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

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

Corresponding operator expectation values

Two point functions


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


Variation of the generating functional

After integration by part, one can ob-tain 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 mo-mentum space


After the wick rotationWard identity with the Minkowski sig-

nature


For more specific cases Turning on spatial indices in the Green’s

functions i 0i i

)


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 conductivi-ties into the WI,,,


The Ward identity for the conductivi-ties

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

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.


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


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)


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

Axion model without magnetic field

Superconducting state(Keun-young's talk)-Holographic superconductor with momentum relaxation

+ fluctuation


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.


Let us consider the first Ward identity

With

WI in terms of coefficients of asymp-totic 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 be-havior in the conductivities


Electric conductivities (Normal state)


Electric conductivity


Thermo-electric coefficient


Thermal conductivity


The other conductivities


The 1st Ward identity and numerical confir-mation


The 2nd Ward identity

The numeri-cal confir-mation


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.


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)

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 transforma-tion.


DC formula for the 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.


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

Thus it is natural to consider pole structure of the Ward identity.

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

Uemura’s law

Homes’s law and Uemura’s law

Homes’s law

Homes’ law and Uemura’s law

This model is good for the under-doped regime.

Homes’ law and Uemura’s law

In progressNumerical confirmation < 10^-16

Test with Axion model in the magnetic field

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

Thank you for your attention!

Documents

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