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Holography of Incoherent Metal
Sang-Jin Sin (Hanyang U.)
2014.11.26@Sogang
1
Based on arXiv:1409.8346
collaboratorsKeun-Young Kim*, Kyung-Kyu Kim*, Yunseok Seo#
*GwangJu Institute of Science and technology, #Hanyang univ.
2
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.
5
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.
12
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;
16
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
17
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
18
RN AdS BH
19
‘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
20
- 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)
21
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
23
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?
24
RN AdS black holes + scalar
Actions
EOMs
Bardoux, Caldarelli, Charmousis (2012), Andrade, Withers(2013)
RN-AdS solution + two scalars
25
RN AdS black holes + scalar (continue)
Fluctuations
EOMs
Boundary action
26
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
27
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
30
Main Result : AC electric conductivity
DC limit
Andrade and Withers
1311.5157
Drude peak
31
Drude model
Ward identity
Fitting
Drude peak
32
Relaxation time
33
Coherent to incoherent transition
‘Clean’ region
‘Dirty’ region
Drude
Drude
Coherent metal
Incoherent metal
34
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
36
In our model (theory)
37
Intermediate frequency scaling
The best we’ve found so far
General feature
See also 1406.4870, Taylor and Woodhead
38
Thermal and thermoelectric conductivity
DC results:
Donos and Gauntlett
1406.4742
Drude-like? Intermediate scaling?
Wiedermann-Franz Law
40
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.-