aainstitutetext: Department of Physics and Astronomy, University of Alabama,
514 University Boulevard, Tuscaloosa, AL 35487, USAbbinstitutetext: Institut
für Theoretische Festkörperphysik (TP1)
Julius-Maximilians-Universität Würzburg, Am Hubland, D-97074 Würzburg, Germany

#  Topological confinement in Skyrme holography

Casey Cartwright<EMAIL_ADDRESS>a Benjamin Harms<EMAIL_ADDRESS>a Matthias Kaminski<EMAIL_ADDRESS>b Ronny Thomale<EMAIL_ADDRESS>wuerzburg.de

###### Abstract

We study phase transitions in five-dimensional Einstein Gravity with a
negative cosmological constant, coupled to a Skyrme matter field. These
transitions are topological generalizations of the Hawking-Page transition
between thermal Anti de Sitter (AdS) spacetime and an AdS black hole. Phases
are characterized by a topological number associated with the Skyrme field
configuration. Depending on that topological number and on the Skyrme coupling
strength, there occur transitions between those phases at two, one, or no
value(s) of the temperature. Through the holographic (AdS/CFT) correspondence,
these solutions are dual to topologically non-trivial states in a conformal
field theory (CFT) with an SU(2)-symmetry, which support either confined or
deconfined (quasi-)particles at strong coupling. We compare to similar known
phase transitions, and discuss potential applications to confinement in
topological phases of condensed matter and the quark-gluon plasma.

##  1 Introduction

The study of phase transitions in the context of topological black holes in
five-dimensional anti-de Sitter spacetime is important because of the duality
between such black holes and the four-dimensional conformal field theories
which live on the boundary. An understanding of the dynamics of phase
transitions within gravitational systems in the bulk can potentially elucidate
the dual transitions which occur in the strongly-coupled system on the
boundary. In Cartwright:2020yoc  an analytic solution of the Einstein
equations for such a black hole endowed with a Skyrme field was obtained and
shown to be holographically Maldacena:1997re  dual to an $\mathcal{N}\,=\,4$
Super-Yang-Mills (SYM) gauge field theory. For a large Skyrme coupling
constant the Skyrme field can significantly alter the temperature at which the
gravitational system undergoes a Hawking - Page (H - P) transition between
thermal AdS and an AdS black hole. As we show in section 2, the size of the
coupling also determines whether the number of H - P transitions is zero, one,
or two at a given topological charge value. Via the holographic (AdS/CFT)
correspondence, the H - P transition was found to be dual to the
(de-)confinement transition of the $\mathcal{N}=4$ SYM field theory
Witten:1998zw . In Cartwright:2020yoc , the Skyrme coupling and Skyrme field
were shown to correspond to $SU(2)$-gauge fields and their coupling to
$\mathcal{N}=4$ SYM theory, respectively.

The results of our work form the foundation of a more generalized treatment of
topological phases of matter at strong coupling, as our technique allows for
an analytic study of such transitions. The analytic solutions we have found
can, via AdS/CFT duality, be applied to other physical systems in which
topological phase transitions occur. Our mapping of a topological black hole
to its CFT analogue allows the analysis of (de-)confinement transitions which
are decorated with a non-Abelian gauge field texture of potentially
topologically non-trivial character. The intertwining of confinement and
topology is ubiquitous in condensed matter. For quantum spin chains, the
spinons as elementary quasiparticles can develop confinement to form a fully
symmetric yet topologically non-trivial ground state for integer spins, or
retain their deconfined character in the spin fluid phase of half-integer spin
chains haldane2016ground . Likewise, certain spin liquids in 2+1 dimensional
Mott insulators can be interpreted as the deconfined limit of an SU(2) gauge
theory PhysRevB.65.165113 , bearing similarities to certain parton mean fields
describing fractional quantum Hall systems. Quantum phase transitions further
allow for emergent gauge fields in their critical theory that cannot be
reconciled by the adjacent order parameter doi:10.1126/science.1091806 , but
instead embody new elementary excitations with potentially topological
features.

Another application could be (de-)confined topological phases and transitions
between them, potentially occurring in heavy ion collisions. In that particle
physics context, topological field configurations are believed to play a key
role in the generation of axial charge asymmetry in heavy ion collisions, and
potentially to the matter/anti-matter asymmetry of the universe
Kharzeev:2004ey ; Ghosh:2021naw ; Cartwright:2021maz .

In this paper we carry out a thermodynamic analysis of the phase transitions
which can occur in our topological black hole. We begin with a brief
discussion in section 2 where describe our topological black hole. We analyze
the H - P transitions which can occur in our model and discuss the stability
our gravitational system. The holographic interpretation of the results of
section 2 is given in section 3. In section 4, we compare and contrast the
phase transitions described by our model to the phase transitions which occur
in other physical systems.

##  2 Topological black holes with Skyrmion hair

###  2.1 Our model

The Skyrmion action in five dimensions is given by111In our conventions, Greek
indices $\mu,\,\nu,\,...\in\\{0,1,2,3,4\\}$ range over the five-dimensional
gravity bulk coordinates, Latin indices $m,\,n,...\in\\{0,1,2,3\\}$ range over
the four-dimensional boundary field theory coordinates.

$\displaystyle S=\int d^{5}x\sqrt{-g}\left(\frac{R-2\Lambda}{16\pi
G_{5}}+\frac{f_{\pi}}{16\pi}K_{\mu}K^{\mu}+\frac{1}{32\,\tilde{e}^{2}}\text{Tr}\left(\left[K_{\mu},K_{\nu}\right]\left[K^{\mu},K^{\nu}\right]\right),\right),$
(1)

in which $R$ is the Ricci scalar, $\Lambda=-6/L^{2}$ is the cosmological
constant in $AdS_{5}$, $L$ is the $AdS_{5}$ radius and $G_{5}$ is the five-
dimensional gravitational constant, while $\tilde{e}$ is the Skyrme coupling
constant. The Skyrme field $U$ is an $SU(2)$-valued Lorentz scalar and has
been introduced into the action via the $SU(2)$-valued Lorentz four-vector
${K_{\mu}\,=\,U^{-1}\partial_{\mu}U}$. The Skyrme field tensor is defined as

$\displaystyle F_{\mu\nu}\,=\,\left[K_{\mu},K_{\nu}\right]$ (2)

In the following sections we will work with $\tilde{e}=8\pi G_{5}e$ where $e$
is a dimensionless coupling, measuring the strength of the Skyrme coupling
compared to the gravitational coupling.

