# Negative Barnett effect, negative moment of inertia of (quark-)gluon plasma
and thermal evaporation of chromomagnetic condensate

Victor V. Braguta<EMAIL_ADDRESS>Bogoliubov Laboratory of
Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna,
Russia  Moscow Institute of Physics and Technology, 141700 Dolgoprudny, Russia
Maxim N. Chernodub<EMAIL_ADDRESS>Institut Denis Poisson UMR
7013, Université de Tours, 37200 Tours, France  Nordita, Stockholm University,
Roslagstullsbacken 23, SE-106 91 Stockholm, Sweden  Ilya E. Kudrov
<EMAIL_ADDRESS>Institute for High Energy Physics, NRC “Kurchatov
Institute”, 142281 Protvino, Russia  Artem A. Roenko<EMAIL_ADDRESS>Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear
Research, 141980 Dubna, Russia  Dmitrii A. Sychev<EMAIL_ADDRESS>Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear
Research, 141980 Dubna, Russia  Moscow Institute of Physics and Technology,
141700 Dolgoprudny, Russia

###### Abstract

We discuss the negativity of the moment of inertia of (quark-)gluon plasma in
a window of “supervortical” range of temperatures above the deconfining phase
transition, $T\simeq(1\dots 1.5)T_{c}$ found recently in numerical Monte Carlo
simulations by two independent methods. In our work, we confirm numerically
that the origin of this effect is rooted in the thermal evaporation of the
non-perturbative chromomagnetic condensate. We argue that the negative moment
of inertia of gluon plasma indicates the presence of a novel effect, the
negative spin-vortical coupling for gluons resulting in a negative gluonic
Barnett effect: the spin polarization of gluons exceeds the total angular
momentum of rotating plasma thus forcing the orbital angular momentum to take
negative values in the supervortical range of temperatures.

## Introduction.

The moment of inertia $I$ of a physical body quantifies the angular momentum
$\boldsymbol{L}$ carried by the body when it is set in rotation with an
angular velocity $\boldsymbol{\Omega}$ Landau and Lifshitz (1976a). For slowly
rotating rigid mechanical systems, these quantities are related to each other
via the linear relation, ${\boldsymbol{L}}={\hat{I}}{\boldsymbol{\Omega}}$,
where the moment of inertia takes a form of the tensor of the second rank,
$\hat{I}$. If the body rotates around one of its principal axis of inertia,
the above relation simplifies to:

$\displaystyle L=I\Omega,$ (1)

where $I$ is the corresponding eigenvalue of the tensor $\hat{I}$.

In classical systems residing in thermal equilibrium, the moment of inertia is
always a positive quantity, $I>0$. This intuitively clear statement originates
from the fact that a system of physical particles has an everywhere-positive
energy density, implying that the momentum of any small subsystem of a rigidly
rotating body points out in the direction of its velocity, expressed, in turn,
via the angular velocity $\boldsymbol{\Omega}$. As all physical objects we
know have a positive energy density, our intuition tells us that they must
have also a positive moment of inertia.111An academic counterexample to this
assertion is provided by the Casimir effect Casimir and Polder (1948), where
the moment of inertia associated with the negative Casimir energy is a
negative quantity Chernodub (2012, 2013) in consistency with the equivalence
principle established by the gravitational response of the Casimir energy in
the background gravitational field Fulling  _et al._ (2007); Milton  _et al._
(2007). We do not consider the Casimir effect since it does not play a role in
the effect discussed in our article.

However, the numerical simulations performed by two different methods both for
static Braguta  _et al._ (2023a) and rotating Braguta  _et al._ (2023b) gluon
plasmas show that the assertion of the positivity of the mechanical moment of
inertia cannot be applied to plasma of gluons. On the contrary, the gluon
plasma possesses a negative moment of inertia in a wide window of temperatures
above the deconfinement transition Braguta  _et al._ (2023a, b). This
conclusion drastically contradicts our intuition since (quark-)gluon plasma
has a positive energy density in every point while possessing at the same time
a negative moment of inertia. According to Ref. Braguta  _et al._ (2023b), the
origin of the negative moment of inertia lies in the particularities of the
evaporation of the chromomagnetic gluon condensate at high temperatures.

A signature of the opposite direction of the angular momentum of the QCD
vacuum with respect to the angular velocity, which would be consistent with
$I<0$ in Eq. (1), has also been noticed in non-renormalized data of Ref.
Yamamoto and Hirono (2013) about a decade ago. In that work, this property has
been attributed to a classical frame-dependence effect as a particle that
rests in the laboratory seems oppositely rotating from the point of view of
the rotating frame. This observation – made for the QCD vacuum which is
characterized by the gluon and chiral condensates Shifman  _et al._ (1979a, b)
– lies in the line of our earlier results Braguta  _et al._ (2023a, b) that
highlight the importance of the chromomagnetic condensate for rotation in
(quark-)gluon plasma.

In our work, we study the contribution of the gluon condensate to the moment
of inertia of gluon plasma, pointing out to the important role of the
chromomagnetic component as compared to the ordinary mechanical term.

## Rotation of quark-gluon plasma.

Rotating quark-gluon plasma (QGP) is routinely produced in relativistic heavy-
ion collisions et al. (2007); Adamczyk  _et al._ (2017). The appropriate
quantity in this case is vorticity Florkowski  _et al._ (2019); Huang  _et
al._ (2021); Becattini (2022) which describes the local circular motion of
plasma constituents at a given point of spacetime. While simulations show in
vortical plasma, vorticity does not generally correspond to a rigid rotation
Huang  _et al._ (2021), the rigid rotation is a useful probe of the response
of the quark-gluon plasma to vorticity in most analytical Ambruş and
Winstanley (2014); Chen  _et al._ (2016); Chernodub and Gongyo (2017); Wang
_et al._ (2019); Zhang  _et al._ (2020); Sadooghi  _et al._ (2021); Fujimoto
_et al._ (2021); Golubtsova  _et al._ (2022); Chen  _et al._ (2022); Zhao  _et
al._ (2023); Chernodub (2021); Mameda and Takizawa (2023); Sun  _et al._
(2023); Satapathy (2023); Braga and Junqueira (2023) and numerical Yamamoto
and Hirono (2013); Braguta  _et al._ (2020, 2021, 2023b); Yang and Huang
(2023) approaches to its thermodynamic properties. Our knowledge of this
system can be tested by confronting the analytical results with numerical
simulations for a rigid rotation. We follow the same strategy in our paper.

