Scaling Anomaly and Atomic Collapse Collective Energy …levitov/Scaling_Anomaly_2013.pdf · 2013....
Transcript of Scaling Anomaly and Atomic Collapse Collective Energy …levitov/Scaling_Anomaly_2013.pdf · 2013....
05/03/2013 Electron Interactions in Graphene
Scaling Anomaly and Atomic Collapse
Collective Energy Propagation at Charge Neutrality
Leonid Levitov (MIT)
Electron Interactions in GrapheneFTPI, University of Minnesota 05/04/2013
05/03/2013 Electron Interactions in Graphene
Scaling symmetry: a primer
For similar shapes areas increase as squares of the corresponding sides
A∼L2Observables:Invariants: angles
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Scaling symmetry: a primer
Use scaling to prove Pythagorean theorem1. Similar triangles2. Areas add up3. (the corresponding sides)^2 add up
s c2=s a2
+sb2
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Scaling symmetry: a primer
c2=a2
+b2
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Dirac-Kepler problem at large Z, atomic collapseW. Greiner, B. Muller & J. Rafelski, QED of Strong Fields (Springer, 1985).I. Pomeranchuk and Y. Smorodinsky, J. Phys. USSR 9, 97 (1945)Y. B. Zeldovich and V. S. Popov, Usp. Fiz. Nauk 105, 403 (1971)V. S. Popov, “Critical Charge in Quantum Electrodynamics,” in: A. B. Migdal memorial volume, Yadernaya Fizika 64, 421 (2001) [Physics of Atomic Nuclei 64, 367 (2001)] Coulomb impurity in grapheneK. Nomura & A. H. MacDonald, PRL 96, 256602 (2006).T. Ando, J. Phys. Soc. Jpn. 75, 074716 (2006).R. R. Biswas, S. Sachdev, and D. T. Son, PRB76, 205122 (2007)D. S. Novikov, PRB 76, 245435 (2007) Atomic collapse in grapheneA. V. Shytov, M. I. Katsnelson, and L. S. Levitov, PRL 99, 236801 (2007)V. M. Pereira, J. Nilsson, and A. H. Castro Neto, PRL 99, 166802 (2007)A. V. Shytov, M. I. Katsnelson, and L. S. Levitov, PRL 99, 246802 (2007)M. M. Fogler, D. S. Novikov, and B. I. Shklovskii, PRB76, 233402 (2007).I. S. Terekhov, A. I. Milstein, V. N. Kotov, O. P. Sushkov, PRL 100, 076803 (2008)V. N. Kotov, B. Uchoa, and A. H. Castro Neto, PRB 78, 035119 (2008). ExperimentY. Wang ... and M. F. Crommie, Nat. Phys. 8, 653 (2012); Science March 2013E. Y. Andrei, this conference
Atomic collapse
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The Dirac-Kepler problem
r '=a−1r , ∇ '=a∇ , H '=a H
H=−i ℏ v σ⃗ ∇⃗−Ze2
rScaling symmetry
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The Dirac-Kepler problem
r '=a−1r , ∇ '=a∇ , H '=a H
H=−i ℏ v σ⃗ ∇⃗−Ze2
rScaling symmetry
Invariant: β=Ze2/ℏ v
05/03/2013 Electron Interactions in Graphene
The Dirac-Kepler problem
r '=a−1r , ∇ '=a∇ , H '=a H
H=−i ℏ v σ⃗ ∇⃗−Ze2
rScaling symmetry
Invariant: β=Ze2/ℏ v
Scaling for observables:a) scattering phases energy-independentb) conductivityc)
σ∼EF2∼n
05/03/2013 Electron Interactions in Graphene
The Dirac-Kepler problem
r '=a−1r , ∇ '=a∇ , H '=a H
H=−i ℏ v σ⃗ ∇⃗−Ze2
rScaling symmetry
Invariant: β=Ze2/ℏ v
Scaling for observables:a) scattering phases energy-independentb) conductivityc) no resonances for any , no collapse?
σ∼EF2∼n
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Anomalies in Quantum PhysicsRegularized quantum theory can violate a theory's classical symmetry
Ubiquitous: chiral A, conformal A, gauge A... Scaling A responsible for the large strength of nuclear interactions and quark confinement (asymptotic freedom)
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Scaling anomaly:
1. “Global effect” of a short-range cutoff alters the behavior at all energy scales
2. Bound states formation, violation of scaling
3. Essential in theory of quark confinement in QCD developed in 1980's by Gribov(as an alternative to asymptotic freedom)
05/03/2013 Electron Interactions in Graphene
Scaling anomaly:
1. “Global effect” of a short-range cutoff alters the behavior at all energy scales
2. Bound states formation, violation of scaling
3. Essential in theory of quark confinement in QCD developed in 1980's by Gribov(as an alternative to asymptotic freedom)
Novel hep-condmat connection?