Topological solution: In Cartwright:2020yoc  an analytic solution to the
Einstein-Skyrme equations of motions was discovered. There, the authors
demonstrated that the solution could be mapped to a solution of the massless
$SU(2)$ Einstein-Yang-Mills theory. Furthermore it was demonstrated that the
configuration of the equivalent $SU(2)$ gauge field is topologically non-
trivial, characterized by a non-vanishing topological charge. This charge
counts the number of times the mapping wraps the internal $SU(2)$ group
manifold. Here, we briefly review this solution, which in general can be
written as

$U=e^{in\chi v^{i}\tau^{i}},$ (3)

where $v^{i}v^{i}=1$ and $n\in\mathbb{Z}$ and $\tau_{a}$ are the Pauli
matrices

$\tau_{1}=\begin{pmatrix}0&1\\\
1&0\end{pmatrix},\quad\tau_{2}=\begin{pmatrix}0&-i\\\
i&0\end{pmatrix},\quad\tau_{3}=\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix},$ (4)

such that $\text{tr}\\{\tau_{1}\tau_{2}\tau_{3}\\}=2i$ . The integer $n$
quantifies how many times the mapping wraps the internal $S^{3}$. The
coordinates chosen here are $(t,\psi,\theta,\phi,r)$, where $\psi\in(0,\pi)$,
$\theta\in(0,\pi)$, $\phi\in(0,2\pi)$, $t$ is the temporal coordinate and $r$
is the bulk $AdS$ radial coordinate. A particularly useful form of the unit
vector $v$ is given by

$v=(\cos(\theta),\sin(\theta)\cos(\phi),\sin(\theta)\sin(\phi)),$ (5)

and we choose the Skyrmion field $\chi$ such that $\chi=\psi$. For $n=1$ this
results in a standard mapping of the unit three-sphere into the $SU(2)$ gauge
manifold. The spacetime metric is taken to be

$ds^{2}=\frac{1}{A(r)}\mathrm{d}r^{2}-A(r)\mathrm{d}t^{2}+r^{2}\left(h_{1}(\psi)\mathrm{d}\psi^{2}+h_{2}(\psi)(\mathrm{d}\theta^{2}+\sin(\theta)^{2}\mathrm{d}\phi^{2})\right).$
(6)

The Einstein-Skyrme equations of motion at vanishing pion coupling,
$f_{\pi}=0$, result in three independent equations for $h_{1},h_{2}$ and $A$.
The solutions of these equations are given by

$h_{1}(\psi)=n^{2},\quad h_{2}(\psi)=\sin(n\psi),\quad
A(r)=\frac{1}{e^{2}r^{2}}+\frac{\log(r)}{e^{2}r^{2}}+\frac{r^{2}}{L^{2}}-\frac{m_{t}}{r^{2}}+1\,,$
(7)

where $m_{t}$ may be thought of as the black hole mass. As stated in
Cartwright:2020qov , using the solutions in eq. (7) along with
$K_{\mu}=U\partial_{\mu}U$ for the $U$ given in eq. (3) and eq. (5) the Skyrme
equations of motion, along with the Yang-Mills equations of motion can be
shown to be trivially satisfied.

Temperature: The solution has a horizon at the location $A(r_{h})=0$. That
equation is transcendental and can only be solved for $r_{h}$ numerically. The
horizon radius of solutions (at various values of $e$) is displayed in the
left panel of figure 1 as a function of the mass $m_{t}$. The temperature of
such solutions is given by the standard formula

$T=\frac{1}{4\pi}|A^{\prime}(r_{h})|\,,$ (8)

where the temperature as a function of the horizon radius is displayed in the
right panel of figure 1. Notably, there exists a minimal temperature
$T_{\text{min}}$ for solutions of each winding number $n$ and Skyrme coupling
$e$, indicating that only solutions with temperatures $T\geq T_{\text{min}}$
exist.

The limit $e\rightarrow 0$ corresponds to an infinitely strong effect of the
Skyrmion as a matter source in the bulk geometry. We note that the black hole
of this configuration has a finite horizon radius determined by

$A(r_{h})=0=\frac{1}{e^{2}r_{h}^{2}}+\frac{\log(r_{h})}{e^{2}r_{h}^{2}}+\frac{r_{h}^{2}}{L^{2}}-\frac{m_{t}}{r_{h}^{2}}+1\hookrightarrow\frac{1}{r_{h}^{2}}+\frac{\log(r_{h})}{r_{h}^{2}}+\frac{e^{2}r_{h}^{2}}{L^{2}}-\frac{e^{2}\,m_{t}}{r_{h}^{2}}+e^{2}=0\,,$
(9)

Taking the limit $e\rightarrow 0$, we find

$\frac{1}{r_{h}^{2}}+\frac{\log(r_{h})}{r_{h}^{2}}=0\rightarrow
r_{h}=\text{exp({-1})}\approx 0.368\,.$ (10)

The black hole attains a finite horizon radius as $e\rightarrow 0$, and as
this occurs, $T\rightarrow\infty$.

Figure 1: Left: The horizon radius of the solution as a function of the black
hole mass $m_{t}$. Right: The temperature $T$ of the solution as a function of
the horizon radius.

Entropy: The entropy of the black hole is given by the area of the horizon,
which is found to be quantized

$S=\frac{1}{4G_{5}}\int\mathrm{d}^{3}x\sqrt{\det(g_{ij})}=\frac{2\pi^{2}n}{4\,G_{5}}r_{h}^{3}\,,\quad
n\in\mathbb{N}\,.$ (11)

This is in line with the fact that the energy of this solution gains a quantum
of energy each time the winding number $n$ of the topological Skyrmion
solution is increased. Figure 2 displays the dependence of the entropy on the
temperature for various Skryme couplings. The solutions can be seen to be
multi-valued.

Figure 2:  Entropy density versus Temperature. The entropy density displayed
for various winding numbers as a function of the temperature of the solution.

Free energy: According to the AdS/CFT dictionary the free energy of the dual
$(3+1)$-dimensional field theory may be computed by means of the on-shell
$(4+1)$-dimensional action by $\beta F=S_{ren}$. One may note that the action
in eq. (1) is infinite and so must be renormalized by subtracting a
counterterm. The renormalized action is given by

$S_{ren}=\lim_{\epsilon\rightarrow
0}\left(\int\mathrm{d}^{4}x\left(\int_{r_{h}}^{1/\epsilon}\mathrm{d}r\sqrt{-g}\left(\frac{(R-2\Lambda)}{16\pi
G_{5}}+\frac{\text{Tr}(F^{2})}{16\pi\gamma^{2}}\right)\right)+S_{ct}\right),$
(12)

where $S_{ct}$ is the counter-term action. This action is given as the usual
Gibbons-Hawking-York boundary term needed to make the variational problem well
defined plus additional contributions needed to cancel both $1/\epsilon^{n}$
and $\log(\epsilon)$ divergences Taylor:2000xw ,

$S_{ct}=\frac{1}{8\pi
G_{5}}\int\mathrm{d}^{4}x\sqrt{\gamma}\left(K-\frac{1}{2L}\left(2(1-d)-\frac{L^{2}}{d-2}R(\gamma)\right)\right)+\frac{L}{16\pi\gamma^{2}}\log(\epsilon)\int\mathrm{d}^{4}x\sqrt{\gamma_{0}}\text{Tr}(F_{0}^{2})\,,$
(13)

where $K$ is the trace of the extrinsic curvature, $\gamma$ is the induced
metric on a constant $r=1/\epsilon$ hypersurface, $\gamma_{0}$ is the metric
of the dual field theory and $F_{0}$ is the external field strength of the
gauge field $A$ in the dual theory.

