Sang-Jin Sin (Hanyang)2016.8.23@APCTP

Holography of Dirac Fluid

0

Based on 1608.xxxxxY.Seo, G.Song, P.Kim, S.Sachdev, + sjs

1. Introduction

Physics after particles are gone

Quantum criticality

1

Look at the Spectral function as we increases the coupling:Pole resonance background

Strong coupling Loss of particle character

Effect of strong coupling

2

After disappearance of (quasi-) particles?

3

Quantum Critical Point, Fermi arc, Strange metal, ….. Characteristic phenomena of strongly interacting system Non-Fermi Liquid

Quantum Critical point

Fermi arcT-linear Resistivity

Quantum Criticality

4

Divergence after regularization (at g_c)

• Meaning of divergence: Presence of phase transition

If second order, QCP

1. loss of original degree of freedom

2. appearance of new scale free field.

• Classification of QCP:

• key: if massless particle has a relevant

interaction

No solution in field theory QCD

Need new field theory Holography, but ….

Z, θ: ω=kz, [s]=D-θ g

Energy

5

AdS/CFT (Maldacena 1997)

QCP(z,θ)

CFT

Reminder : wishful thinking

1 Int roduct ion

The AdS/CFT correspondence, sometimes called as gauge/gravity duality, is a conjectured relationship

between quantum field theory and gravity. Precisely, in thiscorrespondence, quantum physicsof strongly

correlated many-body systems is related to the classical dynamics of gravity which lives in one higher

dimension. On the other hand according to the AdS/CFT dictionary, an AdS geometry at the gravity

side could only address the conformal symmetry of the dual field theory. However, the generalization of

gauge/gravity correspondence to geometries which are not asymptotically AdS seems to be important

as long as such extension may be related to the invariance under a certain scaling of dual field theory

which does not even have conformal symmetry. Such generalization of AdS is actually motivated by

consideration of gravity toy models in condensed matter physics (the application of such generalization

can be found, for example, in [2]). A prototype of this generalization is a theory with the Lifshitz fixed

point in which the spatial and time coordinates of a field theory have been scaled as

t → ζz t, x⃗ → ζx⃗, r → ζr, (1.1)

where z is the critical dynamical exponent. From the holographic duality point of view, for a (D + 1)-

dimensional theory, the corresponding (D + 2)-dimensional gravity dual can be defined by the following

metric

ds2D + 2 =−dt2

r 2z+dr 2

r 2+1

r 2

D

i= 1

dx2i , (1.2)

where in this paper the AdS radius is set to be one. Due to the anisotropy between spaceand time, it is

clear that this metric can not be an ordinary solution of the Einstein equation, in fact one needs some

sortsof matter fields to break the isotropy, e.g., by adding a massive vector field or a gaugefield coupled

to a scalar field [3–6]. In general by adding a dilaton with non trivial potential and an abelian gauge

field to Einstein-Hilbert action (Einstein-Maxwell-Dilaton theory), one can find even more interesting

metrics, in particular the followingmetric has been used frequently [7]

ds2D + 2 = r2θD

−dt2

r 2z+dr 2

r 2+1

r 2

D

i= 1

dx2i , (1.3)

where θ is hyperscaling violation exponent. Thismetric under the scale-transformation (1.1) transforms

as ds→ ζθD ds. In a theory with hyperscaling violation, the thermodynamic parameters behave in such

a way that they are stated in D −θ dimensions; Moreprecisely, in a (D + 1)-dimensional theorieswhich

are dual to background (1.2), the entropy scales with temperature as TD /z, however, in the presence of

θ namely dual to (1.3), it scales as T (D−θ)/ z [7,8]. Therefore one may associate an effective dimension

to the theory and this becomes important in studying the log behavior of the entanglement entropy of

system with Fermi surface in condense matter physics, explicitly it was shown that for θ = D − 1 for

any z, the entanglement entropy exhibits a logarithmic violation of the area law [9]. On the other hand

for such backgrounds time-dependency can be achieved by Vaidya metric with a hyperscaling violating

factor. It is the main aim of this paper to investigate how entanglement (mutual information) spreads

in time-dependent hyperscaling violating backgrounds.

Basically, the AdS-Vaidya metric is used to describe a gravitational collapse of a thin shell of matter

in formation of the black hole. This metric in D + 2 dimensions is given by

ds2 =1

ρ2− f (ρ,v)dv2 −2dρdv +

D

i= 1

dx2i , f (ρ,v) = 1−m(v)ρD + 1, (1.4)

where ρ is the radial coordinate, x i ’s (i = 1,..,D) are spatial boundary coordinates and, here, the

2

AdS

6

Def of AdS/CMT

QCP(z,θ) Geometry (z,θ)

1 Int roduct ion

The AdS/CFT correspondence, sometimes called as gauge/gravity duality, is a conjectured relationship

between quantum field theory and gravity. Precisely, in thiscorrespondence, quantum physicsof strongly

correlated many-body systems is related to the classical dynamics of gravity which lives in one higher

dimension. On the other hand according to the AdS/CFT dictionary, an AdS geometry at the gravity

side could only address the conformal symmetry of the dual field theory. However, the generalization of

gauge/gravity correspondence to geometries which are not asymptotically AdS seems to be important

as long as such extension may be related to the invariance under a certain scaling of dual field theory

which does not even have conformal symmetry. Such generalization of AdS is actually motivated by

consideration of gravity toy models in condensed matter physics (the application of such generalization

can be found, for example, in [2]). A prototype of this generalization is a theory with the Lifshitz fixed

point in which the spatial and time coordinates of a field theory have been scaled as

t → ζz t, x⃗ → ζx⃗, r → ζr, (1.1)

where z is the critical dynamical exponent. From the holographic duality point of view, for a (D + 1)-

dimensional theory, the corresponding (D + 2)-dimensional gravity dual can be defined by the following

metric

ds2D + 2 =−dt2

r 2z+dr 2

r 2+1

r 2

D

i= 1

dx2i , (1.2)

where in this paper the AdS radius is set to be one. Due to the anisotropy between spaceand time, it is

clear that this metric can not be an ordinary solution of the Einstein equation, in fact one needs some

sortsof matter fields to break the isotropy, e.g., by adding a massive vector field or a gaugefield coupled

to a scalar field [3–6]. In general by adding a dilaton with non trivial potential and an abelian gauge

field to Einstein-Hilbert action (Einstein-Maxwell-Dilaton theory), one can find even more interesting

metrics, in particular the followingmetric has been used frequently [7]

ds2D + 2 = r2θD

−dt2

r 2z+dr 2

r 2+1

r 2

D

i= 1

dx2i , (1.3)

where θ is hyperscaling violation exponent. Thismetric under the scale-transformation (1.1) transforms

as ds→ ζθD ds. In a theory with hyperscaling violation, the thermodynamic parameters behave in such

a way that they are stated in D −θ dimensions; Moreprecisely, in a (D + 1)-dimensional theorieswhich

are dual to background (1.2), the entropy scales with temperature as TD /z, however, in the presence of

θ namely dual to (1.3), it scales as T (D−θ)/ z [7,8]. Therefore one may associate an effective dimension

to the theory and this becomes important in studying the log behavior of the entanglement entropy of

system with Fermi surface in condense matter physics, explicitly it was shown that for θ = D − 1 for

any z, the entanglement entropy exhibits a logarithmic violation of the area law [9]. On the other hand

for such backgrounds time-dependency can be achieved by Vaidya metric with a hyperscaling violating

factor. It is the main aim of this paper to investigate how entanglement (mutual information) spreads

in time-dependent hyperscaling violating backgrounds.

Basically, the AdS-Vaidya metric is used to describe a gravitational collapse of a thin shell of matter

in formation of the black hole. This metric in D + 2 dimensions is given by

ds2 =1

ρ2− f (ρ,v)dv2 −2dρdv +

D

i= 1

dx2i , f (ρ,v) = 1−m(v)ρD + 1, (1.4)

where ρ is the radial coordinate, x i ’s (i = 1,..,D) are spatial boundary coordinates and, here, the

2

Dual?

