YITP-23-125

On thermal transition in QCD

Masanori Hanadaa and Hiromasa Watanabeb

aSchool of Mathematical Sciences, Queen Mary University of London
Mile End Road, London, E1 4NS, United Kingdom

bYukawa Institute for Theoretical Physics, Kyoto University
Kitashirakawa Oiwakecho, Sakyo-ku, Kyoto 606-8502, Japan

Email<EMAIL_ADDRESS><EMAIL_ADDRESS>
Abstract

We describe how the general mechanism of partial deconfinement applies to
large-$N$ QCD and the partially-deconfined phase inevitably appears between
completely-confined and completely-deconfined phases. Furthermore, we propose
how the partial deconfinement can be observed in the real-world QCD with the
SU(3) gauge group. For this purpose, we employ lattice configurations obtained
by the WHOT-QCD collaboration and examine our proposal numerically. In the
discussion, the Polyakov loop plays a crucial role in characterizing the
phases, without relying on center symmetry, and hence, we clarify the meaning
of the Polyakov loop in QCD at large $N$ and finite $N$.

Both at large $N$ and finite $N$, the complete confinement is characterized by
the Haar-random distribution of the Polyakov line phases. Haar-randomness,
which is stronger than unbroken center symmetry, indicates that Polyakov loops
in any nontrivial representations have vanishing expectation values and
deviation from the Haar-random distribution at higher temperatures is
quantified with the loops. We discuss that the transitions separating the
partially-deconfined phase are characterized by the behaviors of Polyakov
loops in various representations. The lattice QCD data provide us with the
signals exhibiting two different characteristic temperatures: deconfinement of
the fundamental representation and deconfinement of higher representations. As
a nontrivial test for our proposal, we also investigate the relation between
partial deconfinement and instanton condensation and confirm the consistency
with the lattice data.

To make the presentation more easily accessible, we provide a detailed review
of the previously known aspects of partial deconfinement.

###### Contents

  1. 1 Introduction
  2. 2 Partial deconfinement in large-$N$ Hermitian matrix model
    1. 2.1 Partial deconfinement at weak coupling
    2. 2.2 Underlying mechanism
    3. 2.3 Partial deconfinement at nonzero coupling
    4. 2.4 Euclidean path integral and Monte Carlo simulation
    5. 2.5 Introducing fermions
  3. 3 Partial deconfinement in large-$N$ QFT
    1. 3.1 Yang-Mills theory
      1. 3.1.1 Strong coupling on lattice
      2. 3.1.2 Continuum limit (weak coupling on lattice)
    2. 3.2 Large-$N$ QCD
    3. 3.3 Some physical consequences
      1. 3.3.1 Flux tube in confined sector
      2. 3.3.2 Chiral symmetry breaking
      3. 3.3.3 CP symmetry breaking at $\theta=\pi$
  4. 4 Partial deconfinement at finite $N$
    1. 4.1 Distribution of Polyakov line phases at finite $N$
    2. 4.2 Characerization via global symmetry and instanton condensation
  5. 5 Partial deconfinement in real-world QCD ($N=3$)
    1. 5.1 Poylakov line phases vs. partial deconfinement
    2. 5.2 Instanton condensation
  6. 6 Conclusion and discussion
  7. A Canonical ensemble and microcanonical ensemble
  8. B Phase coexistence in color space vs. phase coexistence in coordinate space
  9. C Operator formalism and Euclidean path integral
  10. D SU($N$) Haar random distribution
  11. E Characters of SU(3) group
    1. E.1 List of characters for irreducible representations
    2. E.2 Multiply-wound Polyakov loops and characters

##  1 Introduction

Confinement/deconfinement transition in gauge theory [1, 2] has been
investigated for a long time for the understanding of the quark-gluon plasma
in QCD under extreme conditions and the description of black holes via
gauge/gravity duality. Focusing on real-world QCD with three colors, it is
widely believed that there is no phase transition at zero chemical potential,
rather confined and deconfined phases are believed to be connected by a rapid
crossover. This belief is based on lattice QCD simulations, e.g., Ref. [3],
and the apparent lack of symmetry that could characterize a possible phase
transition. In this paper, we argue that this belief may be wrong and there
might be a phase transition of third or higher order.

To motivate this seemingly bizarre proposal, let us consider the ’t Hooft
large-$N$ limit of SU($N$) gauge theory. Many large-$N$ gauge theories exhibit
confinement/deconfinement transition. In theories such as pure Yang-Mills or
4d $\mathcal{N}=4$ super Yang-Mills theory, the Polyakov loop is commonly used
as an order parameter associated with the $\mathbb{Z}_{N}$ center symmetry.
The $N$ phases of Polyakov loop $\theta_{1},\cdots,\theta_{N}$ form a
continuous distribution $\rho(\theta)$ between $-\pi$ and $+\pi$ in the
large-$N$ limit. The $\mathbb{Z}_{N}$ center acts on $\theta$ as a constant
shift, and hence the confined phase is characterized by the uniform
distribution $\rho(\theta)=\frac{1}{2\pi}$, while the deconfinement phase is
characterized by non-uniform distribution. An interesting observation [4, 5]
is that there are two types of non-uniform distributions — jointed
($\rho(\theta)>0$ everywhere) and disjointed ($\rho(\theta)=0$ in a finite
interval) — separated by the ‘Gross-Witten-Wadia (GWW) transition’. 111The
original setup of GWW is the two-dimensional lattice gauge theory at large
$N$, which has a phase transition associated with the formation of a gap in
the distribution of phases of the plaquette. The GWW model can describe
confinement/deconfinement transition in a class of weakly-coupled systems
including pure Yang-Mills on S3, modulo the difference of the interpretation
of the unitary variable (plaquette or Polyakov loop). Even when the
confinement/deconfinement transition is not precisely described in the GWW
model, we use the word ‘GWW transition’ for the formation of a gap, although
it may be a little bit of abuse of terminology. Because the jointed and
disjointed phases cannot be distinguished in terms of the center symmetry, it
is reasonable to expect that the Polyakov loop has some meaning that is
independent of the center symmetry. This expectation is strengthened by
noticing that the Polyakov loop can characterize the phase transition in QCD
despite the explicit breaking of center symmetry due to quarks in the
fundamental representation [6]. As we will explain in detail, the answer is
given by partial deconfinement; see Fig. 1, Fig. 2, and Fig. 3.

Figure 1: Completely-confined, partially-deconfined, and completely-deconfined
phases correspond to uniform, non-uniform but jointed, and disjointed
distributions of Polyakov line phases, respectively. This figure is a slight
modification of the one from Ref. [7].

Another convenient way to characterize confined and deconfined phases is to
use the $N$-dependence of energy and entropy. In the confined phase, because
individual color degrees of freedom are not visible (they are ‘confined’), the
energy and entropy typically scale as $N^{0}$ up to zero-point contributions.
On the other hand, in the deconfined phase, the scaling is typically of order
$N^{2}$ because color degrees of freedom are ‘deconfined’. What if the energy
is set to $\epsilon N^{2}$, where $\epsilon$ is small but of order $N^{0}$?
The energy is too high to be in the confined phase, but not high enough to
deconfine all color degrees of freedom. In many cases including large-$N$ QCD,
partial deconfinement takes place, i.e., an SU($M$) subgroup of SU($N$)
deconfines, where, roughly speaking, $M\sim\sqrt{\epsilon}N$ (see Fig. 2).
Originally, partial deconfinement was introduced for 4d maximal super Yang-
Mills theory compactified on S3, in order to explain the QFT-dual of the small
black hole in AdS${}_{5}\times$S5 [8]. Later, it was suggested that partial
deconfinement is a rather generic phenomenon in many large-$N$ gauge theories
[9, 10]. Subsequent investigations provided analytic [11, 12, 13] and
numerical [14, 15, 13, 16] evidence, and clarified the underlying mechanism
[17, 18, 19].

Figure 2: Partial deconfinement in the gauge sector and adjoint matters. Only
the $M\times M$-block shown in red is deconfined. The consistency of this
cartoon picture with the gauge singlet constraint is discussed in the main
text. This figure is taken from Ref. [11].

Figure 3: Basic three patterns of confinement/deconfinement transition. Blue,
red, and orange lines correspond to completely-confined, completely-
deconfined, and partially-deconfined phases, respectively. Solid and dotted
lines are the minimum and maximum of free energy at each temperature.

Intuitively, partial deconfinement can be phrased as the coexistence of two
phases (confined phase and deconfined phase) in the internal space
(specifically, color space), rather than the coordinate space. A common
example of the coexistence of two phases in coordinate space is observed in
the first-order transition between water (liquid water) and ice (solid water)
at 1 atm (Fig. 4). Supercooled water below 0℃ and superheated ice above 0℃ are
not stable, and they turn to a mixture of water and ice just by a small
perturbation as shown by arrows in Fig. 4. The temperature of the mixture is
0℃ due to the local nature of the interaction.222Because of the locality, each
small region is described by the canonical thermodynamics. Free energy is
defined locally. They are minimized such that water or ice is favored. The
interaction at the interface of two phases can be ignored if the interaction
is local and the spatial volume is large. However, we can recognize that the
coexistence occurs in color space as well and the above picture does not work
there anymore. The interactions in the color space are all-to-all, and this
nonlocal feature makes the system robust against the small perturbation.
Furthermore, the way of splitting into two sectors depends on the
representations, as depicted in Fig. 5 for the case of adjoint and fundamental
representations in large-$N$ QCD. Taking these into account, the temperature
is not necessarily fixed during the process of transition as depicted in Fig.
3, and the variety of equations of states is allowed in the intermediate
region. In retrospect, such two-phase-coexistence in gauge theory has been
known for a long time: a system of $N$ indistinguishable bosons is described
by gauge theory with SN-gauge symmetry that permutes bosons, and two phases —
superfluid (Bose-Einstein condensate) and normal fluid — coexist in the
internal space. In fact, these two kinds of coexistence take place due to the
same mechanism [17].333 In this sense, Einstein was the first person who
considered non-Abelian gauge theory, took the large-$N$ limit, and took the
weak-coupling limit. We can also see that Feynman introduced an equivalent of
the Polyakov loop and characterized the Bose-Einstein condensation by the
uniform distribution of the phases. See Sec. 2.2 for details.

Figure 4: First-order transition between water (liquid water) and ice (solid
water) at 1 atm. At 0℃, water and ice coexist. Supercooled water below 0℃ and
superheated ice above 0℃ are not stable, and they turn to a mixture of water
and ice just by a small perturbation as shown by arrows in Fig. 4. (Energy is
fixed because the change takes place quickly.) This figure is taken from Ref.
[10].

Figure 5: Partial deconfinement in weakly-coupled QCD on S3 [11]. [Left] the
gauge sector and matters in the adjoint representation, [Right] matters in the
fundamental representation. $M$ colors shown in red are deconfined. The
consistency of this cartoon picture with the gauge singlet constraint is
discussed in the main text. This figure is taken from Ref. [11]

Due to partial deconfinement, there are three phases: completely-confined
phase ($M=0$), partially-deconfined phase ($0<M<N$), and completely-deconfined
phase ($M=N$). Even if the theory possesses the center symmetry or not, three
phases can be distinguished by the distribution of the phases of Polyakov line
[17, 10]: completely-confined, partially-deconfined, and completely-deconfined
phases correspond to uniform, non-uniform but jointed, and disjointed
distributions of Polyakov line phases, respectively, as depicted in Fig. 1.
When the system possesses center symmetry, the transition from the completely-
confined phase to the partially-deconfined phase is associated with the
breaking of center symmetry, which is a traditional definition of
deconfinement for theories in this class. Since, at the critical point, the
formation of long strings is involved, we call it the Hagedorn transition. The
transition from the partially-deconfined phase to the completely-deconfined
phase is associated with the formation of a gap in the distribution of
Polyakov line phases. We call it the Gross-Witten-Wadia (GWW) transition [20,
21].

In this paper, we explain how partial deconfinement can take place large-$N$
QCD. Furthermore, we suggest that partial deconfinement takes place in SU(3)
QCD that describes our universe. To make the presentation as self-contained as
possible, we provide a detailed review of the previously known aspects of
partial deconfinement.

The organization of this paper is as follows. In Sec. 2, we use gauged
Hermitian matrix models as the simplest examples to explain the essence of
partial deconfinement. Then, we apply the same idea to QFT in Sec. 3.
Potential generalization to finite $N$ is discussed in Sec. 4. The majority of
these three sections review the results provided in several references in the
past. Because crucial results appeared in different papers and there was no
systematic review, we hope that the review parts make the topic more
accessible. Sec. 3.1.2 and Sec. 4 contain new materials. In Sec. 5, we analyze
the configurations from lattice gauge theory simulations and discuss the
consistency with our conjecture.

We have included several appendices in this paper to supplement the main text.
Appendix D and Appendix E have large overlap with the letter version [22].
They are included here to make this paper self-contained.

An important part of this paper is based on collaboration with Hiroki Ohata
and Hidehiko Shimada. They contributed significantly to the analysis of
lattice QCD data, including a key idea of the use of the character.

##  2 Partial deconfinement in large-$N$ Hermitian matrix model

In this section, we review partial deconfinement by taking a large-$N$
Hermitian matrix model as an example. This example contains all the essences
that can also be applied to other theories. As a concrete example, we start
with a bosonic SU($N$) matrix model with the following Hamiltonian:

$\displaystyle\hat{H}={\rm
Tr}\left(\frac{1}{2}\hat{P}_{I}^{2}+\frac{1}{2}\hat{X}_{I}^{2}-\frac{g_{\rm
YM}^{2}}{4}[\hat{X}_{I},\hat{X}_{J}]^{2}\right).$ (1)

Here $I,J=1,2,\cdots,D$, where $D\geq 2$. Each $\hat{X}_{I}$ has $N^{2}$
components $\hat{X}_{I,ij}$, where $i,j=1,2,\cdots,N$, that satisfy the
Hermiticity condition $(\hat{X}_{I,ij})^{\dagger}=\hat{X}_{I,ji}$. We do not
impose the traceless condition. The operator $\hat{P}_{I,ij}$ is the conjugate
momentum of $\hat{X}_{I,ji}$.

We introduce the adjoint index running from 1 to $N^{2}$ by using the
generators $\tau_{\alpha}$ of U($N$) normalized as ${\rm
Tr}(\tau_{\alpha}\tau_{\beta})=\delta_{\alpha\beta}$,
$\sum_{\alpha}\tau_{\alpha,ij}\tau_{\alpha,kl}=\delta_{il}\delta_{jk}$. Then,
the operators $\hat{P}_{I}$ and $\hat{X}_{I}$ are expressed as

$\displaystyle\hat{P}_{I,ij}=\sum_{\alpha=1}^{N^{2}}\hat{P}_{I}^{\alpha}\tau_{\alpha,ij},\qquad\hat{X}_{I,ij}=\sum_{\alpha=1}^{N^{2}}\hat{X}_{I}^{\alpha}\tau_{\alpha,ij}.$
(2)

These $\hat{X}_{I}^{\alpha}$ and $\hat{P}_{I}^{\alpha}$ are real, i.e.,
$(\hat{X}_{I}^{\alpha})^{\dagger}=\hat{X}_{I}^{\alpha}$ and
$(\hat{P}_{I}^{\alpha})^{\dagger}=\hat{P}_{I}^{\alpha}$. The canonical
commutation relation is444 Equivalently,
$\displaystyle[\hat{X}_{I,ij},\hat{P}_{J,kl}]=i\delta_{IJ}\delta_{il}\delta_{jk}\,.$

$\displaystyle[\hat{X}^{\alpha}_{I},\hat{P}^{\beta}_{J}]=i\delta_{IJ}\delta^{\alpha\beta}\,.$
(3)

The Hamiltonian (1) is invariant under the adjoint action of SU($N$) defined
by

$\displaystyle\hat{X}_{I,ij}\to(U\hat{X}_{I}U^{-1})_{ij}=\sum_{k,l}U_{ij}\hat{X}_{I,kl}U^{-1}_{lj},\qquad\hat{P}_{I,ij}\to(U\hat{P}_{I}U^{-1})_{ij}=\sum_{k,l}U_{ik}\hat{P}_{I,kl}U^{-1}_{lj}.$
(4)

We consider gauged matrix model, i.e., this SU($N$) symmetry is gauged. One
way of gauging SU($N$) is to restrict the Hilbert space to SU($N$)-singlet
states. By using the singlet Hilbert space (gauge-invariant Hilbert space)
${\cal H}_{\rm inv}$, the canonical partition function at temperature $T$ can
be written as

$\displaystyle Z(T)$ $\displaystyle={\rm Tr}_{\mathcal{H}_{\rm
inv}}\left(e^{-\hat{H}/T}\right).$ (5)

Equivalently, we can select the extended Hilbert space with non-singlet
states, which we denote by ${\cal H}_{\rm ext}$. The extended space is spanned
by, for example, the coordinate eigenstates $\ket{X}$ that satisfy
$\hat{X}_{I,ij}\ket{X}=X_{I,ij}\ket{X}$:

$\displaystyle\mathcal{H}_{\rm
ext}=\textrm{Span}\\{\ket{X}|X\in\mathbb{R}^{DN^{2}}\\}.$ (6)

Note that $X$ consists of $DN^{2}$ real numbers
$X_{I=1,\cdots,D}^{\alpha=1,\cdots,N^{2}}$. We can also use the momentum
eigenstates $\ket{P}$ that satisfy $\hat{P}_{I,ij}\ket{P}=P_{I,ij}\ket{P}$:

$\displaystyle\mathcal{H}_{\rm
ext}=\textrm{Span}\\{\ket{P}|P\in\mathbb{R}^{DN^{2}}\\}.$ (7)

The coordinate basis and momentum basis are related with each other by Fourier
transformation. Gauge transformation acts on $\mathcal{H}_{\rm ext}$ as

$\displaystyle\ket{X}\to\ket{U^{-1}XU}\,,\qquad\ket{P}\to\ket{U^{-1}PU}\,.$
(8)

The extended Hilbert space is convenient because the operators $\hat{X}$ and
$\hat{P}$ are defined on this space. On the other hand, these operators are
not defined on the singlet Hilbert space, because they map a singlet to a non-
singlet. In the extended Hilbert space, the gauged SU($N$) enforces that the
states related by an SU($N$) transformation should be identified. This is the
same as in the usual treatment in the classical theory, and also in path-
integral formalism: Field configurations that are related by gauge
transformation should be identified. The canonical partition function can be
written as

$\displaystyle Z(T)$ $\displaystyle=\frac{1}{{\rm vol}G}\int_{G}dg{\rm
Tr}_{{\cal H}_{\rm ext}}\left(\hat{g}e^{-\hat{H}/T}\right).$ (9)

Here $G=\textrm{SU}(N)$ is the gauge group, $g$ is a group element, and
$\hat{g}$ is the representation of $g$ acting on the extended Hilbert space
${\cal H}_{\rm ext}$ (in the case under consideration, the adjoint
representation). The integral is taken with respect to the Haar measure.555
Some readers may be familiar with a similar expression for the case of the
discrete gauge group. The insertion of $\hat{g}$ is regarded as a twisted
boundary condition, and the discrete group is gauged by taking a sum over all
twisted boundary conditions.  As shown in Appendix C, this expression is
equivalent to the other expression (5) and this is directly related to the
path integral with temporal gauge field $A_{t}$. The group element $g$ can be
translated into the Polyakov line. It is easy to see that

$\displaystyle\hat{\pi}\equiv\frac{1}{\textrm{vol}G}\int_{G}dg\hat{g}$ (10)

is a projection operator from ${\cal H}_{\rm ext}$ to ${\cal H}_{\rm inv}$. By
using this, (9) can also be written as

$\displaystyle Z(T)$ $\displaystyle={\rm Tr}_{{\cal H}_{\rm
ext}}\left(\hat{{\pi}}e^{-\hat{H}/T}\right).$ (11)

From a state $|\phi\rangle\in\mathcal{H}_{\rm ext}$, which can be non-singlet,
we can obtain a singlet $\hat{\pi}|\phi\rangle\in\mathcal{H}_{\rm inv}$. By
using this correspondence, we can describe the same physics by using
$\mathcal{H}_{\rm ext}$ or $\mathcal{H}_{\rm inv}$. As we will see, the use of
$\mathcal{H}_{\rm ext}$ can make physics much more transparent.

