#  Spin Green’s Function Approach to Understanding Localized Majorana Fermions
in the Kitaev Model

Hibiki Takegami<EMAIL_ADDRESS>Course of Studies on
Materials Science, Graduate School of Human and Environmental Studies, Kyoto
University, Kyoto 606-8501, Japan  Takao Morinari
<EMAIL_ADDRESS>Course of Studies on Materials Science,
Graduate School of Human and Environmental Studies, Kyoto University, Kyoto
606-8501, Japan

###### Abstract

The Kitaev model, defined on a honeycomb lattice, features an exactly solvable
ground state with fractionalized Majorana fermion excitations, which can
potentially form non-Abelian anyons crucial for fault-tolerant topological
quantum computing. Although Majorana fermions are essential for obtaining the
exact ground state, their physical interpretation in terms of spin operators
remains unclear. In this study, we employ a Green’s function approach that
maintains SU(2) symmetry to address this issue and explore the model’s finite
temperature properties. Our results demonstrate that the computed temperature
dependence of the correlation functions closely approximates the exact values
at zero temperature, confirming the accuracy of our method. We also present
several exact results concerning the spin Green’s function and spin-spin
correlation functions that are specific to the Kitaev model. Furthermore, the
spin Green’s function effectively describes the behavior of $\mathbb{Z}_{2}$
fluxes, which are formed by the product of $\mathbb{Z}_{2}$ gauge fields
around the hexagon loop, where the gauge field has the form of the product of
two localized Majorana fermions. However, the contribution of itinerant
Majorana fermions remains an open challenge for future research.

Quantum spin liquids have emerged as a central topic in condensed matter
physics, capturing significant attention due to their complex and intriguing
properties Savary and Balents (2017). Among the various theoretical models,
the Kitaev model is particularly notable for its exactly solvable ground
state, which features fractionalized Majorana fermions Kitaev (2006). This
model is not only analytically tractable but also holds potential for real-
world applications in materials with significant spin-orbit coupling Jackeli
and Khaliullin (2009); Takagi  _et al._ (2019). Majorana fermions play a
crucial role in the development of topological quantum computers, offering a
platform for fault-tolerant quantum computation Kitaev (2003); Nayak  _et al._
(2008).

While investigating the Kitaev model in terms of Majorana fermions provides
significant mathematical elegance, a deeper understanding of the physical
nature of Majorana fermions may be achieved through the study of spin
operators. However, the Kitaev model presents substantial challenges due to
its highly frustrated and strongly correlated nature.

In this paper, we employ the spin Green’s function approach for the spin
operators while preserving SU(2) symmetry and solve its equation of motion.
This method, which maintains SU(2) symmetry Kondo and Yamaji (1972), is
particularly suitable because the system does not exhibit magnetic long-range
order even at zero temperature. Our spin Green’s function describes the
propagation of a pair of $\mathbb{Z}_{2}$ fluxes. Another critical observation
is that the spin-spin correlation function in the ground state is finite only
between nearest neighbor sites Baskaran  _et al._ (2007), suggesting that the
spin-spin correlation length remains short-ranged even at finite temperatures.
Since the equation of motion for the Green’s function Zubarev (1960) relies on
finite-range correlation functions, it is well-suited for exploring the finite
temperature properties of the Kitaev model. Additionally, this approach aligns
with the high-temperature expansion Kondo and Yamaji (1972), enhancing the
accuracy of results at higher temperatures. Our results demonstrate that the
computed temperature dependence of the correlation functions yields a value at
zero temperature that is quite close to the exact value. This comprehensive
analysis provides new insights into the physical nature of Majorana fermions
in the Kitaev model.

The Hamiltonian of the isotropic Kitaev model on the honeycomb lattice is
given by Kitaev (2006):

${\cal H}=K\sum_{\gamma=x,y,z}\sum_{{\langle
i,j\rangle}_{\gamma}}S_{i}^{\gamma}S_{j}^{\gamma},$ (1)

where $S_{i}^{\gamma}$ is the $\gamma$-component of the spin-1/2 operator at
site $i$. The summation of ${\langle i,j\rangle}_{\gamma}$ represents the sum
over the nearest neighbor sites connected by a $\gamma$ bond, as shown in Fig.
1. In the honeycomb lattice, each unit cell comprises two sites corresponding
to the A and B sublattices, as illustrated in Fig. 1. Additionally, Fig. 1
shows a plaquette $p$ consisting of six sites. For this plaquette, we define
the following plaquette operator Kitaev (2006):

$W_{p}=\sigma_{1}^{x}\sigma_{2}^{y}\sigma_{3}^{z}\sigma_{4}^{x}\sigma_{5}^{y}\sigma_{6}^{z},$
(2)

where $\sigma_{i}^{\gamma}$ represents the $\gamma$-component of the Pauli
matrix at site $i$. $\sigma_{i}^{\gamma}$ can be expressed as
$\sigma_{i}^{\gamma}=ib_{i}^{\gamma}c_{i}$ in terms of two types of Majorana
fermions Kitaev (2006), where this equality holds by restricting the extended
Hilbert space of Majorana fermions to the physical Hilbert space of the
original spin operators. The product of two Majorana fermions, $b_{i}$ and
$b_{j}$, residing at nearest neighbor sites forms a $\mathbb{Z}_{2}$ gauge
field. In this formulation, $W_{p}$ represents the $\mathbb{Z}_{2}$ gauge
flux. Meanwhile, the Majorana fermions $c_{i}$ are itinerant under these
$\mathbb{Z}_{2}$ gauge fields.

Figure 1:  (Color online) Kitaev model on the honeycomb lattice, showing two
interpenetrating sublattices, A and B. Sublattice A is represented by red
circles, while sublattice B is represented by blue circles. The direction-
dependent interactions in Eq. (1) are labeled as $x$, $y$, and $z$. The
lattice vectors are denoted as ${\bf n}_{1}=(\sqrt{3}/2,3/2)a_{0}$ and ${\bf
n}_{2}=(-\sqrt{3}/2,3/2)a_{0}$, where $a_{0}$ is the nearest neighbor
distance. The plaquette operator $W_{p}$ for plaquette $p$ is defined by Eq.
(2).

An intriguing aspect of the Kitaev model is that it has been rigorously
demonstrated that its ground state is a spin liquid state without long-range
magnetic order. Because of the absence of the magnetic long-range order, we
may apply SU(2) invariant formalsim of the Green’s function methodKondo and
Yamaji (1972). We define the Matsubara Green’s function:

$G_{n{\alpha_{1}},m{\alpha_{2}}}^{\gamma}\left(\tau\right)=-\left\langle{{T_{\tau}}S_{n{\alpha_{1}}}^{\gamma}\left(\tau\right)S_{m{\alpha_{2}}}^{\gamma}\left(0\right)}\right\rangle\equiv{\left\langle{{S_{n{\alpha_{1}}}^{\gamma}}}\mathrel{\left|{\vphantom{{S_{n{\alpha_{1}}}^{\gamma}}{S_{m{\alpha_{2}}}^{\gamma}}}}\right.\kern-1.2pt}{{S_{m{\alpha_{2}}}^{\gamma}}}\right\rangle_{\tau}}.$
(3)

Here $\tau$ represents the imaginary time and $T_{\tau}$ represents the
imaginary time ordering operator. The notation $S_{n{\alpha}}^{\gamma}$
represents the $\gamma$-th component of the spin operator for the $\alpha$-th
sublattice within the $n$-th unit cell.

Before delving into the analysis of the Green’s function, we first discuss its
physical meaning. Specifically, we consider the Green’s function (3) with
$\gamma=x$, $\alpha_{1}=B$, and $\alpha_{2}=A$. This Green’s function
describes a process where the $\mathbb{Z}_{2}$ flux values at adjacent hexagon
plaquettes are flipped at imaginary time $0$ and then flipped again at
imaginary time $\tau$. This process is schematically illustrated in Fig. 2. We
observe that any flipped $\mathbb{Z}_{2}$ flux values must be flipped again;
otherwise, the thermal average will vanish.

Figure 2:  (Color online) The physical meaning of the Green’s function (3) for
the case of $\gamma=x$, $\alpha_{1}=B$, and $\alpha_{2}=A$, with the $n$-th
unit cell shifted from the $m$-th unit cell by ${\bf n}_{1}$. In the left
panel, the red circle denotes the A sublattice in the $m$-th unit cell. At
imaginary time $0$, the $\mathbb{Z}_{2}$ flux values in plaquettes $p_{1}$ and
$p_{2}$ are flipped by the spin operator $S_{m{\rm A}}^{x}$. These flipped
$\mathbb{Z}_{2}$ flux values, indicated by the filled hexagons in the right
panel, are flipped again at imaginary time $\tau$ by the spin operator
$S_{n{\rm B}}^{x}$.

Denoting the inverse temperature by $\beta=1/(k_{\rm B}T)$, where $T$ is the
temperature and $k_{\rm B}$ is the Boltzmann constant, the Fourier
representation of Eq. (3) in the Matsubara frequency, $\omega_{n}=2\pi
n/\beta$ with $n$ being an integer, is given by

$\displaystyle
G_{n{\alpha_{1}},m{\alpha_{2}}}^{\gamma}\left({i{\omega_{n}}}\right)$
$\displaystyle=$
$\displaystyle\int_{0}^{\beta}{d\tau}{e^{i{\omega_{n}}\tau}}G_{n{\alpha_{1}},m{\alpha_{2}}}^{\gamma}\left(\tau\right)$
(4) $\displaystyle\equiv$
$\displaystyle{\left\langle{{S_{n{\alpha_{1}}}^{\gamma}}}\mathrel{\left|{\vphantom{{S_{n{\alpha_{1}}}^{\gamma}}{S_{m{\alpha_{2}}}^{\gamma}}}}\right.\kern-1.2pt}{{S_{m{\alpha_{2}}}^{\gamma}}}\right\rangle_{i{\omega_{n}}}}$

To find the expression for the Green’s function, we first take the derivative
of Eq. (3) with respect to $\tau$ and then perform the Fourier transform. This
process yields

$\displaystyle
i{\omega_{n}}{\left\langle{{S_{n{\alpha_{1}}}^{\gamma}}}\mathrel{\left|{\vphantom{{S_{n{\alpha_{1}}}^{\gamma}}{S_{m{\alpha_{2}}}^{\gamma}}}}\right.\kern-1.2pt}{{S_{m{\alpha_{2}}}^{\gamma}}}\right\rangle_{i{\omega_{n}}}}$
$\displaystyle=$ $\displaystyle{\left\langle{{\left[{{\cal
H},S_{n{\alpha_{1}}}^{\gamma}}\right]}}\mathrel{\left|{\vphantom{{\left[{{\cal
H},S_{n{\alpha_{1}}}^{\gamma}}\right]}{S_{m{\alpha_{2}}}^{\gamma}}}}\right.\kern-1.2pt}{{S_{m{\alpha_{2}}}^{\gamma}}}\right\rangle_{i{\omega_{n}}}}$
(5)
$\displaystyle+\left\langle{\left[{S_{n{\alpha_{1}}}^{\gamma},S_{m{\alpha_{2}}}^{\gamma}}\right]}\right\rangle.$