###  2.2 Stability of our topological black hole

Because gravitational interactions are attractive, any ensemble of matter is
unstable against gravitational collapse. Even an ensemble of matter in thermal
equilibrium has gravitational instabilities due to the fact that there is no
way to shield against gravity. A typical gas compressed below a critical
volume will collapse to undergo a phase transition, for example water vapor
condensing to liquid water. Classical thermal gravitational systems also
develop instabilities due to quantum effects. For example, quantum black holes
can nucleate in flat spaces above a critical temperature, making such spaces
unstable Gross:1982cv . However, black hole systems can be made
thermodynamically stable with the addition of special boundary conditions or
when placed in spacetimes with a negative cosmological constant
Hawking:1975vcx ; Gibbons:1976ue ; York:1986yv .

Stability conditions: A system is stable against thermal fluctuations if the
second variation of the entropy with respect to the temperature is negative,
or equivalently if the heat capacity is positive Chamblin:1999hg

$c_{V}=\left(\frac{\partial E}{\partial T}\right)_{V}\geq 0\,.$ (14)

Likewise a system is stable against charge density fluctuations if the second
variation of the entropy with respect to the charge density is negative, or
equivalently if the charge susceptibility, which is defined to be the
variation of the chemical potential with respect to the charge density, is
positive

$\chi_{\rho}=\left(\frac{\partial\mu}{\partial\rho}\right)_{V}\geq 0\,.$ (15)

For the topological black hole to be discussed in section 3.2 with isospin
“charge” the charge susceptibility $\chi_{\rho}$ is a tensor in isospin space,
${(\chi_{\rho})}^{a}_{b}$, defined as

${(\chi_{\rho})}^{a}_{b}=\frac{\partial Q^{a}}{\partial\mu^{b}}\,,$ (16)

where $\rho$ here is a label and $a,b=1,2,3$ are the directions in isospin
space. This then defines a relation between the charge density and chemical
potential $Q^{a}=\braket{J^{a}}={(\chi_{\rho})}^{a}_{b}\mu^{b}$. Using this
definition we can extract the charge susceptibility $(\chi_{\rho})^{a}_{b}$
for which we require all elements to be positive definite for the solution to
be stable against isospin charge fluctuations.

Conserved Charges: The Komar integrals define conserved charges of the
solutions Balasubramanian:1999re . Given a Killing vector $\xi$ and a time-
like unit vector $u^{\mu}$ normal to a space-like hypersurface $\Sigma$ there
is an associated conserved charge

$Q=-\int_{\Sigma}\sqrt{\text{det}(g_{(0)})}\xi_{i}u_{j}\braket{T^{ij}}.$ (17)

For the thermal stability we consider a time-like Killing vector associated
with time translation $\xi^{t}$, whose resulting conserved charge is the mass
of the spacetime

$M=-\int_{\Sigma}\sqrt{\text{det}(g_{(0)})}\xi^{t}_{i}u_{j}\braket{T^{ij}}.$
(18)

To study the thermal stability of the solution given in eq. (7) we first
compute the Komar integral. Inserting the energy momentum tensor as found in
Cartwright:2020yoc  into this expression we find

$M=\frac{3\pi L^{2}n}{32G_{5}}+\frac{3\pi\,m_{t}\,n}{8G_{5}}+\frac{3\pi
n}{8e^{2}G_{5}}\log\left(\Lambda L\right)\,.$ (19)

To compute the heat capacity we consider the following chain rule

$\displaystyle c_{V}$ $\displaystyle=\left(\frac{\partial M}{\partial
m_{t}}\right)\left(\frac{\partial m_{t}}{\partial
r_{h}}\right)\left(\frac{\partial r_{h}}{\partial T}\right)$ (20)
$\displaystyle=\frac{3\pi^{2}nr_{h}^{3}\left(L^{2}\left(2e^{2}r_{h}^{2}+1\right)+4e^{2}r_{h}^{4}\right)}{2G_{5}\left(L^{2}\left(-2e^{2}r_{h}^{2}+3\right)+4e^{2}r_{h}^{4}\right)}\,.$
(21)

It is clear from this expression that the heat capacity would only be negative
if

$L^{2}\left(2e^{2}r_{h}^{2}+3\right)-4e^{2}r_{h}^{4}<0\,,$ (22)

which places a constraint on the horizon radius for a particular value of the
Skyrme coupling. The bound is saturated when

$r_{h}=r_{h}^{*}=\frac{1}{2}\left[L^{2}\pm\frac{1}{e}\sqrt{L^{2}\left(e^{2}L^{2}+12\right)}\right]^{1/2}\,,$
(23)

at this point the heat capacity diverges (as it switches its sign), i.e.
$\lim_{r\rightarrow r_{h}^{*-}}c_{V}\rightarrow\infty$. Otherwise we find that
our solutions have a positive heat capacity ($c_{V}>0$) provided that,
$r_{h}<r_{h}^{*}$. Furthermore there is a particular value implied by eq.
(23); for the mass of the black hole it is given in terms of the Skyrme
coupling and the $AdS$ radius as

$m_{t}=\frac{3e^{2}L^{2}+3\sqrt{e^{2}L^{2}\left(e^{2}L^{2}+12\right)}+4\log\left(\frac{\sqrt{e^{2}L^{2}\left(e^{2}L^{2}+12\right)}}{e^{2}}+L^{2}\right)+14-8\log(2)}{8e^{2}}\,.$
(24)

Figure 3: Heat capacity. Here we display the heat capacity as calculated via
eq. (21) as a function of temperature. The different colors represent
different values of the Skyrme coupling $e$. We display this result for
various winding numbers with the brightest color representing $n=1$ while the
darkest color represents $n=8$.

The charge density stability of the solution in eq. (7) is trivial for this
solution to the Einstein equations. The values of all isospin charge densities
is zero as can be seen in Cartwright:2020yoc  hence $\chi_{\rho}=0$.

