# Phase transitions of black strings in dynamical Chern-Simons modified
gravity

Cristóbal Corral<EMAIL_ADDRESS>Instituto de Ciencias Exactas y Naturales
and Facultad de Ciencias, Universidad Arturo Prat, Avenida Arturo Prat Chacón
2120, 1110939, Iquique, Chile  Cristián Erices<EMAIL_ADDRESS>Universidad Central de Chile, Vicerrectoría Académica, Toesca 1783, Santiago,
Chile.  Universidad Católica del Maule, Av. San Miguel 3605, Talca, Chile.
Daniel Flores-Alfonso<EMAIL_ADDRESS>Departamento de Física,
CINVESTAV-IPN, A.P. 14-740, C.P. 07000, Ciudad de México, Mexico  Kristiansen
Lara<EMAIL_ADDRESS>Departamento de Física, Universidad de
Santiago de Chile, Avenida Ecuador 3493, Santiago, Chile

###### Abstract

We study conserved charges and thermodynamics of analytic rotating anti-de
Sitter black holes with extended horizon topology—also known as black
strings—in dynamical Chern-Simons modified gravity. The solution is supported
by a scalar field with an axionic profile that depends linearly on the
coordinate that spans the string. We compute conserved charges by making use
of the renormalized boundary stress-energy tensor. Then, by adopting the
Noether-Wald formalism, we compute the black string entropy and obtain its
area law. Indeed, the reduced Euclidean Hamiltonian approach shows that these
methods yield a consistent first law of thermodynamics. Additionally, we
derive a Smarr formula using a radial conservation law associated to the scale
invariance of the reduced action and obtain a Cardy formula for the black
string. A first-order phase transition takes place at a critical temperature
between the ground state and the black string, above which the black string is
the thermodynamically favored configuration.

##  I Introduction

Chern–Simons modified gravity (CSMG) is a well-known scalar-tensor theory in
four dimensions which was first proposed in Ref. Jackiw and Pi (2003). It is
motivated by anomaly cancellation in curved spacetimes, particle physics, and
by the low-energy limit of string theory (for a review see Alexander and Yunes
(2009)). The theory is endowed with a nonminimal coupling between a scalar
field and the Chern-Pontryagin density. Quite notably, this interaction
captures parity-violating features in the strong gravity regime. Moreover,
under certain conditions (see Ref. Ahmedov and Aliev (2010)) its equations of
motion effectively reduce to those of topologically massive gravity Deser  _et
al._ (1982a, b)—a 3D gravity theory constructed from a Chern–Simons form. This
theory does not belong to the Horndeski family of theories Horndeski (1974),
since the Chern–Simons coupling generates higher-order field equations. Thus,
in order to avoid drawbacks with the Cauchy initial-value problem, it should
be considered as an effective theory arising from the UV completion of general
relativity (GR) Delsate  _et al._ (2015).

Different physical consequences of this theory have been explored since its
conception. For instance, the Chern–Simons coupling yields a nontrivial
contribution to the post-Newtonian parameter related to frame dragging and
gyroscopic precession Alexander and Yunes (2007a); observables that could be
tested in the near future Alexander and Yunes (2007b). When the scalar field
is nondynamical and invariant under $SO(3)$, all static and spherically
symmetric solutions of GR remain solutions of CSMG since the Pontryagin
density vanishes identically; this is not true for axially symmetric
configurations, where only slowly rotating and numerical black hole solutions
with and without NUT charge are known Konno  _et al._ (2007); Grumiller and
Yunes (2008); Alexander and Yunes (2007b); Alexander  _et al._ (2008); Konno
_et al._ (2008); Yunes and Pretorius (2009); Cambiaso and Urrutia (2010);
Brihaye and Radu (2017). Shadows of the latter were analyzed in Amarilla  _et
al._ (2010) by integrating null geodesics, showing how the Chern–Simons
coupling modifies the shape of the shadow. Perturbations and quasinormal modes
around Schwarzschild and slowly rotating black holes have been recently
attracted a lot of interest and they have been studied in Refs. Bhattacharyya
and Shankaranarayanan (2019); Okounkova  _et al._ (2019, 2020); Wagle  _et
al._ (2021); Srivastava  _et al._ (2021). On the other hand, CSMG is the only
parity-violating extension of GR compatible with luminal propagation of
gravitational waves Nishizawa and Kobayashi (2018). However, the nonminimal
coupling does affect the polarization of the latter, producing an amplitude
birefringence in their propagation Alexander and Yunes (2018). Regarding
holographic applications, the scalar-Chern-Simons coupling generates
spontaneous angular momentum on the conformal field at the boundary, even for
static configurations in the bulk Liu  _et al._ (2013). Moreover, this theory
provides a simple model to study Hall viscocity of holographic fluids; the
latter depending only on near-horizon information of the background Saremi and
Son (2012); Chen  _et al._ (2012).

Recently, analytic rotating solutions were found in CSMG by relaxing the
symmetry assumptions on the scalar field Cisterna  _et al._ (2019) (see for
instance Refs. Grumiller and Yunes (2008); Bardoux  _et al._ (2012); Cisterna
and Oliva (2018)). These solutions represent rotating black holes with an
extended horizon. They correspond to the uplift of the
Bañados–Teitelboim–Zanelli (BTZ) black hole Banados  _et al._ (1992, 1993) to
four dimensions supported by scalar fields with axionic profile: a homogeneous
BTZ black string. Black strings are related to line mass gravitational sources
in the same way black holes relate to point mass sources. Homogeneous black
strings are product spaces, i.e., a black hole times a one-dimensional space.
In this work, we strictly consider homogeneous black strings, in particular
those which have locally $\mbox{AdS}_{3}\times\mathbb{R}$ asymptotics (see
Gregory (2000) for the flat case). These spaces bear some resemblance with
Nariai and Bertotti-Robinson spacetimes Nariai (1951); Bertotti (1959);
Robinson (1959) and solutions in flux compactification of string theory
Douglas and Kachru (2007). Indeed, the Chern–Simons modification to GR yields
a highly nontrivial contribution to the field equations in presence of
rotation, rendering this class of solution scarce. Moreover, when metric and
connection are treated as independent variables in the tetrad formulation, BTZ
black strings with nontrivial torsion were found in Cisterna  _et al._ (2019).
Thus, these are valuable configurations for the study of conserved charges and
thermodynamics of rotating objects in CSMG, as we do here.

Generically, black strings are higher-dimensional asymptotically flat vacuum
solutions with extended topology of their horizon that represent interesting
counterexamples to topological censorship Friedman  _et al._ (1993) and to
uniqueness theorems in higher-dimensional GR Israel (1968); Carter (1971);
Wald (1971). They are also known in the presence of nonlinear matter content
and in theories beyond GR (for an incomplete list of developments see Chan and
Mann (1995); Giribet  _et al._ (2006); Astorino  _et al._ (2018); Cisterna
_et al._ (2019, 2018); Nakas  _et al._ (2019); Giacomini and Ortaggio (2019);
Cisterna  _et al._ (2020a); Hendi  _et al._ (2021); Hennigar  _et al._ (2021);
Arratia  _et al._ (2021); Cisterna  _et al._ (2021); Canfora  _et al._
(2021)). Nevertheless, it is well-known that these solutions usually suffer
from the Gregory–Laflamme instability: a long-wavelength linear instability
driven by an unstable mode traveling along the extended direction Gregory and
Laflamme (1993, 1994); Gregory (2000). In fact, numerical evolution suggests
that this instability remains at the non-linear regime Choptuik  _et al._
(2003); Lehner and Pretorius (2010), as well as perturbatively in Lovelock
theory Giacomini  _et al._ (2015, 2016). On the other hand, constructing
homogeneous black strings in GR with a nonvanishing cosmological constant is a
nontrivial task, due to the dynamics. However, this issue can be circumvented
by introducing scalar fields with an axionic profile, i.e., with a linear
dependence on the extended flat coordinates Cisterna  _et al._ (2019);
Cisterna and Oliva (2018); Cisterna  _et al._ (2018). Interestingly enough,
this approach allows one to find black strings in four dimensions, despite the
fact that these solutions were first believed to be strictly higher-
dimensional objects. Additionally, there exists evidence that axionic fields
might stabilize the Gregory–Laflame instability at the perturbative level in
four dimensions and they could play an important role in the final state of
black strings with axionic charge Cisterna  _et al._ (2020b); something that
it is not guaranteed in higher dimensions Henríquez-Báez (2021); Dhumuntarao
and Mahbub (2021). Nevertheless, a complete proof of stabilization of axionic
black strings in four dimensions at the full nonlinear level is still lacking.