Now we have a new Green’s function, represented by the first term on the
right-hand side. The equation of motion for this Green’s function is given by

$\displaystyle i{\omega_{n}}{\left\langle{{\left[{{\cal
H},S_{n{\alpha_{1}}}^{\gamma}}\right]}}\mathrel{\left|{\vphantom{{\left[{{\cal
H},S_{n{\alpha_{1}}}^{\gamma}}\right]}{S_{m{\alpha_{2}}}^{\gamma}}}}\right.\kern-1.2pt}{{S_{m{\alpha_{2}}}^{\gamma}}}\right\rangle_{i{\omega_{n}}}}$
$\displaystyle=$ $\displaystyle{\left\langle{{\left[{{\cal H},\left[{{\cal
H},S_{n{\alpha_{1}}}^{\gamma}}\right]}\right]}}\mathrel{\left|{\vphantom{{\left[{{\cal
H},\left[{{\cal
H},S_{n{\alpha_{1}}}^{\gamma}}\right]}\right]}{S_{m{\alpha_{2}}}^{\gamma}}}}\right.\kern-1.2pt}{{S_{m{\alpha_{2}}}^{\gamma}}}\right\rangle_{i{\omega_{n}}}}$
(6) $\displaystyle+\left\langle{\left[{\left[{{\cal
H},S_{n{\alpha_{1}}}^{\gamma}}\right],S_{m{\alpha_{2}}}^{\gamma}}\right]}\right\rangle.$

We first compute $\left[{{\cal H},S_{n{\alpha 1}}^{\gamma}}\right]$, and then
compute $\left[{{\cal H},\left[{{\cal
H},S{n{\alpha_{1}}}^{\gamma}}\right]}\right]$. From the latter calculation,
terms like
${\left\langle{{S_{nA}^{\gamma}S_{{n_{1}}B}^{x}S_{{n_{2}}A}^{\gamma}}}\mathrel{\left|{\vphantom{{S_{nA}^{\gamma}S_{{n_{1}}B}^{x}S_{{n_{2}}A}^{\gamma}}{S_{m\alpha}^{x}}}}\right.\kern-1.2pt}{{S_{m\alpha}^{x}}}\right\rangle_{i{\omega_{n}}}}$,
${\left\langle{{S_{n}^{\gamma}S_{{n_{1}}B}^{\gamma}S_{{n_{2}}A}^{x}}}\mathrel{\left|{\vphantom{{S_{n}^{\gamma}S_{{n_{1}}B}^{\gamma}S_{{n_{2}}A}^{x}}{S_{m\alpha}^{x}}}}\right.\kern-1.2pt}{{S_{m\alpha}^{x}}}\right\rangle_{i{\omega_{n}}}}$,
etc., appear. Instead of considering the equation of motion for such terms, we
approximate them as follows:

$\displaystyle{\left\langle{{S_{nA}^{\gamma}S_{{n_{1}}B}^{x}S_{{n_{2}}A}^{\gamma}}}\mathrel{\left|{\vphantom{{S_{nA}^{\gamma}S_{{n_{1}}B}^{x}S_{{n_{2}}A}^{\gamma}}{S_{m\alpha}^{x}}}}\right.\kern-1.2pt}{{S_{m\alpha}^{x}}}\right\rangle_{i{\omega_{n}}}}$
$\displaystyle\simeq$ $\displaystyle\alpha
c_{AA}^{\gamma}\left({{{\bf{R}}_{{n_{2}}}}-{{\bf{R}}_{n}}}\right)$ (7)
$\displaystyle\times{\left\langle{{S_{{n_{1}}B}^{x}}}\mathrel{\left|{\vphantom{{S_{{n_{1}}B}^{x}}{S_{m\alpha}^{x}}}}\right.\kern-1.2pt}{{S_{m\alpha}^{x}}}\right\rangle_{i{\omega_{n}}}},$
$\displaystyle{\left\langle{{S_{nA}^{\gamma}S_{{n_{1}}B}^{\gamma}S_{{n_{2}}A}^{x}}}\mathrel{\left|{\vphantom{{S_{nA}^{\gamma}S_{{n_{1}}B}^{\gamma}S_{{n_{2}}A}^{x}}{S_{m\alpha}^{x}}}}\right.\kern-1.2pt}{{S_{m\alpha}^{x}}}\right\rangle_{i{\omega_{n}}}}$
$\displaystyle\simeq$ $\displaystyle\alpha
c_{AB}^{\gamma}\left({{{\bf{R}}_{{n_{1}}}}-{{\bf{R}}_{n}}}\right)$ (8)
$\displaystyle\times{\left\langle{{S_{{n_{2}}A}^{x}}}\mathrel{\left|{\vphantom{{S_{{n_{2}}A}^{x}}{S_{m\alpha}^{x}}}}\right.\kern-1.2pt}{{S_{m\alpha}^{x}}}\right\rangle_{i{\omega_{n}}}},$

where $\alpha$ is a parameter to be determined from the constraint. Here, the
correlation functions are defined by

$c_{{\alpha_{1}}{\alpha_{2}}}^{\gamma}\left({{{\bf{R}}_{{n_{2}}}}-{{\bf{R}}_{{n_{1}}}}}\right)=\left\langle{S_{{n_{1}}{\alpha_{1}}}^{\gamma}S_{{n_{2}}{\alpha_{2}}}^{\gamma}}\right\rangle$
(9)

with ${\bf R}_{n}$ being the coordinate vector of the $n$-th unit cell.

After a tedious but straightforward calculation followed by a Fourier
transform to momentum space, we obtain

$\left[{{{\left({i{\omega_{n}}}\right)}^{2}}-\frac{K^{2}}{2}}\right]{{G}^{x}}\left({{\bf{q}},i{\omega_{n}}}\right)-{K^{2}}M\left({\bf{q}}\right){{G}^{x}}\left({{\bf{q}},i{\omega_{n}}}\right)\simeq
KN\left({\bf{q}}\right).$ (10)

The matrix form of the Green’s function,
${{G}^{x}}\left({{\bf{q}},i{\omega_{n}}}\right)$, is given by:

${{G}^{x}}\left({{\bf{q}},i{\omega_{n}}}\right)=\left({\begin{array}[]{*{20}{c}}{G_{AA}^{x}\left({{\bf{q}},i{\omega_{n}}}\right)}&{G_{AB}^{x}\left({{\bf{q}},i{\omega_{n}}}\right)}\\\
{G_{BA}^{x}\left({{\bf{q}},i{\omega_{n}}}\right)}&{G_{BB}^{x}\left({{\bf{q}},i{\omega_{n}}}\right)}\end{array}}\right).$
(11)

The components of the matrices $M({\bf q})$ and $N({\bf q})$ are:

$\displaystyle{M_{AA}}\left({\bf{q}}\right)/\alpha$ $\displaystyle=$
$\displaystyle
c_{AB}^{y}\left(0\right){e^{i{\bf{q}}\cdot{{\bf{n}}_{1}}}}+c_{AB}^{z}\left({{{\bf{n}}_{2}}}\right){e^{i{\bf{q}}\cdot\left({{{\bf{n}}_{1}}-{{\bf{n}}_{2}}}\right)}},$
(12) $\displaystyle{M_{AB}}\left({\bf{q}}\right)/\alpha$ $\displaystyle=$
$\displaystyle-
c_{AA}^{y}\left({-{{\bf{n}}_{2}}}\right)-\left[{c_{AB}^{z}\left(0\right)+c_{AB}^{y}\left({{{\bf{n}}_{2}}}\right)}\right]{e^{-i{\bf{q}}\cdot{{\bf{n}}_{1}}}}$
(13) $\displaystyle-
c_{AA}^{z}\left({{{\bf{n}}_{2}}}\right){e^{-i{\bf{q}}\cdot{{\bf{n}}_{2}}}},$
$\displaystyle{M_{BA}}\left({\bf{q}}\right)/\alpha$ $\displaystyle=$
$\displaystyle-
c_{BB}^{y}\left({{{\bf{n}}_{2}}}\right)-\left[{c_{BA}^{z}\left(0\right)+c_{BA}^{y}\left({-{{\bf{n}}_{2}}}\right)}\right]{e^{i{\bf{q}}\cdot{{\bf{n}}_{1}}}}$
(14) $\displaystyle-
c_{BB}^{z}\left({-{{\bf{n}}_{2}}}\right){e^{i{\bf{q}}\cdot{{\bf{n}}_{2}}}},$
$\displaystyle{M_{BB}}\left({\bf{q}}\right)/\alpha$ $\displaystyle=$
$\displaystyle c_{BA}^{y}\left(0\right){e^{-i{\bf{q}}\cdot{{\bf{n}}_{1}}}}$
(15)
$\displaystyle+c_{BA}^{z}\left({-{{\bf{n}}_{2}}}\right){e^{-i{\bf{q}}\cdot\left({{{\bf{n}}_{1}}-{{\bf{n}}_{2}}}\right)}}.$

and

$\displaystyle{N_{AA}}\left({\bf{q}}\right)$ $\displaystyle=$ $\displaystyle
c_{AB}^{z}\left(0\right)+c_{AB}^{y}\left({{{\bf{n}}_{2}}}\right)$
$\displaystyle{N_{AB}}\left({\bf{q}}\right)$ $\displaystyle=$ $\displaystyle-
c_{AB}^{y}\left(0\right)-c_{AB}^{z}\left({{{\bf{n}}_{2}}}\right){e^{-i{{\bf{n}}_{2}}\cdot{\bf{q}}}}$
$\displaystyle{N_{BA}}\left({\bf{q}}\right)$ $\displaystyle=$ $\displaystyle-
c_{BA}^{y}\left(0\right)-c_{BA}^{z}\left({-{{\bf{n}}_{2}}}\right){e^{i{{\bf{n}}_{2}}\cdot{\bf{q}}}}$
$\displaystyle{N_{BB}}\left({\bf{q}}\right)$ $\displaystyle=$ $\displaystyle
c_{BA}^{z}\left(0\right)+c_{BA}^{y}\left({-{{\bf{n}}_{2}}}\right),$ (16)

respectively. We obtain a similar formula for
${{G}^{y}}\left({{\bf{q}},i{\omega_{n}}}\right)$ and
${{G}^{z}}\left({{\bf{q}},i{\omega_{n}}}\right)$. However, due to the symmetry
of the honeycomb lattice, we do not need these for the isotropic case.