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Scaling Anomaly and Atomic Collapse
YES for ||<Zc – regularization insensitive
NO for ||>Zc – for all energies at once!
Scaling symmetry for charge impurity (regularized Dirac-Kepler problem):
Tunable/switchable anomaly
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Scattering phases and resonances
Shytov, Katsnelson and LL, PRL 99, 236801 (2007), PRL 99, 246802 (2007)
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Scattering phases (tuning )
Shytov, Katsnelson and LL, PRL 99, 236801 (2007), PRL 99, 246802 (2007)
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Caught in the act
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Observation of Atomic Collapse Resonances (Wang et al Science March 7 2013)
Ca Dimers are Moveable Charge Centers
Fabricate Artificial Nuclei
+ +
++++
+ +
Tune Z above Zc
n=3 dimer cluster
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Tuning Z by Building Artificial Nuclei from Ca Dimers
-0.6 -0.4 -0.2 0.0 0.2 0.4 0.60
1
2
3
Nor
mal
ized
dI/d
V
Sample Bias (eV)
1.9nm 2.7nm 3.6nm 4.5nm 16.9nm
-0.4 -0.2 0.0 0.2 0.4
Sample Bias (eV)
2.4nm 3.3nm 4.2nm 6.0nm 17.1nm
-0.4 -0.2 0.0 0.2 0.4
Sample Bias (eV)
2.6nm 3.4nm 4.3nm 6.0nm 17.1nm
-0.4 -0.2 0.0 0.2 0.4
Sample Bias (eV)
3.2nm 4.1nm 4.9nm 6.7nm 17.6nm
-0.4 -0.2 0.0 0.2 0.4
Sample Bias (eV)
3.7nm 4.6nm 5.5nm 7.3nm 18.9nm
3 nm
Dirac pt.
n=1 n=2 n=3 n=4 n=5
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Compare to Simulated Behavior
-0.6 -0.4 -0.2 0.0 0.2 0.4 0.60
1
2
3
Nor
mal
ized
dI/d
V
Sample Bias (eV)
1.9nm 2.7nm 3.6nm 4.5nm 16.9nm
-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6
Sample Bias (eV)
3.7nm 4.6nm 5.5nm 7.3nm 18.9nm
-0.6 -0.4 -0.2 0.0 0.2 0.4 0.60
1
2
3
Nor
mal
ized
dI/d
V
Sample Bias (eV)
1.9nm 2.7nm 3.6nm 4.5nm 16.9nm
-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6
Sample Bias (eV)
2.6nm 3.4nm 4.3nm 6.0nm 17.1nm
-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6
Sample Bias (eV)
3.7nm 4.6nm 5.5nm 7.3nm 18.9nm
-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6
Sample Bias (eV)
2.6nm 3.4nm 4.3nm 6.0nm 17.1nm
Experiment:n=1
Theory:Z/Zc = 0.6
Theory:Z/Zc = 1.4
Theory:Z/Zc = 2.2
Experiment:n=3
Experiment:n=5
Theory fit parameter: Z/Zc
Atomic Collapse
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What about scaling symmetry?
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Directly probe scaling symmetry (and its violation)?
H=v σ⃗ (−i ℏ ∇⃗−e A⃗ (r ))−Ze2
r
Adding magnetic field A⃗(r)=B⃗× r⃗ /2
05/03/2013 Electron Interactions in Graphene
Directly probe scaling symmetry (and its violation)?
H=v σ⃗ (−i ℏ ∇⃗−e A⃗ (r ))−Ze2
r
Adding magnetic field A⃗(r)=B⃗× r⃗ /2
sets a length scale and an energy scale:
λB=(ℏ /eB)1 /2 , E (B)=v (ℏ e B)1 /2
05/03/2013 Electron Interactions in Graphene
Directly probe scaling symmetry (and its violation)?
H=v σ⃗ (−i ℏ ∇⃗−e A⃗ (r ))−Ze2
r
Adding magnetic field A⃗(r)=B⃗× r⃗ /2
sets a length scale and an energy scale:
λB=(ℏ /eB)1 /2 , E (B)=v (ℏ e B)1 /2
Prediction: all energies must scale with B as sqrt(B)
05/03/2013 Electron Interactions in Graphene
Directly probe scaling symmetry (and its violation)?
