetic coupling
($J/k_{B}\\!=\\!748$ K) yielded by the $syn-syn$ bound, which shields the
compound from environment fluctuations Reis  _et al._ (2012); Čenčariková and
Strečka (2020); Cruz and Anka (2020); Souza  _et al._ (2008); Cruz  _et al._
(2017). Thus, from Eq. (S6)-(S9), for any $B_{z}\ll B_{c}$ ($E_{0}\ll J$), the
compound behaves as an effective two-level system.

Figure S1: (a) Populations, Eqs. (S2)–(S5), as a function of the magnetic
field for different values of temperature. The inset shows the the magnetic
field dependence of the energy levels. Vertical (orange) dotted line
represents the value of the crossing field, as can be seen the population
crossover occur at $B_{c}=556$ T.

##  Appendix B Ergotropy of dinuclear spin-1/2 metal complexes

Quantum systems can be defined as a quantum batteries when a finite amount of
work, stored in its quantum states can be extracted via unitary process Liu
_et al._ (2019); Alicki and Fannes (2013); Çakmak (2020). Considering the
carboxylate-based metal complex described by Eq. (S1), one can define the
battery Hamiltonian as

$H=H_{0}+H_{int},$ (S10)

where $H_{0}=E_{0}\left(S_{1}^{(z)}+S_{2}^{(z)}\right)$ is the battery self-
Hamiltonian, which defines the energy scale
$\epsilon_{1}\\!\leq\\!\epsilon_{2}\\!\leq\\!\cdots\\!\leq\\!\epsilon_{N}$ and
eigenstates $\ket{\epsilon_{i}}$ of the quantum battery, satisfying the
eigenvalue equation
$H_{0}\ket{\epsilon_{i}}\\!=\\!\epsilon_{i}\ket{\epsilon_{i}}$; and
$H_{int}=J\left(\vec{S}_{1}\cdot\vec{S}_{2}\right)$ defines an internal
interaction of the system. The system has a corresponding density matrix,
defined in thermal equilibrium as:

$\displaystyle\rho(T,B_{z})$ $\displaystyle=$
$\displaystyle\frac{e^{{\zeta}}}{2Z}\left[\begin{matrix}2e^{\beta
E_{0}}&0&0&0\\\ 0&{1+e^{-4{\zeta}}}&{1-e^{-4{\zeta}}}&0\\\
0&{1-e^{-4{\zeta}}}&{1+e^{-4{\zeta}}}&0\\\ 0&0&0&2e^{-\beta
E_{0}}\end{matrix}\right]\leavevmode\nobreak\ ,$ (S11)

where $Z(T,B_{z})\\!=\\!e^{{\zeta}}+e^{-3{\zeta}}+2e^{{\zeta}}\cosh\left(\beta
E_{0}\right)$ is the partition function.

For non-pure states ($Tr\\{\rho^{2}\\}\\!\neq\\!1$), the amount of available
work which can be extracted from unitary operations $V$ is not given by the
internal energy of the system $U\\!=\\!Tr\\{H_{0}\rho\\}$, but rather by the
maximum amount of available work extractable via unitary processes, defined by
the ergotropy Allahverdyan  _et al._ (2004); Liu  _et al._ (2019); Alicki and
Fannes (2013); Çakmak (2020):

$\mathcal{E}(T,B_{z})\\!=\\!\mbox{Tr}\left[\rho(T,B_{z})H_{0}\right]-\min_{V\in\mathcal{V}}\left\\{\mbox{Tr}\left[V\rho(T,B_{z})V^{\dagger}H_{0}\right]\right\\},$
(S12)

where the minimization, taken over the set $\mathcal{V}$ of all unitary
operators which acts on the system Allahverdyan  _et al._ (2004) is obtained
by writing Eq. (S11) in its spectral decomposition, with the populations
$\varrho_{n}$, Eq. (S6)-(S9), in decreasing order
$\varrho_{1}\\!\geq\\!\varrho_{2}\\!\geq\\!\cdots\\!\geq\\!\varrho_{N}$, and
the eigenvalues of the self-Hamiltonian in the increasing order
$\epsilon_{1}\\!\leq\\!\epsilon_{2}\\!\leq\\!\cdots\\!\leq\\!\epsilon_{N}$.
Then, the maximum amount of available work which can be extracted from a
quantum state, is given by Allahverdyan  _et al._ (2004); Francica  _et al._
(2017); Alicki and Fannes (2013):

$\displaystyle\mathcal{E}=\sum\nolimits_{i,n}^{N,N}\varrho_{n}\epsilon_{i}\left(|\langle\varrho_{n}|\epsilon_{i}\rangle|^{2}-\delta_{ni}\right).$
(S13)

In particular, as mentioned before, due to the characteristics quantum level
crossing presented by quantum antiferromagnets Chakraborty and Mitra (2019);
Cruz and Anka (2020); Breunig  _et al._ (2017), it is possible to establish
two orderings of the quantities $\varrho_{n}$. Consequently, for dinuclear
spin-1/2 metal complexes, the ergotropy can defined in two physical regimes:
one for $E_{0}\\!<\\!J$, corresponding to $B_{z}<B_{c}$; and other for
$E_{0}\\!\geq\\!J$, corresponding to $B_{z}\geq B_{c}$. Thus, from Eq. (S13),
we obtain the ergotropy per molecule in thermal equilibrium with a reservoir
at temperature $T$ under presence of a static reference magnetic field $B_{z}$
as

$\mathcal{E}=\left\\{\begin{aligned}
E_{0}\left\\{\frac{1-e^{{\zeta}}\left[\cosh\left(\beta
E_{0}\right)-3\sinh\left(\beta
E_{0}\right)\right]}{e^{{\zeta}}\left[2\cosh\left(\beta
E_{0}\right)+1\right]+1}\right\\},\leavevmode\nobreak\ \leavevmode\nobreak\
E_{0}<{J}\\\ \frac{4E_{0}e^{{\zeta}}\sinh\left(\beta
E_{0}\right)}{e^{{\zeta}}\left(2\cosh\left(\beta
E_{0}\right)+1\right)+1},\leavevmode\nobreak\ \leavevmode\nobreak\
E_{0}\geq{J}\end{aligned}\right.\leavevmode\nobreak\ .$ (S14)

In particular, for the carboxylate-based metal complex described in the main
text, the regime $B_{z}\geq B_{c}$ are not accessible by usual experimental
techniques, due to the huge value of crossing field ($556$ T), consequence of
the syn-syn bound. Therefore, the experimentally feasible available work,
stored in the metal-carboxylate compound, can be found only in the regime
$E_{0}\\!<\\!J_{\text{int}}$.

##  Appendix C Ergotropy and magnetic susceptibility

From the system Hamiltonian, Eq. (S1), we obtain the canonical partition
function
$Z(T,B_{z})\\!=\\!e^{{\zeta}}+e^{-3{\zeta}}+2e^{{\zeta}}\cosh\left(\beta
E_{0}\right)$. Then, one can obtain the magnetization of a dinuclear spin-1/2
metal complex as Reis (2013)

$\displaystyle\mathcal{M}_{z}(T,B_{z})$
$\displaystyle=\frac{Nk_{B}T}{Z(T,B_{z})}\frac{\partial}{\partial
B_{z}}Z(T,B_{z})\leavevmode\nobreak\ ,$
$\displaystyle=\frac{Ng\mu_{B}\left(e^{2\beta E_{0}}-1\right)}{1+e^{\beta
E_{0}}\left(1+e^{-\zeta}+e^{\beta E_{0}}\right)}\leavevmode\nobreak\ .$ (S15)

Thus, the regime of magnetic susceptibility of this system is reached under
low values of magnetic field in which the magnetization has the linear
dependence $\mathcal{M}_{z}=B_{z}\chi$ Reis (2013). In particular, for the
described carboxylate-based metal complex, with the experimental parameters
$J/k_{B}\\!=\\!748$ K and $g=2$ Cruz  _et al._ (2016), we obtain the
theoretical magnetization $\mathcal{M}_{z}(T,B_{z})$ as a function of the
magnetic field for different temperatures (Fig. S2). As can be seen in Fig.
S2, for the temperature regime under which the magnetic susceptibility is
measured, the system remains in the limit of small magnetic fields even for
values up to dozens of Teslas. In a general way, the enormous gap between the
ground (singlet - entangled) and the first excited (triplet - separable) state
provided by the syn-syn bound, allows us to address the system in the regime
of magnetic susceptibility at $E_{0}\ll k_{B}T$ when $B_{z}\ll B_{c}$.

Figure S2: Magnetization $\mathcal{M}_{z}$ as a function of the magnetic field
$B_{z}$ for the temperature range under which the magnetic susceptibility of
the described metal complex is measured. As can be seen, in the entire
temperature range the system remains in the regime of magnetic susceptibility
(linear magnetization) up to dozens of Tesla.

Thus, applying the limit $E_{0}\ll k_{B}T$ in Eq. (S15) we can obtain the the
Bleaney-Bowers equation for magnetic susceptibility for two spin-$1/2$
interacting systems Bleaney and Bowers (1952)

$\displaystyle\chi(T)$ $\displaystyle=\lim_{E_{0}\ll
k_{B}T}\frac{\mathcal{M}_{z}}{B_{z}}=\frac{2N(g\mu_{B})^{2}}{k_{B}T}\left(\frac{1}{3+e^{-\zeta}}\right).$
(S16)

From Eq. (S14) is possible to write the ergotropy in the regime of magnetic
susceptibility ($E_{0}\ll k_{B}T$ ) as

$\mathcal{E}_{E_{0}\ll k_{B}T}=\left\\{\begin{aligned}
E_{0}\left(\frac{1-e^{{\zeta}}}{3e^{{\zeta}}+1}\right),\leavevmode\nobreak\
\leavevmode\nobreak\ E_{0}<{J}\\\ 0,\leavevmode\nobreak\ \leavevmode\nobreak\
E_{0}\geq{J}\end{aligned}\right.\leavevmode\nobreak\ .$ (S17)

As expected, there is no ergotropy for $E_{0}\geq{J}$, since in this region
$B_{z}\geq B_{c}$ and the system cannot be found under the magnetic
susceptibility regime. Thus, from Eqs. (S16) and (S17) it is possible to write
the ergotropy in terms of the magnetic susceptibility of the compound as

$\mathcal{E}_{E_{0}\ll k_{B}T}(T)=\
E_{0}\frac{k_{B}T\chi(T)}{2Ng^{2}\mu_{B}^{2}}\left[e^{-\zeta}-1\right]\leavevmode\nobreak\
.$ (S18)

Therefore, we establish a relationship between the available work stored in
the quantum states of the reported metal complex and one of its thermodynamic
properties. This results enables the measurement of the ergotropy of the
system without disturbing the stored energy by any destructive readout.