## Decomposing gluonic moment of inertia.

Our experience in classical mechanics tells us that we do not need to set a
physical system in motion for the purpose of evaluating its moment of inertia.
In other words, the moment of inertia can be calculated also in the static
limit, when the object does not rotate at all. This observation simplifies the
calculation of the moment of inertia since it does not require introduction of
the curved space with a nontrivial metric (the latter method has been actively
used, for example, in Refs. Yamamoto and Hirono (2013); Braguta  _et al._
(2020, 2021, 2023b); Yang and Huang (2023)). In our paper, we analyze the
moment of inertia of (quark-)gluon plasma using standard, non-rotating
lattices for which the model can conveniently be formulated in a flat
Euclidean spacetime following the strategy of Ref. Braguta  _et al._ (2023a).

The moment of inertia $I\equiv I_{\boldsymbol{n}}$ of a physical system with
respect to global rotations with the angular velocity
${\boldsymbol{\Omega}}=\Omega{\boldsymbol{n}}$ around the principal axis of
inertia $\boldsymbol{n}$ (with ${\boldsymbol{n}}^{2}=1$) is determined via its
free energy $F$ in the co-rotating reference frame as follows Landau and
Lifshitz (1976b):

$\displaystyle I(T)=\frac{J(T,\Omega)}{\Omega}{\biggl{|}}_{\Omega\to
0}\equiv-\frac{1}{\Omega}{\left(\frac{\partial
F(T,\Omega)}{\partial\Omega}\right)}_{T}{\biggl{|}}_{\Omega\to 0}\,.$ (2)

In this expression, we implicitly implied that the size of the body in the
direction perpendicular to the axis is bounded,
$R_{\perp}<R_{\mathrm{light}}$, by the radius of light cylinder,
$R_{\mathrm{light}}=1/|\Omega|$, so that the causality is not violated.
Equation (2) also allows us to work in the thermodynamic limit
$R_{\perp}\to\infty$, provided it is taken with a proper order of limits,
$\lim_{R_{\perp}\to\infty}\lim_{\Omega\to 0}$, or taking
$R_{\perp}=C_{0}/\Omega$ with the real-valued parameter $0<C_{0}<1$.

The moment of inertia (2) of the gluon plasma can be decomposed into two parts
Braguta  _et al._ (2023b):

$\displaystyle
I^{\mathrm{gl}}=I^{\mathrm{gl}}_{\text{mech}}+I^{\mathrm{gl}}_{\text{magn}}\,.$
(3)

This result can be readily obtained from Eq. (2) using the free energy
$F=-T\ln\int DAe^{iS}$ of the rotating gluon gas described by Yang-Mills
theory with the action

$\displaystyle S=-\frac{1}{2g^{2}}\int
d^{4}x\,\sqrt{-g}\,g^{\mu\nu}g^{\alpha\beta}F_{\mu\alpha}^{a}F_{\nu\beta}^{a}\,,$
(4)

where the rotation (taken to be around the $z$-axis) is defined by the
following metric tensor:

$\displaystyle g_{\mu\nu}=\begin{pmatrix}1-r_{\perp}^{2}\Omega^{2}&\Omega
y&-\Omega x&0\\\ \Omega y&-1&0&0\\\ -\Omega x&0&-1&0\\\
0&0&0&-1\end{pmatrix}\,,$ (5)

with $r_{\perp}=\sqrt{x^{2}+y^{2}}$.

The first term in the right-hand side of Eq. (3):

$\displaystyle
I^{\mathrm{gl}}_{\text{mech}}=\frac{1}{T}\left\llangle\bigl{(}{\boldsymbol{n}}\cdot{\boldsymbol{J}}^{\mathrm{gl}}\bigr{)}^{2}\right\rrangle_{T},\quad$
(6)

is the mechanical contribution which accounts for the fluctuations of the
total angular momentum of gluons,

$\displaystyle
J_{i}^{\mathrm{gl}}=\frac{1}{2}\int_{V}d^{3}x\,\epsilon_{ijk}M^{jk}_{\mathrm{gl}}({\boldsymbol{x}})\,,\qquad
i,j=1,2,3\,,$ (7)

projected on the axis of rotation $\boldsymbol{n}$. The local angular momentum
of gluons,

$\displaystyle
M^{ij}_{\mathrm{gl}}({\boldsymbol{x}})=x^{i}T^{j0}_{\mathrm{gl}}({\boldsymbol{x}})-x^{j}T^{i0}_{\mathrm{gl}}({\boldsymbol{x}})\,,$
(8)

is expressed through the gluonic stress-energy tensor:
$T^{\mu\nu}_{\mathrm{gl}}=G^{a,\mu\alpha}G^{a,\nu}_{\quad\,\alpha}-(1/4)\eta^{\mu\nu}G^{a,\alpha\beta}G^{a}_{\alpha\beta}$
with the metric $\eta_{\mu\nu}={\text{diag}}(+1,-1,-1,-1)$ of flat Minkowski
spacetime. Here, we use notations

$\displaystyle\left\llangle{\mathcal{O}}\right\rrangle_{T}=\left\langle{\mathcal{O}}\right\rangle_{T}-\left\langle{\mathcal{O}}\right\rangle_{T=0}\,,$
(9)

to represent the thermal part of the expectation value of an operator
$\mathcal{O}$. We also notice that
$\langle{\boldsymbol{n}}\cdot{\boldsymbol{J}}\rangle\equiv 0$ at
${\boldsymbol{\Omega}}=0$.

