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

Scaling symmetry: a primer

For similar shapes areas increase as squares of the corresponding sides

A∼L2Observables:Invariants: angles

Use scaling to prove Pythagorean theorem1. Similar triangles2. Areas add up3. (the corresponding sides)^2 add up

s c2=s a2

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

The Dirac-Kepler problem

r '=a−1r , ∇ '=a∇ , H '=a H

H=−i ℏ v σ⃗ ∇⃗−Ze2

rScaling symmetry

r '=a−1r , ∇ '=a∇ , H '=a H

H=−i ℏ v σ⃗ ∇⃗−Ze2

rScaling symmetry

Invariant: β=Ze2/ℏ v

r '=a−1r , ∇ '=a∇ , H '=a H

H=−i ℏ v σ⃗ ∇⃗−Ze2

rScaling symmetry

Scaling for observables:a) scattering phases energy-independentb) conductivityc)

σ∼EF2∼n

r '=a−1r , ∇ '=a∇ , H '=a H

H=−i ℏ v σ⃗ ∇⃗−Ze2

rScaling symmetry

Scaling for observables:a) scattering phases energy-independentb) conductivityc) no resonances for any , no collapse?

σ∼EF2∼n

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)

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)

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?

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

Scattering phases and resonances

Shytov, Katsnelson and LL, PRL 99, 236801 (2007), PRL 99, 246802 (2007)

Scattering phases (tuning )

Shytov, Katsnelson and LL, PRL 99, 236801 (2007), PRL 99, 246802 (2007)

Caught in the act

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

Tuning Z by Building Artificial Nuclei from Ca Dimers

-0.6 -0.4 -0.2 0.0 0.2 0.4 0.60

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

Dirac pt.

n=1 n=2 n=3 n=4 n=5

Compare to Simulated Behavior

-0.6 -0.4 -0.2 0.0 0.2 0.4 0.60

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

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

What about scaling symmetry?

Directly probe scaling symmetry (and its violation)?

H=v σ⃗ (−i ℏ ∇⃗−e A⃗ (r ))−Ze2

Adding magnetic field A⃗(r)=B⃗× r⃗ /2

H=v σ⃗ (−i ℏ ∇⃗−e A⃗ (r ))−Ze2

sets a length scale and an energy scale:

λB=(ℏ /eB)1 /2 , E (B)=v (ℏ e B)1 /2

H=v σ⃗ (−i ℏ ∇⃗−e A⃗ (r ))−Ze2

λB=(ℏ /eB)1 /2 , E (B)=v (ℏ e B)1 /2

Prediction: all energies must scale with B as sqrt(B)

H=v σ⃗ (−i ℏ ∇⃗−e A⃗ (r ))−Ze2

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!

Energy spectrum in a B fieldDirac Landau levels(no charge):

Subcritical charge(DOS near impurity):

all levels show sqrt(B) scaling

Infrared spectroscopy of Landau levels: interactions shifting transitions the sqrt(B) scaling unaffected

Jiang et al PRL98 197403 (2007)

Scaling anomaly

Subcritical charge: Supercritical charge:

Scaling upheld Scaling violated

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

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

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?

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

Energy waves

Relativistic hydrodynamics: energy-momentum conservation (4x4 stress tensor)

∂iT ij=0, T=(E j⃗j⃗ P) , P=

j⃗=c2 p⃗

Energy flux and momentum density related by

giving(∂t

13c2∇

Energy waves

Relativistic hydrodynamics: energy-momentum conservation (4x4 stress tensor)

∂iT ij=0, T=(E j⃗j⃗ P) , P=

j⃗=c2 p⃗

Energy flux and momentum density related by

(∂t2−

13c2∇

Cosmic sound in graphene?

giving

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

03/18/13

Hydrodynamics at CN

“Inertial mass”: E-flux proportional to P-density j⃗w=v2 p⃗

∂tW+∇⃗ j⃗W=0, ∂t p i+∇ iσij=0, σij=12W δij

03/18/13

Hydrodynamics at CN

(∂t2−

12v2∇ 2)W=0, v '=

√2≈0.71⋅106 m / s

03/18/13

Hydrodynamics at CN

(∂t2−

12v2∇ 2)W=0, v '=

√2≈0.71⋅106 m / s

Electronic thermal waves x1000 faster than phonon 2nd sound, however x250

slower than cosmic sound

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

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 '

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 '

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)=

(∂t−D∇2)W +v 2

∇⃗ p⃗=0, D=κ/C

(∂t−ν∇2+γp) p⃗+

12∇W=0

03/18/13

Validity and frequency range

τϵ≈ℏ

α2 kT

For T=100K find

carrier-carrier scattering timeKashuba (2008), Fritz et al (2008)

τϵ≈60 fs

03/18/13

(ω+ iDk2)(ω+i γp+ i νk 2

τϵ≈ℏ

α2 kT

For T=100K find

τϵ≈60 fsHydrodynamics valid for ω≪1/ τϵ

03/18/13

(ω+ iDk2)(ω+i γp+ i νk 2

τϵ≈ℏ

α2 kT

For T=100K find

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 '=

√d, d=1

03/18/13

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

03/18/13

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

Textbook: sound is the form of energy that travels in waves

scaling anomaly and atomic collapse collective energy …levitov/scaling_anomaly_2013.pdf · 2013....

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