Relativistic magnetotransport in graphene, at quantum criticality and in black holes

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Relativistic magnetotransport in graphene, at
quantum criticality and in black holes
Markus Müller in collaboration with Sean
Hartnoll (Harvard) Pavel Kovtun (Victoria)
Subir Sachdev (Harvard) Lars Fritz
(Harvard) Jörg Schmalian (Iowa)
The Abdus Salam ICTP Trieste
AdS/CFT Strongly coupled systems and exact
results - Paris 26 Nov, 2009
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Outline

• Relativistic physics in graphene, quantum
  critical systems and conformal field theories
• Strong coupling features in collision-dominated
  transport as inspired by AdS-CFT results
• Comparison with strongly coupled fluids
• (via AdS-CFT)
• Graphene an almost perfect quantum liquid

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Quantum critical systems in condensed matterA
few examples

• Graphene
• High Tc
• Superconductor-to-insulator transition
  (interaction driven)

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Dirac fermions in graphene
(Semenoff 84, Haldane 88)
Honeycomb lattice of C atoms
Tight binding dispersion
2 massless Dirac cones in the Brillouin
zone (Sublattice degree of freedom ? pseudospin)
Close to the two Fermi points K, K
Fermi velocity (speed of light)
Coulomb interactions Fine structure constant
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Relativistic fluid at the Dirac point
D. Sheehy, J. Schmalian, Phys. Rev. Lett. 99,
226803 (2007).

• Relativistic plasma physics of interacting
  particles and holes!

Crossover
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Relativistic fluid at the Dirac point
D. Sheehy, J. Schmalian, Phys. Rev. Lett. 99,
226803 (2007).

• Relativistic plasma physics of interacting
  particles and holes!
• Strongly coupled, nearly quantum critical fluid
  at m 0

Crossover
Strong coupling! Quantum critical
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Other relativistic fluids

• Systems close to quantum criticality (with z
  1)
• Example Superconductor-insulator transition
  (Bose-Hubbard model)
• Conformal field theories
• E.g. strongly coupled Yang-Mills theories
• ? Exact treatment via AdS-CFT correspondence

Damle, Sachdev (1996, 1997) Bhaseen, Green,
Sondhi (2007). Hartnoll, Kovtun, MM, Sachdev
(2007)
Maximal possible relaxation rate!
C. P. Herzog, P. Kovtun, S. Sachdev, and D. T.
Son (2007) Hartnoll, Kovtun, MM, Sachdev (2007)
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Strongly correlated electrons Cuprate high Tc
Thermoelectric measurements.
Example Anomalously large Nernst
Effect! (thermal analogue of the Hall effect)
Conformal field theory?
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Simplest example exhibiting quantum critical
features Graphene
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Questions

• Transport characteristics in the strongly coupled
  relativistic plasma?
• Response functions, nearly universal transport
  coefficients
• Graphene as a nearly perfect and possibly
  turbulent quantum fluid
• (like the quark-gluon plasma)

Relativistic, Strong coupling regime
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Graphene Fermi liquid?
1. Tight binding kinetic energy ? massless
Dirac quasiparticles
2. Coulomb interactions Unexpectedly strong! ?
nearly quantum critical!
Coulomb only marginally irrelevant for m 0!
Strong coupling!
RG (m 0)
(m gt 0)
Screening kicks in, short ranged Cb irrelevant
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Strong coupling in undoped graphene
MM, L. Fritz, and S. Sachdev, PRB 08.
Inelastic scattering rate (Electron-electron
interactions)
m gtgt T standard 2d Fermi liquid
Relaxation rate T, like in quantum critical
systems! Fastest possible rate!
m lt T strongly coupled relativistic liquid
Heisenberg uncertainty principle for
well-defined quasiparticles
As long as a(T) 1, energy uncertainty is
saturated, scattering is maximal ? Nearly
universal strong coupling features in transport,
similarly as at the 2d superfluid-insulator
transition Damle, Sachdev (1996, 1997)
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Consequences for transport
1. -Collisionlimited conductivity s in clean
undoped graphene -Collisionlimited spin
diffusion Ds at any doping 2. Graphene - a
perfect quantum liquid very small viscosity h!

• 3. Emergent relativistic invariance at low
  frequencies!
• Despite the breaking of relativistic
  invariance by
• finite T,
• finite m,
• instantaneous 1/r Coulomb interactions

Collision-dominated transport ? relativistic
hydrodynamics a) Response fully determined by
covariance, thermodynamics, and s, h b)
Collective cyclotron resonance in small magnetic
field (low frequency)
Hydrodynamic regime (collision-dominated)
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Collisionlimited conductivities
Damle, Sachdev, (1996). Fritz et al. (2008),
Kashuba (2008)
Finite charge or spin conductivity in a pure
system (for m 0 or B 0) !

• Key Charge or spin current without momentum
• Finite collision-limited conductivity!
• Finite collision-limited spin diffusivity!