Solving Eq. (11), we obtain

$\displaystyle{G^{x}}\left({{\bf{q}},i{\omega_{n}}}\right)$ $\displaystyle=$
$\displaystyle\frac{K}{{\left[{{{\left({i{\omega_{n}}}\right)}^{2}}-{{\left({\omega_{\bf{q}}^{\left(+\right)}}\right)}^{2}}}\right]\left[{{{\left({i{\omega_{n}}}\right)}^{2}}-{{\left({\omega_{\bf{q}}^{\left(-\right)}}\right)}^{2}}}\right]}}$
(19)
$\displaystyle\times\left({\begin{array}[]{*{20}{c}}{{{\left({i{\omega_{n}}}\right)}^{2}}-{K^{2}}/2-a_{\bf{q}}^{*}}&{{b_{\bf{q}}}}\\\
{b_{\bf{q}}^{*}}&{{{\left({i{\omega_{n}}}\right)}^{2}}-{K^{2}}/2-{a_{\bf{q}}}}\end{array}}\right){N_{\bf{q}}},$

The energy dispersion $\omega_{\bf{q}}^{(\pm)}$ is given by

$\omega_{\bf{q}}^{\left(\pm\right)}=\sqrt{\frac{{{K^{2}}}}{2}+{\mathop{\rm
Re}\nolimits}{a_{\bf{q}}}\pm{\lambda_{\bf{q}}}},$ (20)

with
${\lambda_{\bf{q}}}=\sqrt{{{\left|{{b_{\bf{q}}}}\right|}^{2}}-{{\left|{{\mathop{\rm
Im}\nolimits}{a_{\bf{q}}}}\right|}^{2}}}$. The terms $a_{\bf q}$ and $b_{\bf
q}$ are defined by

${a_{\bf{q}}}=+\alpha{K^{2}}{e^{i{\bf{q}}\cdot{{\bf{n}}_{1}}}}\left[{c_{AB}^{y}\left(0\right)+c_{AB}^{z}\left({{{\bf{n}}_{2}}}\right){e^{-i{\bf{q}}\cdot{{\bf{n}}_{2}}}}}\right],$
(21) $\displaystyle{b_{\bf{q}}}$ $\displaystyle=$
$\displaystyle-\alpha{K^{2}}\left[c_{AB}^{z}\left(0\right){e^{-i{\bf{q}}\cdot{{\bf{n}}_{1}}}}+c_{AB}^{y}\left({{{\bf{n}}_{2}}}\right){e^{-i{\bf{q}}\cdot{{\bf{n}}_{1}}}}\right.$
(22)
$\displaystyle\left.+c_{AA}^{y}\left({-{{\bf{n}}_{2}}}\right)+c_{AA}^{z}\left({{{\bf{n}}_{2}}}\right){e^{-i{\bf{q}}\cdot{{\bf{n}}_{2}}}}\right],$

In our Green’s function approach, we determine the correlation functions in a
self-consistent manner. The correlation functions are expressed in terms of
the Green’s function as

$c_{{\alpha_{1}}{\alpha_{2}}}^{x}\left({\bf{r}}\right)=-\frac{1}{N}\sum\limits_{\bf{q}}{{e^{i{\bf{q}}\cdot{\bf{r}}}}}{\left[{{G^{x}}\left({{\bf{q}},\tau={0^{+}}}\right)}\right]_{{\alpha_{1}}{\alpha_{2}}}},$
(23)

where $N$ is the number of unit cells. In the numerical calculations below, we
take $N=45\times 45$. There are relationships between the correlation
functions due to the absence of magnetic long-range order and any symmetry
breaking. By utilizing these symmetries, we define

${c_{1}}\equiv
c_{AB}^{z}\left(0\right)=c_{AB}^{x}\left({{{\bf{n}}_{1}}}\right)=c_{AB}^{y}\left({{{\bf{n}}_{2}}}\right),$
(24) $\displaystyle{c^{\prime}_{1}}$ $\displaystyle\equiv$ $\displaystyle
c_{AB}^{x}\left(0\right)=c_{AB}^{y}\left({{{\bf{n}}_{1}}}\right)=c_{AB}^{z}\left({{{\bf{n}}_{2}}}\right)$
(25) $\displaystyle=$ $\displaystyle
c_{AB}^{y}\left(0\right)=c_{AB}^{z}\left({{{\bf{n}}_{1}}}\right)=c_{AB}^{x}\left({{{\bf{n}}_{2}}}\right),$
$\displaystyle{c_{2}}$ $\displaystyle\equiv$ $\displaystyle
c_{AA}^{y}\left({{{\bf{n}}_{2}}}\right)=c_{AA}^{x}\left({{{\bf{n}}_{1}}-{{\bf{n}}_{2}}}\right)=c_{AA}^{z}\left({{{\bf{n}}_{1}}}\right)$
(26) $\displaystyle=$ $\displaystyle
c_{AA}^{z}\left({{{\bf{n}}_{2}}}\right)=c_{AA}^{y}\left({{{\bf{n}}_{1}}-{{\bf{n}}_{2}}}\right)=c_{AA}^{x}\left({{{\bf{n}}_{1}}}\right).$

The self-consistent equations to be solved are then given by

$\displaystyle{c_{1}}$ $\displaystyle=$ $\displaystyle\frac{1}{{\beta
N}}\sum\limits_{\bf{q}}{\sum\limits_{s=\pm}{{e^{i{\bf{q}}\cdot{{\bf{n}}_{1}}}}}}\frac{{Z_{AB}^{s}}}{{{{\left({\omega_{\bf{q}}^{s}}\right)}^{2}}}}g\left({\frac{{\beta\omega_{\bf{q}}^{s}}}{2}}\right)$
(27) $\displaystyle c_{1}^{\prime}$ $\displaystyle=$
$\displaystyle\frac{1}{{\beta
N}}\sum\limits_{\bf{q}}{\sum\limits_{s=\pm}{\frac{{Z_{AB}^{s}}}{{{{\left({\omega_{\bf{q}}^{s}}\right)}^{2}}}}g\left({\frac{{\beta\omega_{\bf{q}}^{s}}}{2}}\right)}}$
(28) $\displaystyle{c_{2}}$ $\displaystyle=$ $\displaystyle\frac{1}{{\beta
N}}\sum\limits_{\bf{q}}{\sum\limits_{s=\pm}{{e^{i{\bf{q}}\cdot\left({{{\bf{n}}_{1}}-{{\bf{n}}_{2}}}\right)}}}}\frac{{Z_{AA}^{s}}}{{{{\left({\omega_{\bf{q}}^{s}}\right)}^{2}}}}g\left({\frac{{\beta\omega_{\bf{q}}^{s}}}{2}}\right)$
(29) $\displaystyle\frac{1}{4}$ $\displaystyle=$ $\displaystyle\frac{1}{{\beta
N}}\sum\limits_{\bf{q}}{\sum\limits_{s=\pm}{\frac{{Z_{AA}^{s}}}{{{{\left({\omega_{\bf{q}}^{s}}\right)}^{2}}}}g\left({\frac{{\beta\omega_{\bf{q}}^{s}}}{2}}\right)}}$
(30)

where $g(x)=x\coth x$. The last equation is obtained from
$c_{AA}^{x}\left(0\right)=1/4$. $Z_{AA}^{\pm}$ and $Z_{AB}^{\pm}$ are given by

$Z_{AA}^{\pm}=\pm\frac{2\left({i{\rm
Im}{a_{\bf{q}}}\pm{\lambda_{\bf{q}}}}\right){c_{1}}+{b_{\bf{q}}}\left({{e^{i{{\bf{n}}_{2}}\cdot{\bf{q}}}}+1}\right){c_{1}^{\prime}}}{{2{\lambda_{\bf{q}}}}},$
(31)
$Z_{AB}^{\pm}=\mp\frac{{\left({{e^{-i{{\bf{n}}_{2}}\cdot{\bf{q}}}}+1}\right)\left({i{\rm
Im}{a_{\bf{q}}}\pm{\lambda_{\bf{q}}}}\right){c_{1}^{\prime}}+2{b_{\bf{q}}}{c_{1}}}}{{2{\lambda_{\bf{q}}}}}.$
(32)

The temperature dependencies of $c_{1}$, $c^{\prime}_{1}$, and $c_{2}$ are
shown in the left panel of Fig. 3. The value of $4c_{1}$ converges to 0.4859
as $T$ approaches zero, closely approximating the exact valueBaskaran  _et
al._ (2007) of $0.5249\equiv 4c_{1}^{\rm exact}$. At high temperatures, the
behavior of $c_{1}$ aligns well with the high-temperature expansion result,
expressed as $c_{1}=\beta K/16+O(\beta^{3})$. Meanwhile, $c^{\prime}_{1}$ and
$c_{2}$ remain zero across all temperatures, a finding rigorously
confirmedBaskaran  _et al._ (2007) at $T=0$ and consistent even at finite
temperatures. The vanishing of $c^{\prime}_{1}$ and $c_{2}$ can be understood
as follows. Consider, for instance, $c_{AB}^{x}(0)=\left\langle
S_{0A}^{x}S_{0B}^{x}\right\rangle$, which corresponds to $c^{\prime}_{1}$. In
this correlation function, the spin operator $S_{0B}^{x}$ flips the
$\mathbb{Z}_{2}$ flux values of the adjacent hexagons that share the $x$-bond
emanating from the B sublattice in the 0-th unit cell. Similarly, the spin
operator $S_{0A}^{x}$ flips the $\mathbb{Z}_{2}$ flux values of the adjacent
hexagons that share the $x$-bond emanating from the A sublattice in the 0-th
unit cell. Since the hexagons affected by $S_{0B}^{x}$ and $S_{0A}^{x}$ are
different and the Hamiltonian $H$ does not alter the $\mathbb{Z}_{2}$ flux
values, the thermal average of their product is identically zero. This
reasoning also applies to the other correlation functions, highlighting a
unique characteristic of the Kitaev model. The same argument leads to
$\left\langle{{T_{\tau}}S_{n{\alpha_{1}}}^{\gamma_{1}}\left(\tau\right)S_{m{\alpha_{2}}}^{\gamma_{2}}\left(0\right)}\right\rangle\equiv
0$ for $\gamma_{1}\neq\gamma_{2}$. The flux values flipped by
$S_{m{\alpha_{2}}}^{\gamma_{2}}$ at imaginary time 0 cannot be flipped again
by $S_{n{\alpha_{1}}}^{\gamma_{1}}$ at imaginary time $\tau$. We note that
this result is exact. From this observation, we may restrict the two-spin
Green’s function to the form of Eq. (3). We also note that the vanishing of
$c^{\prime}_{1}$ and $c_{2}$ results in $\omega_{\bf{q}}^{\pm}$ forming a flat
band, given by $K\sqrt{\left(1\pm 4\alpha{c_{1}}\right)/2}$. Consequently, the
Green’s function describes a localized field.