Transport coefficients

Holography model calculation30 years of data

How holography works in E&M where both attraction and repulsion?

7

1/r r

Long ShortExpansion Falling

Repulsive Coulomb Attractive gravity

No dictionary for lattice-electron interaction

1997 discovery of AdS/CFT

1. 2002 Heavy Ion(QGP ~ perfect fluid) Son et al.

2. 2007 first ads/CMT paper by Sachdev et.al (Magneto-Hydro)

3. 2008 Superconductivity: Horowitz,Hartnoll,Herzog; Gubser

4. 2008 Non-fermi liquid : Sung-Sik Lee; Hong Liu

5. 2013 Metal-Insulator transition [Donos/Hartnoll],

6. 2015 Phase diagram of cuperate[Kiritsis, Li Li]

……

7. 2013-2014 Holography as a method of experimental science

Donos, Hartnoll, Gaunttlet, Tong, Gauteraux, Kiritsis, Yi Ling/J.Wu(hep),

X.Ge, S. Wu, KKK+S,….

Calculational transport (still for z=1 QCP)8

A Brief history of holography

2. Transport in Graphene

For what material holography is good?from my condensed matter friends

9

Theoretical evidence of strong correlation in graphene

10

Experimental evidence of strong correlation i

11

Science 4 March 2016

originally published online February 11, 2016 (6277), 1055-1058. [doi: 10.1126/science.aad0201]351Science

M. Polini (February 11, 2016) Novoselov, I. V. Grigorieva, L. A. Ponomarenko, A. K. Geim andTomadin, A. Principi, G. H. Auton, E. Khestanova, K. S. D. A. Bandurin, I. Torre, R. Krishna Kumar, M. Ben Shalom, A.grapheneNegative local resistance caused by viscous electron backflow in

Editor's Summary

, this issue p. 1061, 1055, 1058; see also p. 1026Sciencetransport in graphene, a signature of so-called Dirac fluids.

observed a huge increase of thermalet al.fluid flowing through a small opening. Finally, Crossno found evidence in graphene of electron whirlpools similar to those formed by viscouset al.Bandurin

had a major effect on the flow, much like what happens in regular fluids.2fluid in thin wires of PdCoO found that the viscosity of the electronet al.counterexamples (see the Perspective by Zaanen). Moll

flow rarely resembles anything like the familiar flow of water through a pipe, but three groups describe Electrons inside a conductor are often described as flowing in response to an electric field. This

Electrons that flow like a fluid

This copy is for your personal, non-commercial use only.

Article Tools

http://science.sciencemag.org/content/351/6277/1055article tools: Visit the online version of this article to access the personalization and

Permissionshttp://www.sciencemag.org/about/permissions.dtlObtain information about reproducing this article:

is a registered trademark of AAAS. ScienceAdvancement of Science; all rights reserved. The title Avenue NW, Washington, DC 20005. Copyright 2016 by the American Association for thein December, by the American Association for the Advancement of Science, 1200 New York

(print ISSN 0036-8075; online ISSN 1095-9203) is published weekly, except the last weekScience

on

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7,

201

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ttp

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Science 4 March 2016

Experimental evidence of strong correlation ii

12

originally published online February 11, 2016 (6277), 1055-1058. [doi: 10.1126/science.aad0201]351Science

M. Polini (February 11, 2016) Novoselov, I. V. Grigorieva, L. A. Ponomarenko, A. K. Geim andTomadin, A. Principi, G. H. Auton, E. Khestanova, K. S. D. A. Bandurin, I. Torre, R. Krishna Kumar, M. Ben Shalom, A.grapheneNegative local resistance caused by viscous electron backflow in

Editor's Summary

, this issue p. 1061, 1055, 1058; see also p. 1026Sciencetransport in graphene, a signature of so-called Dirac fluids.

observed a huge increase of thermalet al.fluid flowing through a small opening. Finally, Crossno found evidence in graphene of electron whirlpools similar to those formed by viscouset al.Bandurin

had a major effect on the flow, much like what happens in regular fluids.2fluid in thin wires of PdCoO found that the viscosity of the electronet al.counterexamples (see the Perspective by Zaanen). Moll

flow rarely resembles anything like the familiar flow of water through a pipe, but three groups describe Electrons inside a conductor are often described as flowing in response to an electric field. This

Electrons that flow like a fluid

This copy is for your personal, non-commercial use only.

Article Tools

http://science.sciencemag.org/content/351/6277/1055article tools: Visit the online version of this article to access the personalization and

Permissionshttp://www.sciencemag.org/about/permissions.dtlObtain information about reproducing this article:

is a registered trademark of AAAS. ScienceAdvancement of Science; all rights reserved. The title Avenue NW, Washington, DC 20005. Copyright 2016 by the American Association for thein December, by the American Association for the Advancement of Science, 1200 New York

(print ISSN 0036-8075; online ISSN 1095-9203) is published weekly, except the last weekScience

on

Au

gust

7,

201

6h

ttp

://s

cien

ce.s

cien

cem

ag.o

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Theory vs. experiment

13

Hydrodynamic model(Harvard group) w/ or wo/ the effect of puddle

Holography (two current) HYU-Harvard: physical mechanism

Model: U(1) x U(1)

14

Transport of U(1) x U(1) model

15

Origin of two U(1) model : imbalance effect

• Due to Kinematics, Imbalance exist for a time electron and hole currents have independent chemical potential. (at equilibrium: )

suppressed kinematically due to E,p, conservation.

16

2

FIG. 1: K inematical constraints for imbalance relaxation.The left figure shows the Feynman diagram for the typi-cal two-particle decay process given by Eq. (1.1a); εi andpi respectively denote the energy and momentum of the i

t h

electron or hole. The right figure depicts momentum con-servation for this process. The length of the dashed pathL f ≡ |p2|+ |p3|+ |p4|traced out by the sum of “decay prod-uct” momenta is always greater than or equal to the lengthL i ≡ |p1|of the“parent” particlemomentum: L f ≥ L i . If theparticle energy spectrum takes the form ε(p) = |p|β , then thedepicted decay process is kinematically forbidden for β < 1.For β = 1, only forward-scattering is allowed.

h+ ↔ h+ + h+ + e−. (1.1b)

In graphene, however, these processes are kinematicallysuppressed by the conservation of energy ε and momen-tum p, because the spectrum ε(p) of thequasiparticles isnot decaying

d2ε(p)

d2p≤ 0.

(Negative curvature of the spectrum at T = 0K arises due to the logarithmic renormalization ofthe Fermi velocity vF attributed to electron-electroninteractions.)13,14,15 As explicated in Fig. 1, the linearspectrum allows only decay products with collinear mo-menta (i.e. pure forward scattering), but these processesmake a negligible contribution to the imbalance relax-ation. The sublinear spectrum forbids even this forwardscattering decay. In clean graphene, higher order (e.g.three particle collision) imbalance relaxation processesarealready allowed,whileimpurity-assistedcollisionswillcontribute in a disordered sample; τQ is therefore likelyfinite, although it may significantly exceed other relax-ation times in the system.In the limit of zero relaxation, a graphene monolayerprobed through thermally conducting, electrically insu-lating contacts would possess electron and hole popu-lations that are strictly conserved. In direct analogywith a single component, non-relativistic classical gas,16

the electron-hole plasma in clean graphene with vanish-ing imbalance relaxation would exhibit a finiteelectronicthermal conductivity κ for arbitrarily large system sizes.(The imbalance relaxation length lQ , introduced above,diverges as τQ →∞.) In this regime, interparticle colli-sions facilitate heat conduction without particle numberconvection. In this paper, we will demonstrate that thesamebehavior obtains for non-zero imbalance relaxation(1/lQ > 0) in the limit of short samples, L ≪ lQ . Bycomparison, prior work10 effectively assumed infinite re-

laxation of population imbalance (lQ → 0). We demon-strate that the results previously obtained in Ref. 10 forboth κ and the thermoelectric power α at non-zero dop-ing and temperature emerge in the limit of asymptoti-cally large system sizes L ≫ lQ for a finite rate of im-balance relaxation, (lQ > 0). We will show that whenthe ends of a graphene slab of length L lQ are held atdisparate temperaturesand noelectriccurrent ispermit-ted to flow, steady state particle convection does nev-ertheless occur; carrier flux is created or destroyed byimbalance relaxation processes near the terminals of thedevice. Thermopower measurements require the junc-tion of the graphene slab with metallic contacts. Weincorporate into our calculations carrier exchange withnon-ideal contacts, which also relax the imbalance, andwecarefully delimit the regimesin which deviationsfromthe infinite relaxation limit should be observable in ex-periments. The effects of weak quenched disorder areincluded in all of our computations.