###  2.1 Partial deconfinement at weak coupling

We start with the gauged Gaussian matrix model, i.e., the free limit ($g_{\rm
YM}^{2}=0$) of the theory with the Hamiltonian (1), following Ref. [11].
Despite the lack of interactions, confinement/deconfinement transition can
take place in the large-$N$ limit due to the SU($N$)-singlet constraint and
some qualitative features of strong coupling region can be captured [5, 4].666
The same applies to Bose-Einstein condensation [17]: the transition can take
place in the large-$N$ limit due to the SN-singlet constraint as pointed out
by Einstein, and some qualitative features of strong coupling region can be
captured as pointed out by Feynman.  The Hamiltonian is

$\displaystyle\hat{H}_{\rm free}={\rm
Tr}\sum_{I=1}^{D}\left(\frac{1}{2}\hat{P}_{I}^{2}+\frac{1}{2}\hat{X}_{I}^{2}\right)={\rm
Tr}\sum_{I=1}^{D}\left(\hat{A}_{I}^{\dagger}\hat{A}_{I}+\frac{1}{2}\right)\,,$
(12)

where $\hat{A}_{I}^{\dagger}=\frac{\hat{X}_{I}-i\hat{P}_{I}}{\sqrt{2}}$ and
$\hat{A}_{I}=\frac{\hat{X}_{I}+i\hat{P}_{I}}{\sqrt{2}}$ are creation and
annihilation operators. Equivalently,

$\displaystyle\hat{H}_{\rm free}={\rm
Tr}\sum_{I=1}^{D}\left(\frac{1}{2}(\hat{P}_{I}^{\alpha})^{2}+\frac{1}{2}(\hat{X}_{I}^{\alpha})^{2}\right)=\sum_{I=1}^{D}\sum_{\alpha=1}^{N^{2}}\hat{A}_{I}^{\alpha\dagger}\hat{A}_{I}^{\alpha}+\frac{DN^{2}}{2}.$
(13)

There are $DN^{2}$ harmonic oscillators that are subject to the gauge-singlet
constraint, or equivalently, the identification of non-singlets that transform
to each other via SU($N$) transformation. The ground state is the Fock vacuum
$\ket{0}$ that is annihilated by any annihilation operator:

$\displaystyle\hat{A}_{I}^{\alpha}\ket{0}=0.$ (14)

Generic states in the extended Hilbert space $\mathcal{H}_{\rm ext}$ can be
obtained by acting creation operators. States in the singlet Hilbert space
$\mathcal{H}_{\rm inv}$ can be obtained by acting the projection operator
$\hat{\pi}$, or equivalently, by multiplying the traces of products of
creation operators to the ground state.

Due to the above construction in $\mathcal{H}_{\rm ext}$, SU($M$)-deconfined
states can be obtained by acting $\hat{A}_{I,ij}^{\dagger}$ with $1\leq
i,j\leq M$ to the ground state. Such states are invariant under
$\textrm{SU}(N-M)\subset\textrm{SU}(N)$ that does not act nontrivially on the
SU($M$)-sector. This unbroken symmetry in the extended Hilbert space plays a
crucial role for partial deconfinement, as we will see in Sec. 2.2. By
definition, SU($M$)-deconfined states explain at least a part of the entropy
of the system. The nontrivial point is that these states can explain the
entire entropy, and hence, thermodynamically dominant. We can show this as
follows:

  1. 1.

We calculate the entire entropy at energy $E$ without assuming partial
deconfinement.

  2. 2.

Besides, we calculate the entropy of the SU($M$)-deconfined states at energy
$E$.

  3. 3.

We confirm that the entire entropy comes from the SU($M$)-deconfined states.

Let us start with the step 1. The free energy can be calculated by using the
path-integral formulation and repeating similar computations in Refs. [5, 4].
The free energy is expressed as [11]

$\displaystyle\beta
F=\frac{DN^{2}\beta}{2}+N^{2}\sum_{n=1}^{\infty}\frac{1-Dx^{n}}{n}|u_{n}|^{2},$
(15)

in terms of the Polyakov loops $u_{n}=\frac{1}{N}{\rm Tr}{\cal P}^{n}$, where
${\cal P}$ is the Polyakov line, and $x=e^{-\beta}$. According to the saddle
point condition, the stable configuration of $u_{n}$’s of the model should
minimize the free energy. At $T<T_{\rm c}=\frac{1}{\log D}$, $|u_{n}|=0$
($n\geq 1$) is favored. At $T=T_{\rm c}$, $|u_{1}|$ can take any value from 0
to $\frac{1}{2}$, while $u_{2},u_{3},\cdots$ remain zero, without changing the
free energy. This is a first-order deconfinement transition. In Fig. 6, we
have shown how the Polyakov loop $P$, entropy $S$ and energy $E$ depend on
temperature. Notice that the energy and entropy are determined by the values
of $u_{n}$’s, just as the free energy is. The point $P=0$ is the onset of
deconfinement in the sense that center symmetry breaks and entropy and energy
(excluding the zero-point energy) become of order $N^{2}$. The point
$P=\frac{1}{2}$ is called the Gross-Witten-Wadia (GWW) transition point.

Let us focus on the deconfinement transition at $T=T_{\rm c}$.

  * •

The Polyakov loop $P=|\frac{1}{N}{\rm Tr}{\cal P}|=|u_{1}|$ can take any value
between 0 and $\frac{1}{2}$.777 Strictly speaking, there is an ambiguity
associated with the center symmetry, namely the constant shift of all the
phases. We fixed the center symmetry such that the Polyakov loop becomes real
and non-negative, i.e. $P=|P|$.  Each value of $P$ corresponds to the
different saddle point in the path integral. From the microcanonical
viewpoint, $P=0$ is the transition point from confinement to deconfinement,
and $P=\frac{1}{2}$ is the GWW transition point. We will interpret them as the
transitions from confinement to partial deconfinement, and from partial
deconfinement to complete deconfinement.

  * •

The energy can be written as

$\displaystyle E(T=T_{\rm c},P,N)=\frac{DN^{2}}{2}+N^{2}P^{2}.$ (16)

The first term is the zero-point energy.

  * •

The entropy $S=\beta(E-F)$ is

$\displaystyle S(T=T_{\rm c},P,N)=N^{2}P^{2}\log D.$ (17)

  * •

The distribution of the Polyakov line phases at the critical temperature is

$\displaystyle\rho(\theta)=\frac{1}{2\pi}\left(1+2P\cos\theta\right)=\left(1-2P\right)\cdot\frac{1}{2\pi}+2P\cdot\frac{1}{2\pi}\left(1+\cos\theta\right),$
(18)

up to $1/N$ corrections. In particular, the distribution at the GWW transition
is

$\displaystyle\rho_{\rm
GWW}(\theta;N)=\frac{1}{2\pi}\left(1+\cos\theta\right).$ (19)

Figure 6:  Cartoon pictures of the temperature dependence of the Polyakov loop
$P$, entropy $S$, and energy $E$ in the gauged Gaussian matrix model. Blue,
orange, and red lines are identified with the completely-confined, partially-
deconfined, and completely-deconfined phases, respectively. These pictures
appeared originally in Ref. [11].

Let us move on to step 2. Let us consider SU($N$) theory and SU($N^{\prime}$)
theory, with $N^{\prime}<N$. To obtain the energy and entropy in the
SU($N^{\prime}$)-theory, we only have to replace $N$ with $N^{\prime}$ in (16)
and (17). The plots of the energy and entropy are shown in Fig. 7. A seemingly
trivial but crucial observation is that the SU($N$)-theory and
SU($N^{\prime}$)-theory behave in the same way below the GWW point of the
SU($N^{\prime}$)-theory, $E-E_{0}=\frac{N^{\prime 2}}{4}$ (here, $E_{0}$ is
the zero-point energy) and $S=\frac{\log D}{4}N^{\prime 2}$. Indeed,

$\displaystyle S=\log D\cdot(E-E_{0})$ (20)

holds for both SU($N$)- and SU($N^{\prime}$)-theories. What does it mean?

Suppose, for a given state in the SU($N$)-theory, all the excitations are in
the upper-left $N^{\prime}\times N^{\prime}$ block up to gauge transformation.
Because we are considering the free limit, we can cut out the
$N^{\prime}\times N^{\prime}$ block and obtain a counterpart in the
SU($N^{\prime}$)-theory with the same energy.888Such a truncation would change
the energy if the coupling constant is nozero. Note that, as long as
$N^{\prime}\times N^{\prime}$ contains all the excitation, the size of $N’$ is
arbitrary below $N$. The agreement of behaviors pointed out above means that,
at $E-E_{0}\leq\frac{N^{\prime 2}}{4}$, the entropy of the SU($N$)-theory can
be reproduced just by considering such states. Namely, the excitations in the
SU($N$)-theory are taking place only in the $N^{\prime}\times N^{\prime}$
block up to gauge transformation. In addition, the excitations in the
SU($N$)-theory cannot fit into the $N^{\prime}\times N^{\prime}$ block beyond
the GWW point of the SU($N^{\prime}$)-theory. It reasonably tells us that the
SU($N$)-theory at $E-E_{0}=\frac{N^{\prime 2}}{4}$ is dominated by the
SU($N^{\prime}$)-partially-deconfined states and the deconfined sector should
be identified with the GWW point of the SU($N^{\prime}$)-theory.

By switching the letter $N^{\prime}$ to $M$, we obtain the precise picture of
the SU($M$)-deconfined states in the free theory: SU($M$)-deconfined states
are identified with the states at the GWW point of the SU($M$)-theory. See
also Fig. 8. The Polyakov loop $P$ in the SU($N$)-theory should be related to
$M$ by $N^{2}P^{2}=\frac{M^{2}}{4}$, or equivalently,

$\displaystyle P=\frac{M}{2N}\,.$ (21)

This is consistent with (18): $M$ of the $N$ phases correspond to the GWW
point, while the rest correspond to the confined phase (uniform distribution).
A more precise understanding regarding this point is given in Sec. 2.2.

Figure 7:  With the absence of the interaction, SU($N$)-theory and
SU($N^{\prime}$)-theory behave in the same way until the GWW point of the
SU($N^{\prime}$), up to zero-point energy $E_{0}$. From this, it follows that
the SU($M$)-deconfined sector in the SU($N$)-theory corresponds to the GWW
point of the SU($M$)-theory.

Figure 8:  With the absence of the interaction, the deconfined sector of
SU($N$)-theory and SU($N^{\prime}$)-theory behave in the same way up to
$M=N^{\prime}$. From this, it follows that the SU($M$)-deconfined sector in
the SU($N$)-theory corresponds to the GWW-transition point of the SU($M$)
theory. These pictures appeared originally in Ref. [7].

Step 3 is rather obvious from the above discussion. We can see the
contributions from confined and deconfined sectors in $E$, $S$, and
$\rho(\theta)$ as follows:

$\displaystyle E(T=T_{\rm
c},P=\frac{M}{2N},N)={\color[rgb]{0,0,1}\frac{D}{2}\cdot(N^{2}-M^{2})}+{\color[rgb]{1,0,0}\left(\frac{D}{2}+\frac{1}{4}\right)\cdot
M^{2}}\,,$ (22) $\displaystyle S(T=T_{\rm
c},P=\frac{M}{2N},N)={\color[rgb]{0,0,1}0\cdot(N^{2}-M^{2})}+{\color[rgb]{1,0,0}\frac{\log
D}{4}\cdot M^{2}}\,,$ (23) $\displaystyle\left.\rho(\theta)\right|_{T=T_{\rm
c},P=\frac{M}{2N}}={\color[rgb]{0,0,1}\frac{1}{2\pi}\cdot\left(1-\frac{M}{N}\right)}+{\color[rgb]{1,0,0}\frac{1}{2\pi}\left(1+\cos\theta\right)\cdot\frac{M}{N}}\,.$
(24)

Blue and red are contributions from the confined and deconfined sectors,
respectively. The SU($M$)-deconfined states dominate microcanonical
thermodynamics because they explain the entropy precisely.

###  2.2 Underlying mechanism

In the previous subsection, we confirmed the formation of the
SU($M$)-deconfined block by direct computations. Admittedly, such computations
are not tractable for generic interacting theories. Furthermore, although we
could confirm partial deconfinement, the physics principle behind it was not
clear. In this subsection, we explain the underlying mechanism of partial
deconfinement that generalizes to the interacting theories with generic matter
content. The key is a deep connection between Bose-Einstein condensation (BEC)
and color confinement [17]. In retrospect, historically the first example of
non-Abelian gauge theory was the system of $N$ identical bosons, which has SN
permutation group as the gauge group. It was also the first example of the
large-$N$ gauge theory because the thermodynamic limit $N\to\infty$ was
studied. As discovered by Einstein, Bose-Einstein condensation is formed at
low temperatures. As we will see, BEC and confinement are essentially the same
phenomenon.

Let us consider a system of $N$ indistinguishable bosons in the harmonic
potential. The Hamiltonian is given by

$\displaystyle\hat{H}=\frac{1}{2}\sum_{i=1}^{N}\left(\hat{p}_{xi}^{2}+\hat{x}_{i}^{2}+\hat{p}_{yi}^{2}+\hat{y}_{i}^{2}+\hat{p}_{zi}^{2}+\hat{z}_{i}^{2}\right).$
(25)

This Hamiltonian is invariant under the SN permutation group which exchanges
particles, $i\to\sigma(i)$ ($\sigma\in\textrm{S}_{N}$). That $N$ bosons are
“indistinguishable” means that this SN symmetry is gauged.

It is convenient to use the Fock states to describe the system. We use
$\vec{n}_{i}=(n_{x,i},n_{y,i},n_{z,i})$ to denote the excitation level of the
$i$-th boson. Then, the Fock states are defined as

$\displaystyle\ket{\vec{n}_{1},\vec{n}_{2},\cdots,\vec{n}_{N}}\equiv\prod_{j=x,y,z}\frac{\hat{a}_{1j}^{\dagger
n_{1j}}}{\sqrt{n_{1j}!}}\frac{\hat{a}_{2j}^{\dagger
n_{2j}}}{\sqrt{n_{2j}!}}\cdots\frac{\hat{a}_{Nj}^{\dagger
n_{Nj}}}{\sqrt{n_{Nj}!}}\ket{0}.$ (26)

These states span the extended Hilbert space $\mathcal{H}_{\textrm{ext}}$. The
Fock vacuum $\ket{0}=\ket{\vec{0},\cdots,\vec{0}}$ is the ground state.

