YITP-14-35, SU-ITP-14/10, IPMU14-0106

Deconfinement transition as black hole formation by the condensation of QCD strings
Masanori Hanadaa,b,c , Jonathan Maltzd , and Leonard SusskindaaarXiv:1405.1732v1 [hep-th] 7 May 2014

Stanford Institute for Theoretical Physics, Stanford University, Stanford CA 94305, USAb
Yukawa Institute for Theoretical Physics, Kyoto University,
Kitashirakawa Oiwakecho, Sakyo-ku, Kyoto 606-8502, Japanc
The Hakubi Center for Advanced Research, Kyoto University,
Yoshida-Ushinomiya-cho, Sakyo-ku, Kyoto 606-8501, Japan anddKavli Institute for the Physics and Mathematics of the Universe,
Todai Institutes for Advanced Study, The University of Tokyo, Kashiwa, Chiba 277-8582, JapanWe argue that the deconfinement transition of large-N Yang-Mills theory is the condensationof very long and self-intersecting chromo-electric flux strings (QCD string), which is analogous
to the formation of a black hole in string theory. We do this by using lattice gauge theory andmatrix models. As evidence, we derive an analytic formula for the deconfinement temperature in
the strong coupling limit and confirm it numerically. Dual gravity descriptions interpreted in thismanner should make it possible to understand the details of the formation of black holes in terms offundamental strings. We argue that very simple matrix models capture the essence of the formationof black holes.
INTRODUCTION

In the gauge/gravity duality [1], the deconfinementtransition of the gauge theory is dual to the formationof a black hole in the gravity side [2]. In this paper, wegive an intuitive way of understanding this correspondence, without referring to a sophisticated dictionary ofthe duality. Our argument does not assume the dual
gravity description, and hence it is applicable to genericgauge theories including QCD. We pay attention to thestringy degrees of freedom in gauge theory – the Wilson
loops – and see how their behaviors change across thedeconfinement transition. The authors believe that thisresult will be useful in understanding the formation andthermalization of the quark-gluon plasma in QCD, andthe formation of a black hole in the gravitational picture,from a unified perspective.
Although our argument applies to rather generic gauge
theories and matrix models, our initial motivation comesfrom a simple matrix model for the black hole [3, 4] tostudy the duality [2] between the deconfinement transition in 4d N = 4 super Yang-Mills theory on a threesphere and the Hawking-Page transition [5] of a blackhole in the AdS space. At the Hawking-Page transition,a small black hole whose Schwarzschild radius is of order
AdS radius is formed. Such a small black hole can naturally be identified with a long and winding string [7, 8].
One of the authors (L. S.) has argued that it can be modeled by using a lattice gauge theory with a continuum
time and a few spatial lattice sites, e.g. a tetrahedron(Fig. 1) sitting at the center of the AdS space, by identifying the string and chromo-electric flux string. Our
result in this paper justifies this argument.
As a concrete example, let us consider pure U (N )Yang-Mills theory. The Hilbert space is spanned by Wilson loops acting on the vacuum |0i, WC WC 0 · · · |0i, where
WC represent the Wilson loop along a closed contour C.

FIG. 1.
Tetrahedron matrix model (left) and single-site
model (Eguchi-Kawai model) with three links (right).

In the standard identification of the Feynman diagramsand the string world-sheet [6], the Wilson loop is naturally interpreted as the creation operator of the string.
Therefore, WC1 WC2 · · · WCk |0i is the state which consistsof k strings, and C1 , C2 , · · · , Ck represent the shapes ofthe strings.
In the large-N limit, the energy of the string is approximated by its length. In the confinement phase, theenergy is of order N 0 per unit volume, and hence a typicalstate is a finite-density gas of loops with finite length. In
other words, it is gas of glue-balls. In this gas, two loops
can intersect with each other and combined to form alonger string. Alternatively, when a loop intersects withitself, it can be split into two shorter loops. However suchjoining and splitting are suppressed at large-N .
In the deconfinement phase, the energy density is oforder N 2 . In this phase the loops necessarily intersect
O(N 2 ) times. Although the interaction at each intersection is 1/N -suppressed, small interactions at many
intersections accumulate to a non-negligible amount.
As we will explain shortly, a typical state consists offinitely many very long and self-intersecting strings,
whose lengths are of O(N 2 ). In the string theory, it isnatural to interpret such very long and self-intersecting
strings as a black hole [7, 8]. In this sense, the deconfinement transition can be understood as the formation of a
‘black hole’ through condensation of QCD strings.

