Partially Deconfined Phase in Gauge Theories

• Partial deconfinement is an intermediate large-N phase in which only an SU(M) subgroup deconfines while the remainder stays confined, exhibiting O(M^2) scaling.
• The analysis uses the Polyakov loop eigenvalue distribution, which deviates from uniformity and signals phase transitions through gap formation.
• Holographically, this phase connects graviton gas and large black hole regimes, offering key insights into gauge/gravity duality and black hole thermodynamics.

Partial deconfinement is an intermediate large-$N$ phase of gauge theory in which only a subset of the color degrees of freedom behaves as deconfined, while the rest remains confined. Instead of the whole $\mathrm{SU}(N)$ color space being either fully confined or fully deconfined, the system dynamically splits into an effectively deconfined $\mathrm{SU}(M)\subset \mathrm{SU}(N)$ subsector with $0$N-M$. This gives a concrete large-$N$ answer to the question of how gauge theories interpolate between the usual $O(N^0)$ confined regime and the $O(N^2)$ fully deconfined regime, and in gauge/gravity duality it is proposed as the field-theory description of the intermediate small-black-hole or Hagedorn-string regime between graviton gas and large black hole (Hanada et al., 2022, Hanada et al., 2018).

1. Large-$N$ definition and thermodynamic organization

The basic thermodynamic statement is that confinement, partial deconfinement, and complete deconfinement differ by how many adjoint color degrees of freedom are active. In the confined phase, entropy and energy scale as

$S,E\sim O(N^0),$

whereas in the completely deconfined phase they scale as

$\mathrm{SU}(N)$0

Partial deconfinement fills the interval between these limits: if only an $\mathrm{SU}(N)$1 block is excited, then the number of active adjoint degrees of freedom is of order $\mathrm{SU}(N)$2, so

$\mathrm{SU}(N)$3

with $\mathrm{SU}(N)$4 itself varying with temperature or total energy (Hanada et al., 2022).

Regime	Deconfined rank	Large-$\mathrm{SU}(N)$5 scaling
Confined	$\mathrm{SU}(N)$6	$\mathrm{SU}(N)$7
Partially deconfined	$\mathrm{SU}(N)$8	$\mathrm{SU}(N)$9
Completely deconfined	$\mathrm{SU}(M)\subset \mathrm{SU}(N)$0	$\mathrm{SU}(M)\subset \mathrm{SU}(N)$1

This organization is not ordinary phase coexistence in position space. The coexistence is in color space or internal space: an $\mathrm{SU}(M)\subset \mathrm{SU}(N)$2 deconfined block coexists with a confined remainder. In the large-$\mathrm{SU}(M)\subset \mathrm{SU}(N)$3 limit, the distinction between $\mathrm{SU}(M)\subset \mathrm{SU}(N)$4, $\mathrm{SU}(M)\subset \mathrm{SU}(N)$5, and $\mathrm{SU}(M)\subset \mathrm{SU}(N)$6 becomes thermodynamically sharp, because these sectors carry parametrically different $\mathrm{SU}(M)\subset \mathrm{SU}(N)$7-scalings (Hanada et al., 2019).

A useful heuristic relation follows from intermediate-energy states. If

$\mathrm{SU}(M)\subset \mathrm{SU}(N)$8

then partial deconfinement suggests

$\mathrm{SU}(M)\subset \mathrm{SU}(N)$9

This expresses that the system is too energetic to remain in an $0Hanada et al., 2019).

2. Polyakov loop, eigenvalue densities, and the deconfined fraction

The principal diagnostic is the Polyakov loop and, more fundamentally, the distribution of its phases. The Polyakov loop is

$0

with $0Hanada et al., 2022).

The three regimes are encoded directly in $0

$0

so the phases are uniformly distributed. In partial deconfinement, the distribution becomes non-uniform but remains nonzero everywhere on $0Hanada et al., 2022).

A particularly sharp formula is

$N-M$0

Equivalently, the deficit from the uniform distribution measures the deconfined fraction $N-M$1. This permits a direct extraction of $N-M$2 from Polyakov-line data without specifying a microscopic gauge-fixed block (Hanada et al., 2022, Watanabe et al., 2020).