In thermodynamic sense, the moment of inertia of a system is associated with
its susceptibility with respect to global rotations. For example, Eq. (6) has
a form of a susceptibility, which estimates how the system is susceptible to
rotational motion.

Thermodynamic susceptibilities are associated with the second-order
derivatives of the free energy with respect to the corresponding conjugate
variables (for example, with respect to a chemical potential if we study
density fluctuations). Since rotation is defined by the shift of the
Hamiltonian, $H_{0}\to H=H_{0}-{\boldsymbol{\Omega}}\cdot{\boldsymbol{J}}$,
the second-order derivative of the corresponding free energy222Equation (2)
can be rewritten as a second derivative since $\lim\nolimits_{x\to
0}f^{\prime}(x)/x=f^{\prime\prime}(0)$ for any function $f(x)$, continuous in
the origin and possessing a vanishing first derivative, $f^{\prime}(0)=0$.
$F=-T\ln{\mathrm{Tr}}\,e^{-H/T}$ with respect to the angular frequency
$\Omega$, gives us susceptibility (6) of the angular momentum
${\boldsymbol{J}}$ provided the latter does not depend on the angular velocity
${\boldsymbol{\Omega}}$. The particularity of a vector field, such as the
gluon field considered in our article, is that the angular momentum does
depend on the angular velocity ${\boldsymbol{\Omega}}$. This property leads to
the second term in Eq. (2).

The second contribution to the total momentum (2) is given by the fluctuation
of chromomagnetic field,

$\displaystyle
I^{\mathrm{gl}}_{\text{magn}}=\int_{V}d^{3}x\Bigl{[}\left\llangle({\boldsymbol{B}}^{a}\cdot{\boldsymbol{x}}_{\perp})^{2}\right\rrangle_{T}+\left\llangle({\boldsymbol{B}}^{a}\cdot{\boldsymbol{n}})^{2}\right\rrangle_{T}{\boldsymbol{x}}_{\perp}^{2}\Bigr{]}\,,$
(10)

where $B^{a}_{i}=\frac{1}{2}\epsilon^{ijk}G_{jk}^{a}$ is the chromomagnetic
field and
${\boldsymbol{x}}_{\perp}={\boldsymbol{x}}-{\boldsymbol{n}}({\boldsymbol{n}}\cdot{\boldsymbol{x}})$
is the vector between the point $\boldsymbol{x}$ and the axis of rotation set
by the unit vector ${\boldsymbol{n}}^{2}=1$. The normalization of Eq. (2) is
chosen in such a way that cold ($T=0$) vacuum has no moment of inertia with
both mechanical (6) and magnetic (10) contributions vanishing.333Therefore, a
proper renormalization of the data of Ref. Yamamoto and Hirono (2013) would
give us a zero moment of inertia for the QCD vacuum at zero temperature as
nothing (i.e., vacuum) cannot move or rotate.

While the mechanical correlator term (6) always gives a positive contribution
to the moment of inertia, the local magnetic term (10) is proportional to the
magnetic gluonic condensate which does change the sign above the deconfinement
phase transition Boyd  _et al._ (1996). It is the last term which is
responsible for the negative moment of inertia of the gluon plasma Braguta
_et al._ (2023a, b).

Both terms in Eq. (2) are calculated in the static limit,
${\boldsymbol{\Omega}}\equiv 0$, which possesses the $\mathrm{SO}(3)$
rotational symmetry. For the magnetic term (10), the space isotropy implies

$\displaystyle\left\llangle
B^{a}_{i}B^{a}_{j}\right\rrangle_{T}=\frac{1}{3}\delta_{ij}\left\llangle({\boldsymbol{B}}^{a})^{2}\right\rrangle_{T}\,,$
(11)

thus allowing us to link the magnetic contribution (10) to the magnetic
component of the gluon condensate:

$\displaystyle\hskip-2.84526ptI^{\mathrm{gl}}_{\text{magn}}=\frac{2}{3}\int_{V}d^{3}x\,x_{\perp}^{2}\left\llangle({\boldsymbol{B}}^{a})^{2}\right\rrangle_{T}$
(12)

Surprisingly, the chromomagnetic contribution (12) has the same form as the
moment of inertia of a classical body with the mass density
$\rho({\boldsymbol{x}})$:

$\displaystyle
I_{\mathrm{class}}=\int_{V}d^{3}x\,\rho({\boldsymbol{x}})\,{\boldsymbol{x}}_{\perp}^{2}\,.$
(13)

Indeed, one recovers the non-perturbative quantum contribution of the gluon
condensate (12) from the classical expression (13) with the use the formal
substitution
$\rho(T)\to\frac{2}{3}\left\llangle({\boldsymbol{B}}^{a})^{2}\right\rrangle_{T}$.
While this relation might seem natural, we will see below that
$\left\llangle({\boldsymbol{B}}^{a})^{2}\right\rrangle_{T}<0$ in a substantial
range of temperatures above $T_{c}$, which, in the classical formula (13),
would correspond to an unphysical negative mass density,
$\rho({\boldsymbol{x}})<0$. Of course, it is well known that the gluon plasma
does not have a negative mass (energy) density Boyd  _et al._ (1996) and
still, we show Braguta  _et al._ (2023a, b) the moment of inertia of this
plasma becomes a negative quantity in a range of temperatures. We will discuss
this would-be paradox below.

## Thermodynamic and continuum limits.

Before discussing the results of our numerical simulations, we would like to
comment on their reliability to describe physics in the continuum limit.