In this work, we study conserved charges and thermodynamics of black strings
in dynamical Chern–Simons modified gravity sourced by a scalar field with
axionic profile. To this end, we first compute conserved charges using the
renormalized boundary stress-energy tensor. This allows us to obtain the mass
and angular momentum of the solution. Then, we compute the black string
entropy via the Noether-Wald formalism and obtain the corresponding area law.
Moreover, we carry out the Hamiltonian approach and show explicitly that the
conserved charges and the black string entropy satisfy the first law of
thermodynamics. The Euclidean on-shell action is computed by adding to it
standard boundary terms that render its value finite while defining a well-
posed variational principle. To lowest order in the saddle-point
approximation, we obtain the free energy, mass, entropy, and angular momentum
of the solution, and show that the first law of thermodynamics is satisfied.
We also obtain a consistent Smarr formula following the approach of Ref.
Banados and Theisen (2005). Additionally, we find the ground state and a non-
Einstein product space that belong to the space of solutions. We compute the
mass of the former through standard Euclidean methods and, as it might be
expected, they turn out to be negative, similar to the hairy solitons found in
Refs. Correa  _et al._ (2011, 2012); Gonzalez  _et al._ (2011); Anabalon  _et
al._ (2016); Ayón-Beato  _et al._ (2019). Remarkably, by analyzing global
thermodynamic stability, we find that the black string can develop a phase
transition below a critical temperature. Critical phenomena of black objects
has become an active area of research since the seminal works of Hawking-Page
Hawking and Page (1983) and it has been extensively studied until today. Just
to mention the well-known cases in the literature, in the four dimensional
canonical approach, i.e. with fixed cosmological constant, phase transitions
of Reissner-Nördstrom-AdS and Kerr-Newman-AdS have been analysed. In the
former, the black hole free energy exhibits a ‘cusp’ behavior respect to the
temperature, which means that there are two branches of black holes, large and
small ones Chamblin  _et al._ (1999). In the latter, phase transitions between
the large and small black holes can disappear for sufficiently large electric
charge or angular momentum Caldarelli  _et al._ (2000). From a qualitative
perspective, the kind of phase transitions that the black string configuration
suffers at a critical temperature, resembles the three-dimensional case, where
the critical temperature at which thermal AdS collapse to the BTZ black hole,
is parametrized by the AdS radius. There is only one branch of BTZ black
holes. In our black string, this behavior is similar, but the critical
temperature is determined by the axionic charge.

The article is organized as follows. In Sec. II we review Chern–Simons
modified gravity as well as the rotating black string solutions with
nontrivial scalar field obtained in Ref. Cisterna  _et al._ (2019). In Sec.
III we compute the asymptotic charges through the renormalized boundary
stress-energy tensors. Then, in Sec. IV we compute the black string entropy
and the renormalized Euclidean on-shell action to obtain the relevant
thermodynamic quantities. We derive the Smarr formula for these extended
solutions in Sec. IV.3. Section V is devoted to presenting the ground state
and a new non-Einstein conformally flat product space as solutions to CSMG.
Both are supported by scalar fields with linear dependence on the flat and
time coordinate, respectively. We show explicitly that the mass of the ground
state is negative and the non-Einstein conformally flat product space is
topologically $\mathbb{R}\times S^{3}$. In Sec. VI we show that, below a
certain critical temperature, the ground state is thermodynamically favored
over the black string, developing a first order Hawking-Page phase transition.
Finally, in Sec. VII we present our conclusions and further remarks.

##  II Rotating Black strings in Chern–Simons modified gravity

Let us review the black string solutions found in Ref. Cisterna  _et al._
(2019), as they are the main subject of this work. The theory under
consideration is given by the Einstein–Hilbert action modified by the
nonminimal coupling between the Chern–Pontryagin density and a dynamical
scalar field. This is to say, we focus on the action principle

$\displaystyle I$ $\displaystyle=I_{\rm bulk}+I_{\rm GHY}+I_{\rm ct}\,,$ (1)

where

$\displaystyle I_{\rm bulk}$
$\displaystyle=\frac{1}{2\kappa}\int_{\mathcal{V}}\text{d}^{4}x\sqrt{-g}\left(R-2\Lambda+\frac{\alpha\phi}{4}\,^{*}RR-\kappa
g^{\mu\nu}\nabla_{\mu}\phi\nabla_{\nu}\phi\right)\equiv\int_{\mathcal{V}}\text{d}^{4}x\sqrt{-g}\;\mathscr{L}_{\rm
bulk}\,,$ (2) $\displaystyle I_{\rm GHY}$
$\displaystyle=\frac{1}{\kappa}\int_{\partial\mathcal{V}}\text{d}^{3}x\sqrt{-h}\left(K+\frac{\alpha\phi}{2}\,n_{\mu}\epsilon^{\mu\nu\lambda\rho}K_{\nu}{}^{\sigma}\nabla_{\lambda}K_{\rho\sigma}\right)\equiv\int_{\partial\mathcal{V}}\text{d}^{3}x\sqrt{-h}\;\mathscr{L}_{\rm
GHY},$ (3) $\displaystyle I_{\rm ct}$
$\displaystyle=\int_{\partial\mathcal{V}}\text{d}^{3}x\sqrt{-h}\left(\zeta_{1}+\zeta_{2}\mathcal{R}+\zeta_{3}h^{\mu\nu}\nabla_{\mu}\phi\nabla_{\nu}\phi+...\right)\equiv\int_{\partial\mathcal{V}}\text{d}^{3}x\sqrt{-h}\;\mathscr{L}_{\rm
ct}.$ (4)

Here, the gravitational constant is defined as $\kappa=8\pi G_{N}$, with
$G_{N}$ being Newton’s constant. The nonminimal coupling is measured by the
dimensionful parameter $\alpha$ and the Chern–Pontryagin density is defined as

$\displaystyle^{*}RR\equiv\frac{1}{2}\epsilon^{\lambda\rho\sigma\tau}R^{\mu}{}_{\nu\lambda\rho}R^{\nu}{}_{\mu\sigma\tau}=2\nabla_{\mu}\left[\epsilon^{\mu\nu\alpha\beta}\Gamma^{\sigma}_{\nu\lambda}\left(\partial_{\alpha}\Gamma^{\lambda}_{\beta\sigma}+\frac{2}{3}\Gamma^{\lambda}_{\alpha\gamma}\Gamma^{\gamma}_{\beta\sigma}\right)\right],$
(5)

with $\epsilon^{\mu\nu\lambda\rho}$ being the Levi-civita tensor. The induced
metric on the boundary $\partial\mathcal{V}$ is given by
$h_{\mu\nu}=g_{\mu\nu}-n_{\mu}n_{\nu}$, where $n^{\mu}$ is a space-like normal
unit vector, and $h=\det h_{\mu\nu}$ is the determinant of the induced metric.
Moreover,
$K_{\mu\nu}=h^{\rho}{}_{\mu}h^{\lambda}{}_{\nu}\nabla_{\rho}n_{\lambda}$
denotes the extrinsic curvature with $K$ being its trace. The first boundary
term, $I_{\rm GHY}$, defines a well-posed variational principle with Dirichlet
boundary conditions Grumiller  _et al._ (2008). It is composed by the standard
Gibbons–Hawking–York term Gibbons and Hawking (1977a); York (1972) and an
additional piece that involves the nonminimal coupling between the scalar
field and the Chern–Simons form for the extrinsic curvature at the boundary
Grumiller  _et al._ (2008). The second boundary term, $I_{\rm ct}$, guarantees
that the on-shell action remains finite in spacetimes with AdS asymptotics,
provided a particular choice of the $\zeta$-couplings Emparan  _et al._
(1999); Balasubramanian and Kraus (1999);111In GR, the renormalization
prescription used here is equivalent to the Kounterterms proposed in Ref. Olea
(2007) and to early proposals in Aros  _et al._ (2000a, b); Mora  _et al._
(2004) when the Weyl tensor vanishes at the AdS boundary (see Ref. Anastasiou
_et al._ (2020)). in this case, $\mathcal{R}^{\mu\nu}_{\lambda\rho}$ is the
intrinsic curvature of the boundary,
$\mathcal{R}_{\mu\nu}=\mathcal{R}^{\lambda}{}_{\mu\lambda\nu}$, and
$\mathcal{R}=h^{\mu\nu}\mathcal{R}_{\mu\nu}$. Notice that the
$\zeta$-couplings have different length units, namely $[\zeta_{1}]=L^{-3}$,
$[\zeta_{2}]=L^{-2}$ and $[\zeta_{3}]=L$. Hence, the term $I_{\rm ct}$ is
constructed solely from intrinsic quantities in the boundary, unaffecting the
bulk dynamics.

The field equations are obtained by performing variations of the action (1)
with respect to the metric and scalar field, giving

$\displaystyle\mathcal{E}_{\mu\nu}$ $\displaystyle\equiv
R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R+\Lambda g_{\mu\nu}+\alpha C_{\mu\nu}-\kappa
T_{\mu\nu}=0,$ (6a) $\displaystyle\mathcal{E}$
$\displaystyle\equiv\Box\phi+\frac{\alpha}{8\kappa}\,^{*}RR=0,$ (6b)

respectively. We use the notation $\Box=g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}$
and

$\displaystyle T_{\mu\nu}$
$\displaystyle=\nabla_{\mu}\phi\nabla_{\nu}\phi-\frac{1}{2}g_{\mu\nu}\nabla_{\lambda}\phi\nabla^{\lambda}\phi,$
(7) $\displaystyle C^{\mu\nu}$
$\displaystyle=\nabla_{\rho}\phi\,\epsilon^{\rho\sigma\lambda(\mu}\nabla_{\lambda}R^{\nu)}{}_{\sigma}+\nabla_{\rho}\nabla_{\sigma}\phi\,^{*}R^{\sigma(\mu\nu)\rho},$
(8)

with
${}^{*}R^{\mu}{}_{\nu}{}^{\lambda\rho}=\tfrac{1}{2}\epsilon^{\lambda\rho\sigma\tau}R^{\mu}{}_{\nu\sigma\tau}$.
The tensor $C_{\mu\nu}$ is traceless and it satisfies the property Jackiw and
Pi (2003); Alexander and Yunes (2009)

$\displaystyle\nabla_{\mu}C^{\mu\nu}$
$\displaystyle=-\frac{1}{8}\,^{*}RR\nabla^{\nu}\phi.$ (9)

We dub $C_{\mu\nu}$ as the $C$-tensor from hereon. Notice that its
contribution to the Einstein field equations involves covariant derivatives of
the Riemann tensor, yielding third order equations for the metric. Thus, in
order to avoid issues with the initial value formulation, Chern–Simons
modified gravity should be considered as an effective field theory coming from
their ultraviolet completion Delsate  _et al._ (2015).