###  2.3 Phase transition in our model

The well-known Hawking-Page transition hawking1982  from thermal $AdS$ to an
$AdS$ black hole spacetime occurs in our solution in the limit
$e\rightarrow\infty$. In this limit the critical horizon value is $r_{h}=L$,
which determines the critical temperature $T_{c}=3/(2\pi L)$. By computing the
free energy associated with our topological Skyrme black hole solution and
comparing it with that of thermal $AdS$ we discern the effect the Skyrmion has
on the transition. The free energy difference between our topological $AdS$
black hole and thermal $AdS$ can be calculated from the renormalized action in
eq. (12). This difference is Cartwright:2020yoc 222Note that eq. (3.69) of the
published version of Cartwright:2020yoc  contains a typo. The factor $\beta$
on the right hand side should be removed. In addition the factors of
$\tilde{e}$ should be $e$. However we note that in the choice of units used in
that work $\kappa^{2}=8\pi G=1$, hence in figure 4. $\tilde{e}=e$. Also, the
factor $1/\beta$ in figure 4 of Cartwright:2020yoc  should be removed from the
vertical axis label.

$\Delta
F=F_{\text{BH}}-F_{\text{thermal}}=\frac{\pi^{2}nr_{h}^{2}}{\kappa^{2}}\left(-\frac{e^{2}L^{4}L^{2}\left(\frac{3}{n}-3\right)+4e^{2}r_{h}^{4}+4L^{2}}{4e^{2}L^{2}r_{h}^{2}}+\frac{3\log(r_{h})}{e^{2}r_{h}^{2}}+1\right)\,,$
(25)

where $F_{\text{thermal}}=3\pi^{2}L^{2}/(4\kappa^{2})$ and $\kappa^{2}=8\pi
G_{5}$. If we take the limit $e\rightarrow\infty$, the horizon radius
asymptotes to the horizon radius of the standard global $AdS_{5}$ black hole,
$r_{h}\to r_{h}(e\to\infty)$, and leads to the free energy difference (for
$n=1$),

$\lim_{e\rightarrow\infty}\frac{\kappa^{2}}{\pi^{2}r_{h}^{2}}\Delta
F=\left(1-\frac{r_{h}^{2}}{L^{2}}\right)\,.$ (26)

This restores the standard Hawking-Page transition temperature defined by
$r_{h}=L$. However, if $\infty>e>0$ and we consider e.g. $n=1$, then the
$\text{log}(r_{h})$-term and the $1/(e^{2}r_{h}^{2})$-terms compete with
$r_{h}^{2}/L^{2}$. The free energy difference (25) is displayed in figure 4 as
a function of the horizon radius.

Figure 4:  The free energy difference plotted for winding numbers n = 1, 2, 3.
For each value of the Skyrmion coupling parameter $e$ the free energy
difference vanishes for two values of the horizon radius $r_{h}$. Thus for
each value of $e$ two phase transitions are possible, a transition to a large
black hole or one to a smaller black hole. The origin of the “focal points”,
where the curves for all values of $n$ cross is described in the text. Here,
we chose $e=\sqrt{10}$.

Figure 5:  Free energy difference versus horizon radius. The free energy
plotted for various winding numbers as a function of the horizon radius. In
the top left image the free energy difference intersects the line $\Delta F=0$
again at a larger radius, $r_{h}$, outside the displayed plot range.

These results are generated at a value of the Skyrme coupling $e=\sqrt{10}$.
The lowest (blue) curve is virtually identical to the standard free energy for
a thermal $AdS_{5}$ to global $AdS_{5}$ black hole transition at a critical
temperature fixed by $r_{h}=L$. Increasing the winding number of our solution,
this critical horizon radius increases by a $\Delta r_{h}$ which is a discrete
function of $n$, shifting the Hawking-Page phase transition. This trend
continues for larger $n>2$ as well. Below the transition, the configurations
with larger $n$ have a larger free energy. However, there exists a “focal
point” at a certain $r_{h}$ where all the free energies intersect. That means
there is no energetically favored topology at that horizon radius (or
corresponding temperature). Above that horizon radius, larger winding numbers
are energetically preferred. As can be seen from (25), the contribution from
the thermal $AdS$ free energy, $F_{thermal}$, (the $3/n$-term), which is
multiplied by a factor of $n$, is independent of the winding number $n$. The
remaining terms in the expression are proportional to $n$, which can be
factored out, leaving an expression which is a polynomial in $r_{h}$. The
polynomial vanishes at $r_{h}\,=\,0.113$ and $r_{h}\,=\,0.22$ for
$e\,=\,\sqrt{10}$ and $\kappa\,=\,1$, leaving $F_{thermal}$ as the only
contribution to $\Delta F$. Thus the parameters of the Skyrmion field can be
chosen such that the free energy of the topological black hole is zero for all
winding numbers. In other words, in figure 4 at $r_{h}\,=\,0.113$ and
$r_{h}=1.22$ there is only one contribution to $\Delta F$, which is
independent of $n$, hence the curves for different $n$ intersect at the focal
points at those horizon values.

In figure 5, the difference $\Delta F$ between the free energy corresponding
to the topological black hole and that corresponding to thermal AdS, according
to eq. (25), is shown as a function of the horizon radius $r_{h}$. Similarly,
in figure 6, this quantity is displayed as a function of the black hole
temperature. Just like in figure 4, increasing the temperature, for some
combinations of $n$ and $e$ we observe two transitions: first a confinement
and then a subsequent deconfinement transition. As the free energy of the dual
field theory in figure 5 suggests, at high temperatures corresponding to large
values of $r_{h}$, the free energy difference is negative $\Delta F<0$,
indicating that the deconfined phase (topological black hole) is preferred.
Decreasing the temperature, for any given $n$ we find a confinement transition
to (H - P transition to thermal AdS) at a transition radius corresponding to
the H - P transition temperature. For large values of the Skyrme coupling
(lower right plot in figure 5), most values of $n$ produces only this single
transition analogous to the original confinement transition dual to the H - P
transition Witten:1998zw , with the latter case reproduced exactly at
$e\to\infty$ for $n=1$. For the large but finite value, $e=10$, shown in the
lower right plot of figure 5, the case $n=1$ undergoes a second transition at
a smaller radius $r_{h}$. Since this corresponds to a transition from the
confined to the deconfined phase, we refer to this second H - P transition as
the deconfinement transition. For small values of $e\leq 0.1$, the top right
plot in figure 5 shows that with stronger influence of the Skyrme sector
(smaller $e$) all the windings $n\geq 1$ undergo two transitions, one forming
a large black hole and one forming a small black hole. This is still true for
the top left plot in figure 5, the first (large black hole) transition is
pushed to very large values of $r_{h}$ beyond the plot range. Furthermore,
there exists an intermediate regime around $e\approx 1.$, where no H - P
transition occurs, and the theory remains in its deconfined phase for all
radii as $\Delta F<0\forall T$ at all $n$.
In order to understand this phase transition in field theory terms, we
consider the free energy as a function of temperature 6. Increasing the
temperature from $T=0$, we are first located in the confined phase (thermal
AdS solution). No black hole solution exists there. For any given winding
number, $n$, there is a minimal temperature $T_{\text{min}}$, at which the
black hole solutions start to exist. In the top left panel of figure 6, we
find that for all values of $n\geq 1$ the black hole phase is not preferred at
that $T_{\text{min}}$, because $\Delta F>0$. Increasing the temperature
further, we can follow one of two branches and find it intersects $\Delta F=0$
at a low critical transition temperature, $T_{c,L}$. This is the new
transition facilitated by the coupling to the Skyrmion (dual to the
topological gauge field configuration in the field theory). Following the
other branch, one arrives at a version of the original H - P transition at a
high critical temperature $T_{c,H}$. Note that for the top panel cases
$e=0.01$ and $0.1$, the critical temperatures $T_{c,L/H}$ are virtually
independent from $n$. In contrast to that, the bottom right plot in figure 6
shows that the critical temperatures in the case $e=10$ increase with
increasing $n$.

