A new perspective on thermal transition in QCD

Motivated by the picture of partial deconfinement developed in recent years for large-$N$ gauge theories, we propose a new way of analyzing and understanding thermal phase transition in QCD. We find nontrivial support for our proposal by analyzing the WHOT-QCD collaboration's lattice configurations for SU(3) QCD in $3+1$ spacetime dimensions with up, down, and strange quarks. We find that the Polyakov line (the holonomy matrix around a thermal time circle) is governed by the Haar-random distribution at low temperatures. The deviation from the Haar-random distribution at higher temperatures can be measured via the character expansion, or equivalently, via the expectation values of the Polyakov loop defined by the various nontrivial representations of SU(3). We find that the Polyakov loop corresponding to the fundamental representation and loops in the higher representation condense at different temperatures. This suggests that there are (at least) three phases, one intermediate phase existing in between the completely-confined and the completely-deconfined phases. Our identification of the intermediate phase is supported also by the condensation of instantons: by studying the instanton numbers of the WHOT-QCD configurations, we find that the instanton condensation occurs for temperature regimes corresponding to what we identify as the completely-confined and intermediate phases, whereas the instantons do not condense in the completely-deconfined phase. Our characterization of confinement based on the Haar-randomness explains why the Polyakov loop is a good observable to distinguish the confinement and the deconfinement phases in QCD despite the absence of the $\mathbb{Z}_3$ center symmetry.


Introduction
Confinement/deconfinement transition in gauge theory [1,2] is an important phenomenon that has applications ranging from the quark-gluon plasma phase in QCD under extreme conditions to the description of black holes via gauge/gravity duality.However, the definition of confinement and deconfinement has been somewhat unclear in real-world QCD with three colors.The biggest issue was the lack of the strict notion of symmetry characterizing confinement and deconfinement.Specifically, the Z 3 center symmetry that provides us with a good characterization for pure Yang-Mills theory (i.e., unbroken and broken center symmetry correspond to confinement and deconfinement, respectively) does not exist for QCD due to the presence of quarks in the fundamental representation.The Polyakov loop1 no longer plays the role of the order parameter associated with the center symmetry.Nonetheless, the Polyakov loop has been empirically used as an "order parameter" of the deconfinement transitions.
Investigations of large-N gauge theories have been successful in elucidating the nontrivial nature of QCD such as the occurrence of deconfinement transition.It has been known that the phase transition can be detected by the Polyakov line (i.e., the holonomy matrix whose trace gives the Polyakov loop) in an independent way to center symmetry. 2The reason that the Polyakov line captures the phase transition turned out to be its connection to gauge symmetry [6].
In this letter and a companion paper [7], we explore whether and how this idea in large-N theory can be applied to the real-world QCD, i.e., SU(N = 3) QCD with dynamical quarks.Note that, although we are considering finite N , the phase transition is not prohibited because the thermodynamic limit is realized at infinite volume.
We take a bottom-up approach in this letter, by looking into lattice data generated by WHOT-QCD collaboration 3 in Sec. 2 and then giving the interpretation based on the connection to the theoretical understanding of large-N theory in Sec. 3. The presentation in the companion paper [7] follows a top-down approach, i.e., we start with the large-N theories, make conjectures on SU(3) QCD, and then confirm these conjectures based on lattice QCD data.In fact, we took such a top-down approach in our investigation.