In this paper, we restrict our attention to the hy-drodynamic (or “interaction-limited”) transport regime,whereinelasticinterparticlecollisionsdominateover elas-tic impurity scattering. Prior work addressing ther-moelectric transport in the opposite, “disorder-limited”regime, in which real carrier-carrier scattering processesmay be neglected, includes that of Refs. 11,17,18. Inthe disorder-limited case, κ and α are determined bythe energy-dependenceof the electrical conductivity, viathe “generalized” Wiedemann-Franz law and Mott re-lation, respectively.19 The effect of the slow imbalancerelaxation upon the dc conductivity in graphene undernon-equilibrium interband photoexcitation hasalso beenaddressed.21

What essential new physics emerges through the in-corporation of imbalance relaxation effects into the de-scription of thermoelectric transport, and how can itbe extracted from experiments? The entirety of lineartransport phenomena in graphene within the hydrody-namic regime is essentially quantified by four intrinsicparameters. A finite rate of imbalance relaxation meansthat electrons and holes respond independently to ex-ternal forces; the single “quantum critical” conductiv-ity identified previously in Refs. 8,9,10 generalizes to a2x2 tensor of coefficients, with diagonal elements σee,σhh and off-diagonal elements σeh = σhe; all are me-diated entirely by inelastic interparticle collisions. Thedescription is similar to that of Coulomb drag:22 the di-agonal element σee (σhh ) characterizes the response ofthe conduction band electrons (valence band holes) to a(gedanken) electricfield that couplesonly to that carriertype, whereas σeh characterizes the “drag” exerted byone carrier species upon the other (due to electron-holecollisions) under the application of such a field. σee andσhh are related by particle-hole symmetry, leaving twoindependent parameters which we can take as σeh andσmin ≡ (σee+ σhh−2σeh ); the latter combination isequalto thebulk dcelectrical conductivity σ at theDiracpointin thehydrodynamicregime.12 The imbalance relaxation

Origin of two current 2: E-p conservation

2

FIG. 1: K inematical constraints for imbalance relaxation.The left figure shows the Feynman diagram for the typi-cal two-particle decay process given by Eq. (1.1a); εi andp i respectively denote the energy and momentum of the i

t h

electron or hole. The right figure depicts momentum con-servation for this process. The length of the dashed pathL f ≡ |p2|+ |p3|+ |p4|traced out by the sum of “decay prod-uct” momenta is always greater than or equal to the lengthL i ≡ |p1|of the“parent” particlemomentum: L f ≥ L i . I f theparticle energy spectrum takes the form ε(p) = |p|β , then thedepicted decay process is kinematically forbidden for β < 1.For β = 1, only forward-scattering is allowed.

h+ ↔ h+ + h+ + e−. (1.1b)

In graphene, however, these processes are kinematicallysuppressed by the conservation of energy ε and momen-tum p, because the spectrum ε(p) of the quasiparticles isnot decaying

d2ε(p)

d2p≤ 0.

(Negative curvature of the spectrum at T = 0K arises due to the logarithmic renormalization ofthe Fermi velocity vF attributed to electron-electroninteractions.)13,14,15 As explicated in Fig. 1, the linearspectrum allows only decay products with collinear mo-menta (i.e. pure forward scattering), but these processesmake a negligible contribution to the imbalance relax-ation. The sublinear spectrum forbids even this forwardscattering decay. In clean graphene, higher order (e.g.three particle collision) imbalance relaxation processesarealready allowed,whileimpurity-assisted collisionswillcontribute in a disordered sample; τQ is therefore likelyfinite, although it may significantly exceed other relax-ation times in the system.In the limit of zero relaxation, a graphene monolayer

probed through thermally conducting, electrically insu-lating contacts would possess electron and hole popu-lations that are strictly conserved. In direct analogywith a single component, non-relativistic classical gas,16

the electron-hole plasma in clean graphene with vanish-ing imbalance relaxation would exhibit a finite electronicthermal conductivity κ for arbitrarily large system sizes.(The imbalance relaxation length lQ , introduced above,diverges as τQ → ∞.) In this regime, interparticle colli-sions facilitate heat conduction without particle numberconvection. In this paper, we will demonstrate that thesamebehavior obtains for non-zero imbalance relaxation(1/lQ > 0) in the limit of short samples, L ≪ lQ . Bycomparison, prior work10 effectively assumed infinite re-

laxation of population imbalance (lQ → 0). We demon-strate that the results previously obtained in Ref. 10 forboth κ and the thermoelectric power α at non-zero dop-ing and temperature emerge in the limit of asymptoti-cally large system sizes L ≫ lQ for a finite rate of im-balance relaxation, (lQ > 0). We will show that whenthe ends of a graphene slab of length L lQ are held atdisparate temperaturesand no electriccurrent ispermit-ted to flow, steady state particle convection does nev-ertheless occur; carrier flux is created or destroyed byimbalance relaxation processes near the terminals of thedevice. Thermopower measurements require the junc-tion of the graphene slab with metallic contacts. Weincorporate into our calculations carrier exchange withnon-ideal contacts, which also relax the imbalance, andwecarefully delimit the regimes in which deviations fromthe infinite relaxation limit should be observable in ex-periments. The effects of weak quenched disorder areincluded in all of our computations.

In this paper, we restrict our attention to the hy-drodynamic (or “interaction-limited”) transport regime,whereinelastic interparticlecollisionsdominateover elas-tic impurity scattering. Prior work addressing ther-moelectric transport in the opposite, “disorder-limited”regime, in which real carrier-carrier scattering processesmay be neglected, includes that of Refs. 11,17,18. Inthe disorder-limited case, κ and α are determined bythe energy-dependence of the electrical conductivity, viathe “generalized” Wiedemann-Franz law and Mott re-lation, respectively.19 The effect of the slow imbalancerelaxation upon the dc conductivity in graphene undernon-equilibrium interband photoexcitation hasalso beenaddressed.21

What essential new physics emerges through the in-corporation of imbalance relaxation effects into the de-scription of thermoelectric transport, and how can itbe extracted from experiments? The entirety of lineartransport phenomena in graphene within the hydrody-namic regime is essentially quantified by four intrinsicparameters. A finite rate of imbalance relaxation meansthat electrons and holes respond independently to ex-ternal forces; the single “quantum critical” conductiv-ity identified previously in Refs. 8,9,10 generalizes to a2x2 tensor of coefficients, with diagonal elements σee,σhh and off-diagonal elements σeh = σhe; all are me-diated entirely by inelastic interparticle collisions. Thedescription is similar to that of Coulomb drag:22 the di-agonal element σee (σhh ) characterizes the response ofthe conduction band electrons (valence band holes) to a(gedanken) electric field that couples only to that carriertype, whereas σeh characterizes the “drag” exerted byone carrier species upon the other (due to electron-holecollisions) under the application of such a field. σee andσhh are related by particle-hole symmetry, leaving twoindependent parameters which we can take as σeh andσmin ≡ (σee+ σhh−2σeh ); the latter combination is equalto thebulk dcelectrical conductivity σ at theDiracpointin the hydrodynamic regime.12 The imbalance relaxation

17

2

FIG. 1: K inematical constraints for imbalance relaxation.The left figure shows the Feynman diagram for the typi-cal two-particle decay process given by Eq. (1.1a); εi andpi respectively denote the energy and momentum of the i

t h

electron or hole. The right figure depicts momentum con-servation for this process. The length of the dashed pathL f ≡ |p2|+ |p3|+ |p4|traced out by the sum of “decay prod-uct” momenta is always greater than or equal to the lengthL i ≡ |p1|of the“parent” particlemomentum: L f ≥ L i . I f theparticleenergy spectrum takesthe form ε(p) = |p|β , then thedepicted decay process is kinematically forbidden for β < 1.For β = 1, only forward-scattering is allowed.

h+ ↔ h+ + h+ + e−. (1.1b)

In graphene, however, these processes are kinematicallysuppressed by the conservation of energy ε and momen-tum p, because thespectrum ε(p) of thequasiparticles isnot decaying

d2ε(p)

d2p≤ 0.

