Relativistic magnetotransport in graphene, at quantum ... - ICTP

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

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

Quantum critical systems in
condensed matter
A few examples
• Graphene
• High Tc
• Superconductor-to-insulator
transition (interaction driven)

Dirac fermions in graphene
Honeycomb lattice of C atoms
2 massless Dirac cones in
the Brillouin zone:
(Sublattice degree of
freedom ↔ pseudospin)
Fermi velocity (speed of light”)
Close to the two
Fermi points K, K’:
Coulomb interactions: Fine structure constant
v
F
6 c
≈1.
1⋅10
m s ≈
300
(Semenoff ’84, Haldane ‘88)
Tight binding dispersion
H
→
≈
p
( p − K)
⋅
r
v p − K
=
r
F
2
e
α ≡
ε hv
F
r
r
σ
vF sublattice
E
= O
( 1)

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:
T = μ
Timp
~ ρimp
μ

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 μ = 0
Strong coupling!
“Quantum critical”
Crossover:
T = μ
Timp
~ ρimp
μ

Other relativistic fluids:
• Systems close to quantum criticality (with z = 1)
Example: Superconductor-insulator transition (Bose-Hubbard model)
rel ≈
−1
τ
kBT h
Maximal possible relaxation rate!
• 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)
C. P. Herzog, P. Kovtun, S. Sachdev, and D. T. Son (2007)
Hartnoll, Kovtun, MM, Sachdev (2007)

Strongly correlated electrons: Cuprate high T c
Thermoelectric
measurements.
Example: Anomalously
large Nernst Effect!
(“thermal analogue” of
the Hall effect)
Conformal
field theory?

Simplest example exhibiting “quantum
critical” features:
Graphene

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
T = μ
T
imp
μ
~ ρ
imp

Graphene – Fermi liquid?
1. Tight binding kinetic energy
→ massless Dirac quasiparticles
2. Coulomb interactions:
Unexpectedly strong!
→ nearly quantum critical!
Coulomb only marginally irrelevant for μ = 0!
RG:
(μ = 0)
(μ > 0)
< μ :
2
e
α ≡
εhv
F
= O
T Screening kicks in, short ranged Cb irrelevant
( 1)
Strong
coupling!

Strong coupling in undoped graphene
Inelastic scattering rate
(Electron-electron interactions)
Relaxation rate ~ T,
like in quantum critical systems!
Fastest possible rate!
MM, L. Fritz, and S. Sachdev, PRB ‘08.
μ >> T: standard 2d
Fermi liquid
μ < T: strongly
coupled relativistic
liquid
“Heisenberg uncertainty principle for well-defined quasiparticles”
( ) 1 2
~ k T E τ ~ α k T
−
≥ Δ = h
Eqp B
int ee B
As long as α(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)]
−
τ
1
ee
−1
τ ee
=
2
~ α
( k T )
C B
hμ
kBT h
2

Consequences for transport
1. -Collisionlimited conductivity σ in clean undoped graphene;
-Collisionlimited spin diffusion D s at any doping
2. Graphene - a perfect quantum liquid: very small viscosity η!
3. Emergent relativistic invariance at low frequencies!
Despite the breaking of relativistic invariance by
• finite T,
• finite μ,
• instantaneous 1/r Coulomb interactions
Collision-dominated transport → relativistic hydrodynamics:
a) Response fully determined by covariance, thermodynamics, and σ, η
b) Collective cyclotron resonance in small magnetic field (low frequency)
Hydrodynamic regime:
(collision-dominated)
τ
− 1
ee >>
τ
−1
imp
, ω , ω
typ
c
AC

Collisionlimited conductivities
Damle, Sachdev, (1996).
Fritz et al. (2008), Kashuba (2008)
Finite charge or spin conductivity in a pure system (for μ = 0 or B = 0) !
• Key: Charge or spin current without momentum
(particle/spin up)
(hole/spin down)
σ
Drude
J s,
c ≠
r
0,
• Finite collision-limited conductivity!
e
= ρτ → σ
m
but
P r
( μ = 0)
~
k
e
T v
Marginal irrelevance of Coulomb:
=
0
B
2
Pair creation/annihilation
leads to current decay
⎛
⎜
e
⎝
σ
( μ = 0) < ∞ ; σ ( μ ≠ 0)
= ∞
( μ;
B
2
0)
∝ v < ∞,
• Finite collision-limited spin diffusivity! Ds = Fτ
ee
• Only marginal irrelevance of Coulomb:
Maximal possible relaxation rate ~ T
−1
2 kBT τ ee ≈ α
h
→ Nearly universal conductivity at strong coupling
2
2
( kBT
) h 1 e
~
2 2
2
( hv)
⎟α k T α h
⎟
⎞
4
α ≈
log
⎠
B
( ) 1 <
Λ T
Expect saturation
as α →1, and
eventually phase
transition to
insulator

Boltzmann approach
Boltzmann equation in Born approximation
⎛
⎜
⎝
a.) +, i +, i
−, i
∂ ⎞
+ eE⋅
⎟ f
∂k
⎠
Cb
2
( k,
t)
= I [ k,
t | { f ( k′
, t)
} ] ∝ α ( T )
∂ t
±
coll
±
b.)
−, i
1
2
−
+, i +, i
−, i
+, i +, i
+, i
+, i
−, i
−
2
L. Fritz, J. Schmalian, MM, and S. Sachdev, PRB 2008
+(N − 1)
+, i +, i
−, j
−, j
2
+(N − 1)
+, i +, i +, i +, i
2
+, i
Collision-limited conductivity:
+, i
+(N − 1)
+, j
0.
76
σ ( μ = 0)
= 2
α
+, j
e
2
( T ) h
+, i +, j
−, j
2
−, i
2