Given the extensivity of the moment of inertia $I$ and, consequently, its
dependence on the shape of the system, it is convenient to consider an
associated non-extensive quantity $K_{2}=-I/(F_{0}R_{\perp}^{2})$, which
represents a dimensionless moment of inertia. This expression, written for a
cylinder-shaped object, is normalized by its size (squared) in transverse
dimensions $R_{\perp}$ as well as by the free energy in the nonrotating limit,
$F_{0}=\lim_{\Omega\to 0}F<0$. For a non-interacting system of massless
particles, one gets the exact result: $K_{2}=2$ Ambruş and Chernodub (2023).
This result is also recovered, within the numerical accuracy, in high-
temperature limit of quark-gluon plasma Braguta  _et al._ (2023b), which shows
the consistency of our understanding of the numerical data.

The independence of the dimensionless moment of inertia $K_{2}$ on the spatial
volume of the system (with various spatial sizes of the lattice, $N_{s}$),
established in Ref. Braguta  _et al._ (2023b), implies that our results are
very close to the thermodynamic limit. In addition, the excellent scaling
towards the continuum limit, taken as a sequence of various $N_{t}$, has been
established Braguta  _et al._ (2023a, b). Given these observation, we choose
below to work only with a single lattice spacing (corresponding to the
temporal extension $N_{t}=6$) and tree-level improved Symanzik lattice gauge
action, which is adequately close to continuum physics.

The moment of inertia is an extensive quantity which behaves as $I\propto
VR_{\perp}^{2}$, where $V$ is the volume of the system and $R_{\perp}$ is the
size of the system with respect to the axis of rotation. Since we work on
square lattices with the transverse extension $R_{\perp}=L_{s}/2$, we will
present the results of the specific moment of inertia defined as

$\displaystyle{\mathsf{i}}_{2}=\frac{I}{VR_{\perp}^{2}}\,,\qquad
R_{\perp}=\frac{L_{s}}{2}\,.$ (14)

Here the specific (size-independent) notations are introduced as
${\mathsf{i}}_{n}=I_{n}/(VR_{\perp}^{n})$, with $n=2,4,\dots$, in order to
highlight the size-dependence of the expansion of the free energy $F$, defined
in the co-rotating reference frame, over the angular frequencies $\Omega$:

$\displaystyle F(T,R_{\perp},\Omega)$
$\displaystyle=F_{0}(T,R_{\perp})-V\sum_{k=1}^{\infty}\frac{{\mathsf{i}}_{2k}(T)}{(2k)!}R_{\perp}^{2k}\Omega^{2k}$
$\displaystyle\equiv F_{0}-\frac{I}{2}\Omega^{2}+O(\Omega^{4})\,,$ (15)

(see, e.g., Ref. Ambruş and Chernodub (2023)). The usual moment of inertia,
$I_{2}\equiv I$, corresponding to the static limit $\Omega\to 0$, appears in
the second term in the series in the second line of Eq. (15). It is expressed
via the specific moment of inertia ${\mathsf{i}}_{2}$ defined in Eq. (14). The
higher-order terms in Eq. (15) correspond to the nonlinear contributions to
the moment of inertia due to deformation of the system caused by rotation. We
will not consider them in this article. For a rotating system of free bosons,
these terms were computed analytically in Ref. Ambruş and Chernodub (2023).

## Monte Carlo results for the specific moment of inertia

of SU(3) gluon plasma (quenched QCD) are presented in Fig. 1. These data were
obtained on the finite temperature lattice $6\times 24\times 31^{2}$ with the
zero-temperature subtraction performed at $N_{t}=24$, while other details of
the lattice simulations may be found in Ref. Braguta  _et al._ (2023a). We
plot separately the mechanical term (6), the magnetic term (10), and the total
specific moment of inertia (3):

$\displaystyle{\mathsf{i}}_{2}={\mathsf{i}}_{2}^{\mathsf{mech}}+{\mathsf{i}}_{2}^{\mathsf{magn}}.$
(16)

How should we understand these results?

Figure 1: The mechanical (6) and chromomagnetic (10) contribution to the total
moment of inertia (16) along with the fits (25) and (26), respectively (given
by the solid lines). The dashed lines are plotted to guide the eye.

Deeply in the confinement (hadronic) phase, the moment of inertia of the
system is zero. This property has a simple explanation since at temperature
substantially lower than $T_{c}$, there is no gluonic matter that can be set
in motion: gluons are confined into glueballs which are too heavy to be
excited thermally. Indeed, intuitively, we cannot move or rotate nothing
(i.e., vacuum). The vanishing of the moment of inertia is well seen in Fig. 1
for the temperature point $T\simeq 0.70T_{c}$. This result also demonstrates
the correctness of the additive normalization procedure (9) used in our
calculations.

As soon as temperature raises and we are approaching the deconfining
temperature, glueballs become thermally activated and they form an interacting
dilute gas. In addition, since the phase transition in SU(3) Yang-Mills theory
is of a (weak) first order, the thermal fluctuations should produce small
bubbles of the high-temperature phase. Thus, slightly below the critical
temperature $T_{c}$, we have a warm gluon matter which is expected to be
sensitive to rotation and possess a small moment of inertia. Our results in
Fig. 1 at $T\simeq 0.94T_{c}$ confirm this anticipated fact which, however,
possesses one surprising property that the total moment of inertia has a
tendency to become a negative quantity in agreement with Braguta  _et al._
(2023a, b).

Figure 1 shows that this extraordinary property extends also to the gluon
plasma regime above the critical temperature. The total moment of inertia $I$
is negative up to the supervortical critical temperature Braguta  _et al._
(2023a, b),

$\displaystyle T_{s}=1.50(10)T_{c}\,,$ (17)

at which it vanishes and turns to positive values at $T>T_{s}$. The positivity
of $I$ in the high-temperature regime fits well our understanding that plasma
becomes overwhelmingly perturbative at $T\to\infty$.