(Negative curvature of the spectrum at T = 0K arises due to the logarithmic renormalization ofthe Fermi velocity vF attributed to electron-electroninteractions.)13,14,15 As explicated in Fig. 1, the linearspectrum allows only decay products with collinear mo-menta (i.e. pure forward scattering), but theseprocessesmake a negligible contribution to the imbalance relax-ation. The sublinear spectrum forbids even this forwardscattering decay. In clean graphene, higher order (e.g.three particle collision) imbalance relaxation processesarealready allowed,whileimpurity-assistedcollisionswillcontribute in a disordered sample; τQ is therefore likelyfinite, although it may significantly exceed other relax-ation times in the system.In the limit of zero relaxation, a graphene monolayerprobed through thermally conducting, electrically insu-lating contacts would possess electron and hole popu-lations that are strictly conserved. In direct analogywith a single component, non-relativistic classical gas,16

the electron-hole plasma in clean graphenewith vanish-ing imbalancerelaxation would exhibit a finiteelectronicthermal conductivity κ for arbitrarily largesystem sizes.(The imbalance relaxation length lQ , introduced above,diverges as τQ →∞.) In this regime, interparticle colli-sions facilitate heat conduction without particle numberconvection. In this paper, we will demonstrate that thesamebehavior obtains for non-zero imbalance relaxation(1/lQ > 0) in the limit of short samples, L ≪ lQ . Bycomparison, prior work10 effectively assumed infinite re-

laxation of population imbalance (lQ → 0). We demon-strate that the results previously obtained in Ref. 10 forboth κ and the thermoelectric power α at non-zero dop-ing and temperature emerge in the limit of asymptoti-cally large system sizes L ≫ lQ for a finite rate of im-balance relaxation, (lQ > 0). We will show that whenthe ends of a grapheneslab of length L lQ are held atdisparatetemperaturesand noelectriccurrent ispermit-ted to flow, steady state particle convection does nev-ertheless occur; carrier flux is created or destroyed byimbalance relaxation processesnear the terminals of thedevice. Thermopower measurements require the junc-tion of the graphene slab with metallic contacts. Weincorporate into our calculations carrier exchange withnon-ideal contacts, which also relax the imbalance, andwecarefully delimit theregimesin which deviationsfromthe infinite relaxation limit should be observable in ex-periments. The effects of weak quenched disorder areincluded in all of our computations.

In this paper, we restrict our attention to the hy-drodynamic (or “interaction-limited”) transport regime,whereinelasticinterparticlecollisionsdominateover elas-tic impurity scattering. Prior work addressing ther-moelectric transport in the opposite, “disorder-limited”regime, in which real carrier-carrier scattering processesmay be neglected, includes that of Refs. 11,17,18. Inthe disorder-limited case, κ and α are determined bythe energy-dependenceof the electrical conductivity, viathe “generalized” Wiedemann-Franz law and Mott re-lation, respectively.19 The effect of the slow imbalancerelaxation upon the dc conductivity in graphene undernon-equilibrium interband photoexcitation hasalsobeenaddressed.21

What essential new physics emerges through the in-corporation of imbalance relaxation effects into the de-scription of thermoelectric transport, and how can itbe extracted from experiments? The entirety of lineartransport phenomena in graphene within the hydrody-namic regime is essentially quantified by four intrinsicparameters. A finite rate of imbalance relaxation meansthat electrons and holes respond independently to ex-ternal forces; the single “quantum critical” conductiv-ity identified previously in Refs. 8,9,10 generalizes to a2x2 tensor of coefficients, with diagonal elements σee,σhh and off-diagonal elements σeh = σhe; all are me-diated entirely by inelastic interparticle collisions. Thedescription is similar to that of Coulomb drag:22 the di-agonal element σee (σhh ) characterizes the response ofthe conduction band electrons (valence band holes) to a(gedanken) electricfield that couplesonly to that carriertype, whereas σeh characterizes the “drag” exerted byone carrier species upon the other (due to electron-holecollisions) under the application of such a field. σee andσhh are related by particle-hole symmetry, leaving twoindependent parameters which we can take as σeh andσmin ≡ (σee+ σhh−2σeh ); the latter combination isequalto thebulk dcelectrical conductivity σ at theDiracpointin thehydrodynamicregime.12 The imbalancerelaxation

2

FIG. 1: K inematical constraints for imbalance relaxation.The left figure shows the Feynman diagram for the typi-cal two-particle decay process given by Eq. (1.1a); εi andpi respectively denote the energy and momentum of the i

t h

electron or hole. The right figure depicts momentum con-servation for this process. The length of the dashed pathL f ≡ |p2|+ |p3|+ |p4|traced out by the sum of “decay prod-uct” momenta is always greater than or equal to the lengthL i ≡ |p1|of the“parent” particlemomentum: L f ≥ L i . I f theparticle energy spectrum takes the form ε(p) = |p|β , then thedepicted decay process is kinematically forbidden for β < 1.For β = 1, only forward-scattering is allowed.

h+ ↔ h+ + h+ + e−. (1.1b)

In graphene, however, these processes are kinematicallysuppressed by the conservation of energy ε and momen-tum p, because thespectrum ε(p) of thequasiparticles isnot decaying

d2ε(p)

d2p≤ 0.

(Negative curvature of the spectrum at T = 0K arises due to the logarithmic renormalization ofthe Fermi velocity vF attributed to electron-electroninteractions.)13,14,15 As explicated in Fig. 1, the linearspectrum allows only decay products with collinear mo-menta (i.e. pure forward scattering), but these processesmake a negligible contribution to the imbalance relax-ation. The sublinear spectrum forbids even this forwardscattering decay. In clean graphene, higher order (e.g.three particle collision) imbalance relaxation processesarealready allowed,whileimpurity-assistedcollisionswillcontribute in a disordered sample; τQ is therefore likelyfinite, although it may significantly exceed other relax-ation times in the system.In the limit of zero relaxation, a graphene monolayerprobed through thermally conducting, electrically insu-lating contacts would possess electron and hole popu-lations that are strictly conserved. In direct analogywith a single component, non-relativistic classical gas,16

the electron-hole plasma in clean graphenewith vanish-ing imbalance relaxation would exhibit a finiteelectronicthermal conductivity κ for arbitrarily largesystem sizes.(The imbalance relaxation length lQ , introduced above,diverges as τQ →∞.) In this regime, interparticle colli-sions facilitate heat conduction without particle numberconvection. In this paper, we will demonstrate that thesamebehavior obtains for non-zero imbalance relaxation(1/lQ > 0) in the limit of short samples, L ≪ lQ . Bycomparison, prior work10 effectively assumed infinite re-

laxation of population imbalance (lQ → 0). We demon-strate that the results previously obtained in Ref. 10 forboth κ and the thermoelectric power α at non-zero dop-ing and temperature emerge in the limit of asymptoti-cally large system sizes L ≫ lQ for a finite rate of im-balance relaxation, (lQ > 0). We will show that whenthe ends of a grapheneslab of length L lQ are held atdisparatetemperaturesand noelectriccurrent ispermit-ted to flow, steady state particle convection does nev-ertheless occur; carrier flux is created or destroyed byimbalance relaxation processesnear the terminals of thedevice. Thermopower measurements require the junc-tion of the graphene slab with metallic contacts. Weincorporate into our calculations carrier exchange withnon-ideal contacts, which also relax the imbalance, andwecarefully delimit the regimesin which deviationsfromthe infinite relaxation limit should be observable in ex-periments. The effects of weak quenched disorder areincluded in all of our computations.

