String theory and the String theory and the mysterious quantum - - PowerPoint PPT Presentation

String theory and the String theory and the mysterious quantum matter of mysterious quantum matter of condensed matter physics. condensed matter physics.

Jan Zaanen

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String theory: what is it really good for?

• Hadron (nuclear) physics: quark-gluon plasma in RIHC.
• Quantum matter: quantum criticality in heavy fermion

systems, high Tc superconductors, … Started in 2001, got on steam in 2007.

Son Hartnoll Herzog Kovtun McGreevy Liu Schalm

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Quantum critical matter

Quantum critical

Heavy fermions High Tc superconductors Iron superconductors (?) Quark gluon plasma

Quantum critical

High-Tc Has Changed Landscape of Condensed Matter Physics High-resolution ARPES Spin-polarized Neutron Magneto-optics

STM

Transport-Nernst effect High Tc Superconductivity

Angle-resolved MR/Heat Capacity

Inelastic X-Ray Scattering

?

Photoemission spectrum

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Hairy Black holes …

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Holography and quantum matter

Reissner Nordstrom black hole: “critical Fermi-liquids”, like high Tc’s normal state (Hong Liu, John McGreevy). Dirac hair/electron star: Fermi-liquids emerging from a non Fermi liquid (critical) ultraviolet, like overdoped high Tc (Schalm, Cubrovic, Hartnoll). Scalar hair: holographic superconductivity, a new mechanism for superconductivity at a high temperature (Hartnoll, Herzog,Horowitz) . “Planckian dissipation”: quantum critical matter at high temperature, perfect fluids and the linear resistivity (Son, Policastro, …, Sachdev).

But first: crash course in holography

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General relativity “=“ quantum field theory

Gravity Quantum fields Maldacena 1997

=

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Anti de Sitter-conformal quantum field theory correspondence

AdS geometry

(“near” the boundary)

Conformal quantum field theory (at ‘high’ energies) Another word for:

Quantum criticality!

Not like our universe …

Holography with lasers

Three dimensional image Encoded on a two dimensional photographic plate

Gravity - quantum field holography

Einstein world “AdS” = Anti de Sitter universe Quantum fields in flat space “CFT”= quantum critical

Hawking radiation

1 1

1

1 1 1 1 1 1 1 1 1 1 1 1

0 0 1

1 1 1

1

1

1 1 1 1 1 1 1

1

1

1

1 1 1 1

Hawking Temperature:

g = acceleration at horizon

A = area of horizon

‘t Hooft’s holographic principle

BH entropy:

Number of degrees of freedom (field theory) scales with the area and not with the volume (gravity)

The bulk: Anti-de Sitter space

Extra radial dimension

• f the bulk <=> scaling

“dimension” in the field theory Bulk AdS geometry = scale invariance of the field theory UV IR

Fractal Cauliflower (romanesco)

Quantum critical cauliflower

Quantum critical cauliflower

Quantum critical cauliflower

Quantum critical cauliflower

Fermion sign problem

Imaginary time path-integral formulation Boltzmannons or Bosons:

• integrand non-negative
• probability of equivalent classical

system: (crosslinked) ringpolymers Fermions:

• negative Boltzmann weights
• non probablistic: NP-hard

problem (Troyer, Wiese)!!!

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Renormalization group for quantum critical matter

Wilson-Fisher RG: based on Boltzmannian statistical physics boundary: d-dim space-time Hawking radiation gluons Black holes strings quarks

The Magic of AdS/CFT!

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Black hole hair codes the quantum matter

“Hairy black holes” code for (un)stable states of quantum matter emerging from the quantum critical stuff.