The mechanical contribution (6) possesses a mundane constant behaviour with
$I_{\mathrm{mech}}^{\mathrm{gl}}>0$, which gets established right above the
critical temperature. The chromomagnetic part (10) of the moment of inertia
is, however, a non-trivial function of temperature. It takes a negative value
in range of temperatures which makes the total momentum of inertia negative as
well in the supervortical region of temperatures $T_{c}\lesssim T<T_{s}$.

We work in the formalism of Ref. Braguta  _et al._ (2023a) which does not
require introduction of the metric tensor thus giving more robustness to our
calculations. Another way to calculate the moment of inertia on the lattice
involves the simulation of the rotating Yang-Mills theory in curved space and
analytic continuation procedure similarly to Refs. Braguta  _et al._ (2020,
2021); Yang and Huang (2023). The results of such calculations, presented in
Ref. Braguta  _et al._ (2023b), firmly overlap with them from non-rotating
lattices thus demonstrating the consistency of our approaches.

Now let us address the question of how can it be that the moment of inertia of
a gluon gas is negative while the energy density of the very same gluon gas is
positive everywhere? The last statement contradicts our experience of
classical mechanics, Eq. (13) which implies that the moment of inertia should
both be a positive quantity as the mass density is a positive quantity as
well.

## Evaporation of the chromomagnetic condensate.

The negative moment of inertia originates from the thermal properties of
(chromo)magnetic condensate (12). The chromomagnetic,
$\left\llangle{\mathcal{B}}^{2}\right\rrangle_{T}$, and chromoelectric,
$\left\llangle{\mathcal{E}}^{2}\right\rrangle_{T}$, condensates enter the
(thermal part of) the total gluon condensate:

$\displaystyle\left\llangle
G^{2}\right\rrangle_{T}=\left\llangle{\mathcal{B}}^{2}\right\rrangle_{T}+\left\llangle{\mathcal{E}}^{2}\right\rrangle_{T}\,,$
(18)

which determines, via the trace anomaly Boyd  _et al._ (1996),

$\displaystyle\varepsilon-3p=\left\langle T^{\mu}_{\
\mu}\right\rangle=\frac{\beta(\alpha_{s})}{4\pi}\left\llangle(G^{a}_{\mu\nu})^{2}\right\rrangle_{T}\equiv-\left\llangle
G^{2}\right\rrangle_{T}\,.$ (19)

the equation of state which relates the energy density $\varepsilon$ with
pressure $p$. Eq. (19) involves the QCD beta function,

$\displaystyle\beta\bigl{(}\alpha_{s}(\mu)\bigr{)}=\frac{{\mathrm{d}}\alpha_{s}(\mu)}{{\mathrm{d}}\ln\mu}<0\,,$
(20)

which reflects the running of the QCD coupling constant
$\alpha_{s}=g^{2}/(4\pi)$ with the energy scale $\mu$.

Our Monte Carlo results for all gluonic condensates (19) are shown in Fig.
2.444This figure essentially reproduces Fig. 10 (taken with a minus sign) of
Ref. Boyd  _et al._ (1996) re-calculated for the improved Symanzik action.
These results were calculated within standard lattice technique Boyd  _et al._
(1996); Borsanyi  _et al._ (2012) on the lattice $6\times 36\times 145^{2}$
with performed zero-temperature subtraction at $N_{t}=36$. The magnetic
condensate becomes negative above the phase transition and then it changes the
sign to a positive domain at $T\simeq 2T_{c}$ Boyd  _et al._ (1996).

Figure 2: The total gluon condensate (19), and, separately, its electric
$\left\llangle{\mathcal{E}}^{2}\right\rrangle_{T}$ and magnetic
$\left\llangle{\mathcal{B}}^{2}\right\rrangle_{T}$ components (18), as
functions of temperature $T$.

How to understand the thermal behaviour of the magnetic condensate? One can
associate its behaviour with the evaporation of the magnetic component of the
gluon plasma Chernodub and Zakharov (2007); Liao and Shuryak (2008) and the
associated string dynamics Chernodub and Zakharov (2007); Glozman (2023).
Moreover, the magnetic condensate is an infrared non-perturbative property of
the vacuum. As any other condensate, it can be associated with the infrared
coherence of the vacuum state. Strong thermal fluctuations add large
ultraviolet noise to the system and, consequently, disorder this condensate.
Therefore, the condensate quickly evaporates as temperature raises above the
critical point, $T=T_{c}$. This behaviour is clearly seen in Fig. 2 in the
temperature range $1\lesssim T/T_{c}\lesssim 1.1$. Notice that the magnetic
condensate is not an order parameter of the deconfinement phase transition
and, therefore, nothing special happens with it at the phase transition
$T=T_{c}$ apart from the quick drop caused by the evaporation.

Finite temperature environment generates also essentially perturbative thermal
fluctuations in the system which contribute positively to the magnetic
condensate. As temperature raises to about $T\simeq 1.1T_{c}$, the raise of
the magnetic thermal fluctuations overcome the evaporation of the remaining
magnetic condensate, and the slope changes: at this temperature, the quantity
$\left\llangle{\mathcal{B}}^{2}\right\rrangle_{T}$ starts to rise.555At
$T\gtrsim 1.1T_{c}$, the quantity
$\left\llangle{\mathcal{B}}^{2}\right\rrangle_{T}$ includes an essential
contribution coming from thermal fluctuations so that it is not justified,
strictly speaking, to call it a condensate. We will, however, continue to use
the word “condensate” to keep our narration smooth. At $T\simeq 2T_{c}$, the
thermal contribution becomes larger than the drop due to the evaporation of
the $T=0$ condensate, and $\left\llangle{\mathcal{B}}^{2}\right\rrangle_{T}$
becomes a positive quantity at higher temperatures, approaching the expected
$\left\llangle{\mathcal{B}}^{2}\right\rrangle_{T}\propto T^{4}$ Stefan-
Boltzmann law at $T\to\infty$.