Figure 6:  Free energy versus temperature. The free energy plotted for various
winding numbers as a function of temperature. In the top panel all curves
intersect the $\Delta F=0$ axis twice, once at a lower critical temperature
$T_{c,L}$ and once at a higher one $T_{c,H}$. For the top left plot this
intersection at $T_{c,H}$ is outside the displayed plot range.

##  3 Holographic interpretation

In this section, we remind the reader of an interpretation of our solution
from section 2 in the holographically dual field theory as given in
Cartwright:2020yoc , where this topic is expounded upon in much further
detail. In summary, the Skyrme coupling and Skyrme field correspond to
$SU(2)$-gauge fields and their coupling to $\mathcal{N}=4$ SYM theory,
respectively.

Skyrmions as merons: In Cartwright:2020yoc  we discussed that one can consider
Skyrmion solutions as being equivalent to specific $SU(2)$ gauge field
configurations known as merons Canfora:2013osa ; Ayon-Beato:2015eca ;
Canfora:2017yio ; Canfora:2018ppu ; Ayon-Beato:2019tvu ; Ipinza:2020xgc ,
classical topological soliton solutions RevModPhys.51.461 . Merons were
originally considered in Euclidean space-time and interpreted as half-
instantons. By definition $U^{-1}\partial U$ is a pure gauge solution, hence
merons with $A=\frac{1}{2}K=\frac{1}{2}U^{-1}\partial U$ are half of pure
gauge solutions. In particular the action of this meron theory is given by

$S_{\text{meron}}=\int d^{5}x\sqrt{-g}\left(\frac{1}{16\pi
G}(R-2\Lambda)+\frac{1}{16\pi\gamma^{2}}\text{Tr}[F_{\mu\nu}F^{\mu\nu}-2m^{2}A^{\mu}A_{\nu}]\right)\,,$
(27)

with the Yang-Mills coupling constant $\gamma$, the cosmological constant
$\Lambda=-6/L^{2}$ with $L$ the $AdS$ radius and the Proca mass $m$
Kunimasa:1967GT ; Shizuya:1975ek . If we identify particular values of the
parameter $\lambda$, the meron and the Skyrmion theory have the same
solutions. The exact mapping is given by,

$A_{\mu}=\lambda K_{\mu}\,,\quad
F_{\mu\nu}=\lambda(\lambda-1)[K_{\mu},K_{\nu}]\,,\quad\frac{m^{2}\lambda^{2}}{\pi\gamma^{2}}=\frac{f_{\pi}^{2}}{2}\,,\quad\frac{\lambda^{2}(\lambda-1)^{2}}{\pi\gamma^{2}}=\frac{1}{2\tilde{e}^{2}}\,,$
(28)

where it is crucially important that $|\lambda|=1/2$ and $f_{\pi}$ is the
coupling constant multiplying the kinetic Skyrmion term in the action eq. (1).
When evaluated at vanishing Proca mass, $m=0$, the meron action (27) coincides
with the Einstein-Yang-Mills action with an $SU(2)$ gauge field, $A_{\mu}$.
This represents a consistent truncation of the bosonic part of minimal gauged
type IIB supergravity in five dimensions. While our analysis does not require
the mass term, such a term can be generated within type IIB supergravity
through spontaneous symmetry breaking in the bulk Klebanov:2002gr . In other
words, the massless meron theory corresponds to $\mathcal{N}=4$ Super-Yang-
Mills theory (SYM) in flat Minkowski space coupled to an external $SU(2)$
gauge field $F$ associated with an $SU(2)$ subgroup of the $SU(4)$ R-symmetry
of the $\mathcal{N}=4$ SYM theory, see e.g. Son:2006em ; Behrndt:1998jd ;
Cvetic:1999ne ; Gubser:1998jb ; Chamblin:1999tk .

In the AdS/CFT correspondence, black branes/holes are generally regarded to be
dual to a state of a strongly coupled quantum many-body system at non-zero
temperature. One may imagine the quark-gluon-plasma generated in heavy ion
collisions, or a strongly-correlated electron system. Meanwhile, the Skyrme
hair of the branes is dual to the gauge field sourcing the $SU(2)$ current
$J^{\mu}$. In the case at hand the dual current operator is restricted in that
it is dual only to merons, pure gauge solutions. In Cartwright:2020yoc  the
topological configurations of the associated $SU(2)$ gauge field, $A_{m}$, on
the field theory side were determined from the near-boundary exponents of the
normalizable and non-normalizable modes of our meron gauge field $A_{\mu}$, in
the gravity bulk given by the solution in eq. (3) and eq. (7). These
topological configurations are characterized by the integer winding number

$n=-i\frac{e^{3}}{12\pi^{2}}\oint\limits_{S^{3}}\mathrm{d}^{3}x\,\hat{n}_{i}\epsilon^{ijkm}\text{tr}(A_{j}A_{k}A_{m})\,.$
(29)

In summary, the two key features of our holographic model are confinement and
the topological winding number $n$. Various systems share these key features
with our model, for example there is a notion of confinement on spin chains
which are also topologically non-trivial Bera:2017 ; Lake:2009 ;
Vestergren:2005in . As we alluded to in section 1, application of our model to
such condensed matter systems provides an exciting prospect for the future.

