Download Ward identity and Thermo-electric conductivities

Holographic duality for condensed matter physics
From 2015-07-06 To 2015-07-31, KITPC, Beijing, China
Kyung Kiu Kim(GIST;Gwangju Institute of Science and Technology)
Based on 1507.xxxxx, 1502.05386 , 1502.02100, 1501.00446 and
1409.8346
With
Keun-young Kim(GIST) , Yunseok Seo(Hanyang Univ.), Miok
Park(KIAS) and Sang-Jin Sin(Han yang Univ.)
Motivation
Summary of our other works
Derivation of the Ward identity
Test with a Holographic Model
-Why momentum relaxation?
-Toward holographic models of the real matters
 Axion model without Magnetic field
-Complex scalar condensation
-Real scalar condensation(q=0 case)
-Superfluid density
-Home's law and Uemura's law
 Test with Axion model in a magnetic field
 Summary
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 Ward identity is a nonperturbative property of field theories.
 Gauge/gravity correspondence is another nonperturbative desciption of
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field theories.
Computation through gauge/gravity duality should respect the Ward
identity.
Recently computation of conductivities has been developed by regarding
momentum relaxation.
Now, we can produce more realistic conductivities through appropriate
holographic models.
So we test Ward identity by computing the conductivities.
The ward identity helps for us to understand the pole structure of
conductivities.
Such a pole structure is related to the superfluild density.
We may study the Universal laws like Homes' law and Uemura's law
through the Ward identity.
Also, in the magnetic field, the Ward identity is important to study
behaviors of the strange metal.
 Sang-Jin Sin’s talk
-An important issue of gauge degrees of freedom between
the bulk theory and the boundary theory and advent of the
Ward identity when we compute the AC conductivity.
 Keun-young Kim’s talk
- Phase transition between metal and superconducting state
through the holographic superconductor model with
momentum relaxation
 Yunseok Seo’s talk
-AC and DC conductivities in a magnetic field and magnetic
impurity.
 I will contain all results in the context of the Ward identity.
 Our derivation is based on Herzog(09),
[arXiv:0904.1975 [hep-th]]
 We modify the derivation with scalar sources
 Let us start with a class of field theory system with
non-dynamical sources, metric, U(1) gauge field, a
complex scalar field and a real scalar field with an
internal index I.
 Generating functional of Green’s functions with the
sources
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Corresponding operator expectation values
Two point functions
We assume that this system has diffeomorphism invariance
and gauge invariance related to the background metric
and the external gauge field.
The transformations
Variation of the generating functional
After integration by part, one can obtain a
Ward identity( The first WI )
For gauge transformation
 Taking one more functional derivative
 More Assumptions:
 Constant 1-pt functions and constant
external fields
 Then, we can go to the momentum space.
 Euclidean Ward identities in the
momentum space
 After the wick rotation
 Ward identity with the Minkowski signature
 For more specific cases
 Turning on spatial indices in the Green’s
functions
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i
0i
i
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 Practical form of the identity( The 2nd WI)
 So far the derivation has nothing to do with holography.
 Conditions
- 2+1 d, diffeomorphism invariance and gauge invariance
- Special choice of the sources
- non-vanishing correlation among the spatial vector currents
 Let us consider the ward identity without
the magnetic field
 B=0 and i = x
 The Ward identity for the two point
functions
 Plugging thermo-electric conductivities
into the WI,,,
 The Ward identity for the conductivities
 We need subtraction
 Let’s consider WI in the magnetic field
 Previous form
of the W I
 The ward identity in the magnetic field B
 With
 Let us find consistent holographic models
with the condition of the Ward Identity.
 2+1 d, diffeomorphism invariance and gauge
invariance
- special choice of the sources
- Non-vanishing correlations among only the
spatial vector currents
 We will restrict our case to a model
with momentum relaxation.
 Why momentum relaxation?
 For the realistic conductivity
 In AdS/CMT the charged black hole
is regarded as a normal state of
superconductors.
 The electric conductivity of the charged black
black brane.
 Infinite DC conductivity: This shows the ideal
conductor behavior.
 The matter corresponding to the charged
black brane is structureless.
