Partial deconfinement for some boson matrix models
Hiromasa Watanabe / 渡辺 展正[arXiv:2005.04103](Univ. Tsukuba)

G. Bergner (U. Jena), N. Bodendorfer (U. Regensburg), S. S. Funai(OIST), M. Hanada (U. Surrey), E. Rinaldi(Arithmer Inc. & RIKEN), A. Schäfer (U. Regensburg), P.Vranas (LLNL, LBNL) & M. Knaggs (U. Southampton)

Review of partial deconfinement

In terms of distribution of the Polyakov line phases, P = 1 ∑N eiθjj=1N

Partial Deconﬁnement

ρ (P)(θ)

P=0

Around the critical temperature, the physical degrees
of freedom those behave as in confined/deconfined phases
are coexisting in internal space (color space).

−ππ

E/N 2 & | P |E/N 2 & | P |E/N 2 & | P |π−πpartially deconfined

Hagedorn transition[Aharony, Marsano, Minwalla, Papadodimas & Raamsdonk, (2003)/Schnitzer, (2004)]

ρ (P)(θ)π−π

confined

Possibilities of phase structure in (large N) gauge theories.

ρ (P)(θ)

P≠0completely deconfined

Gross-Witten-Wadia transition

In partially-deconfined phase,[Hanada, Ishiki & HW, (2018) / Hanada, Jevicki, Peng & Wintergerst, (2019)]

MM (P)(P)ρ (θ) = 1 −⋅ ρcon(θ) +⋅ ρdec (θ)
(N)
N(P)

T1 = T2

T

T1 T2

M
P=,2N

Note that due to the large N limit as thermodynamic limit andnonlocal interaction, the situation for thermodynamic properties changes.

・Intuitive picture

・counterpart of black hole with negative specific heat() Deconfined(2E(T ∈ Tdec) ∼ O(N ))02O(N)<E<O(N)?
What happens if∼ ∼N−M)

M∼

S = S(M),εN : order parameter

11 2XI : N × N Hermitian mat.
2S=N
dt Tr(Dt XI ) + XI∑∫{2} D X = ∂ X − i A,X
20[ t I]t It I
I=1At critical temperature Tc = 1/log D

N
D 2
D 122
⟨E(Tc)⟩ =dt Tr XI= (N − M ) ++M2
⟨β ∫( 2 4)
2T=Tc⟩
222= DN ⟨x ⟩ = D(N − M )⟨x ⟩con + DM ⟨x ⟩dec

12 (X)⟨x ⟩con = dx x ρcon(x) = ,∫21122S=N
dt Tr(Dt XI ) − [XI, XJ]∑∫{}
24I=1 0Showing the hysteresis in the narrow region
We apply the same method as Gaussian MMIntroducing a term to try to constrain configs.

ΔS =2 (γ2PN−M − δ)

2M

1PM =eiθjM∑
j=1( PM< 1 −2 δ )( PN−M > δ)N

1PN−M =eiθj∑
N − M j=M+1M{

N − M {(MN−M

1| PM | ≈ ,2{PM(2− 1 −2 δ )γ( PM1+δ> 2 ){PM(21+δ− 2 )2)| PN−M | ≈ 0can be realized     (M, M′)
= (16, 24)       



(X)
(X)
・ρcon(x), ρdec(x) are independent on M or M′ (up to finite-size effect.)

・The values of their variances are 1/2 and 5/8 (D = 2).
・They do not coincide the Gaussian distribution with above variances.
At low-temp.(i.e. confined phase), the distribution approaches to it.
Also possible to see the coexistence in a different manner.(discussed in paper)β 

(⋆)c.f.) [Bergner, Bodendorfer, Hanada, Rinaldi, Schafer & Vranas, (2019)]

  1 D 12 (X)⟨x ⟩dec = dx x ρdec (x) =+∫D ( 2 4)

2. Yang-Mills matrix model (bosonic-BFSS)

γ





(X)
ρdec (x)

2⋯  (X)ρcon(x)

22MM(X)(X)
ρ (X)(x; M) = 1 − ( N ) ρcon(x) + ( N ) ρdec(x)()( as M = 2NP )N=32 @T=Tc  

We expectD[Hanada, Shimada & Wintergerst, (2019)]

     

with x ≡ { NXI,ii, 2NReXI,ij, 2NImXI,ij} (i > j)2

N=32 @T=Tc



2Bose-Einstein-Condensate

By solving simultaneous eq. (★) with different M and M′

β

2

Bulk

Since they should be uniquely determined.

[Hanada, Jevicki, Peng & Wintergerst (2019)]

2Bulk(X)(X)
Estimation of ρcon(x), ρdec (x) numerically; (for D = 2 case)

1. gauged Gaussian matrix model2)

Permutation symmetry among N identical bosons is gauge symmetry!
Polyakov loop
Oﬀ-Diagonal Long Range Order (ODLRO)of partial deconfinement

Analysis of matrix models()BHc.f.) Confinement at large N& its gauge-equivalents

D(BH

E = E(M) ∼ εN 2,{{{N − M {(Mfrom deconfined sector

N=64  




(X)
ρcon(x)


        

(X)ρdec(x)

N=64      (X)(X)
ρdec(x) − ρcon(x)) Confined0E(T ∈ Tcon) ∼ O(N )M

S = SGWW(M)・Originally motivated from gauge/gravity duality [Hanada & Maltz, (2016)]e.g) large N theories whose gauge group is U(N) with adjoint matters(E = EGWW(M),The contribution comes only T1T



T2

T(P)
ρdec(θ) is defined at GWW-point   
















Note : small nontrivial M dependence may be found.
It is the diﬀerence from Gaussian MM (due to interaction?)

・Energy contribution only comes from deconfined sector;
23N βEcon ≡ ⟨− 4β ∫0 dt Tr [XconI, XconJ] ⟩,Edec ≡ E − Econ(0)Econ
is computed by the configs. with | P | ≈ 0 and | PM | ≈ 0        