H=v σ⃗ (−i ℏ ∇⃗−e A⃗ (r ))−Ze2
r
Adding magnetic field A⃗(r)=B⃗× r⃗ /2
sets a length scale and an energy scale:
λB=(ℏ /eB)1 /2 , E (B)=v (ℏ e B)1 /2
Prediction: all energies must scale with B as sqrt(B)
True for both el-imp and el-el interactionsmind running coupling!
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Energy spectrum in a B fieldDirac Landau levels(no charge):
Subcritical charge(DOS near impurity):
all levels show sqrt(B) scaling
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Infrared spectroscopy of Landau levels: interactions shifting transitions the sqrt(B) scaling unaffected
Jiang et al PRL98 197403 (2007)
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Scaling anomaly
Subcritical charge: Supercritical charge:
Scaling upheld Scaling violated
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B-field as an experimental knob
1) Scaling symmetry for massless Dirac fermions with 1/r interactions (both el-imp and el-el)
2) Scaling anomaly: scaling violated for supercritical Coulomb potential
3) sqrt(B) scaling breakdown as anomaly litmus test
4) Other knobs (calibration?): pseudo-B field, strain, curvature, gating
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Collective Energy Propagation at Charge Neutrality
ArXiv:1306.4972 TV Phan, JCW Song & LL
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How fast can heat propagate?
Typically, heat propagates through diffusion:
∂tW=κ∇2T , L∼√Dt
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How fast can heat propagate?
Typically, heat propagates through diffusion:
∂tW=κ∇2T , L∼√Dt
In special cases, a compressional wave (second sound)
reaching speeds as high as 20 m/s (superfluid liquid, He II), 780 m/s (solid, Bi)
(∂t2−v2
∇2)T=0
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How fast can heat propagate?
Typically, heat propagates through diffusion:
∂tW=κ∇2T , L∼√Dt
In special cases, a compressional wave (second sound)
reaching speeds as high as 20 m/s (superfluid liquid, He II), 780 m/s (solid, Bi)
(∂t2−v2
∇2)T=0Is this fast enough?
05/03/2013 Electron Interactions in Graphene
Cosmic soundFluid mechanics predicts isentropic waves in a relativistic gas propagating with the velocity
c '=c /√3In the early universe “baryon acoustic oscillations” arise due to gravity and radiation pressure. Key in interpreting Cosmic Microwave Background
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Energy waves
Relativistic hydrodynamics: energy-momentum conservation (4x4 stress tensor)
∂iT ij=0, T=(E j⃗j⃗ P) , P=
13
Eδij
j⃗=c2 p⃗
3×3
Energy flux and momentum density related by
giving(∂t
2−
13c2∇
2)E=0
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Energy waves
Relativistic hydrodynamics: energy-momentum conservation (4x4 stress tensor)
∂iT ij=0, T=(E j⃗j⃗ P) , P=
13
Eδij
j⃗=c2 p⃗
3×3
Energy flux and momentum density related by
(∂t2−
13c2∇
2)E=0
Cosmic sound in graphene?