In this paper, we restrict our attention to the hy-drodynamic (or “interaction-limited”) transport regime,whereinelasticinterparticlecollisionsdominateover elas-tic impurity scattering. Prior work addressing ther-moelectric transport in the opposite, “disorder-limited”regime, in which real carrier-carrier scattering processesmay be neglected, includes that of Refs. 11,17,18. Inthe disorder-limited case, κ and α are determined bythe energy-dependenceof the electrical conductivity, viathe “generalized” Wiedemann-Franz law and Mott re-lation, respectively.19 The effect of the slow imbalancerelaxation upon the dc conductivity in graphene undernon-equilibrium interband photoexcitation hasalsobeenaddressed.21

What essential new physics emerges through the in-corporation of imbalance relaxation effects into the de-scription of thermoelectric transport, and how can itbe extracted from experiments? The entirety of lineartransport phenomena in graphene within the hydrody-namic regime is essentially quantified by four intrinsicparameters. A finite rate of imbalance relaxation meansthat electrons and holes respond independently to ex-ternal forces; the single “quantum critical” conductiv-ity identified previously in Refs. 8,9,10 generalizes to a2x2 tensor of coefficients, with diagonal elements σee,σhh and off-diagonal elements σeh = σhe; all are me-diated entirely by inelastic interparticle collisions. Thedescription is similar to that of Coulomb drag:22 the di-agonal element σee (σhh ) characterizes the response ofthe conduction band electrons (valence band holes) to a(gedanken) electricfield that couplesonly to that carriertype, whereas σeh characterizes the “drag” exerted byone carrier species upon the other (due to electron-holecollisions) under the application of such a field. σee andσhh are related by particle-hole symmetry, leaving twoindependent parameters which we can take as σeh andσmin ≡ (σee+ σhh−2σeh ); the latter combination isequalto thebulk dcelectrical conductivity σ at theDiracpointin thehydrodynamicregime.12 The imbalance relaxation

2

FIG. 1: K inematical constraints for imbalance relaxation.The left figure shows the Feynman diagram for the typi-cal two-particle decay process given by Eq. (1.1a); εi andpi respectively denote the energy and momentum of the i

t h

electron or hole. The right figure depicts momentum con-servation for this process. The length of the dashed pathL f ≡ |p2|+ |p3|+ |p4|traced out by the sum of “decay prod-uct” momenta is always greater than or equal to the lengthL i ≡ |p1|of the“parent” particlemomentum: L f ≥ L i . I f theparticleenergy spectrum takesthe form ε(p) = |p|β , then thedepicted decay process is kinematically forbidden for β < 1.For β = 1, only forward-scattering is allowed.

h+ ↔ h+ + h+ + e−. (1.1b)

In graphene, however, these processes are kinematicallysuppressed by the conservation of energy ε and momen-tum p, because thespectrum ε(p) of thequasiparticles isnot decaying

d2ε(p)

d2p≤ 0.

(Negative curvature of the spectrum at T = 0K arises due to the logarithmic renormalization ofthe Fermi velocity vF attributed to electron-electroninteractions.)13,14,15 As explicated in Fig. 1, the linearspectrum allows only decay products with collinear mo-menta (i.e. pure forward scattering), but theseprocessesmake a negligible contribution to the imbalance relax-ation. The sublinear spectrum forbids even this forwardscattering decay. In clean graphene, higher order (e.g.three particle collision) imbalance relaxation processesarealready allowed,whileimpurity-assistedcollisionswillcontribute in a disordered sample; τQ is therefore likelyfinite, although it may significantly exceed other relax-ation times in the system.In the limit of zero relaxation, a graphene monolayerprobed through thermally conducting, electrically insu-lating contacts would possess electron and hole popu-lations that are strictly conserved. In direct analogywith a single component, non-relativistic classical gas,16

the electron-hole plasma in clean graphenewith vanish-ing imbalancerelaxation would exhibit a finiteelectronicthermal conductivity κ for arbitrarily largesystem sizes.(The imbalance relaxation length lQ , introduced above,diverges as τQ →∞.) In this regime, interparticle colli-sions facilitate heat conduction without particle numberconvection. In this paper, we will demonstrate that thesamebehavior obtains for non-zero imbalance relaxation(1/lQ > 0) in the limit of short samples, L ≪ lQ . Bycomparison, prior work10 effectively assumed infinite re-

laxation of population imbalance (lQ → 0). We demon-strate that the results previously obtained in Ref. 10 forboth κ and the thermoelectric power α at non-zero dop-ing and temperature emerge in the limit of asymptoti-cally large system sizes L ≫ lQ for a finite rate of im-balance relaxation, (lQ > 0). We will show that whenthe ends of a grapheneslab of length L lQ are held atdisparatetemperaturesand noelectriccurrent ispermit-ted to flow, steady state particle convection does nev-ertheless occur; carrier flux is created or destroyed byimbalance relaxation processesnear the terminals of thedevice. Thermopower measurements require the junc-tion of the graphene slab with metallic contacts. Weincorporate into our calculations carrier exchange withnon-ideal contacts, which also relax the imbalance, andwecarefully delimit theregimesin which deviationsfromthe infinite relaxation limit should be observable in ex-periments. The effects of weak quenched disorder areincluded in all of our computations.

In this paper, we restrict our attention to the hy-drodynamic (or “interaction-limited”) transport regime,whereinelasticinterparticlecollisionsdominateover elas-tic impurity scattering. Prior work addressing ther-moelectric transport in the opposite, “disorder-limited”regime, in which real carrier-carrier scattering processesmay be neglected, includes that of Refs. 11,17,18. Inthe disorder-limited case, κ and α are determined bythe energy-dependenceof the electrical conductivity, viathe “generalized” Wiedemann-Franz law and Mott re-lation, respectively.19 The effect of the slow imbalancerelaxation upon the dc conductivity in graphene undernon-equilibrium interband photoexcitation hasalsobeenaddressed.21

What essential new physics emerges through the in-corporation of imbalance relaxation effects into the de-scription of thermoelectric transport, and how can itbe extracted from experiments? The entirety of lineartransport phenomena in graphene within the hydrody-namic regime is essentially quantified by four intrinsicparameters. A finite rate of imbalance relaxation meansthat electrons and holes respond independently to ex-ternal forces; the single “quantum critical” conductiv-ity identified previously in Refs. 8,9,10 generalizes to a2x2 tensor of coefficients, with diagonal elements σee,σhh and off-diagonal elements σeh = σhe; all are me-diated entirely by inelastic interparticle collisions. Thedescription is similar to that of Coulomb drag:22 the di-agonal element σee (σhh ) characterizes the response ofthe conduction band electrons (valence band holes) to a(gedanken) electricfield that couplesonly to that carriertype, whereas σeh characterizes the “drag” exerted byone carrier species upon the other (due to electron-holecollisions) under the application of such a field. σee andσhh are related by particle-hole symmetry, leaving twoindependent parameters which we can take as σeh andσmin ≡ (σee+ σhh−2σeh ); the latter combination isequalto thebulk dcelectrical conductivity σ at theDiracpointin thehydrodynamicregime.12 The imbalancerelaxation

2

FIG. 1: Kinematical constraints for imbalance relaxation.The left figure shows the Feynman diagram for the typi-cal two-particle decay process given by Eq. (1.1a); εi andpi respectively denote the energy and momentum of the i

t h

electron or hole. The right figure depicts momentum con-servation for this process. The length of the dashed pathL f ≡ |p2|+ |p3|+ |p4|traced out by thesum of “decay prod-uct” momenta is always greater than or equal to the lengthL i ≡ |p1|of the“parent” particlemomentum: L f ≥ L i . If theparticleenergy spectrum takestheform ε(p) = |p|β , then thedepicted decay process is kinematically forbidden for β < 1.For β = 1, only forward-scattering is allowed.

h+ ↔ h+ + h+ + e−. (1.1b)

In graphene, however, these processes are kinematicallysuppressed by the conservation of energy ε and momen-tum p, becausethespectrum ε(p) of thequasiparticles isnot decaying

d2ε(p)

d2p≤0.