The canonical partition function can be written in two ways, like in (5) and
(9). Here we use the latter. By replacing $G$ and integral with SN and sum, we
obtain

$\displaystyle
Z=\frac{1}{N!}\sum_{\sigma\in\mathrm{S}_{N}}\textrm{Tr}_{\mathcal{H}_{\rm
ext}}\left(\hat{\sigma}e^{-\beta\hat{H}}\right).$ (27)

Here, $N!$ is the number of the elements of SN, which corresponds to the
volume of $G$. Trace over extended Hilbert space is the sum over Fock states,
and hence

$\displaystyle Z$
$\displaystyle=\frac{1}{N!}\sum_{\sigma\in\mathrm{S}_{N}}\sum_{\vec{n}_{1},\cdots,\vec{n}_{N}}\langle\vec{n}_{1},\cdots,\vec{n}_{N}|\hat{\sigma}e^{-\beta\hat{H}}|\vec{n}_{1},\cdots,\vec{n}_{N}\rangle$
$\displaystyle=\frac{1}{N!}\sum_{\vec{n}_{1},\cdots,\vec{n}_{N}}e^{-\beta\left(E_{\vec{n}_{1}}+\,\cdots\,+E_{\vec{n}_{N}}\right)}\sum_{\sigma\in\mathrm{S}_{N}}\langle\vec{n}_{1},\cdots,\vec{n}_{N}|\vec{n}_{\sigma(1)},\cdots,\vec{n}_{\sigma(N)}\rangle$
$\displaystyle=\frac{1}{N!}\sum_{\vec{n}_{1},\cdots,\vec{n}_{N}}e^{-\beta\left(E_{\vec{n}_{1}}+\,\cdots\,+E_{\vec{n}_{N}}\right)}\sum_{\sigma\in\mathrm{S}_{N}}\delta_{\vec{n}_{1}\vec{n}_{\sigma(1)}}\delta_{\vec{n}_{2}\vec{n}_{\sigma(2)}}\cdots\delta_{\vec{n}_{N}\vec{n}_{\sigma(N)}}$
$\displaystyle\equiv\frac{1}{N!}\sum_{\vec{n}_{1},\cdots,\vec{n}_{N}}e^{-\beta\left(E_{\vec{n}_{1}}+\,\cdots\,+E_{\vec{n}_{N}}\right)}V_{\ket{\vec{n}_{1},\cdots,\vec{n}_{N}}}.$
(28)

Here,

$\displaystyle
V_{\ket{\vec{n}_{1},\cdots,\vec{n}_{N}}}\equiv\sum_{\sigma\in\mathrm{S}_{N}}\delta_{\vec{n}_{1}\vec{n}_{\sigma(1)}}\delta_{\vec{n}_{2}\vec{n}_{\sigma(2)}}\cdots\delta_{\vec{n}_{N}\vec{n}_{\sigma(N)}}$
(29)

is the volume of the stabilizer of the state
$\ket{\vec{n}_{1},\cdots,\vec{n}_{N}}$ in the extended Hilbert space
$\mathcal{H}_{\rm ext}$, i.e.,

$\displaystyle V_{\ket{\vec{n}_{1},\cdots,\vec{n}_{N}}}$
$\displaystyle=\left(\textrm{\\#\ of}\ \sigma\in\textrm{S}_{N}\ \textrm{s.t.}\
\ket{\vec{n}_{1},\cdots,\vec{n}_{N}}=\hat{\sigma}\ket{\vec{n}_{1},\cdots,\vec{n}_{N}}\right)$
$\displaystyle=\left(\textrm{The\ volume\ of\ the\ stabilizer\ of\
}\ket{\vec{n}_{1},\cdots,\vec{n}_{N}}\ \textrm{in}\ \mathcal{H}_{\rm
ext}\right)\,.$ (30)

If all $N$ particles are in different states, only $\sigma=\textbf{1}$ gives
rise to a nonzero contribution, and hence, the volume of the stabilizer is 1.
On the other hand, if all $N$ particles are in the same state, all $\sigma$’s
return the same nonzero contribution, and hence, the volume of the stabilizer
is $N!$. For a state of the form
$\ket{\vec{n}_{1},\cdots,\vec{n}_{M},\vec{0},\cdots,\vec{0}}$, where
$\vec{n}_{1},\cdots,\vec{n}_{M}$ are generic excitations and most of them are
different from each other, the BEC-sector consisting of $N-M$ particles in the
ground state leads to the enhancement factor $(N-M)!$. BEC is formed, i.e.,
many particles fall into the ground state, due to such an enhancement factor.

The same mechanism explains partial deconfinement. An SU($M$)-deconfined state
$\ket{\Phi}$ is invariant under the action of
$\textrm{SU}(N-M)\subset\textrm{SU}(N)$, and hence999 We use $\sim$ in (31)
because the deconfinement sector may have a small stabilizer.

$\displaystyle V_{\ket{\Phi}}\sim\textrm{vol}(\textrm{SU}(N-M))\sim
e^{(N-M)^{2}}.$ (31)

The excitations are rolled up to form the $M\times M$ block structure because
of this enhancement factor.

Note that such an enhancement can exist for more complicated states that
consist of the low-energy regime of the interacting theories. The explicit
construction for the matrix model at nonzero coupling is given in Sec. 2.3.

As explained in Appendix C, $g\in G$ in (9) is the Polyakov line. Then,
$\sigma\in\textrm{S}_{N}$ in (27) can also be regarded as the Polyakov line.
Let us use $\mathcal{P}$ to denote them. When BEC or confinement set in,
typical states dominating thermodynamics are invariant under
$\text{S}_{N-M}\subset\text{S}_{N}$ or
$\textrm{SU}(N-M)\subset\textrm{SU}(N)$. Specifically, by choosing the
SU($N-M$) to correspond to the lower-right $(N-M)\times(N-M)$-block, we can
choose the SU($M$)-deconfined sector to be the upper-left $M\times M$-block as
in Fig. 2. This choice of embedding of SU($N-M$) into SU($N$) fixes SU($N$)
down to $\mathrm{SU}(M)\times\mathrm{SU}(N-M)$. The Polyakov line
$\mathcal{P}$ takes the following form:

$\displaystyle\mathcal{P}=\left(\begin{array}[]{cc}\mathcal{P}_{\rm dec}&0\\\
0&\mathcal{P}_{\rm con}\end{array}\right)\,.$ (34)

Here, $\mathcal{P}_{\rm con}$ can be any element of SU($N-M$), and the generic
phase distribution in this part is uniform in the limit of $N-M\to\infty$. The
phases of $\mathcal{P}_{\rm dec}$ and $\mathcal{P}_{\rm con}$ are
$\theta_{1},\cdots,\theta_{M}$ and $\theta_{M+1},\cdots,\theta_{N}$,
respectively. We can determine the distribution of the phases $\rho_{\rm
dec}(\theta)=\rho_{\rm GWW}(\theta)$ and $\rho_{\rm con}(\theta)$. The latter
is constant,

$\displaystyle\rho_{\rm con}(\theta)=\frac{1}{2\pi}\,,$ (35)

while the former is not and its smallest value is zero. As we will discuss in
Sec. 3 and Sec. 4.1, this constant distribution is tightly connected to the
concept of Haar randomness. For the model under consideration, we can fix
center symmetry in such a way that

$\displaystyle\rho_{\rm GWW}(\pm\pi)=0\,.$ (36)

The full distribution is

$\displaystyle\rho(\theta)$
$\displaystyle=\left(1-\frac{M}{N}\right)\cdot\rho_{\rm
con}(\theta)+\frac{M}{N}\cdot\rho_{\rm GWW}(\theta)$
$\displaystyle=\frac{1}{2\pi}\cdot\left(1-\frac{M}{N}\right)+\frac{M}{N}\cdot\rho_{\rm
GWW}(\theta)\,.$ (37)

Figure 9: Sketches of the distribution of the phases of the Polyakov line in
the partially-deconfined phase. Constant offset comes from the confined
Polyakov loop $\tr\mathcal{P}_{\rm con}$ while the non-uniform part comes from
the deconfined Polyakov loop $\tr\mathcal{P}_{\rm dec}$. This picture is taken
from Ref. [16].

See Fig. 9. An important consequence is the relationship between the phase
distribution and the size of the deconfined sector, which is

$\displaystyle{\rm The\ constant\
offset}=\frac{1}{2\pi}\left(1-\frac{M}{N}\right).$ (38)

This relation holds for the system of indistinguishable bosons as well. The
counterpart of $\mathcal{P}_{\rm con}$ can be any element of SN-M, which leads
to the same constant offset. Indeed, Feynman used a similar characterization
for superfluid. The ‘Polakov line’ Feynman studied was the permutation matrix
associated with the twisted boundary condition in the finite-temperature
Euclidean path integral. If Feynman had known the Polyakov line and Gross-
Witten-Wadia transition in 1953, he could have identified the onset of BEC
with the GWW point!

###  2.3 Partial deconfinement at nonzero coupling

In free theories, some degrees of freedom can be excited without affecting
others. We can split the Hilbert space into confined and deconfined sectors
without caring about the interaction between them (because there is no
interaction!) and perform simple analyses as we have done in Sec. 2.1. In this
subsection, we explain how the characterization based on the unbroken symmetry
in the extended Hilbert space and the relation (38) can be used even for the
interacting theories [18, 19].

The key concept is the wave packet in the space in which the matrix components
live. Let us first recall the wave packet in a one-particle system, and then
generalize it to a gauged matrix model. We will formulate SU($M$)-deconfined
wave packets which are invariant under SU($N-M$). Generic SU($M$)-deconfined
states can be written as linear combinations of such wave packets.

#### Wave packet in one-particle system

Let us consider a particle moving in a one-dimensional space. The Hamiltonian
is

$\displaystyle\hat{H}=\frac{\hat{p}^{2}}{2}+V(\hat{x}).$ (39)

Usually, energy eigenstates are far from our classical intuition. To connect
classical and quantum descriptions, we use a wave packet localized both in
coordinate and momentum bases. We can obtain a low-energy wave packet
$\ket{y,q}$ localized about $y$ in the coordinate basis and $q$ in the
momentum basis by minimizing the energy $\bra{y,q}\hat{H}\ket{y,q}$ with
constraints $\bra{y,q}\hat{x}\ket{y,q}=y$ and $\bra{y,q}\hat{p}\ket{y,q}=q$.
When the size of the wave packet is negligible compared to the typical values
of $q$ and $y$, such a wave packet can be interpreted as a classical particle
at location $y$ moving with momentum $q$.

As a concrete example, let us consider the harmonic potential,
$V(\hat{x})=\frac{\hat{x}^{2}}{2}$. The low-energy wave packet mentioned above
is the coherent state

$\displaystyle\ket{y,q}=e^{i(q\hat{x}-y\hat{p})}\ket{0,0},$ (40)

where $\ket{0,0}$ is the same as the Fock vacuum. When $q=0$, it is merely a
Gaussian wave packet in the coordinate basis,

$\displaystyle\bra{x}\ket{y,q=0}=\frac{e^{-(x-y)^{2}/2}}{\pi^{1/4}}.$ (41)

In terms of the creation and annihilation operators
$\hat{a}^{\dagger}=\frac{\hat{x}-i\hat{p}}{\sqrt{2}}$ and
$\hat{a}=\frac{\hat{x}+i\hat{p}}{\sqrt{2}}$, the coherent state is expressed
as

$\displaystyle\ket{y,q}=e^{\xi\hat{a}^{\dagger}-\xi^{\ast}\hat{a}}\ket{0}=e^{-\frac{1}{2}\xi^{\ast}\xi}e^{\xi\hat{a}^{\dagger}}e^{-\xi^{\ast}\hat{a}}\ket{0}\,,\qquad\xi=\frac{y+iq}{\sqrt{2}}\,,$
(42) $\displaystyle
e^{i(q\hat{x}-y\hat{p})}\ket{0}=e^{-\frac{1}{2}\xi^{\ast}\xi}\sum_{n=0}^{\infty}\frac{\xi^{n}}{\sqrt{n!}}\ket{n}$
(43)

and

$\displaystyle\hat{a}\ket{y,q}=\xi\ket{y,q}\,.$ (44)

Combined with $\hat{H}=\hat{a}^{\dagger}\hat{a}+\frac{1}{2}$, we obtain

$\displaystyle\bra{y,q}\hat{H}\ket{y,q}=\xi\xi^{\ast}+\frac{1}{2}=\frac{y^{2}+q^{2}+1}{2}\,.$
(45)

Time evolution is simply

$\displaystyle e^{-i\hat{H}t}\ket{\xi}$
$\displaystyle=e^{-\frac{1}{2}\xi^{\ast}\xi}\sum_{n=0}^{\infty}\frac{e^{-i(n+1/2)t}\xi^{n}}{\sqrt{n!}}\ket{n}$
$\displaystyle=e^{-it/2}e^{-\frac{1}{2}\xi^{\ast}\xi}\sum_{n=0}^{\infty}\frac{(e^{-it}\xi)^{n}}{\sqrt{n!}}\ket{n}\,.$
(46)

Therefore, $\xi=\frac{y+iq}{\sqrt{2}}$ evolves as $e^{-it}\xi$, i.e., the
center of the wave packet behaves as a classical harmonic oscillator. (Note
that the overall phase factor $e^{-it/2}$ does not affect the interpretation.)

#### Wave packet in matrix space

Now we consider the extended Hilbert space of the matrix model. The only
difference from the one-particle example is that the coordinate space and
momentum space are $\mathbb{R}^{DN^{2}}$ rather than $\mathbb{R}$. We consider
a wave packet around $Y\in\mathbb{R}^{DN^{2}}$ in the coordinate basis and
$Q\in\mathbb{R}^{DN^{2}}$ in the momentum basis. A natural way to obtain such
a low-energy wave packet is to find $\ket{Y,Q}$ that minimizes the energy

$\displaystyle\langle Y,Q|\hat{H}|Y,Q\rangle$ (47)

satisfying the constraints

$\displaystyle\langle Y,Q|\hat{X}_{I}|Y,Q\rangle=Y_{I},\qquad\langle
Y,Q|\hat{P}_{I}|Y,Q\rangle=Q_{I}.$ (48)

For the free theory, such a wave packet is the coherent state:

$\displaystyle\ket{Y,Q}=e^{i\textrm{Tr}(Q_{I}\hat{X}_{I}-Y_{I}\hat{P}_{I})}\ket{0,0}.\qquad\textrm{(for
free theory)}$ (49)

The energy is (again, for the free theory)

$\displaystyle\bra{Y,Q}\hat{H}\ket{Y,Q}=\frac{1}{2}\left(\sum_{I}\textrm{Tr}(Y_{I}^{2})+\sum_{I}\textrm{Tr}(Q_{I}^{2})+DN^{2}\right).$
(50)

The ground state $\ket{Y=0,Q=0}$ is the Fock vacuum, which is
SU($N$)-invariant. Coherent states are transformed as

$\displaystyle\ket{Y,Q}\to\ket{U^{-1}YU,U^{-1}QU}.$ (51)

See Fig. 10. To see it by using equations, it is instructive to see a Gaussian
wave packet in the coordinate basis

$\displaystyle\bra{X}\ket{Y,Q=0}=\frac{e^{-\sum_{I=1}^{D}\textrm{Tr}(X_{I}-Y_{I})^{2}/2}}{\pi^{DN^{2}/4}}\,.$
(52)

From this, we can see that

$\displaystyle\ket{Y,Q=0}=\int d^{DN^{2}}X\ket{X}\bra{X}\ket{Y,Q=0}$
$\displaystyle\qquad\to\int
d^{DN^{2}}X\ket{U^{-1}XU}\frac{e^{-\sum_{I=1}^{D}\textrm{Tr}(X_{I}-Y_{I})^{2}/2}}{\pi^{DN^{2}/4}}$
$\displaystyle\qquad\quad=\int
d^{DN^{2}}X\ket{X}\frac{e^{-\sum_{I=1}^{D}\textrm{Tr}(UX_{I}U^{-1}-Y_{I})^{2}/2}}{\pi^{DN^{2}/4}}$
$\displaystyle\qquad\quad=\int
d^{DN^{2}}X\ket{X}\frac{e^{-\sum_{I=1}^{D}\textrm{Tr}(X_{I}-U^{-1}Y_{I}U)^{2}/2}}{\pi^{DN^{2}/4}}\,.$
(53)

Note that the transformation (51) is not affected by the center-transformation
$U\to U^{\prime}=e^{i\delta}U$, where $\delta$ is any phase factor for U($N$)
theory and $\mathbb{Z}_{N}$-phase factor $\frac{2\pi in}{N}(n=1,2,\cdots,N-1)$
for SU($N$) theory, respectively. When we study the distribution of Polyakov
line phases, we can use the center transformation to make
$P=\textrm{Tr}\mathcal{P}$ as close to real and positive as
possible.101010This is not the case for theories without center symmetry such
as QCD; see Sec. 5.1.

Because of the symmetry in the minimization problem, the same transformation
law (51) applies to the low-energy wave packets in the interacting theory,
while the shape of the wave packet, such as the width along each direction,
depends on the details of the interaction. States that transform to each other
via gauge transformation should be identified. The corresponding element in
$\mathcal{H}_{\rm inv}$ can be obtained by acting the projector $\hat{\pi}$
defined in (10), up to a normalization constant:

$\displaystyle\ket{Y,Q}\to\mathcal{N}^{-1/2}\hat{\pi}\ket{Y,Q},\qquad\mathcal{N}\equiv\bra{Y,Q}\hat{\pi}\ket{Y,Q}.$
(54)

See Fig. 11. A gauge singlet corresponds to a gauge orbit in $\mathcal{H}_{\rm
ext}$. We do not expect that the expectation value of $\hat{H}$ changes
significantly,111111 This is not the case when localized wave packets are
constructed from the coordinate eigenstates by taking linear combinations,
because the coordinate eigenstates are not localized in the momentum space and
a significant reduction of the kinetic energy can be achieved.  because the
overlap between $\ket{Y,Q}$ and $\ket{U^{-1}YU,U^{-1}QU}$ is very small unless
$U$ is very close to the stabilizer of $\ket{Y,Q}$. The normalization factor
$\mathcal{N}$ is essentially the volume of the stabilizer of $\ket{Y,Q}$. If
the stabilizer is large, then the enhancement factor in the partition function
is large.

Each wave packet smoothly spreads in the coordinate space
$\mathbb{R}^{DN^{2}}$, and also in the momentum space $\mathbb{R}^{DN^{2}}$.
Along each of $DN^{2}$ dimensions, the size is at most of order $N^{0}$. Two
wave packets centered around $Y$ and $Y^{\prime}$ in $\mathbb{R}^{DN^{2}}$
have vanishingly small overlap if the centers of wave packets are well
separated, i.e., $\sqrt{{\rm Tr}(Y-Y^{\prime})^{2}}\gg 1$.

Figure 10:  Wave packets in the coordinate space $\mathbb{R}^{DN^{2}}$. Each
wave packet smoothly spreads in $\mathbb{R}^{DN^{2}}$. Along each of $DN^{2}$
dimensions, the size of the wave packet is at most of order $N^{0}$. Black
points are the centers of wave packets ($Y$, $Y^{(U)}$ and
$Y^{(U^{\prime})}$), and gray disks are the wave packets (more precisely, the
regions where the wave functions are not vanishingly small). Under the gauge
transformation, the center of the wave packet moves as $Y\to
Y^{(U)}=U^{-1}YU$, but the shape of the wave packet does not change. The
center of the wave packet gives ‘slow modes’ that describe the geometry
consisting of D-branes and strings. This figure is a slight modification of a
similar one in Ref. [18].  Figure 11:  The gauge-singlet that is obtained by
acting the projection operator $\hat{\pi}$ on the wave packet $\ket{Y,Q}$. The
wave function is localized about the gauge orbit of $\\{Y_{I,\alpha}\\}$ in
$\mathbb{R}^{DN^{2}}$. This figure is a slight modification of the one in Ref.
[18].