Looking into lattice data
We consider thermal circles defined at each spatial point ⃗ x and the Polyakov lines P ⃗ x ∈ SU(3).A technical but crucial idea which facilitates our analysis is to consider an ensemble (probability distribution) of the Polyakov line, counting P ⃗ x for each ⃗ x as a sample. 4n Table 1, we show a short profile of the WHOT-QCD lattice configurations at seven different values of temperature which we use, together with our identification of the phases associated with the temperatures which will be explained below.The spatial volume is 32 3 , and hence, we obtain 32 3 = 32768 P ⃗ x 's from each lattice QCD configuration. 5attice size Temperature Phase 4 × 32 3  697 MeV CD 6 × 32 3  464 MeV CD 8 × 32 3  348 MeV PD or CD 10 × 32 3  279 MeV PD 12 × 32 3  232 MeV PD 14 × 32 3  199 MeV PD 16 × 32 3  174 MeV CC or PD  At each spatial point ⃗ x, P ⃗ x is a 3×3 matrix with eigenvalues e iθ 1 , e iθ 2 and e iθ 3 , with θ 1 +θ 2 +θ 3 ≡ 0 mod 2π.There are 3 × 32 3 = 98304 eigenvalues per configuration.We can estimate the distribution ρ(θ) (−π < θ ≤ π) by combining many configurations.The results are shown in Fig 1 .We compare this distribution against that would arise from the Haar-random distribution on SU(3), (For completeness, we give the derivation of this formula in appendix B.) The plots show that ρ(θ) deviates from the Haar-random distribution for higher temperatures T = 348 MeV, 464 MeV, but appears indistinguishable for lower temperatures, with our naked eyes, from ρ Haar (θ) at T = 174 MeV, 232 MeV.We will discuss a method to measure the deviation from the Haar-random distribution systematically shortly below, which shows that the deviation decreases rapidly toward T = 174 MeV.
The agreement with the Haar-random distribution at low temperatures is a crucial feature that has been theoretically understood in the large-N theories based on analysis focused on the redundancy of the states under gauge transformations. 6We provide a short summary of this point in Appendix A.
We can measure the deviation from the Haar-randomness quantitatively by using character expansion. 7(We collect some properties of character expansion together with explicit formulae for characters of SU(3) in Appendix C.) We write the probability distribution of the Polyakov line P ∈ SU(3) by ρ(P ).Let χ r (P ) be the character associated with an irreducible representation r.By the completeness of the characters, one can expand ρ(P ) in terms of χ r (P ) as ρ(P ) = r ρ r χ r (P ), where the expansion coefficients are ρ r = dP ρ(P )χ * r (P ) because of the orthonormality of the characters dP χ r (P )χ * r ′ (P ) = δ rr ′ .By construction, ρ r coincides with the expectation value of the Polyakov loop in the representation r.
For the exact Haar-random distribution, ρ(P ) is completely dominated by the trivial representation, i.e., ρ r vanishes for any nontrivial representation r.Hence, the Polyakov loops in nontrivial representations give good measures of the deviation of the Polyakov line phases from the Haarrandom distribution.Note that ρ r contains all statistical information of the distribution of the Polyakov line, including the correlations between three eigenvalues of the Polyakov line such as the level repulsion.
Fig. 2 and Fig. 3 show the expectation values of the Polyakov loops in several nontrivial representations. 8These plots show, firstly, that the Polyakov loops do disappear at T ≲ 174 MeV.In particular, Fig. 3 shows that the Polyakov loop in the fundamental representation is suppressed exponentially in the low-temperature regime. 9Secondly, the expectation values of the Polyakov loops in higher representations become nonzero at different temperatures, T ≳ 348 MeV.
The simplest possibility consistent with our observations is that there are three phases: (i) T < T 1 ∼ 174 MeV where the Polyakov line is governed by the Haar-random distribution, (ii) T 1 < T < T 2 ∼ 348MeV where the fundamental Polyakov loop is non-zero but the Polyakov loops associated with higher representations vanish, and (iii) T 2 < T where Polyakov loop in all representations are non-zero.
It is natural to interpret the nonzero values of the Polyakov loops in nontrivial representations as indicating that the degrees of freedom associated with the representations are deconfined.As such, we shall call the regimes, (i) the completely-confined, (ii) partially-deconfined (or equivalently, partially-confined), and (iii) completely-deconfined phases, respectively.The identified phases are shown in Table 1.
Remarkably, further support for this identification of the phases is obtained by studying the condensation of instantons.Namely, we find that the instanton condensation occurs for temperature regimes corresponding to what we identify as the completely-confined and partially-confined phases, whereas the instanton does not condense in the completely-deconfined phase.The quantity we use to detect the instanton condensation is the topological charge of each lattice configuration Q computed by the WHOT-QCD collaboration [12].Because the topological charge is sensitive to the ultraviolet cutoff, it should be evaluated from a smeared lattice configuration, for example, by using the gradient flow [13].After smearing, each configuration returns an integer value, or more precisely speaking, the histogram peaks at integer values.Fig. 4 shows the distributions of the topological charge at various temperatures.At T ≤ 279 MeV, we clearly see multiple peaks including the ones at Q ̸ = 0 that signal the instanton condensation.(That the peaks gradually become lower and eventually disappear can be understood as the finite-volume effect.)The condensation melts as the temperature goes up.At T = 348 MeV, we observe one sharp peak at Q = 0 and almost vanishing peaks at Q = ±1.The instantons cease to condense around this temperature.
Figure 1: Distribution of Polyakov line phases, N t × 32 3 lattice, obtained from the WHOT-QCD configurations, is shown as the solid line.The lattice size is N t × 32 3 (N t = 6, 8, 12, 16, and correspondingly T = 464, 348, 232, and 174 MeV.) and 599 configurations were used.These histograms are drawn with 992 bins.Although the agreement with the Haar-random distribution shown in the dash-dot line (see eq. ( 1)) seems to be good at T = 232 MeV, more careful investigation reveals a small deviation and hence the onset of partial deconfinement; see Figs. 2 and 3 and main text.At 174 MeV, agreement with the Haar-random distribution is much better.

