emergent gauge fields and the high temperature...

Emergent gauge fields and the

high temperature superconductors Nambu Memorial Symposium

University of ChicagoMarch 12, 2016

Subir Sachdev

HARVARD

Talk online: sachdev.physics.harvard.edu

Nambu and superconductivity

P H YSI CAL REVI EW VOLUM E 117, NUMBER 3 FEBRUARY 1, 1960

Quasi-Particles and Gauge Invariance in the Theory of Superconductivity*YoicHmo NAMsU

The Enrico Fermi Institute for Nuclear Studies and the Department of Physics, The University of Chicago, Chicago, Illinois(Received July 23, 1959)

Ideas and techniques known in quantum electrodynamics havebeen applied to the Bardeen-Cooper-Schrieffer theory of super-conductivity. In an approximation which corresponds to ageneralization of the Hartree-Pock 6elds, one can write down anintegral equation defining the self-energy of an electron in anelectron gas with phonon and Coulomb interaction. The form ofthe equation implies the existence of a particular solution whichdoes not follow from perturbation theory, and which leads to theenergy gap equation and the quasi-particle picture analogous toBogoliubov's.The gauge invariance, to the erst order in the external electro-

magnetic field, can be maintained in the quasi-particle picture bytaking into account a certain class of corrections to the charge-current operator due to the phonon and Coulomb interaction. Infact, generalized forms of the Ward identity are obtained betweencertain vertex parts and the self-energy. The Meissner effect cal-culation is thus rendered strictly gauge invariant, but essentiallykeeping the BCS result unaltered for transverse 6elds.It is shown also that the integral equation for vertex parts

allows homogeneous solutions which describe collective excitationsof quasi-particle pairs, and the nature and effects of such col-lective states are discussed.

1. INTRODUCTION which is given by the expressionNUMBER of papers have appeared on variousaspects of the Bardeen-Cooper-SchrieGer' theory

of superconductivity. On the whole, the BCS theory,which leads to the existence of an energy gap, presentsus with a remarkably good understanding of the generalfeatures of superconducivity. A mathematical for-mulation based on the BCS theory has been developedin a very elegant way by Bogoliubov, ' who introducedcoherent mixtures of particles and holes to describe asuperconductor. Such "quasi-particles" are not eigen-states of charge and particle number, and reveal a verybold departure, inherent in the BCS theory, from theconventional approach to many-fermion problems.This, however, creates at the same time certain theo-retical difhculties which are matters of principle. Thusthe derivation of the Meissner effect in the original BCStheory is not gauge-invariant, as is obvious from theviewpoint of the quasi-particle picture, and poses aserious problem as to the correctness of the resultsobtained in such a theory.This question of gauge invariance has been taken up

by many people. ' In the Meissner effect one deals witha linear relation between the Fourier components of theexternal vector potential 3 and the induced current J,

f'(q) =2&"(c)~'(v),with

j(ol j,(q) In)(nl j;(—q) IO)Jt;, (q) =——(0I p I o&5;;++I

m

oI jt(-~) I )( I j.(v) lo)iz. i (& &)

p and j are the charge-current density, and IO) refersto the superconductive ground state. In the BCSmodel,the second term vanishes in the limit q~0, leavingthe first term alone to give a nongauge invariant result.It has been pointed out, however, that there is a sig-nificant difference between the transversal and longi-tudinal current operators in their matrix elements.Namely, there exist collective excited states of quasi-particle pairs, as was first derived by Bogoliubov, whichcan be excited only by the longitudinal current.As a result, the second term does not vanish for a

longitudinal current, but cancels the first term (thelongitudinal sum rule) to produce no physical effect;whereas for a transversal Geld, the original result willremain essentially correct.If such collective states are essential to the gauge-

invariant character of the theory, then one might arguethat the former is a necessary consequence of thelatter. But this point has not been c1.ear so far.Another way to understand the BCS theory and its

problems is to recognize it as a generalized Hartree-Pockapproximation. ' We will develop this point a littlefurther here since it is the starting point of what followslater as the main part of the paper.

*This work was supported by the U. S. Atomic Energy Com-.mission.' Bardeen, Cooper, and SchrieA'er, . Phys. Rev. 106, 162 (1957);108, 1175 (1957).