#### Partial deconfinement at nonzero coupling

The SU($M$)-deconfined states can be constructed as linear combinations of
wave packets of the form

$\displaystyle Y_{I}=\left(\begin{array}[]{cc}\tilde{Y}_{I}&0\\\
0&0\end{array}\right),\qquad Q_{I}=\left(\begin{array}[]{cc}\tilde{Q}_{I}&0\\\
0&0\end{array}\right).$ (59)

Here, $\tilde{Y}_{I}$ and $\tilde{Q}_{I}$ are $M\times M$ matrices. By
construction, such wave packets are invariant under SU($N-M$)-transformations
that do not touch the upper-right $M\times M$ block, i.e.,

$\displaystyle\left(\begin{array}[]{cc}\textbf{1}_{M}&0\\\
0&\Omega^{-1}\end{array}\right)\left(\begin{array}[]{cc}\tilde{Y}_{I}&0\\\
0&0\end{array}\right)\left(\begin{array}[]{cc}\textbf{1}_{M}&0\\\
0&\Omega\end{array}\right)=\left(\begin{array}[]{cc}\tilde{Y}_{I}&0\\\
0&0\end{array}\right)$ (68)

and

$\displaystyle\left(\begin{array}[]{cc}\textbf{1}_{M}&0\\\
0&\Omega^{-1}\end{array}\right)\left(\begin{array}[]{cc}\tilde{Q}_{I}&0\\\
0&0\end{array}\right)\left(\begin{array}[]{cc}\textbf{1}_{M}&0\\\
0&\Omega\end{array}\right)=\left(\begin{array}[]{cc}\tilde{Q}_{I}&0\\\
0&0\end{array}\right)$ (77)

hold for any $\Omega\in{\rm SU}(N-M)$. Let us write a generic linear
combination of such states as

$\displaystyle|\Phi\rangle=\sum_{a=1}^{K}c_{a}|Y_{[a]},Q_{[a]}\rangle\,,\qquad
K=e^{S}\,,\qquad|c_{a}|\sim\frac{1}{\sqrt{K}}\,.$ (78)

Here, $S$ is the coarse-grained entropy that counts independent wave packets,
which is $\sim M^{2}$ at weak coupling. By construction, such linear
combinations are invariant under the same SU($N-M$)-transformations.
Therefore, the mechanism explained in Sec. 2.2 is applicable even with the
presence of the interactions. The invariance under the
$SU(N-M)$-transformation leads to the enhancement factor $\sim e^{(N-M)^{2}}$.
The value of $M$ is determined in such a way that the product of this
enhancement factor and $K\sim e^{S}$ is maximized.

Such states are quite intriguing also from the D-brane point of view: the
SU($M$)-deconfined states are interpreted through the holographic perspective
by the situation that $M$ D-branes and open strings between them are excited
and form an extended object. Other $N-M$ D-branes are sitting at the origin
and no open strings are excited between them, or between those $N-M$ D-branes
and the other $M$ D-branes. Note that all D-branes contribute to the exterior
geometry. Such states can describe a small black hole, with no radiation
emitted yet [18, 23, 8].

###  2.4 Euclidean path integral and Monte Carlo simulation

Euclidean path integral and Monte Carlo simulation are powerful and, at least
at present, practically the only tools to study strongly-coupled
nonperturbative dynamics of matrix models quantitatively. Therefore, it is
important to understand how partial deconfinement can be observed in path-
integral formalism. Specifically, we want a gauge fixing that makes the
splitting of confined and deconfined sectors manifest as in Fig. 2.

The relation between the Polyakov loop and partial deconfinement reviewed in
Sec. 2.2 plays the central role for this purpose. If the splitting depicted in
Fig. 2 is realized, then the Polyakov loop and its phase distribution take the
forms shown in Fig. 9 and Eqs. (34) — (37). Therefore, we take the static-
diagonal gauge, i.e.,

$\displaystyle
A_{t}(t)=T\times\textrm{diag}(\theta_{1},\cdots,\theta_{N})\,,\qquad-\pi\leq\theta_{i}<\pi\,,$
(79)

where $\theta_{1},\cdots,\theta_{N}$ are the phases of the Polyakov line. They
fix the residual SN permutation symmetry down to
$\mathrm{S}_{M}\times\mathrm{S}_{N-M}$ in such a way that
$\theta_{M+1},\cdots,\theta_{N}$ are distributed uniformly and describe the
constant offset in the density distribution function [15, 16]. Associated with
the static diagonal gauge, the Faddeev-Popov term of the following form is
added [24, 25]:

$\displaystyle S_{\rm
F.P.}=-\sum_{i<j}2\log\left|\sin\left(\frac{\theta_{i}-\theta_{j}}{2}\right)\right|\,.$
(80)

Note that the Polyakov loop takes the same form everywhere on the temporal
circle, and hence, the same separation of the form shown in Fig. 2 is realized
everywhere.

This gauge fixing was applied to the matrix model [15] and partial
deconfinement was demonstrated by using lattice Monte Carlo simulations. Ref.
[16] studied the flux tube in the confined sector of the partially-deconfined
saddle of Yang-Mills theory, as we will review in Sec. 3.3.1.

###  2.5 Introducing fermions

In this subsection, we add fermions to the analysis. Let $\psi$ be a complex
adjoint fermion as a concrete example. We regard $\hat{\psi}^{\dagger}$ and
$\hat{\psi}$ as the creation and annihilation operators, respectively. The
anticommutation relation is

$\displaystyle\\{\hat{\psi}_{\alpha,a},\hat{\psi}^{\dagger}_{\beta,b}\\}=\delta_{\alpha\beta}\delta_{ab}$
(81)

where $\alpha,\beta$ are spinor indices and $a,b$ are adjoint indices.

Let us start with the free theory, with the fermionic part of the Hamiltonian
$\mu{\rm Tr}(\hat{\psi}^{\dagger}\hat{\psi})$, $\mu>0$. Then, the Fock vacuum
$\ket{0}=\otimes_{\alpha,a}\ket{0}_{\alpha,a}$ characterized by
$\hat{\psi}_{\alpha,a}\ket{0}=0$ is the ground state. Note that the Fock
vacuum is SU($N$)-invariant. The highest excited level
$\otimes_{\alpha,a}\ket{1}_{\alpha,a}$ is also SU($N$)-invariant. Fermions are
excited ($\hat{\psi}^{\dagger}$ can be multiplied) only in the deconfined
sector.

For bosons, we used the wave-packet description for the generalization to
finite coupling. The reason was that the coordinate eigenstates were not low-
energy states. Ultimately, it was because the dimension of the Hilbert space
for bosons is infinite. The situation is different for fermions, and it is not
clear if we need to use such a thing like a ‘wave packet’. Furthermore, it is
difficult to make sense of the ‘wave packet’ of fermionic degrees of freedom.
In fact, a coherent state for fermion is tricky because it is written as
$\ket{\eta}=\ket{0}+\eta\ket{1}$ where $\eta$ is a Grassmann number, and
hence, it is not a quantum state in the usual sense. Note also that Fock
states $\ket{0}$ and $\ket{1}$ are already analogous to wave packets, in that
they are low-energy Fock states. (The Fock vacuum for boson is the Gaussian
wave packet around the origin.)

Below, we use wave packets only for the bosonic sector. Let $\mathcal{H}_{\rm
ext}=\mathcal{H}_{\rm ext,b}\otimes\mathcal{H}_{\rm ext,f}$. For each
coordinate eigenstate $\ket{X}\in\mathcal{H}_{\rm ext,b}$, we can associate
$\ket{\psi_{X}}\in\mathcal{H}_{\rm ext,f}$ and obtain a generic state in
$\mathcal{H}_{\rm ext}$ as $\int dXf(X)\ket{X}\otimes\ket{\psi_{X}}$. Such a
state is transformed as

$\displaystyle\ket{\Phi}=\int
dXf(X)\ket{X}\otimes\ket{\psi_{X}}\longrightarrow\hat{U}\ket{\Phi}$
$\displaystyle=\int dXf(X)\ket{U^{-1}XU}\otimes(\hat{U}\ket{\psi_{X}})$
$\displaystyle=\int
dXf(UXU^{-1})\ket{X}\otimes(\hat{U}\ket{\psi_{UXU^{-1}}})\,.$ (82)

The stabilizer $G_{\ket{\Phi}}$ is

$\displaystyle G_{\ket{\Phi}}=\\{U\in{\rm SU}(N)|f(UXU^{-1})=f(X)\ {\rm and}\
\hat{U}\ket{\psi_{UXU^{-1}}}=\ket{\psi_{X}}\\}\,.$ (83)

If we define a low-energy wave packet by minimizing the energy fixing the one-
point function of $\hat{X}_{I}$ and $\hat{P}_{I}$ to $Y_{I}$ and $Q_{I}$, it
is natural to expect that the stabilizer of $Y_{I}$ and $Q_{I}$ satisfy (83)
due to the symmetry of the problem.

#### Analogy to BEC with fermionic degrees of freedom

The analogy between confinement and BEC is not affected by the existence of
fermions. Let us consider Fock space for fermionic oscillators,

$\displaystyle\ket{\vec{n}_{1},\vec{n}_{2},\cdots,\vec{n}_{N}}\equiv\prod_{j}\hat{c}_{1j}^{\dagger
n_{1j}}\hat{c}_{2j}^{\dagger n_{2j}}\cdots\hat{c}_{Nj}^{\dagger
n_{Nj}}\ket{0}.$ (84)

Here, $j$ represents the spinor index, flavor index, etc. Each $n_{kj}$ is 0
or 1. Let us see how the state changes by permutation. If
$\sigma\in\textrm{S}_{N}$ exchanges only 1 and 2 (i.e.,
$\sigma(1)=2,\sigma(2)=1,\sigma(j)=j$ for $j\geq 3$), then

$\displaystyle\hat{\sigma}\ket{\vec{n}_{1},\vec{n}_{2},\vec{n}_{3},\cdots,\vec{n}_{N}}$
$\displaystyle\equiv\ket{\vec{n}_{2},\vec{n}_{1},\vec{n}_{3},\cdots,\vec{n}_{N}}$
$\displaystyle=\prod_{j}\hat{c}_{2j}^{\dagger n_{1j}}\hat{c}_{1j}^{\dagger
n_{2j}}\cdots\hat{c}_{Nj}^{\dagger n_{Nj}}\ket{0}$
$\displaystyle=\prod_{j}(-1)^{n_{1j}n_{2j}}\hat{c}_{1j}^{\dagger
n_{2j}}\hat{c}_{2j}^{\dagger n_{1j}}\cdots\hat{c}_{Nj}^{\dagger
n_{Nj}}\ket{0}$
$\displaystyle=(-1)^{\sum_{j}n_{1j}n_{2j}}\ket{\vec{n}_{1},\vec{n}_{2},\vec{n}_{3},\cdots,\vec{n}_{N}}.$
(85)

Here, we assumed that the Fock vacuum is permutation invariant. This
assumption is consistent with (85), by setting all $\vec{n}_{i}$ to be
$\vec{0}$. If $\vec{n}_{1}=\vec{n}_{2}=\vec{m}$,

$\displaystyle\hat{\sigma}\ket{\vec{m},\vec{m},\vec{n}_{3},\cdots,\vec{n}_{N}}=(-1)^{\sum_{j}m_{j}}\ket{\vec{m},\vec{m},\vec{n}_{3},\cdots,\vec{n}_{N}}.$
(86)

Therefore, $\ket{\vec{m},\vec{m},\vec{n}_{3},\cdots,\vec{n}_{N}}$ and
$\hat{\sigma}\ket{\vec{m},\vec{m},\vec{n}_{3},\cdots,\vec{n}_{N}}$ cancel out
by the symmetrization if $\sum_{j}m_{j}$ is odd, but they survive if
$\sum_{j}m_{j}$ is even. In this case, an even number of fermion creation
operators are combined to form a bosonic excitation.

A similar argument holds for SU($N$). Suppose the fermions are in the
fundamental representation, and consider the SU(2) subgroup acting on the
first two elements as
$\hat{c}^{\dagger}_{1j}\to\sum_{k=1}^{2}U_{1k}\hat{c}^{\dagger}_{kj}$ and
$\hat{c}^{\dagger}_{2j}\to\sum_{k=1}^{2}U_{2k}\hat{c}^{\dagger}_{kj}$. Then,

$\displaystyle\hat{c}_{1j}^{\dagger}\hat{c}_{2j}^{\dagger}\to\left(\sum_{k=1}^{2}U_{1k}\hat{c}^{\dagger}_{kj}\right)\left(\sum_{k=1}^{2}U_{2k}\hat{c}^{\dagger}_{kj}\right)=\hat{c}_{1j}^{\dagger}\hat{c}_{2j}^{\dagger}\left(U_{11}U_{22}-U_{12}U_{21}\right)=\hat{c}_{1j}^{\dagger}\hat{c}_{2j}^{\dagger}.$
(87)

Because $m_{j}$ is 0 or 1, $\hat{c}_{1j}^{\dagger m_{j}}\hat{c}_{2j}^{\dagger
m_{j}}$, and hence $\ket{\vec{m},\vec{m},\vec{n}_{3},\cdots,\vec{n}_{N}}$, are
invariant under SU(2).

##  3 Partial deconfinement in large-$N$ QFT

Let us consider gauge theory living in nonzero spatial dimensions, i.e., QFT.
Intuitively, the mechanism introduced for matrix models should apply locally.
The precise meaning of this is discussed below. In short, the same argument
can be used for slowly-varying SU($N$) transformations.

In Sec. 3.1, we consider Yang-Mills theory. To make the discussion precise, we
adopt lattice regularization. We use the Kogut-Susskind Hamiltonian
formulation. We start with studying the strong-coupling limit in Sec. 3.1.1.
The ground state is gauge invariant in $\mathcal{H}_{\rm ext}$, and the
mechanism discussed for the case of the matrix model is applicable without
modification. In Sec. 3.1.2, we consider the continuum limit that requires
taking weak coupling. In this case, the ground state is a localized wave
packet which is not gauge-invariant in $\mathcal{H}_{\rm ext}$. Still,
essentially the same mechanism is still applicable, by considering the slowly-
varying SU($N$) transformations. The generalization to the large-$N$ QCD is
provided in Sec. 3.2 and some physical consequences are discussed in Sec. 3.3.

Note that we consider the bare Polyakov loop without renormalization, which
corresponds to the group element in the projector from $\mathcal{H}_{\rm ext}$
to $\mathcal{H}_{\rm inv}$.

###  3.1 Yang-Mills theory

The meaning of partial deconfinement is clearer if we look directly at quantum
states by working in the operator formalism. We describe the partially-
deconfined phase by using the Hamiltonian formulation by Kogut and Susskind
[26]. We consider the (3+1)-d Yang-Mills theory with the SU($N$) gauge group.

The Kogut-Susskind Hamiltonian consists of the electric and magnetic terms,

$\displaystyle\hat{H}=\hat{H}_{\rm E}+\hat{H}_{\rm B}.$ (88)

The electric part of the Hamiltonian is

$\displaystyle\hat{H}_{\rm
E}=\frac{a^{3}}{2}\sum_{\vec{n}}\sum_{\mu=1}^{3}\sum_{\alpha=1}^{N^{2}}\left(\hat{E}_{\mu,\vec{n}}^{\alpha}\right)^{2}.$
(89)

Here $a$ is the lattice spacing, and $\hat{E}_{\mu,\vec{n}}^{\alpha}$ is the
electric field. The magnetic part is

$\displaystyle\hat{H}_{\rm
B}=-\frac{1}{2ag^{2}}\sum_{\vec{n}}\sum_{\mu<\nu}\left(\hat{U}_{\mu,\vec{n}}\hat{U}_{\nu,\vec{n}+\hat{\mu}}\hat{U}^{\dagger}_{\mu,\vec{n}+\hat{\nu}}\hat{U}^{\dagger}_{\nu,\vec{n}}+\textrm{h.c.}\right).$
(90)

At weak coupling, the link variable $U_{\mu}$ is regarded as
$U_{\mu}=e^{iagA_{\mu}}$. The commutation relations are

$\displaystyle\left[\hat{U},\hat{U}\right]=\left[\hat{U},\hat{U}^{\dagger}\right]=\left[\hat{U}^{\dagger},\hat{U}^{\dagger}\right]=0.$
(91)

In addition,

$\displaystyle\left[\hat{E}_{\mu,\vec{n}}^{\alpha},\hat{U}_{\nu,\vec{n}^{\prime}}\right]=\delta_{\mu\nu}\delta_{\vec{n}\vec{n}^{\prime}}\tau_{\alpha}\hat{U}_{\nu,\vec{n}^{\prime}},\qquad\left[\hat{E}_{\mu,\vec{n}}^{\alpha},\hat{U}^{\dagger}_{\nu,\vec{n}^{\prime}}\right]=-\delta_{\mu\nu}\delta_{\vec{n}\vec{n}^{\prime}}\hat{U}_{\nu,\vec{n}^{\prime}}^{\dagger}\tau_{\alpha},$
(92)
$\displaystyle\left[\hat{E}_{\mu,\vec{n}}^{\alpha},\hat{E}_{\nu,\vec{n}^{\prime}}^{\beta}\right]=-if^{\alpha\beta\gamma}\delta_{\mu\nu}\delta_{\vec{n}\vec{n}^{\prime}}\hat{E}^{\gamma}_{\nu,\vec{n}^{\prime}},$
(93)

where $f^{\alpha\beta\gamma}$ is the structure constant of SU($N$). 121212 We
can choose the generators in such a way that
$\textrm{Tr}(\tau^{\alpha}\tau^{\beta})=\delta^{\alpha\beta}$ and
$\sum_{\alpha=1}^{N^{2}-1}\tau^{\alpha}_{pq}\tau^{\alpha}_{rs}=\delta_{ps}\delta_{qr}-\frac{\delta_{pq}\delta_{rs}}{N}$
are satisfied.

The operator $\hat{U}_{\mu,\vec{n}}$ is interpreted as the coordinate of the
group manifold SU($N$) for the link variable on the site $\vec{n}$ in the
$\mu$-direction. The operators $\hat{U}$ and $\hat{E}$ are defined on the
extended Hilbert space $\mathcal{H}_{\rm ext}$ that contains gauge non-singlet
modes. A convenient basis of $\mathcal{H}_{\rm ext}$ is the coordinate
representation,

$\displaystyle{\cal H}_{\rm ext}=\otimes_{\mu,\vec{n}}{\cal
H}_{\mu,\vec{n}}\sim\otimes_{\mu,\vec{n}}\left(\oplus_{g\in{\rm
SU}(N)}|g\rangle_{\mu,\vec{n}}\right),$ (94)

where

$\displaystyle\hat{U}_{\mu,\vec{n}}|g\rangle_{\mu,\vec{n}}=g|g\rangle_{\mu,\vec{n}}\qquad
g\in{\rm SU}(N).$ (95)

More precisely, we will consider only the Hilbert space of square-integrable
wave functions.