Comparison to the large-N partial deconfinement
The observations in the previous section fit nicely within the framework of partial deconfinement [14][15][16][17] in large-N theories.The close connection between the large-N partial deconfinement and the behavior of QCD justifies the use of the terms we used to designate the phases: completely-confined, partially-deconfined, and completely-deconfined.
To understand the meaning of partial deconfinement and Polyakov loop, the amount of gauge   redundancy plays an important role [6].This is explained in Appendix A. See also the companion paper [7] which explains more details.
In seminal papers [3,4], it was pointed out that the confinement/deconfinement transition consists of two phase transitions based on the weak coupling analysis.A more explicit understanding of the physical interpretation and the mechanism of the emergence of the intermediate phase was developed in a series of papers [6,7,11,16,17].Specifically, this phase was identified as the coexistence of confined and deconfined degrees of freedom in the space of colors (internal space) rather than in the usual coordinate space.
We can summarize the connection between the large-N partial deconfinement and the new perspectives discussed above as follows.
1.The completely-confined phase is governed by the Haar-random distribution of the Polyakov lines [6].This is common to both large-N theories and finite-N theories including QCD.
2. The transition from the partially-deconfined phase to the completely-deconfined phase is identified with the Gross-Witten-Wadia (GWW) transition in the large-N theory [18,19].
The GWW transition can be captured by using the expectation values of the multiply-wound Polyakov loops u n = ⟨tr P n ⟩.In particular, after the GWW transition, all u n become nonzero. 10For the QCD, we find that the transition from the partially-deconfined phase to the completely-deconfined phase is associated with the onsets of the Polyakov loops in the higher representations.The Polyakov loops in the higher representations and u n 's (with n ≥ 1) play analogous roles and some of them are directly related.In Appendix C, we give examples of direct relations between u n and Polyakov loops in higher representations.
3. As discussed and observed in Refs.[11,20], it is natural to expect that the chiral symmetry breaking takes place at the GWW point when quarks are massless.(Intuitively, quarks in the confined sector should form a chiral condensate; otherwise, the 't Hooft anomaly would not be preserved.)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. 11Since in QCD the quark mass breaks the chiral symmetry explicitly, the instanton condensation is a natural probe for the finite-N analog of the GWW transition.Indeed, we find that the phase structure suggested by instantons is consistent with that obtained from the Polyakov loops.

