Introduction to
Holographic Superconductors
Bin Wang
Fudan University
为什么在LHC要出成果的前夜，高能物理和引力物理学家要来思考能标比他们通常研
究的量级低很多的物理现象？
第一，AdS/CFT对应性是对强耦合场理论进行研究的特殊方法，运用这个方法强
耦合场的一些问题可以被计算处理，概念也变得更清晰。凝聚态物理中有很多强
场系统，这些系统对基于弱相互作用和对称破缺理论的传统凝聚态理论是个挑战。
第二，凝聚态系统也许能提供一个舞台从实验上来实现很多有趣的高能物理的理
论想法。标准模型的拉氏量和它假定的完备性在我们的宇宙中是特别的。在凝聚
态物理中有很多有效哈密顿，更多的哈密顿能被诱导出来。也许最终能产生和已
知的AdS对偶的新兴场论，在实验上实现AdS/CFT对应性。
第三，AdS/CFT对应性改变了对自然的看法，传统物理中按照场的能级分类变得
不是很重要了。如果量子引力理论能对偶于很多和量子极端电子相同的特性，那么
哪个更基本的问题就没有意义了。 取而代之的是强调两者之间的对偶更有意义。
这个观点有实际的效果，比如寻找超导表述的对偶，人们意识到黑洞的无毛定理也
许有漏洞，人们也许能找到新的黑洞解。
There is a long long long way to go…
PART I:
1) Introduction to superconductivity
2) Simple model for a holographic superconductor
3) Probe limit in SAdS, RNAdS backgrounds
(condensate and conductivity)
 PART II: signature of the phase transition
 PART III: Probe limit GBAdS background
 PART IV: magnetic field discussion
 PART V: more on phase transition

Review articles:
1) S. Hartnoll, 0903.3246
2) C. Herzog, 0904.1975
3) G. Horowitz, 1002.1722


a.
b.
c.
d.
e.
Our works:
Q. Pan, B. Wang, et al, arXiv:0912.2475 PRD(10)
X He, B Wang, R-G Cai, C-Y Lin, arXiv:1002.2679 PLB(10)
X-H Ge, Bin Wang, et al, arXiv:1002.4901 JHEP(10)
R-G Cai, Z-Y Nie, B Wang, H-Q Zhang, arXiv:1005.1233
Q. Pan, B. Wang, 1005.4743

In conventional superconductors, when T<Tc

Landau-Ginzburg: superconductivity is a second order phase transition
with a complex scalar field ϕ as order parameter,
 Electrical resistivity->0
 Meissner effect, magnetic field is expelled
Free energy:
T>Tc, F(min) is at ϕ=0, T<Tc, F(min) at ϕ not 0

BCS theory: pairs of elections with opposite spin can bind to
form a charged boson called a Cooper pair.
Below a critical temperature Tc, there is a second order phase
transition and these bosons condense.
microscopic theory of superconductivity
It was once thought that the highest Tc for a BCS superconductor was
around 30K. But in 2001, MgB2 was found to be superconducting at
40K and is believed to be described by BCS. Some people now speculate
that BCS could describe a superconductor with Tc = 200K.
The new high Tc superconductors were discovered in 1986. These cuprates
(e.g. YBaCuO) are layered and superconductivity is along CuO2 planes.
Highest Tc today (HgBaCuO) is Tc = 134K.
Another class of superconductors discovered March 08 based on iron and
not copper FeAs(…) Highest Tc = 56K.
The pairing mechanism is not well understood. Unlike BCS theory, it
involves strong coupling.
AdS/CFT is an ideal tool to study strongly coupled field theories.
Sachdev, 0907.0008
How do we go about constructing a holographic dual
for a superconductor?
Superconductor ------ temperature
Gravity {black hole} ---- temperature (Hawking)
Gauge/Gravity duality, BH T ~ dual field theory T

Why do we need AdS space?
Gauge/Gravity dual in AdS space
AdS BH is thermodynamically stable C>0
Confining box in AdS BH

Why not “dilatonic” black holes with scalar hair?
a result of a coupling
Historic steps to have scalar hair condensation:
Hertog (2006) showed that for a real scalar field with arbitrary potential
V(φ), neutral AdS black holes have scalar hair if AdS is unstable.
Gubser (2008) argued that a charged scalar field around a charged black
hole would have the desired property. Consider
This does not work for
asymptotically flat spacetimes
Will cause the scalar
hair to form at low
temperature
Current remains finite at the horizon
Local gauge
symmetry in the bulk
In the bulk


Similar to BCS theory observed in many materials,
condensate rises quickly as the system is cooled
below the critical temperature and goes to a
constant as T->0.
Near Tc,
(Landau-Ginsberg)
Scalar hair is developed
The qualitative behavior is the same as before.
There is a critical Tc above which the condensate is
zero. Near Tc,
.
 In all but one case, T->0,
condensate grows.

If the condensate is big,
the probe limit breaks.
The limit of the electric filed
in the bulk is the electric field
on the boundary
Expectation value of
the induced current
Real part of the conductivity
T>Tc, the conductivity is constant.
Damping
term
Charge carries mass m, charge e, number density
n in normal conductor satisfying :
Relaxation
time
Current J=env
Number density
in normal
conductor
Always true??