Let $\mathcal{G}=\prod_{\vec{n}}[{\rm SU}(N)]_{\vec{n}}$ be the group of all
local gauge transformations. Gauge transformation by
$\hat{\Omega}=\otimes_{\vec{n}}\hat{\Omega}_{\vec{n}}\in\mathcal{G}$ is
defined by

$\displaystyle\hat{\Omega}\left(\otimes_{\mu,\vec{n}}|g\rangle_{\mu,\vec{n}}\right)=\otimes_{\mu,\vec{n}}\left(\hat{\Omega}_{\vec{n}}|g\rangle_{\mu,\vec{n}}\right)=\otimes_{\mu,\vec{n}}|\Omega_{\vec{n}}^{-1}g\Omega_{\vec{n}}\rangle_{\mu,\vec{n}}.$
(96)

####  3.1.1 Strong coupling on lattice

In the ‘strong coupling limit’ [2], we formally take the limit of
$g^{2}\to\infty$ and drop the magnetic term. Although such a limit contains
lattice artifacts, however, it is qualitatively a good model of
confinement/deconfinement transition. Partial deconfinement in this limit was
studied in Refs. [13, 16]. In the strong coupling limit, $a^{3}$ is merely an
overall factor, so we set $a=1$ for simplicity.

The ground state of this strong-coupling theory $|{\rm g.s.}\rangle$ satisfies
$\hat{E}_{\mu,\vec{n}}^{\alpha}|{\rm g.s.}\rangle=0$ for any $\alpha,\mu$, and
$\vec{n}$. Hence, we introduce the notation $|E=0\rangle$ to denote the ground
state. Because the electric field $\hat{E}_{\mu,\vec{n}}^{\alpha}$ is momentum
on the group manifold, zero modes associated with the electric field is the
constant on the group manifold,

$\displaystyle|E=0\rangle=\otimes_{\mu,\vec{n}}|E=0\rangle_{\mu,\vec{n}}\,,\qquad|E=0\rangle_{\mu,\vec{n}}=\frac{1}{\sqrt{{\rm
volSU}(N)}}\int_{{\rm SU}(N)}dg|g\rangle_{\mu,\vec{n}}\,.$ (97)

Namely, the wave function $\langle g|E=0\rangle$ is constant. The ground state
is a well-localized wave packet in the weak-coupling limit, but as coupling
grows the wave packet gradually spreads out and eventually becomes constant in
the strong-coupling limit.

Note that the ground state is gauge-invariant:

$\displaystyle\hat{\Omega}|E=0\rangle=|E=0\rangle.$ (98)

Let us consider low-energy gauge-invariant states consisting of a small number
of closed strings with a total length $L\sim N^{0}$. Then, there are at most
order-$N^{0}$ number of intersections. The interaction (splitting or joining)
at each intersection is suppressed by $\frac{1}{N}$, and hence the interaction
is negligible at large $N$. The energy of the system is simply $\frac{L}{2}$.
Such states are in the confined phase. If we introduce a probe quark-antiquark
pair connected by the open Wilson line, the energy increases linearly as
$E_{q\bar{q}}=\frac{L}{2}$. This leads to the linear confinement potential.

In the deconfined phase [1, 2], long strings with lengths of order $N^{2}$
condense. There are many intersections, and thus the $1/N$-suppressed
interactions from many intersections pile up and become non-negligible as a
whole. Intuitively, if we introduce a short open string, it interacts with a
condensed long string and forms a long open string, which allows us to
separate quark and antiquark without making the string longer.

Finally, we consider partial deconfinement. Let $\hat{U}_{\rm dec}$ be the
SU($M$)-subsector. The SU($M$)-deconfined states can be constructed by acting
with long traces of $\hat{U}_{\rm dec}$’s on $|E=0\rangle$ [11]. By using the
Wilson loops restricted to the SU($M$)-subsector

$\displaystyle\hat{W}_{{\rm dec},C}={\rm Tr}(\hat{U}_{{\rm
dec};\mu,\vec{n}}\hat{U}_{{\rm dec};\nu,\vec{n}+\hat{\mu}}\cdots),$ (99)

we can construct multi-string states

$\displaystyle\hat{W}_{{\rm dec},C}\hat{W}_{{\rm
dec},C^{\prime}}\cdots|E=0\rangle,$ (100)

and then we can take a linear combination of such states. Such states are
SU($N-M$)-invariant, but not SU($N$)-invariant. If we want to get an
SU($N$)-invariant state, we can act with the projector $\hat{\pi}$.

####  3.1.2 Continuum limit (weak coupling on lattice)

Now we discuss how the wave-packet picture generalizes to the continuum limit
of lattice-regularized Yang-Mills, which was not explicitly discussed in the
past.

To obtain Yang-Mills theory from Wilson’s plaquette action, one usually
considers fluctuation about $U_{\mu,\vec{x}}=\textbf{1}$, which corresponds to
$A_{\mu,\vec{x}}=\textbf{0}$. Strictly speaking, this is not a gauge-invariant
statement, because this configuration transforms as
$\textbf{1}\to\Omega_{\vec{n}}^{-1}\Omega_{\vec{n}+\hat{\mu}}$ by a gauge
transformation. More precisely, one considers fluctuations around a pure-gauge
configuration

$\displaystyle
U_{\mu,\vec{n}}=\Omega_{\vec{n}}^{-1}\Omega_{\vec{n}+\hat{\mu}}\,.$ (101)

Because both the plaquette action and the continuum action are
SU($N$)-invariant, this ambiguity is not a problem at all.

In terms of Hilbert space, we should have a wave packet localized around
(101). The width of the wave packet should shrink as the lattice coupling
becomes smaller. Such a state is not invariant under local SU($N$)
transformation. More generally, no localized wave packet is SU($N$)-invariant
because the center of a wave packet, which is a unitary matrix, cannot be
SU($N$)-invariant. Then, how can the mechanism of partial deconfinement
introduced in Sec. 2.2, which is based on the invariance of the wave packet
under a subgroup of SU($N$), work?

To resolve this issue, let us consider the ground state first. The ground
state should be the wave packet localized around a pure-gauge configuration
(101). Such a state is invariant under the ‘global’ SU($N$) transformation by
$\Omega^{\prime}_{\vec{n}}\equiv\Omega_{\vec{n}}^{-1}V\Omega_{\vec{n}}$, where
$V$ is independent of $\vec{n}$:

$\displaystyle\Omega_{\vec{n}}^{\prime-1}(\Omega_{\vec{n}}^{-1}\Omega_{\vec{n}+\hat{\mu}})\Omega^{\prime}_{\vec{n}+\hat{\mu}}=\Omega_{\vec{n}}^{-1}\Omega_{\vec{n}+\hat{\mu}}\,.$
(102)

The invariance under this ‘global’ SU($N$) transformation can lead to the
enhancement factor discussed in Sec. 2.2. Furthermore, essentially the same
enhancement factor can appear if $V$ is sufficiently slowly varying. Namely,
for
$\Omega^{\prime}_{\vec{n}}\equiv\Omega_{\vec{n}}^{-1}V_{\vec{n}}\Omega_{\vec{n}}$,

$\displaystyle\Omega_{\vec{n}}^{\prime-1}(\Omega_{\vec{n}}^{-1}\Omega_{\vec{n}+\hat{\mu}})\Omega^{\prime}_{\vec{n}+\hat{\mu}}=\Omega_{\vec{n}}^{-1}V_{\vec{n}}^{-1}V_{\vec{n}+\hat{\mu}}\Omega_{\vec{n}+\hat{\mu}}\,,$
(103)

and if $V_{\vec{n}}^{-1}V_{\vec{n}+\hat{\mu}}$ is sufficiently close to 1 we
can still have a significant enhancement factor.

Next, we consider the excited states. Because we consider the continuum limit,
we write generic link variables as

$\displaystyle
U_{\mu,\vec{n}}=\Omega_{\vec{n}}^{-1}e^{iaA_{\mu,\vec{n}}}\Omega_{\vec{n}+\hat{\mu}}$
(104)

and assume that $aA_{\mu,\vec{n}}$ converges to zero in the continuum limit
($a\to 0$). By expanding the lattice theory with respect to this
$A_{\mu,\vec{n}}$, we get the continuum theory. Suppose that we could take
$A_{\mu,\vec{n}}$ to be invariant under the adjoint action of
$\textrm{SU}(N-M)\subset\textrm{SU}(N)$, i.e.,
$V^{-1}A_{\mu,\vec{n}}V=A_{\mu,\vec{n}}$ for
$V\in\textrm{SU}(N-M)\subset\textrm{SU}(N)$. Then, such a state acquires the
enhancement factor under the ‘global’ SU($N-M$) transformation by
$\Omega^{\prime}_{\vec{n}}=\Omega_{\vec{n}}^{-1}V\Omega_{\vec{n}}$. Now we let
$V=e^{ia\Lambda}$ depend on $\vec{n}$, but we assume it changes slowly. Then,
by ignoring the higher-order terms with respect to the lattice spacing $a$,
the gauge transformation is

$\displaystyle A_{\mu,\vec{n}}\to
A_{\mu,\vec{n}}+i[A_{\mu,\vec{n}},\Lambda_{\vec{n}}]+\frac{\Lambda_{\vec{n}+\hat{\mu}}-\Lambda_{\vec{n}}}{a}.$
(105)

By assumption, $[A_{\mu,\vec{n}},\Lambda_{\vec{n}}]$ is zero. The last term
corresponds to the derivative $\partial\Lambda$ in the continuum limit. If
this term is much smaller than the size of the wave packet (specifically if
the shift of the center of wave packet from the origin,
$\sqrt{\textrm{Tr}(\partial\Lambda)^{2}}$, is much smaller than the spread of
the wave packet along each direction of $\mathbb{R}^{3(N^{2}-1)}$), then a
sufficiently large overlap exists between the wave packet before and after the
gauge transformation. This gives a substantial enhancement factor compared to
a generic wave packet localized around a nonzero value of $A_{\mu}$. Note that
the requirement of small $\partial\Lambda$ removes the potential ultraviolet
divergence of the contribution to free energy from the enhancement factor.

When partial deconfinement takes place, the plaquette
$U_{\square,\mu,\nu,\vec{n}}$ would be centered around

$\displaystyle\left(\begin{array}[]{cc}U^{\prime}_{\square,\mu,\nu,\vec{n}}&\textbf{0}\\\
\textbf{0}&\textbf{1}\end{array}\right)$ (108)

up to gauge transformation, and $F_{\mu\nu,\vec{n}}$ would be centered around

$\displaystyle\left(\begin{array}[]{cc}F^{\prime}_{\mu\nu,\vec{n}}&\textbf{0}\\\
\textbf{0}&\textbf{0}\end{array}\right).$ (111)

Such $V=e^{ia\Lambda}$ that is slowly varying and leaves this block structure
invariant gives the enhancement factor.

For the confined vacuum which is genuinely gauge-invariant, such a $V$ can be
arbitrary, modulo the slowly-varying nature. This fact tells us that the
distribution of Polyakov line phases must be Haar-random. We emphasize that
this is a stronger condition than the unbroken center symmetry, and it can be
generalized to QCD, even at finite $N$.

###  3.2 Large-$N$ QCD

Let us consider SU($N$) QCD with $N_{\rm f}$ flavors of fundamental quarks.
The only difference from Yang-Mills theory is the existence of fermions in the
fundamental representation. Therefore, the mechanism discussed above is
applicable to QCD.

We assume that quark chemical potential is zero so that the fermion
determinant is real and positive. Then, due to the Vafa-Witten theorem [27],
the vector part of the flavor symmetry should not be broken. Therefore, we
expect the deconfinement pattern depicted in Fig. 5. This was confirmed for
the weakly-coupled QCD on three-sphere, by re-interpreting the results
obtained by Schnitzer [6]. The temperature dependence is depicted
schematically in Fig. 12. For details, see [12].

Figure 12:  A schematic picture of the temperature dependence of the size of
the deconfined sector $M$ (see Fig. 5 for the definition) in weakly-coupled
QCD on S3 [11]. Complete confinement is realized only at zero temperature.

###  3.3 Some physical consequences

####  3.3.1 Flux tube in confined sector

In pure Yang-Mills theory, the formation of the flux tube is a good
characterization of confinement. When partial deconfinement takes place, we
naturally expect the existence of the flux tube in the confined sector.

This idea was tested for the case of large-$N$ strong-coupling limit of
lattice gauge theory [16]. To separate confined and deconfined sectors, the
gauge-fixing condition discussed in Sec. 2.4 was used. Then, the Polyakov loop
in those sectors can be defined by

$\displaystyle P_{\rm dec}=\frac{1}{M}\sum_{j=1}^{M}e^{i\theta_{j}}\,,\qquad
P_{\rm con}=\frac{1}{N-M}\sum_{j=M+1}^{N}e^{i\theta_{j}}\,.$ (112)

See also Fig. 9. String tension can be obtained from the two-point function of
Polyakov loops. Modulo mild assumptions (including that partial deconfinement
is taking place), a few exact results can be obtained and they were reproduced
by lattice simulations. For details, see Ref. [16].

####  3.3.2 Chiral symmetry breaking

There is a famous argument based on ’t Hooft anomaly that relates chiral
symmetry breaking and confinement [28]. At high energy, free quarks carry some
’t Hooft anomaly. If confinement takes place, free quarks are gone (because of
the flux tube discussed in Sec. 3.3.1), and hence some other degrees of
freedom must reproduce the same anomaly. If chiral symmetry is broken, pions
can carry the correct amount of ’t Hooft anomaly.

Nothing forbids us from applying the same idea at the intermediate energy
scale where partial deconfinement is taking place. In the confined sector,
free quarks are gone, and hence, it is natural to expect that the chiral
symmetry is broken so that pions in the confined sector can account for the
anomaly [12]. If this is the case, the chiral-symmetry breaking should set in
at or above the GWW point. Given the absence of other characteristic energy
scales, there is no strong motivation to believe that the chiral-symmetry
breaking and the GWW point do not coincide, and hence, it is natural to
conjecture that the chiral symmetry breaks at the GWW transition point. This
conjecture was confirmed for the case of large-$N$ strongly-coupled lattice
gauge theory by lattice simulations [13].

####  3.3.3 CP symmetry breaking at $\theta=\pi$

Yang-Mills theory admits the $\theta$-term that is defined by the second term
in the following Lagrangian density:

$\displaystyle\mathcal{L}=\frac{1}{4g^{2}}\textrm{Tr}(F_{\mu\nu}F^{\mu\nu})-\frac{i\theta}{8\pi^{2}}\textrm{Tr}(F_{\mu\nu}\tilde{F}^{\mu\nu})\,.$
(113)

The theory possesses the CP symmetry at $\theta=0$ and $\theta=\pi$.
Interestingly, at $\theta=\pi$, the CP symmetry breaks spontaneously in the
confined phase [29]. This motivates us to speculate that, similarly to the
chiral symmetry breaking, CP symmetry breaking coincides with the GWW
transition.

The CP-symmetry breaking at $\theta=\pi$ can be studied systematically for the
case of mass-deformed $\mathcal{N}=1$ SYM, by focusing on the weakly-coupled
region [30]. One can explicitly check that, in the large-$N$ limit, the CP
symmetry breaks at the GWW point [13]. Furthermore, one can also investigate
finite-$N$ theories. At $N\geq 3$, we can find the unstable saddle that can be
identified with the partially-deconfined phase and stable saddles that
correspond to the completely-deconfined and completely-confined phases. The
CP-symmetry breaking takes place precisely at the border between the partiall-
deconfined and completely-deconfined saddles.

##  4 Partial deconfinement at finite $N$

Crisp characterization of phase transitions associated with partial
deconfinement in the large-$N$ limit was given by using the distribution of
Polyakov line phases: the departure from the constant distribution is the
transition from the completely-confined phase to partially-deconfined phase,
and the GWW transition is the transition from the partially-deconfined phase
to completely-deconfined phase. These characterizations rely on the specific
features of the large-$N$ limit.

In this section, we discuss how partial deconfinement could be observed in
finite-$N$ theories. On top of that, we will discuss the application to SU(3)
QCD in Sec. 5. We will consider two possibilities: to find the counterparts of
constant distribution and GWW transition at finite $N$ from the Polaykov loop
(Sec. 4.1) or to introduce different characterizations that do not require
large $N$ based on global symmetry and instanton (Sec. 4.2). We can compare
them to check the consistency of our analyses of the SU(3) theory in Sec. 5.

###  4.1 Distribution of Polyakov line phases at finite $N$

Symmetry in extended Hilbert space and the amount of gauge redundancy are
well-defined notions both at large $N$ and at finite $N$. The Polyakov loop
$P_{\vec{x}}$ defined at each point $\vec{x}$ provides us with a convenient
way to estimate them. The set of $P_{\vec{x}}$ specifies an element in a big
group $\mathcal{G}=\prod_{\vec{x}}[{\rm SU}(N)]_{\vec{x}}$. We can diagonalize
them and get $N$ eigenvalues at each point.

#### Complete confinement and Haar-random distribution

For theories with a center symmetry such as SU($N$) pure Yang-Mills theory, an
unbroken center symmetry provides a simple way to detect the complete
confinement. However, the characterization by center symmetry does not apply
to QCD, because such a symmetry does not exist. A much better characterization
is based on the genuine gauge invariance of the confined vacuum [17], i.e.,
the gauge invariance in the extended Hilbert space. As discussed in Sec. 3, it
follows that the distribution of Polyakov line phases must be Haar-random. The
Haar-random distribution was constant in the large-$N$ limit. As we discuss in
Appendix D, the Haar-random distribution is given by

$\displaystyle\rho_{\rm
Haar}(\theta)=\frac{1}{2\pi}\left(1-(-1)^{N}\cdot\frac{2}{N}\cos(N\theta)\right)$
(114)

for SU($N$). Deviation from this Haar-random distribution can be a good
criterion to observe the transition from the completely-confined phase to the
partially-deconfined phase even at finite $N$. At large $N$, it is indeed true
for theories we have investigated the partial deconfinement.