In the right panel of Fig. 3, we present the temperature dependence of
$\alpha$. As temperature increases, $\alpha$ approaches one, indicating that
the decoupling approximation in Eqs. (7) and (8) becomes exact without the
need for the correction parameter $\alpha$. However, deviations of $\alpha$
from one at lower temperatures suggest the necessity of this correction
parameter in the equations.

Figure 3:  (Color online) Temperature dependence of the correlation functions
and the correction parameter $\alpha$. Left panel: The temperature
dependencies of $c_{1}$, $c^{\prime}_{1}$, and $c_{2}$ are shown. The value of
$4c_{1}$ approaches the exact theoretical valueBaskaran  _et al._ (2007) of
0.5249 at $T=0$ and aligns with high-temperature expansion results at elevated
temperatures, while $c^{\prime}_{1}$ and $c_{2}$ remain zero across the
temperature range. The dashed line represents the high-temperature expansion.
Right panel: The variation of the parameter $\alpha$ with temperature,
demonstrating the exactness of the decoupling approximation at high
temperatures (where $\alpha$ approaches 1) and the necessity of this
correction parameter at lower temperatures, where $\alpha$ deviates from 1.

From the temperature dependence of $c_{1}$, the internal energy can be
computed using the relation

$E=\left\langle H\right\rangle=-\frac{3}{4}NK{c_{1}}.$ (33)

We compared the temperature dependence of the energy with the results shown in
Fig. 15(a) of Ref. Motome and Nasu, 2020, as shown in Fig. 4(a). We found an
almost perfect match for $T>T_{H}$ with $T_{H}\simeq 0.375K$, but our energy
values are slightly higher for $T<T_{H}$. This discrepancy likely originates
from the error in the $c_{1}$ value at $T=0$. Although the relative error is
$|(c_{1}^{\rm exact}-c_{1})/c_{1}^{\rm exact}|\simeq 0.07$, minor details and
significant physics, such as the fractionalization of spins, may be hidden
within this discrepancy.

Figure 4 shows the temperature dependencies of the specific heat and entropy.
The specific heat is calculated from the temperature derivative of the
internal energy $E$ and compared with the high-temperature expansion (HTE)
result, $C_{\rm HTE}=3Nk_{\rm B}K^{2}\beta^{2}/16$. The entropy is computed
using the formula:

$S=S\left(\infty\right)-\int_{T}^{T_{0}}dT\frac{C(T)}{T}-\int_{T_{0}}^{\infty}dT\frac{C_{\rm
HTE}(T)}{T},$ (34)

where $S\left(\infty\right)=2N{k_{\rm B}}\ln 2$ and $T_{0}$ is a sufficiently
large value at which the Green’s function result for the specific heat closely
approximates the HTE result (here, we take $T_{0}/K=10$). The specific heat
exhibits a single broad peak around $T/K\simeq 0.4$, consistent with mean
field calculations Saheli  _et al._ (2024). We likely fail to observe the low-
temperature peak around $T_{L}\simeq 0.012K$ reported in Ref. Nasu  _et al._ ,
2015 due to the discrepancy mentioned above. The entropy indicates the
presence of residual entropy, but if fractionalization were captured, this
residual entropy should vanish, as it has been rigorously shown that there are
no $\mathbb{Z}_{2}$ fluxes in the ground state and the system must satisfy the
third law of thermodynamics. However, we note that the residual entropy is
less than one-half of the entropy expected for itinerant Majorana fermions.

Figure 4:  (Color online) Temperature dependencies of (a) the energy
$E=-3NKc_{1}/4$, (b) specific heat, and (c) entropy. The red circles in (a)
are data taken from Ref. Motome and Nasu, 2020. The specific heat, calculated
from the temperature derivative of the internal energy $E$, is compared with
the high-temperature expansion result $C_{\rm HTE}=3Nk_{\rm
B}K^{2}\beta^{2}/16$. It displays a broad peak around $T/K\simeq 0.4$ without
any lower-temperature peaks. The entropy is computed based on Eq. (34).

In our Green’s function approach, the behavior of $\mathbb{Z}_{2}$ fluxes is
effectively described. The $\mathbb{Z}_{2}$ flux defined at each hexagon
plaquette is formed by the product of the $\mathbb{Z}_{2}$ gauge field around
the sides of that hexagon Kitaev (2006). In terms of Majorana fermions Kitaev
(2006), the $\mathbb{Z}_{2}$ gauge field consists of the product of two
localized Majorana fermions, $ib_{i}b_{j}$. In this sense, our Green’s
function approach is related to the localized Majorana fermions. However, our
approach likely does not account for the itinerant Majorana fermions, $c_{i}$.
To include itinerant Majorana fermions, it is necessary to consider a
multiplicative flipping process of $\mathbb{Z}_{2}$ flux values by spin
operators. Achieving the itinerant nature probably requires incorporating
infinite orders of spin operators. In other words, we need to consider the
Green’s function with an infinite number of spin operators. It is plausible
that such a Green’s function has a simple functional form in terms of spin
operators, like an exponential function, although its explicit form is
currently unknown. The challenge of finding a concrete formula associated with
itinerant Majorana fermions in terms of spin operators remains a task for
future study.

In conclusion, our Green’s function approach effectively describes the
behavior of $\mathbb{Z}_{2}$ fluxes in the Kitaev model, revealing their
relationship with the localized components of Majorana fermions. Our results
demonstrate that the computed temperature dependence of the correlation
functions gives a value at zero temperature that is quite close to the exact
value. However, the current approach likely omits the contribution of
itinerant Majorana fermions. To incorporate itinerant Majorana fermions, it is
necessary to consider a multiplicative flipping process of $\mathbb{Z}_{2}$
flux values by spin operators, which requires incorporating infinite orders of
spin operators. Future work is needed to derive a concrete formula for
itinerant Majorana fermions in terms of spin operators, which remains an open
challenge.

###### Acknowledgements.

The authors thank K. Harada and D. Sasamoto for helpful discussions.

## References

  * Savary and Balents (2017) L. Savary and L. Balents, Rep. Progr. Phys. 80, 016502 (2017).
  * Kitaev (2006) A. Kitaev, Ann. Phys. 321, 2 (2006).
  * Jackeli and Khaliullin (2009) G. Jackeli and G. Khaliullin, Phys. Rev. Lett. 102, 017205 (2009).
  * Takagi  _et al._ (2019) H. Takagi, T. Takayama, G. Jackeli, G. Khaliullin, and S. E. Nagler, Nat. Rev. Phys. 1, 264 (2019).
  * Kitaev (2003) A. Kitaev, Ann. Physics 303, 2 (2003).
  * Nayak  _et al._ (2008) C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, Rev. Mod. Phys. 80, 1083 (2008).
  * Kondo and Yamaji (1972) J. Kondo and K. Yamaji, Progr. Theoret. Phys. 47, 807 (1972).
  * Baskaran  _et al._ (2007) G. Baskaran, S. Mandal, and R. Shankar, Phys. Rev. Lett. 98, 247201 (2007).
  * Zubarev (1960) D. N. Zubarev, Sov. Phys. Usp. 3, 320 (1960).
  * Motome and Nasu (2020) Y. Motome and J. Nasu, J. Phys. Soc. Japan 89, 012002 (2020).
  * Saheli  _et al._ (2024) S. G. Saheli, J. Lin, H. Hu, and F. Krüger, Phys. Rev. B 109, 014407 (2024).
  * Nasu  _et al._ (2015) J. Nasu, M. Udagawa, and Y. Motome, Phys. Rev. B 92, 115122 (2015).


# A black hole solution in conformal supergravity

Pedro D. Alvarez<EMAIL_ADDRESS>Departamento de Física, Universidad
de Antofagasta, Aptdo. 02800, Chile  Cristóbal Corral<EMAIL_ADDRESS>Instituto de Ciencias Exactas y Naturales, Universidad Arturo Prat, Playa
Brava 3256, 1111346, Iquique, Chile  Facultad de Ciencias, Universidad Arturo
Prat, Avenida Arturo Prat Chacón 2120, 1110939, Iquique, Chile  Jorge Zanelli
<EMAIL_ADDRESS>Centro de Estudios Científicos (CECs), Arturo Prat 514,
Valdivia, Chile,  Universidad San Sebastián, General Lagos 1163, Valdivia,
Chile

###### Abstract

We present a three-parameter family of analytic black-hole solutions in the
bosonic sector of a four-dimensional supersymmetric model with matter fields
in the adjoint representation. The solutions are endowed with a curvature and
torsional singularities which are both surrounded by an event horizon. They
are asymptotically Lorentz flat, representing the torsional generalization of
the Riegert black hole in conformal gravity. We compute the partition function
to first order in the saddle-point approximation which turns out to be finite
without any reference to boundary counterterms. We find a non-maximmally
symmetric thermalized ground state, whose existence is relevant when studying
Hawking-Page phase transitions. Finally, we discuss future directions
regarding its extended phase space.

##  I Introduction

Supersymmetry (SUSY) is an appealing approach to address different problems in
theoretical physics. From the viewpoint of quantum field theory, it attracted
considerable interest as a mechanism to cancel divergences arising at the one-
loop level coming from fermions and bosons. It is also a robust candidate for
solving the hierarchy problem in the Standard Model of particle physics Martin
(2010); Cohen  _et al._ (1996); Dimopoulos and Giudice (1995). Moreover, by
combining spacetime and internal symmetries in a graded Lie algebra, SUSY
circumvents the Coleman-Mandula theorem Coleman and Mandula (1967), offering a
natural framework to unify those hitherto segregated symmetries. Supergravity,
on the other hand, provides an interesting gravitational setup where the one-
loop divergences are renormalized, in contrast to what happens in general
relativity (for a review see Van Nieuwenhuizen (1981)). The former represents
the gauge theory of SUSY and it appears as the low-energy limit of string
theory Blumenhagen  _et al._ (2013); Ortin (2015).

In recent years, a novel method of implementing SUSY, inspired by ideas from
Yang-Mills theories, supergravity and Einstein-Cartan gravity, has been
proposed Alvarez  _et al._ (2012, 2014a, 2015). In this approach –dubbed
unconventional supersymmetry (USUSY)–, the spin connection, the gauge
potentials and matter fields belong to a gauge connection for a superextension
of the anti-de Sitter (AdS) or Poincaré groups (for a review see Alvarez  _et
al._ (2021a)). What is unconventional about this approach is that matter
fields are in the adjoint representation of the superalgebra. This feature is
faithful to the spirit of supersymmetry, treating bosons and fermions on equal
footing as parts of the same gauge connection. Fermionic fields representing
matter are directly incorporated in the gauge connection by means of a
Clifford algebra-valued soldering form that provides a metric structure and,
eventually, the inclusion of gravity.