2¹ N. Bogoliubov, I.Exptl. Theoret. Phys. U.S.S.R. 34, 58, 73

(1958) I translation: Soviet Phys. 34, 41, 51 (1958)g; Bogoliubov,To!machev, and Shirkov, A Nezo method in the Theory of Supercondzsct&ity (Academy of Sciences of U.S.S.R., Moscow, 1958).See also J. G. Valatin, Nuovo cimento 7, 843 (1958).' M. J. Buckingam, Nuovo cimento 5, 1763 (1957).J. Bardeen,Nuovo cimento 5, 1765 (1957).M. R. Schafroth, Phys. Rev. 111,72 (1958.P. W. Anderson, Phys. Rev. 110, 827 (1958);112, 1900(1958). G. Rickayzen, Phys. Rev. 111, 817 (1958); Phys. Rev.Letters 2, 91 (1959).D. Pines and R. Schrieffer, Nuovo ciment10, 496 (1958); Phys. Rev. Letters 2, 407 (1958). G. WentzelPhys. Rev. 111, 1488 (1958); Phys. Rev. Letters 2, 33 (1959)J.M. Blatt and T. Matsubara, Progr. Theoret. Phys. (Kyoto) 20781 (1958). Blatt, Matsubara, and May, Progr. Theoret. Phys(Kyoto) 21, 745 (1959).K. Yosida, ibid. 731.

o 4 Recently ¹ ¹ Bogoliubov, Uspekhi Fiz, Nauk 67, 549 (1959)[translation: Soviet Phys.—Uspekhi 67, 236 (1959)g, has alsoreformulated his theory as a Hartree-Fock approximation, anddiscussed the gauge invariance collective excitations from thisviewpoint. The author is indebted to Prof. Bogoliubov for sendinghim a preprint.

648

P H YSI CAL R EVI EW VOLUME &22, NUMBER AI RII,

Dynamical Model of Elementary Particles Based on an Analogywith Superconductivity. PY. NAMBU AND G. JONA-LASINIoj'

The Enrico terms Institute for Nuclear StuCkes and the Department of Physics, The University of Chicago, Chicago, Illinois(Received October 27, 1960)

It is suggested that the nucleon mass arises largely as a self-energy of some primary fermion field throughthe same mechanism as the appearance of energy gap in the theory of superconductivity. The idea can be putinto a mathematical formulation utilizing a generalized Hartree-Fock approximation which regards realnucleons as quasi-particle excitations. We consider a simplified model of nonlinear four-fermion interactionwhich allows a p5-gauge group. An interesting consequence of the symmetry is that there arise automaticallypseudoscalar zero-mass bound states of nucleon-antinucleon pair which may be regarded as an idealized pion.In addition, massive bound states of nucleon number zero and two are predicted in a simple approximation.The theory contains two parameters which can be explicitly related to observed nucleon mass and the

pion-nucleon coupling constant. Some paradoxical aspects of the theory in connection with the p5 trans-formation are discussed in detail.

I. INTRODUCTION" 'N this paper we are going to develop a dynamical- theory of elementary particles in which nucleons and

mesons are derived in a unified way from a fundamentalspinor field. In basic physical ideas, it has thus thecharacteristic features of a compound-particle model,but unlike most of the existing theories, dynamicaltreatment of the interaction makes up an essential partof the theory. Strange particles are not yet considered.The scheme is motivated by the observation of an

interesting analogy between the properties of Diracparticles and the quasi-particle excitations that appearin the theory of superconductivity, which was originatedwith great success by Bardeen, Cooper, and Schrieffer, 'and subsequently given an elegant mathematical forlnu-lation by Bogoliubov. ' The characteristic feature of theBCS theory is that it produces an energy gap betweenthe ground state and the excited states of a supercon-ductor, a fact w'hich has been confirmed experimentally.The gap is caused due to the fact that the attractivephonon-mediated interaction between electrons producescorrelated pairs of electrons with opposite momenta andspin near the Fermi surface, and it takes a finite amountof energy to break this correlation.Elementary excitations in a superconductor can be

conveniently described by means of a coherent mixtureof electrons and holes, which obeys the following