As shown in Ref. Cisterna  _et al._ (2019), the field equations can be solved
by assuming the following _Ansätze_ for the line element and scalar field

$\displaystyle\text{d}s^{2}=-N^{2}(r)f(r)\text{d}t^{2}+\frac{\text{d}r^{2}}{f(r)}+r^{2}\left(N^{\varphi}(r)\text{d}t+\text{d}\varphi\right)^{2}+\text{d}z^{2}\;\;\;\;\;\mbox{and}\;\;\;\;\;\phi=\phi(z),$
(10)

respectively. Under this assumption, the Pontryagin density vanishes
identically. This is a consequence of the isometry group used in Eq. (10)
rather than an imposition to simplify the field equations. Then, the Klein-
Gordon equation is directly solved by

$\displaystyle\phi=\phi_{0}+\lambda z,$ (11)

where $\lambda$ is an integration constant usually referred to as the axionic
charge. Additionally, the shift symmetry in field space, i.e.
$\phi\to\phi-\phi_{0}$, allows one to set $\phi_{0}=0$ without loss of
generality.

When $N(r)=1$,222The condition $N(r)=1$ will be justified a posteriori by
Hamiltonian analysis (see Eq. (39) below). and assuming that the solution of
the scalar field in Eq. (11) holds, the components of the field equations for
the metric, namely
$g^{\mu\rho}\mathcal{E}_{\rho\nu}=\mathcal{E}^{\mu}_{\nu}=0$, become

$\displaystyle\mathcal{E}^{t}_{t}$
$\displaystyle=2r^{3}N^{\varphi\,\prime\prime}N^{\varphi}+r^{3}\left(N^{\varphi\,\prime}\right)^{2}+2r\kappa\lambda^{2}+6r^{2}N^{\varphi\,\prime}N^{\varphi}+4r\Lambda+2f^{\prime}$
$\displaystyle\quad+\lambda\alpha\bigg{[}6r^{4}N^{\varphi\,\prime\prime}N^{\varphi\,\prime}N^{\varphi}+2r^{4}\left(N^{\varphi\,\prime}\right)^{3}+12r^{3}\left(N^{\varphi\,\prime}\right)^{2}N^{\varphi}+2r^{2}N^{\varphi\,\prime\prime\prime}f-r^{2}f^{\prime\prime\prime}N^{\varphi}$
$\displaystyle\quad+r^{2}N^{\varphi\,\prime\prime}f^{\prime}-r^{2}f^{\prime\prime}N^{\varphi\,\prime}+10rN^{\varphi\,\prime\prime}f+4rN^{\varphi\,\prime}f^{\prime}+6N^{\varphi\,\prime}f\bigg{]}=0\,,$
(12) $\displaystyle\mathcal{E}^{r}_{r}$
$\displaystyle=r^{3}\left(N^{\varphi\,\prime}\right)^{2}+2r\kappa\lambda^{2}+4r\Lambda+2f^{\prime}+\lambda\alpha\bigg{[}2r^{4}\left(N^{\varphi\,\prime}\right)^{3}+r^{2}N^{\varphi\,\prime\prime}f^{\prime}-r^{2}f^{\prime\prime}N^{\varphi\,\prime}$
$\displaystyle\quad-2rN^{\varphi\,\prime\prime}f+4rN^{\varphi\,\prime}f^{\prime}-6N^{\varphi\,\prime}f\bigg{]}=0\,,$
(13) $\displaystyle\mathcal{E}^{\varphi}_{\varphi}$
$\displaystyle=2r^{3}N^{\varphi\,\prime\prime}N^{\varphi}+3r^{3}\left(N^{\varphi\,\prime}\right)^{2}-2r\kappa\lambda^{2}+6r^{2}N^{\varphi\,\prime}N^{\varphi}-4r\Lambda-2rf^{\prime\prime}$
$\displaystyle\quad+\lambda\alpha\bigg{[}6r^{4}N^{\varphi\,\prime\prime}N^{\varphi\,\prime}N^{\varphi}+4r^{4}\left(N^{\varphi\,\prime}\right)^{3}+12r^{3}\left(N^{\varphi\,\prime}\right)^{2}N^{\varphi}+2r^{2}N^{\varphi\,\prime\prime\prime}f$
$\displaystyle\quad-r^{2}f^{\prime\prime\prime}N^{\varphi}+2r^{2}N^{\varphi\,\prime\prime}f^{\prime}-2r^{2}f^{\prime\prime}N^{\varphi\,\prime}+8rN^{\varphi\,\prime\prime}f+8rN^{\varphi\,\prime}f^{\prime}\bigg{]}=0\,,$
(14) $\displaystyle\mathcal{E}^{t}_{\varphi}$
$\displaystyle=2r^{3}N^{\varphi\,\prime\prime}+6r^{2}N^{\varphi\,\prime}+\lambda\alpha\left[6r^{4}N^{\varphi\,\prime\prime}N^{\varphi\,\prime}+12r^{3}\left(N^{\varphi\,\prime}\right)^{2}-r^{2}f^{\prime\prime\prime}\right]=0\,,$
(15) $\displaystyle\mathcal{E}^{z}_{z}$
$\displaystyle=-r^{3}\left(N^{\varphi\,\prime}\right)^{2}-2r\kappa\lambda^{2}+4\Lambda
r+2rf^{\prime\prime}+4f^{\prime}=0\,,$ (16)

where prime denotes differentiation with respect to the radial coordinate $r$.
Additionally, since the trace of the C-tensor vanishes, we have

$\displaystyle g^{\mu\nu}\mathcal{E}_{\mu\nu}$
$\displaystyle=-r^{3}\left(N^{\varphi\,\prime}\right)^{2}+2r\kappa\lambda^{2}+8\Lambda
r+2rf^{\prime\prime}+4f^{\prime}=0\,.$ (17)

The equation $\mathcal{E}^{z}_{z}-g^{\mu\nu}\mathcal{E}_{\mu\nu}=0$ implies
immediately that the integration constant associated to the scalar field is
fixed in terms of the cosmological constant through
$\lambda^{2}=-\Lambda/\kappa$. Thus, in order for the scalar field to be real,
the condition $\Lambda<0$ must be met in order to avoid problems with
unitarity. Additionally, the equation

$\displaystyle\mathcal{E}^{t}_{t}-\mathcal{E}^{r}_{r}-N^{\varphi}\mathcal{E}^{t}_{\varphi}=2\lambda\alpha
f\left[r^{2}N^{\varphi\,\prime\prime\prime}+6rN^{\varphi\,\prime\prime}+6N^{\varphi\,\prime}\right]=0\,,$
(18)

implies that, for $\lambda\alpha\neq 0$, the metric function $N^{\varphi}(r)$
can be integrated as

$\displaystyle N^{\varphi}(r)=j_{0}+\frac{j_{1}}{r}+\frac{j}{2r^{2}}.$ (19)

Indeed, the condition $\lambda\alpha\neq 0$ shows that the rotating solution
is supported only if the axionic charge or Chern-Simons coupling $\alpha$ are
nonvanishing. It is worth mentioning that, in general relativity, the
asymptotically (A)dS cylindrical black hole of Ref. Lemos (1995) does not
require any scalar field. This is related to its horizon’s topology,
$\mathbb{T}^{2}$; in such a case, the equations of motion does not impose any
constraints on the cosmological constant whatsoever. Nevertheless, the
topology of the homogeneous black string’s horizon studied here,
$\mathbb{S}^{1}\times\mathbb{R}$, requires the existence of a nontrivial
scalar field $\phi=\lambda z$ alongside the condition $\lambda\alpha\neq 0$,
due to the presence of the scalar-Chern-Simons coupling.