This way of implementing SUSY does not produce a pairing of boson-fermion
states as in standard SUSY Sohnius (1985); Alvarez  _et al._ (2012, 2014a) and
therefore, USUSY models greatly differ from standard SUSYs and supergravities
Stelle and West (1978). The lack of superpartners remarkably coincides with
the absence of supporting evidence for superpartners in the LHC. Recently, we
have constructed chiral gauge models for $SU(2,2|2)$ Alvarez  _et al._ (2020)
and $SU(2,2|N)$ Alvarez  _et al._ (2021b), and we have also described the
embedding of the bi-fundamental spinor representation in the superconformal
algebras Alvarez  _et al._ (2022). These developments can be seen as a step
towards defining a grand unified theory based on the conformal superalgebras.

Conformal supergravity is an interesting gauge theory for the conformal
superalgebra studied by many authors in the last decades Kaku  _et al._
(1977); Ferrara  _et al._ (1977); Kaku and Townsend (1978); Kaku  _et al._
(1978); Bergshoeff  _et al._ (1981, 1983); Fradkin and Tseytlin (1985, 1985);
Liu and Tseytlin (1998); Butter (2010); Ferrara  _et al._ (2018); D’Auria and
Ravera (2021). Indeed, by a proper gauge fixing, it is possible to arrive from
conformal supergravity to the standard $\mathcal{N}=1$ supergravity in four
dimensions Freedman and Van Proeyen (2012). In Refs. Alvarez  _et al._ (2014b,
2020, 2021c), we discussed a conformal supergravity theory based on the
$\mathfrak{su}(2,2|N)$ gauge algebra written in a Townsend-MacDowell Mansouri
MacDowell and Mansouri (1977); Townsend (1977) form, which can be regarded as
the $3+1$ generalization of the proposal in Ref. Horne and Witten
(1989).111For a similar approach see Ref. Andrianopoli and D’Auria (2014),
whose applications in holography have been studied in Ref. Andrianopoli  _et
al._ (2021a). Its novelty relies on the implementation of supersymmetry in the
adjoint representation, whose partially broken conformal symmetry is realized
explicitly in the action. Remarkably, Dirac supercharges renders the full
$R$-symmetry manifest Trigiante (2017), where the latter is identified with
the internal $SU(N)\times U(1)$ group.

In the present paper, we explore some gravitational vacuum solutions of the
theory, where “vacuum” means absence of fermionic matter fields. Similar
configurations have been found in Chern-Simons AdS5 supergravity models
Giribet  _et al._ (2014); Andrianopoli  _et al._ (2021b). Here, we focus on
four-dimensional topological black holes with constant-curvature transverse
sections possessing torsional hair in unconventional conformal supergravity.
We found a three-parameter family of analytic asymptotically locally Lorentz-
flat solutions that represents the torsional generalization of the Riegert
metric Riegert (1984) in conformal gravity. We compute its temperature and
partition function to first order in the saddle-point approximation. In a
certain limit, we demonstrate that the black hole becomes a particular two-
parameter family of solutions, possessing a central singularity dressed by a
horizon, a nonvanishing temperature, but its free energy, mass, and entropy
are identically zero. This class of spacetimes has been found in Weyl gravity
as well and it has been regarded as a thermalized ground state Lu  _et al._
(2012). Here, we show their existence in unconventional conformal supergravity
as well.

The paper is organized as follows: In Sec. II, we define the basics of the
model, namely, the field content, action principle, and field equations;
essential ingredients for searching new vacuum solutions. In Sec. III, we
present the novel topological black hole solution with torsion, alongside
their properties. In Sec. IV, we find the thermalized ground state as a
limiting case of the three-parameter family of black holes. Finally, in Sec. V
we summarize our results.

##  II The model

Since we are interested in studying of vacuum solutions, we consider the
purely bosonic sector of the theory discussed in Alvarez  _et al._
(2021c).222A general construction of this theory including spinors can be
found in Ref. Alvarez  _et al._ (2021c). The bosonic part of the gauge
connection $1$-form is then given by

$\mathds{A}=\frac{1}{2}\omega^{ab}\mathds{J}_{ab}+f^{a}\mathds{J}_{a}+g^{a}\mathds{K}_{a}+h\mathds{D}+A^{I}\mathds{T}_{I}+A\mathds{Z}\,,$
(1)

where $\mathds{J}_{a},\mathds{K}_{a},\mathds{J}_{ab}$, and $\mathds{D}$ denote
the generators of the conformal group $SO(3,2)$, while $\mathds{T}_{I}$ and
$\mathds{Z}$ represent the generators of the internal $SU(N)$ and $U(1)$
groups, respectively. Hereon, upper and lowercase Latin indices label internal
$SU(N)$ and Lorentz generators, respectively, while Greek indices denote
spacetime coordinates. Each of the generators is contracted with its
respective $1$-form compensating field and the field strength associated to
this gauge connection is

$\mathds{F}=\text{d}\mathds{A}+\mathds{A}\wedge\mathds{A}=\frac{1}{2}\mathcal{R}^{ab}\mathds{J}_{ab}+F^{a}\mathds{J}_{a}+G^{a}\mathds{K}_{a}+H\mathds{D}+F^{I}\mathds{T}_{I}+F\mathds{Z}\,,$
(2)

where d is the exterior derivative, $\wedge$ is the exterior product of
differential forms. The explicit expressions of the curvature components are

$\displaystyle\mathcal{R}^{ab}$ $\displaystyle=R^{ab}+f^{a}\wedge
f^{b}-g^{a}\wedge g^{b}\,,$ $\displaystyle F^{a}$
$\displaystyle=\text{D}f^{a}+g^{a}\wedge h\,,$ $\displaystyle G^{a}$
$\displaystyle=\text{D}g^{a}+f^{a}\wedge h\,,$ (3) $\displaystyle H$
$\displaystyle=\text{d}h+f^{a}\wedge g_{a}\,,$ $\displaystyle F^{I}$
$\displaystyle=\text{d}A^{I}+\frac{1}{2}f_{JK}^{I}A^{J}\wedge A^{K}\,,$
$\displaystyle F$ $\displaystyle=\text{d}A\,,$ (4)

with $R^{ab}=\text{d}\omega^{ab}+\omega^{a}_{\ c}\wedge\omega^{cb}$ being the
Lorentz curvature $2$-form and D denotes the covariant derivative with respect
to the Lorentz connection.

The dynamics of the theory is described by a generalization of the MacDowell-
Mansouri action MacDowell and Mansouri (1977) for the bosonic sector of the
superconformal group, that is,

$\mathcal{S}=-\int\langle\mathds{F}\wedge\circledast\ \mathds{F}\rangle\,,$
(5)

where $\langle\ldots\rangle$ denotes trace over the internal indices and the
dual operator $\circledast$ is defined through

$\circledast\mathds{F}=S\left(\frac{1}{2}\mathcal{R}^{ab}\mathds{J}_{ab}+F^{a}\mathds{J}_{a}+G^{a}\mathds{K}_{a}\right)+(\varepsilon_{1}\ast)H\mathds{D}+(\varepsilon_{2}\ast)F^{I}\mathds{T}_{I}+(\varepsilon_{3}\ast)F\mathds{Z}\,.$
(6)

Here $\varepsilon_{i}=\pm 1$, with $i=1,2,3$, $\ast$ is the Hodge dual, and
$S=i\gamma_{5}$ with $\gamma_{5}$ being the chiral gamma matrix defined such
that $S^{2}=-\mathbb{I}$. The ambiguity in $\varepsilon_{i}$ can be eliminated
by demanding the correct sign of the kinetic terms of the bosonic fields in
the action. This, in turn, will depend on the details of the algebra
representation. More details of this operator can be found in Alvarez  _et
al._ (2021b).

The action functional (5) can be thought of as a Yang-Mills theory for an
embedding $G^{+}\hookrightarrow SU(2,2|N)$. Using the properties of the
supertrace (cf. Ref. Alvarez  _et al._ (2021b)), we can express the the
bosonic sector of the Lagrangian along the curvature components as

$\displaystyle\mathscr{L}_{\text{bos}}$
$\displaystyle=\frac{1}{4}\epsilon_{abcd}\mathcal{R}^{ab}\wedge\mathcal{R}^{cd}-\varepsilon_{1}(H+f^{a}\wedge
g_{a})\wedge\ast(H+f^{b}\wedge g_{b})$
$\displaystyle-\frac{1}{2}\varepsilon_{2}F^{I}\wedge\ast
F^{I}-4\left(\frac{4}{N}-1\right)\varepsilon_{3}F\wedge\ast F\,.$ (7)

A ghost-free action demands $\varepsilon_{1}=\varepsilon_{2}=1$. The value of
$\varepsilon_{3}$, on the other hand, depends on $N$: the absence of ghosts
demands $\varepsilon_{3}=\pm 1$ for $N<4$ and $N>4$, respectively, such that
$\varepsilon_{3}(N-4)<0$. For $N=4$, however, the $U(1)$ sector drops out from
the Lagrangian (7).

Varying the Lagrangian (7) with respect to $f^{a}$ and $g^{a}$ yields

$\displaystyle\epsilon_{abcd}f^{b}\wedge\mathcal{R}^{cd}-2g_{a}\wedge\ast(H+f^{b}\wedge
g_{b})$ $\displaystyle=0\,,$ (8)
$\displaystyle\epsilon_{abcd}g^{b}\wedge\mathcal{R}^{cd}-2f_{a}\wedge\ast(H+f^{b}\wedge
g_{b})$ $\displaystyle=0\,,$ (9)

respectively. We will search for solutions of these equations (8) and (9)
under the assumption of Lorentz invariance, which is naturally guaranteed if
both $f^{a}$ and $g^{a}$ are conformally related to a soldering form $e^{a}$
that provides the metric structure for the theory. This means
$f^{a}=\rho(x)e^{a}$, $g^{a}=\sigma(x)e^{a}$, where $\rho(x)$ and $\sigma(x)$
are arbitrary scalar functions and $e^{a}=e^{a}_{\mu}\text{d}x^{\mu}$ with
$g_{\mu\nu}=\eta_{ab}\,e^{a}_{\mu}\,e^{b}_{\nu}$. This choice breaks the
conformal invariance and, therefore, the Weyl rescaling is no longer a gauge
symmetry. Then, in the simplest version of this theory, we can take $h=0$.
These assumptions imply $\mathcal{R}^{ab}=R^{ab}-\Lambda(x)e^{a}\wedge e^{b}$,
where we have defined $\Lambda(x):=\sigma^{2}(x)-\rho^{2}(x)$. Note that
asymptotically locally AdS solutions are allowed provided $\Lambda(x)<0$.