Discussion
We have discussed the thermal phase transitions on 4d QCD, i.e., SU(3) gauge group and dynamical quarks, as a natural extension of partial deconfinement in the large-N gauge theories.
Employing the gauge configuration generated by the WHOT-QCD collaboration, we found numerically that the Polyakov loops in various representations start to have nonzero expectation values at different temperatures. 12Our main point is that these should play an important role in characterizing the phases of QCD at finite temperatures.While we found that some expectation values are consistent with zero at the current numerical precision, their true behavior may be ⟨χ r (P )⟩ ∼ e −mr/T where m r is the mass gap from the vacuum (at zero temperature) for the degrees of freedom associated with the representation r. 13 This is so in particular because it is not forbidden by center symmetry.Our interpretation is that deviation from this scaling detects the deconfinement of the modes in the corresponding representation.
In this letter, we focused on the bare, rather than the renormalized, Polyakov loops to characterize the phases of thermal QCD.It is crucial that we study the theory defined at the identical UV cutoff for various temperatures; the configurations use fixed lattice spacing and the temperature is controlled by changing the number of lattice points along the Euclidean time direction.As such, renormalization is not necessary for our characterization.
However, we should emphasize that this is based on the viewpoint of a cutoff field theory.To compare configurations with different cutoff scales and to take the continuum limit, i.e., to confirm that our characterization of phases works properly in QFTs, it is necessary to rephrase our analysis in terms of renormalized Polyakov loops.If the expectation values of the Polyakov loops are multiplicatively renormalized without mixing, we may expect the characterization to be unchanged even if we use the renormalized Polyakov loops instead of the bare Polyakov loops.There are various prescriptions for defining renormalized Polyakov loops.See, for example, [22][23][24][25][26][27] and references therein.However, it is nontrivial whether one can apply appropriate renormalization schemes that satisfy the multiplicative renormalizability and also preserve the "exponential smallness" of the expectation values in the (completely and partially) confined phases.This is an important problem open for questions and it is worthwhile to explore as a future direction.
To overcome the issue of renormalization, it would be useful to make use of other quantities to distinguish phases.In this letter, we focused on the one-point function of the Polyakov loops.It is important to consider other observables, such as multi-point correlation functions, from the point of view of the Haar-randomness and the deviation from it.See Ref. [28] for development along this line, deriving the so-called Casimir scaling of Polyakov loops in various representations from this point of view.The smearing and gradient flow might be useful in this context.
Our conjectured phase structure of thermal QCD is as follows; The first 'transition' at T 1 may well be a crossover.For example, the Polyakov loops are allowed to have small nonzero values that become exponentially small when the sizes of the representations are increased.It would be more natural to expect a transition with non-analyticity at T 2 , given the connection to the condensation of instantons.This can be the case even if the Polyakov loops in large representations are not exactly zero at T < T 2 ; a natural possibility would be that there is a transition between the exponential decay with respect to the dimension of the representation at T < T 2 and the power-law decay at T > T 2 .Note that the possible presence of the intermediate phase in the region T c ≤ T ≲ 3T c (where T c denotes the usual QCD critical temperature) has been discussed from various perspectives (see, e.g., [29][30][31]).
It is clearly important to further verify the new perspective we advocated in this letter.In particular, studying the Polyakov loop in larger representations and its behavior when departing from Haar-randomness, identifying the transition temperature and the order of the phase transitions, and establishing suitable renormalization schemes are valuable future problems.Practically, SU(3) QCD with finite quark mass and pure Yang-Mills are the most tractable targets because many sets of lattice configurations are available for SU(3) QCD and the simulation of pure Yang-Mills is not costly.
An obstacle to generalizing partial deconfinement to finite N had been the meaning of the size of the deconfined sector M (0 ≤ M ≤ N ).Even in the large-N limit, 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 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 also has the advantage of being manifestly gauge invariant.Although in large N -theory the use of the parameter M is shown to have gauge invariant meaning, it may be worthwhile to revisit the analysis of the 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.[32,33].A recent work [34] employs the character expansion to study the deconfinement transition in large-N theory.
In this letter, we have explored the finite-N counterpart of the large-N partial deconfinement.The original motivation for partial deconfinement [14] was to study black hole geometry via holography.Since 1/N corrections should play crucial roles in black hole physics, we hope that our work would be useful in obtaining important intuition into quantum gravitational phenomena such as black hole evaporation from the QFT side.
This work was supported by the Japan Lattice Data Grid (JLDG) constructed over the SINET5 of NII and by the Center for Gravitational Physics and Quantum Information (CGPQI) at Yukawa Institute for Theoretical Physics.M. H. thanks for the STFC grants ST/R003599/1 and ST/X000656/1.H. O. is supported by a Grant-in-Aid for JSPS Fellows (Grant No.22KJ1662).H. S. is supported by the Japan Society for the Promotion of Science (JSPS) KAKENHI Grant number 21H05182.H. W. is supported by the Japan Society for the Promotion of Science (JSPS) KAKENHI Grant number 22H01218.