Let us consider the Fourier expansion of the phase distribution,

$\displaystyle\rho_{\rm Polyakov}(\theta)$
$\displaystyle=\frac{1}{2\pi}+\frac{1}{2\pi}\sum_{n>0}\left(\tilde{\rho}_{n}e^{-in\theta}+\tilde{\rho}_{-n}e^{in\theta}\right)$
$\displaystyle=\frac{1}{2\pi}+\frac{1}{2\pi}\sum_{n>0}2\tilde{\rho}_{n}\cos(n\theta)\,.$
(115)

Here, we assumed $\tilde{\rho}_{n}=\tilde{\rho}_{-n}\in\mathbb{R}$ for
simplicity, which is indeed the case for QCD at zero chemical potential. The
coefficient $\tilde{\rho}_{n}$ is the expectation values of the $n$-wound
Polyakov loop $u_{n}\equiv\frac{1}{N}\textrm{Tr}(\mathcal{P}^{n})$:

$\displaystyle\tilde{\rho}_{n}=\langle
u_{n}\rangle=\frac{1}{N}\left\langle\textrm{Tr}(\mathcal{P}^{n})\right\rangle\,.$
(116)

For the Haar-random distribution, these coefficients are

$\displaystyle\tilde{\rho}_{{\rm
Haar},n}=\left\\{\begin{array}[]{cc}\frac{(-1)^{N}}{N}&(n=\pm N)\\\
0&(n\neq\pm N)\,.\end{array}\right.$ (119)

Therefore, $\langle\textrm{Tr}\mathcal{P}^{n}\rangle$ is nonzero only for
$n=\pm N$. This is consistent with the existence of baryon as a bound state of
$N$ quarks.

One may not find the appearance of the Haar-random distribution too surprising
in the strict zero-temperature limit, because the Polyakov loop in the path-
integral formalism is the product of infinitely many unitary link variables.
However, that the Haar-randomness should survive even at finite temperature
modulo small corrections due to hadron gas is a highly nontrivial
statement.131313 The contribution from hadron gas is $1/N$-suppressed and
hence it is hard to see as long as the ’t Hooft expansion is valid. However,
if we consider the parameter region where ’t Hooft $1/N$ expansion is not
good, we may see a large correction to the Haar-random distribution. An
example is the weakly-coupled regime of mass-deformed $\mathcal{N}=1$ SYM
discussed in Sec. 3.3.3, where the phases are frozen at the multiples of
$\frac{2\pi}{N}$. In this particular setup, the phase transition takes place
in the weakly-coupled regime, or equivalently, at rather high temperatures.

#### Finite-$N$ counterpart of GWW

One may also imagine that the onset of the gap in the phase distribution,
which is by definition the GWW point in the large-$N$ limit, is somehow
related to the transition between complete deconfinement and partial
confinement. However, as we will see in Sec. 5, $N=3$ does not seem to be
large enough to admit such a naive interpretation.

A better criterion comes from the behavior of the multiply-wound Polyakov
loops. At large $N$, the phase distribution $\rho(\theta)$ in the completely-
deconfined phase is “gapped”, i.e., $\rho(\theta)$ is zero in a certain finite
interval. This requires that the expansion (115) does not terminate at finite
order.141414 If a given function is identically zero at a finite interval, its
derivatives at arbitrary order are also identically zero. It gives infinitely
many constraints to the Fourier coefficients that cannot be solved if the
expansion terminates at a finite order.  Therefore, multiply-wound loops with
arbitrarily large winding numbers can have a nonzero expectation value. In
particular, in the infinite-temperature limit, the distribution is the delta
function and all loops contribute equally. Intuitively, a state consisting of
any number of probe quarks can be excited rather easily, which is a natural
consequence of deconfinement.

The absence of the gap in the distribution does not immediately guarantee the
absence or suppression of the loops with large wingding numbers. Still, in the
partially-confined phase of the large-$N$ theories we studied analytically or
numerically in the past, we did not observe significant contributions from
such loops. It would be a natural feature given that the deconfined sector is
not sufficiently excited; when more energy is added to the system, it is used
to make the deconfined sector larger (to increase $M$), and hence, the
excitations are diluted across the deconfined degrees of freedom. Because this
intuition does not require the large-$N$ limit, we conjecture that the
condensation of multiply-wound Polyakov loops with large winding numbers can
be regarded as a finite-$N$ counterpart of the GWW point.

Equivalently, we can consider the characters $\chi_{r}(\mathcal{P})$
corresponding to irreducible representations $r$. The character $\chi_{r}$ is
the Polyakov loop in the representation $r$. (The usual Polyakov loop is for
the fundamental representation.) That the expectation value
$\langle\chi_{r}\rangle$ is nonzero would mean that the excitation in the
representation $r$ can be excited without costing free energy. In Sec. 5.1, we
will examine lattice QCD configurations and find that the fundamental
representation deconfines earlier than other representations.

###  4.2 Characerization via global symmetry and instanton condensation

Another natural characterization of partial confinement at finite $N$ is to
make use of global symmetry [13]. As discussed in Sec. 3.3, the onset of
partial deconfinement (i.e., the transition between complete deconfinement and
partial deconfinement) can be associated with the breaking of global symmetry
such as chiral symmetry at least in the concrete examples studied in Ref.
[13]. Breaking of global symmetry is a well-defined concept even at finite
$N$, and hence, it is natural to adopt it to detect partial deconfinement at
finite $N$.

When the confinement/deconfinement transition is of first order, the maximum
of free energy provides us with a natural counterpart of the partially-
deconfined phase that can be generalized to finite $N$. In the plot of the
free energy as a function of temperature, the cusp can be regarded as the
transition between the completely-deconfined phase and the partially-
deconfined phase (Fig. 3, center-left).151515 This identification may not
always be valid. Specifically, superconformal index with imaginary potential
can give counter-example [31]: at large $N$, the cusp does not coincide with
the GWW point. Whether a similar disagreement can be seen in a non-
supersymmetric setup is currently not clear. We thank Jack Holden and Seok Kim
for the discussions.  In Ref. [13], such a cusp was studied for mass-deformed
$\mathcal{N}=1$ super Yang-Mills on $\mathbb{R}^{3}\times\mathrm{S}^{1}$ with
periodic boundary condition. When $\theta$-angle is $\pi$, the CP breaking
takes place at this point even at finite $N$. This provides us with a further
motivation to use the breaking of global symmetry to detect partial
confinement.

Chiral symmetry is a well-defined concept only for massless quarks, which is
different from the real-world QCD. Although we can still consider the chiral
symmetry of massless probe quark, such a quantity is admittedly ad-hoc. A
natural alternative that could capture partial deconfinement regardless of the
mass of quarks is instanton condensation. Since the instantons are intimately
connected with the chiral symmetry, it is natural to expect the behavior of
instantons to change across the GWW transition. This is indeed the case for
the large-$N$ two-dimensional lattice gauge theory (i.e., the original Gross-
Witten-Wadia model) [32]. In Sec. 5, we will use instanton condensation as a
probe for the finite-$N$ analog of the GWW transition and compare it to the
analyses based on the Polyakov line.

##  5 Partial deconfinement in real-world QCD ($N=3$)

As discussed in the previous section, it is natural to expect that partial
deconfinement takes place in the real-world QCD, with SU(3) gauge and infinite
volume, unless the intermediate-energy region is spatially inhomogeneous. In
this section, we want to address how we can detect partial deconfinement,
adopting the ideas provided in Sec. 4.

We use lattice QCD configurations generated by WHOT-QCD collaboration [33]. In
the collaboration, $O(a)$-improved Wilson quark action [34] coupled with the
RG-improved Iwasaki gauge action [35] was used to simulate $N_{\rm f}=2+1$
QCD, with lattice spacing is $a\simeq 0.07$ fm, up- and down-quark mass
heavier than physical mass (i.e., $\frac{m_{\pi}}{m_{\rho}}\simeq 0.63$ is
larger than the real-world value $\frac{m_{\pi}}{m_{\rho}}\simeq 0.18$) and
approximately physical strange-quark mass. Although up and down quarks are
heavier than in the real world, they are light enough so that there is no
first-order phase transition, similar to the case of physical quark mass. The
scale was set in such a way that the $\rho$-meson mass coincides with the
physical value, i.e., $m_{\rho}=775.45$ MeV.

Configurations used in our study are listed in Table 1. The lattice size is
$N_{t}\times 32^{3}$, where the size along the imaginary-time direction is
$N_{t}=4,6,\cdots,16$. The third column of the table shows the phase (complete
deconfinement, partial deconfinement, or complete confinement) determined
based on the following criteria:

  * •

Polyakov loop. Departure from Haar-random distribution indicates partial
deconfinement. Condensation of the loops in the higher representations
indicates complete deconfinement.

  * •

Instantons. Instanton condensation indicates partial deconfinement or complete
confinement. The absence of the instanton condensation indicates complete
deconfinement.

The details will be discussed below.

Lattice size | Temperature | Phase
---|---|---
$4\times 32^{3}$ | 697 MeV | CD
$6\times 32^{3}$ | 464 MeV | CD
$8\times 32^{3}$ | 348 MeV |  PD or CD
$10\times 32^{3}$ | 279 MeV | PD
$12\times 32^{3}$ | 232 MeV | PD
$14\times 32^{3}$ | 199 MeV | PD
$16\times 32^{3}$ | 174 MeV |  CC or PD
Table 1: WHOT-QCD configurations of $N_{\rm f}=2+1$ QCD [33]. Lattice spacing
$a\simeq 0.07$ fm, with up and down quarks heavier than physical mass (i.e.,
$\frac{m_{\pi}}{m_{\rho}}\simeq 0.63$ is larger than the real-world value
$\frac{m_{\pi}}{m_{\rho}}\simeq 0.18$) and approximately physical strange-
quark mass. The third column is our estimate of the phase discussed in the
main text. CD, PD, and CC denote Complete Deconfinement, Partial
Deconfinement, and Complete Confinement.

###  5.1 Poylakov line phases vs. partial deconfinement

The Polyakov loop $P_{\vec{x}}$ can be defined at each spatial point
$\vec{x}$. They characterize an element in the group of local transformations
$\mathcal{G}=\prod_{\vec{x}}[{\rm SU}(N)]_{\vec{x}}$ that describes the local
gauge transformation. We can diagonalize them at each point and obtain
$3n_{\rm space}$ eigenvalues, where $n_{\rm space}$ is the number of spatial
points on the lattice ($n_{\rm space}=32^{3}=32768$ for the configurations
used below). In the limit of $n_{\rm space}\to\infty$, we can define
continuous phase distribution with respect to the $3n_{\rm space}$ eigenvalues
for each configuration. When the physical volume goes to infinity, a phase
transition can possibly take place, even if $N$ is finite.

In this section, we consider the bare Polyakov loop without renormalization,
because it captures the symmetry of the quantum states. Note also that we show
only the real part of $\tilde{\rho}_{n}$’s because, as we explained, they are
real from the theoretical point of view. We confirmed that the imaginary part
is consistent with zero.

#### No emergent center symmetry at high temperatures

At asymptotically high temperatures, we expect that typical states are not
invariant under any element of $\mathcal{G}$ but the identity. The phase
distribution should be peaked at $\theta=0$.

In the context of the Euclidean path integral with imaginary time, people
often say ‘fermions decouple at high temperatures’. This is because fermions
have the anti-periodic boundary condition and hence have parametrically large
Kaluza-Klein momentum (or Matsubara frequency) with respect to the imaginary
time direction which is a circle with circumference $\frac{1}{T}$. Because of
this, one may think the high-temperature limit reduces to pure Yang-Mills and
the $\mathbb{Z}_{3}$-center symmetry emerges. Although this intuition
correctly explains the dominance of gluonic contributions to many observables,
fermions still contribute to the Polyakov loop because they are transformed by
$\mathcal{G}$. At high temperature, the fermionic part of typical quantum
states are a generic superposition of the ground state and excited states in
the Fock basis, i.e.,
$\otimes_{\vec{n},\alpha,c,f}\left(c^{(0)}_{\vec{n},\alpha,c,f}\ket{0}_{\vec{n},\alpha,c,f}+c^{(1)}_{\vec{n},\alpha,c,f}\ket{1}_{\vec{n},\alpha,c,f}\right)$,
where
$|c^{(0)}_{\vec{n},\alpha,c,f}|^{2}\sim|c^{(1)}_{\vec{n},\alpha,c,f}|^{2}\sim\frac{1}{2}$.
The indices $\vec{n},\alpha,c$, and $f$ represent lattice site, spinor, color,
and flavor. Because fermions are in the fundamental representation, the
nontrivial elements of $\mathbb{Z}_{3}$-transformation act on them as
multiplication of phase $e^{\pm\frac{2\pi i}{3}}$. The Fock vacuum does not
change, $\ket{0}_{\vec{n},\alpha,c,f}\to\ket{0}_{\vec{n},\alpha,c,f}$, but the
excited state transforms as $\ket{1}_{\vec{n},\alpha,c,f}\to e^{\pm\frac{2\pi
i}{3}}\ket{1}_{\vec{n},\alpha,c,f}$. This leads to a substantial suppression
factor in the partition function, and hence, the $\mathbb{Z}_{3}$-center
symmetry does not emerge. Therefore, there is no ambiguity regarding the phase
distribution analogous to the one in pure Yang-Mills associated with the
spontaneous breaking of the center symmetry.

The situation is different in the ’t Hooft large-$N$ limit ($N_{\rm f}$ fixed
and $N\to\infty$), where such an effect is negligible compared to the
contributions from the gluon sector due to the $1/N$-suppression. The ordering
of large-$N$ and high-temperature limits can matter, because the ‘decoupling’
of fermions takes place for different reasons.

#### Haar-random distribution at low temperatures

Next, let us consider low temperatures. We expect that the ground state is an
element in the extended Hilbert space which is invariant under the slowly-
varying part of $\mathcal{G}$. In particular, the phase distribution coincides
with that of Haar-random SU(3). As we show in Appendix D, the Haar-random
distribution is

$\displaystyle\rho_{\rm
Haar}(\theta)=\frac{1}{2\pi}\left(1+\frac{2}{3}\cos(3\theta)\right)\,.$ (120)

Therefore, $\langle\textrm{Tr}\mathcal{P}^{n}\rangle$ is nonzero for $n=\pm
3$, that is consistent with the existence of baryon.161616 One may wonder why
other multiples of 3 cannot have nonzero expectation values. A more precise
statement is that only $\textrm{Tr}\mathcal{P}^{3}$ couples to SU(3)-singlet
and $\textrm{Tr}\mathcal{P}^{6}$, $\textrm{Tr}\mathcal{P}^{9}$, etc, do not
couple to two-, three-, or more-baryon states. See Appendix E.2 for details.

We emphasize that the center symmetry does not play any role in our argument.
We can use the Polyakov loop to describe confinement and deconfinement even
for theories without center symmetry including QCD. The vanishing expectation
value of the Polyakov loop in QCD at zero temperature is a consequence of the
Haar-randomness. In Fig. 13, we compared the Haar-random distribution and the
phase distribution obtained from the WHOT-QCD collaboration’s configurations
at 174 MeV. We can see a good agreement.

Figure 13: Haar-random distribution vs. Polyakov line phase distribution at
174 MeV, obtained from the WHOT-QCD configurations. The lattice size is
$16\times 32^{3}$ and 599 configurations were used. These histograms are drawn
with 992 bins.

#### Deviation from Haar-random distribution and Hagedorn point

The Hagedorn point, i.e., the transition between completely-confined and
partially-confined phases, can be seen as the departure from the Haar-random
distribution. In fact, that the phase distribution becomes Haar random is
already somewhat nontrivial.

In Fig. 14, we plot the distribution of Polyakov line phases obtained from the
configurations created by the WHOT-QCD collaboration. We can see larger
deviations from the Haar-random distribution at higher temperatures. To see
the deviations quantitatively, we plot the Fourier coefficients
$\tilde{\rho}_{n}$ in Fig. 15. The Haar-random distribution specifies
$\tilde{\rho}_{3}=\frac{1}{3}$, $\tilde{\rho}_{n}=0$ ($n=1,2,4,5,\cdots$). We
can see that the onset of the departure from Haar-random around the lowest
temperature in our configuration set, $T=174$ MeV. This implies that $T=174$
MeV is the completely-confined or partially-deconfined phase, and $T\geq 199$
MeV should be partially deconfined.

Figure 14: Distributions of Polyakov line phases, $N_{t}\times 32^{3}$
lattice, obtained from the WHOT-QCD configurations. The lattice size is
$N_{t}\times 32^{3}$ ($N_{t}=4,6,8,12$ and correspondingly $T=$ 697, 464, 348,
and 232 MeV.) and 599 configurations were used. Although the agreement with
Haar-random distribution seems to be good at $T=232$ MeV, more careful
investigation shows the small deviation and hence the onset of partial
deconfinement; see Fig. 15, Fig. 16, and main text. See also Fig. 13 for $T=$
174 MeV.

Figure 15: Expectation values of multiply-wound Polyakov loops
$\tilde{\rho}_{n}=\langle\frac{1}{N}{\rm Tr}\mathcal{P}^{n}\rangle$ obtained
from WHOT-QCD configurations. The lattice size is $N_{t}\times 32^{3}$
($N_{t}=4,\cdots,16$ and correspondingly, $T=174,\cdots,697$ MeV) and 599
configurations were used. These histograms are drawn with 992 bins. [Top]
$\tilde{\rho}_{n}$ vs. temperature. [Middle] $|\tilde{\rho}_{n}|$ vs.
temperature in the log scale. [Bottom] $\frac{1}{3}-\tilde{\rho}_{3}$ vs.
temperature and $\left|\frac{1}{3}-\tilde{\rho}_{3}\right|$ vs. temperature in
the log scale. The error bars are not shown in the log plots.

This ‘transition’ may well be a crossover. We could imagine that the Polyakov
loops have small nonzero values that are exponential with respect to the size
of the representation. We cannot exclude this possibility with our numerical
precision, and this is indeed the case for large-$N$ QCD on a small three-
sphere [12]. Intuitively, a small deviation from Haar-randomness could be
understood as contributions from sparse hadron gas.

#### Condensation of Multiply-wound loops and characters

At $N=3$, the Haar-random distribution is rather far from the large-$N$ limit
(i.e., the constant distribution). Therefore, it is hard to make sense of the
‘constant offset’ that was the key to connecting the GWW point (onset of the
gap) and partial deconfinement/complete deconfinement transition in the
large-$N$ limit. In Fig. 14, even though we can observe the onset of the gap
at $697$ MeV, this temperature is unnaturally high compared to other scales in
QCD.

