Holography of Incoherent Metal

Sang-Jin Sin (Hanyang U.)

2014.11.26@Sogang

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Based on arXiv:1409.8346

collaboratorsKeun-Young Kim*, Kyung-Kyu Kim*, Yunseok Seo#

*GwangJu Institute of Science and technology, #Hanyang univ.

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References

Introduction

3

4

Physics Goal

understand the Cuprate phase diagram Quantitatively

A Challenging problem of 21C physics

Based on Royal Society publishing, D. Galanakis et.al

Non-Fermi Liquid

Linear Registivity

Fermi Liquid

T^2 registivity

Wiederman-Frant Law

Mott Insulator

d-wave condensation

Normal state first SC by the instability of it.

(Non)Fermi liquid of Landau

1. Free fermions and Fermi Sea

2. Interaction weak dressed particles (q-p) Fermi liquid (stable: irrelevance of most perturbation

& SC is an instability channel.3. Strong interaction fermi surface disappearNFL4. System property is not reduced to that of individual particles.

So Band picture lose meaning. Pauli principle ?5. System move semi collectively. Gap generation by int. 6. Strange dispersion Emergent un-particle at QCP.

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Linear Registivity in Cuprate

Fermi Liquid theory

e-e : T^2

E-phonon: T^5

So

T^1 is strange !

What will happen to Metal when ee

interaction become strong?

Meta-insulator transition

1p vs 2p Green function

Pseudo Gap

Strong coupling + impurity

increase impurity, what will happen?

Coherent vs. Incoherent Metal

Can Drude peak still exist?

In the absence of Quasi-particle

How about Wiederman-Frantz Law?

2particle Green function will answer

2p Green function DC/AC conductivity

1. Metal/insulator, Good/Bad metal2. Normal/Strange metal (T dependence of

conductivity),3. Coherent and incoherent metal (presence/absence

of Drude peak in AC cd4. Gap and pseudo gap

5. 1,410,000 image for optical conductivity in google image. standard probe of materials in experiment. Photo electric effect.

6. Comes from Two point function. Precisely defined.

10Remark: what about the ARPES?

Strong interaction in e-e int.?

1. For QCD: yes g^2 ~ 1 for low energy.

2. But in condensed matter, Isn’t it e^2 ~ 1/137<<1

3. How ee interaction can be strong …?

4. Lippman-Schiwinger eq.

11GoV=V/K: Slow electrons interact strongly.

0

5

10

15

20

Ba

nd

wid

th

(eV

)

Atomic Number

5d

4d

3d

5f4f

Electrons in the unfilled shell become progressively more localized in the sequence 5d 4d 3d 5f 4f

Such electrons are slower in this order.

itinerant and localized state – of 3d, 5f, 4f electrons.

Or spin and charge separation, .

Slow electrons are interacting strongly.

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Hard to calculate in strongly interaction regime in field theory.

mimic Holographic calculation of N=4 Gauge theory

However, out of Three pillars of proving duality

(Susy, large N, conformality) at least two should be broken

Analogy: 1st law of thermodynamics

or Schroedinger eq. without/with potential

Only Experiment can tell us the validity!

Meanwhile we practice and develop new intuitition about model

and phenomena.

Strongly interacting many body system.

Present status:

What has been achieved by such scheme so far? I can list very biased topics.

1. Transport near quantum critical point [15]

2. General formalism to construct finite temperature retarded green functions and

transport coefficients [4].

3. Holography of Non-Fermi Liquid [18,19,20]

4. Mean field theory of superconductivity with s-wave condensation without explici

t Higgs potential [5]. Similarly models with p-wave [6] as well as d-wave [7] co

ndensation were constructed.

5. models generating the registivity linear in temperature [8,9,10].

6. models showing metal-insulate transition generated by interaction. It is probed

by the behavior of AC conductivity [14].

7. Models with Fermion coupling that induces Mott gap [16].

8. Understanding easy thermalization in strong coupling [17].

References

[1] J. Maldacena, \The large N limit of superconformal _eld theories and supergravity," Adv.

Theor. Math. Phys. 2, 231 (1998) [Int. J. Theor. Phys. 38, 1113 (1999)] [arXiv:hep-th/9711200];

[2] S. S. Gubser, I. R. Klebanov, and A. M. Polyakov, \Gauge theory correlators from non-critical string theory," Phys. Lett. B 428, 105 (1998) [arXiv:hep-th/9802109];

[3] E. Witten, \Anti-de Sitter space and holography," Adv. Theor. Math. Phys. 2, 253 (1998)

[arXiv:hep-th/9802150].