2
In this paper, we establish this picture by using latticegauge theory and matrix models as concrete examples.
Our interpretation does not rely on details of the theory
and hence is applicable to any gauge theory. When it
is applied to theories with gravity duals, the ‘analogy’
turns to an equivalence. That is to say that the ‘QCD
string’ is naturally identified with the fundamental string,and their condensation is equivalent to the formation ofan actual black hole in superstring theory. Therefore
we believe our picture is useful for understanding theblack hole from the degrees of freedom of the fundamental
string.

the same link variable. (With this definition, even if twoloops go through the same point ~x, they do not necessarily ‘intersect’. Although this definition may look strange
at first glance, this is actually a natural generalizationof the intersection between strings in continuum space,because two strings represented by the Wilson loops on
the lattice can interact only when they share the same
link, as we will see shortly.)
When the loop WC does not self-intersect, the electric
term K acts on the state |WC i ≡ WC |0i as
K|WC i =
λL
λN X  α  αEµ,~x , Eµ,~x , WC |0i =|WC i.
2
2
µ,~
x,α

(4)

QUANTITATIVE ARGUMENT ON LATTICE

Let us consider (D + 1)-dimensional U (N ) Yang-Mills
lattice gauge theory in the Hamiltonian formulation [9].
(We consider D ≥ 2, which goes through a first-order deconfinement transition as the temperature is raised.) The
time direction is continuous, and D-dimensional spaceis regularized by a D-dimensional square lattice. The
Hamiltonian is given by the kinetic term (or electricterm) K and the potential term (magnetic term) V respectively, H = K + V , where
N2

λN X X X α 2
Eµ,~x
K=
2
µ α=1

(1)

~
x

and
V =


N XX
†
†
N − Tr(Uµ,~x Uν,~x+µ̂ Uµ,~U).(2)x+ν̂ ν,~
x
λ
µ<ν
~
x

Here Uµ (~x) is the unitary link variable connecting ~x andα~x + µ̂, where µ̂ is the unit vector along µ-direction. Eµ,~x
is defined via its commutation relation which is given by,
α
α
[Eµ,~
y ] = δµν δ~x~y · τ Uν,~y,x , Uν,~†[Eµ,~x , Eν,~y ] = [Uµ,~x , Uν,~y ] = [Uµ,~x , Uν,~y ] = 0.

(3)τα (α = 1, 2, · · · , N 2 ) are generators of the U (N ) group,
which are N × N Hermitian matrices normalized as
PN 2 α α2α=1 τij τkl = δil δjk /N . The vacuum |0i is annihilatedααby Eµ,~x , Eµ,~x |0i = 0.
The Hilbert space consists of gauge-invariant states, inwhich Wilson loops acting on |0i form an over-complete
basis. In the lattice gauge theory, the Wilson loop is
the trace of the product of link variables along a contour
C, WC = T r (Uµ,~x Uν,~x+µ̂ · · · Uρ,~x−ρ̂ ). The Wilson loop
sources a set of interacting gluons. The correspondence
between the Feynman diagram expansion and the string
world-sheet leads us to interpret the Wilson loop as thecreation operator of the closed string.
Let us say the Wilson loop is not self-intersecting wheneach link variable appears only once. Similarly, two loops
WC and WC 0 do not intersect when they do not share

where L is the length of the contour C in the lattice unit.
In the same manner, for a multi-loop
state |WC , WC 0 , · · · i = WC WC 0 · · · |0i, K acts as
K|WC , WC 0 , · · · i = λ2 (L + L0 + · · · )|WC , WC 0 , · · · i when
there is no intersection.
When two loops WC and WC 0 intersect once by sharing
Uµ,~x , they can be joined to form a longer loop as follows.
Let us write WC and WC 0 as WC = Tr(Uµ,~x MC ) and
WC 0 = Tr(Uµ,~x MC 0 ), where MC and MC 0 are product ofother link variables. Then,
K|WC , WC 0 i
λ(L + L0 )|WC , WC 0 i=
2X+λN
Tr(τ α Uµ,~x MC ) · Tr(τ α Uµ,~x MC 0 )|0iα=λ(L + L0 )λ|WC , WC 0 i + Tr(Uµ,~x MC Uµ,~x MC 0 )|0i.
2N(5)