 The holographic superconductor by HHH
Ideal conductor
Infinite conductivity
superconductor
Infinite conductivity
 By momentum relaxation(or Breaking translation
invariance)
Keun-Young’s talks
Metal
Finite conductivity
superconductor
Infinite conductivity
 Explicit lattice : We have to solve PDE
Santos, Horowitz and Tong(2012)
 Massive Gravity
-Breaking diffeomorphism invariance by mass terms of
graviton
-Final state of gravitational Higgs mechanism
-Vegh(2013)
 Q-Lattice model
-It is possible to avoid PDE.
-Finite DC conductivity.
 Donos and Gauntlett(2013)
 Axion-model(Andrade,Withers 2014)
-The simplest model in the momentum relaxation models.
-A special case of the Q Lattice model.
-This model shares a same solution with the massive gravity
model.
We will discuss the Ward Identities with this model.
 Normal state(Charged black hole with
momentum relaxation)
 + fluctuation
 Superconducting state(Keun-young's talk)
-Holographic superconductor with
momentum relaxation
 + fluctuation
 This bulk geometry is dual to a field theory system,
which satisfies some conditions.
 In the bulk the axion is a massless field.
 So it corresponds to a dimension 3 operator .
 Thus the field theory system has metric, external
gauge field and scalars(axions) as sources.
 We can apply the Ward identity to this system.
 Let us consider the first Ward identity
 With
 WI in terms of coefficients of asymptotic
expansion
 The first Ward identity
 Without the momentum relaxation and
with DC applied electric field.
 Momentum current is linear in t.
 This is the origin of the delta function in
DC conductivity.
 With momentum relaxation
 Drude model
 One can expect Drude model like behavior
in the conductivities
 Electric conductivities (Normal state)
 Electric conductivity
 Thermo-electric coefficient
 Thermal conductivity
 The other conductivities
 The 1st Ward identity and numerical
confirmation
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 The 2nd Ward
identity
 The numerical
confirmation
 From this result, it seems that our
approach is following a healthy direction.
 The ward identity can be a powerful tool
of the holographic approach.
 When there is a real scalar instead of a
complex scalar, we can consider another kind
of broken phase.
 Without momentum relaxation, however, it is
not clear whether we can distinguish the real
scalar condensation from complex scalar
condensation.
 Because both cases give infinite DC
conductivity in the broken phases.
 Broken U(1) symmetry vs Broken Z_2
symmetry.
 No delta function and no pole
 Furthermore, since there is no pole in the
conductivities, we may use the method in
Yunseok’s talk.
 One can calculate DC conductivities by a
simple coordinate transformation.
 DC formula for the q=0 case
 With momentum relaxation
 Complex scalar vs Real scalar
 U(1) vs Z_2
 (Metal-Superconductor) vs (Metal-Metal)
 Therefore, the real scalar model is not a
holographic superconductor.
 Let’s come back to the complex scalar
condensation.
 The Ac conductivity satisfies FGT sum rule
(Keun-Young’s talk)
K_s
 Thus it is natural to consider pole
structure of the Ward identity.
 By small frequency behavior of the Ward identity
 We can identify the superfluid density with other
correlation function.
If we define
The normal fluid density
 Physical meaning from the bulk theory
 From the Maxwell equation and the
boundary current
 In the Fluctuation level
 The hairy configuration of a complex
scalar field generates super fluid density.
 There are universal behaviors in
superconductors
(Meyer, Erdmenger, KY Kim)
 Homes’s law(Broad class of material)
 Uemura’s law(Underdoped case)
 Uemura’s law
 Homes’s law
 This model is good for the underdoped
regime.
 In progress
 Numerical confirmation < 10^-16
 We derived the WI in a 2+1 system.
 Using the Axion model, we showed that the
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holographic conductivities satisfy the WI very well.
We found that the real scalar condensation model is
not a superconductor model.
The superfluid density can be represented by another
correlation function.
We found that physical meaning of the superfluid
density from the bulk hairy configuration.
The Axion model is a good model for the Uemura’s law,
but it is not good for the Homes’s law.
We showed that the WI is satisfied even in the
magnetic field.
Thank you for your attention!