In weakly coupled large-$N-M$3 Yang–Mills on $N-M$4, the eigenvalue density between the Hagedorn and GWW transitions is explicitly a mixture,

$N-M$5

with

$N-M$6

$N-M$7 at the Hagedorn transition, and $N-M$8 at the GWW transition. In that example the Polyakov loop takes the characteristic values $N-M$9 at the confinement $N$0 partial-deconfinement transition and $N$1 at the partial $N$2 complete-deconfinement transition (Hanada et al., 2019).

This eigenvalue-density formulation is broader than the center-symmetry criterion. It can be applied even when center symmetry is absent or explicitly broken, including theories with fundamental matter and QCD-like theories (Hanada et al., 2022).

3. Microscopic $N$3 subsectors and phase transitions

A concrete microscopic realization appears in the gauged Gaussian matrix model and related weak-coupling examples. Gauge-invariant excitations of the full $N$4 theory are built from traces over all color indices, while a partially deconfined sector is isolated by restricting those indices to $N$5. Full $N$6 gauge invariance is then restored by averaging over the gauge group,

$N$7

so the dominant states can still be interpreted as arising from an excited $N$8 block embedded in $N$9 (Hanada et al., 2022).

This construction underlies the thermodynamic ansatz

$O(N^0)$0

which states that the partially deconfined $O(N^0)$1 state behaves thermodynamically like the GWW-critical state of a truncated $O(N^0)$2 theory. In analytically tractable weak-coupling examples, the entropy at the relevant energy is saturated by states whose excitations live entirely in an $O(N^0)$3 submatrix (Hanada et al., 2019).

Large $O(N^0)$4 also organizes the transition sequence. Two transitions naturally delimit the intermediate regime:

1. Hagedorn transition: onset of deconfinement, $O(N^0)$5.
2. Gross–Witten–Wadia transition: completion of deconfinement, $O(N^0)$6.

Accordingly, partial deconfinement is not merely a smooth interpolation in observables; at large $O(N^0)$7 it is an intermediate phase or branch bounded by identifiable transitions (Hanada et al., 2022).

The ensemble dependence is important. In the microcanonical ensemble, the intermediate branch is naturally interpreted as a genuine regime between low-energy confinement and high-energy complete deconfinement. In theories with first-order confinement/deconfinement transitions, such as pure Yang–Mills, the partially deconfined configuration is typically a saddle of the canonical free energy rather than the dominant canonical phase. In other theories it can be a stable intermediate phase (Hanada et al., 2018, Gautam et al., 2022).

The physical intuition developed in the review literature is that the system “uses only part of its color space” to thermalize. This is repeatedly compared with Bose–Einstein condensation: some degrees of freedom remain in a confined, condensate-like sector while only a subset is excited (Hanada et al., 2022).

4. Holographic interpretation: small black holes, Hagedorn strings, and negative specific heat

One of the main motivations for partial deconfinement is holography. In the standard AdS/CFT dictionary, the confined and fully deconfined phases correspond to thermal AdS or graviton gas, and to the large AdS black hole, respectively. Gravity, however, also contains intermediate configurations, especially small black holes and Hagedorn strings. Partial deconfinement is proposed as their gauge-theory dual (Hanada et al., 2022, Hanada et al., 2019).

The canonical scaling puzzle is the small AdS black hole in AdS$O(N^0)$8/CFT$O(N^0)$9, for which

$O(N^2)$0

Because energy decreases as temperature increases, this is a negative-specific-heat object. A fully deconfined $O(N^2)$1 plasma has a fixed $O(N^2)$2 number of active degrees of freedom, so it does not naturally reproduce this behavior. Partial deconfinement resolves the mismatch by making the active rank $O(N^2)$3 energy-dependent: the black hole is identified not with all $O(N^2)$4 colors deconfining, but with an $O(N^2)$5 block, $O(N^2)$6 (Hanada et al., 2022, Hanada et al., 2019).

In the D-brane picture, $O(N^2)$7 D-branes form the bound state dual to the small black hole. The effective ’t Hooft coupling of that bound state is

$O(N^2)$8

and for strong coupling one obtains

$O(N^2)$9

which leads to

$N$0

The varying size of the deconfined block is therefore the microscopic mechanism for the small-black-hole equation of state (Hanada et al., 2018).