* Supported by the U. S. Atomic Energy Commission.f' Fulbright Fellow, on leave of absence from Instituto di Fisica

dell Universita, Roma, Italy and Istituto Nazionale di FisicaNucleare, Sezione di Roma, Italy.'A preliminary version of the work was presented at the

Midwestern Conference on Theoretical Physics, April, 1960 (un-published). See also Y. Nambu, Phys. Rev. Letters 4, 380 (1960);and Proceedings of the Tenth Annual Rochester Conference onHigh-Energy Nuclear Physics, 1960 (to be published).' J.Bardeen, L. N. Cooper, and J.R. Schrieffer, Phys. Rev. 106,162 (1957).3 N. N. Bogoliubov, J. Exptl. Theoret. Phys. (U.S.S.R.) 34, 58,

73 (1958) Ltranslation: Soviet Phys. -JETP 34, 41, 51 (1958)g;N. N. Sogoliubov, V. V. Tolmachev, and D. V. Shirkov, A %emMethodin the Theory of Supercondlctivity (Academy of Sciences ofU, S.S.R., Moscow, 1958).

equations' 4:E4~= e lto~+40 (1.1

E0 ~*= eA ~*—+44~,near the Fermi surface. 11„+ is the component of theexcitation corresponding to an electron state of mo-mentum P and spin +(up), andri ~*corresponding toa hole state of momentum p and spin +, which meansan absence of an electron of momentum —p and spin—(down). eo is the kinetic energy measured from theFermi surface; g is a constant. There will also be anequation complex conjugate to Eq. (1), describinganother type of excitation.Equation (1) gives the eigenvalues

E„=a (e,'+y')-*'. (1.2)The two states of this quasi-particle are separated inenergy by 2

~E„~.In the ground state of the system all

the quasi-particles should be in the lower (negative)energy states of Eq. (2), and it would take a finiteenergy 2)E„~ )~2~&~ to excite a particle to the upperstate. The situation bears a remarkable resemblance tothe case of a Dirac particle. The four-component Diracequation can be split into two sets to read

EP,=o"Pter+ res,Egs———o"Pigs+ nell r,E„=W (p'+nt') l,

where tPt and Ps are the two eigenstates of the chiralityoperator ys——yjy2y3y4.According to Dirac's original interpretation, the

ground state (vacuum) of the world has all the electronsin the negative energy states, and to create excitedstates (with zero particle number) we have to supply anenergy &~2m.In the BCS-Bogoliubov theory, the gap parameter @,

which is absent for free electrons, is determined es-sentially as a self-consistent (Hartree-Fock) representa-tion of the electron-electron interaction eGect.

4 J. G. Valatin, Nuovo cimento 7, 843 (1958).345

The problems of Nambu-Goldstone bosons linked to the

spontaneous breaking of a global symmetryand

the Higgs phase with a massive photon in a weakly-coupled gauge theory

are closely connected

High temperature superconductors:

Electrons in crystals provide a novel “vacuum”, and their interactions can lead to quantum ground states with long-range quantum entanglement.

The dynamics of such states is described by “emergent” gauge fields, usually at strong coupling.

YBa2Cu3O6+x

High temperature superconductors

CuO2 plane

Cu

O

SM

FL

Figure: K. Fujita and J. C. Seamus Davis

YBa2Cu3O6+x

SM

FL


YBa2Cu3O6+x

Insulating Antiferromagnet

“Undoped”insulating

anti-ferromagnet

Anti-ferromagnet

with p mobile holes

per square

FilledBand

But relative to the band

insulator, there are 1+ p holes

per square

Anti-ferromagnet

with p mobile holes

per square

But relative to the band

insulator, there are 1+ p holes

per square

Anti-ferromagnet

with p mobile holes

per square

In a conventional metal (a Fermi liquid), with no

broken symmetry, the area enclosed by the Fermi surface must be 1+p

SM

FL


YBa2Cu3O6+x

A conventional metal:

the Fermi liquid

M. Plate, J. D. F. Mottershead, I. S. Elfimov, D. C. Peets, Ruixing Liang, D. A. Bonn, W. N. Hardy,S. Chiuzbaian, M. Falub, M. Shi, L. Patthey, and A. Damascelli, Phys. Rev. Lett. 95, 077001 (2005)

SM

FL

SM

FL

• Fermi surface separates empty andoccupied states in momentum space.