The second term in (5) is a longer string whose lengthis L + L0 . The state is then a superposition of a state
containing two loops of length L and L0 and a state of asingle long string of length L + L0 . In the same manner,
a self-intersecting string can be split into two strings. If
there are multiple intersections, such joinings and splittings take place at all intersections.
In this manner, the electric term K measures the total length of the strings, and also joins and splits strings.
The magnetic term V is the plaquette, which is the smallest possible Wilson loop; this term adds a very shortstring (one plaquette) to the states.

Confinement phase as gas of free strings

In the confining phase, the energy density is of orderN 0 . Therefore, the length of the strings is also of order N 0 . In this phase joining and splitting coming fromthe electric term is negligible at large-N , because there
are only O(N 0 ) intersections, and interaction at each3intersection is 1/N -suppressed. The confinement phasecan therefore be understood as a gas of non-interacting
strings. (The left of Fig. 2)

Deconfinement phase as black holeIn the deconfinement phase, the energy density is of order N 2 . Therefore the total length of the strings is alsoof order N 2 . The strings must intersect O(N 2 ) times,
and then the 1/N -suppressed interaction at each intersection accumulates and becomes non-negligible (Fig. 2).
Strings are joining, splitting, as well as changing theirshapes very rapidly in this phase. This is true even in thestrong coupling limit λ = ∞, where the magnetic term
V vanishes. The deconfinement phase is highly dynamical, unlike the common folklore stating that “the strong
coupling limit is free string theory.

Then the energy of the deconfinement phase is proportional to the total length of the strings Ltotal (T ), which
is an unknown function of the temperature T , because
the electric term measures the length:
E=K=

λ
Ltotal (T ).
2

The entropy S is also proportional to Ltotal (T ),S = Ltotal log(2D − 1).
Therefore the free energy F = E − T S is given by
λF = Ltotal (T )− T log(2D − 1) .
2

There are many string states with total length Ltotal ∼N 2 . One extreme, are cases with configurations containing a single long string, while the other extreme, arestates only containing the O(N 2 ) shortest strings (plaquettes). A single long string can possess a lot of states;
after Uµ,~x , one can multiply any link variables originat†
ing from the point ~x + µ̂ except Uµ,~x , and hence at eachpoint there are 2D −1 choices, and the entropy is roughlylog((2D − 1)Ltotal ) = Ltotal log(2D − 1) ∼ N 2 . The states
which consist of several long strings can carry the sameamount of entropy. On the other hand, if we consider
a bunch of very short strings, say a bunch of plaquetts,
then the entropy is much smaller, because the state canbe specified by the number of each type of plaquette.
Therefore, it is natural to conclude that typical states
contain a few long strings. In string theory, this is exactly
how the black hole (or more precisely a D-dimensional
black brane) is formed from the fundamental string.
In order to justify this picture, we derive the following
analytic predictions, and then confirm them numerically.
Let us consider the strong coupling limit (λ = ∞, V = 0).

(7)

(8)

The free energy of the confinement phase is zero up to
a 1/N correction, and hence the confinement phase isfavored when (8) is positive and the deconfinement transition takes place when (8) crosses zero. Therefore, although we do not know the T -dependence of the length
Ltotal , we can easily determine the critical temperature
Tc ,Tc = λ/(2 log(2D − 1)).

FIG. 2. The intuitive picture of the deconfinement transitionin terms of QCD strings. In the confining phase, the strings
do not intersect often, interaction between them is negligible,
and the free-string picture is valid (left). As the system isheated up, more strings are excited, they intersect more often(middle). When the density of strings becomes large, theinteraction becomes non-negligible and a long string is formed(right). This is the same as the formation of a black hole instring theory.