##  Appendix D Quantumness of the battery

One of the main tasks of this work is to explore the quantumness of our
carboxylate based quantum battery, which is characterized by the set of
physical quantities given by entanglement of formation and quantum discord:

##  D.0.1 Entanglement of Formation

Entanglement of formation is an entanglement quantifier which corresponds to
the amount of quantum pure states needed in order to create a mixed entangled
state. In this sense, entanglement in a statistical mixture would be the
average of the mixture of pure states Hill and Wootters (1997); Wootters
(1998). The entanglement of formation for a two-qubit system is defined as
Cruz  _et al._ (2016, 2017)

$\mathcal{F}_{E}=-\mathbb{E}_{+}-\mathbb{E}_{-}\leavevmode\nobreak\
\leavevmode\nobreak\ ,$ (S19)

where

$\mathbb{E}_{\pm}=\frac{1\pm\sqrt{1-\mathbbm{C}^{2}}}{2}\log_{2}\left(\frac{1\pm\sqrt{1-\mathbbm{C}^{2}}}{2}\right)$
(S20)

and $\mathbbm{C}$ is the concurrence Hill and Wootters (1997); Wootters
(1998), which is given by
$\mathbbm{C}(\tau)=\max\\{0,\lambda_{1}-\lambda_{2}-\lambda_{3}-\lambda_{4}\\}$,
where $\lambda_{1},\cdots,\lambda_{4}$ are the eigenvalues in decreasing order
of the matrix
$\hat{R}\\!=\\!(\hat{\rho}^{1/2}\hat{\tilde{\rho}}\hat{\rho}^{1/2})^{1/2}$,
where
$\hat{\tilde{\rho}}\\!=\\!(\hat{\sigma}_{y}\otimes\hat{\sigma}_{y})\hat{\rho}^{*}(\hat{\sigma}_{y}\otimes\hat{\sigma}_{y})$,
with $\hat{\rho}^{*}$ being the complex conjugate of $\hat{\rho}$ taken in the
standard basis, which for our case is the battery energy basis. Thus, for our
carboxylate based metal complex, concurrence can be analytically computed as

$\displaystyle\mathbbm{C}=\max\left[0,\frac{e^{E_{0}\beta}\left(1-3e^{{\zeta}}\right)}{4\left(e^{{\zeta}}+e^{E_{0}\beta}+e^{{\zeta}+E_{0}\beta}+e^{{\zeta}+2E_{0}\beta}\right)}\right]\leavevmode\nobreak\
.$ (S21)

##  D.0.2 Quantum discord

Although entanglement provides a way to determine pure quantum correlations,
it does not encompass all possible types of quantum correlations of a system
Cruz  _et al._ (2016); Ciccarello  _et al._ (2014); Obando  _et al._ (2015);
Yurishchev (2011). In this sense, quantum discord turns out to be a powerful
tool for quantifying the quantumness of correlations Ollivier and Zurek
(2001). We calculated the quantum discord from the trace distance discord,
based on Schatten 1-norm ($||X||_{1}\\!=\\!\tr[\sqrt{X^{\dagger}X}]$), defined
as the minimum distance between the state $\rho$ of a bipartite system and the
set ($\Sigma$) of closest classical-quantum state $\sigma$, given by
$\mathcal{D}(\rho)\\!=\\!\min_{\sigma\in\Sigma}||\rho-\sigma||_{1}$ Ciccarello
_et al._ (2014). For the system of interest, Eq. (S11), the Schatten 1-norm
quantum discord is given by

$\displaystyle\mathcal{D}=\frac{1}{2}\left|\coth(\frac{\zeta}{2})+\cosh(E_{0}\beta)\left(1+\coth(\frac{\zeta}{2})\right)\right|^{-1}.$
(S22)

##  D.0.3 Ergotropy and Quantum Correlations

Fig. S3 shows all the quantities above described: entanglement of formation,
quantum discord, and the dimensionless ergotropy
($\mathcal{E}/\mathcal{E}^{\text{max}}$) as a function of temperature. The
theoretical curves are plotted by Eqs. (S19), (S22) and (S14), respectively,
considering the Earth’s magnetic field of $B_{z}\\!=\\!10^{-4}$ T, and the
experimental parameters $g=2.$ ($d^{9}$ ions) and $J/k_{B}\\!=\\!-748.5$ K, in
order to represent the laboratory environment in which batteries should work.
As can be seen, the asymptotic decay of ergotropy with temperature coincides
with the decay of Schatten 1-norm quantum discord, showing a strong
relationship between these quantities, beyond entanglement. Hence, by the
theoretically increasing of temperature, one finds that above the maximum
temperature $T_{\text{e}}=J/(\ln(3)k_{B})$, under which there is no
entanglement ($\mathcal{F}_{E}(T\\!\geq\\!T_{\text{e}})\\!=\\!0$), the
ergotropy of the system is $16.6\%$ of the maximum one. This result suggests
that quantum correlations beyond entanglement needs to taken into account to
describe the amount of available work stored in metal complexes.

Figure S3: (color online) (a) Entanglement of Formation (dot-dashed red line),
Schatten 1-norm Quantum Discord (dashed blue line) and theoretical
dimensionless Ergotropy $\mathcal{E}/\mathcal{E}^{\text{max}}$(solid black
line), as a function of the temperature for Earth’s magnetic field
$B_{z}\\!=\\!10^{-4}$ T. Vertical (orange) dotted line represents the
temperature of entanglement $T_{\text{e}}=J/(\ln(3)k_{B})$ above which there
is no entanglement ($\mathcal{F}_{E}(T\\!\geq\\!T_{\text{e}})\\!=\\!0$).

As pointed before, we can address the available work of our carboxylate-based
quantum battery in the experimentally feasible region $E_{0}\\!<\\!J_{int}$ in
the magnetic susceptibility regime, Eq. (6). Applying the limit $E_{0}\ll
k_{B}T$ in Eq. (S22) we obtain

$\displaystyle\mathcal{D}=\frac{1}{2}\left|\frac{1-e^{\zeta}}{3e^{\zeta}+1}\right|\leavevmode\nobreak\
.$ (S23)

Thus, comparing Eq. (S17) with Eq. (S23), we obtain

$\mathcal{E}_{E_{0}\ll k_{B}T}=2E_{0}\mathcal{D}\leavevmode\nobreak\
\leavevmode\nobreak\ \forall\leavevmode\nobreak\ \leavevmode\nobreak\
J>0\leavevmode\nobreak\ \leavevmode\nobreak\ ,$ (S24)

where $2E_{0}$ is the maximum energy stored, obtained from the spectral
decomposition of the reference Hamiltonian $H_{0}$.

Therefore, since ${\zeta}\\!=\\!-\beta J_{\text{int}}$, for any spin-$1/2$
quantum antiferromagnet ($J>0$), in the regime of magnetic susceptibility
($E_{0}\ll k_{B}T$), the existence of a finite amount of available work will
be directly related to the presence of genuine quantum correlations between
the cells of the battery. Then, the energy is stored from physical quantities
and correlations that cannot be modeled by a classical system, i.e., we have a
genuine QB.

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[19, p.111], the group of holomorphic automorphisms of $\mathcal{D}_{n}$ is
$\textrm{Aut}(\mathcal{D}_{n})=PSO^{*}(2n)$, so
$\textrm{Aut}(\mathcal{D}_{n})$ is a quotient of $G_{0}$ by a discrete central
subgroup. From the list on [24, p.341], the complexified Lie algebra
$\mathfrak{g}$ of $G_{0}$ has complex dimension $n(2n-1)$, and can be
expressed as the set of complex $2n\times 2n$ trace zero matrices
$\begin{bmatrix}A&B\\\ C&-A^{T}\end{bmatrix}$, where $A$ is a complex $n\times
n$ matrix, and $B$ and $C$ are complex skew-symmetric $n\times n$ matrices.
Since $G_{0}$ is a Hodge group of Hermitian type, the Lie algebra
$\mathfrak{g}$ has a Hodge decomposition given by
$\mathfrak{g}=\mathfrak{g}^{-1,1}\oplus\mathfrak{g}^{0,0}\oplus\mathfrak{g}^{1,-1}$,
where $\mathfrak{g}^{-1,1}$ consists of matrices of the form
$\begin{bmatrix}0&B\\\ 0&0\end{bmatrix}$, $\mathfrak{g}^{0,0}$ consists of
matrices of the form $\begin{bmatrix}A&0\\\ 0&-A^{T}\end{bmatrix}$, and
$\mathfrak{g}^{1,-1}$ consists of matrices of the form $\begin{bmatrix}0&0\\\
C&0\end{bmatrix}$. The Lie algebras $\mathfrak{g}^{-1,1}$,
$\mathfrak{g}^{0,0}$, and $\mathfrak{g}^{1,-1}$ have dimensions $n(n-1)/2$,
$n^{2}$, and $n(n-1)/2$ respectively.
The compact dual of $\mathcal{D}_{n}$ is the Isotropic Grassmannian $I_{n}$,
which is a homogeneous complex projective variety of dimension $n(n-1)/2$ for
the action of $G=SO^{*}(2n,\mathbb{C})$. It parametrizes all isotropic
subspaces of a complex vector space of dimension $2n$ equipped with an inner
product. Just as in the case of the Lagrangian Grassmannian, there is an open
embedding $j:\mathcal{D}_{n}\hookrightarrow I_{n}$. The tangent bundle of
$I_{n}$ is given by $\mathcal{T}_{I_{n}}\cong\bigwedge^{2}(\mathcal{E})$,
where $\mathcal{E}$ is the tautological vector bundle on $I_{n}$, and
corresponds to the standard representation of $K_{0}=U(n)$. It follows that
the tangent bundle of $\mathcal{D}_{n}$ can be expressed as
$\mathcal{T}_{\mathcal{D}_{n}}\cong\bigwedge^{2}(\mathcal{E}^{\prime})$, where
$\mathcal{E}^{\prime}$ denotes the restriction of $\mathcal{E}$ to
$\mathcal{D}_{n}$.