Since the right hand side of the anomaly equation (19), given by the sum of
electric and magnetic condensates is always a positive quantity, therefore
thermodynamic relations imply that the energy density is always a positive
quantity as well Boyd  _et al._ (1996). Therefore, the moment of inertia,
substantially altered by the chromomagnetic component (10) can take negative
values while the energy density, given by the sum of both electric and
magnetic contributions (18), is always a positive quantity. In other words,
one can have a negative moment of inertia for an object with an everywhere-
positive energy density.

## Rotating photon gas.

It is instructive to compare our numerical results for gluonic plasma,
extrapolated to the high-temperature limit, with thermodynamics of non-
interacting massless vector fields, that is, photons. The interest in this
comparison originates from the fact that the generic relation (2) equally
applies to rotating photon gas while the vector nature of the photon field
assures – similarly to the gluon field – that the photon gas receives two
contributions to its moment of inertia (3), both mechanical (6) and magnetic
(10) ones.

One could naively expect that at high temperatures, the physics of gluons
becomes perturbative due to asymptotic freedom: in a hot gluon plasma, the
mean energy of gluons is high and for such high-energy gluons, the Yang-Mills
coupling constant is asymptotically vanishing. However, this expectation does
not work for all gluons since the magnetic gluon component always contains a
non-perturbative part, regardless how high temperature is Karsch (2002).

Moreover, in the studied range of temperatures, the equation of state of the
gluon plasma contains a strong correction from a non-perturbative sector
Karsch (2002) and therefore any relation with free photodynamics are at least
far fetched. In addition, it is the chromomagnetic contribution (10) which
makes the negative contribution to the moment of inertia. However, contrary to
the gluon case, the magnetic part of photons is not non-perturbative and
easily calculable analytically, which makes the comparison of high-temperature
gluodynamics with high-temperature photodynamics especially valuable. Below,
we will make the comparison, normalized by degrees of freedom, to see how
different the thermodynamics of these rotating vector fields are.

To evaluate the total moment of inertia of rotating photons, one can take the
result for a free scalar field Ambruş and Chernodub (2023) and rescale it by
the factor of $2$ corresponding to degrees of freedom of a photon. One gets
for a photon in a cylinder of the radius $R$ and the height $L$:

$\displaystyle I^{\mathrm{ph}}_{\mathrm{total}}\equiv
I^{\mathrm{ph}}_{\mathrm{mech}}+I^{\mathrm{ph}}_{\mathrm{magn}}=\frac{2\pi^{2}}{45}LR^{4}T^{4}\,.$
(21)

This expression excellently agrees with the numerical results of Ref.
Chernodub  _et al._ (2018) for the moment of inertia of hot photon gas
rotating in a cylinder, $I_{\mathrm{ph}}/(LR^{4}T^{4})\simeq 0.44$.

Similarly to gluons (3), the total moment of inertia (21) of the photon gas
possesses two contributions which are easily calculable. The mechanical part,
given by the susceptibility of the angular momentum (6),

$\displaystyle
I^{\mathrm{ph}}_{\mathrm{mech}}=\frac{\pi^{2}}{90}(4-\pi)LR^{4}T^{4}\,,$ (22)

can be obtained by subtraction of the contribution from thermal fluctuations
of magnetic field (10),

$\displaystyle
I^{\mathrm{ph}}_{\mathrm{magn}}\equiv\left\langle{\boldsymbol{B}}^{2}\right\rangle_{T}\cdot\frac{2}{3}\int_{V}d^{3}r\,r_{\perp}^{2}=\frac{\pi^{3}}{90}LR^{4}T^{4}\,,$
(23)

from the full expression (21). In order to calculate Eq. (23), we took into
account the Stefan-Boltzmann value of the thermal fluctuations of the magnetic
field, $\left\langle{\boldsymbol{B}}^{2}\right\rangle_{T}=\pi^{2}T^{4}/15$.
The mechanical contribution (22), obtained by the difference between the total
value (21) and the magnetic contribution (23), gives only a small fraction of
the total moment of inertia of the photon gas:

$\displaystyle\frac{I^{\mathrm{ph}}_{\mathrm{mech}}}{I^{\mathrm{ph}}_{\mathrm{total}}}=1-\frac{\pi}{4}\simeq
0.215,\quad{\mathrm{or}}\quad\frac{I^{\mathrm{ph}}_{\mathrm{magn}}}{I^{\mathrm{ph}}_{\mathrm{mech}}}=\frac{\pi}{4-\pi}\simeq
3.66\,.$ (24)

This observation should be compared with the behaviour of gluon plasma shown
in Fig. 1, where the magnetic contribution is indeed bigger than the
mechanical part which, however, still takes an essential part of the total
moment of inertia at our higher temperature $T\simeq 3.6T_{c}$.

The asymptotic values of the mechanical and magnetic parts of the moment of
inertia in the gluon plasma, can be obtained by suitable fits. The mechanical
part is almost insensitive to temperature in deconfinement phase and it can be
well described in the region $T\gtrsim 1.35T_{c}$, with $\chi^{2}/d.o.f.\simeq
0.1$, by the constant function,

$\displaystyle\frac{I_{\mathrm{mech}}^{\mathrm{gl}}(T)}{T^{4}}=\frac{I_{\mathrm{mech}}^{\mathrm{gl},\infty}}{T^{4}}=1.21(6)\,.$
(25)

The chromomagnetic contribution, as well as the total specific moment of
inertia, has a more involved behaviour which can be fitted at $T\gtrsim
1.35T_{c}$ by the following function (with $\chi^{2}/{\mathrm{d.o.f.}}\simeq
1.9$ for magnetic part and $\chi^{2}/{\mathrm{d.o.f.}}\simeq 0.1$ for total
values):

$\frac{I_{\mathrm{magn/total}}^{\mathrm{gl,fit}}(T)}{T^{4}}={}\\\
{}=\frac{I_{\mathrm{magn/total}}^{\mathrm{gl},\infty}}{T^{4}}\left[1+c_{1}\frac{T_{c}}{T-T_{c}}+c_{2}\left(\frac{T_{c}}{T-T_{c}}\right)^{2}\right]\,,$
(26) $\displaystyle I_{\mathrm{magn}}^{\mathrm{gl},\infty}/T^{4}$
$\displaystyle=2.474(15),\quad c_{1}=-1.001(5),\quad c_{2}=0.114(3),$
$\displaystyle I_{\mathrm{total}}^{\mathrm{gl},\infty}/T^{4}$
$\displaystyle=3.70(23),\quad c_{1}=-0.672(61),\quad c_{2}=0.076(26).$

These fits are shown by the solid lines in Fig. 1.