Instead, we pay attention to the behavior of multiply-wound Polyakov loops. As
we can see from Fig. 15, the loops with winding numbers $n=1,2,4$ start to
contribute at around 174 MeV, while ones with $n=5,6$ set in at around 348
MeV, and $n=3$ starts to decrease around there. The latter is likely to signal
the transition from partial deconfinement to complete deconfinement.

An interesting feature is that $\tilde{\rho}_{1}$, $\tilde{\rho}_{2}$, and
$\tilde{\rho}_{4}$ take the same magnitude at $174$ MeV $\lesssim T\lesssim$
$348$ MeV. This means only the fundamental representation is deconfined in
this temperature range, as we will see shortly.

To make the physical interpretation clearer, we also calculate the characters
$\chi_{r}(P)$ corresponding to several irreducible representations. The
character $\chi_{r}$ is the Polyakov loop in the representation $r$. (The
usual Polyakov loop is for the fundamental representation. See Appendix E for
some properties of the characters.) For SU(3), we have

$\displaystyle\chi_{\rm fund.}=\sum_{j=1}^{3}e^{i\theta_{j}}$ (121)

for the fundamental representation,

$\displaystyle\chi_{\rm adj.}=2+\sum_{j\neq k}e^{i(\theta_{j}-\theta_{k})}$
(122)

for the adjoint representation,

$\chi_{\rm 2\mathchar 45\relax
sym.}=\sum_{j=1}^{3}e^{2i\theta_{j}}+\sum_{j=1}^{3}e^{-i\theta_{j}}\,.$ (123)

for the rank-2 symmetric representation, and

$\displaystyle\chi_{\rm 3\mathchar 45\relax
sym.}=1+\sum_{j=1}^{3}e^{3i\theta_{j}}+\sum_{j\neq
k}e^{i(\theta_{j}-\theta_{k})}$ (124)

for the rank-3 symmetric representation. Notice when $\langle\chi_{r}\rangle$
is nonzero, the excitation in representation $r$ is deconfined. The characters
are related to the multiply-wound loops
$u_{n}\equiv\frac{1}{N}\textrm{Tr}(\mathcal{P}^{n})$ as

$\displaystyle 3u_{1}$ $\displaystyle=\chi_{\rm fund.}\,,$ (125)
$\displaystyle 3u_{2}$ $\displaystyle=\chi_{\rm 2\mathchar 45\relax
sym.}-\left(\chi_{\rm fund.}\right)^{\ast}\,,$ (126) $\displaystyle 3u_{3}$
$\displaystyle=\chi_{\rm 3\mathchar 45\relax sym.}-\chi_{\rm adj.}+1\,,$ (127)

and so on.

In Fig. 16, $\langle\chi_{\rm fund.}\rangle$, $\langle\chi_{\rm adj.}\rangle$,
$\langle\chi_{\rm 2\mathchar 45\relax sym.}\rangle$ and $\langle\chi_{\rm
3\mathchar 45\relax sym.}\rangle$ are plotted. We can see that
$\langle\chi_{\rm fund.}\rangle$ departs from zero at around $T\sim 174~{}{\rm
MeV}$. We can also see $\langle\chi_{\rm adj.}\rangle$, $\langle\chi_{\rm
2\mathchar 45\relax sym.\rangle}$ and $\langle\chi_{\rm 3\mathchar 45\relax
sym.}\rangle$ depart from zero at higher temperatures ($T\gtrsim 348$ MeV). It
would be natural to interpret that the onset of nonzero values of
$\langle\chi_{\rm adj.}\rangle$, $\langle\chi_{\rm 2\mathchar 45\relax
sym.}\rangle$ and $\langle\chi_{\rm 3\mathchar 45\relax sym.}\rangle$ is
associated with the transition from partial deconfinement to complete
deconfinement.

In summary, we estimate that $174$ MeV $\lesssim T\lesssim$ $348$ MeV is
partially deconfined.

The reason $\tilde{\rho}_{1}$, $\tilde{\rho}_{2}$, and $\tilde{\rho}_{4}$ take
the same value up to sign at $174$ MeV $\lesssim T\lesssim$ $348$ MeV is that
only the fundamental representation is deconfined. Indeed, from the equations
above, we can see that $\tilde{\rho}_{1}=\langle
u_{1}\rangle=\frac{1}{3}\langle\chi_{\rm fund.}\rangle$, and
$\tilde{\rho}_{2}=\langle u_{2}\rangle=\frac{1}{3}\langle\chi_{\rm 2\mathchar
45\relax sym.}\rangle-\frac{1}{3}\langle\chi_{\rm
fund.}\rangle^{\ast}=-\frac{1}{3}\langle\chi_{\rm fund.}\rangle$, and hence,
$\tilde{\rho}_{1}=-\tilde{\rho}_{2}$ in this temperature range. The agreement
with $\tilde{\rho}_{4}$ can be shown by using the relations in Appendix E,
assuming only the fundamental representation is deconfined.

Our numerical observation does not exclude the possibility that Polyakov loops
in large representations are very small but not exactly zero in the partially-
deconfined phase. If this is the case, a natural possibility would be that the
suppression is exponential with respect to the size of the representation in
the partially-deconfined phase and power in the completely-deconfined phase.

Figure 16:  The expectation values of characters vs. temperature for the
fundamental, adjoint, rank-2 symmetric, and rank-3 symmetric representations,
obtained from WHOT-QCD configurations. The bottom shows all at once in the log
scale. Note that the expectation values are real.

###  5.2 Instanton condensation

As discussed in Sec. 4.2, we expect that the instanton condensation is
associated with the partial deconfinement. To detect the instanton
condensation, we use the topological charge of each lattice configuration
computed by the WHOT-QCD collaboration [36]. Since lattice configurations are
sensitive to the ultraviolet cutoff, they must be treated by the smearing
with, for example, the gradient flow [37]. After smearing, each configuration
returns an integer value, or more precisely, the histogram of the topological
charge has peaks at integer values as shown in Fig. 17. As we can see from
Fig. 18, at $T\leq 279$ MeV, clear multiple peaks appear at charge including
the ones at $Q\neq 0$ that signal the instanton condensation. (Because lattice
volume is finite, the peaks gradually become lower and eventually disappear.)
We can see wider distributions at lower temperatures. Peaks at $Q\neq 0$
disappear $T\gtrsim 348$ MeV. Therefore, we estimate that $T\lesssim$ $348$
MeV is the partially- or completely-confined phase. Remarkably, we have
obtained the same estimate from the Polyakov loops and instantons!

Figure 17: Histogram of the topological charge taken from Ref. [36]. Each
lattice configuration gives an integer value after smearing by the gradient
flow. The parameter $t$ is the flow time. After sufficient smearing, charges
peak at integer values (the third row). At $T=232$ MeV, peaks at nonzero
values signal the condensation of instantons. Flow time $t_{1/2}$ is defined
in Ref. [36] and chosen in such a way that unphysical effects from too much
smearing can be avoided.

Figure 18: Histogram of the topological charge provided by the authors of Ref.
[36]. Flow time $t_{1/2}$ is defined in Ref. [36] and chosen in such a way
that unphysical effects from too much smearing can be avoided.

##  6 Conclusion and discussion

In this paper, we described how the general mechanism of partial deconfinement
[17, 18, 19] works for QCD and clarified the meaning of the Polyakov loop that
does not rely on center symmetry. We studied the large-$N$ limit first and
then provided a reasonable picture at finite $N$ combining inputs from the
large-$N$ limit and lattice QCD simulation (WHOT-QCD collaborations’
configurations). The distribution of the phases of the Polyakov line was a
convenient tool to detect three phases of QCD at finite temperature: complete
confinement, partial deconfinement, and complete deconfinement. Both at large
$N$ and finite $N$, Haar-random distribution can give a good criterion for the
complete confinement. In the WHOT-QCD configuration set, a remarkable
agreement with the Haar-random distribution was observed at 174 MeV. In the
large-$N$ limit, the partially-deconfined phase and completely-deconfined
phases are separated by the GWW transition. We suggested that a natural
counterpart of the GWW transition at finite $N$ is the condensation of
Polyakov loops in the higher representations. In the set of WHOT-QCD
configurations, only the Polyakov loop in the fundamental representation is
condensed at $174~{}\textrm{MeV}\lesssim T\lesssim 348~{}\textrm{MeV}$, while
other loops are condensed at $T\gtrsim 348~{}\textrm{MeV}$. Hence, we
conjectured that the temperature range $174~{}\textrm{MeV}\lesssim T\lesssim
348~{}\textrm{MeV}$ is partially deconfined. As a nontrivial check, we studied
the topological charge. We argued that instanton condensation should be
observed in partially-deconfined and completely-confined phases and confirmed
the nonzero topological charge at $T\lesssim 348~{}\textrm{MeV}$ which
indicates the instanton condensation.

There are various directions to be explored. First and foremost, it is
desirable to study other lattice configurations and check consistency with our
results based on WHOT-QCD collaboration’s configurations. Note that partial
deconfinement is a generic property of gauge theories that does not rely on
detail of gauge group or matter content. Therefore, it is significant to
investigate various theories and confirm the consistency of our scenario. The
computation of Polyakov loops would be straightforward. (At this moment, the
authors are not aware of the study of the Polyakov loops in larger
representations except for pure SU($N$) Yang-Mills theory for $N=2,3,4,5$ and
$6$ [38, 39] in the context of Casimir scaling in the deconfined phase.171717
Ref. [38] reported the nonzero value of the Polyakov loop in the adjoint
representation in the confined phase of pure SU(3) Yang-Mills theory on a
coarse lattice. It is interesting to see if the nonzero value survives in the
continuum limit. ) When one considers the case a quark is massless, it is
reasonable to anticipate that the chiral transition takes place at (the
finite-$N$ analog of) the GWW point. In the current analysis, we focused on
the one-point function of the Polyakov loops. It is interesting to consider
whether one can also study other observables from the point of view of the
Haar-randomness and the deviation from it.

An obstacle to applying partial deconfinement to finite $N$ had been the
meaning of the size of the deconfined sector $M$ ($0\leq M\leq N$), as shown
in Fig. 2. Even in the large-$N$ limit, the size $M$ is not literally an
integer. It could have an uncertainty of order $N^{0}$ which is negligible at
large $N$. Admittedly, such an ambiguity makes the use of “$M$” very subtle at
finite $N$, say, $N=3$. To circumvent this issue, we avoided the use of $M$
and relied on characters (the Polyakov loops in various representations). The
use of the character $\chi_{r}$ also has the advantage of being manifestly
gauge invariant. Although in large $N$-theory using the size of the deconfined
sector $M$ is shown to have gauge invariant meaning, as discussed in this
paper, it may be worthwhile to revisit the analysis of partial deconfinement
in the large-$N$ theory using the character expansion. Character expansion
played an important role in the large-$N$ theory, for example in Refs. [40,
41]. A recent work [42] employs the character expansion to study the
deconfinement transition in large-$N$ theory.

Once we understand the nature of phase transitions in real-world QCD better,
it would be fascinating if we could find experimental or cosmological signals.
Recent studies of the large-$N$ perspective of the confinement/deconfinement
phase transition in the context of cosmology can be seen, for instance, in
Refs. [43, 44].

Last but not least, application to quantum gravity is important. Indeed, the
original motivation for partial deconfinement [8] was to study black hole
geometry via holography. Specifically, the partially-deconfined phase is
proposed as the counterpart of the small black hole state that appears in the
context of the AdS/CFT correspondence [45, 46, 47, 48]. It is interesting and
worthwhile to investigate the finite-$N$ prospects of the large-$N$ (partial)
deconfinement in the context of holography since $1/N$ corrections should play
crucial roles in black hole physics as the quantum effects on the gravity
side. Hopefully, we can gain some intuition into quantum gravitational
phenomena such as black hole evaporation from the QFT side.

Data management

No additional research data beyond the data presented and cited in this work
are needed to validate the research findings in this work.

Acknowledgement

Many of the new results in this paper were obtained in collaboration with
Hiroki Ohata and Hidehiko Shimada. The letter version is Ref. [22]. We would
like to thank the members of the WHOT-QCD collaboration, including Shinji
Ejiri, Kazuyuki Kanaya, Masakiyo Kitazawa, and Takashi Umeda, for providing us
with their lattice configurations and many plots and having stimulating
discussions with us. The analysis of topological charge in Ref. [36], which
was crucial in Sec. 5.2, was led by Yusuke Taniguchi who passed away in 2022.
Kazuyuki Kanaya collected the data and plots created by Yusuke Taniguchi for
us. We deeply thank Yusuke Taniguchi and Kazuyuki Kanaya. We also thank Sinya
Aoki, Hidenori Fukaya, Vaibhav Gautam, Yui Hayashi, Jack Holden, Seok Kim,
Tamas Kovacs, Atsushi Nakamura, Marco Panero, Robert Pisarski, Enrico Rinaldi,
Yuya Tanizaki, and Jacobus Verbaarschot for discussions and comments. Shohei
Ohtani’s unbelievable performance gave us the energy to survive the hot and
humid summer of 2023. This work was supported by the Japan Lattice Data Grid
(JLDG) constructed over the SINET5 of NII. M. H. thanks for the STFC grant
ST/R003599/1. H. W. is supported by the Japan Society for the Promotion of
Science (JSPS) KAKENHI Grant number 22H01218.

##  Appendix A Canonical ensemble and microcanonical ensemble

Let us consider an isolated system in which the energy is conserved. Let
$\Omega(E)dE$ be the number of microscopic states in the energy range between
$E$ and $E+dE$. The entropy $S(E)$ is defined by

$\displaystyle\Omega(E)=e^{S(E)}\,.$ (128)

In the microcanonical ensemble, the energy is fixed in a narrow range
$[E,E+dE]$, and all states in this energy range are allowed with equal
probability. Many microscopic states correspond to the same macroscopic state.
More precisely, many microscopic states give almost the same values as
macroscopic observables. A macroscopic state can be determined by seeing the
entropy as a function of macroscopic observables and maximizing it. The size
of the deconfined sector $M$ is determined in the same manner, by seeing the
entropy as a function of $E$ and $M$ and then maximizing it with respect to
$M$. When there is more than one maxima, the one with the largest entropy is
dominant.

Temperature in the microcanonical ensemble is defined by

$\displaystyle T=\left(\frac{dS}{dE}\right)^{-1}\,.$ (129)

Note that temperature is a function of energy: $T=T(E)$. The specific heat is
defined by $C=\frac{dE}{dT}$. It can be positive or negative; we will discuss
more regarding this point shortly.

In the canonical ensemble, the system is attached to a heat bath with
temperature$T$, and the energy is allowed to change. Temperature is the
controlling parameter, and energy is determined as a function of temperature:
$E=E(T)$. The partition function is given by

$\displaystyle Z(T)=\int dE\Omega(E)e^{-E/T}=\int dEe^{-F(E,T)/T}\,,$ (130)

where $e^{-E/T}$ is the Boltzmann factor and

$\displaystyle F(E,T)=E-TS(E)$ (131)

is the free energy. The energy $E(T)$ is determined by minimizing the free
energy. When there are multiple minima, the one with the smallest free energy
is dominant.

Let us see the meaning of minimum and maximum of the free energy more
precisely. By taking the derivative of free energy with respect to the energy,
we obtain

$\displaystyle\left.\frac{\partial F}{\partial E}\right|_{T_{\rm
canonical}}=1-T_{\rm canonical}\frac{dS}{dE}=1-\frac{{T_{\rm
canonical}}}{{T_{\rm microcanonical}}}\,.$ (132)

Therefore, at the minimum and maximum of the free energy, canonical
temperature and microcanonical temperature coincide. Due to this property, the
same energy-versus-temperature plot (like Fig. 3) is obtained from the
canonical ensemble and microcanonical ensemble. The same holds for other
quantities including the size of the deconfined sector.

By taking the second derivative, we obtain

$\displaystyle\left.\frac{\partial^{2}F}{\partial E^{2}}\right|_{T_{\rm
canonical}}=\frac{{T_{\rm canonical}}}{{T_{\rm
microcanonical}}^{2}}\frac{dT_{\rm microcanonical}}{dE}=\frac{{T_{\rm
canonical}}}{{T_{\rm microcanonical}}^{2}}\cdot C^{-1},$ (133)

where $C$ is the specific heat in the microcanonical ensemble. Therefore,

Positive specific heat $\displaystyle\leftrightarrow\textrm{Free energy
minimum}$ Negative specific heat $\displaystyle\leftrightarrow\textrm{Free
energy maximum}$

From this, we see that the phase in the microcanonical ensemble with negative
specific heat is not dominant in the canonical ensemble. Whether a phase with
negative specific heat is stable or not in the microcanonical ensemble depends
on the details of the system, as we discuss next.

##  Appendix B Phase coexistence in color space vs. phase coexistence in
coordinate space

Let us consider a first-order transition between water (liquid water) and ice
(solid water) at 1 atm (Fig. 4). Supercooled water below 0℃ and superheated
ice above 0℃are not stable, and they turn to a mixture of water and ice just
by a small perturbation as shown by arrows in Fig. 4. (Energy is fixed because
the change takes place quickly.) Due to the local nature of the interaction,
the temperature of the mixture is 0℃.

Gauge theories admit two-phase-coexistence in the color space as well. Color
space is nonlocal in the sense that the interactions are all-to-all.
Furthermore, details of the splitting into two phases are different for
different representations, as depicted in Fig. 5 for the case of adjoint and
fundamental representations. Therefore, the temperature is not necessarily
fixed, as depicted in Fig. 3.

When the transition is of the first order, splitting in color space or in the
coordinate space can happen depending on the details of the model. For matrix
models, for example, only the splitting in color space is allowed because the
‘coordinate space’ is, by definition, just a point. Another important case
without splitting in the coordinate space is the four-dimensional maximally
supersymmetric Yang-Mills theory on the three-sphere. In such cases, the
partially-deconfined phase is stable in a microcanonical ensemble. Pure Yang-
Mills theory admits splitting in the coordinate space if the volume is
sufficiently large. In such a case, the partially-deconfined phase is not
stable even in a microcanonical ensemble.

##  Appendix C Operator formalism and Euclidean path integral

In this appendix, we show how the partition function (9) can be rewritten into
the form of the path integral. This section is based on the appendix A of
[49]. Although the authors are not aware of an older reference that pointed
out the connection, they think this was known by many people. Presumably,
however, the fact that $g$ in (9) is the Polyakov line was not appreciated.