We want to formulate necessary and sufficient conditions for a complex
projective variety of dimension $n(n-1)/2$ with klt singularities and ample
canonical divisor to be uniformized by the Hermitian symmetric space
$\mathcal{D}_{n}$ of type DIII.

###  7.1 Sufficient conditions

A version of Proposition 6.1 also holds in this case, and the proof is
essentially the same. For completeness, we repeat the proof.

######  Proposition 7.1.

Let $X$ be a projective, klt variety of dimension $n(n-1)/2$ for some
$n\in\mathbb{Z}_{\geq 1}$ with $K_{X}$ ample. Suppose that the tangent bundle
of the regular locus $X_{reg}$ of $X$ satisfies
$\mathcal{T}_{X_{reg}}\cong\bigwedge^{2}(\mathcal{E})$, where $\mathcal{E}$ is
a vector bundle of rank $n$ on $X_{reg}$. Let $\mathcal{E}^{\prime}$ denote
the reflexive extension of $\mathcal{E}$ to $X$, and suppose that the Chern
class equality

$\displaystyle[2\widehat{c}_{2}(X)-\widehat{c}_{1}(X)^{2}+2n\widehat{c}_{2}(\mathcal{E}^{\prime})-(n-1)\widehat{c}_{1}(\mathcal{E}^{\prime})^{2}]\cdot[K_{X}]^{n-2}=0$
(11)

holds on $X$. Then $X\cong\mathcal{D}_{n}/\Gamma$, where $\Gamma\subset
PSO^{*}(2n,\mathbb{R})$ is a discrete, cocompact subgroup, and acts fixed
point freely in codimension one on the domain $\mathcal{D}_{n}$ of type
$DIII$.

###### Proof.

Let $j:X_{reg}\to X$ denote the natural inclusion. Then we have
$\mathcal{T}_{X}\cong
j_{*}\mathcal{T}_{X_{reg}}\cong\bigwedge^{[2]}(\mathcal{E}^{\prime})=(\bigwedge^{2}(\mathcal{E}^{\prime}))^{**}$
by the uniqueness of reflexive extension. Let $G_{0}=SO^{*}(2n,\mathbb{R})$,
$K_{0}=U(n)$ a maximal compact subgroup of $G_{0}$. Then we choose
complexifications $G$ and $K$ of $G_{0}$ and $K_{0}$ respectively, to be
$G=SO^{*}(2n,\mathbb{C})$ and $K=GL(n,\mathbb{C})$.
Let $P$ be the frame bundle of $\mathcal{E}$. Then $P$ is a principal
$K=GL(n,\mathbb{C})$-bundle on $X_{reg}$ and we can write $\mathcal{E}\cong
P\times_{K}V$, where $V\cong\mathbb{C}^{n}$ denotes the typical fiber of
$\mathcal{E}$. It follows that
$\mathcal{T}_{X_{reg}}\cong\bigwedge^{2}(P\times_{K}V)=P\times_{K}\bigwedge^{2}(V)$,
so in particular $\mathcal{T}_{X_{reg}}$ admits a reduction in structure group
from $GL(n(n-1)/2,\mathbb{C})$ to $K$. Note that $\bigwedge^{2}(V)$ and
$\mathfrak{g}^{-1,1}$ are isomorphic as $K$-representations, again by [31,
Table 2]. Thus there is an isomorphism

$\displaystyle\theta:\mathcal{T}_{X_{reg}}\to P\times_{K}\mathfrak{g}^{-1,1}$

such that $[\theta(u),\theta(v)]=0$ for all local sections $u,v$ of
$\mathcal{T}_{X_{reg}}$. It follows that $(P,\theta)$ is a uniformizing system
of Hodge bundles on $X_{reg}$.
Hence we can form the system of Hodge bundles
$E=P\times_{K}\mathfrak{g}=P\times_{K}\mathfrak{g}^{-1,1}\oplus
P\times_{K}\mathfrak{g}^{0,0}\oplus P\times_{K}\mathfrak{g}^{1,-1}$, where we
have
$P\times_{K}\mathfrak{g}^{1,-1}\cong\bigwedge^{2}(\mathcal{E}^{\vee})\cong\Omega^{1}_{X_{reg}}$,
and $P\times_{K}\mathfrak{g}^{0,0}\cong\mathcal{E}nd(\mathcal{E})$. The last
isomorphism follows from observing, as in Proposition 6.1, that
$\mathfrak{g}^{0,0}$ and $\textrm{End}(V)$ are isomorphic as
$K$-representations. Hence we have $P\times_{K}\mathfrak{g}^{0,0}\cong
P\times_{K}\textrm{End}(V)=\mathcal{E}nd(P\times_{K}V)$.
Note that the system of Hodge bundles
$E=P\times_{K}\mathfrak{g}\cong\mathcal{T}_{X_{reg}}\oplus\mathcal{E}nd(\mathcal{E})\oplus\Omega^{1}_{X_{reg}}$
is in particular a Higgs bundle with Higgs field $\widehat{\theta}:E\to
E\otimes\Omega^{1}_{X_{reg}}$ given by sending a local section $u$ of $E$ to
the local section $v\mapsto[\theta(v),u]$ of
$\mathcal{H}om(\mathcal{T}_{X_{reg}},E)\cong E\otimes\Omega^{1}_{X_{reg}}$. It
is clear that $E$ has slope zero with respect to $K_{X}$. We know from
Proposition 4.4 that $E$ is $K_{X}$-polystable as a Higgs bundle on $X_{reg}$.
In fact, $E$ is $K_{X}$-stable as a Higgs bundle on $X_{reg}$ because again
$\mathfrak{g}$ is a simple Lie algebra.
Let $\mathcal{F}_{X}$ denote the reflexive extension of $E$ to $X$, then
$\mathcal{F}_{X}\cong\mathcal{T}_{X}\oplus\mathcal{E}nd(\mathcal{E}^{\prime})\oplus\Omega^{[1]}_{X}$.
It is clear that $\widehat{c}_{1}(\mathcal{F}_{X})\cdot[K_{X}]^{n-1}=0$.
Moreover, we compute

$\displaystyle\widehat{c}_{2}(\mathcal{F}_{X})=2\widehat{c}_{2}(X)-\widehat{c}_{1}(X)^{2}+2n\widehat{c}_{2}(\mathcal{E}^{\prime})-(n-1)\widehat{c}_{1}(X)^{2},$

so by the assumption of the Proposition we have
$\widehat{c}_{2}(\mathcal{F}_{X})\cdot[K_{X}]^{n-2}=\widehat{ch}_{2}(\mathcal{F}_{X})\cdot[K_{X}]^{n-2}=0$.
Again we see that $X$ satisfies the two conditions of Theorem 1.1, and we
conclude that $X\cong\mathcal{D}_{n}/\Gamma$, where $\Gamma\subset
PSO^{*}(2n,\mathbb{R})$ is a discrete, cocompact subgroup, acting fixed point
freely in codimension one on $\mathcal{D}_{n}$. ∎

###  7.2 Necessary conditions

Since $\textrm{Aut}(\mathcal{D}_{n})=PSO^{*}(2n)$ is a connected Lie group, we
take the associated Hodge group to be $G_{0}=SO^{*}(2n)$, and its
complexification to be $G=SO^{*}(2n,\mathbb{C})$, which is also connected.
Moreover, $K_{0}=U(n)$ and its complexification $K=GL(n,\mathbb{C})$ are
connected too, so the sufficient conditions of Proposition 7.1 are also
necessary conditions for a projective klt variety with ample canonical divisor
to be uniformized by $\mathcal{D}_{n}$. The proof is essentialy the same as
that of Proposition 6.2.

######  Proposition 7.2.

Let $X$ be a projective klt variety of dimension $n(n-1)/2$ with $K_{X}$
ample, such that $X=\mathcal{D}_{n}/\widehat{\Gamma}$, where
$\widehat{\Gamma}\subset\textrm{Aut}(\mathcal{D}_{n})$ acts discretely,
cocompactly, and fixed point freely in codimension one on $\mathcal{D}_{n}$.
Then the tangent bundle of the regular locus of $X$ satisfies
$\mathcal{T}_{X_{reg}}\cong\bigwedge^{2}(\mathcal{E})$, where $\mathcal{E}$ is
a rank $n$ vector bundle on $X_{reg}$. Moreover, $X$ satisfies the Chern class
equality 11.