Replacing Eq. (19) into the remaining components of the field equations, one
finds that $j_{1}=0$ and the metric function $f(r)$ is found to be

$\displaystyle
f(r)=-m+\frac{r^{2}}{\ell^{2}}+\frac{j^{2}}{4r^{2}}\;\;\;\;\;\mbox{with}\;\;\;\;\;\ell^{-2}=-\frac{\Lambda}{2}\,.$
(20)

The line element (10), alongside the metric functions (19) and (20)
($j_{1}=0$), illustrates the rotating black string found in Ref. Cisterna  _et
al._ (2019). It corresponds to the uplifting of the BTZ black hole Banados
_et al._ (1992, 1993) with a nontrivial scalar field into Chern-Simons
modified gravity. It is locally equivalent to
$\mbox{AdS}_{3}\times\mathbb{R}$, cf. Ayón-Beato  _et al._ (2004). Its
horizons are determined by the condition $f(r_{\pm})=0$ and are located at

$\displaystyle
r_{\pm}=\ell\left[\frac{m\pm\sqrt{m^{2}-j^{2}/\ell^{2}}}{2}\right]^{1/2}.$
(21)

Thus, for their existence, the condition $m^{2}-j^{2}/\ell^{2}\geq 0$ must be
met. The extremal case is obtained when the latter bound is saturated. As in
the BTZ black hole, all curvature invariants are constant and depend solely on
the AdS radius. However, identifications of points of AdS spacetime by a
discrete subgroup of $SO(2,2)$ yields a singularity in the causal structure
where closed time-like curves arise Banados  _et al._ (1993). The same type of
singularity appears in the BTZ black string above. Finally, notice that the
AdS radius is neither $\ell^{-2}=-\Lambda$ nor $\ell^{-2}=-\Lambda/3$, as in
dimensions $D=3$ and $D=4$, respectively. This is due to the presence of
axionic fields that generate an effective cosmological constant.

##  III Conserved charges

Identifying the conserved charges of a gravitational configuration is an
important issue in standard gravity. In extensions to it, the issue becomes
even more cumbersome. There have been many techniques developed for this
purpose, each approach has its benefits and drawbacks. In Chern-Simons
modified gravity, conserved charges have been studied in the context of
background Killing vector symmetries Tekin (2008) and quasilocal methods Liu
_et al._ (2013). In this section, we use the latter to compute the mass and
angular momentum of the black string. Intuitively, since the transverse
section of the black string are BTZ black holes, we expect the mass and
angular momentum per unit length of the black string to be related to these
quantities. In what follows we sort out the specifics of the computations in
CSMG.

First, it is well-known that, in spacetimes with AdS asymptotics, the standard
Brown–York stress tensor Brown and York (1993) needs to be suplemented by
boundary counterterms to render conserved charges finite Balasubramanian and
Kraus (1999). The boundary stress-energy tensor of Chern–Simons modified
gravity is

$\displaystyle T^{\rm(bdy)}_{\mu\nu}$
$\displaystyle=-\frac{1}{2\kappa}\left(2K_{\mu\nu}-2h_{\mu\nu}K+T^{\rm(CS)}_{\mu\nu}+T^{\rm(GMM)}_{\mu\nu}\right)$
$\displaystyle\quad+\zeta_{1}h_{\mu\nu}-2\zeta_{2}\left(\mathcal{R}_{\mu\nu}-\frac{1}{2}h_{\mu\nu}\mathcal{R}\right)-2\zeta_{3}T^{(\phi)}_{\mu\nu}\,,$
(22)

where $T^{\rm(CS)}_{\mu\nu}$ arises from the variation with respect to the
induced metric of the scalar-Chern–Simons term (for instance see Liu  _et al._
(2013)). On the other hand, and $T^{\rm(GMM)}_{\mu\nu}$ appears from the
second piece of Eq. (3), proposed by Grumiller, Mann, and McNees (GMM) in Ref.
Grumiller  _et al._ (2008). However, it is known that in asymptotically AdS
spacetimes these two contributions decay fast enough towards the boundary and
they do not contribute to the quasilocal charges Liu  _et al._ (2013).
Additionally, the last four terms of Eq. (22) renormalize the stress-energy
tensor in asymptotically AdS spacetimes and they arise from variations of
$I_{\rm ct}$ with respect to the induced metric Balasubramanian and Kraus
(1999). Here, $T^{(\phi)}_{\mu\nu}$ denotes the energy-momentum tensor of the
scalar fields constructed out of the induced metric and we do not include the
other contributions in $I_{\rm ct}$ since they decay sufficiently fast as
$r\to\infty$ (see also Caldarelli  _et al._ (2017); Flores-Alfonso and Quevedo
(2019); Flores-Alfonso and Larios (2020)).

The quasilocal charge associated to a Killing vector field
$\xi=\xi^{\mu}\partial_{\mu}$ is defined through Balasubramanian and Kraus
(1999)

$\displaystyle
Q\left[\xi\right]=\int_{\Sigma}\text{d}^{2}x\sqrt{\sigma}\;T^{\rm(bdy)}_{\mu\nu}u^{\mu}\xi^{\nu}\,,$
(23)

where $u^{\mu}$ is the vector that generates the flow of time in $\partial V$
and $\sigma$ is the determinant of the induced metric on $\Sigma$ (see
Balasubramanian and Kraus (1999) for details). Renormalization of the boundary
stress-energy tensor (22) on asymptotically $\mbox{AdS}_{3}\times\mathbb{R}$
spacetimes demands that the counterterms must satisfy the relation

$\displaystyle\zeta_{1}$
$\displaystyle=-\frac{2\zeta_{3}+\ell}{\kappa\ell^{2}}.$ (24)

This value can be obtained by expanding (23) at large but fixed radius while
choosing $\zeta_{1}$ such that the divergent terms vanish. Afterward, one can
take the limit $r\to\infty$ safely and obtain a finite asymptotic charge.

The mass, angular, and linear momentum per unit of length $L$ of the solution
are obtained from the Killing vectors field that generate the temporal,
rotational, and translational symmetry, respectively; these are

$\displaystyle Q\left[\partial_{t}\right]$ $\displaystyle\equiv
M=\frac{mL}{8G},$ (25) $\displaystyle Q\left[\partial_{\varphi}\right]$
$\displaystyle\equiv J=\frac{jL}{8G},$ (26) $\displaystyle
Q\left[\partial_{z}\right]$ $\displaystyle\equiv P_{z}=0\,.$ (27)

Thus, this method shows that $m$, and $j$, are integration constants related
to the mass and angular momentum of the solution. Moreover, homogeneity of the
black string along the $z$-direction yields a vanishing conserved charge.
Thus, as long as the axionic charge remains fixed in terms of the cosmological
constant as $\lambda^{2}=-\Lambda/\kappa$, it cannot be associated to a
conserved quantity. This could provide a first glimpse of a counterexample to
the stronger form of correlated stability; at least in higher-dimensions
Dhumuntarao and Mahbub (2021). Nevertheless, it is worth mentioning that there
exist different approaches where the cosmological constant arises as an
integration constant Buchmuller and Dragon (1988); Unruh (1989); Henneaux and
Teitelboim (1989); Ng and van Dam (1991). In this case, the axionic charge can
be interpreted truly as a conserved quantity associated to volume preserving
diffeomorphisms, providing a natural scenario to introduce a thermodynamic
pressure in the extended phase space approach Kubiznak and Mann (2015).

##  IV Black string thermodynamics

Our current understanding of black hole thermodynamics is a deep-seated
consequence of semiclassical gravity Hawking (1975); Wald (1975). It is the
means by which we know horizon surface area is not analogous to thermodynamic
entropy, rather, it is the black hole’s entropy Bekenstein (1973). This has
been proved by Noether charge analysis Wald (1993) and through Euclidean
methods Gibbons and Hawking (1977a). Indeed, many advances have been made
possible thanks to the latter approach taken from quantum physics Gibbons and
Hawking (1993). Moreover, black hole thermodynamics has been the inspiration
of a plethora of works in theoretical physics and is a key sector of quantum
gravity. Even the simplest black hole distinguishes itself from ordinary
thermodynamic substances, such as quantum fluids and ferromagnetic matter
Callen (1985). This is due to the degree of homogeneity of a black hole’s
fundamental relation. Such a number can be read off from a system’s
Gibbs–Duhem equation, which highlights the significance of a black hole’s
Smarr formula Smarr (1973) (see for example Bravetti  _et al._ (2017)).

In this section, we obtain the black string entropy using the Noether-Wald
formalism. Then, we consider the analytic continuation of the BTZ black string
to calculate the Euclidean on-shell action, which corresponds to the free
energy of the system. The partition function is identified with the Euclidean
path integral at first order in the saddle-point approximation around the
classical solution Gibbons and Hawking (1977b). Fixing the spacetime boundary
conditions is akin to choosing a thermodynamic ensemble. Indeed, in AdS, the
boundary conditions for minimally coupled scalar fields with radial profile
have been discussed in Ref. Henneaux  _et al._ (2007). For homogeneous black
strings, the situation is different since the asymptotic behavior is
$\mbox{AdS}_{3}\times\mathbb{R}$ rather than $\mbox{AdS}_{4}$. Nevertheless,
since the axion generates only an effective AdS radius in the transverse
section, the standard Henneaux-Teitelboim boundary conditions for AdS gravity
Henneaux and Teitelboim (1985) should be enough to ensure well-defined
asymptotic charges in this particular case. Still, one should consider all
infilling geometries in the path integral and sum over all possible
topologies. This is only sensible when enormous symmetry restrictions are
enforced, as we consider here. This strategy allows for the thermodynamics of
black holes to be completely determined via the Euclidean approach. Then, we
employ the reduced Hamiltonian formalism on the Euclidean versions of the
black string to calculate its conserved charges. By construction, these
charges satisfy the first law of thermodynamics. Indeed, we find that
conserved charges and entropy obtained through the Hamiltonian approach
coincide with those found in previous sections. Additionally, we derive a
Smarr formula using the scale invariance of the reduced action following Ref.
Banados and Theisen (2005).