[4] Dam T. Son, Andrei O. Starinets, \Viscosity, Black Holes, and Quantum Field Theory,

arXiv:0704.0240 [hep-th], Ann.Rev.Nucl.Part.Sci. 57 (2007) 95-118.

[5] Sean A. Hartnoll, Christopher P. Herzog, Gary T. Horowitz, Holographic Superconductors, arXiv:0810.1563 [hep-th], JHEP 0812 (2008) 015;

[6] S.S. Gubser and S.S. Pufu, ``The Gravity dual of a p-wave superconductor,''

JHEP {\bf 0811}, 033 (2008) [arXiv:0805.2960 [hep-th]].

[7] F.~Benini, C.~P.~Herzog and A.~Yarom, ``Holographic Fermi arcs and a d-wave gap,'' Phys.\ Lett.\ B {\bf 701}, 626 (2011) [arXiv:1006.0731 [hep-th]].

[8] S.~A.~Hartnoll, J.~Polchinski, E.~Silverstein and D.~Tong, ``Towards strange metallic holography,''

JHEP {\bf 1004}, 120 (2010) [arXiv:0912.1061 [hep-th]].

[9] T.~Faulkner, N.~Iqbal, H.~Liu, J.~McGreevy and D.~Vegh,

``Strange metal transport realized by gauge/gravity duality,'' Science {\bf 329}, 1043 (2010).

[10] Richard A. Davison, Koenraad Schalm, Jan Zaanen , Holographic duality and the resistivity of strange metals, arXiv:1311.2451 [hep-th].

[11] R.~A.~Janik and R.~B.~Peschanski, ``Asymptotic perfect fluid dynamics as a consequence of Ads/CFT,'' Phys.\ Rev.\ D {\bf 73}, 045013 (2006) [hep-th/0512162].

[12] S.~Nakamura and S.~J.~Sin,

``A Holographic dual of hydrodynamics,'' JHEP {\bf 0609}, 020 (2006) [hep-th/0607123].

[13] S.~Bhattacharyya, V.~E.~Hubeny, S.~Minwalla and M.~Rangamani,

``Nonlinear Fluid Dynamics from Gravity,' JHEP {\bf 0802} (2008) 045 [arXiv:0712.2456 [hep-th]].

[14] A.~Donos and S.~A.~Hartnoll,

``Interaction-driven localization in holography,'' Nature Phys.\ {\bf 9}, 649 (2013) [arXiv:1212.2998].

[15] S.~A.~Hartnoll, P.~K.~Kovtun, M.~Muller and S.~Sachdev,

``Theory of the Nernst effect near quantum phase transitions in condensed matter, and in dyonic black holes,'' Phys.\ Rev.\ B {\bf 76}, 144502 (2007) [arXiv:0706.3215 [cond-mat.str-el]].

[16] M.~Edalati, R.~G.~Leigh, K.~W.~Lo and P.~W.~Phillips,

``Dynamical Gap and Cuprate-like Physics from Holography,'' Phys.\ Rev.\ D {\bf 83}, 046012 (2011) [arXiv:1012.3751 [hep-th]].

[17] Eunseok Oh, Sang-Jin Sin, Non-spherical collapse in AdS and Early Thermalization in RHIC

Phys.Lett. B726 (2013) 456-460, arXiv:1302.1277 [hep-th]; Sang-Jin Sin, The physical mechanism of AdS instability and Holographic Thermalization, arXiv:1310.7179.

[18] S.~S.~Lee, ``A Non-Fermi Liquid from a Charged Black Hole: A Critical Fermi Ball,''

Phys.\ Rev.\ D {\bf 79}, 086006 (2009) [arXiv:0809.3402 [hep-th]]

[20] H.~Liu, J.~McGreevy and D.~Vegh, %``Non-Fermi liquids from holography,''

Phys.\ Rev.\ D {\bf 83}, 065029 (2011) [arXiv:0903.2477 [hep-th]];

Nabil Iqbal, Hong Liu, Mark Mezei, Lectures on holographic non-Fermi liquids and quantum phase transitions, arXiv:1110.3814 [hep-th].

[19] M.~Cubrovic, J.~Zaanen and K.~Schalm,

``String Theory, Quantum Phase Transitions and the Emergent Fermi-Liquid,''

Science {\bf 325}, 439 (2009) [arXiv:0904.1993 [hep-th]].