(Negative curvature of the spectrum at T = 0K arises due to the logarithmic renormalization ofthe Fermi velocity vF attributed to electron-electroninteractions.)13,14,15 As explicated in Fig. 1, the linearspectrum allowsonly decay productswith collinear mo-menta (i.e. pure forward scattering), but theseprocessesmake a negligible contribution to the imbalance relax-ation. The sublinear spectrum forbids even this forwardscattering decay. In clean graphene, higher order (e.g.three particle collision) imbalance relaxation processesarealready allowed,whileimpurity-assistedcollisionswillcontribute in a disordered sample; τQ is therefore likelyfinite, although it may significantly exceed other relax-ation times in the system.In the limit of zero relaxation, a graphenemonolayerprobed through thermally conducting, electrically insu-lating contacts would possess electron and hole popu-lations that are strictly conserved. In direct analogywith a single component, non-relativistic classical gas,16

the electron-holeplasma in clean graphenewith vanish-ing imbalancerelaxation would exhibit a finiteelectronicthermal conductivity κ for arbitrarily largesystem sizes.(The imbalance relaxation length lQ , introduced above,divergesas τQ →∞.) In this regime, interparticle colli-sions facilitate heat conduction without particle numberconvection. In this paper, wewill demonstrate that thesamebehavior obtainsfor non-zero imbalancerelaxation(1/lQ > 0) in the limit of short samples, L ≪ lQ. Bycomparison, prior work10 effectively assumed infinite re-

laxation of population imbalance (lQ → 0). We demon-strate that the results previously obtained in Ref. 10 forboth κ and the thermoelectricpower α at non-zerodop-ing and temperature emerge in the limit of asymptoti-cally large system sizes L ≫ lQ for a finite rate of im-balance relaxation, (lQ > 0). We will show that whenthe endsof a grapheneslab of length L lQ areheld atdisparatetemperaturesand noelectriccurrent ispermit-ted to flow, steady state particle convection does nev-ertheless occur; carrier flux is created or destroyed byimbalance relaxation processesnear the terminalsof thedevice. Thermopower measurements require the junc-tion of the graphene slab with metallic contacts. Weincorporate into our calculations carrier exchange withnon-ideal contacts, which also relax the imbalance, andwecarefully delimit theregimesin which deviationsfromthe infinite relaxation limit should be observable in ex-periments. The effects of weak quenched disorder areincluded in all of our computations.

In this paper, we restrict our attention to the hy-drodynamic (or “interaction-limited”) transport regime,whereinelasticinterparticlecollisionsdominateover elas-tic impurity scattering. Prior work addressing ther-moelectric transport in the opposite, “disorder-limited”regime, in which real carrier-carrier scattering processesmay be neglected, includes that of Refs. 11,17,18. Inthe disorder-limited case, κ and α are determined bythe energy-dependenceof the electrical conductivity, viathe “generalized” Wiedemann-Franz law and Mott re-lation, respectively.19 The effect of the slow imbalancerelaxation upon the dc conductivity in graphene undernon-equilibrium interband photoexcitation hasalsobeenaddressed.21

What essential new physics emerges through the in-corporation of imbalance relaxation effects into the de-scription of thermoelectric transport, and how can itbe extracted from experiments? The entirety of lineartransport phenomena in graphene within the hydrody-namic regime is essentially quantified by four intrinsicparameters. A finite rateof imbalance relaxation meansthat electrons and holes respond independently to ex-ternal forces; the single “quantum critical” conductiv-ity identified previously in Refs. 8,9,10 generalizes to a2x2 tensor of coefficients, with diagonal elements σee,σhh and off-diagonal elements σeh = σhe; all are me-diated entirely by inelastic interparticle collisions. Thedescription is similar to that of Coulomb drag:22 the di-agonal element σee (σhh) characterizes the response ofthe conduction band electrons (valence band holes) to a(gedanken) electricfield that couplesonly to that carriertype, whereas σeh characterizes the “drag” exerted byone carrier species upon the other (due to electron-holecollisions) under the application of such a field. σee andσhh are related by particle-hole symmetry, leaving twoindependent parameters which we can take as σeh andσmin ≡ (σee+ σhh−2σeh); thelatter combination isequaltothebulk dcelectrical conductivity σ at theDiracpointin thehydrodynamicregime.12 Theimbalancerelaxation

2

FIG. 1: K inematical constraints for imbalance relaxation.The left figure shows the Feynman diagram for the typi-cal two-particle decay process given by Eq. (1.1a); εi andpi respectively denote the energy and momentum of the i

t h

electron or hole. The right figure depicts momentum con-servation for this process. The length of the dashed pathL f ≡ |p2|+ |p3|+ |p4|traced out by thesum of “decay prod-uct” momenta is always greater than or equal to the lengthL i ≡ |p1|of the“parent” particlemomentum: L f ≥ L i . If theparticleenergy spectrum takestheform ε(p) = |p|β , then thedepicted decay process is kinematically forbidden for β < 1.For β = 1, only forward-scattering is allowed.

h+ ↔ h+ + h+ + e−. (1.1b)

In graphene, however, these processes are kinematicallysuppressed by the conservation of energy ε and momen-tum p, becausethespectrum ε(p) of thequasiparticles isnot decaying

d2ε(p)

d2p≤0.

(Negative curvature of the spectrum at T = 0K arises due to the logarithmic renormalization ofthe Fermi velocity vF attributed to electron-electroninteractions.)13,14,15 As explicated in Fig. 1, the linearspectrum allowsonly decay productswith collinear mo-menta (i.e. pure forward scattering), but theseprocessesmake a negligible contribution to the imbalance relax-ation. The sublinear spectrum forbids even this forwardscattering decay. In clean graphene, higher order (e.g.three particle collision) imbalance relaxation processesarealready allowed,whileimpurity-assistedcollisionswillcontribute in a disordered sample; τQ is therefore likelyfinite, although it may significantly exceed other relax-ation times in the system.In the limit of zero relaxation, a graphenemonolayerprobed through thermally conducting, electrically insu-lating contacts would possess electron and hole popu-lations that are strictly conserved. In direct analogywith a single component, non-relativistic classical gas,16

the electron-hole plasma in clean graphenewith vanish-ing imbalancerelaxationwould exhibit a finiteelectronicthermal conductivity κ for arbitrarily largesystem sizes.(The imbalance relaxation length lQ , introduced above,diverges as τQ →∞.) In this regime, interparticle colli-sions facilitate heat conduction without particle numberconvection. In this paper, wewill demonstrate that thesamebehavior obtains for non-zero imbalancerelaxation(1/lQ > 0) in the limit of short samples, L ≪ lQ. Bycomparison, prior work10 effectively assumed infinite re-

laxation of population imbalance (lQ → 0). We demon-strate that the results previously obtained in Ref. 10 forboth κ and the thermoelectricpower α at non-zero dop-ing and temperature emerge in the limit of asymptoti-cally large system sizes L ≫ lQ for a finite rate of im-balance relaxation, (lQ > 0). We will show that whenthe endsof a grapheneslab of length L lQ areheld atdisparatetemperaturesand noelectriccurrent ispermit-ted to flow, steady state particle convection does nev-ertheless occur; carrier flux is created or destroyed byimbalance relaxation processesnear the terminals of thedevice. Thermopower measurements require the junc-tion of the graphene slab with metallic contacts. Weincorporate into our calculations carrier exchange withnon-ideal contacts, which also relax the imbalance, andwecarefully delimit theregimesin which deviationsfromthe infinite relaxation limit should be observable in ex-periments. The effects of weak quenched disorder areincluded in all of our computations.