###  IV.1 Wald entropy

A useful and covariant method to compute the black string entropy is the
Noether-Wald formalism Wald (1993); Iyer and Wald (1994). This is based on the
diffeomorphism invariance of a gravitational action principle, generated by a
vector field $\xi=\xi^{\mu}\partial_{\mu}$, which yields an on-shell conserved
current, $J^{\mu}$, satisfying $\nabla_{\mu}J^{\mu}=0$. The Poincaré lemma, in
turn, implies that the latter can be written locally as
$J^{\mu}=\nabla_{\nu}q^{\mu\nu}$, where

$\displaystyle
q^{\mu\nu}=-2\left(E^{\mu\nu}_{\lambda\rho}\nabla^{\lambda}\xi^{\rho}+2\xi^{\lambda}\nabla^{\rho}E^{\mu\nu}_{\lambda\rho}\right)=-q^{\nu\mu}\,,$
(28)

is known as the Noether prepotential. Here, $E^{\mu\nu}_{\lambda\rho}$ stands
for the functional derivative of the bulk Lagrangian with respect to the
Riemann tensor and derivatives thereof (see Iyer and Wald (1994) for details).
In particular, for the action in Eq. (1), we obtain

$\displaystyle E^{\mu\nu}_{\lambda\rho}\equiv\frac{\partial\mathscr{L}_{\rm
bulk}}{\partial
R^{\lambda\rho}_{\mu\nu}}=\frac{1}{8\kappa}\left[2\delta_{\lambda\rho}^{\mu\nu}-\,\alpha\phi\left({}^{*}R^{\mu\nu}{}_{\lambda\rho}+\,^{*}R_{\lambda\rho}{}^{\mu\nu}\right)\right]\,,$
(29)

where
$\delta^{\mu_{1}\ldots\mu_{p}}_{\nu_{1}\ldots\nu_{p}}=p!\delta_{[\nu_{1}}^{[\mu_{1}}\ldots\delta_{\nu_{p}]}^{\mu_{p}]}$
is the generalized Kronecker delta. Conserved charges associated to Killing
vector fields $\xi$ are obtained by integrating the Noether prepotential on a
codimension-2 hypersurface, $\Sigma$, namely,

$\displaystyle\mathcal{Q}\left[\xi\right]=\frac{1}{2}\int_{\Sigma}\epsilon_{\mu\nu\lambda\rho}q^{\mu\nu}\text{d}x^{\lambda}\wedge\text{d}x^{\rho}\equiv\int_{\Sigma}\mathcal{Q}_{\mu\nu}\text{d}x^{\mu}\wedge\text{d}x^{\nu}\,,$
(30)

where $\wedge$ is the wedge product of differential forms. Wald showed that
the black hole entropy is the Noether charge (30) evaluated at the horizon
$\mathcal{H}$ Wald (1993). More generally, it is obtained by integrating the
Noether prepotential over all codimension-2 hypersurfaces that produce
obstructions to foliation by functions that define the unitary Hamiltonian
evolution of the system Hawking and Hunter (1999); Garfinkle and Mann (2000);
Ciambelli  _et al._ (2021).

Since the black string in CSMG is stationary, we consider the isometry
generated by the asymptotically time-like Killing vector field that vanishes
at the horizon, that is,

$\displaystyle\xi$ $\displaystyle=\partial_{t}+\Omega\partial_{\varphi}\,.$
(31)

Here, $\Omega$ denotes the angular velocity at the horizon—see Eq. (45) below.
The relevant components of the dual Noether prepotential for computing the
entropy are

$\displaystyle\mathcal{Q}_{\varphi
z}=\frac{r}{4\kappa}\left[2f^{\prime}-\alpha\lambda
f\left(rN^{\varphi\,\prime\prime}+3N^{\varphi\,\prime}\right)\right],$ (32)

where $\phi=\lambda z$ has been used. In the case of the black string, the
horizon $\mathcal{H}$ is defined as the codimension-2 hypersurface at constant
$t-r$ coordinates, where the metric function $f(r)$ vanishes, i.e.
$f(r_{+})=0$. Direct integration leads to

$\displaystyle
S=\beta_{\tau}\int_{0}^{2\pi}\text{d}\phi\int_{0}^{L}\text{d}z\,\mathcal{Q}_{\varphi
z}\bigg{|}_{r=r_{+}}=\frac{\pi r_{+}L}{2G}=\frac{\mathcal{A}}{4G}\,,$ (33)

where $\mathcal{A}=2\pi r_{+}L$ is the area of the horizon and $\beta_{\tau}$
is related to the surface gravity $\mathscr{K}$ through
$\beta_{\tau}^{-1}=\mathscr{K}/(2\pi)$. Indeed, $T=\beta_{\tau}^{-1}$ is the
Hawking temperature of the black string. Thus, the system studied above
satisfies the black hole area law, which is an unexpected result in gravity
theories beyond GR (for instance see Jacobson and Myers (1993)).

###  IV.2 Hamiltonian approach

For the purposes of this work, it is sufficient to consider the analytic
continuation of the black hole metric (10) with a base manifold of topology
$S^{1}\times\mathbb{R}$, namely,

$\text{d}s_{E}^{2}=N^{2}(r)f(r)\text{d}\tau^{2}+\frac{\text{d}r^{2}}{f(r)}+r^{2}\left(N^{\varphi}(r)\text{d}\tau+\text{d}\varphi\right)^{2}+\text{d}z^{2}\,,$
(34)

where the Euclidean time range is given by $0\leq\tau\leq\beta_{\tau}$, the
angular coordinate satisfies $0\leq\varphi\leq\beta_{\varphi}$ and the flat
coordinate goes as $0\leq z\leq L$. Additionally, we consider a scalar field
with axionic profile, i.e., $\phi=\phi(z)$. Since the black string has an
event horizon, its Euclidean version possesses a degenerate submanifold in its
place, called a bolt. The absence of conical singularities at the bolt demands
that the period of the Euclidean time, $\beta_{\tau}$, and the period of the
angular coordinate, $\beta_{\varphi}$ are fixed according to the relations
$N(r_{+})f^{\prime}(r_{+})=4\pi/\beta_{\tau}$ and
$\beta_{\varphi}=-\beta_{\tau}N^{\varphi}(r_{+})$, respectively. Moreover,
these quantities are related to the Hawking temperature and angular velocity
of the black string at the horizon as $T=\beta_{\tau}^{-1}$ and
$\Omega=-\beta_{\varphi}/\beta_{\tau}$, respectively.

The Euclidean action is related to the free energy $\mathcal{G}$ by
$I_{E}=\beta_{\tau}\mathcal{G}$. Then, the reduced action in Hamiltonian form
is

$I_{E}=\beta_{\tau}\beta_{\varphi}\int(N\mathcal{H}+N^{\varphi}\mathcal{H}_{\varphi})\text{d}r\text{d}z+B_{E}\
.$ (35)

Here, $B_{E}$ denotes a boundary term whose variation cancels out all the
contributions coming from variations of the bulk action, defining a well-posed
variation principle. Additionally, the constraints and the nonvanishing
components of the conjugated momenta are respectively given by

$\displaystyle\mathcal{H}$
$\displaystyle=\frac{r\Lambda}{\kappa}+\frac{f^{\prime}}{2\kappa}-4\kappa
r\left(\pi^{r\varphi}\right)^{2}+\frac{1}{2}r\left(\frac{\text{d}\phi}{\text{d}z}\right)^{2}\
,$ (36) $\displaystyle\mathcal{H}_{\varphi}$
$\displaystyle=-2\,\left(r^{2}\pi^{r\varphi}\right)^{\prime}\ ,$ (37)
$\displaystyle\pi^{r\varphi}$
$\displaystyle=\frac{rN^{\varphi\,\prime}}{4\kappa N}\ .$ (38)

Performing stationary variations of the action with respect to
$\\{N,f,N^{\varphi},\pi^{r\varphi},\phi\\}$, we obtain a set of differential
equations equivalent to the ones of Chern-Simons modified gravity in Euclidean
signature [see Eq. (6)]. These equations can be solved analytically,
recovering the black string solution of Sec. II, namely,

$\displaystyle N(r)$ $\displaystyle=1\ ,$ $\displaystyle\phi(z)$
$\displaystyle=\lambda z\ ,$ $\displaystyle\pi^{r\varphi}(r)$
$\displaystyle=-\frac{j}{4\kappa r^{2}}\ ,$ (39) $\displaystyle
N^{\varphi}(r)$ $\displaystyle=j_{0}+\frac{j}{2r^{2}}\ ,$ $\displaystyle f(r)$
$\displaystyle=\frac{r^{2}}{\ell^{2}}-m-\frac{j^{2}}{4r^{2}}\ ,$
$\displaystyle\frac{1}{\ell^{2}}$ $\displaystyle=\frac{\kappa\lambda^{2}}{2}\
,$

where $\lambda,m,j_{0}$ and $j$ are integration constants. Moreover, the
scalar field is rigid since $\lambda$ is a constant without variation, fixed
in terms of the cosmological constant. It is worth mentioning that, although
the scalar field is endowed with a linear dependence on the $z$ coordinate,
the shift symmetry protects the theory for any stability problem related to
the unboundedness of the $z$ coordinate. This can be seen by noticing that all
physical quantities depends purely on derivatives of the scalar field. On the
other hand, one could impose periodicity on the $z$ coordinate such that it
becomes compact; in that case, the topology of the horizon would be
$\mathbb{S}^{1}\times\mathbb{S}^{1}$, representing a black ring rather than a
black string.