(6)

(9)

Note that this derivation is formally the same as thederivation of the Hagedorn temperature in free string
theory on a lattice. See e.g. [10].

Matrix models

Strictly speaking, the (D + 1)-dimensional Yang-Mills
theory is analogous to a D-dimensional black branerather than a black hole, because the condensation of
the strings fills entire D-dimensional space. In order todescribe a black hole (0-brane), let us consider the dimensionally reduced D-matrix model H = K + V , where
P PN 2α 2is the kinetic term (or electric
K = λNµα=1 Eµ2Pterm) and V = N
N− Tr(Uµ Uν Uµ† Uν† ) . Now theµ6=ν
λspace is reduced to a single point, and D link variables areattached to that point (see Fig. 1 for the case of D = 3).
This is the Eguchi-Kawai model [11] with a continuous
time direction. At strong coupling the U (1)D center symmetry along the spatial directions, Uµ → eiθµ Uµ , is not
spontaneously broken. Then this theory is known to beequivalent to the (D+1)-dimensional theory at large-N in
the sense that translationally invariant observables in thelatter, for example the energy density and entropy density, are reproduced from the former to the leading orderin the 1/N -expansion [11]. At weak coupling, this modelis analogous to the bosonic part of the matrix model ofM-theory [12], which is dual to the black zero-brane in
type IIA supergravity in the ’t Hooft large-N limit [13].
For D ≥ 2, this theory exhibits the deconfinement transition, which is characterized by the non-vanishing expectation value of the Polyakov loop. At sufficiently strong

4
coupling, the transition is of first order. In the deconfinement phase, the energy and the entropy is of order N 2 ,and typical state should be described by long, winding
strings such as T r(U1 U2 U1† U1† U2 U2 U1 · · · ). All of thesearguments parallel the case of the (D + 1)-dimensionallattice, and (9) should hold in this case as well.
We can also consider other matrix models such as thetetrahedron model (Fig. 1). In this model, space consists of four points labelled by 1, 2, 3, and 4. There
are six link variables Umn (m 6= n) which satisfy†
Umn= Unm . We have shown numerically that thismodel also possesses a first-order deconfinement transition. The long string can be described with a termsuch as T r(U12 U23 U31 U14 U42 · · · ). The entropy is S =Ltotal log 2, and (9) is modified as Tc = λ/(2 log 2).
NUMERICAL CONFIRMATIONIn this section we confirm the analytic predictions bylattice Monte Carlo simulation. There exists a wide literature on Monte Carlo algorithms and a nice review can befound in [14]. Thanks to the Eguchi-Kawai equivalence[11], we do not have to study the (D+1)-dimensional YM
theory, and hence we consider the matrix models. In order to study the thermodynamics, we consider the theory
in Euclidean time, and compactify the time direction toa circle with circumference β = 1/T . The Lagrangian in
Euclidean signature is given by P
L = K + V , where V isN†
the same as before and K = 2λx · (Dt U~
x)~x Tr Dt U~
with Dt U~x = ∂t U~x − i[At , U~x ]. We regularize the modelby introducing nt lattice sites along the temporal direction. The lattice spacing a is given by ant = β. The
action isN X †Slattice = −Tr Vt Uµ,t+a Vµ,t
Uµ,t + c.c.
2aλ µ,t
aN X 
††+N − Tr(Uµ,t Uν,t Uµ,tUν,t) (10)λFor P
the tetrahedron matrix model the loop is Ptet =4
1
m=1 Tr(Vm,t=a Vm,t=2a · · · Vm,t=nt a ). The expecta4N
tion value of P itself vanishes trivially due to the U (1)phase. Therefore we consider the expectation value ofthe absolute value, h|P |i. We performed hot and cold
start simulations, in which the temperature is gradually
decreased and increased, respectively. In Fig. 3, h|P |iin the tetrahedron model is plotted. We can see a clear
hysteresis at 0.66 ≤ (T /λ) ≤ 0.74, which means the transition is of first order as expected. The theoretically predicted critical temperature (Tc /λ) = 1/(2 log 2) ' 0.721
is in this range. We observed similar hystereses also in
the Eguchi-Kawai model with various values of D. In
the right panel of Fig. 3 we plot the temperature rangeof the hystereses for D = 2, 3, · · · , 10. We can see thatthe theoretical predictions for Tc is consistent with thesimulation results. Note that Tc goes close to the lower
edge of the hysteresis as D becomes large; it would beinteresting we can understand the mechanism behind it.