In this paper, we restrict our attention to the hy-drodynamic (or “interaction-limited”) transport regime,whereinelasticinterparticlecollisionsdominateover elas-tic impurity scattering. Prior work addressing ther-moelectric transport in the opposite, “disorder-limited”regime, in which real carrier-carrier scattering processesmay be neglected, includes that of Refs. 11,17,18. Inthe disorder-limited case, κ and α are determined bythe energy-dependenceof the electrical conductivity, viathe “generalized” Wiedemann-Franz law and Mott re-lation, respectively.19 The effect of the slow imbalancerelaxation upon the dc conductivity in graphene undernon-equilibrium interband photoexcitation hasalsobeenaddressed.21

What essential new physics emerges through the in-corporation of imbalance relaxation effects into the de-scription of thermoelectric transport, and how can itbe extracted from experiments? The entirety of lineartransport phenomena in graphene within the hydrody-namic regime is essentially quantified by four intrinsicparameters. A finite rateof imbalance relaxation meansthat electrons and holes respond independently to ex-ternal forces; the single “quantum critical” conductiv-ity identified previously in Refs. 8,9,10 generalizes to a2x2 tensor of coefficients, with diagonal elements σee,σhh and off-diagonal elements σeh = σhe; all are me-diated entirely by inelastic interparticle collisions. Thedescription is similar to that of Coulomb drag:22 the di-agonal element σee (σhh) characterizes the response ofthe conduction band electrons (valence band holes) to a(gedanken) electricfield that couplesonly to that carriertype, whereas σeh characterizes the “drag” exerted byone carrier species upon the other (due to electron-holecollisions) under the application of such a field. σee andσhh are related by particle-hole symmetry, leaving twoindependent parameters which we can take as σeh andσmin ≡ (σee+ σhh−2σeh); thelatter combination isequaltothebulk dcelectrical conductivity σ at theDiracpointin thehydrodynamicregime.12 Theimbalancerelaxation

2

FIG. 1: K inematical constraints for imbalance relaxation.The left figure shows the Feynman diagram for the typi-cal two-particle decay process given by Eq. (1.1a); εi andpi respectively denote the energy and momentum of the i

t h

electron or hole. The right figure depicts momentum con-servation for this process. The length of the dashed pathL f ≡ |p2|+ |p3|+ |p4|traced out by thesum of “decay prod-uct” momenta is always greater than or equal to the lengthL i ≡ |p1|of the“parent” particlemomentum: L f ≥ L i . If theparticleenergy spectrum takes the form ε(p) = |p|β , then thedepicted decay process is kinematically forbidden for β < 1.For β = 1, only forward-scattering is allowed.

h+ ↔ h+ + h+ + e−. (1.1b)

In graphene, however, these processes are kinematicallysuppressed by the conservation of energy ε and momen-tum p, because thespectrum ε(p) of thequasiparticles isnot decaying

d2ε(p)

d2p≤ 0.

(Negative curvature of the spectrum at T = 0K arises due to the logarithmic renormalization ofthe Fermi velocity vF attributed to electron-electroninteractions.)13,14,15 As explicated in Fig. 1, the linearspectrum allows only decay productswith collinear mo-menta (i.e. pure forward scattering), but theseprocessesmake a negligible contribution to the imbalance relax-ation. The sublinear spectrum forbids even this forwardscattering decay. In clean graphene, higher order (e.g.three particle collision) imbalance relaxation processesarealready allowed,whileimpurity-assistedcollisionswillcontribute in a disordered sample; τQ is therefore likelyfinite, although it may significantly exceed other relax-ation times in the system.In the limit of zero relaxation, a graphene monolayerprobed through thermally conducting, electrically insu-lating contacts would possess electron and hole popu-lations that are strictly conserved. In direct analogywith a single component, non-relativistic classical gas,16

the electron-hole plasma in clean graphenewith vanish-ing imbalancerelaxation would exhibit a finiteelectronicthermal conductivity κ for arbitrarily largesystem sizes.(The imbalance relaxation length lQ , introduced above,diverges as τQ →∞.) In this regime, interparticle colli-sions facilitate heat conduction without particle numberconvection. In this paper, we will demonstrate that thesamebehavior obtains for non-zero imbalance relaxation(1/lQ > 0) in the limit of short samples, L ≪ lQ . Bycomparison, prior work10 effectively assumed infinite re-

laxation of population imbalance (lQ → 0). We demon-strate that the results previously obtained in Ref. 10 forboth κ and the thermoelectric power α at non-zero dop-ing and temperature emerge in the limit of asymptoti-cally large system sizes L ≫ lQ for a finite rate of im-balance relaxation, (lQ > 0). We will show that whenthe ends of a grapheneslab of length L lQ are held atdisparatetemperaturesand noelectriccurrent ispermit-ted to flow, steady state particle convection does nev-ertheless occur; carrier flux is created or destroyed byimbalance relaxation processesnear the terminals of thedevice. Thermopower measurements require the junc-tion of the graphene slab with metallic contacts. Weincorporate into our calculations carrier exchange withnon-ideal contacts, which also relax the imbalance, andwecarefully delimit theregimesin which deviationsfromthe infinite relaxation limit should be observable in ex-periments. The effects of weak quenched disorder areincluded in all of our computations.

In this paper, we restrict our attention to the hy-drodynamic (or “interaction-limited”) transport regime,whereinelasticinterparticlecollisionsdominateover elas-tic impurity scattering. Prior work addressing ther-moelectric transport in the opposite, “disorder-limited”regime, in which real carrier-carrier scattering processesmay be neglected, includes that of Refs. 11,17,18. Inthe disorder-limited case, κ and α are determined bythe energy-dependenceof the electrical conductivity, viathe “generalized” Wiedemann-Franz law and Mott re-lation, respectively.19 The effect of the slow imbalancerelaxation upon the dc conductivity in graphene undernon-equilibrium interband photoexcitation hasalsobeenaddressed.21

What essential new physics emerges through the in-corporation of imbalance relaxation effects into the de-scription of thermoelectric transport, and how can itbe extracted from experiments? The entirety of lineartransport phenomena in graphene within the hydrody-namic regime is essentially quantified by four intrinsicparameters. A finite rate of imbalance relaxation meansthat electrons and holes respond independently to ex-ternal forces; the single “quantum critical” conductiv-ity identified previously in Refs. 8,9,10 generalizes to a2x2 tensor of coefficients, with diagonal elements σee,σhh and off-diagonal elements σeh = σhe; all are me-diated entirely by inelastic interparticle collisions. Thedescription is similar to that of Coulomb drag:22 the di-agonal element σee (σhh) characterizes the response ofthe conduction band electrons (valence band holes) to a(gedanken) electricfield that couplesonly to that carriertype, whereas σeh characterizes the “drag” exerted byone carrier species upon the other (due to electron-holecollisions) under the application of such a field. σee andσhh are related by particle-hole symmetry, leaving twoindependent parameters which we can take as σeh andσmin ≡ (σee+ σhh−2σeh); the latter combination isequalto thebulk dcelectrical conductivity σ at theDiracpointin thehydrodynamicregime.12 The imbalancerelaxation

For beta=2, three momentum 2,3,4 on the sphere of diameter p1.

In graphenebeta<1 due to logarithmic renormalization of fermi velocity v*: E=v* p

Foster+Almeiner (0810.4342)

Why two charge density proportional?

• Still not a resolved issue.

18

A few physics questions

• i) Why Graphene works for holography? Z=1

• ii) Why axion works? Conductivity does not see inhomogeneity.

• Iii) Why hydrodynamics works for strong coupling?

19

i) Why Graphene works for Holography?