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Quantum critical dynamics: classical waves in AdS

WCFT J

( ) = SAdS φ ( )φx0 →0= J

gYM

2 N = R4

α gYM

2

= gs

E-field transverse E-field <=> 3d electric field radial E-field <=> 3d charge density B-field radial B-field <=> 3d magnetic field transverse B-field <=> 3d current density spatial metric perturb. transverse gradient <=> 3d distortion radial gradient <=> 3d stress tensor temporal metric perturb. transverse gradient <=> temperature gradient radial gradient <=> heat flow

SUSY Einstein-Maxwell in AdS <==> SUSY Yang-Mills CFT

The AdS/CFT dictionary

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Holography and quantum matter

Reissner Nordstrom black hole: “critical Fermi-liquids”, like high Tc’s normal state (Hong Liu, John McGreevy). Dirac hair/electron star: Fermi-liquids emerging from a non Fermi liquid (critical) ultraviolet, like overdoped high Tc (Schalm, Cubrovic, Hartnoll). Scalar hair: holographic superconductivity, a new mechanism for superconductivity at a high temperature (Hartnoll, Herzog,Horowitz) . “Planckian dissipation”: quantum critical matter at high temperature, perfect fluids and the linear resistivity (Son, Policastro, …, Sachdev).

But first: crash course in holography

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The Schwarzschild Black Hole is the heater

GR in Anti de Sitter space Quantum-critical fields on the boundary:

Black hole temperature entropy

• at the Hawking temperature
• entropy = black hole entropy

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Dissipation = absorption of classical waves by Black hole!

Viscosity: absorption cross section of gravitons by black hole Entropy density s: Bekenstein-Hawking BH entropy = area of horizon

η = σ abs 0

( )

16πG

= area of horizon (GR theorems)

Universal viscosity-entropy ratio for CFT’s with gravitational dual limited in large N by:

η s = 1 4π h kB

Policastro-Son-Starinets (2002):

4πkBη hs

AdS/CFT viscosity

Kovtun-Son-Starinets (2005)

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The quark-gluon plasma

Relativistic Heavy Ion Collider Quark-gluon ‘fireball’

The tiny viscosity of the Quark- Gluon plasma

QG plasma: within 20% of the AdS/CFT viscosity!

4πkBη hs

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Quantum critical hydrodynamics: Planckian dissipation & viscosity

Planckian dissipation:

h kBT

Viscosity, entropy density:

Planckian viscosity: η = ε + p

( )τ, s = ε + p

T ⇒ η s = Tτ

τ = τ h ≈ h kBT

η s ≈ h kB

In a finite temperature quantum critical state the time it takes to convert work in heat (relaxation time) has to be

1 ω

Sachdev, 1992

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Twenty five years ago …

Mueller Bednorz

Ceramic CuO’s, likeYBa2Cu3O7

Superconductivity jumps to ‘high’ temperatures

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Graveyard of Theories

Schrieffer Anderson Mueller Bednorz Laughlin Abrikosov Leggett Wilczek Mott Ginzburg De Gennes Yang Lee

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Phase diagram high Tc superconductors

JZ, Science 315, 1372 (2007)

Mystery quantum critical metal

‘Stripy stuff’, spontaneous currents, phase fluctuations ..

ΨBCS = Πk uk + vkck↑

+ c−k↓ +

( ) vac.

The return of normalcy

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Quantum Phase transitions

Quantum scale invariance emerges naturally at a zero temperature continuous phase transition driven by quantum fluctuations:

JZ, Science 319, 1205 (2008)

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A universal phase diagram

Quantum critical

Heavy fermions High Tc superconductors Iron superconductors (?)

Quantum critical

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Divine resistivity

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Critical Cuprates are Planckian Dissipators

A= 0.7: the normal state of optimallly doped cuprates is a

Planckian dissipator!

σ1(ω,T) = 1 4π ω pr

2 τ r

1+ ω 2τ r

2 ,

τ r = A h kBT

van der Marel, JZ, … Nature 2003: Optical conductivity QC cuprates Frequency less than temperature:

⇒ [ h kBTσ1 ] = const.(1+ A2[ hω kBT ]2)

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Divine resistivity = Planckian Dissipation!