The variation of the boundary term, on the other hand, yields

$\displaystyle\delta B_{E}$
$\displaystyle=-\beta_{\tau}\beta_{\varphi}L\left[\frac{N\delta
f}{2\kappa}-2r^{2}N^{\varphi}\delta\pi^{r\varphi}\right]^{r=\infty}_{r=r_{+}}\,.$
(40)

To compute the latter, we first evaluate the variation of the boundary term on
the solution given in Eq. (39). Then, taking into account that the variation
of the fields at the bolt are

$\displaystyle\delta f|_{r_{+}}=-\frac{4\pi}{\beta_{\tau}}\delta
r_{+}\;\;\;\;\;\mbox{and}\;\;\;\;\;\delta\pi^{r\varphi}|_{r_{+}}=-\frac{\delta
j}{4\kappa r_{+}^{2}}\ ,$ (41)

then it follows that the variation of the boundary term at the bolt and
infinity are respectively given by

$\displaystyle\delta B_{E}(\infty)$
$\displaystyle=\frac{\beta_{\tau}\beta_{\varphi}L}{2\kappa}\left[\delta{m}-j_{0}\delta
j\right]\,,$ (42) $\displaystyle\delta B_{E}(r_{+})$
$\displaystyle=\frac{\beta_{\tau}\beta_{\varphi}L}{2\kappa}\left[\frac{4\pi\delta
r_{+}}{\beta_{\tau}}-\left(j_{0}+\frac{j}{2r_{+}^{2}}\right)\delta
j\right]\,.$ (43)

Thus, we conclude that the total variation of the boundary term, evaluated at
the solution (39), becomes

$\delta B_{E}=\delta B_{E}(\infty)-\delta
B_{E}(r_{+})=\beta_{\tau}\delta\left(\frac{\sigma
m}{2\kappa}\right)-\frac{2\pi}{\kappa}\delta
A_{+}+\beta_{\tau}\frac{j}{2r_{+}^{2}}\delta\left(\frac{\sigma
j}{2\kappa}\right)\ ,$ (44)

where $\sigma=\beta_{\varphi}L$ stands for the volume of the base manifold and
$A_{+}=\sigma r_{+}$ for the area of the black string horizon. From this
expression, we identify

$\displaystyle\Omega=\frac{j}{2r_{+}^{2}}\,,$ (45)

as the conjugated thermodynamic variable associated to the angular momentum:
the angular velocity of the black string.

Hereafter, we work in the grand canonical ensemble, where the temperature and
the angular velocity are kept fixed at the bolt. This allows us to integrate
the variation of the boundary term. Its value determines the reduced on-shell
action, giving

$I_{E}=B_{E}=\beta_{\tau}\left(\frac{\sigma
m}{2\kappa}\right)-\frac{2\pi}{\kappa}A_{+}-\beta_{\tau}\Omega\left(-\frac{\sigma
j}{2\kappa}\right)\ ,$ (46)

up to an arbitrary additive constant without variation. Since the Gibbs free
energy and the Euclidean action are related by
$I_{E}=\beta_{\tau}\mathcal{G}=\beta_{\tau}M-S-\beta_{\tau}\Omega J$, the
conserved charges and entropy can be obtained through the standard
thermodynamical relations

$\displaystyle M$
$\displaystyle=\left(\frac{\partial}{\partial\beta_{\tau}}-\beta_{\tau}^{-1}\Omega\frac{\partial}{\partial\Omega}\right)I_{E}=\frac{\sigma
m}{2\kappa}\ ,$ (47a) $\displaystyle J$
$\displaystyle=-\frac{1}{\beta_{\tau}}\frac{\partial
I_{E}}{\partial\Omega}=-\frac{\sigma j}{2\kappa}\ ,$ (47b) $\displaystyle S$
$\displaystyle=\left(\beta_{\tau}\frac{\partial}{\partial\beta_{\tau}}-1\right)I_{E}=\frac{A_{+}}{4G}\
.$ (47c)

It is reassuring to verify that the Noether-Wald formalism and the Hamiltonian
approach lead to the Bekenstein-Hawking entropy, considering that non-
minimally coupled scalar fields can produce additional contributions such as
in other scalar-tensor theories Ashtekar  _et al._ (2003); Barlow  _et al._
(2005); Martinez  _et al._ (2006). The sign flip in the angular momentum is a
consequence of the computation in Euclidean signature. By construction, the
first law of black hole thermodynamics is satisfied, namely,

$\displaystyle\delta M=T\delta S+\Omega\delta J\,.$ (48)

Notice that $j_{0}$ plays no role in the thermodynamics. This is just a
reflection of the fact that we can consider a rotating asymptotic geometry
$\mbox{AdS}_{3}\times\mathbb{R}$ where $j_{0}$ emerges as the boost parameter
along the $t-\varphi$ plane in this region. In consequence, since the angular
momentum of the black string is measured respect to this background, without
loss of generality, we can always choose the comoving frame where $j_{0}=0$.
Therefore, we conclude that the Hamiltonian approach leads to the same
conserved quantities as the renormalized boundary stress tensor.

###  IV.3 Smarr formula

This section is devoted to deriving a Smarr formula for the black string using
a Euclidean Hamiltonian approach, as in Ref. Banados and Theisen (2005). To
this end, we compute a radial conservation law associated to the scale
invariance of the reduced action. Let us recall that only regular solutions
may dominate the gravitational path integral. Thus, we impose that matter
fields must be finite at the bolt and additionally that they vanish at
infinity, which is satisfied along a finite section of the base manifold.

Let us consider a minisuperspace determined by the metric (10). Then, the
reduced action of Chern-Simons modified gravity [see Eq. (35)] is invariant
under the set of transformations $\Phi(x)\to\bar{\Phi}(\bar{x})$ generated by
the rescaling along the radial coordinate $\bar{r}=\delta r$, where $\delta$
is a positive constant,
$\Phi=\left\\{N,f,N^{\varphi},\pi^{r\varphi},\phi\right\\}$, and

$\displaystyle\bar{N}(\bar{r})$ $\displaystyle=\delta^{-2}N(r)\ ,$
$\displaystyle\bar{f}(\bar{r})$ $\displaystyle=\delta^{2}f(r)\ ,$
$\displaystyle\bar{\pi}^{r\varphi}(\bar{r})$ $\displaystyle=\pi^{r\varphi}(r)\
,$ (49) $\displaystyle\bar{\phi}(z)$ $\displaystyle=\phi(z)\ ,$
$\displaystyle\bar{N}^{\varphi}(\bar{r})$
$\displaystyle=\delta^{-2}N^{\varphi}(r)\ .$

The Noether theorem, in turn, implies that there exists a conserved quantity
$C(r)$ associated to radial rescalings, such that $C^{\,\prime}(r)=0$. Using
the constraint $\mathcal{H}=0$, we rewrite the conserved quantity as

$C(r)=\left[N\left(\frac{f^{\prime}}{2\kappa}r-\frac{f}{\kappa}\right)+N^{\varphi}(4r^{2}\pi^{r\varphi})\right]\sigma\
.$ (50)

Since the latter represents a conserved quantity along the radial direction,
it must satisfy $C(\infty)=C(r_{+})$. A direct computation using the black
string solution (10) with (11), (19) and (20), yields

$\displaystyle
C(r_{+})=\left(r_{+}f^{\prime}(r_{+})+\frac{j^{2}}{r_{+}^{2}}\right)\frac{\sigma}{2\kappa}=TS+2\Omega
J\;\;\;\;\;\mbox{and}\;\;\;\;\;C(\infty)=\frac{m\sigma}{\kappa}=2M\ ,$ (51)

where the definition of thermodynamic variables in Sec. IV.2 have been used.
Thus, by comparing these two expressions, we obtain the Smarr relation for the
black string, namely,

$M=\frac{1}{2}TS+\Omega J\ .$ (52)

Morover, the very same relation can be obtained by direct application of the
Euler theorem for homogeneous functions, since the entropy can be regarded as
a homogeneous function of degree $1/2$ in $(M,J)$, i.e. $S(\lambda M,\lambda
J)=\lambda^{1/2}S(M,J)$. Then,

$\frac{1}{2}S=\left.M\frac{\partial S}{\partial
M}\right|_{J}+\left.J\frac{\partial S}{\partial J}\right|_{M}\ .$ (53)

Using the first law $T\delta S=\delta M-\Omega\delta J$ we obtain,

$\left.\frac{\partial S}{\partial M}\right|_{J}=\frac{1}{T}\
,\quad\left.\frac{\partial S}{\partial J}\right|_{M}=-\frac{\Omega}{T}\ .$
(54)

It is straightforward to show that Eqs. (53) and (54) lead to (52). This
relation is the same as the Smarr formula of a rotating BTZ black hole in
three dimensional gravity Erices  _et al._ (2018). Of course, the Smarr
relation is independent of the Euclidean or Lorentzian signature.