A Polyakov loop and the amount of redundancy at N = ∞
The purpose of this appendix is to give a short summary of the essential mechanism of the large-N partial deconfinement.In particular, we explain the relevance of the redundancy of states under gauge transformations and its relation to the Polyakov line.
That the amount of gauge redundancy has important consequences is not really new.In fact, it has been known for a century since the theoretical discovery of Bose-Einstein condensation [35], although the connection to Polyakov line and confinement was pointed out only recently [6].To see how the Polyakov line and gauge redundancy are related, we consider three theories with increasing levels of complexity: N indistinguishable bosons, SU(N ) Hermitian matrix model, and SU(N ) QCD.

N indistinguishable bosons and Bose-Einstein condensation
To describe N indistinguishable bosons in R 3 , we use coordinate operators ⃗ x i = (x i , ŷi , ẑi ) and momentum operators ⃗ p i = (p x,i , py,i , pz,i ), where i = 1, 2, • • • , N labels bosons.The Hamiltonian Ĥ is invariant under the permutation of the labels, e.g., Ĥ = x 2 i ) in the weakcoupling limit.That the bosons are indistinguishable means the S N permutation symmetry is gauged.
We can use the coordinate eigenstates |x⟩ = |⃗ x 1 , • • • , ⃗ x N ⟩ that satisfy ⃗ x i |x⟩ = ⃗ x i |x⟩ to describe quantum states.The coordinate eigenstates span the extended Hilbert space H ext that contains S N -non-singlets: For a permutation σ ∈ S N , we define σ by σ . From this, we can define the projection operator π = 1 N !σ∈S N σ that maps H ext to the S N -invariant Hilbert space H inv .Canonical partition function at temperature T can be written in two ways, in terms of H ext and H inv : To see how states in H inv and H ext are related, let us consider the weak-coupling limit and consider a product of one-particle states, |Φ⟩, whose wave function is written as ⟨x|Φ⟩ = N i=1 ϕ i (⃗ x i ).The symmetric group S N acts on |Φ⟩ as ⟨x| σ |Φ⟩ = N i=1 ϕ i (⃗ x σ(i) ).If all one-particle states are the same, i.e., ϕ 1 = • • • = ϕ N , the product state is invariant under S N .On the other hand, if then the product state is invariant under S N −M ⊂ S N .Such an unbroken symmetry acting on H ext leads to an enhancement factor in the partition function.Specifically, when the overlap of different one-particle states can be neglected, we obtain This enhancement factor is responsible for the Bose-Einstein condensation.Many particles fall into the one-particle ground state assisted by this enhancement factor.From this, one can learn that less redundant states are preferred at low energy.We will next discuss the SU(N ) Hermitian matrix model and will see that the permutation σ is nothing but the Polyakov line.This implies that the typical Polyakov line in the path integral is determined in such a way that it leaves typical states dominating partition function invariant, i.e., σ |typical⟩ = |typical⟩.We will elaborate on this statement shortly.