ρ ∝ 1 τ h ∝ kBT

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Holography and quantum matter

Reissner Nordstrom black hole: “critical Fermi-liquids”, like high Tc’s normal state (Hong Liu, John McGreevy). Dirac hair/electron star: Fermi-liquids emerging from a non Fermi liquid (critical) ultraviolet, like heavy fermions (Schalm, Cubrovic, Hartnoll). Scalar hair: holographic superconductivity, a new mechanism for superconductivity at a high temperature (Hartnoll, Herzog,Horowitz) . “Planckian dissipation”: quantum critical matter at high temperature, perfect fluids and the linear resistivity (Son, Policastro, …, Sachdev).

But first: crash course in holography

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Holographic quantum critical fermion state

Liu McGreevy

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The quantum in the kitchen: Landau’s miracle

Kinetic energy k=1/wavelength

Electrons are waves Pauli exclusion principle: every state occupied by one electron

Fermi momenta Fermi energy Fermi surface of copper

Unreasonable: electrons strongly interact !! Landau’s Fermi-liquid: the highly collective low energy quantum excitations are like electrons that do not interact.

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Watching electrons: photoemission

Kinetic energy k=1/wavelength Fermi momenta Fermi energy Fermi surface of copper Electron spectral function: probability to create or annihilate an electron at a given momentum and energy. k=1/wavelength Fermi energy energy

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ARPES: Observing Fermi liquids

‘MDC’ at EF in conventional 2D metal (NbSe2) Fermi-liquids: sharp Quasiparticle ‘poles’

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Cuprates: “Marginal” or “Critical” Fermi liquids

Fermi ‘arcs’ (underdoped) closing to Fermi-surfaces (optimally-, overdoped). EDC lineshape: ‘branch cut’ (conformal), width propotional to energy

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Breaking fermionic criticality with a chemical potential

‘Dirac waves’

Electrical monopole k E

µ µ

Fermi-surface??

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AdS/ARPES for the Reissner- Nordstrom non-Fermi liquids

Critical FL Marginal FL Non Landau FL

Fermi surfaces but no quasiparticles!

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Horizon geometry of the extremal black hole: ‘emergent’ AdS2 => IR of boundary theory controlled by emergent temporal criticality Gravitational ‘mechanism’ for marginal (critical) Fermi-liquids:

G−1 = ω − vF k − kF

( ) − Σ k,ω ( )

Σ"∝ω

2ν kF

Fermi-surface “discovered” by matching UV-IR: like Mandelstam “fermion insertion” for Luttinger liquid! Temporal scale invariance IR “lands” in probing fermion self energy!

Gravitationally coding the fermion propagators (Faulkner et al. Science 329, 1043, 2010)

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Holography and quantum matter

Reissner Nordstrom black hole: “critical Fermi-liquids”, like high Tc’s normal state (Hong Liu, John McGreevy). Dirac hair/electron star: Fermi-liquids emerging from a non Fermi liquid (critical) ultraviolet, like heavy fermions (Schalm, Cubrovic, Hartnoll). Scalar hair: holographic superconductivity, a new mechanism for superconductivity at a high temperature (Hartnoll, Herzog,Horowitz) . “Planckian dissipation”: quantum critical matter at high temperature, perfect fluids and the linear resistivity (Son, Policastro, …, Sachdev).

But first: crash course in holography

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Phase diagram high Tc superconductors

JZ, Science 315, 1372 (2007)

Mystery quantum critical metal

‘Stripy stuff’, spontaneous currents, phase fluctuations ..

ΨBCS = Πk uk + vkck↑

+ c−k↓ +

( ) vac.

The return

• f normalcy

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“AdS-to-ARPES”: Fermi-liquid (?) emerging from a quantum critical state.

Schalm Cubrovic

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The zero temperature extensive entropy ‘disaster’

AdS-CFT The ‘extremal’ charged black hole with AdS2 horizon geometry has zero Hawking temperature but a finite horizon area. The ‘seriously entangled’ quantum critical matter at zero temperature should have an extensive ground state entropy (?*##!!)