##  V Ground state and conformally flat non-Einstein product spaces

The conformal nature of quantum gravity on AdS3 Brown and Henneaux (1986) and
the Cardy formula for the asymptotic growth of the number states in 2D
conformal field theories Cardy (1986) has allowed for the microscopic
derivation of the BTZ black hole entropy Strominger (1998). This established
AdS3 as the ground state of the theory. In higher dimensions, the AdS soliton
was later found to play the same role Horowitz and Myers (1998); Galloway  _et
al._ (2002). Currently, Cardy formulae are written so as to relate the entropy
of a black hole with its mass and the mass of its associated ground state even
for more general asymptotic behavior, see for instance Correa  _et al._ (2011,
2012); Gonzalez  _et al._ (2011). This motivates us to establish the ground
state of the BTZ black string under consideration.

By noticing that the homogeneous foliation of the four dimensional metric (10)
admits an analytic continuation along the symmetries generated by its Killing
vector fields $\partial_{t}$ and $\partial_{\varphi}$, we propose an _Ansatz_
of the form

$\text{d}s^{2}=-N^{2}(r)f(r)\text{d}t^{2}+\frac{\text{d}r^{2}}{f(r)}+h(r)r^{2}\text{d}\varphi^{2}+\text{d}z^{2}\,.$
(55)

The metric (55) solves the field equations of Chern-Simons modified gravity,
recall Eq. (6), when

$\displaystyle f(r)$ $\displaystyle=-\mu+\frac{r^{2}}{\ell^{2}}\,,$
$\displaystyle N^{2}(r)$ $\displaystyle=\frac{r^{2}}{f(r)\ell^{2}}\,,$
$\displaystyle h(r)$ $\displaystyle=\frac{f(r)\ell^{2}}{r^{2}}\,,$
$\displaystyle\phi(z)$ $\displaystyle=\lambda\,z\,,$ (56)

where $\ell^{-2}=-\Lambda/2$ is the AdS radius, $\mu$ is an integration
constant, and the axionic charge $\lambda$ is fixed in terms of the
cosmological constant through $\lambda^{2}=-\Lambda/\kappa$. Reality of the
scalar field, on the other hand, implies that the cosmological constant must
be negative. Positivity of the Riemannian section of (55) demands that the
radial coordinate must be restricted to $r\geq r_{s}\equiv\ell\sqrt{\mu}$. The
absence of conical singularities imposes that $\mu>0$ and
$\varphi\sim\varphi+\beta_{\varphi}^{(s)}$, where
$\beta^{(s)}_{\varphi}=\tfrac{4\pi}{f^{\prime}(r_{s})\ell}=\tfrac{2\pi}{\sqrt{\mu}}$.
With these conditions, the line element (55) with metric functions (56)
represents the ground state with topology $\mbox{AdS}_{3}\times\mathbb{R}$.

Its mass can be computed directly through standard Euclidean methods of Sec.
IV. To do so, we consider the Gibbs-Duhem relation in absence of entropy.
Moreover, the temperature is arbitrary due to their lack of horizon. Thus, the
mass of the ground state (56) is given by

$M_{0}=-\frac{L}{8G}\,.$ (57)

It is clear that the mass of the ground state in Eq. (57) is negative. The
negative-energy contribution can be interpreted as the Casimir energy that is
generated in the dual field theory when the fermions are antiperiodic on the
compact coordinate Coussaert and Henneaux (1994). Additionally, it is direct
to see that all curvature invariants associated to this solution remain
finite. Therefore, it represents a negative-energy regular solution without
horizon. These configuration are relevant when studying phase transitions, as
we show further below.

The Casimir energy interpretation stems from the Cardy formula. Before writing
it here, let us remark that we have used in Eq. (56) the same notation as for
the black string. In the Cardy formula, states of the same configuration are
compared. Thus, the ground state and the black string must be comparable,
i.e., they have the same boundary conditions and global symmetries.
Consequently, the entropy of the black string can be written as

$S=4\pi
l\left(\sqrt{\frac{-M_{0}\Delta^{+}}{2}}+\sqrt{\frac{-M_{0}\Delta^{-}}{2}}\right),$
(58)

where $\Delta^{\pm}=(M\pm J/l)/2$, cf. Bravo Gaete  _et al._ (2017). This is
the Cardy formula for the analytic rotating black string presented in Sec. II.

Interestingly enough, metric (10) admits another analytic continuation; this
one along the symmetries generated by the Killing fields $\partial_{t}$ and
$\partial_{z}$. Hence, we now explore a metric _Ansatz_ of the form

$\text{d}s^{2}=-\text{d}t^{2}+\frac{\text{d}\rho^{2}}{u(\rho)}+\rho^{2}\left(v(\rho)\text{d}z+\text{d}\varphi\right)^{2}+u(\rho)\text{d}z^{2}\,,$
(59)

which solves the field equations (6) provided the metric functions and scalar
field are

$\displaystyle u(\rho)$
$\displaystyle=\mu_{1}-\frac{\gamma^{2}}{4\rho^{2}}-\frac{\Lambda\rho^{2}}{2},$
$\displaystyle v(\rho)$ $\displaystyle=\frac{\gamma}{2\rho^{2}},$
$\displaystyle\phi(t)$ $\displaystyle=\lambda_{1}\,t\,,$ (60)

where $\mu_{1}$, $\gamma$ and $\lambda_{1}$ are integration constants; the
latter satisfying the condition $\lambda_{1}^{2}=\Lambda/\kappa$. Notice that,
in contrast to the black strings in Sec. II, this solution is supported only
by a positive cosmological constant, such that the time-dependent scalar field
remains real. The Lorentzian signature of this solution is preserved provided
$\mu_{1}>\sqrt{\Lambda\gamma^{2}/2}$ and the range for the radial coordinate
is restricted to $\rho\in[\rho_{-},\rho_{+}]$, where

$\rho_{\pm}=\left[\frac{\mu_{1}\pm\sqrt{\mu_{1}^{2}-\Lambda\gamma^{2}/2}}{\Lambda}\right]^{1/2}\
.$ (61)

The absence of conical singularities demands that the $z$-coordinate has to be
identified as $z\sim z+\beta_{z}^{(\pm)}$ at $\rho_{\pm}$, whose period is
$\beta_{z}^{(\pm)}=4\pi/u^{\prime}(\rho_{\pm})$. This, in turn, implies that
the periodicity of the angular coordinate is
$\varphi\sim\varphi+\beta_{\varphi}^{(\pm)}$, where
$\beta_{\varphi}^{(\pm)}=-\beta_{z}^{(\pm)}v(\rho_{\pm})$.

It is clear that this configuration is a product space $\mathbb{R}\times
M^{3}$, where $M^{3}$ is compact (as displayed above) and has positive
definite geometry. Moreover, a straightforward calculation shows that $M^{3}$
has positive curvature. Then, a theorem by Richard S. Hamilton implies that
$M^{3}$ is diffeomorphic to $S^{3}$, considering that we want spacetime to be
simply-connected, see Ziller (2014) and references therein. Indeed, one can
check that it is a non-Einstein conformally flat spacetime supported by a
time-dependent scalar field. This can be seen by computing its traceless Ricci
and Weyl tensor, respectively, and noticing that the former is nonvanishing
while the latter is zero.

In summary, by using the Cardy formula we have established
$\mbox{AdS}_{3}\times\mathbb{R}$ as the ground state of the black string. We
have found that both $\mbox{AdS}_{3}\times\mathbb{R}$ and $\mathbb{R}\times
S^{3}$ are non-trivial solutions of CSMG. These spaces are conformally flat
non-Einstein product spaces sourced by nontrivial scalar fields. These spaces
are analogs of the Plebański-Hacyan spacetimes, which have geometries
$\mbox{AdS}_{2}\times\mathbb{R}^{2}$ and $\mathbb{R}^{2}\times S^{2}$
Plebański and Hacyan (1979). However, $\mathbb{R}\times S^{3}$ is incompatible
with the symmetries of the black string and they do not share the same
boundary conditions. This implies that they cannot be compared as
configurations that belong to the same ensemble. We postpone a deeper study of
its properties for future investigations. Thus, from hereon, we only consider
the ground state in a thermodynamic context.

##  VI Thermodynamic Stability and Phase Transitions

It is well known that the second law of thermodynamics is equivalent to the
weakly-convex property of the internal energy function Callen (1985). In our
framework, what is relevant is to analyze the concavity of the free energy
$\mathcal{G}(T,\Omega)$. Phase transitions occur when this function is not
analytic. A function with multiple branches corresponds to a system with
various phases. The branch of the function which has less free energy is
statistically preferred and it represents a stable state of the system under
fluctuations.

From previous sections, we know that the free energy of the black string is

$\mathcal{G}(T,\Omega)=-\frac{2\pi^{2}l^{2}\sigma
T^{2}}{\kappa(1-\Omega^{2}l^{2})}\ .$ (62)

Notice that it satisfies the concavity conditions

$\frac{\partial^{2}\mathcal{G}}{\partial T^{2}}\leq
0,\quad\frac{\partial^{2}\mathcal{G}}{\partial\Omega^{2}}\leq
0,\quad\frac{\partial^{2}\mathcal{G}}{\partial
T^{2}}\frac{\partial^{2}\mathcal{G}}{\partial\Omega^{2}}-\left(\frac{\partial^{2}\mathcal{G}}{\partial
T\partial\Omega}\right)^{2}\geq 0,$ (63)

since the existence of event horizons is determined by $|j|<ml$ which requires
$\Omega^{2}\ell^{2}<1$.