20

Fermi level ~ near Dirac point : imperfect screening strong correlation

Dirac Fluid (i.e QCP with z=1)

Iii) Why hydrodynamics work for strong coupling?

• This is question of Rapid thermalization for strong interaction.

• Question of Plankian dissipation:

• Extreme example: in RHIC , fireball equilibrate just in passing time:

21

AdS dual of the question

• Thermalization Formation of AdS BH

• Does a generic (non-spherical) collapse in AdS give a black hole in one dynamical time.

In the talk of Bin Wang,

In flat space or de Sitter, it is impossible to makea BH in one dynamical time from generic initial configuration.

However, In AdS space

• Claim: Generic collapse will give a black hole. Oh+ SJS 1302.1277; SJS 1310.7179

period is initial position independent!

Figure 2. Left: r (t) v.s t: Fallingof twoshellswith di↵erent energy scales. Each shell has thickness

of 10 % of its average energy. The higher the average position, the sooner the thickness reduces;

Right: Hydro-nization time as function of initial height of the shell. The higher is the energy, the

sooner it becomes isotropic.

but never overtakes. Therefore, we can conclude that soft modes thermalize first and then

hard modes thermalize on top of the former.

The thermalization time is simply given by:

tT H =

s1

r 2H−1

r 20=1

⇡T·

r

1−(⇡Tm)2

E2. (5.2)

where rH , and T aretheradiusof the ‘would-be-horizon’ and itscorresponding temperature

which will be reached by the system after thermalization. r0 is the initial height of the

particle with highest energy. If we consider the initial energy distribution of particles in

thesystem such that it isnot concentrated on specificenergy scale, then thehighest energy

scale is much bigger than the temperature. If rH < < r0 the thermalization time is given

by the inverse temperature,

tT h '1

rH=1

⇡T, (5.3)

Otherwise, thermalization time should be less than this.

Now look at the falling of two particles whose initial heights are order of magnitude

di↵erent. Although there is no passing, particles with higher radial position almost catch

up the lower particles within half of the thermalization time. Such phenomena strongly

suggests that approximate isotropization happens much before the actual thermalization.

See figure 2.

To an external observer, after rapid process of isotropization, not much change will

happen although the shell is still falling. Only residual process of isotropization is under

progress. We can identify such stage as hydronized state. This is the regime where the

metric is still time dependent but hydrodynamics works e↵ectively. One can quantify the

hydro-nization time thy as follows: Take a shell of thickness ∆ r0 := r0,max − r0,mi n which

is, say, 10% of its average height r0. This amounts to taking the particles whose initial

energy distribution width is∆ E/ E = 10% around itsaveragevalueE . Wecan say that the

– 7 –

Figure 2. Left: r (t) v.s t: Fallingof twoshellswith di↵erent energy scales. Each shell has thickness

of 10 % of its average energy. The higher the average position, the sooner the thickness reduces;

Right: Hydro-nization time as function of initial height of the shell. The higher is the energy, the

sooner it becomes isotropic.

but never overtakes. Therefore, we can conclude that soft modes thermalize first and then

hard modes thermalize on top of the former.

The thermalization time is simply given by:

tT H =

s1

r 2H−1

r 20=1

⇡T·

r

1−(⇡Tm)2

E2. (5.2)

where rH , andT aretheradiusof the ‘would-be-horizon’ and itscorresponding temperature

which will be reached by the system after thermalization. r0 is the initial height of the

particle with highest energy. If we consider the initial energy distribution of particles in

thesystem such that it isnot concentrated on specificenergy scale, then thehighest energy

scale is much bigger than the temperature. If rH < < r0 the thermalization time is given

by the inverse temperature,

tT h '1

rH=1

⇡T, (5.3)

Otherwise, thermalization time should be less than this.

Now look at the falling of two particles whose initial heights are order of magnitude

di↵erent. Although there is no passing, particles with higher radial position almost catch

up the lower particles within half of the thermalization time. Such phenomena strongly

suggests that approximate isotropization happens much before the actual thermalization.

See figure 2.

To an external observer, after rapid process of isotropization, not much change will

happen although the shell is still falling. Only residual process of isotropization is under

progress. We can identify such stage as hydronized state. This is the regime where the

metric is still time dependent but hydrodynamics works e↵ectively. One can quantify the

hydro-nization time thy as follows: Take a shell of thickness ∆ r0 := r0,max − r0,min which

is, say, 10% of its average height r0. This amounts to taking the particles whose initial

energy distribution width is∆ E/ E = 10% around itsaveragevalueE . Wecan say that the

– 7 –

time

3. Graphite Magnetization

24

How to characterize a specific system in holography?

Based on

25

Diamagnetism in annealed graphite

SEO, KY.Kim, KK.Kim, Sin arXiv:1512.08916

Data from

26

magnetization

• ee int is already included by gravity but

e-lattice should be encoded explicitly.

• 1. Integrate out massive fermion in 2+1 Chern-Simon 2. Lift it to the bulk F^F 3. Axionize for dynamical effect

time reversal invariant case time reversal Sym broken case

• Focus on T-broken case.

Magnetization/Anomalous Hall effect /Pseudo gap/

• Basic interaction is known to be Spin-Orbit interaction

How to encode the system specific information?

27

Model

• Chern-Simon Axion term (with TRS broken) can represent magneto-electric phenomena

• Dopping controls magnetic property as well as charge transport.

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Background solution (exact!)

29

4. Anomalous Hall Effect

Experimental result is shown in right figure

For Intrinsic deflection / Side jump

For skew scattering

30

Known Theory for anmalous Hall effect (from Nagaosa et.al,

)

31

Holographic calculation of Exact DC transports

32

Our results from holography

33

Sharper transition at higher T

34

5. Pseudo Gap in optical conductivity

35

36

Fluctuations

EOMs in

Boundary action

Independent

Fluctuation dynamics

Boundary Condition

Infalling

Dirichlett

Boudnary value problem

Determined by horizon regularity by the first three

37

Use Method developed in Kim,Kim,Seo,Sin(1409.8346)

for AC (electric,thermoelectric,thermal) conductivities

Figure 4. Relaxation time⌧at small ! as a function of µ/ T and β/ T . We do not plot the rangeβ/ T < 1 since⌧diverges quickly as β goes to zero.

production, σQ, which is also a↵ected by charge density. Once we assume (4.19), three

parameters K ,⌧, and σQ can be fixed by considering two limits.13

First, in the limit ⌧! 1 (β ! 0) σQ and K can be read o↵ from (3.17)

σQ =

0

@3− µ2

4r 20

3+ 3µ2

4r 20

1

A

2

, K = r0

µ2

r 20

3+ 3µ2

4r 20

, (4.20)

where

r0 =2⇡

3

⇣T +

pT2 + 3(µ/ 4⇡ )2 + 6(β/ 4⇡ )2

⌘, (4.21)

which is defined in (2.19).

Next, in the limit ! ! 0 with finite β

σ(! ! 0) = K ⌧+ σQ = 1+µ2

β2, (4.22)

where (2.29) is used. Therefore, the relaxation time⌧reads

⌧=1+ µ2

β2− σQ

K(4.23)

=1

4⇡T·45β̃4 + 36µ̃4 + 2(1+ ∆ ) + 6β̃2(4+ 12µ̃2 + 3∆ ) + 3µ̃2(5+ 4∆ )

β̃2(1+ ∆ )(1+ 3β̃2 + 6µ̃2 + ∆ ), (4.24)

where σQ and K is given in (4.20) and

∆ ⌘q

1+ 3µ̃2 + 6β̃2 , µ̃ ⌘µ

4⇡T, β̃ ⌘

β

4⇡T. (4.25)

13This form of Drude conductivity implicit ly appeared in [48]with shifted pole due to themagnetic field.

– 18 –

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Conclusion

• A NEW field theory method for strongly correlated system based on holography began.

• To identify the system we need to identify key interactions in bulk local field theories.

• We considered magneto-electric term. The result is surprisingly good.

39

Thank you.

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