The remaining field equations are obtained by extremizing the Lagrangian (7)
with respect to the dynamical fields in the AdS sector. Then, the full set of
field equations is

$\displaystyle\epsilon_{abcd}\text{D}\mathcal{R}^{cd}$ $\displaystyle=0\,,$
$\displaystyle\epsilon_{abcd}e^{b}\wedge\mathcal{R}^{cd}$ $\displaystyle=0\,,$
$\displaystyle\mathfrak{D}\ast F^{I}$ $\displaystyle=0\,,$
$\displaystyle\text{d}\ast F$ $\displaystyle=0\,,$ (10)

where $\mathfrak{D}\equiv D_{(A^{J}\mathds{T}_{J})}$ is the covariant
derivative for the internal gauge connection. These equations, alongside (8)
and (9), are second-order dynamics equations for the gauge connection
$\mathds{A}$ that determine the states of the theory. For the sake of
simplicity, we henceforth focus in the sector with vanishing $SU(N)$ and
$U(1)$ gauge fields.

In order to look for the ground state of the theory, we notice that these
equations admit $\mathcal{R}^{ab}=0$ as solution. Choosing
$\rho(x)^{2}-\sigma(x)^{2}=\pm\ell^{-2}$ to be constant, locally AdS or dS
spacetimes are obtained, that is,

$R^{ab}\pm\ell^{-2}e^{a}\wedge e^{b}=0\,.$ (11)

These geometries include maximally symmetric spaces with up to $10$ globally
defined Killing vectors, which can be genuinely interpreted as vacuum
configurations. The symmetric case $\rho(x)^{2}=\sigma(x)^{2}$, on the other
hand, yields

$\epsilon_{abcd}e^{b}\wedge R^{cd}=0\,,$ (12)

which includes locally Lorentz-flat configurations, $R^{ab}=0$, as well as
nontrivial asymptotically locally Lorentz-flat solutions. This can be achieved
if the torsion $2$-form is convariantly conserved. We present a solution of
the former class in the next section.

##  III Static spherically symmetric solution

Here, we look for black hole solutions to the field equations (10). We work in
the first-order formalism where the vielbein $1$-form,
$e^{a}=e^{a}{}_{\mu}\text{d}x^{\mu}$, and the Lorentz connection $1$-form,
$\omega^{ab}=\omega^{ab}{}_{\mu}\text{d}x^{\mu}$, define the Lorentz curvature
and torsion $2$-forms through the Cartan structure equations as

$\displaystyle R^{ab}$
$\displaystyle=\text{d}\omega^{ab}+\omega^{a}{}_{c}\wedge\omega^{cb}=\tfrac{1}{2}R^{ab}_{\
\mu\nu}\text{d}x^{\mu}\wedge\text{d}x^{\nu}\,,$ (13) $\displaystyle T^{a}$
$\displaystyle=\text{d}e^{a}+\omega^{a}{}_{b}\wedge e^{b}=\tfrac{1}{2}T^{a}_{\
\mu\nu}\text{d}x^{\mu}\wedge\text{d}x^{\nu}\,,$ (14)

respectively. These quantities satisfy the Bianchi identities
$\text{D}R^{ab}=0$ and $\text{D}T^{a}=R^{a}{}_{b}\wedge e^{b}$. The Lorentz
connection $1$-form can be decomposed as
$\omega^{ab}=\mathring{\omega}^{ab}+\kappa^{ab}$, where
$\mathring{\omega}^{ab}$ denotes its torsion-free part satisfying
$\text{d}e^{a}+\mathring{\omega}^{a}{}_{b}\wedge e^{b}=0$ and $\kappa^{ab}$ is
the contorsion $1$-form defined through $T^{a}=\kappa^{a}{}_{b}\wedge e^{b}$.
This decomposition allows rewriting the Lorentz curvature $2$-form as
$R^{ab}=\mathring{R}^{ab}+\mathring{\text{D}}\kappa^{ab}+\kappa^{a}{}_{c}\wedge\kappa^{cb}$,
where $\mathring{R}^{ab}=\tfrac{1}{2}\mathring{R}^{ab}_{\
\mu\nu}\text{d}x^{\mu}\wedge\text{d}x^{\nu}$ represents its torsion-free part
related to the Riemann tensor through $\mathring{R}^{\lambda\rho}_{\
\mu\nu}=e^{\lambda}{}_{a}e^{\rho}{}_{b}\mathring{R}^{ab}_{\ \mu\nu}$ with
$e^{\mu}{}_{a}$ being the inverse vielbein, while
$\mathring{\text{D}}\kappa^{ab}$ denotes the covariant derivative of the
contorsion $2$-form with respect to the torsion-free Lorentz connection. From
hereon, ringed quantities denote torsion-free geometric objects constructed
out of $\mathring{\omega}^{ab}$.

In order to search for static topological black-hole solutions to the field
equations (10), we focus on a metric ansatz possessing a constant-curvature
base manifold, namely,

$\text{d}s^{2}=g_{\mu\nu}\text{d}x^{\mu}\otimes\text{d}x^{\nu}=-f(r)\text{d}t^{2}+\frac{\text{d}r^{2}}{f(r)}+r^{2}\text{d}\Sigma^{2}_{(k)}\,,$
(15)

where the line element of the two-dimensional base manifold is parametrized as

$\displaystyle\text{d}\Sigma_{(k)}^{2}=\bar{g}_{\bar{\mu}\bar{\nu}}\text{d}\bar{x}^{\bar{\mu}}\otimes\text{d}\bar{x}^{\bar{\nu}}=\delta_{\bar{a}\bar{b}}\bar{e}^{\bar{a}}\otimes\bar{e}^{\bar{b}}\,.$
(16)

Here, barred quantities are intrinsically defined on the two-dimensional
transverse section $\text{d}\Sigma_{(k)}^{2}$ and $k=\pm 1,0$ stands for
spherical, hyperbolic or flat topology, respectively. According to the
relation $g_{\mu\nu}=\eta_{ab}e^{a}_{\mu}e^{b}_{\mu}$, we choose a vielbein
basis

$\displaystyle e^{0}$ $\displaystyle=\sqrt{f(r)}\,\text{d}t\,,$ $\displaystyle
e^{1}$ $\displaystyle=\frac{\text{d}r}{\sqrt{f(r)}}\,,$ $\displaystyle
e^{\bar{a}}$ $\displaystyle=r\,\bar{e}^{\bar{a}}\,.$ (17)

Demanding invariance under the same isometry group of the metric (15) to the
remaining bosonic fields,333When $k=\pm 1,0$, the metric (15) is locally
invariant under the action of $\mbox{SO}(3)\times\mathbb{R}$,
$\mbox{SO}(1,2)\times\mathbb{R}$, or $\mbox{ISO}(2)\times\mathbb{R}$ isometry
groups, respectively. To obtain the ansatz (18), we decompose
$\omega^{ab}=\mathring{\omega}^{ab}+\kappa^{ab}$ and we read off their
independent components from the invariance of $\mathring{\omega}^{ab}$ and
$\kappa^{ab}$ under these isometry groups. we find $\Lambda=\Lambda(r)$, and

$\displaystyle\omega^{01}$
$\displaystyle=\omega_{1}(r)e^{0}+\omega_{2}(r)e^{1}\,,$
$\displaystyle\omega^{02}$
$\displaystyle=\omega_{3}(r)e^{2}+\omega_{4}(r)e^{3}\,,$ (18a)
$\displaystyle\omega^{03}$
$\displaystyle=-\omega_{4}(r)e^{2}+\omega_{3}(r)e^{3}\,,$
$\displaystyle\omega^{12}$
$\displaystyle=\omega_{5}(r)e^{2}+\omega_{6}(r)e^{3}\,,$ (18b)
$\displaystyle\omega^{13}$
$\displaystyle=-\omega_{6}(r)e^{2}+\omega_{5}(r)e^{3}\,,$
$\displaystyle\omega^{23}$
$\displaystyle=\omega_{7}(r)e^{0}+\omega_{8}(r)e^{1}+\mathring{\omega}^{23}\,.$
(18c)

In this case, $\omega_{i}(r)$, with $i=1,\dots,8$, are indeterminate functions
depending on the radial coordinate only. The nontrivial components of the
Lorentz curvature and torsion can be easily found from Eq. (13) alongside Eqs.
(17) and (18).

Inserting these ansätze into the field equations (10), we find

$\displaystyle\omega_{2}=\omega_{3}=\omega_{4}=\omega_{6}=\omega_{7}=\omega_{8}$
$\displaystyle=0\,,$ (19a)
$\displaystyle\Lambda^{\prime}fr+2f\Lambda+2\sqrt{f}\,r\Lambda\omega_{5}$
$\displaystyle=0\,,$ (19b)
$\displaystyle\left(fr^{2}\Lambda^{2}\right)^{\prime}+2\sqrt{f}\,r^{2}\Lambda^{2}\left(\omega_{5}-\omega_{1}\right)$
$\displaystyle=0\,,$ (19c) $\displaystyle
2\omega_{5}^{\prime}f\,r^{2}+2f\omega_{5}\,r+k\sqrt{f}+3\sqrt{f}\,\Lambda\,r^{2}-\sqrt{f}\omega_{5}^{2}\,r^{2}$
$\displaystyle=0\,,$ (19d)
$\displaystyle\frac{k}{r^{2}}+3\Lambda+2\omega_{1}\omega_{5}-\omega_{5}^{2}$
$\displaystyle=0\,,$ (19e)

where prime denotes differentiation with respect to $r$. This nonlinear system
of first-order ordinary differential equations is solved by

$\displaystyle f(r)$
$\displaystyle=\gamma-\frac{2m}{r}-\frac{(\gamma^{2}-k^{2})r}{6m}+a\,r^{2}\,,$
$\displaystyle\omega_{5}(r)$
$\displaystyle=\frac{\sqrt{f}(k+\frac{1}{2}f^{\prime\prime}r^{2}-f^{\prime}r+f)}{r\left(f^{\prime}r-2f\right)}\,,$
(20a) $\displaystyle\omega_{1}(r)$
$\displaystyle=\sqrt{f}\,\frac{\text{d}}{\text{d}r}\ln(\omega_{5}\,r)\,,$
$\displaystyle\Lambda(r)$
$\displaystyle=\frac{k-\omega_{5}\,r^{2}\left(\omega_{5}-2\omega_{1}\right)}{3r^{2}}\,.$
(20b)