###### Proof.

From Theorem 1.1, it follows that $X_{reg}$ admits a uniformizing system of
Hodge bundles $(P,\theta)$ corresponding to the Hodge group
$G_{0}=SO^{*}(2n)$, such that the system of Hodge bundles
$E=P\times_{K}\mathfrak{g}$ is $K_{X}$-polystable as a Higgs bundle on
$X_{reg}$. Moreover, the equality
$\widehat{c}_{2}(E^{\prime})\cdot[K_{X}]^{n-2}=0$. holds, where $E^{\prime}$
denotes the extension of $E$ to $X$ as a reflexive sheaf. Hence there is an
isomorphism $\theta:\mathcal{T}_{Y}\cong P\times_{K}\mathfrak{g}^{-1,1}$.
Recall from the proof of Proposition 7.1 that $\mathfrak{g}^{-1,1}$ is
isomorphic as a $K$-representation to $\bigwedge^{2}(V)$, where $V$ is a
complex $n$-dimensional vector space. So we can write
$\mathcal{T}_{X_{reg}}\cong
P\times_{K}\bigwedge^{2}(V)=\bigwedge^{2}(\mathcal{E})$, where
$\mathcal{E}=P\times_{K}V$ is a rank $n$ vector bundle on $Y$. We also have
$\Omega^{1}_{X_{reg}}\cong\bigwedge^{2}(\mathcal{E}^{\vee})\cong
P\times_{K}\mathfrak{g}^{1,-1}$, and $\mathcal{E}nd(\mathcal{E})\cong
P\times_{K}\mathfrak{g}^{0,0}$, from which it follows that the system of Hodge
bundles $P\times_{K}\mathfrak{g}$ is isomorphic to
$\mathcal{T}_{X_{reg}}\oplus\mathcal{E}nd(\mathcal{E})\oplus\Omega^{1}_{X_{reg}}$.
Let $\mathcal{E}^{\prime}$ denote the reflexive extension of $\mathcal{E}$ to
$X$. Then
$E^{\prime}=\mathcal{T}_{X}\oplus\mathcal{E}nd(\mathcal{E}^{\prime})\oplus\Omega^{[1]}_{X}$,
and the Chern class equality $\widehat{c}_{2}(E^{\prime})\cdot[K_{X}]^{n-2}=0$
is equivalent to

$\displaystyle\widehat{c}_{2}(\mathcal{T}_{X}\oplus\mathcal{E}nd(\mathcal{E}^{\prime})\oplus\Omega^{[1]}_{X})\cdot[K_{X}]^{n-2}=0.$

Using the formula for the second $\mathbb{Q}$-Chern class of a direct sum of
vector bundles, the above equality can be rephrased as

$\displaystyle[2\widehat{c}_{2}(X)-\widehat{c}_{1}(X)^{2}+2n\widehat{c}_{2}(\mathcal{E})-(n-1)\widehat{c}_{1}(\mathcal{E})^{2}]\cdot[K_{X}]^{n-2}=0.$

Thus it follows that $X$ satisfies the Chern class equality (11), and we are
done. ∎

Putting together Propositions 7.1 and 7.2, we arrive at the following
necessary and sufficient condition for a projective klt variety $X$ with ample
canonical divisor to be uniformized by the Hermitian symmetric space
$\mathcal{D}_{n}$ of type $DIII$.

######  Theorem 7.3.

Let $X$ be a projective klt variety of dimension $n(n-1)/2$ such that the
canonical divisor $K_{X}$ is ample. Then $X\cong\mathcal{D}_{n}/\Gamma$, where
$\Gamma\subset PSO^{*}(2n,\mathbb{R})$ is a discrete, cocompact subgroup, and
acts fixed point freely in codimension one on $\mathcal{D}_{n}$, if and only
if $X$ satisfies

  * •

$\mathcal{T}_{X_{reg}}\cong\bigwedge^{2}(\mathcal{E})$

  * •

$[2\widehat{c}_{2}(X)-\widehat{c}_{1}(X)^{2}+2n\widehat{c}_{2}(\mathcal{E}^{\prime})-(n-1)\widehat{c}_{1}(\mathcal{E}^{\prime})^{2}]\cdot[K_{X}]^{n-2}=0$,

where $\mathcal{E}$ is a vector bundle of rank $n$ on $X_{reg}$, and
$\mathcal{E}^{\prime}$ denotes the reflexive extension of $\mathcal{E}$ to
$X$.

##  8 Uniformization by Hermitian symmetric space of type $BDI$

The Hermitian symmetric space of type $BDI$, which we denote by
$\mathcal{B}_{n}$, can be expressed as the quotient $G_{0}/K_{0}$, where we
may take the associated Hodge group to be $G_{0}=SO(2,n)$, and
$K_{0}=SO(2)\times SO(n)$ a maximal compact subgroup of $G_{0}$, for
$n\in\mathbb{Z}_{\geq 2}$. The bounded symmetric domain realization of $BDI$
is more involved than the other examples, we refer the reader to [33, p.75]
for details. When $n=2$, the domain $\mathcal{B}_{2}$ is isomorphic to the
bidisk $\mathbb{H}^{2}$, which we have already studied. Thus we may assume
$n\geq 3$. The Lie group $G_{0}$ has two connected components, and the
automorphism group $PSO(2,n)$ of $\mathcal{B}_{n}$ (see again [19]) is a
quotient of $SO(2,n)$ by a discrete central subgroup. The dimension of
$\mathcal{B}_{n}$ as a complex manifold is $n$. The complexified Lie algebra
of $G_{0}$ is $\mathfrak{g}=\mathfrak{so}(2+n,\mathbb{C})$. The Lie algebra
$\mathfrak{g}$ has complex dimension $(2+n)(1+n)/2$, and using [24, p.341],
can be expressed as the set of complex $(n+2)\times(n+2)$ matrices
$\begin{bmatrix}A&B\\\ C&D\end{bmatrix}$, where $A$ is a $2\times 2$
orthogonal matrix, $D$ is a $n\times n$ orthogonal matrix, and $B$ and $C$ are
$2\times n$ and $n\times 2$ matrices respectively. The group $G_{0}=SO(2,n)$
is in fact a Hodge group of Hermitian type, therefore the Lie algebra
$\mathfrak{g}$ has a Hodge decomposition given by
$\mathfrak{g}=\mathfrak{g}^{-1,1}\oplus\mathfrak{g}^{0,0}\oplus\mathfrak{g}^{1,-1}$,
where $\mathfrak{g}^{-1,1}$ consists of those matrices of $\mathfrak{g}$ of
the form $\begin{bmatrix}0&B\\\ 0&0\end{bmatrix}$, $\mathfrak{g}^{0,0}$
consists of matrices of the form $\begin{bmatrix}A&0\\\ 0&D\end{bmatrix}$, and
$\mathfrak{g}^{1,-1}$ consists of matrices of the form $\begin{bmatrix}0&0\\\
C&0\end{bmatrix}$. The Lie algebras $\mathfrak{g}^{-1,1}$,
$\mathfrak{g}^{0,0}$, and $\mathfrak{g}^{1,-1}$ are of dimensions $n$,
$n(n-1)/2+1$, and $n$ respectively.
By Lemma 3.4, we know there is be a principal $K=SO(2,\mathbb{C})\times
SO(n,\mathbb{C})$-bundle $P$ on $\mathcal{B}_{n}$ such that there is an
isomorphism $\theta:\mathcal{T}_{\mathcal{B}_{n}}\to
P\times_{K}\mathfrak{g}^{-1,1}$. Then there is an orthogonal vector bundle
$\mathcal{W}$ of rank $n$, and a line bundle $\mathcal{L}$ such that the
system of Hodge bundles $P\times_{K}\mathfrak{g}$ is isomorphic to
$\bigwedge^{2}(\mathcal{W}\oplus\mathcal{L}\oplus\mathcal{L}^{\vee})$. From
the Hodge decomposition of the Lie algebra $\mathfrak{g}$, it follows that
$P\times_{K}\mathfrak{g}^{-1,1}\cong\mathcal{T}_{\mathcal{B}_{n}}\cong\mathcal{H}om(\mathcal{W},\mathcal{L})$,
$P\times_{K}\mathfrak{g}^{0,0}\cong\bigwedge^{2}(\mathcal{W})\oplus\mathcal{O}_{\mathcal{B}_{n}}$,
and
$P\times_{K}\mathfrak{g}^{1,-1}\cong\Omega^{1}_{\mathcal{B}_{n}}\cong\mathcal{H}om(\mathcal{W},\mathcal{L}^{\vee})$.
Note that, since $\mathcal{W}$ is an orthogonal bundle, it is self-dual.

We want to determine necessary and sufficient conditions for a projective, klt
variety of dimension $n$ with ample canonical divisor to be uniformized by
$\mathcal{B}_{n}$.

###  8.1 Sufficient conditions

We apply Theorem 1.1 to prove the following.

######  Proposition 8.1.

Let $X$ be a projective, klt variety of dimension $n\geq 3$ with $K_{X}$
ample. Suppose that the tangent bundle of the regular locus $X_{reg}$ of $X$
satisfies $\mathcal{T}_{X_{reg}}\cong\mathcal{H}om(\mathcal{W},\mathcal{L})$,
where $\mathcal{W}$ is an orthogonal vector bundle of rank $n$, and
$\mathcal{L}$ is a line bundle on $X_{reg}$. Let $\mathcal{W}^{\prime}$ and
$\mathcal{L}^{\prime}$ denote the reflexive extensions of $\mathcal{W}$ and
$\mathcal{L}$ respectively to $X$, and suppose that the Chern class equality

$\displaystyle[\widehat{c}_{2}(\mathcal{W}^{\prime}\wedge\mathcal{W}^{\prime})+2\widehat{c}_{2}(\mathcal{W}^{\prime})-n\widehat{c}_{1}(\mathcal{L}^{\prime})^{2}]\cdot[K_{X}]^{n-2}=0$
(12)

holds on $X$. Then $X\cong\mathcal{B}_{n}/\Gamma$, where
$\Gamma\subset\textrm{Aut}(\mathcal{B}_{n})$ is a discrete, cocompact
subgroup, and acts fixed point freely in codimension one on $\mathcal{B}_{n}$.