(particle/spin up)
Pair creation/annihilation leads to current decay
(hole/spin down)
but

• Only marginal irrelevance of Coulomb
• Maximal possible relaxation rate T

? Nearly universal conductivity at strong
coupling
Expect saturation as a ?1, and eventually phase
transition to insulator
Marginal irrelevance of Coulomb
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Boltzmann approach
L. Fritz, J. Schmalian, MM, and S. Sachdev, PRB
2008
Boltzmann equation in Born approximation
Collision-limited conductivity
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Transport and thermoelectric response at low
frequencies?
Hydrodynamic regime (collision-dominated)
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Hydrodynamics
Hydrodynamic collision-dominated regime
Long times, Large scales

• Local equilibrium established
• Study relaxation towards global equilibrium
• Slow modes Diffusion of the density of
  conserved quantities
• Charge
• Momentum
• Energy

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Relativistic Hydrodynamics
S. Hartnoll, P. Kovton, MM, and S. Sachdev, Phys.
Rev. B 76, 144502 (2007).
Irrelevant at small k
Conservation laws (equations of motion)
Charge conservation
Energy/momentum conservation
1. Construct entropy current
2. Second law of thermodynamics
3. Covariance
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Thermoelectric response
S. Hartnoll, P. Kovton, MM, and S. Sachdev, Phys.
Rev. B 76, 144502 (2007).
Thermo-electric response
etc.
Transverse thermoelectric response (Nernst)
Charge and heat current
Recipe i) Solve linearized hydrodynamic
equations ii) Read off the response functions
(Kadanoff Martin 1960)
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Collective cyclotron resonance
MM, and S. Sachdev, 2008
S. Hartnoll and C Herzog, 2007
Relativistic magnetohydrodynamics pole in AC
response
Pole in the response
Collective cyclotron frequency of the
relativistic plasma
Broadening of resonance
Observable at room temperature in the GHz regime!
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Relativistic hydrodynamics from microscopics
Does relativistic hydro really apply to graphene
even though Coulomb interactions break
relativistic invariance?
Yes! Key point There is a zero (momentum)
mode of the collision integral due to
translational invariance of the interactions The
dynamics of the zero mode under an AC driving
field (coupling it to other modes) reproduces
relativistic hydrodynamics at low frequencies
exactly. Condition Relativistic, weak-coupling
quasiparticle picture applies
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Beyond weak coupling approximationGraphene ?
Very strongly coupled, critical relativistic
liquids?AdS CFT !
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AuAu collisions at RHIC
Quark-gluon plasma is described by QCD (nearly
conformal, critical theory) _ Low viscosity
fluid!
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Compare graphene to Strongly coupled
relativistic liquids
S. Hartnoll, P. Kovtun, MM, S. Sachdev (2007)
Response for special strongly coupled
relativistic fluids (maximally supersymmetric
SU(N) Yang Mills theory with
colors) By mapping to weakly coupled gravity
problem AdS
(gravity) ? CFT21 SU(Ngtgt1)
weak coupling ? strong coupling Obtain
exact results for transport via the AdSCFT
correspondence
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SU(N) transport from AdS/CFT
Gravitational dual to SUSY SU(N)-CFT21
Einstein-Maxwell theory
(embedded in M theory as
)
It has a black hole solution (with electric and
magnetic charge)
z 0
AdS31
Electric charge q and magnetic charge h ? m and B
for the CFT
Black hole
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SU(N) transport from AdS/CFT
Gravitational dual to SUSY SU(N)-CFT21
Einstein-Maxwell theory
(embedded in M theory as
)
It has a black hole solution (with electric and
magnetic charge)
z 0
Background ? Equilibrium
AdS31
Transport ? Perturbations in .
Response via Kubo formula from .
Black hole
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Compare graphene to Strongly coupled
relativistic liquids
S. Hartnoll, P. Kovtun, MM, S. Sachdev (2007)
Obtain exact results via string theoretical
AdSCFT correspondence

• Confirm the results of hydrodynamics response
  functions s(w), resonances
• Calculate the transport coefficients for a
  strongly coupled theory!

SUSY - SU(N)

effective degrees of freedom, strongly coupled
tT O(1)
Interpretation
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Graphene a nearly perfect liquid!
MM, J. Schmalian, and L. Fritz, (PRL 2009)
Anomalously low viscosity (like quark-gluon
plasma)
Measure of strong coupling
Heisenberg
Conjecture from AdS-CFT
Doped Graphene Fermi liquids (Khalatnikov etc)
Undoped Graphene
Exact (Boltzmann-Born Approx)
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Graphene
T. Schäfer, Phys. Rev. A 76, 063618 (2007). A.
Turlapov, J. Kinast, B. Clancy, Le Luo, J.
Joseph, J. E. Thomas, J. Low Temp. Phys. 150, 567
(2008)
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Electronic consequences of low viscosity?
MM, J. Schmalian, L. Fritz, (PRL 2009)
Electronic turbulence in clean, strongly coupled
graphene? (or at quantum criticality!) Reynolds
number
Strongly driven mesoscopic systems (Kims group
Columbia)
Complex fluid dynamics! (pre-turbulent flow) New
phenomenon in an electronic system!
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Summary
Strong coupling

• Undoped graphene is strongly
• coupled in a large temperature window!
• Nearly universal strong coupling features in
  transport many similarities with strongly
  coupled critical fluids (described by AdS-CFT)
• Emergent relativistic hydrodynamics at low
  frequency
• Graphene Nearly perfect quantum liquid!
• ? Possibility of complex (turbulent?) current
  flow at high bias