This solution represents a three-parameter family of topological black holes
characterized by the integration constants $\gamma$, $m$, and $a$. It can be
checked that, as $r\to\infty$, the metric function behaves $f\sim
ar^{2}+\mathcal{O}(r)$, and therefore $\omega_{1}\sim\omega_{5}\to 0$. Then,
from Eq. (20b), one finds that $\Lambda\to 0$. Thus, even though the solution
represents a weakened asymptotically AdS metric, the geometry is
asymptotically Lorentz flat from a Riemann-Cartan viewpoint. A similar feature
has been observed for locally AdS configurations in three dimensions that
represents Lorentz-flat spacetimes in presence of covariantly constant torsion
Alvarez  _et al._ (2014c). Indeed, the solution (20) resembles this behavior
asymptotically in four dimensions. The nonvanishing components of the Lorentz
curvature and torsion are given by

$\displaystyle R^{01}$ $\displaystyle=R_{I}(r)\,e^{0}\wedge e^{1}\,,$
$\displaystyle R^{0\bar{a}}$ $\displaystyle=R_{II}(r)\,e^{0}\wedge
e^{\bar{a}}\,,$ $\displaystyle R^{1\bar{a}}$
$\displaystyle=R_{II}(r)\,e^{1}\wedge e^{\bar{a}}\,,$ (21a) $\displaystyle
R^{23}$ $\displaystyle=R_{I}(r)\,e^{2}\wedge e^{3}\,,$ $\displaystyle T^{a}$
$\displaystyle=T_{I}(r)\,e^{1}\wedge e^{a}\,,$ $\displaystyle T^{\bar{a}}$
$\displaystyle=T_{II}(r)\,\epsilon^{\bar{a}}{}_{\,\bar{b}\bar{c}}\,e^{\bar{b}}\wedge
e^{\bar{c}}\;,$ (21b)

where the particular values for the solution (20) are

$\displaystyle R_{I}(r)$
$\displaystyle=\frac{\left[k\left(\gamma-k\right)^{2}-36am^{2}\right]r^{3}+6m(\gamma-k)^{2}r^{2}-36m^{2}(\gamma-k)r+72m^{3}}{\left[(\gamma-k)r-6m\right]^{2}r^{3}}\,,$
(22a) $\displaystyle R_{II}(r)$
$\displaystyle=\frac{\left[(\gamma+k)(\gamma-k)^{2}-72am^{2}\right]r^{3}-6m(\gamma-k)^{2}r^{2}+36m^{2}(\gamma-k)r-72m^{3}}{2\left[(\gamma-k)r-6m\right]^{2}r^{3}}\,,$
(22b) $\displaystyle T_{I}(r)$
$\displaystyle=\frac{\gamma-k}{(\gamma-k)r-6m}\sqrt{ar^{2}-\frac{(\gamma^{2}-k^{2})}{6m}r+\gamma-\frac{2m}{r}}\,,$
(22c) $\displaystyle T_{II}(r)$ $\displaystyle=0\,.$ (22d)

We shall discuss the main properties of this three-parameter family of black-
hole solutions in the next section.

##  IV Features of the torsional black hole

Here, we analyze different aspects of the solution including the structure of
singularities and their thermodynamic properties in presence of torsion.
Moreover, we derive a particular limit where it becomes a thermalized ground
state.

###  IV.1 Curvature

The linear term in $r$ of the metric function $f(r)$ sources the non-Einstein
mode of the solution. This can be seen as follows: First, consider the
torsion-free Riemann tensor $\mathring{R}^{\mu\nu}_{\ \lambda\rho}$
constructed out of the Levi-Civita connection associated to the line element
(15) with metric function (20a). In general, the traceless Ricci tensor
$\mathring{H}_{\mu\nu}\equiv\mathring{R}_{\mu\nu}-\frac{1}{4}g_{\mu\nu}\mathring{R}$
vanishes identically for Einstein spaces. The curvature invariant

$\displaystyle\mathring{H}_{\mu\nu}\mathring{H}^{\mu\nu}=\frac{(\gamma-k)^{2}[(\gamma+k)r-6m]^{2}}{36m^{2}r^{4}}\,,$
(23)

shows that, in this case, $\mathring{H}_{\mu\nu}$ is nonvanishing. Thus, we
conclude that Eq. (20) represents a non-Einstein space. It is worth mentioning
that Maldacena showed in Ref. Maldacena (2011) that imposing Neumann boundary
conditions on the Fefferman-Graham expansion of the weakened asymptotically
locally AdS metric allows one to remove the linear mode present in conformal
gravity. This procedure selects Einstein spaces from the space of solutions of
conformal gravity, whose renormalization is guaranteed for asymptotically
locally AdS spaces due to conformal invariance Grumiller  _et al._ (2014).

The solution is Bach-flat, as it can be checked by noticing that the Bach
tensor,

$\displaystyle\mathring{B}_{\mu\nu}=\nabla^{\lambda}\mathring{C}_{\mu\nu\lambda}-\mathring{S}^{\lambda\rho}\mathring{W}_{\mu\lambda\rho\nu}\,,$
(24)

vanishes identically for the solution (20), where
$\mathring{C}_{\mu\nu\lambda}=2\nabla_{[\lambda}\mathring{S}_{\nu]\mu}$ and
$\mathring{S}_{\mu\nu}=\tfrac{1}{2}\left(\mathring{R}_{\mu\nu}-\tfrac{1}{6}g_{\mu\nu}\mathring{R}\right)$
are the torsion-free Cotton and Schouten tensor, respectively. Thus, we
conclude that spacetime (20) can be regarded as a torsional generalization of
the Riegert metric found in the second-order (purely metric) formulation of
conformal gravity Riegert (1984). Notice that, in the presence of the extra
fields required by USUSY, torsion is switched on, so that the first and second
order formulations are not equivalent: the torsional features of the geometry
do not vanish. These torsional contributions can be moved to the right hand
side of the field equations (10), acting as a source for the torsion-free part
of the geometry. This occurs in many situations in which the first and second
order formalisms are not equivalent Toloza and Zanelli (2013); Espiro and
Vásquez (2016); Castillo-Felisola  _et al._ (2015, 2016); Barrientos  _et al._
(2017); Cid  _et al._ (2018); Cisterna  _et al._ (2019); Barrientos  _et al._
(2019).

For $\gamma\neq k$, the solution (20) has a curvature singularity at $r=0$ as
it can be seen from the torsion-free Ricci scalar [see also Eq. (23)]

$\displaystyle\mathring{R}$ $\displaystyle=\mathring{R}^{\mu\nu}_{\
\mu\nu}=-12a+\frac{\left(\gamma^{2}-k^{2}\right)}{m\,r}-\frac{2(\gamma-k)}{r^{2}}\,.$
(25)

The central singularity is surrounded by a horizon located at $r=r_{h}$,
defined by the condition $f(r_{h})=0$. This is a cubic equation for $r_{h}$
having one real root and two complex conjugate ones. The real one represents
the black hole’s horizon whose explicit form is rather cumbersome and not very
illuminating. The conditions on the integration constants that make $r_{h}>0$
are discussed in Eq. (28) below.

###  IV.2 Torsion

Besides the singularity at $r=0$, there exists a torsional singularity at
$r_{s}\equiv 6m(\gamma-k)^{-1}$, as can be seen from the invariant

$\displaystyle T_{\mu\nu\lambda}T^{\mu\nu\lambda}$
$\displaystyle=\frac{6}{r_{s}(r-r_{s})^{2}}\,\left[ar_{s}\,r^{2}-(\gamma+k)r+\gamma\,r_{s}-\frac{2mr_{s}}{r}\right]\,,$
(26)

where $T^{\lambda}_{\ \mu\nu}=e^{\lambda}{}_{a}T^{a}_{\ \mu\nu}$. The
conditions on the parameters of the solution that ensure that the torsional
singularity lies inside the black hole’s event horizon, $r_{h}>r_{s}$, are
discussed below.

From (22) it is apparent that the Lorentz curvature $R^{ab}$ and the torsion
components $T^{a}$ diverge both at $r=r_{s}$ and at $r=0$. In fact, it is also
the case for all polynomial invariants obtained by contracting these tensors.
However, it is a remarkable feature of the solution that torsion-free
quantities, like the Ricci scalar (25), are regular at $r=r_{s}$. This is true
of other torsion-free geometric quantities, like the Kretchsmann or the
quadratic curvature invariant (23). In particular, the torsion-free Weyl
tensor
$\mathring{W}^{\mu\nu}_{\lambda\rho}=\mathring{R}^{\mu\nu}_{\lambda\rho}-4\delta^{[\mu}_{[\lambda}\mathring{S}^{\nu]}_{\rho]}$,

$\displaystyle\mathring{W}^{\mu\nu}_{\lambda\rho}$
$\displaystyle=-\frac{(\gamma-k)r-6m}{3r^{2}}\delta^{\mu\nu}_{\lambda\rho}\,,$
(27)

is regular everywhere except at $r=0$. In fact, this solution is conformally
flat at $r=r_{s}$ $\mathring{W}^{\mu\nu}_{\lambda\rho}$ where (27) vanishes.

The torsional singularity does not affect the motion of spinless test
particles since they couple to the torsion-free Christoffel connection and
follow geodesics rather than autoparallels Hehl  _et al._ (1976, 1995).
Spinning particles and fermions in general, on the other hand, couple to
torsion Hehl and Datta (1971); Hehl (1976); Chandia and Zanelli (1998).
Therefore, their propagation must be sensitive to the torsional singularity in
this background. This should be taken into account when studying the geodesic
completeness of this solution for spinning test particles.

The existence of a horizon plus the condition $r_{s}<r_{h}$, that enforces the
cosmic censorship principle, provides a restriction on the parameters. For the
sake of simplicity, consider the particular case of weakened asymptotically
AdS solution with $m>0$, $a>0$, $k=1$, and $-1\leq\gamma\leq 1$. In this case,
the conditions $0<r_{s}<r_{h}$ translate into

$\displaystyle-1\leq\gamma<1\;\;\;\;\;\wedge\;\;\;\;\;\frac{2-3\gamma+\gamma^{3}}{108m^{2}}\leq
a\;\;\;\;\;\vee\;\;\;\;\;\gamma=1\;\;\;\;\;\wedge\;\;\;\;\;0<a\,.$ (28)

Similar restrictions on parameter space are found for the asymptotically de
Sitter case so that the curvature and torsional singularities are hidden by
the event horizon. Nevertheless, if $a<0$, the solution is endowed with a
cosmological horizon as well.

The solution (20) possesses a covariantly constant torsion, which can be
checked by using the Bianchi identity $\text{D}T^{a}=R^{a}{}_{b}\wedge e^{b}$
and noticing that the Lorentz curvature $2$-form (22) satisfies
$R^{a}{}_{b}\wedge e^{b}=0$. Geometries with covariantly constant torsion
include constant Lorentz curvature solutions and Lorentz-flat geometries as
particular cases. In three dimensions, the most general solution to
$\text{D}T^{a}=0$ has at least one pseudoscalar degree of freedom Alvarez  _et
al._ (2014c) and admits a Proca-like excitation Andrianopoli  _et al._ (2022).
In four or higher dimensions additional degrees of freedom could be present.