###### Proof.

Let $j:X_{reg}\to X$ denote the natural inclusion. Then we have
$\mathcal{T}_{X}\cong j_{*}\mathcal{T}_{X_{reg}}\cong
j_{*}\mathcal{H}om(\mathcal{W},\mathcal{L})\cong\mathcal{H}om(\mathcal{W}^{\prime},\mathcal{L}^{\prime})$
by the uniqueness of reflexive extension. Let $G_{0}=SO(2,n)$,
$K_{0}=SO(2)\times SO(n)$ the maximal compact subgroup of $G_{0}$, and let $G$
and $K$ denote complexifications of $G_{0}$ and $K_{0}$ respectively. We
choose $G=SO(2+n,\mathbb{C})$, then $K=SO(2,\mathbb{C})\times
SO(n,\mathbb{C})$.
We can deduce again from [31, Table 2] that the Lie algebra
$\mathfrak{g}=\mathfrak{so}(2+n,\mathbb{C})$ and the typical fiber of the
vector bundle
$\bigwedge^{2}(\mathcal{W}\oplus\mathcal{L}\oplus\mathcal{L}^{\vee})$ are
isomorphic as $K$-representations. Moreover, the vector bundle
$\bigwedge^{2}(\mathcal{W}\oplus\mathcal{L}\oplus\mathcal{L}^{\vee})$ admits a
reduction in structure group from $GL((n+2)(n+1)/2,\mathbb{C})$ to
$K=SO(2,\mathbb{C})\times SO(n,\mathbb{C})$. Thus there is a principal
$K$-bundle $P$ on $X_{reg}$ such that
$P\times_{K}\mathfrak{g}\cong\bigwedge^{2}(\mathcal{W}\oplus\mathcal{L}\oplus\mathcal{L}^{\vee})$.
Recall from the earlier discussion that the Hodge decomposition of
$\mathfrak{g}$ gives
$P\times_{K}\mathfrak{g}^{-1,1}\cong\mathcal{H}om(\mathcal{W},\mathcal{L})$,
$P\times_{K}\mathfrak{g}^{0,0}\cong\bigwedge^{2}\mathcal{W}\oplus\mathcal{O}_{X_{reg}}$,
and
$P\times_{K}\mathfrak{g}^{1,-1}\cong\mathcal{H}om(\mathcal{W},\mathcal{L}^{\vee})$.
Since $\mathcal{T}_{X_{reg}}\cong\mathcal{H}om(\mathcal{W},\mathcal{L})$ by
assumption, we have the following isomorphism

$\displaystyle\theta:\mathcal{T}_{X_{reg}}\to P\times_{K}\mathfrak{g}^{-1,1}$

such that $[\theta(u),\theta(v)]=0$ for all local sections $u,v$ of
$\mathcal{T}_{X_{reg}}$. It follows that $(P,\theta)$ gives a uniformizing
system of Hodge bundles on $X_{reg}$.
The system of Hodge bundles
$E=P\times_{K}\mathfrak{g}\cong\mathcal{T}_{X_{reg}}\oplus\bigwedge^{2}\mathcal{W}\oplus\mathcal{O}_{X_{reg}}\oplus\Omega^{1}_{X_{reg}}$
is in particular a Higgs bundle with Higgs field $\widehat{\theta}:E\to
E\otimes\Omega^{1}_{X_{reg}}$ given by sending a local section $u$ of $E$ to
the local section $v\mapsto[\theta(v),u]$ of
$\mathcal{H}om(\mathcal{T}_{X_{reg}},E)\cong E\otimes\Omega^{1}_{X_{reg}}$.
Since $\mathcal{W}$ is self-dual, we have that
$\textrm{det}(\mathcal{W})\cong\mathcal{O}_{X_{reg}}$, and hence
$c_{1}(E)=c_{1}(P\times_{K}\mathfrak{g})=0$. We know again from Proposition
4.4 that the system of Hodge bundles $E$ is $K_{X}$-polystable as a Higgs
bundle on $X_{reg}$. In fact, $E$ is $K_{X}$-stable as a Higgs bundle on
$X_{reg}$ because $\mathfrak{so}(2+n,\mathbb{C})$ is a simple Lie algebra for
$n\geq 3$.
Let $\mathcal{F}_{X}$ denote the reflexive extension of $E$ to $X$. It is
clear that $\widehat{c}_{1}(\mathcal{F}_{X})\cdot[K_{X}]^{n-1}=0$. Moreover,
we have

$\displaystyle\widehat{c}_{2}(\mathcal{F}_{X})\cdot[K_{X}]^{n-2}=\widehat{c}_{2}(\bigwedge^{2}(\mathcal{W}^{\prime}\oplus\mathcal{L}^{\prime}\oplus\mathcal{L}^{\prime\vee}))\cdot[K_{X}]^{n-2}.$

Expanding the right hand side of the above equality gives
$[\widehat{c}_{2}(\mathcal{W}^{\prime}\wedge\mathcal{W}^{\prime})+2\widehat{c}_{2}(\mathcal{W}^{\prime})-n\widehat{c}_{1}(\mathcal{L}^{\prime})^{2}]\cdot[K_{X}]^{n-2}$,
which by assumption is zero. Therefore, we have
$\widehat{ch}_{2}(\mathcal{F}_{X})\cdot[K_{X}]^{n-2}=0$.
It follows from Theorem 1.1 that $X\cong\mathcal{B}_{n}/\Gamma$, where
$\Gamma\subset\textrm{Aut}(\mathcal{B}_{n})$ is a discrete, cocompact
subgroup, and acts fixed point freely in codimension one on $\mathcal{B}_{n}$.
This concludes the proof. ∎

###  8.2 Necessary conditions

Note that the full automorphism group of $\mathcal{B}_{n}$ is a quotient of
$G_{0}=SO(2,n)$ by a discrete central subgroup, and thus the complexification
of the maximal compact subgroup of $\textrm{Aut}(\mathcal{B}_{n})$ is a
quotient of $K=SO(2,\mathbb{C})\times SO(n,\mathbb{C})$ by a discrete central
subgroup. Since $K$ is connected, the sufficient criteria for uniformization
by $\mathcal{B}_{n}$ are also necessary.

######  Proposition 8.2.

Let $X$ be a projective klt variety of dimension $n\geq 3$ with $K_{X}$ ample,
such that $X=\mathcal{B}_{n}/\widehat{\Gamma}$, where $\widehat{\Gamma}\subset
Aut(\mathcal{B}_{n})$ is a discrete, cocompact subgroup which acts fixed point
freely in codimension one on $\mathcal{B}_{n}$. Then the tangent bundle of the
regular locus of $X$ satisfies
$\mathcal{T}_{X_{reg}}\cong\mathcal{H}om(\mathcal{V},\mathcal{M})$, where
$\mathcal{V}$ is an orthogonal vector bundle of rank $n$, and $\mathcal{M}$ is
a line bundle on $X_{reg}$. Moreover, $X$ satisfies the Chern class equality
12.

###### Proof.

Again from Theorem 1.1, it follows that the smooth locus $X_{reg}$ of $X$
admits a uniformizing system of Hodge bundles $(P,\theta)$ corresponding to
the Hodge group $G_{0}=SO(2,n)$, such that $E=P\times_{K}\mathfrak{g}$ is
$K_{X}$-polystable as a Higgs bundle on $X_{reg}$. Furthermore, the equality
$\widehat{c}_{2}(E^{\prime})\cdot[K_{X}]^{n-2}=0$ is satisfied, where
$E^{\prime}$ denotes the reflexive extension of $E$ to $X$. Hence there is an
isomorphism $\theta:\mathcal{T}_{X_{reg}}\cong
P\times_{K}\mathfrak{g}^{-1,1}$. Recall that from the proof of Proposition 8.1
that $\mathfrak{g}^{-1,1}$ is isomorphic as a $K$-representation to
$\textrm{Hom}(W,L)$, where $W$ and $L$ are complex $n$ and 1-dimensional
vector spaces respectively. So we can write $\mathcal{T}_{X_{reg}}\cong
P\times_{K}\textrm{Hom}(W,L)=\mathcal{H}om(\mathcal{W},\mathcal{L})$, where
$\mathcal{W}$ is an orthogonal rank $n$ vector bundle, and $\mathcal{L}$ is a
line bundle on $X_{reg}$. We also have
$\Omega^{1}_{X_{reg}}\cong\mathcal{H}om(\mathcal{W},\mathcal{L}^{\vee})\cong
P\times_{K}\mathfrak{g}^{1,-1}$, and
$\bigwedge^{2}\mathcal{W}\oplus\mathcal{O}_{X_{reg}}\cong
P\times_{K}\mathfrak{g}^{0,0}$, from which it follows that the system of Hodge
bundles $P\times_{K}\mathfrak{g}$ is isomorphic to
$\bigwedge^{2}(\mathcal{W}\oplus\mathcal{L}\oplus\mathcal{L}^{\vee})$.
Let $E^{\prime}$, $\mathcal{W}^{\prime}$, and $\mathcal{L}^{\prime}$ denote
the reflexive extensions of $E$, $\mathcal{W}$, and $\mathcal{L}$ respectively
to $X$. Then the equality $\widehat{c}_{2}(E^{\prime})\cdot[K_{X}]^{n-2}=0$ is
equivalent to

$\displaystyle\widehat{c}_{2}(\bigwedge^{2}(\mathcal{W}^{\prime}\oplus\mathcal{L}^{\prime}\oplus\mathcal{L}^{\prime\vee}))\cdot[K_{X}]^{n-2}=0.$

Expanding the above expression using formulae for $\mathbb{Q}$-Chern classes,
we arrive at the following expression.