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Black hole hair can be fermionic!

Schalm, Cubrovic, JZ (arXiv:1012.5681) ‘Hydrogen atom’: quantum mechanical probability density ‘atmosphere’ of one fermion/surface area of black brane. AdS-CFT Stable Fermi liquid on the boundary!

The Fermi-liquid VEV: Hair profile vs. statistics

Fermionic hair: the probability distribution along the radial direction of the AdS “hydrogen atom” wave function.

Position of the maximum determines the Fermi energy

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Fermionic hair: stability and equation of state.

Strongly renormalized EF Single Fermion spectral function: non Fermi-liquid Fermi surfaces have disappeared!

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Holography and quantum matter

Reissner Nordstrom black hole: “critical Fermi-liquids”, like high Tc’s normal state (Hong Liu, John McGreevy). Dirac hair/electron star: Fermi-liquids emerging from a non Fermi liquid (critical) ultraviolet, like heavy fermions (Schalm, Cubrovic, Hartnoll). Scalar hair: holographic superconductivity, a new mechanism for superconductivity at a high temperature (Hartnoll, Herzog,Horowitz) . “Planckian dissipation”: quantum critical matter at high temperature, perfect fluids and the linear resistivity (Son, Policastro, …, Sachdev).

But first: crash course in holography

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BCS theory: fermions turning into bosons

Fermi-liquid + attractive interaction

Bardeen Cooper Schrieffer

Quasiparticles pair and Bose condense: D-wave SC: Dirac spectrum

ΨBCS = Πk uk + vkck↑

+ c−k↓ +

( ) vac.

Ground state

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Superglue !

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The zero temperature extensive entropy ‘disaster’

AdS-CFT The ‘extremal’ charged black hole with AdS2 horizon geometry has zero Hawking temperature but a finite horizon area. The ‘seriously entangled’ quantum critical matter at zero temperature should have an extensive ground state entropy (?*##!!)

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The holographic superconductor

Hartnoll, Herzog, Horowitz, arXiv:0803.3295 (Scalar) matter ‘atmosphere’ AdS-CFT Condensate (superconductor, … ) on the boundary! ‘Super radiance’: in the presence of matter the extremal BH is unstable => zero T entropy always avoided by low T order!!!

The Bose-Einstein hair cut.

Black hole scalar hair coding for the holographic superconductor

Scalar matter accumulates at the horizon

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Holographic superconductivity: stabilizing the fermions.

Fermion spectrum for scalar-hair back hole (Faulkner et al., 911.340;

Chen et al., 0911.282):

‘BCS’ Gap in fermion spectrum !! Temperature dependence as expected for ‘quantum-critical’ superconductivity (She, JZ, 0905.1225)

Excessive temperature dependence ‘pacified’ !

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Spielberg Thorne Hartnoll Herzog Horowitz Fisk ThomsonRonning

MacKenzie

Grigeria

Los Alamos St Andrews

Nature Nov 5 2009

Fermionic quantum phase transitions in the heavy fermion metals

Paschen et al., Nature (2004)

JZ, Science 319, 1205 (2008)

m* = 1 EF EF → 0 ⇒ m* → ∞

QP effective mass ‘bad actors’

Coleman Rutgers

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Experimentalists: back to the entropic drawing board ..

Grigeria

MacKenzie

Thomson Ronning

Nailing down T=0 entropy hidden by last minute order: high precision entropy balance needed.

ΔSorder = ΔC T

Tc

∫

dT

Lanthanides, actinides: Los Alamos Ruthenates:

• St. Andrews

L in e

• f

c r i ti c a l s p

• i

n t

p

q

?