The same interpolation is formulated as

• graviton gas $N$1,
• small black hole / Hagedorn string $N$2,
• large black hole $N$3.

The review literature also emphasizes a geometric interpretation in terms of string length and operator length: very long strings and long-trace operators correspond to deconfined blocks, whereas short strings such as gravitons correspond to short traces and tiny deconfined blocks. This suggests that emergent bulk geometry is encoded in how color degrees of freedom cluster into deconfined subblocks (Hanada et al., 2022).

In BFSS and BMN matrix models, the same logic is used to interpret the partially deconfined phase as an intermediate Schwarzschild black hole phase, in particular as an 11d Schwarzschild black hole in the M-theory picture (Hanada et al., 2022).

5. Evidence across analytic models, lattice simulations, and protected observables

The simplest analytic examples are the gauged Gaussian matrix model and weakly coupled large-$N$4 gauge theories on $N$5. In these systems one can count states explicitly, derive the Polyakov-loop eigenvalue density, and show that the interval between the Hagedorn and GWW transitions is naturally interpreted as a partially deconfined regime (Hanada et al., 2019, Hanada et al., 2019).

Strong-coupling evidence comes from matrix quantum mechanics on the lattice. In the bosonic Yang–Mills matrix model, lattice Monte Carlo studies in static diagonal gauge observed that, after appropriate gauge fixing, an $N$6 submatrix carries the deconfined behavior while the complementary sector remains confined. In the interacting $N$7 model at $N$8, the Polyakov-line distribution and extensive observables were fit by coexistence formulas of the form

$N$9

together with

$S,E\sim O(N^0),$0

and

$S,E\sim O(N^0),$1

The central claim was that the excess energy above confinement comes almost entirely from the $S,E\sim O(N^0),$2 block (Watanabe et al., 2020).

A complementary strong-coupling result concerns pure Yang–Mills. In the partially deconfined saddle of finite-temperature strong-coupling lattice gauge theory, the confined sector still supports flux tubes and a linear confinement potential with the same string tension as in the completely confined phase. In the strong-coupling normalization,

$S,E\sim O(N^0),$3

and the confined and mixed sector Wilson loops decay with that same $S,E\sim O(N^0),$4, while the deconfined subsector does not show confining exponential decay. The claim is therefore sector-dependent confinement: the deconfined $S,E\sim O(N^0),$5 sector is nonconfining, but the confined complement remains linearly confining (Gautam et al., 2022).

The bosonic plane-wave matrix model provides further support. Near the upper transition $S,E\sim O(N^0),$6, the Polyakov-line eigenvalue histogram is well fit by a gapped GWW-type distribution with fitted

$S,E\sim O(N^0),$7

slightly above the GWW value $S,E\sim O(N^0),$8, consistent with being just above the partial-deconfinement regime and close to complete deconfinement (Hanada et al., 2019).

A more specialized realization appears in the Cardy-like asymptotics of the 4d $S,E\sim O(N^0),$9 superconformal index. There, the dominant saddle can break $\mathrm{SU}(N)$00 to $\mathrm{SU}(N)$01, and the leading large-$\mathrm{SU}(N)$02 growth is conjectured to organize as an infinite tower of $\mathrm{SU}(N)$03-center saddles. The entropy of the $\mathrm{SU}(N)$04-th saddle obeys

$\mathrm{SU}(N)$05

so the index exhibits a precise notion of partial deconfinement with reduced asymptotic growth relative to the fully deconfined black-hole saddle (Ardehali et al., 2019).

6. QCD, symmetry criteria, and finite-$\mathrm{SU}(N)$06 limitations

The large-$\mathrm{SU}(N)$07 QCD-oriented formulation argues for a three-stage thermal structure—completely confined, partially deconfined, completely deconfined—and proposes that the intermediate regime is generic rather than exceptional. One route to this claim uses the Polyakov-loop eigenvalue density: at $\mathrm{SU}(N)$08, the confined limit is

$\mathrm{SU}(N)$09

whereas the infinitely hot limit is

$\mathrm{SU}(N)$10

If the distribution changes continuously with temperature, then a gap must open somewhere along the interpolation, and that gap-opening is the GWW transition. In this sense, the GWW point separates partial from complete deconfinement (Hanada et al., 2023).