As a concrete example, we consider the bosonic Hermitian matrix model.
Generalizations to other cases are straightforward. The expression (9) can be
rewritten as

$\displaystyle Z(T)$ $\displaystyle=\frac{1}{[{\rm
vol}(G)]^{K}}\int\left(\prod_{k=1}^{K}dU_{(k)}\right){\rm Tr}_{{\cal H}_{\rm
ext}}\Bigl{(}\hat{U}_{(K)}e^{-\frac{H(\hat{P},\hat{X})}{TK}}\hat{U}_{(K-1)}^{-1}\hat{U}_{(K-1)}$
$\displaystyle\quad
e^{-\frac{H(\hat{P},\hat{X})}{TK}}\hat{U}_{(K-2)}^{-1}\hat{U}_{(K-2)}\cdots\hat{U}_{(1)}^{-1}\hat{U}_{(1)}e^{-\frac{H(\hat{P},\hat{X})}{TK}}\Bigl{)}$
$\displaystyle=\frac{1}{[{\rm
vol}(G)]^{K}}\int\left(\prod_{k=1}^{K}dU_{(k)}\right)\int\left(\prod_{k=1}^{K}dX_{(k)}\right)$
$\displaystyle\qquad\langle
X_{(K)}|\hat{U}_{(K)}e^{-\frac{H(\hat{P},\hat{X})}{TK}}\hat{U}_{(K-1)}^{-1}|X_{(K-1)}\rangle$
$\displaystyle\qquad\times\langle
X_{(K-1)}|\hat{U}_{(K-1)}e^{-\frac{H(\hat{P},\hat{X})}{TK}}\hat{U}_{(K-2)}^{-1}|X_{(K-2)}\rangle$
$\displaystyle\qquad\times\cdots\times\langle
X_{(1)}|\hat{U}_{(1)}e^{-\frac{H(\hat{P},\hat{X})}{TK}}|X_{(K)}\rangle\ .$
(134)

At the first line, we simply inserted $\hat{U}^{-1}\hat{U}=\hat{1}$ at many
places. $\hat{U}_{(K)}$ corresponds to $\hat{g}$. At the next line, we
inserted $\int dX_{(k)}|X_{(k)}\rangle\langle X_{(k)}|=\hat{1}$ at many
places. Each term in the product can be rewritten as

$\displaystyle\langle
X_{(k)}|\hat{U}_{(k)}e^{-\frac{H(\hat{P},\hat{X})}{TK}}\hat{U}_{(k-1)}^{-1}|X_{(k-1)}\rangle$
$\displaystyle\quad=\langle
U_{(k)}X_{(k)}U_{(k)}^{-1}|e^{-\frac{H(\hat{P},\hat{X})}{TK}}|U_{(k-1)}X_{(k-1)}U_{(k-1)}^{-1}\rangle$
$\displaystyle\quad=\int dP\langle
U_{(k)}X_{(k)}U_{(k)}^{-1}|e^{-\frac{H(\hat{P},\hat{X})}{TK}}|P\rangle\langle
P|U_{(k-1)}X_{(k-1)}U_{(k-1)}^{-1}\rangle$ $\displaystyle\quad=\int dPe^{i{\rm
Tr}[P(U_{(k)}X_{(k)}U_{(k)}^{-1}-U_{(k-1)}X_{(k-1)}U_{(k-1)}^{-1})]}e^{-H(P,U_{(k)}X_{(k)}U_{(k)}^{-1})/(TK)}$
$\displaystyle\quad=e^{-KT{\rm
Tr}[(U_{(k)}X_{(k)}U_{(k)}^{-1}-U_{(k-1)}X_{(k-1)}U_{(k-1)}^{-1})^{2}]}e^{-V(U_{(k)}X_{(k)}U_{(k)}^{-1})/(TK)}$
$\displaystyle\quad\simeq
e^{-L[D_{t}(U_{(k)}X_{(k)}U_{(k)}^{-1}),(U_{(k)}X_{(k)}U_{(k)}^{-1})]/(TK)}$
$\displaystyle\quad=e^{-L[D_{t}X_{(k)},X_{(k)}]/(TK)}\ ,$ (135)

where $L[D_{t}X,X]$ is the Lagrangian with the Euclidean signature. We used

$\displaystyle U_{(k-1)}^{-1}U_{(k)}\equiv e^{iA_{(K)}/(KT)}$ (136)

and

$\displaystyle
X_{(k)}-(U_{(k-1)}U_{(k)}^{-1})^{-1}X_{(k-1)}(U_{(k-1)}U_{(k)}^{-1})\simeq\frac{D_{t}X_{(k)}}{KT}\,.$
(137)

In the limit of $K\to\infty$, we obtain

$\displaystyle Z(T)$ $\displaystyle=\int[dA][dX]e^{-\int dtL[D_{t}X,X]}\ .$
(138)

Therefore,

$\displaystyle g$ $\displaystyle=U_{K}$
$\displaystyle=U_{K}(U_{K-1}^{-1}U_{K-1})(U_{K-2}^{-1}U_{K-2})\cdots(U_{1}^{-1}U_{1})$
$\displaystyle=(U_{K}U_{K-1}^{-1})(U_{K-1}U_{K-2}^{-1})\cdots(U_{2}U_{1}^{-1})U_{1}$
$\displaystyle={\rm P}e^{i\int_{0}^{1/T}dtA_{t}}$ (139)

is the Polyakov line. Here ${\rm P}$ stands for the path-ordered product.

### Equivalence between (5) and (9)

Let us also see why (5) and (9) are equivalent. As a simple example, let us
start with the case of $N$ indistinguishable bosons. The gauge group $G$ is
the symmetric group $S_{\rm N}$. From a Fock state
$\ket{\phi}\equiv\ket{\vec{n}_{1},\cdots,\vec{n}_{N}}\in\mathcal{H}_{\rm
ext}$, we can obtain $\ket{\phi}_{\rm
inv}\equiv\mathcal{N}_{\ket{\phi}}^{-1/2}\hat{\pi}\ket{\phi}\in\mathcal{H}_{\rm
inv}$, where the normalization factor is

$\displaystyle\mathcal{N}_{\ket{\phi}}\equiv\bra{\phi}\hat{\pi}\ket{\phi}=\frac{{\rm
Vol}G_{\ket{\phi}}}{{\rm Vol}G}=\frac{{\rm Vol}G_{\ket{\phi}}}{N!}\,$ (140)

where $G_{\ket{\phi}}$ is the stabilizer of the state $\ket{\phi}$, i.e.,
$G_{\ket{\phi}}=\\{\sigma\in G\ |\ \ket{\phi}=\hat{\sigma}\ket{\phi}\\}$.
Because this is a finite group, the ‘volume’ ${\rm Vol}G_{\ket{\phi}}$ is the
number of the elements ($V_{\ket{\vec{n}_{1},\cdots,\vec{n}_{N}}}$ in Sec.
2.2). The number of states on the gauge orbit $\\{\hat{\sigma}|\ket{\phi}\ |\
\sigma\in G\\}$ is $\mathcal{N}_{\ket{\phi}}^{-1}$. If we naively took trace
over the extended Hilbert space, we would count gauge-equivalent states
$\mathcal{N}_{\ket{\phi}}^{-1}$ times. To relate the trace over the extended
Hilbert $\textrm{Tr}_{\mathcal{H}_{\rm ext}}$ to the trace over the gauge-
invariant Hilbert $\textrm{Tr}_{\mathcal{H}_{\rm inv}}$, we must remove such
an over counting. Therefore,

$\displaystyle\sum_{\ket{\phi}_{\rm inv}\in\mathcal{H}_{\rm
inv}}\bra{\phi}_{\rm inv}e^{-\hat{H}/T}\ket{\phi}_{\rm inv}$
$\displaystyle=\sum_{\ket{\phi}\in\mathcal{H}_{\rm
ext}}\mathcal{N}_{\ket{\phi}}\bra{\phi}_{\rm inv}e^{-\hat{H}/T}\ket{\phi}_{\rm
inv}$ $\displaystyle=\sum_{\ket{\phi}\in\mathcal{H}_{\rm
ext}}\bra{\phi}\hat{\pi}e^{-\hat{H}/T}\hat{\pi}\ket{\phi}$
$\displaystyle=\sum_{\ket{\phi}\in\mathcal{H}_{\rm
ext}}\bra{\phi}\hat{\pi}e^{-\hat{H}/T}\ket{\phi}\,,$ (141)

and hence,

$\displaystyle\mathrm{Tr}_{\mathcal{H}_{\rm
inv}}e^{-\hat{H}/T}=\mathrm{Tr}_{\mathcal{H}_{\rm
ext}}(\hat{\pi}e^{-\hat{H}/T})\,.$ (142)

The same argument applies to the SU($N$) matrix model. As
$\ket{\phi}\in\mathcal{H}_{\rm ext}$ we can take a wave packet or Fock state,
for example. The projection to $\mathcal{H}_{\rm inv}$ is defined by

$\displaystyle\ket{\phi}_{\rm
inv}\equiv\mathcal{N}_{\ket{\phi}}^{-1/2}\hat{\pi}\ket{\phi},\qquad\mathcal{N}_{\ket{\phi}}\equiv\bra{\phi}\hat{\pi}\ket{\phi}\,,$
(143)

where $\mathcal{N}_{\ket{\phi}}$ is the inverse of the over-counting factor.
An intuitive way to understand it is that $\int dg\bra{\phi}\hat{g}\ket{\phi}$
is, roughly speaking, the volume of the stabilizer of $\ket{\phi}$. This is
somewhat imprecise because $G=\textrm{SU}(N)$ is continuous and an
infinitesimal transformation away from $\ket{\phi}$ can have a nonzero
contribution to $\int dg\bra{\phi}\hat{g}\ket{\phi}$. A more precise way is to
notice that the over-counting factor $C_{\ket{\phi}}$ is determined so that
all gauge orbits $\\{\hat{g}\ket{\phi}\ |\ g\in G\\}$ give equal contribution
to the partition function at infinite temperature. Namely,

$\displaystyle C_{\ket{\phi}}\int dg\int
dg^{\prime}(\bra{\phi}\hat{g}^{-1})(\hat{g}^{\prime}\ket{\phi})=C_{\ket{\phi}}\cdot(\textrm{Vol}G)^{2}\cdot\mathcal{N}_{\ket{\phi}}$
(144)

should be independent of $\ket{\phi}$. Gauge-invariant states such as the
confining vacuum satisfy $C_{\ket{\phi}}=\mathcal{N}_{\ket{\phi}}=1$, and
hence, $C_{\ket{\phi}}=\mathcal{N}_{\ket{\phi}}^{-1}$.

##  Appendix D SU($N$) Haar random distribution

In this appendix, we show the analytic formula for the Haar-random
distribution of the phases in SU($N$),

$\displaystyle\rho_{\rm
Haar}(\theta)=\frac{1}{2\pi}\left(1-(-1)^{N}\cdot\frac{2}{N}\cos(N\theta)\right)\,.$
(145)

For other details on random matrix theory, see also Ref. [50] and references
therein.

The starting point is

$\quantity[\differential
U]=\rho(\theta_{1},\cdots,\theta_{N})\prod_{j=1}^{N}\differential\theta_{j}\,,$
(146)

where

$\displaystyle\rho(\theta_{1},\cdots,\theta_{N})=C\prod_{j<k}\sin^{2}\left(\frac{\theta_{j}-\theta_{k}}{2}\right)\times\sum_{n=-\infty}^{\infty}\delta\left(\sum_{j=1}^{N}\theta_{j}-2\pi
n\right)\,.$ (147)

Here, $C$ is the normalization factor and
$\prod_{j<k}\sin^{2}\left(\frac{\theta_{j}-\theta_{k}}{2}\right)$ is the
Vandermonde determinant. Our task is to integrate out
$\theta_{2},\cdots,\theta_{N}$ and estimate

$\displaystyle\rho(\theta_{1})=\int\differential\theta_{2}\cdots\int\differential\theta_{N}\
\rho(\theta_{1},\theta_{2},\cdots,\theta_{N})\,.$ (148)

We can easily generate $\rho(\theta_{1})$ by using the Monte Carlo method such
as the Gibbs sampling algorithm; see Fig. 19.181818 See Ref. [51] for a
numerical study in the past and Ref. [52] for an introduction to the Monte
Carlo method.  The agreement with (145) can be confirmed numerically. When $N$
is not so large, we can perform the integral analytically and confirm (145),
as we will do for $N=2$ and $N=3$ below.

$0$$0.05$$0.1$$0.15$$0.2$$0.25$$0.3$$0.35$$-3$$-2$$-1$$0$$1$$2$$3$

$\rho(\theta)$

$\theta$SU(2)

$0$$0.05$$0.1$$0.15$$0.2$$0.25$$0.3$$-3$$-2$$-1$$0$$1$$2$$3$

$\rho(\theta)$

$\theta$SU(3)

$0$$0.05$$0.1$$0.15$$0.2$$0.25$$-3$$-2$$-1$$0$$1$$2$$3$

$\rho(\theta)$

$\theta$SU(4)

$0$$0.05$$0.1$$0.15$$0.2$$0.25$$-3$$-2$$-1$$0$$1$$2$$3$

$\rho(\theta)$

$\theta$SU(5)

$0$$0.02$$0.04$$0.06$$0.08$$0.1$$0.12$$0.14$$0.16$$0.18$$0.2$$-3$$-2$$-1$$0$$1$$2$$3$

$\rho(\theta)$

$\theta$SU(10)

$0$$0.02$$0.04$$0.06$$0.08$$0.1$$0.12$$0.14$$0.16$$0.18$$-3$$-2$$-1$$0$$1$$2$$3$

$\rho(\theta)$

$\theta$SU(25)

Figure 19: SU($N$) Haar-random distribution for $N=2,3,4,5,10,25$ (purple
line) and $\frac{1}{2\pi}\left(1-(-1)^{N}\cdot\frac{2}{N}\cos(N\theta)\right)$
(green line) are plotted together. They overlap completely.

####  $N=2$ case

If we fix $\theta_{1}$, $\theta_{2}$ is also fixed as $\theta_{2}=-\theta_{1}$
up to the shift by $2\pi n$. By substituting this into (147), we obtain the
distribution for fixed $\theta_{1}$ as

$\displaystyle\rho(\theta_{1})=C\sin^{2}\theta_{1}=C\cdot\frac{1-\cos(2\theta_{1})}{2}\,.$
(149)

The normalization constant $C$ can be determined as

$\int_{-\pi}^{\pi}\differential\theta_{1}\,\rho(\theta_{1})=1\,.$ (150)

The final answer is

$\displaystyle\rho(\theta_{1})=\frac{1-\cos(2\theta_{1})}{2\pi}\,.$ (151)

####  $N=3$ case

Same as the $N=2$ case, $\theta_{3}=-(\theta_{1}+\theta_{2})$ for fixed
$\theta_{1},\theta_{2}$ up to the shift by $2\pi n$. Then, we obtain the
distribution for fixed $\theta_{1},\theta_{2}$ as

$\displaystyle\rho(\theta_{1},\theta_{2})$
$\displaystyle=C\sin^{2}\left(\frac{\theta_{1}-\theta_{2}}{2}\right)\sin^{2}\left(\frac{\theta_{1}+2\theta_{2}}{2}\right)\sin^{2}\left(\frac{\theta_{2}+2\theta_{1}}{2}\right)$
$\displaystyle=\frac{C}{8}\left(1-\cos(\theta_{1}-\theta_{2})\right)\left(1-\cos(\theta_{1}+2\theta_{2})\right)\left(1-\cos(2\theta_{1}+\theta_{2})\right)$
$\displaystyle=\frac{C}{8}\left(\frac{3}{4}+\frac{1}{2}\cos(3\theta_{1})+\cdots\right)\,,$
(152)

where $\cdots$ are terms like $\cos(\theta_{1}-\theta_{2})$ or
$\cos(\theta_{1}+2\theta_{2})$ which disappear after $\theta_{2}$ is
integrated. To go from the second line to the third line, we used $\cos
x\cdot\cos y=\frac{1}{2}\left(\cos(x+y)+\cos(x-y)\right)$ several times.
Therefore,

$\rho(\theta_{1})=\int_{-\pi}^{\pi}\differential\theta_{2}\,\rho(\theta_{1},\theta_{2})\propto
1+\frac{2}{3}\cos(3\theta_{1})\,.$ (153)

The overall factor is determined such that
$\int_{-\pi}^{\pi}\differential\theta_{1}\,\rho(\theta_{1})=1$. The final
result is

$\displaystyle\rho(\theta_{1})=\frac{1}{2\pi}\left(1+\frac{2}{3}\cos(3\theta_{1})\right).$
(154)

##  Appendix E Characters of SU(3) group

The character of $G$ is defined by the trace of the representation matrix
$R_{r}$ in the representation $r$ as

$\chi_{r}(g):=\tr R_{r}(g),$ (155)

where $g\in G$ is the holonomy along the thermal circle. For the irreducible
representations, the orthonormal condition,

$\frac{1}{\mathrm{Vol}(G)}\int_{G}\mathrm{d}g\,\chi_{r}(g)\left(\chi_{s}(g)\right)^{\ast}=\delta_{rs},$
(156)

is satisfied.

When we consider $G=$ SU(3), we can identify $g$ and its representation in the
fundamental representation: $g$ is $3\times 3$ unitary matrix with a
determinant equal to 1. The SU(3) characters are therefore functions only with
respect to the eigenvalues of $g$, denoted by $\lambda_{j}$ $(j=1,2,3)$.
Namely,

$\chi_{r}(g)=\chi_{r}(\\{\lambda\\}).$ (157)

Note that $\lambda_{j}=e^{i\theta_{j}}$ in terms of the Polyakov line phase
$\theta_{j}$, and $\lambda_{1}\lambda_{2}\lambda_{3}=1$ due to the condition
$\det g=1$.

###  E.1 List of characters for irreducible representations

The character is a symmetric polynomial in $\lambda$’s. There should be a nice
way to compute the character from, say, the Young tableaux. Each Young tableau
corresponds to an irreducible representation of SU(3). In the following, the
notation $(n,m)$ represents a Young tableau that has $n$ and $m$ boxes in the
first and second rows, respectively.

(0,0)=1: trivial

Trivial representation is $R_{(0,0)}(g)=1$. The character is

$\chi_{\rm trivial}\equiv\chi_{(0,0)}=1\,.$ (158)

(1,0)=3: fundamental