The torsional Riegert black hole (20) is continuously connected to the
torsion-free Schwarzschild-AdS and dS with constant curvature base manifold in
the limit $\gamma\to k$ for $a>0$ and $a<0$, respectively. When $\gamma=-k$,
however, the linear mode in the metric (20a) is absent and the torsion is
nonvanishing. Nevertheless, this case represents a naked singularity located
at $r=0$, since there is no set of parameters that allows for a horizon. When
$\gamma=k=0$, the metric becomes the cylindrical black hole studied by Lemos
in Ref. Lemos (1995). Thus, a similar large gauge transformation could give
rise to a nonzero angular momentum when $k=0$ and $\gamma\neq 0$ such that the
torsion is nonvanishing. However, in this work we focus on the static case for
the sake of simplicity and we postpone stationary extensions for future
studies.

###  IV.3 Thermodynamics

The thermodynamic properties of these black holes can be derived from the
partition function $\mathcal{Z}$ (see Gibbons and Hawking (1977); Hawking and
Page (1983) for details). This can be computed to first order in the saddle-
point approximation through $\ln\mathcal{Z}\approx-I_{E}$, where $I_{E}$ is
the Euclidean on-shell action. The latter can be obtained by performing the
analytic continuation $t\to-i\tau$ in the line element (15). The absence of
conical singularities when the Euclidean time coordinate is identified as
$\tau\sim\tau+\beta$ fixes thiss period to be $\beta=4\pi/f^{\prime}(r_{h})$.
Then, the Hawking temperature for the torsional Riegert black hole given by
(20) is

$\displaystyle T_{H}=\frac{\gamma+k}{4\pi r_{s}}+\frac{3m-\gamma r_{h}}{2\pi
r_{h}^{2}}\,.$ (29)

In contrast to the Schwarzschild-AdS solution, the temperature of the
torsional Riegert black hole behaves as

$\displaystyle T_{H}\approx\frac{\gamma^{2}-k^{2}}{24\pi
m}+\mathcal{O}(r_{h}^{-1})\;\;\;\;\mbox{when}\;\;\;\;r_{h}\to\infty\,.$ (30)

In the torsion-free limit $\gamma\to k$, however, it behaves as usual, namely,
$T_{H}\approx\mathcal{O}(r_{h}^{-1})$ as $r_{h}\to\infty$.

A direct evaluation of the Euclidean on-shell action (5) on the three-
parameter family of black-hole solutions (20) gives the partition function to
first-order in the saddle point approximation, that is,

$\displaystyle\ln\mathcal{Z}\approx-
I_{E}=\frac{\beta\Omega_{(k)}\left[(\gamma-k)^{2}r_{h}^{2}-6m(\gamma-k)r_{h}+12m^{2}\right]}{3r_{h}^{3}}\,,$
(31)

where $\Omega_{(k)}$ denotes the volume of the base manifold. Remarkably,
conformal invariance renders its value finite without any reference to
boundary counterterms. This feature has been observed in different theories
possessing conformal invariance in the metric formalism Grumiller  _et al._
(2014); Anastasiou  _et al._ (2021a, b); Barrientos  _et al._ (2022) and we
hereby provide additional evidence for conformal supergravity with nontrivial
torsion. Moreover, the cosmological term in the solution (20) arises as an
integration constant, offering a natural scenario to study the extended phase
space of black hole thermodynamics as proposed in Kubiznak  _et al._ (2017),
as well as their critical behavior Kubiznak and Mann (2012). Due to the number
of free parameters, the phase structure will be richer and we postpone a
deeper study of this point for the future.

###  IV.4 Thermalized ground state

An interesting limit of the solution (20) is $\gamma\to k$ and $m\to 0$, with
$k(\gamma-k)(3m)^{-1}\equiv b$ (or $\frac{\gamma-k}{6m}=r_{s}^{-1}$) fixed. In
this limit, the torsional singularity is translated into $r_{s}=2kb^{-1}$.
Then, we obtain

$\displaystyle f(r)=k-br+ar^{2}\,,$ (32)

where $\omega_{1}(r)$, $\omega_{5}(r)$, and $\Lambda(r)$ can be obtained
directly from the relations in Eq. (20). This solution possesses a weakened
AdS or dS asymptotics when $a>0$ or $a<0$, respectively, and it is weakly
asymptotically locally flat when $a=0$. In this case, the Lorentz curvature
and torsion $2$-forms are

$\displaystyle R^{ab}$
$\displaystyle=\frac{k-a\,r_{s}^{2}}{\left(r-r_{s}\right)^{2}}\,e^{a}\wedge
e^{b}\;\;\;\;\;\mbox{and}\;\;\;\;\;T^{a}=\frac{\sqrt{f(r)}}{r-r_{s}}\,e^{1}\wedge
e^{a}\,.$ (33)

Additionally, one can check that this spacetime is both conformally and Bach
flat.

From the Bianchi identity $\text{D}T^{a}=R^{a}{}_{b}\wedge e^{b}$, it is
direct to see that the torsion $2$-form is covariantly conserved as well,
namely, $DT^{a}=0$. Moreover, this solution has curvature and torsional
singularities at $r=0$ and ${\color[rgb]{0,0,1}r=r_{s}}$, respectively. The
horizons are located at the two positive real roots of the metric function
$f(r)$, i.e. $f(r_{\pm})=0$, where

$\displaystyle r_{\pm}=\frac{1}{2a}\left(b\pm\sqrt{b^{2}-4ak}\right)\,.$ (34)

The existence of a horizon requires $b^{2}-4ak\geq 0$, and the bound is
saturated in the extremal case, which can only happen if $ak>0$. In the non-
extremal case, the condition $ak<0$ guarantees that a horizon will always
exist. Additionally, demand the torsional singularity to lie behind the
horizon by imposing $r_{+}>r_{s}$. Note that the torsion vanishes at the
horizon, as it can be seen from Eq. (33). Moreover, in the extremal case,
there is no torsional singularity whatsoever, as it can be checked from the
torsional invariants that become constant, e.g.
$T_{\mu\nu\lambda}T^{\mu\nu\lambda}=\tfrac{3b^{2}}{2k}$. Notice that the
solution (32) is non-Einstein for $b\neq 0$, as can be seen from the torsion-
free traceless Ricci tensor $\mathring{H}$, that is,

$\displaystyle\mathring{H}_{\mu\nu}\mathring{H}^{\mu\nu}=\frac{b^{2}}{r^{2}}\,.$
(35)

The solution (32) has a rather peculiar thermodynamic behavior. First, its
Hawking temperature is nonvanishing and is given by

$\displaystyle T_{H}=\frac{k}{2\pi r_{+}\,r_{s}}(r_{+}-r_{s})\,.$ (36)

The requirement that the torsional singularity must lie behind the black
hole’s horizon, i.e. $r_{+}>r_{s}$, implies that the Hawking temperature is
positive definite. However, a direct computation of the Euclidean on-shell
action shows that it vanishes identically. This implies that its free energy,
mass, and entropy are zero. Therefore, we conclude that this limit represents
a conformally flat ground state of the theory that is not a maximally
symmetric space, although continuously connected to global AdS or dS in the
limit $b\to 0$ for $a>0$ or $a<0$, respectively. Nevertheless, since its
Hawking temperature (36) is nonvanishing, it can be regarded as a thermalized
ground state, similar to the Schwarzschild-AdS solution in critical gravity Lu
and Pope (2011). This ground state has been studied in metric formulation of
conformal gravity Lu  _et al._ (2012) and the solution (32)–(33) represents
its torsional generalization embedded in the bosonic sector of unconventional
conformal supergravity.

##  V Summary

We have explored a sector of the space of solutions of a particular
formulation of conformal supergravity. First, we check that this theory
possesses maximally symmetric spaces as ground states. Additionally, we show
that black-hole solutions also exist; the latter can be regarded as the
torsional extension of the Riegert spacetime Riegert (1984). The compensating
fields for translations and special conformal transformations of the $SO(3,2)$
group generate a backreaction that behaves as a dynamical cosmological
constant. Indeed, we find that the solution is a weakened asymptotically
locally AdS, however, from the Riemann-Cartan viewpoint, it represents a novel
example of an asymptotically Lorentz-flat black hole.

This solution presents a number of interesting features. First, it possesses a
torsional singularity where the metric becomes conformally flat. Spinning test
particles should be sensitive to this kind of singularities, providing an
interesting setup to study autoparallel completeness rather than geodesic
completeness. The conditions on the integration constants of the solution are
found such that the curvature and torsion singularities lie behind the
horizon, avoiding a violation of the cosmic censorship conjecture. When
$SU(N)$ and $U(1)$ connections are turned off, the black-hole solution is
continuously connected to Schwarzschild-(A)dS in the torsion-free limit.
Additionally, the torsion is covariantly conserved, something that could be
relevant for computing conserved charges. In a particular limit, the torsional
black hole becomes a thermalized ground state, namely, a non-maximally
symmetric configuration with positive definite temperature whose free energy
vanishes. This configuration is likely to be of relevance in the study of the
realted Hawking-Page phase transitions.

Other interesting properties of this theory can be explored in the future. For
instance, since the cosmological constant term in Eq. (20) appears as an
integration constant, this theory offers a natural scenario to study the
extended phase space of the torsional Riegert black hole in the Gibbs
canonical ensemble by interpreting the cosmological term as a thermodynamic
pressure Kubiznak  _et al._ (2017). On the other hand, stationary solutions in
unconventional conformal supergravity are certainly of interest to us. We
believe that the role of torsion will be important in that case due to the
axial symmetry involved in this type of configurations and it is expected that
novel topological features associated to torsion could arise (see Chandia and
Zanelli (1997)). Finally, holographic properties of torsion can be studied in
conformal supergravity, since it is interpreted as the source of the spin
density in the dual field theory Banados  _et al._ (2006); Klemm and Tagliabue
(2008); Blagojevic  _et al._ (2013). We expect to return for a deeper study of
these aspects in the near future.

###### Acknowledgements.

We thank Giorgos Anastasiou, Laura Andrianopoli, Ignacio J. Araya, Oscar
Castillo-Felisola, Rodrigo Olea, and Omar Valdivia for useful discussions. P.
A. acknowledges MINEDUC-UA project ANT 1755 and Semillero de Investigación
project SEM18-02 from Universidad de Antofagasta, Chile. The work of C. C. and
J. Z. is supported by Agencia Nacional de Investigación y Desarrollo (ANID)
through FONDECYT No 11200025, 1210500, and 1220862.

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