[21] Sean A. Hartnoll, Lectures on holographic methods for condensed matter physics, arXiv:0903.3246 [hep-th], Class.Quant.Grav. 26 (2009) 224002;

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Einstein-Maxwell system

Reissner-Nordstrom-AdS black hole

~ Boundary field theory at finite temperature and density

Electric conductivity

+

Hartnoll, 1106.4324

Warming UP: charged AdS BH and conductivity

Eq. of Motion + BC@H

How to include chemical pot. My early contri.

Kim,Zahed,sjs; Nakamura,Seo,Yogendran,sjs

…

Conductivity

Kramers-Kronig relation

Motivations: Holographic model

2007

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RN AdS BH

Conductivity

Kramers-Kronig relation

Motivations: Holographic model

Translation invariance + finite density

2007

We do not want perfect conductor without superconductivity

How to make a model without a delta function

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RN AdS BH

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‘Mimic Ionic’ Lattice

Horowitz, Santos, Tong: 1204.0512

Horowitz, Santos, Tong: 1209.1098

Momentum relaxation by explicit x dependence

Breaking Translation Symmetry in Holographic model

Low frequency Intermediate frequency

AC conductivity

Motivations: experiment

Horowitz, Santos, Tong: 1204.0512

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- Fluctuations:11 PDEs in two variables

- Background: 7 PDEs in two variables

‘Ionic’ Lattice

Horowitz, Santos, Tong: 1204.0512

Horowitz, Santos, Tong: 1209.1098

Massive gravity model: Vegh(1301), Davison(1306)

Other methods

Earlier Models for Momentum relaxation (continued)

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Momentum relaxation simplified (ODE): Find and Use a Translation inv. Breaking Exact solution

Andrade and Withers 1311.5157

Metal-Insulator transition in holography: Donos and Hartnoll(1212)

Q-lattice model: Donos and Gauntlett (1311)

….

More Related earlier work

But no AC

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Q2. Drude peak without quasi particle?

Q1. No contribution from pair creation?

1) Weak translation symmetry breaking(coherent metal)

Yes by Hartnoll and Hofman(1201.3917)

2) Strong translation symmetry breaking(incoherent metal)

Q4. finite region scaling?

Q5. Thermoelectric and thermal conductivity?

Main questions

Q3 Transition between with and without Drude peak?

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RN AdS black holes + scalar

Actions

EOMs

Bardoux, Caldarelli, Charmousis (2012), Andrade, Withers(2013)

RN-AdS solution + two scalars

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RN AdS black holes + scalar (continue)

Fluctuations

EOMs

Boundary action

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Fluctuations EOMs

Boundary action

Electric, Thermoelectric, Thermal conductivity:

calculational scheme (illustration with RN)

1. more than one equation

2. identify the sources and currents

Two issues for generalisation

Linear response

Hartnoll 0903.3246

Numerical methods

Fluctuations

Boundary action

Solutions near boundary

Boundary action

Kaminski, Landsteiner, Mas, Shock, Tarrio(2009)

Based on

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Constraint and Gauge invariance (only in bottom up).

Due to constraint eq.

To Reconstruct J by choosing ci properly.

This is doable only by the help of S_0

Checking code with known results

Hartnoll 0903.3234

Ge, Jo, and Sin, 1012.251529

Our results

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Main Result : AC electric conductivity

DC limit

Andrade and Withers

1311.5157

Drude peak

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Drude model

Ward identity

Fitting

Drude peak

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Relaxation time

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Coherent to incoherent transition

‘Clean’ region

‘Dirty’ region

Drude

Drude

Coherent metal

Incoherent metal

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Beyond:

Physics: Drue peak without quasi partilce

coherence is time scale:If beta is small enough, momentum

relaxation is slow and there is a drue peak.

1/mu ~ interparticle distance.

1/beta~ interimpurity distance.

AC conductivity

Scaling law: exp v.s theory

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In our model (theory)

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Intermediate frequency scaling

The best we’ve found so far

General feature

See also 1406.4870, Taylor and Woodhead

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Thermal and thermoelectric conductivity

DC results:

Donos and Gauntlett

1406.4742

Drude-like? Intermediate scaling?

Wiedermann-Franz Law

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Conclusion

• Thermoelectric conductivity has the same relaxation time

• Systematic numerical recipe established.

• No intermediate scaling in our case

•

• In RN black hole with translation symmetry broken by

• AC electric conductivity Coherent /Iincoherent transition

can be discussed by the impurity paparameter beta.-