$\displaystyle[\widehat{c}_{2}(\mathcal{W}^{\prime}\wedge\mathcal{W}^{\prime})+2\widehat{c}_{2}(\mathcal{W}^{\prime})-n\widehat{c}_{1}(\mathcal{L}^{\prime})^{2}]\cdot[K_{X}]^{n-2}=0.$

Therefore, $X$ satisfies the $\mathbb{Q}$-Chern class equality (12). ∎

Putting together Propositions 8.1 and 8.2, we arrive at the following
necessary and sufficient condition for a projective klt variety $X$ with ample
canonical divisor to be uniformized by the Hermitian symmetric space
$\mathcal{B}_{n}$ of type $BDI$.

######  Theorem 8.3.

Let $X$ be a projective klt variety of dimension $n\geq 3$, such that the
canonical divisor $K_{X}$ is ample. Then $X\cong\mathcal{B}_{n}/\Gamma$, where
$\Gamma\subset\textrm{Aut}(\mathcal{B}_{n})$ is a discrete, cocompact subgroup
acting fixed point freely in codimension one on $\mathcal{B}_{n}$, if and only
if $X$ satisfies

  * •

$\mathcal{T}_{X_{reg}}\cong\mathcal{H}om(\mathcal{W},\mathcal{L})$

  * •

$[\widehat{c}_{2}(\mathcal{W}^{\prime}\wedge\mathcal{W}^{\prime})+2\widehat{c}_{2}(\mathcal{W}^{\prime})-n\widehat{c}_{1}(\mathcal{L}^{\prime})^{2}]\cdot[K_{X}]^{n-2}=0$,

where $\mathcal{W}$ is an orthogonal vector bundle of rank $n$, $\mathcal{L}$
is a line bundle on $X_{reg}$, and $\mathcal{W}^{\prime}$ and
$\mathcal{L}^{\prime}$ denote the reflexive extensions of $\mathcal{W}$ and
$\mathcal{L}$ to $X$.

##  9 Uniformization by Hermitian symmetric space of type $AIII$

The Hermitian symmetric space of type $AIII$, denoted $\mathcal{A}_{pq}$, can
be expressed as the quotient $G_{0}/K_{0}$, where we may take $G_{0}=SU(p,q)$,
and $K_{0}=S(U(p)\times U(q))$ its maximal compact subgroup, for
$p,q\in\mathbb{Z}_{\geq 1}$, $pq>1$. The domain $\mathcal{A}_{pq}$ can also be
realized as follows (see [33])

$\displaystyle\mathcal{A}_{pq}=\\{Z\in
M(p,q,\mathbb{C})\cong\mathbb{C}^{pq}:I_{q}-\bar{Z}^{T}Z>0\\},$

where the ”$>0$” again means positive definiteness. The complex dimension of
$\mathcal{A}_{pq}$ is $pq$. The complexified Lie algebra of $SU(p,q)$ is
$\mathfrak{g}=\mathfrak{sl}(p+q,\mathbb{C})$. The Lie algebra $\mathfrak{g}$
has complex dimension $(p+q)^{2}-1$, and again using the list on [24, p.341],
can expressed as the set of complex $(p+q)\times(p+q)$ matrices
$\begin{bmatrix}A&B\\\ C&D\end{bmatrix}$, where $A$ is a $p\times p$ matrix,
$B$ is a $p\times q$ matrix, $C$ is a $q\times p$ matrix, and $D$ is a
$q\times q$ matrix, and $Tr(A)+Tr(D)=0$. Note that $G_{0}=SU(p,q)$ is a Hodge
group of Hermitan type, and $\mathfrak{g}$ has a Hodge decomposition
$\mathfrak{g}=\mathfrak{g}^{-1,1}\oplus\mathfrak{g}^{0,0}\oplus\mathfrak{g}^{1,-1}$
as in Definition 2.15, where $\mathfrak{g}^{-1,1}$ consists of matrices of the
form $\begin{bmatrix}0&B\\\ 0&0\end{bmatrix}$, $\mathfrak{g}^{0,0}$ consists
of matrices of the form $\begin{bmatrix}A&0\\\ 0&D\end{bmatrix}$, and
$\mathfrak{g}^{1,-1}$ consists of matrices of the form $\begin{bmatrix}0&0\\\
C&0\end{bmatrix}$. The Lie algebras $\mathfrak{g}^{-1,1}$,
$\mathfrak{g}^{0,0}$, and $\mathfrak{g}^{1,-1}$ have complex dimensions $pq$,
$p^{2}+q^{2}-1$, and $pq$ respectively.
Let $P$ be a principal $K=S(GL(p,\mathbb{C})\times GL(q,\mathbb{C}))$-bundle
on $\mathcal{A}_{pq}$ such that $\theta:\mathcal{T}_{\mathcal{A}_{pq}}\cong
P\times_{K}\mathfrak{g}^{-1,1}$, which we know exists by Lemma 3.4. Then there
are vector bundles $\mathcal{V}$ and $\mathcal{W}$ of ranks $p$ and $q$ on
$\mathcal{A}_{pq}$, such that the system of Hodge bundles
$P\times_{K}\mathfrak{g}$ is isomorphic to
$\mathcal{E}nd_{0}(\mathcal{V}\oplus\mathcal{W})$, the bundle of trace zero
endomorphisms of $\mathcal{V}\oplus\mathcal{W}$. From the Hodge decomposition
of $\mathfrak{g}$, we get
$P\times_{K}\mathfrak{g}^{-1,1}\cong\mathcal{H}om(\mathcal{V},\mathcal{W})\cong\mathcal{T}_{\mathcal{A}_{pq}}$,
$P\times_{K}\mathfrak{g}^{0,0}\cong(\mathcal{E}nd(\mathcal{V})\oplus\mathcal{E}nd(\mathcal{W}))_{0}$,
and
$P\times_{K}\mathfrak{g}^{1,-1}\cong\mathcal{H}om(\mathcal{W},\mathcal{V})\cong\Omega^{1}_{\mathcal{A}_{pq}}$.
The domain $\mathcal{A}_{q}$ corresponding to $p=1$ is the unit ball
$\mathbb{B}^{q}\subset\mathbb{C}^{q}$.

As usual, the goal is to formulate necessary and sufficient conditions for a
projective, klt variety of dimension $pq$ with ample canonical divisor to be
uniformized by $\mathcal{A}_{pq}$.

###  9.1 Sufficient conditions

We again apply Theorem 1.1 to make the following characterization.

######  Proposition 9.1.

Let $X$ be a projective, klt variety of dimension $pq$ for some
$p,q\in\mathbb{Z}_{\geq 1}$, $pq>1$, with $K_{X}$ ample, such that the tangent
bundle of the regular locus $X_{reg}$ of $X$ satisfies
$\mathcal{T}_{X_{reg}}\cong\mathcal{H}om(\mathcal{V},\mathcal{W})$, where
$\mathcal{V}$ and $\mathcal{W}$ are vector bundles of ranks $p$ and $q$ on
$X_{reg}$. Let $\mathcal{V}^{\prime}$ and $\mathcal{W}^{\prime}$ denote the
reflexive extensions of $\mathcal{V}$ and $\mathcal{W}$ to $X$, and suppose
that the Chern class equality

$\displaystyle[2(p+q)(\widehat{c}_{2}(\mathcal{V}^{\prime})+\widehat{c}_{2}(\mathcal{W}^{\prime}))-(p+q-1)(\widehat{c}_{1}(\mathcal{V}^{\prime})^{2}+\widehat{c}_{1}(\mathcal{W}^{\prime})^{2})+2\widehat{c}_{1}(\mathcal{V}^{\prime})\widehat{c}_{1}(\mathcal{W}^{\prime})]\cdot[K_{X}]^{n-2}=0$
(13)

holds on $X$. Then $X\cong\mathcal{A}_{pq}/\Gamma$, where
$\Gamma\subset\textrm{Aut}(\mathcal{A}_{pq})$ is a discrete, cocompact
subgroup, and acts fixed point freely in codimension one on the domain
$\mathcal{A}_{pq}$ of type $AIII$.

###### Proof.

Let $j:X_{reg}\to X$ denote the natural inclusion. Then we have
$\mathcal{T}_{X}\cong j_{*}\mathcal{T}_{X_{reg}}\cong
j_{*}\mathcal{H}om(\mathcal{V},\mathcal{W})\cong\mathcal{H}om(\mathcal{V}^{\prime},\mathcal{W}^{\prime})$
by the uniqueness of reflexive extension. Let $G_{0}=SU(p,q)$,
$K_{0}=S(U(p)\times U(q))$ the maximal compact subgroup of $G_{0}$, and let
$G$ and $K$ denote complexifications of $G_{0}$ and $K_{0}$ respectively. We
choose $G=SL(p+q,\mathbb{C})$ and $K=S(GL(p,\mathbb{C})\times
GL(q,\mathbb{C}))$.
Let $\mathcal{E}nd_{0}(\mathcal{V}\oplus\mathcal{W})$ denote the bundle of
trace zero endomorphisms of $\mathcal{V}\oplus\mathcal{W}$. We deduce from
[31, Table 2], that the Lie algebra
$\mathfrak{g}=\mathfrak{sl}(p+q,\mathbb{C})$ and the typical fiber
$\textrm{End}_{0}(\mathbb{C}^{p},\mathbb{C}^{q})$ of
$\mathcal{E}nd_{0}(\mathcal{V}\oplus\mathcal{W})$ are isomorphic as
$K$-representations. Moreover, the vector bundle
$\mathcal{E}nd_{0}(\mathcal{V}\oplus\mathcal{W})$ admits a reduction in
structure group from $GL((p+q)^{2}-1,\mathbb{C})$ to
$K=S(GL(p,\mathbb{C})\times GL(q,\mathbb{C}))$. Thus there is a principal
$K$-bundle $P$ on $X_{reg}$ associated to
$\mathcal{E}nd_{0}(\mathcal{V}\oplus\mathcal{W})$ i.e., we have
$P\times_{K}\mathfrak{g}\cong\mathcal{E}nd_{0}(\mathcal{V}\oplus\mathcal{W})$.
The Hodge decomposition of $\mathfrak{g}$ together with identifying isomorphic
$K$-representations using [31, Table 2] gives
$P\times_{K}\mathfrak{g}^{-1,1}\cong\mathcal{H}om(\mathcal{V},\mathcal{W})$,
$P\times_{K}\mathfrak{g}^{0,0}\cong(\mathcal{E}nd(\mathcal{V})\oplus\mathcal{E}nd(\mathcal{W}))_{0}$,
and
$P\times_{K}\mathfrak{g}^{1,-1}\cong\mathcal{H}om(\mathcal{W},\mathcal{V})$.
Since $\mathcal{T}_{X_{reg}}\cong\mathcal{H}om(\mathcal{V},\mathcal{W})$ by
assumption, we have the following isomorphism