Transport and thermoelectric
response at low frequencies?
Hydrodynamic regime:
(collision-dominated)
τ
− 1
ee >>
ω,
τ , ω
−1
imp
th
cyclo

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:
t
• Charge
• Momentum
• Energy
>> τ ee
Tloc r ( ) , μloc( r)
; r
u loc( r)

Relativistic Hydrodynamics
S. Hartnoll, P. Kovton, MM, and S. Sachdev, Phys. Rev. B 76, 144502 (2007).
Conservation laws (equations of motion):
Charge conservation
Energy/momentum conservation
1
T
τ
imp
0ν
μ μ
1. Construct entropy current S = Q T
2. Second law of thermodynamics
3. Covariance
∂
μ
μS ≥
0
Irrelevant at small k

Thermoelectric response
S. Hartnoll, P. Kovton, MM, and S. Sachdev, Phys. Rev. B 76, 144502 (2007).
Thermo-electric response
Charge and heat current:
Transverse thermoelectric response (Nernst)
J
Q
μ
μ
=
μ
ρ −
= u
( ) μ μ
ε + P u − μ J
Recipe: i) Solve linearized hydrodynamic equations
ii) Read off the response functions (Kadanoff & Martin 1960)
ν
μ
etc.

Collective cyclotron resonance
S. Hartnoll and C Herzog, 2007;
Relativistic magnetohydrodynamics: pole in AC response
Pole in the response
c
FL
2
c
( ε + P)
v
m
ω ωc
iγ
− ± =
∗
Collective cyclotron frequency of the relativistic plasma
ρ B c
ω = ↔ ω
Broadening of resonance:
F
MM, and S. Sachdev, 2008
=
e
γ = σ
B
Q
2 ( B c)
2 ( ε + P)
/ v
Observable at room temperature in the GHz regime!
c
F

Yes!
Relativistic hydrodynamics
from microscopics
Does relativistic hydro really apply to graphene
even though Coulomb interactions break relativistic
invariance?
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

Beyond weak coupling
approximation:
Graphene
↔
Very strongly coupled, critical
relativistic liquids?
AdS – CFT !

Au+Au collisions at RHIC
Quark-gluon plasma is described
by QCD (nearly conformal,
critical theory)
_
Low viscosity fluid!

Compare graphene to
Strongly coupled relativistic liquids
Response for special strongly coupled relativistic fluids
(maximally supersymmetric SU(N) Yang Mills theory with N
→ ∞ colors)
By mapping to weakly coupled gravity problem:
AdS (gravity) ↔ CFT 2+1 [SU(N>>1)]
weak coupling ↔ strong coupling
S. Hartnoll, P. Kovtun, MM, S. Sachdev (2007)
Obtain exact results for transport via the AdS–CFT correspondence

SU(N) transport from AdS/CFT
Gravitational dual to SUSY SU(N)-CFT 2+1 : Einstein-Maxwell theory
It has a black hole solution (with electric and magnetic charge):
AdS 3+1
Black hole
(embedded in M theory as )
z = 0
Electric charge q and magnetic charge h
↔ μ and B for the CFT

SU(N) transport from AdS/CFT
Gravitational dual to SUSY SU(N)-CFT 2+1 : Einstein-Maxwell theory
AdS 3+1
Black hole
(embedded in M theory as )
It has a black hole solution (with electric and magnetic charge):
z = 0
Background ↔ Equilibrium
Transport ↔ Perturbations in g tx,
ty , Ax,
y .
( ) 2
2
Response via Kubo formula from δ I δ g,
A .

Compare graphene to
Strongly coupled relativistic liquids
S. Hartnoll, P. Kovtun, MM, S. Sachdev (2007)
Obtain exact results via string theoretical AdS–CFT correspondence
• Confirm the results of hydrodynamics: response functions σ(ω), resonances
• Calculate the transport coefficients for a strongly coupled theory!
3 2
shear
SUSY - SU(N): σ ( μ = 0 ) = N ; ( μ = )
Interpretation:
2
9
e
h
2
η
s
0 =
1 h
4π
k
3 2
N effective degrees of freedom, strongly coupled: τT = O(1)
B

Graphene – a nearly perfect liquid!
Anomalously low viscosity (like quark-gluon plasma)
“Heisenberg”
Doped Graphene &
Fermi liquids:
(Khalatnikov etc)
Undoped Graphene:
Conjecture from
AdS-CFT:
η
η
η
~ E qpτ
≥ 1
s
2
∝ n ⋅mv
⋅τ
Exact (Boltzmann-Born Approx):
→
MM, J. Schmalian, and L. Fritz, (PRL 2009)
shear viscosity
entropy density
n ⋅ E
F
⋅
hE
( ) 2
k T
B
Measure of strong coupling:
η
= >
s
h
kB
1
4π
2
∝ n ⋅mv
⋅τ
→ nth
⋅k
BT
⋅ = n s ∝ k
2
2 th
Bnth
α kBT
α
F
h
T
s ∝ kBn
E
h
F
η h ⎛ EF
⎞
~ ⎜ ⎟
s k ⎝ T ⎠
B
3

Graphene α
= 1
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)

Electronic consequences of low viscosity?
Electronic turbulence in clean, strongly coupled graphene?
(or at quantum criticality!)
Reynolds number:
Strongly driven mesoscopic systems: (Kim’s group [Columbia])
L = 1μm
u
T
typ
=
0.
1v
= 100K
Re ~ 10 −100
MM, J. Schmalian, L. Fritz, (PRL 2009)
Complex fluid dynamics!
(pre-turbulent flow)
New phenomenon in an
electronic system!

Summary
• Undoped graphene is strongly
coupled in a large temperature window!
Strong
coupling
• 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