Holographic duality for condensed matter physics From To , KITPC, Beijing, China Kyung Kiu Kim(GIST;Gwangju Institute of Science and Technology) Based on 1507.xxxxx, , , and 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

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 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: [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

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 i 0i i )

Practical form of the identity( The 2 nd WI) So far the derivation has nothing to do with holography. Conditions 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 superconductor Infinite 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)

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

The 2 nd 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 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!