FIG. 3. (Left)The expectation value of the Polyakov loop in
the tetrahedron matrix model in the strong coupling limit.
There is a strong hysteresis around the theoretically predicted critical temperature (Tc /λ) = 1/(2 log 2) ' 0.721.
N = 64, nt = 12. The temperature range of the hystereses of the Eguchi-Kawai model. (Right)The dashed line isthe analytic prediction for the critical temperature, (Tc /λ) =1/(2 log(2D − 1)). N = 64, nt = 16.

µ6=ν,twhere t = a, 2a, · · · , nt a, and t = nt a is identified tot = 0. Vt is the unitary link variable connecting t and
t+a, and Uµ,t are spatial links at time t. The tetrahedronmatrix model can be regularized in a similar manner withactionN XX
†Tr(Vm,t Umn,t+a Vn,t
Unm,t ) + c.c.
Stet = −2aλ t m<naN X X−((N − Tr(Ulm,t Umn,t Unl,t )) + c.c.) .
λ t

CONCLUSION AND DISCUSSION

In this paper we have visualized the relationship between the deconfinement phase of the large-N YangMills theory and a black hole, by paying attention to
the stringy degrees of freedom (i.e. the Wilson loops)
in Yang-Mills theory. The essence is contained alreadyin the strong coupling limit of the lattice gauge theory. The common folklore that this theory describes freestrings is only correct in the confinement phase. Thel<m<n(11) deconfinement phase is dominated by very dynamical
self-intersecting strings which should be understood as
In the following we concentrate on the strong coua black hole, unlike the free case.
pling limit, where the magnetic terms are omitted.
In the gauge theories with dual gravity descriptions,
The order parameter for the deconfinement transition
the Wilson loops are naturally identified with the funis the Polyakov loop, P = N1 Tr(Vt=a Vt=2a · · · Vt=nt a ).
damental strings. Therefore, the condensation of the5
Wilson loops argued in this paper is equivalent to thecondensation of fundamental strings and formation of an
actual black hole. By following the time evolution of
the loops, it should be possible to see how a black holeforms, thermalizes, and eventually evaporates. As wehave seen, qualitative features are common even in thestrong coupling limit of simple matrix models, and hence
they should serve as good toy models for a black hole.
Among various possible applications of this work is
the fast scrambling conjecture [15]. It has been argued
that a black hole is the fastest scrambler of information,in other words, it exhibits the fastest thermalization innature. Due to the gauge/gravity duality, the large-Ngauge theory is conjectured to scramble the information
as fast as a black hole because they are equivalent. The
intuitive picture discussed in this paper should be useful for understanding the microscopic mechanism of fast
scrambling from gauge theory. Clearly, a huge number ofsimultaneous interactions at various intersections shouldbe the essence of fast scrambling. As we have mentioned
above, it would be possible to show fast scrambling bystudying strong coupling lattice gauge theory or matrix
models. The understanding of fast scrambling in gauge
theory is interesting from the point of view of the quantum information theory, and even more, would be usefulin understanding the very fast thermalization of RHIC
fireball [16]. We hope to report progress in this direction
in near future.

ACKNOWLEDGEMENTS

The authors would like to thank D. Anninos,
E. Berkowitz, P. Hayden, S. Hellerman, A. Hook,
K. Schmitz, S. Shenker, H. Shimada, C. Siegel, D. Stanford, T. Takayanagi and N. Yamamoto for stimulating

discussions and comments. M. H. was supported in partby the Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Young Scientists (B), 25800163.
J. M. was supported by World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan.
Support for the research of L. S. came through NSF grantPhy-1316699 and the Stanford Institute for Theoretical
Physics.

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