SU(N ) Hermitian matrix model and color confinement
Let us consider a bosonic SU(N ) gauged D-matrix model (D ≥ 2) with the Hamiltonian Ĥ = tr 1 2 where tr means the trace as N × N matrix and V ( X) is a potential term such as V ( X) = − g 2 4 [ XI , XJ ] 2 .Each XI has N 2 components XI,ij , where i, j = 1, 2, • • • , N , that satisfy the Hermiticity condition ( XI,ij ) † = XI,ji .We do not impose the traceless condition.The operator PI,ij is the conjugate momentum of XI,ji .They satisfy the canonical commutation relation [ XI,ij , PJ,kl ] = iδ IJ δ il δ jk .
The Hamiltonian is invariant under the adjoint action of SU(N ) defined by We can use the extended Hilbert space with SU(N ) non-singlet states, H ext .The extended space is spanned by the coordinate eigenstates |X⟩ that satisfy XI,ij |X⟩ = X I,ij |X⟩: Note that X consists of DN 2 real numbers X α=1,••• ,N 2 I=1,••• ,D .Gauge transformation acts on H ext as |X⟩ → U −1 XU .By using the SU(N )-invariant Hilbert space H inv , the canonical partition function at temperature T can be written as [6] where G = SU(N ) and π ≡ 1 volG G dg ĝ is a projection operator from H ext to H inv .This figure is taken from Ref. [17].
The standard technique to rewrite the Hamiltonian formulation to the path-integral formulation tells us that g ∈ G is the Polyakov line [6].By comparing ( 8) with (3), we see that σ ∈ S N corresponds to the Polyakov line.The counterpart of the Bose-Einstein condensation is confinement [6].In general, SU(M ) subgroup of SU(N ) can be deconfined [14][15][16][17], leaving SU(N − M ) ⊂ SU(N ) as a symmetry in H ext that leads to the enhancement factor Vol(SU(N − M )) ∼ e (N −M ) 2 [6].The value of M depends on the energy E in a nontrivial manner.

Distribution of Polyakov line phases in the large-N limit
Let us further focus on typical states dominating thermodynamics that are invariant under SU(N − M ) ⊂ SU(N ).We can fix the gauge in such a way that the SU(M )-deconfined sector is the upper-left M × M -block as in Fig. 5.This choice of embedding of SU(M ) into SU(N ) fixes SU(N ) down to SU(M ) × SU(N − M ).Then, the Polyakov line P takes the following form: The eigenvalues of P dec and P con give the phases θ 1 , • • • , θ M and θ M +1 , • • • , θ N , respectively.From them, we can determine the distribution of the phases ρ dec (θ) and ρ con (θ).The latter is constant, This is because P con can be any element of SU(N −M ) (specifically, P con dominating path integral is distributed following the Haar measure) and the generic phase distribution in this part is uniform in the limit of N − M → ∞.The former is not uniform and its smallest value is zero.The full distribution is [6,16,17] We can fix the overall factor from π −π dθ 1 ρ(θ 1 ) = 1.

C 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 χ r (g) := tr R r (g).
For the irreducible representations, the orthonormal condition, is satisfied.
When we consider G = SU(3), we can identify g and its representation in the fundamental representation: g is 3 × 3 unitary matrix with a determinant equal to 1.In our context g is identified with the Polyakov line P .The SU(3) characters are functions only with respect to the eigenvalues of g, denoted by λ j (j = 1, 2, 3).Namely, χ r (g) = χ r ({λ}).(21) Note that λ j = e iθ j in terms of the Polyakov line phase θ j , and λ 1 λ 2 λ 3 = 1 due to the condition det g = 1.

C.1 List of characters for irreducible representations
The character is a symmetric polynomial in λ's.As is well-known, irreducible representations of SU(3) can be labelled by Young diagram with two rows.In the following, the notation (n, m) represents a Young tableau that has n and m boxes in the first and second rows, respectively.

Figure 2 :
Figure 2: The expectation values of characters vs. temperature for the fundamental, adjoint, rank-2 symmetric, and rank-3 symmetric representations, obtained from WHOT-QCD configurations.Note that the expectation values are real.

Figure 3 :
Figure 3: The log plot of the expectation value of the Polyakov loop in fundamental representation.It is very close to zero at the lowest temperature in the set of configurations (174 MeV).

Figure 4 :
Figure 4: Histogram of the topological charge provided by the authors of Ref. [12].

Figure 5 :
Figure 5: Partial deconfinement in the matrix model.The M ×M -block shown in red is deconfined.This figure is taken from Ref. [17].