$\displaystyle\theta:\mathcal{T}_{X_{reg}}\to P\times_{K}\mathfrak{g}^{-1,1}$

such that $[\theta(u),\theta(v)]=0$ for all local sections $u,v$ of
$\mathcal{T}_{X_{reg}}$. It follows that $(P,\theta)$ gives a uniformizing
system of Hodge bundles on $X_{reg}$.
The system of Hodge bundles
$E=P\times_{K}\mathfrak{g}\cong\mathcal{T}_{X_{reg}}\oplus(\mathcal{E}nd(\mathcal{V})\oplus\mathcal{E}nd(\mathcal{W}))_{0}\oplus\Omega^{1}_{X_{reg}}$
is in particular a Higgs bundle with Higgs field $\widehat{\theta}:E\to
E\otimes\Omega^{1}_{X_{reg}}$ given by sending a local section $u$ of $E$ to
the local section $v\mapsto[\theta(v),u]$ of
$\mathcal{H}om(\mathcal{T}_{X_{reg}},E)\cong E\otimes\Omega^{1}_{X_{reg}}$.
Note that $E$ fits into the following short exact sequence

$\displaystyle 0\to
E=\mathcal{E}nd_{0}(\mathcal{V}\oplus\mathcal{W})\to\mathcal{E}nd(\mathcal{V}\oplus\mathcal{W})\to\mathcal{L}\to
0$ (14)

where $\mathcal{L}$ is a line bundle of degree zero. Thus we have
$c_{1}(E)=c_{1}(\mathcal{E}nd_{0}(\mathcal{V}\oplus\mathcal{W}))=0$. From
Proposition 4.4, it follows that $E$ is $K_{X}$-polystable as a Higgs bundle
on $X_{reg}$, and in fact $E$ is $K_{X}$-stable as a Higgs bundle on $X_{reg}$
because $\mathfrak{g}=\mathfrak{sl}(p+q,\mathbb{C})$ is a simple Lie algebra.
Let $\mathcal{F}_{X}$ denote the reflexive extension of $E$ to $X$. Then
$\mathcal{F}_{X}$ also fits into an exact sequence

$\displaystyle
0\to\mathcal{F}_{X}\to\mathcal{E}nd(\mathcal{V}^{\prime}\oplus\mathcal{W}^{\prime})\to\mathcal{L}^{\prime}\to
0$

where $\mathcal{L}^{\prime}$ is a coherent sheaf of degree 0. Thus
$\widehat{c}_{1}(\mathcal{F}_{X})\cdot[K_{X}]^{n-1}=\widehat{c}_{1}(\mathcal{E}nd(\mathcal{V}^{\prime}\oplus\mathcal{W}^{\prime}))\cdot[K_{X}]^{n-1}=0$,
and
$\widehat{c_{2}}(\mathcal{F}_{X})\cdot[K_{X}]^{n-2}=\widehat{c}_{2}(\mathcal{E}nd(\mathcal{V}^{\prime}\oplus\mathcal{W}^{\prime}))\cdot[K_{X}]^{n-2}$.
Moreover, we compute

$\displaystyle\widehat{c}_{2}(\mathcal{F}_{X})\cdot[K_{X}]^{n-2}=[2(p+q)(\widehat{c}_{2}(\mathcal{V}^{\prime})+\widehat{c}_{2}(\mathcal{W}^{\prime}))-(p+q-1)(\widehat{c}_{1}(\mathcal{V}^{\prime})^{2}+\widehat{c}_{1}(\mathcal{W}^{\prime})^{2})+2\widehat{c}_{1}(\mathcal{V}^{\prime})\widehat{c}_{1}(\mathcal{W}^{\prime})]\cdot[K_{X}]^{n-2},$

so by the assumption of the Proposition we have
$\widehat{c}_{2}(\mathcal{F}_{X})\cdot[K_{X}]^{n-2}=\widehat{ch}_{2}(\mathcal{F}_{X})\cdot[K_{X}]^{n-2}=0$.
Thus the conditions of Theorem 1.1 are satisfied, and it follows that
$X\cong\mathcal{A}_{pq}/\Gamma$, where
$\Gamma\subset\textrm{Aut}(\mathcal{A}_{pq})$ is a discrete, cocompact
subgroup acting fixed point freely in codimension one on the domain
$\mathcal{A}_{pq}$. This concludes the proof. ∎

In order to determine the necessary conditions for uniformization by
$\mathcal{A}_{pq}$, we consider two separate cases, namely when (i) $p\neq q$,
and (ii) $p=q$. This is because the group of automorphisms
$\textrm{Aut}(\mathcal{A}_{pq})$ of $\mathcal{A}_{pq}$ is connected when
$p\neq q$, and has two connected components when $p=q$ (see [19]).

###  9.2 Necessary conditions when $p\neq q$

From [19, p.114], we know that the automorphism group of $\mathcal{A}_{pq}$
when $p\neq q$ is $\textrm{Aut}(\mathcal{A}_{pq})=PSU(p,q)$. This is a
connected Lie group and is a quotient of $SU(p,q)$ by a discrete central
subgroup. Thus we can choose the associated Hodge group to be $G_{0}=SU(p,q)$,
and its complexification as $G=SL(p+q,\mathbb{C})$, which is also connected.
Thus the sufficient conditions of Proposition 9.1 are also necessary
conditions for a projective klt variety with ample canonical divisor to be
uniformized by $\mathcal{A}_{pq}$. The proof is essentially the same as that
of Proposition 6.2.

######  Proposition 9.2.

Let $X$ be a projective klt variety of dimension $pq$, $p\neq q$ with $K_{X}$
ample, such that $X=\mathcal{A}_{pq}/\widehat{\Gamma}$, where
$\widehat{\Gamma}\subset Aut(\mathcal{A}_{pq})$ is a discrete, cocompact
subgroup acting fixed point freely in codimension one on $\mathcal{A}_{pq}$.
Then the tangent bundle of the regular locus of $X$ satisfies
$\mathcal{T}_{X_{reg}}\cong\mathcal{H}om(\mathcal{E},\mathcal{F})$, where
$\mathcal{E}$ and $\mathcal{F}$ are vector bundles of ranks $p$ and $q$ on
$X_{reg}$. Moreover, $X$ satisfies the Chern class equality 13.

###### Proof.

By Theorem 1.1, the smooth locus $X_{reg}$ of $X$ admits a uniformizing system
of Hodge bundles $(P,\theta)$ corresponding to the Hodge group
$G_{0}=SU(p,q)$, such that the system of Hodge bundles
$E=P\times_{K}\mathfrak{g}$ is $K_{X}$-polystable as a Higgs bundle on
$X_{reg}$. Moreover, $X$ satisfies the $\mathbb{Q}$-Chern class equality
$\widehat{c}_{2}(E^{\prime})\cdot[K_{X}]^{n-2}=0$, where $E^{\prime}$ denotes
the unique reflexive extension of $E$ to $X$. Hence there is an isomorphism
$\theta:\mathcal{T}_{X_{reg}}\cong P\times_{K}\mathfrak{g}^{-1,1}$. Recall
from the proof of Proposition 9.1 that $\mathfrak{g}^{-1,1}$ is isomorphic as
a $K$-representation to $\textrm{Hom}(V,W)$, where $V$ and $W$ are complex $p$
and $q$-dimensional vector spaces. So we can write $\mathcal{T}_{X_{reg}}\cong
P\times_{K}\textrm{Hom}(V,W)=\mathcal{H}om(\mathcal{V},\mathcal{W})$, where
$\mathcal{V}$ and $\mathcal{W}$ are vector bundles of ranks $p$ and $q$ on
$X_{reg}$. We also have
$\Omega^{1}_{X_{reg}}\cong\mathcal{H}om(\mathcal{W},\mathcal{V})\cong
P\times_{K}\mathfrak{g}^{1,-1}$, and
$(\mathcal{E}nd(\mathcal{V})\oplus\mathcal{E}nd(\mathcal{W}))_{0}\cong
P\times_{K}\mathfrak{g}^{0,0}$, from which it follows that the system of Hodge
bundles $P\times_{K}\mathfrak{g}$ is isomorphic to
$\mathcal{E}nd_{0}(\mathcal{V}\oplus\mathcal{W})$. Let $\mathcal{V}^{\prime}$
and $\mathcal{W}^{\prime}$ denote the reflexive extensions of $\mathcal{V}$
and $\mathcal{W}$ to $X$. Then the Chern class equality
$\widehat{c}_{2}(E^{\prime})\cdot[K_{X}]^{n-2}=0$ is equivalent to

$\displaystyle\widehat{c}_{2}(\mathcal{E}nd_{0}(\mathcal{V}^{\prime}\oplus\mathcal{W}^{\prime}))\cdot[K_{X}]^{n-2}=0.$

Using the short exact sequence 14, and the formula for the second
$\mathbb{Q}$-Chern class of a direct sum of reflexive sheaves, the above
equality can be rephrased as

$\displaystyle[2(p+q)(\widehat{c}_{2}(\mathcal{V}^{\prime})+\widehat{c}_{2}(\mathcal{W^{\prime}}))-(p+q-1)(\widehat{c}_{1}(\mathcal{V}^{\prime})^{2}+\widehat{c}_{1}(\mathcal{W}^{\prime})^{2})+2\widehat{c}_{1}(\mathcal{V}^{\prime})\widehat{c}_{1}(\mathcal{W}^{\prime})]\cdot[K_{X}]^{n-2}=0.$

Thus it follows that $X$ satisfies the Chern class equality (13), which
completes the proof. ∎

Putting together Propositions 9.1 and 9.2, we arrive at the following
necessary and sufficient condition for a projective klt variety $X$ with ample
canonical divisor to be uniformized by the Hermitian symmetric space
$\mathcal{A}_{pq}$ $(p\neq q)$ of type $AIII$.