Second order transition between a non-superconducting state
at high T and a superconducting state at low T.
Whether the QNM can be an effective probe of this phase
transition?
X. He, B.Wang, RG Cai, CY Lin, PLB(10), 1002.2679.
RG Cai, ZY Nie, B Wang, HQ Zhang, 1005.1233

It has been argued that the QNM can reflect the black hole
phase transition
G. Koutsoumbas, S. Musiri, E. Papantonopoulos, and G. Siopsis, JHEP 0610, 006
(2006).
J. Shen, B. Wang, R. K. Su, C.Y. Lin, and R. G. Cai, JHEP 0707, 037 (2007).
X. Rao, B. Wang, and G. H. Yang, Phys. Lett. B 649, 472 (2007).
the Einstein-Maxwell field interacting with a charged scalar field with a
minimal Lagrangian
The electrically charged black hole in d-dimensional AdS space is
described by the metric
The Hawking temperature:
The gauge field ansatz is
Considering the scalar field ψ perturbing the RN-AdS BH:
The radial part of the equation can be separated by setting
the radial wave equation can be expressed as
the effective potential
Negative
Tortoise coordinate
will cause V negative
Choosing scaling symmetry
,, change the value of L
corresponds to vary the temperature of the black hole, L big, T small.
Fix q,
change L
Fix L,
change q
1. Increase L, lower T, the spacetime is easier to be destroyed (for fixed
coupling to EM field)
2. Increase q, the spacetime is easier to be destroyed (for fixed BH
temperature)
3. the topology of the spacetime has the influence on evolution of the
perturbation
The spherical background is the easiest to be destroyed due to the
perturbation
the spacial topology influence on the condensation??
4. the same black hole parameters (r+,Q,L),
the stability of the black hole spacetime can be broken easier in
the high dimensions.
the scalar hair can be formed easier in the higher
dimensional background!
Stability of charged fermions perturbation in a ReissnerNordstrom-anti-de Sitter black hole spacetime has been
studied in
RG Cai, ZY Nie, B Wang, HQ Zhang, 1005.1233

Motivation to study the GB AdS BH:
1. Examine the Mermin-Wagner (MW) theorem in
holographic superconductors.
The MW theorem forbids continuous symmetry breaking in (2+1)-d
because of large fluctuations in lower dimensions. It is possible that
fluctuations in holographic superconductors in 2+1d are suppressed
because classical gravity corresponds to the large N limit. If this is true,
then higher curvature corrections should suppress condensation.
2. Examine the so called universal relation
3. Examine the dimensional influence on condensation
Gregory et al, JHEP(09)
QY Pan, B Wang et al, PRD(10) 0912.2475
The background solution of a neutral GB AdS black hole is
in the asymptotic region
We can define the effective asymptotic AdS scale
the temperature of the CFT:
In the background of the d-dimensional GB AdS black hole, we consider a
Maxwell field and a charged complex scalar field with the action
In probe limit:
Taking the ansatz
only function of r
the equations of motion for
Boundary conditions:
at the horizon
at the asymptotic region
AdS/CFT correspondence,
We can impose boundary conditions that either
vanishes
The condensations of these operators are subjected to curvature corrections
The four lines from bottom to top correspond to increasing mass,
The scalar mass influence on the condensation:
a. for the same
the condensation gap becomes larger if m is less negative
b. The difference caused by the influence of the scalar mass will become
smaller when there is higher curvature correction in the AdS background.
The influence the dimensionality on the scalar condensation.
For the same scalar mass, we see:
as the spacetime dimension increases, the condensation gap
becomes smaller for the same
the scalar hair can be formed easier in the higher-dimensional background.
the difference caused by the curvature corrections are reduced when the
spacetime dimension becomes higher.
The condensation for the scalar operator
for different choices of the mass of the scalar field has completely
different behavior as
is changing.
Selecting the value of
we get the same qualitative dependence of the condensation for
the high curvature corrections really change the expected universal
relation in the gap frequency.
Pan, Wang et al PRD(10)
The holographic superconductor in the presence
of the external magnetic field (Probe limit)
In Superconductor: Meissner effect, magnetic field is expelled
From the Ginzburg-Landau theory, the upper critical magnetic
field
How about in holographic superconductor??
We begin with the 4-dimensional Schwarzschild AdS black hole
the metric can be rewritten
introduce a charged, complex scalar field into the 4-dimensional
Einstein-Maxwell action with negative cosmological constant
In the probe approximation, the Maxwell and scalar field equations obey
To solve these equations analytically we will follow the logic used by Abrikosov
1. consider the weak magnetic field limit, and , obtain the spatially
independent condensate solutions. To the sub-leading order, we can
treat the magnetic field as a small perturbation.
2. consider that the magnetic field is strong enough. We will regard the
scalar field as a perturbation and examine its behavior in the presence
of strong magnetism.
Weak magnetic field limit:
In the weak magnetic field limit,
the equations of motion reduce
Condensations are similar to those found for electric field.
Strong magnetic field limit
We will regard the scalar field as a perturbation and examine its behavior
in the neighborhood of the upper critical magnetic field
the scalar field
AdS/CFT
The vacuum expectation values
at the asymptotic AdS boundary
bulk coordinate
boundary coordinates
To the leading order, it is consistent to set the ansatz
The equation of motion for then becomes
separating the variables, we have
at the horizon z = 1
near the AdS boundary z -> 0
Requiring the solutions to be connected smoothly, we got
the condensation of the scalar field at
critical Tc, and Tc~Bc
Ge,Wang et al (10)
This study has been extended to the GB AdS BH
Weak magnetic field limit:
Strong magnetic field limit
Ge, Wang et al, 1002.4901
First & second order phase transition
 Critical exponent of the second order
transition
 Fluctuations of the system
Franco et al. 0911.1354
Pan & Wang, 1005.4743

Thanks!!!