Now, the local stability of the black string can be analyzed by studying its
response under small perturbations of its thermodynamic variables around
equilibrium. Conditions (63) can be interpreted as manifestations of the
response functions. For instance, the specific heat at constant angular
velocity, $C_{\Omega}=T{(\frac{\partial S}{\partial T})}_{\Omega}$, measures
the heat absorption from a temperature stimulus, while the isothermal
compressibility for the angular momentum,
$\kappa_{T}=\frac{1}{J}{(\frac{\partial J}{\partial\Omega})}_{T}$, measures
the response of the angular momentum respect to an angular velocity stimulus.
Specifically, these quantities are

$\displaystyle C_{\Omega}=\frac{4\pi^{2}l^{2}\sigma
T}{\kappa(1-\Omega^{2}l^{2})}\;\;\;\;\;\mbox{and}\;\;\;\;\;\kappa_{T}=\frac{1+3l^{2}\Omega^{2}}{\Omega(1-\Omega^{2}l^{2})}\
.$ (64)

The stability conditions for these quantities are $C_{\Omega}\geq 0$ as well
as $J\kappa_{T}\geq 0$. The former is clearly satisfied while the latter is

$J\kappa_{T}=\frac{4\pi^{2}l^{2}(1+3l^{2}\Omega^{2})\sigma
T^{2}}{\kappa(1-\Omega^{2}l^{2})^{3}}\geq 0\ .$ (65)

In consequence, the analytic rotating black string is always able to reach a
thermal equilibrium with a heat bath.333The extremal case when $|j|=ml$ has
$\mathcal{G}=0$ and in consequence it is locally stable

We have seen that the action principle of dynamical Chern-Simons modified
gravity admits two solutions with the same boundary conditions for a fixed
temperature. These are the rotating black string presented in Sec. II and the
$\mbox{AdS}_{3}\times\mathbb{R}$ ground state described by the line element
(55) with (56). Both solutions share the same asymptotic geometry and
behavior, representing two phases of the same system. Therefore, global
stability is analyzed by comparing their respective free energies. For
simplicity, we consider only static configurations and emphasize that we
require both configurations to be at the same temperature.

As it was pointed out in Sec. V, the ground state lacks an event horizon and
its temperature can be chosen arbitrarily to match the black string
temperature. Thus, from Eqs. (57) and (62) we have

$\displaystyle\mathcal{G}=-\frac{2\pi^{2}}{\kappa}\ell^{2}\sigma
T^{2}\;\;\;\;\;\mbox{and}\;\;\;\;\;\mathcal{G}_{0}=M_{0}=-\frac{\sigma}{2\kappa}\
.$ (66)

Since $\ell^{-2}=\kappa\lambda^{2}/2$, then the difference between both Gibbs
free energies is

$\Delta\mathcal{G}=\mathcal{G}-\mathcal{G}_{0}=\left(\frac{2\pi}{\kappa\lambda}\right)^{2}\left(\frac{\kappa\lambda^{2}}{8\pi^{2}}-T^{2}\right)\sigma\
.$ (67)

A phase transition between the static black string and the ground state in Eq.
(55) occurs at the critical temperature $T_{c}$, when the condition
$\Delta\mathcal{G}=0$ holds, namely,

$T_{c}=\frac{\lambda}{2\pi}\sqrt{\frac{\kappa}{2}}\,.$ (68)

Therefore, a phase transition is supported by the existence of the scalar
field, as it can be seen from the linear dependence of the critical
temperature on the axionic charge $\lambda$. Moreover, this is a first order
phase transition because the slope of $\Delta\mathcal{G}$ does not vanish at
$T_{c}$. Taking into account Eq. (67), we see that the ground state is
preferred for temperatures below the critial temperature, i.e. $T<T_{c}$, and
the black string is thermodynamically favored when $T>T_{c}$.

##  VII Conclusions

In this work, we study black string spacetimes possessing axionic charge in
dynamical CSMG. We calculate their conserved charges and use them to construct
a Cardy formula to establish the ground state. We later move on to the
thermodynamics of this black string/ground state system, finding phase
transitions between the pair.

The analysis is carry out by considering a scalar field with linear dependence
on the coordinate that spans the string. Although this assumption might break
the isometry group, the shift symmetry in field space renders the energy-
momentum tensor of scalar fields compatible with the symmetries of the metric.
Indeed, this class of scalar fields has been widely used in the context of
holography, providing simple models with momentum relaxation (see Andrade and
Withers (2014) and references therein).

First, we review the analytic rotating BTZ black string solution in CSMG found
in Ref. Cisterna  _et al._ (2019). It is worth recalling that exact rotating
configurations in this theory are scarce since, in general, nontrivial
contribution are introduced by the Chern-Simons coupling in the field
equations Grumiller and Yunes (2008). Nevertheless, we demonstrate that the
presence of the scalar field with axionic profile is crucial for obtaining the
locally $\mbox{AdS}_{3}\times\mathbb{R}$ rotating black string, due to the
addition of nontrivial contributions to the off-diagonal components of the
field equations.

Afterwards, we compute conserved charges by means of the renormalized boundary
stress-energy tensor, obtaining the mass and angular momentum of the black
string. The entropy is obtained through the Wald formalism, by evaluating the
Noether charge at the horizon. The area law is recovered despite the fact that
we are in an effective gravity theory with higher-curvature corrections. Using
the Hamiltonian approach, we obtain the free energy by computing the
renormalized Euclidean on-shell action, augmented by boundary terms that
guarantee finiteness on asymptotically locally
$\mbox{AdS}_{3}\times\mathbb{R}$ spacetimes while defining a well-posed
variational principle. We recover the previous conserved charges as
thermodynamic quantities and find that the first law of thermodynamics is
satisfied. The axionic charge is fixed in terms of the cosmological constant
and it can be only treated as a thermodynamic variable in the extended phase
space (see Kubiznak and Mann (2015)); approach that we do not follow here.
Additionally, exploiting the rescaling symmetry along the radial direction, we
derive a Smarr formula from the Noether theorem following the approach of Ref.
Banados and Theisen (2005). We find that the latter is equivalent to that of
the rotating BTZ black hole in three-dimensional gravity.

Then, we obtain the ground state and a conformally flat non-Einstein product
space with axionic charge by performing the analytic continuation of the black
string along the flow generated by their Killing vector fields. All the
solutions obtained through this method are regular and horizonless. Therefore,
their lack of entropy implies that their energy can be obtained from the
renormalized Euclidean on-shell action with arbitrary value of the Euclidean
time. We compute the energy of the ground state explicitly and find that it is
negative, representing the Casimir energy in the dual field theory when the
fermions are antiperiodic on the compact coordinate Coussaert and Henneaux
(1994).

We study thermodynamic stability and phase transitions in Sec. VI. The Gibbs
free energy of the black string is a concave function of the temperature and
angular velocity implying that this configuration is thermodynamically locally
stable under small perturbations in temperature and angular momentum. We
verify that thermal equilibrium with a heat bath can always be attained by
analyzing the response functions such as the heat capacity and the isothermal
compressibility of the angular momentum. The global stability is a noteworthy
aspect, considering the existence of a ground state in presence of an axionic
scalar field. In particular, we find that a first-order phase transition takes
place at certain critical temperature $T_{c}$ [see Eq. (68)]. For $T<T_{c}$
the gravitational ground state is thermodynamically favored, while for
$T>T_{c}$ the black string becomes the most probable configuration.

Interesting phenomena can be explored in future works. For instance, a
stabilization mechanism of the Gregory–Laflamme instability due to axions in
CSMG is worth pursuing, similar to the one proposed in Cisterna  _et al._
(2020b). This might provide evidence on the dynamical stability of the black
string, as it has been recently analyzed in GR Henríquez-Báez (2021);
Dhumuntarao and Mahbub (2021). Additionally, the extended phase space could
certainly enrich the thermodynamics of phase transitions. This is because the
relation between the axionic charge and the cosmological constant may offer a
scenario where the former, as a free parameter, is interpreted as a pressure.
Finally, a natural extension of these solutions is to include Maxwell fields.
To this end, one could implement the approach of Refs. Liu  _et al._ (2020);
Cisterna  _et al._ (2020c, 2021), rendering this class of solutions compatible
with the presence of electric/magnetic charge.

###### Acknowledgements.

The authors thank I. J. Araya, A. Argandoña, A. Cisterna, O. J. Franca, N.
Merino, R. Olea, M. Pino, R. Troncoso, L. F. Urrutia, and O. Valdivia for
valuable comments, suggestions, and insightful remarks. We also acknowledge
the collaboration of S. del Pino in early stages of this project. The work of
C. C. is supported by Agencia Nacional de Investigación y Desarrollo (ANID)
through FONDECYT No 11200025 and 1210500. C. E. is supported by ANID through
PAI Grant No. 77190046 and Universidad Central de Chile through Proyecto
Interno CIP2020039. D. F.-A. is supported by a CONACYT postdoctoral fellowship
and also acknowledges support from CONACYT Grant No A1-S-11548. K.L. is
financially supported by Beca Doctorado Nacional No 21182110, ANID.

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