giving
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Hydrodynamics at CN
Strongly interacting charge-compensated plasma (Gonzales, Guinea, Vozmediano, Son, Sheehy & Schmalian, Vafek)
E & P conserved in carrier-carrier collisions
Rapid exchange of E and P among colliding particles
Relatively slow P relaxation, very slow E relaxation
“Inertial mass”: E-flux proportional to P-density
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Hydrodynamics at CN
Strongly interacting charge-compensated plasma (Gonzales, Guinea, Vozmediano, Son, Sheehy & Schmalian, Vafek)
E & P conserved in carrier-carrier collisions
Rapid exchange of E and P among colliding particles
Relatively slow P relaxation, very slow E relaxation
“Inertial mass”: E-flux proportional to P-density j⃗w=v2 p⃗
∂tW+∇⃗ j⃗W=0, ∂t p i+∇ iσij=0, σij=12W δij
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Hydrodynamics at CN
Strongly interacting charge-compensated plasma (Gonzales, Guinea, Vozmediano, Son, Sheehy & Schmalian, Vafek)
E & P conserved in carrier-carrier collisions
Rapid exchange of E and P among colliding particles
Relatively slow P relaxation, very slow E relaxation
“Inertial mass”: E-flux proportional to P-density j⃗w=v2 p⃗
∂tW+∇⃗ j⃗W=0, ∂t p i+∇ iσij=0, σij=12W δij
(∂t2−
12v2∇ 2)W=0, v '=
v
√2≈0.71⋅106 m / s
03/18/13
Hydrodynamics at CN
Strongly interacting charge-compensated plasma (Gonzales, Guinea, Vozmediano, Son, Sheehy & Schmalian, Vafek)
E & P conserved in carrier-carrier collisions
Rapid exchange of E and P among colliding particles
Relatively slow P relaxation, very slow E relaxation
“Inertial mass”: E-flux proportional to P-density j⃗w=v2 p⃗
∂tW+∇⃗ j⃗W=0, ∂t p i+∇ iσij=0, σij=12W δij
(∂t2−
12v2∇ 2)W=0, v '=
v
√2≈0.71⋅106 m / s
Electronic thermal waves x1000 faster than phonon 2nd sound, however x250
slower than cosmic sound
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Compare to plasmons
∂tρ+∇⃗ j⃗=0, j⃗=em̃
n0 p⃗
∂t p⃗=ρ E⃗ , E⃗=−∇⃗r∫V (r−r ' )ρ(r ')d3 r '
1. propagating charge density wave2. requires finite doping3. frequencies EF
4. damping: momentum relaxation
ω(ω+i γp)=2e2 EF
ϵ̃ ℏ2 k
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Compare to plasmons
∂tρ+∇⃗ j⃗=0, j⃗=em̃
n0 p⃗
∂t p⃗=ρ E⃗ , E⃗=−∇⃗r∫V (r−r ' )ρ(r ')d3 r '
1. propagating charge density wave2. requires finite doping3. frequencies EF
4. damping: momentum relaxation
ω(ω+i γp)=2e2 EF
ϵ̃ ℏ2 k
03/18/13
Compare to plasmons
∂tρ+∇⃗ j⃗=0, j⃗=em̃
n0 p⃗
∂t p⃗=ρ E⃗ , E⃗=−∇⃗r∫V (r−r ' )ρ(r ')d3 r '
1. propagating charge density wave2. requires finite doping3. frequencies EF
4. damping: momentum relaxation
temperaturezero
^k also Umklapp scattering and viscosity (small)
ω(ω+i γp)=2e2 EF
ϵ̃ ℏ2 k (ω+ iDk2
)(ω+i γp+ i νk 2)=
v2
2k
(∂t−D∇2)W +v 2
∇⃗ p⃗=0, D=κ/C
(∂t−ν∇2+γp) p⃗+
12∇W=0
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Validity and frequency range
τϵ≈ℏ
α2 kT
For T=100K find
carrier-carrier scattering timeKashuba (2008), Fritz et al (2008)
τϵ≈60 fs
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Validity and frequency range
(ω+ iDk2)(ω+i γp+ i νk 2
)=v2
2k2
τϵ≈ℏ
α2 kT
For T=100K find
carrier-carrier scattering timeKashuba (2008), Fritz et al (2008)
τϵ≈60 fsHydrodynamics valid for ω≪1/ τϵ
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Validity and frequency range
(ω+ iDk2)(ω+i γp+ i νk 2
)=v2
2k2
τϵ≈ℏ
α2 kT
For T=100K find
carrier-carrier scattering timeKashuba (2008), Fritz et al (2008)
τϵ≈60 fsHydrodynamics valid for ω≪1/ τϵ
Weak damping, ''', for γp≪ω≪τϵ−1
Few-m mean free paths (high doping, clean G/BN systems)
sqrt(n) dependence extrapolated to DP yields γp−1≈0.5 ps
0.2THz< f <3THz 0.3μm<λ<5μm
03/18/13
How to observe?Nature Nano 2008
03/18/13
How to observe?Nature Nano 2008
Measured velocity distinct from plasmon velocity!Perhaps coincidentally, this equals
≈106 m / sv '=
v
√d, d=1
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How to observe?
J Chen … F H L Koppens, Nature (2012)Visualize localized plasmon modes in real space by scattering-type scanning near-field optical microscopy
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Summary
1) Wavelike (ballistic) heat propagation at CN
2) Charge-neutral mode, characteristic velocity
3) Exist only at CN (switchable/tunable)
4) frequency range
0.2THz< f <3THz 300nm<λ<5μm
v '=v /√2
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Textbook: sound is the form of energy that travels in waves