The same work reframes the Polyakov loop as a probe of gauge-redundancy structure rather than merely an order parameter for center symmetry. In the extended Hilbert-space formulation, a partially deconfined $\mathrm{SU}(N)$11 state retains a large enhancement factor from the confined $\mathrm{SU}(N)$12 complement,

$\mathrm{SU}(N)$13

which thermodynamically favors intermediate-$\mathrm{SU}(N)$14 states in an intermediate regime (Hanada et al., 2023).

At finite $\mathrm{SU}(N)$15, especially $\mathrm{SU}(N)$16, the sharp subgroup picture softens. Complete confinement is then diagnosed by approximate Haar randomness of the Polyakov-line phase distribution. For finite $\mathrm{SU}(N)$17, the Haar benchmark is

$\mathrm{SU}(N)$18

For $\mathrm{SU}(N)$19 QCD, the proposed observational pattern is: low $\mathrm{SU}(N)$20 approximately Haar-random, intermediate $\mathrm{SU}(N)$21 with a growing fundamental Polyakov loop but suppressed higher-representation loops, and higher $\mathrm{SU}(N)$22 where larger-representation loops turn on and topological activity changes (Hanada et al., 2023).

In the WHOT-QCD dataset analyzed in that work, departure from Haar-random behavior begins around $\mathrm{SU}(N)$23 MeV, while for $\mathrm{SU}(N)$24 MeV the system is argued to be partially deconfined. The analogue of the upper transition is inferred from higher-representation Polyakov loops turning on only at

$\mathrm{SU}(N)$25

and from topological-charge histograms in which nonzero-$\mathrm{SU}(N)$26 peaks disappear for

$\mathrm{SU}(N)$27

The nontrivial point is that higher-representation Polyakov loops and topological charge both point to the same temperature range for the upper transition out of partial deconfinement (Hanada et al., 2023).

A separate development addresses the fact that center symmetry alone does not distinguish partial from complete deconfinement, because both break the center. In suitable theories, an additional global symmetry can do so. In softly broken $\mathrm{SU}(N)$28 SYM at $\mathrm{SU}(N)$29, CP is broken in the confined and partially deconfined phases but restored in the completely deconfined phase; in a strongly-coupled lattice gauge theory with probe quarks, chiral symmetry is broken in the confined and partially deconfined phases but restored in the completely deconfined phase. The proposed pattern is therefore:

• confined: center preserved, second global symmetry broken;
• partially deconfined: center broken, second global symmetry still broken;
• completely deconfined: center broken, second global symmetry restored (Hanada et al., 2021).

Several caveats recur in the literature. The large-$\mathrm{SU}(N)$30 inevitability argument relies on continuity of the eigenvalue distribution and on avoiding a direct first-order jump from complete confinement to complete deconfinement. At finite $\mathrm{SU}(N)$31, especially $\mathrm{SU}(N)$32, $\mathrm{SU}(N)$33 is not a sharp variable, $\mathrm{SU}(N)$34 corrections may be substantial, and the three regimes may be separated only by crossovers rather than by nonanalytic transitions. The GWW ansatz for the eigenvalue density is also not universal, and Polyakov-loop renormalization must be handled carefully in finite-$\mathrm{SU}(N)$35 QCD analyses (Hanada et al., 2023, Hanada et al., 2019).

Taken together, these results define the partially deconfined phase as a large-$\mathrm{SU}(N)$36 regime in which an effective $\mathrm{SU}(N)$37 sector deconfines while the complement remains confined, with $\mathrm{SU}(N)$38, $\mathrm{SU}(N)$39, a non-uniform but ungapped Polyakov-loop eigenvalue density, and a natural interpretation as the gauge-theory counterpart of small black holes and Hagedorn strings. The broad conjecture is that this is a generic organizing principle of confinement/deconfinement physics, not a model-specific curiosity (Hanada et al., 2022).