######  Theorem 9.3.

Let $X$ be a projective klt variety of dimension $pq\geq 2$,
$p,q\in\mathbb{Z}_{\geq 1}$, $p\neq q$, such that the canonical divisor
$K_{X}$ is ample. Then $X\cong\mathcal{A}_{pq}/\Gamma$, where
$\Gamma\subset\textrm{Aut}(\mathcal{A}_{pq})$ is a discrete, cocompact
subgroup, and acts fixed point freely in codimension one on
$\mathcal{A}_{pq}$, if and only if $X$ satisfies

  * •

$\mathcal{T}_{X_{reg}}\cong\mathcal{H}om(\mathcal{V},\mathcal{W})$

  * •

$[2(p+q)(\widehat{c}_{2}(\mathcal{V}^{\prime})+\widehat{c}_{2}(\mathcal{W}^{\prime}))-(p+q-1)(\widehat{c}_{1}(\mathcal{V}^{\prime})^{2}+\widehat{c}_{1}(\mathcal{W}^{\prime})^{2})+2\widehat{c}_{1}(\mathcal{V}^{\prime})\widehat{c}_{1}(\mathcal{W}^{\prime})]\cdot[K_{X}]^{n-2}=0$,

where $\mathcal{V}$ and $\mathcal{W}$ are vector bundles of ranks $p$ and $q$
on $X_{reg}$, and $\mathcal{V}^{\prime}$ and $\mathcal{W}^{\prime}$ denote the
reflexive extensions of $\mathcal{V}$ and $\mathcal{W}$ to $X$.

Note that when $p=1$ and $q=n$, the associated domain is the unit ball
$\mathbb{B}^{n}$. In this case we have $\mathcal{V}=\Omega^{1}_{X_{reg}}$,
$\mathcal{W}=\mathcal{O}_{X_{reg}}$, and the system of Hodge bundles is given
by
$P\times_{K}\mathfrak{g}=\mathcal{E}nd_{0}(\Omega^{1}_{X_{reg}}\oplus\mathcal{O}_{X_{reg}})$.
The condition on the structure of the tangent bundle is tautological, and the
Chern class condition is the well known Bogomolov-Miyaoka-Yau equality
satisfied by ball quotients.

###  9.3 Necessary conditions when $p=q$

In this case the group of holomorphic automorphisms has two connected
components. Again from [19] we know that $Z\mapsto Z^{T}$ is an automorphism
of $\mathcal{A}_{pp}$ of order 2 that is not contained in the connected
component of $\textrm{Aut}(\mathcal{A}_{pp})$. This defines a group
homomorphism $\mathbb{Z}_{2}\to\textrm{Aut}(\mathcal{A}_{pp})$, i.e., there is
a splitting
$\textrm{Aut}(\mathcal{A}_{pp})=\textrm{Aut}^{0}(\mathcal{A}_{pp})\rtimes\mathbb{Z}_{2}$,
where $\textrm{Aut}^{0}(\mathcal{A}_{pp})=PSU(p,p)$. It follows that
$\textrm{Aut}(\mathcal{A}_{pp})$ is a quotient of
$SU(p,p)\rtimes\mathbb{Z}_{2}$ by a discrete central subgroup. Let $G$ denote
the chosen complexification of $SU(p,p)\rtimes\mathbb{Z}_{2}$, and $K$ the
complexification of $K_{0}$. Then the tangent bundle of the regular locus of a
projective, klt quotient of $\mathcal{A}_{pp}$ admits a reduction in structure
group to $K$. However, as in the polydisc case, such a reduction in structure
group does not give a meaningful condition on the tangent sheaf. Therefore, we
state the result up to a 2:1 quasi-étale cover. The proof is essentially the
same as that of Proposition 6.2.

######  Proposition 9.4.

Let $X$ be a projective klt variety of dimension $p^{2}$ with $K_{X}$ ample,
such that $X$ is a quotient of $\mathcal{A}_{pp}$ by a discrete, cocompact
subgroup $\Gamma$ of $\textrm{Aut}(\mathcal{A}_{pp})$ acting fixed point
freely in codimension one. Then $X$ admits a 2:1 quasi-étale cover
$X^{\prime}$ such that the tangent bundle of the regular locus
$X^{\prime}_{reg}\subset X^{\prime}$ satisfies
$\mathcal{T}_{X^{\prime}_{reg}}\cong\mathcal{H}om(\mathcal{E},\mathcal{F})$,
where $\mathcal{E}$ and $\mathcal{F}$ are vector bundles of rank $p$ on
$X_{reg}$. Moreover, $X^{\prime}$ satisfies the Chern class equality 13 for
$p=q$.

###### Proof.

Since the automorphism group $\textrm{Aut}(\mathcal{A}_{pp})$ has two
connected components, the group $\Gamma$ can be expressed as an extension of a
normal subgroup $\Gamma^{\prime}$ by a subgroup of $\mathbb{Z}_{2}$. Thus the
quotient map $\mathcal{A}_{pp}\to X$ factors as

$\displaystyle\mathcal{A}_{pp}\to\mathcal{A}_{pp}/\Gamma^{\prime}\to\mathcal{A}_{pp}/\Gamma=X,$

where $X^{\prime}=\mathcal{A}_{pp}/\Gamma^{\prime}$, and the map
$X^{\prime}\to X$ is a Galois, 2:1 quasi-étale cover. Note that $X^{\prime}$
is again projective, klt, and $K_{X^{\prime}}$ is ample. Again from Theorem
1.1, it follows that the smooth locus $X^{\prime}_{reg}$ of $X^{\prime}$
admits a uniformizing system of Hodge bundles $(P,\theta)$ corresponding to
the Hodge group $G_{0}=SU(p,p)$, such that the system of Hodge bundles
$E=P\times_{K}\mathfrak{g}$ is $K_{X^{\prime}}$-polystable as a Higgs bundle
on $X^{\prime}_{reg}$. Moreover, $X^{\prime}$ satisfies the $\mathbb{Q}$-Chern
class equality $\widehat{c}_{2}(E^{\prime})\cdot[K_{X^{\prime}}]^{n-2}=0$,
where $E^{\prime}$ is the reflexive extension of $E$ to $X$. Therefore, there
is an isomorphism $\theta:\mathcal{T}_{X^{\prime}_{reg}}\cong
P\times_{K}\mathfrak{g}^{-1,1}$ of vector bundles. Recall from the proof of
Proposition 9.1 that $\mathfrak{g}^{-1,1}$ is isomorphic as a
$K$-representation to $\textrm{Hom}(V,W)$, where $V$ and $W$ are
$p$-dimensional complex vector spaces. So we can write the tangent bundle as
$\mathcal{T}_{X^{\prime}_{reg}}\cong
P\times_{K}\textrm{Hom}(V,W)=\mathcal{H}om(\mathcal{V},\mathcal{W})$, where
$\mathcal{V}$ and $\mathcal{W}$ are vector bundles of rank $p$ on
$X^{\prime}_{reg}$. Moreover, we have $\Omega^{1}_{X^{\prime}_{reg}}\cong
P\times_{K}\mathfrak{g}^{1,-1}\cong\mathcal{H}om(\mathcal{W},\mathcal{V})$,
and
$P\times_{K}\mathfrak{g}^{0,0}\cong(\mathcal{E}nd(\mathcal{V})\oplus\mathcal{E}nd(\mathcal{W}))_{0}$.
Thus the system of Hodge bundles $P\times_{K}\mathfrak{g}$ is isomorphic to
$\mathcal{E}nd_{0}(\mathcal{V}\oplus\mathcal{W})$.
Let $\mathcal{V}^{\prime}$ and $\mathcal{W}^{\prime}$ denote the reflexive
extensions of $\mathcal{V}$ and $\mathcal{W}$ to $X^{\prime}$. Then
$E^{\prime}\cong\mathcal{E}nd_{0}(\mathcal{V}^{\prime}\oplus\mathcal{W}^{\prime})$,
and the $\mathbb{Q}$-Chern class equality
$\widehat{c}_{2}(E^{\prime})\cdot[K_{X^{\prime}}]^{n-2}=0$ is equivalent to

$\displaystyle[\widehat{c}_{2}(\mathcal{E}nd_{0}(\mathcal{V}^{\prime},\mathcal{W}^{\prime}))]\cdot[K_{X^{\prime}}]^{n-2}=0.$

Expanding this using Chern class formulae, we arrive at the following
expression

$\displaystyle[4p(\widehat{c}_{2}(\mathcal{V})^{\prime}+\widehat{c}_{2}(\mathcal{W}^{\prime}))-(2p-1)(\widehat{c}_{1}(\mathcal{V}^{\prime})^{2})+\widehat{c}_{1}(\mathcal{W}^{\prime})^{2})+2\widehat{c}_{1}(\mathcal{V}^{\prime})\widehat{c}_{1}(\mathcal{W}^{\prime})]\cdot[K_{X^{\prime}}]^{n-2}=0.$

Thus $X^{\prime}$ satisfies the Chern class equality as claimed, and this
completes the proof. ∎

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