Photoemission spectrum

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Further reading

AdS/CMT tutorials:

• J. Mc Greevy, arXiv:0909.0518; S. Hartnoll, arXiv:0909.3553

AdS/CMT fermions: Hong Liu et al., arXiv:0903.2477,0907.2694,1003.0010; M. Cubrovic et al. Science 325,429 (2009), arXiv:1012.5681; T. Faulkner et al., Science 329, 1043 (2010). Condensed matter: High Tc: J. Zaanen et al., Nature 430, 512, arXiv:1012.5461; C.M. Varma et al., Phys. Rep. 361, 267417 Heavy Fermions: J. Zaanen, Science 319, 1205; von Loehneisen et al, Rev.

• Mod. Phys. 79, 1015

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Quantum criticality or ‘conformal fields’

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Fermi-liquid phenomenology

Bare single fermion propagator ‘enumerates the fixed point’: Spectral function:

( ) ( ) ( ) ( ) K

+ − − − = Σʹ″ ʹ″ + Σʹ″ − − − =

F R F

k k v E Z i m k k G ω µ ω ω 2 1 ,

2

ImG(ω,k) = A ω,k

( ) =

ʹ″ ʹ″ Σ ω,k

( )

ω + µ + k − kF

( )

2 2m +

ʹ″ Σ ω,k

( )

2

+ ʹ″ ʹ″ Σ ω,k

( )

2

The Fermi liquid ‘lawyer list’:

• At T= 0 the spectral weight is zero at the Fermi-energy except for the

quasiparticle peak at the Fermi surface:

A EF,k

( ) = Z δ k − kF ( )

• Analytical structure of the self-energy:

( ) ( ) ( ) ( ) K

+ − ∂ Σʹ″ ∂ + − ∂ Σʹ″ ∂ + Σʹ″ = Σʹ″

= = F k k F E F F

k k k E k E k

F F

ω ω ω

ω

, ,

ʹ″ ʹ″ Σ ω,k

( ) ∝ ω − EF ( )

2 +K

• Temperature dependence:

ʹ″ ʹ″ Σ EF,kF,T

( ) ∝T 2 +K

Critical Fermi surfaces in heavy fermion systems

Blue = Fermi liquid Yellow= quantum critical regime

Antiferromagnetic

• rder

FL Fermi surface FL Fermi surface Coexisting critical Fermi surfaces ?

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Marginal Fermi liquid phenomenology.

Fermi-gas interacting by second order perturbation theory with ‘singular heat bath’:

ImP(q,ω) ∝−N(0)ω T , for |ω |< T ∝−N(0)sign ω

( ), for |ω |> T

Directly observed in e.g. Raman ??

G(k,ω) = 1 ω − vF k − kF

( ) − Σ(k,ω)

Σ(k,ω) ∝ g ωc ⎛ ⎝ ⎜ ⎞ ⎠ ⎟

2

ωln max |ω |,T

( )/ωc

( ) − i π

2 max |ω |,T

( )

⎡ ⎣ ⎢ ⎤ ⎦ ⎥

1 τ ∝max |ω |,T

( )

Single electron response (photoemission): Single particle life time is coincident (?!) with the transport life time => linear resistivity.

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Critical fermions at zero density: branchcut propagators

Two point Euclidean correlators: Analytically continue to Minkowski time => susceptibilities

Ψ τ,r r

( ) = φ τ,r

r

( ) φ(0,0)

χ t,r r

( ) = Ψ iτ,r

r

( )

At criticality, conformal invariance: Ψ τ

( ) ∝ 1

τ η ∝ 1 ωn

Δ → χ(ω) ∝

1 iω

( )

Δ

Lorentz invariance:

χ ω,k

( ) ∝

1 −ω 2 + c 2k 2

( )

Δ

Scaling dimension set by mass in AdS Dirac equation.

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AdS/CFT single fermion Spectral functions

ν = 0.1

ν ≈1

Non-Fermi-liquid “Fermi-liquid”

• Scaling metric:
• Scaling fields:
• Scaling relations:

Holographic Pauli-blocking: Lifshitz geometry.