##  4 Relation of phase transitions in our model to phase transitions in other
physical systems

###  4.1 Bekenstein’s atomic black hole model

In Bekenstein:1995ju ; Bekenstein:1997bt  the spectroscopy of a four-
dimensional, quantum black hole was obtained by treating the area of the
horizon of a black hole as an adiabatic invariant. An analogy was made between
non-extremal quantum black holes and the allowed energy levels in an atom by
assuming that the area of a black hole is quantized. The area of a black hole
increases or decreases only by absorbing or emitting area quanta. In this
approach the area of the horizon is found to be linearly proportional to an
integer $N$, which implies that the mass of the black hole is proportional to
the square root of $N$

$\displaystyle M\,\propto\,\sqrt{N}\,.$ (30)

This proportionality leads to a partition function which is not well-defined
if the integer $N$ has the range $[0,\infty]$. The partition function for this
system is

$\displaystyle
Z[T]=\sum_{N=0}^{\infty}\,k^{N}\,\exp^{-\gamma\,\frac{\sqrt{N}}{T}}\,,$ (31)

where $\gamma\,=\,\sqrt{\alpha/(16\,\pi)}$, and
$\alpha\,=\,4\,\ln(k)\,,\,k\,=\,2,4,...$ A well-defined partition function can
be obtained by putting the system in a box of radius $r$ Stephens:2001sd ,
which restricts the range of $N$ to $[0,N_{max}]$ where

$\displaystyle N_{max}\,=\,\frac{r^{2}}{4\gamma^{2}}\,.$ (32)

The partition function is now

$\displaystyle Z(T)\,=\,\sum_{N=0}^{N_{max}}k^{N}\,e^{-\gamma\sqrt{N}/T}\,,$
(33)

where $k$ is the degeneracy of each state. The average energy and entropy
calculated from this partition function are qualitatively similar to the
corresponding quantities which characterize the nucleation of black holes.

In our model the mass of a five-dimensional, topological black hole is
proportional to the winding number, $n$,

$M(T)=\frac{3\pi L^{2}n}{32G_{5}}+\frac{3\pi\,m_{t}(T)\,n}{8G_{5}}+\frac{3\pi
n}{8e^{2}G_{5}}\log\left(\Lambda L\right)\,,$ (34)

repeating eq. (19) for convenience. The quantization of the mass implies the
quantization of the entropy and the specific heat, just as in Bekenstein’s
black-hole-as-an-atom model. These quantities can be obtained from a partition
function, which can be calculated in analytic form

$\displaystyle Z(T)\,=\,\frac{1}{1-e^{-\alpha(T)/T}}\,,$ (35)

where

$\displaystyle\alpha(T)\,=\,\frac{3\pi
L^{2}}{32G_{5}}+\frac{3\pi\,m_{t}(T)\,}{8G_{5}}+\frac{3\pi}{8e^{2}G_{5}}\log\left(\Lambda
L\right)\,.$ (36)

The average energy and entropy calculated from this partition function have
the forms shown in figure 7.

Figure 7:  Energy and entropy of the holographic model. Calculated via the
partition function eq. (35). Left: The average energy. Right: The entropy.

Qualitatively similar graphs can be obtained from the quantum-black-hole-as-
an-atom model if the radius of the box $r$ in eq.32 increases linearly with
the temperature

$\displaystyle r\,=\,\theta\,(T-T_{0})\,,$ (37)

where $\theta$ is a constant, and $T_{0}$ is the value of the temperature for
which $N_{max}\,=\ 0$.

Although our model of a black hole is quite different from Bekenstein’s atomic
model, the thermodynamical quantities of the two models are very similar. In
both models the mass of the black hole is quantized, leading to quantization
of the entropy and specific heat. Furthermore, there is qualitative agreement
between the two models for the variation with respect to temperature of the
average energy and the average entropy, providing that the containment box in
Bekenstein’s model varies linearly with temperature. In both models the black
hole exists inside of a “box”. In the atomic black hole model the black hole
is placed in a box by fiat, a procedure which is necessary in order to render
the partition function finite. In our model the topological black hole exists
in anti-de Sitter space, which due to the gravitational potential of such a
space acts as a box of finite volume Hawking:1975vcx ; Gibbons:1976ue ;
York:1986yv .

####

The major differences between our model and the atomic black hole model are
the forms of the dependence of the masses on an integer and the forms of the
partition functions. In our model the black hole mass depends linearly on an
integer - the winding number. In the atomic black hole model the black hole
horizon area depends linearly on an integer, which means that the black hole
mass in that model is proportional to the square root of the integer. The
canonical ensemble partition function in the atomic black hole model is a
weighted finite sum over powers of even integers. The sum which defines the
partition function for this model must be finite in order to obtain a well-
defined partition function. The necessity of truncating the usual infinite
series which defines a partition function is the reason the black hole in this
model is placed in a box. In our model the partition function can be expressed
as an infinite series, which can be summed to obtain an analytic expression.

Figure 8:  Average energy: The energy of a black hole in a box in Bekenstein’s
black-hole-as-an-atom model for a temperature-dependent box size. This figure
is qualitatively similar to the average black hole energy obtained for our
model (figure 7).

###  4.2 Kosterlitz - Thouless transitions

In a system in a $2+1$-dimensional space with $O(2)$-symmetry the long-range
order of the system is destroyed when the temperature of the system exceeds a
critical value, resulting in the creation of defects (vortices). For example,
the free energy of a global $O(2)$ scalar field model is Stephens:2001sd

$\displaystyle F\,=\,(\pi\,\eta^{2}-2\,T)\,\ln\left(\frac{L}{a}\right)\,,$
(38)

where $a\,\propto\,\frac{1}{\sqrt{\lambda\,\eta}}$, $\lambda$ is the coupling
strength for a quartic interaction term and $\eta$ is the zero point for this
term. In the two-dimensional X-Y model the free energy of a single vortex can
be estimated from the relation

$\displaystyle F\,=\,E-T\,S\,,$ (39)

where the energy of a vortex is

$\displaystyle E\,=\,E_{0}+\pi n^{2}J\ln\left(\frac{L}{a}\right)\,.$ (40)

In this relation $E_{0}$ is the ground state energy of the system, which is
the state when all of the rotors are completely aligned. In this expression
$n$ is a winding number, $J$ is the coupling strength between neighboring
plane rotors on a two-dimensional square lattice, $L$ is the size of the
system and $a$ is the lattice spacing. The entropy can be estimated by
calculating the number of possible positions of a vortex on a square lattice
with $L^{2}$ squares formed by nearest neighbors

$\displaystyle S\,=\,k_{B}\,\ln\left(\frac{L^{2}}{a^{2}}\right)$ (41)

The resulting free energy is

$\displaystyle
F\,=\,E_{0}+\left(\pi\,J-2\,k_{B}\,T\,\right)\ln\left(\frac{L}{a}\right)\,.$
(42)

This heuristic argument shows that at a finite temperature a phase transition
occurs in which vortex - anti-vortex pairs with energy

$\displaystyle
E_{v,\bar{v}}\,=\,2\,E_{c}+\eta\,J\,\ln\left(\frac{L}{a}\right)$ (43)

unbind, producing defects in the system. This is the Kosterlitz - Thouless
transition.