Asymptotic results normalized by degrees of freedom, do not coincide with
those of the hot photonic gas. The reason is in the non-Abelian nature of
magnetic degrees of freedom of gluons which receive non-perturbative
contributions even in the $T\to\infty$ limit Karsch (2002). Still, the
hierarchy of these parts is the same: in the high-temperature regime, the
purely magnetic degrees of freedom provide a dominant contribution to the
total momentum. Still, just above the deconfinement transition at $T=T_{c}$,
up to the supervortical critical temperature $T_{s}\simeq 1.5T_{c}$ , the
negative contribution of the chromomagnetic gluons outweighs the positive
contribution coming from the fluctuation of the angular momentum, and the
plasma appears to carry a negative moment of inertia.

## Strong negative Barnett effect for gluons.

A negative moment of inertia implies that the total angular momentum
${\boldsymbol{J}}={\boldsymbol{L}}+{\boldsymbol{S}}$, which (quark-)gluon
plasma possesses, is pointed out in the opposite direction to the angular
velocity ${\boldsymbol{\Omega}}$ with which the (quark-)gluon plasma rotates.
How can this situation be realized in Nature? We put forward here the
suggestion that the key player here is the gluon spin ${\boldsymbol{S}}$,
which accumulates too much of total angular momentum ${\boldsymbol{J}}$,
forcing, at the same time, the gluons themselves to rotate orbitally in the
opposite direction with respect to the total angular momentum in order to
compensate the over-excess of the angular momentum accumulated in the gluon
spins ${\boldsymbol{S}}$.

We call this effect the “negative Barnett effect” (NBE), which states that
spin polarization of gluons is opposite to the angular velocity of the system
itself. The usual Barnett effect is the manifestation of the spin-orbital-
coupling which implies that the spin of the particles, experiencing a
rotational collective motion, are aligned with the local angular velocity of
the system Barnett (1915).666The effect of the spin-orbital coupling caused by
rotation should be contrasted to the effect in static (non-rotating) bound
systems. In a hydrogen atom, for example, the spin and orbital angular
momentum of an electron tend to be anti-parallel, resulting in a lower energy
of the ${}^{2}P_{1/2}$ state compared to the ${}^{2}P_{3/2}$ state. It is a
fine-structure effect Landau and Lifshitz (1976c) caused by the polarization
of an electron magnetic moment by magnetic field arising in a (momentary) rest
frame of the electron. For an ordinary uncharged substances such as certain
metals Barnett. (1917) or even water Arabgol and Sleator (2019), the spin
polarization can experimentally be measured via rotation-produced
magnetization of the sample.

The NBE, on the contrary, implies that the spin-orbital-coupling has a
reversed sign, so that the orientation of the spins ${\boldsymbol{S}}$ and the
angular velocity ${\boldsymbol{\Omega}}$ is opposite to each other within the
supervortical window of temperatures $T_{c}\lesssim T<T_{s}$. Namely, we
suggest that in a rotating gluon system:

  * (i)

a sizable fraction of the total angular momentum
${\boldsymbol{J}}={\boldsymbol{L}}+{\boldsymbol{S}}$ is accumulated in the
spin of gluons $\boldsymbol{S}$;

  * (ii)

the spin polarization ${\boldsymbol{S}}$ is parallel to the total angular
momentum ${\boldsymbol{J}}$ and anti-parallel to the orbital angular momentum
of plasma $\boldsymbol{L}$.

Yet in other words, projecting all angular momenta onto a single axis
$\boldsymbol{n}$ and assuming without loss of generality $J>0$, the gluonic
spins accumulate too large amount of angular momentum $S>J>0$, so that this
contribution needs to be compensated by a negative orbital motion, $L=J-S<0$.
Therefore, plasma acquires the angular momentum in one direction ($J>0$) and
rotates in the opposite direction ($L<0$), as illustrated in Fig. 3.

Figure 3: (left) The usual (positive) Barnett effect: the system possesses the
positive angular momentum $\boldsymbol{J}$ which is redistributed between the
positive orbital momentum $\boldsymbol{L}$, corresponding to the
counterclockwise rotation and the positive spin polarization $\boldsymbol{S}$
generated by rotation. (right) The negative Barnett effect: the system
possesses the positive angular momentum ${\boldsymbol{J}}=J{\boldsymbol{n}}$
(with ${\boldsymbol{n}}^{2}=1$), which is redistributed between a large
positive spin polarization ${\boldsymbol{S}}=S{\boldsymbol{n}}$ (with $S>J>0$)
and an orbital angular momentum
${\boldsymbol{L}}=L{\boldsymbol{n}}=I_{L}{\boldsymbol{\Omega}}$ that takes a
negative value $L<0$ due to conservation of total angular momentum, $J=S+L$.
Since the moment of inertia stored in orbital motion is positive, $I_{L}>0$,
the system rotates, as a whole, clockwise:
${\boldsymbol{\Omega}}=\Omega{\boldsymbol{n}}$ with $\Omega<0$. The negative
Barnett effect implies that the angular velocity $\boldsymbol{\Omega}$ and
spin polarization $\boldsymbol{S}$ are anti-parallel to each other.