δ γ δ γ γ κ + + + −

∝ ∝ ∝ ∝ Φ

2 2 2 2

, , , z I z J z J z m m 2 1 , , 1 , 1 , 2 1 − = = = = − = δ γ κ β α

2 2 2 2 2 2 2 2

z dz z dy dx z dt ds − + − =

β α

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‘Pseudogap’ fermions in high Tc superconductors

10 K Tc = 82 K 102 K Gap stays open above Tc But sharp quasiparticles disappear in incoherent ‘spectral smears’ in the metal

Shen group, Nature 450, 81 (2007)

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Thermodynamics: where are the fermions?

Hartnoll et al.: arXiv:0908.2657,0912.0008

Large N limit: thermodynamics entirely determined by AdS geometry. Fermi surface dependent thermodynamics, e.g. Haas van Alphen oscillations?

Leading 1/N corrections: “Fermionic one-loop dark energy”

Quantum corrections: one loop using Dirac quasinormal modes: ‘generalized Lifshitz-Kosevich formula’ for HvA oscillations.

χosc. = −∂ 2Ωosc. ∂B2 = πATckF

4

eB3 cosπckF

2

eB e

− cTkF

2

ebµ T µ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟

2ν −1

Fn µ

( )

n= 0 ∞

∑

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Soaked in Entropy ….

S = A + C T d +L F = A T +L

Entropic catastrophe!

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Collective transport: fermion currents

Tedious one loop calculation, ‘accidental’ cancellations:

Hong Liu (MIT)

ρFS ∝ Σ"1− fermion ∝T 2ν

‘Strange coincidence’ of one electron and transport lifetime of marginal fermi liquid finds gravitational explanation!

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‘Shankar/Polchinski’ functional renormalization group

interaction Fermi sphere

UV: weakly interacting Fermi gas Integrate momentum shells: functions of running coupling constants All interactions (except marginal Hartree) irrelevant => Scaling limit might be perfectly ideal Fermi-gas

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The end of weak coupling

interaction

Fermi sphere

Strong interactings: Fermi gas as UV starting point does not make sense! => ‘emergent’ Fermi liquid fixed point remarkably resilient (e.g. 3He) => Non Fermi-liquid/non ‘Hartree- Fock’ (BCS etc) states of fermion matter?

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Numerics and fermionic quantum criticality

Jarrell

DCA results for Hubbard model at intermediate couplings (U = 0.75W):

Non-fermi liquid ‘Mott fluid’ Fermi-liquid at ‘high’ dopings Quantum critical state, very unstable to d-wave superconductivity

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Graphene at the zero density Mott Transition

Herbut, Juricic, Vafek (arXiv:0904.1019): strongly interacting critical point at finite fermion coupling

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Gravitationally coding the fermion propagators (Faulkner et al. Science Aug 27. 2010)

GR ω,k

( ) = F0 k ( ) + F

1 k

( )ω + F2(k)gk ω ( )

| k |≡ kF

GR(ω,k) = h1 k − kF −ω /vF − Σ ω,k

( )

; Σ ω,k

( ) = hgkF ω ( ) = h2e

iγ kF ω 2ν kF

T=0 extremal black hole, near horizon geometry ‘emergent scale invariant’:

AdS2 ⊗ R2 ⇒ gk ω

( ) = c k ( )ω 2ν k

Matching with the UV infalling Dirac waves:

Special momentum shell:

Miracle, this is like critical/marginal Fermi-liquids!!

Space-like: IR-UV matching ‘organizes’ Fermi-surface. Time-like: IR scale invariance picked up via AdS2 self energy

boundary: d-dim space-time Hawking radiation gluons Black holes strings quarks

AdS/CFT correspondence: String theory Magic!

d-dim. gauge theory (d+1)-dim string theory / conformal field theory / gravity theory

Maldacena Witten, Gubser,Klebanov,Polyakov

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Entropic singularities AdS/CFT: black holes and planckian dissipation AdS-to-ARPES Holographic superconductivity quantum critical superconductivity

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