####

In our model the Hawking - Page phase transition is described by eq.25, which
contains a logarithmic term whose argument depends upon the size of the
topological black hole. The free energy expression in the K - T transition,
eq.43, also contains a size-dependent logarithmic term. Another parallel
between the H - P transition in our model and the K-T transition is that
temperature at which the transitions occur depends upon the coupling strength
($r_{h}$ in our model is temperature dependent and coupling strength
dependent).

####

In both models the energy of the primary constituent of the system, a single
vortex in the X - Y model, a black hole in our model, depend on a winding
number. However, in the X - Y model the energy is proportional to $n^{2}$, but
in our model the energy (mass) of the black hole is linearly proportional $n$.
Another difference between the K - T transition and the transition described
in our model is that the number of H - P transitions which can occur depends
on the coupling strength. As shown in Section 4, the number of H - P phase
transitions can be 0, 1, or 2. For the latter case, it is possible to form
either a large black hole (large horizon radius) or a small black hole at a
given temperature.

###  4.3 Hagedorn transitions

The Hagedorn model of strong interactions at high energies is a statistical -
thermodynamical model which treats resonances of strongly interacting
particles as “fireballs”, which are composed of smaller “fireballs”, which are
composed of still smaller “fireballs”, and so on Hagedorn:1965st . This
approach results in an asymptotic bootstrap equation for the density of
states. The solution of this equation predicts that the mass spectrum grows
exponentially

$\displaystyle\rho(m)\,\stackrel{{\scriptstyle
m\to\infty}}{{\longrightarrow}}\,C\,m^{-5/2}\,\exp\left(\frac{m}{T_{H}}\right)$
(44)

where $C$ is a constant and $T_{H}$ is the highest possible temperature for
strong interactions. Above this temperature the partition function diverges.
The physical interpretation of this effect is that at $T_{H}$ a phase
transition occurs in which quarks become deconfined. This is the Hagedorn
transition. The expression for the density of states was generalized in
Frautschi:1971ij

$\displaystyle\rho(m)\,\stackrel{{\scriptstyle
m\to\infty}}{{\longrightarrow}}\,C\,f(m)\,\exp\left(\frac{m^{p}}{T_{H}}\right)$
(45)

where $f(m)$ is a polynomial in $m$. A gas of superstrings has a density of
states with same general form as the one in eq.45 Harms:1992nb  with
$p\,=\,2$. The Hagedorn transition in $SU(N)$ gauge theory for
$N\,\to\,\infty$ is connected to the string theory description of black hole
formation in anti-de Sitter $(AdS)$ space Aharony:1999ti . The deconfinement
phase transition of a supersymmetric gauge theory on the $AdS$ boundary is
dual to the Hawking - Page transition during which a black hole is formed in
the bulk Witten:1998zw . The topological black hole in our model is dual to
$\mathcal{N}=4$ Super-Yang-Mills theory (SYM) in Minkowski space coupled to an
external $SU(2)$ gauge field $F$ associated with an $SU(2)$ subgroup of the
$SU(4)$ R-symmetry of the $\mathcal{N}=4$ SYM theory, see e.g. Son:2006em ;
Behrndt:1998jd ; Cvetic:1999ne ; Gubser:1998jb ; Chamblin:1999tk . The dual
state is a strongly coupled quantum many-body system at non-zero temperature.
Thus the dual to our topological black hole is an example of a system which
can undergo a Hagedorn transition. However, in our model two Hawking - Page
transitions are possible, depending on the strength of the Skyrme coupling
constant. This means that there are two states of deconfinement in the dual
conformal field theory, a lower energy state and a higher energy state. In
quantum chromodynamics the deconfined state is usually considered to be a
‘liquid’ composed of free or nearly free quarks. The interpretation of a
second state of deconfinement has not yet been worked out.

##  5 Discussion

The topological black hole in our model possesses several interesting
features. The mass of the black hole and therefore its energy and its entropy
are quantized. The quantization integer is the winding number, which
parameterizes the solutions of the Einstein-Skyrme field equations. The Skyrme
field possessed by the black hole determines the number of Hawking - Page (H -
P) transitions which the system can undergo. The number of transitions can be
zero, one or two, depending on the strength of the Skyrme field coupling
constant and on the winding number. The fact that two H - P phase transitions
at different temperatures can occur presents an interesting challenge to the
holographic interpretation of these transitions. An H - P transition in the
bulk is usually interpreted as the (de-)confinement transition of a dual,
strongly-interacting system on the $AdS$ boundary. Now the Skyrmion
(holographically dual to the $SU(2)$ gauge sector on the field theory side)
leads to the existence of a second transition at particular values of the
Skyrme coupling $e$ and winding numbers $n$ of the $SU(2)$ field
configurations. This raises the obvious question: What type of (topological)
physical principle in the dual field theory leads to the second phase
transition? It will be interesting to translate the observed phenomenology of
our model into a more concise condensed matter context.

## Acknowledgements

This research was supported in part by DOE grant DE-SC-0012447.