Notice that it is the orbital momentum which encodes rotation of plasma as
whole, while the spin polarization is an intrinsic characteristic of the
system which does necessarily manifest itself in rotation. A relevant case is
represented by a ferromagnet in the broken phase which spontaneously generates
magnetic field due to substantial spin polarization, ${\boldsymbol{S}}\neq 0$,
and, at the same time, does not rotate, ${\boldsymbol{L}}\equiv 0$ Landau and
Lifshitz (1976b). Another example is given by superfluid 3He in a metastable,
but eternally long-lived $p$-wave state where the angular momentum,
accumulated vortices, is opposite to the global rotation of the fluid Autti
_et al._ (2020).

Given the suspected negativity of the spin-orbital coupling, one may ask
ourselves how the moment of inertia is redistributed between spin and orbital
degrees of freedom. Following the analogy with the Barnett effect Barnett
(1915), we assert that the rotation with angular velocity
${\boldsymbol{\Omega}}$ induces the spin polarization of gluons,
${\boldsymbol{S}}=I_{S}{\boldsymbol{\Omega}}$.

Writing similarly for the orbital part,
${\boldsymbol{L}}=I_{L}{\boldsymbol{\Omega}}$, one arrives to Eq. (1) where
the total moment of inertia is the sum of the the spin and orbital
contributions: $I=I_{S}+I_{L}$. Since the energy density is a positive
quantity, the mechanical orbital rotation can only lead to a positive value of
the orbital moment of inertia, $I_{L}>0$. Therefore, the negative moment of
inertia can only be achieved if the rotation polarizes spin negatively,
$I_{S}<-I_{L}<0$, against the rotation direction, so that $I=I_{S}+I_{L}<0$.

## Implications for quark-gluon plasma.

So far, we discussed the gluonic component of the rotating plasma. The
magnetic condensate term will also exist in the presence of rotating fermionic
component, so that our conclusions equally apply to the quark-gluon plasma.
The coupling of quarks to rotation results in a linear term in the free energy
which implies that fermions themselves cannot generate the negative Barnett
effect because their contribution to the total moment of inertia is associated
with the single, purely mechanical term:

$\displaystyle
I^{{\mathrm{(q+gl)}}}_{\text{mech}}=\frac{1}{T}\left\llangle{\boldsymbol{n}}\cdot({\boldsymbol{J}}_{\mathrm{q}}+{\boldsymbol{J}}_{\mathrm{gl}})^{2}\right\rrangle_{T}\,.$
(27)

The angular momentum of quarks ${\boldsymbol{J}}_{\mathrm{q}}$ is given by Eq.
(7) with the substitution $M^{ij}_{\mathrm{gl}}\to
M_{q}^{ij}=i{\bar{\psi}}\gamma^{0}(\gamma^{i}D^{j}-\gamma^{j}D^{i})\psi$. The
covariant derivative $D_{\mu}=\partial_{\mu}+igA_{\mu}$ incorporates the
coupling of the quark spinors $\psi$ to the gluon field $A^{\mu}$.

The mechanical momentum which includes the contribution of quarks (27) is
expected to give a positive addition to the total moment of inertia thus
diminishing the magnitude of its negative value in the supervortical range of
temperatures. However, the evaporation of the gluon condensate also occurs in
the presence of quarks, implying that the negative moment of inertia and the
associated negative Barnett effect will also be realized in the quark-gluon
plasma in relativistic heavy-ion collisions.

## Discussion and Conclusions.

It was recently found that the gluon plasma in the range of temperatures
$T\simeq(1\dots 1.5)T_{c}$ possesses a negative moment of inertia, which may
be interpreted as thermodynamic instability to the global rotation Braguta
_et al._ (2023a, b). The result was obtained using two independent approaches
in static Braguta  _et al._ (2023a) and rotating Braguta  _et al._ (2023b)
plasmas. In this paper, we argued, using thermodynamic arguments and – in
relation to the gluon plasma – the lattice data, that one can have a negative
moment of inertia for an object with an everywhere-positive energy density.

The moment of inertia of a gluon system contains not only the expected
mechanical term corresponding to the total angular momenta, but also a local
term which incorporates the chromomagnetic condensate. This additional term
emerges due to the vector nature of gluons: the latter property leads to the
non-linearity of the gluonic free energy which contains terms both linear and
quadratic in $\Omega$. The linear term produces the standard mechanical
contribution (6) while the quadratic term leads to the novel contribution
which incorporates the chromomagnetic condensate (10). Our Monte Carlo
simulation clearly show that it is the thermal evaporation of the
chromomagnetic condensate that leads to the negative moment of inertia of
rotating plasma.

As a result, we conclude in our paper that the total angular momentum is still
positive, while the plasma mechanically rotates in the opposite direction to
the angular momentum implying the unusual relation
${\boldsymbol{\Omega}}\cdot{\boldsymbol{J}}<0$. We coin this novel phenomenon
as the Negative Barnett Effect. While this effect can have direct consequences
on rotation of gluon plasma, the detailed implications of the emergence of the
negative moment of inertia remains to be explored. The Negative Barnett Effect
for gluons may be one of the possible causes of the thermodynamic instability
reported in Ref. Braguta  _et al._ (2023b).

## Acknowledgements.

This work has been carried out using computing resources of the Federal
collective usage center Complex for Simulation and Data Processing for Mega-
science Facilities at NRC “Kurchatov Institute”, http://ckp.nrcki.ru/ and the
Supercomputer “Govorun” of Joint Institute for Nuclear Research. The work of
VVB, IEK, AAR, and DAS, which consisted in the lattice calculation of the
observables used in the paper, was supported by the Russian Science Foundation
(project no. 23-12-00072). The authors are grateful to Victor Ambrus and
Andrey Kotov for useful discussions.

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