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Ward identity and Thermo-electric conductivitiesHolographic duality for condensed matter physics
From 2015-07-06 To 2015-07-31, KITPC, Beijing, ChinaKyung Kiu Kim(GIST;Gwangju Institute of Science and Technology)Based on 1507.xxxxx, 1502.05386 , 1502.02100, 1501.00446 and 1409.8346WithKeun-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
Contents
Ward identity is a nonperturbative property of field theories. Gauge/gravity correspondence is another nonperturbative desciption of
field theories. Computation through gauge/gravity duality should respect the Ward iden-
tity. 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 conduc-
tivities. 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 behav-
iors of the strange metal.
Motivation
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 im-purity.
I will contain all results in the context of the Ward identity.
Summary of our Works
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
Derivation of the Ward identity
We assume that this system has diffeomorphism in-variance and gauge invariance related to the back-ground metric and the external gauge field.
The transformations
Derivation of the Ward identity
Variation of the generating functional
After integration by part, one can ob-tain a Ward identity( The first WI )
For gauge transformation
Derivation of the Ward identity
More Assumptions:Constant 1-pt functions and constant
external fields
Then, we can go to the momentum space.
Derivation of the Ward identity
For more specific cases Turning on spatial indices in the Green’s
functions i 0i i
)
Derivation of the Ward identity
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
Derivation of the Ward identity
Let us consider the ward identity without the magnetic field
B=0 and i = x
Derivation of the Ward identity
The Ward identity for the two point functions
Plugging thermo-electric conductivi-ties into the WI,,,
Derivation of the Ward identity
Let’s consider WI in the magnetic field Previous form
of the W I
The ward identity in the magnetic field B
With
Derivation of the Ward identity
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
Test with a Holographic Model
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 su-perconductors.
Why momentum relaxation
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.
Why momentum relaxation
The holographic superconductor by HHH
Ideal conductor superconductorInfinite conductivity Infinite conductivity
By momentum relaxation(or Breaking translation invariance)Keun-Young’s talks
Metal superconductor Finite conductivity Infinite conductivity
Why momentum relaxation
Explicit lattice : We have to solve PDE
Santos, Horowitz and Tong(2012)
Toward holographic models of the real matters
Massive Gravity-Breaking diffeomorphism invariance by mass terms of graviton-Final state of gravitational Higgs mechanism-Vegh(2013)
Toward holographic models of the real matters
Q-Lattice model
-It is possible to avoid PDE.-Finite DC conductivity.
Donos and Gauntlett(2013)
Toward holographic models of the real matters
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.
Toward holographic models of the real matters
Normal state(Charged black hole with momentum relaxation)
+ fluctuation
Axion model without magnetic field
Superconducting state(Keun-young's talk)-Holographic superconductor with momentum relaxation
+ fluctuation
Axion model without magnetic field
This bulk geometry is dual to a field theory sys-tem, 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, exter-
nal gauge field and scalars(axions) as sources. We can apply the Ward identity to this system.
Axion model without magnetic field
Let us consider the first Ward identity
With
WI in terms of coefficients of asymp-totic expansion
Axion model without magnetic field
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.
Axion model without magnetic field
With momentum relaxation
Drude model
One can expect Drude model like be-havior in the conductivities
Axion model without magnetic field
From this result, it seems that our approach is following a healthy direc-tion.
The ward identity can be a powerful tool of the holographic approach.
Axion model without magnetic field
When there is a real scalar instead of a com-plex 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 conduc-tivity in the broken phases.
Broken U(1) symmetry vs Broken Z_2 symme-try.
Real scalar condensation(q=0 case)
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 transforma-tion.
Real scalar condensation(q=0 case)
With momentum relaxationComplex scalar vs Real scalarU(1) vs Z_2(Metal-Superconductor) vs (Metal-
Metal)Therefore, the real scalar model is
not a holographic superconductor.
Real scalar condensation(q=0 case)
Let’s come back to the complex scalar condensa-tion.
The Ac conductivity satisfies FGT sum rule (Keun-Young’s talk) K_s
Superfluid density
By small frequency behavior of the Ward iden-tity
We can identify the superfluid density with other correlation function.
If we define
The normal fluid density
Superfluid density
Physical meaning from the bulk the-ory
From the Maxwell equation and the boundary current
Superfluid density
In the Fluctuation level
The hairy configuration of a complex scalar field generates super fluid density.
Superfluid density
There are universal behaviors in su-perconductors(Meyer, Erdmenger, KY Kim)
Homes’s law(Broad class of material)
Uemura’s law(Underdoped case)
Homes’s law and Uemura’s law
We derived the WI in a 2+1 system. Using the Axion model, we showed that the 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 den-
sity 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.
Summary
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