## References

  * (1) C. Cartwright, B. Harms and M. Kaminski, _Topological or Rotational Non-Abelian Gauge Fields from Einstein-Skyrme Holography_ , _JHEP_ 03 (2021) 229 [2010.03578].
  * (2) J. M. Maldacena, _The Large N limit of superconformal field theories and supergravity_ , _Int. J. Theor. Phys._ 38 (1999) 1113 [hep-th/9711200].
  * (3) E. Witten, _Anti-de Sitter space, thermal phase transition, and confinement in gauge theories_ , _Adv. Theor. Math. Phys._ 2 (1998) 505 [hep-th/9803131].
  * (4) F. D. M. Haldane, _Ground state properties of antiferromagnetic chains with unrestricted spin: Integer spin chains as realisations of the o(3) non-linear sigma model_ , 2016.
  * (5) X.-G. Wen, _Quantum orders and symmetric spin liquids_ , _Phys. Rev. B_ 65 (2002) 165113.
  * (6) T. Senthil, A. Vishwanath, L. Balents, S. Sachdev and M. P. A. Fisher, _Deconfined quantum critical points_ , _Science_ 303 (2004) 1490 [https://www.science.org/doi/pdf/10.1126/science.1091806].
  * (7) D. Kharzeev, _Parity violation in hot qcd: Why it can happen, and how to look for it_ , _Phys. Lett. B_ 633 (2006) 260 [hep-ph/0406125].
  * (8) J. K. Ghosh, S. Grieninger, K. Landsteiner and S. Morales-Tejera, _Is the chiral magnetic effect fast enough?_ , _Phys. Rev. D_ 104 (2021) 046009 [2105.05855].
  * (9) C. Cartwright, M. Kaminski and B. Schenke, _Energy dependence of the chiral magnetic effect in expanding holographic plasma_ , 2112.13857.
  * (10) C. Cartwright, _Entropy production far from equilibrium in a chiral charged plasma in the presence of external electromagnetic fields_ , 2003.04325.
  * (11) M. Taylor, _More on counterterms in the gravitational action and anomalies_ , hep-th/0002125.
  * (12) D. J. Gross, M. J. Perry and L. G. Yaffe, _Instability of flat space at finite temperature_ , _Phys. Rev. D_ 25 (1982) 330.
  * (13) S. W. Hawking, _Particle creation by black holes_ , _Commun. Math. Phys._ 43 (1975) 199.
  * (14) G. W. Gibbons and S. W. Hawking, _Action integrals and partition functions in quantum gravity_ , _Phys. Rev. D_ 15 (1977) 2752.
  * (15) J. W. York, _Black-hole thermodynamics and the euclidean einstein action_ , _Phys. Rev. D_ 33 (1986) 2092.
  * (16) A. Chamblin, R. Emparan, C. V. Johnson and R. C. Myers, _Holography, thermodynamics and fluctuations of charged AdS black holes_ , _Phys. Rev. D_ 60 (1999) 104026 [hep-th/9904197].
  * (17) V. Balasubramanian and P. Kraus, _A stress tensor for anti-de sitter gravity_ , _Commun. Math. Phys._ 208 (1999) 413 [hep-th/9902121].
  * (18) S. W. Hawking and D. N. Page, _Thermodynamics of black holes in anti-de sitter space_ , _Comm. Math. Phys._ 87 (1982) 577.
  * (19) F. Canfora and H. Maeda, _Hedgehog ansatz and its generalization for self-gravitating Skyrmions_ , _Phys. Rev._ D87 (2013) 084049 [1302.3232].
  * (20) E. Ayon-Beato, F. Canfora and J. Zanelli, _Analytic self-gravitating Skyrmions, cosmological bounces and AdS wormholes_ , _Phys. Lett. B_ 752 (2016) 201 [1509.02659].
  * (21) F. Canfora, S. H. Oh and P. Salgado-Rebolledo, _Gravitational catalysis of merons in einstein-yang-mills theory_ , _Phys. Rev. D_ 96 (2017) 084038 [1710.00133].
  * (22) F. Canfora, A. Gomberoff, S. H. Oh, F. Rojas and P. Salgado-Rebolledo, _Meronic einstein-yang-mills black hole in 5d and gravitational spin from isospin effect_ , _JHEP_ 06 (2019) 081 [1812.11231].
  * (23) E. Ayón-Beato, F. Canfora, M. Lagos, J. Oliva and A. Vera, _Analytic self-gravitating $4$-baryons, traversable nut-ads wormholes, flat space-time multi-skyrmions at finite volume and a novel transition in the $su(3)$-skyrme model_, _Eur. Phys. J. C_ 80 (2020) 384 [1909.00540].
  * (24) M. Ipinza and P. Salgado-Rebolledo, _Meron-like topological solitons in massive Yang-Mills theory and the Skyrme model_ , 2005.04920.
  * (25) A. Actor, _Classical solutions of $\mathrm{SU}(2)$ yang—mills theories_, _Rev. Mod. Phys._ 51 (1979) 461.
  * (26) T. Kunimasa and T. Gotō, _Generalization of the Stueckelberg Formalism to the Massive Yang-Mills Field_ , _Progress of Theoretical Physics_ 37 (1967) 452.
  * (27) K.-i. Shizuya, _Quantization of the Massive Yang-Mills Field in Arbitrary Gauges_ , _Nucl. Phys. B_ 94 (1975) 260.
  * (28) I. R. Klebanov, P. Ouyang and E. Witten, _A Gravity dual of the chiral anomaly_ , _Phys. Rev. D_ 65 (2002) 105007 [hep-th/0202056].
  * (29) D. T. Son and A. O. Starinets, _Hydrodynamics of r-charged black holes_ , _JHEP_ 03 (2006) 052 [hep-th/0601157].
  * (30) K. Behrndt, M. Cvetic and W. Sabra, _Nonextreme black holes of five-dimensional N=2 AdS supergravity_ , _Nucl. Phys. B_ 553 (1999) 317 [hep-th/9810227].
  * (31) M. Cvetic and S. S. Gubser, _Phases of r charged black holes, spinning branes and strongly coupled gauge theories_ , _JHEP_ 04 (1999) 024 [hep-th/9902195].
  * (32) S. S. Gubser, _Thermodynamics of spinning D3-branes_ , _Nucl. Phys. B_ 551 (1999) 667 [hep-th/9810225].
  * (33) A. Chamblin, R. Emparan, C. V. Johnson and R. C. Myers, _Charged AdS black holes and catastrophic holography_ , _Phys. Rev. D_ 60 (1999) 064018 [hep-th/9902170].
  * (34) A. K. Bera, B. Lake, F. H. L. Essler, L. Vanderstraeten, C. Hubig, U. Schollwöck et al., _Spinon confinement in a quasi-one-dimensional anisotropic heisenberg magnet_ , _Phys. Rev. B_ 96 (2017) 054423.
  * (35) B. Lake, A. M. Tsvelik, S. Notbohm, D. Alan Tennant, T. G. Perring, M. Reehuis et al., _Confinement of fractional quantum number particles in a condensed-matter system_ , _Nature Physics_ 6 (2009) 50–55.
  * (36) A. Vestergren and J. Lidmar, _Topological order and the deconfinement transition in the (2+1) dimensional compact abelian higgs model_ , _Phys. Rev. B_ 72 (2005) 174515 [cond-mat/0502533].
  * (37) J. D. Bekenstein and V. F. Mukhanov, _Spectroscopy of the quantum black hole_ , _Phys. Lett. B_ 360 (1995) 7 [gr-qc/9505012].
  * (38) J. D. Bekenstein, _Quantum black holes as atoms_ , in _8th Marcel Grossmann Meeting on Recent Developments in Theoretical and Experimental General Relativity, Gravitation and Relativistic Field Theories (MG 8)_ , pp. 92–111, 6, 1997, gr-qc/9710076.
  * (39) G. J. Stephens and B. L. Hu, _Notes on black hole phase transitions_ , _Int. J. Theor. Phys._ 40 (2001) 2183 [gr-qc/0102052].
  * (40) R. Hagedorn, _Statistical thermodynamics of strong interactions at high-energies_ , _Nuovo Cim. Suppl._ 3 (1965) 147.
  * (41) S. C. Frautschi, _Statistical bootstrap model of hadrons_ , _Phys. Rev. D_ 3 (1971) 2821.
  * (42) B. Harms and Y. Leblanc, _Statistical mechanics of black holes_ , _Phys. Rev. D_ 46 (1992) 2334 [hep-th/9205021].
  * (43) O. Aharony, S. S. Gubser, J. M. Maldacena, H. Ooguri and Y. Oz, _Large N field theories, string theory and gravity_ , _Phys. Rept._ 323 (2000) 183 [hep-th/9905111].