• Luttinger Theorem: volume (area)enclosed by Fermi surface = theelectron density.

• Hall co-ecientRH = 1/((Fermi volume) e).

Fermisurface

kx

ky

Ordinary metals: the Fermi liquid

Hall effect measurements in YBCO

Badoux, Proust, Taillefer et al., Nature 531, 210 (2016)

14

0 0.1 0.2 0.3p

0

0.5

1

1.5n H

= V

/ e

RH

p

1 + p

SDW CDW FL

p*

a

b

Fermi liquid (FL) with carrier

density 1+p



14

0 0.1 0.2 0.3p

0

0.5

1

1.5n H

= V

/ e

RH

p

1 + p

SDW CDW FL

p*

a

b

Spin density wave (SDW)

breaks translational invariance,

and the Fermi liquid then has carrier density p



14

0 0.1 0.2 0.3p

0

0.5

1

1.5n H

= V

/ e

RH

p

1 + p

SDW CDW FL

p*

a

b

Charge density wave (CDW)

leads to complex Fermi

surface reconstruction and negative

Hall resistance



14

0 0.1 0.2 0.3p

0

0.5

1

1.5n H

= V

/ e

RH

p

1 + p

SDW CDW FL

p*

a

b

Evidence for FL* metal with Fermi surface of

size pand

emergent gauge

fields ?!

A metal with:

• A Fermi surface of electrons enclosing volume p, and not the Luttinger volume of 1+p

• Emergent gauge fields and connections to topological field theories

FL*

T. Senthil, S. Sachdev, and M. Vojta, Phys. Rev. Lett 90, 216403 (2003)

A metal with:

• A Fermi surface of electrons enclosing volume p, and not the Luttinger volume of 1+p

• Emergent gauge fields and connections to topological field theories

There is a general and fundamental relationship between these two characteristics.

T. Senthil, S. Sachdev, and M. Vojta, Phys. Rev. Lett 90, 216403 (2003)

FL*

1. The insulating spin liquid and topological field theory

2. Topology and the size of the Fermi surface

3. Transition between FL* and FL

4. Quantum matter with quasiparticlesstrange metals in superconductors, graphene, the quark-gluon plasma, the superfluid-insulator transition of ultra-cold atoms, and the dynamics of charged black holes horizons

“Undoped”Anti-

ferromagnet

Insulating spin liquid

L. Pauling, Proceedings of the Royal Society London A 196, 343 (1949)P. W. Anderson, Materials Research Bulletin 8, 153 (1973)

= (|"#i |#"i) /p2

The first proposal of a

quantum state with long-range

entanglement

Insulating spin liquid Modern description:

N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991) X.-G. Wen, Phys. Rev. B 44, 2664 (1991) M. Freedman, C. Nayak, K. Shtengel, K. Walker, and Z. Wang, Annals of Physics 310, 428 (2004)

The Z2 spin liquid: Described by the simplest, non-

trivial, topological field theory with time-reversal

symmetry:

L =

1

4

KIJ

Zd

3x a

I ^ da

J

where a

I, I = 1, 2 are U(1) gauge connections, and

the K matrix is

K =

0 2

2 0

Insulating spin liquid Modern description:

N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991) X.-G. Wen, Phys. Rev. B 44, 2664 (1991) M. Freedman, C. Nayak, K. Shtengel, K. Walker, and Z. Wang, Annals of Physics 310, 428 (2004)

The Z2 spin liquid: Described by the simplest, non-

trivial, topological field theory with time-reversal

symmetry:

L =

1

4

KIJ

Zd

3x a

I ^ da

J

where a

I, I = 1, 2 are U(1) gauge connections, and

the K matrix is

K =

0 2

2 0

See also E. Fradkin and S. H. Shenker, “Phase diagrams of lattice gauge theories with Higgs fields,” Phys. Rev. D 19, 3682 (1979); J. M. Maldacena, G. W. Moore and N. Seiberg, “D-brane charges in five-brane backgrounds,” JHEP 0110, 005 (2001).

Place insulator

on a torus:

Ground state degeneracy

Ground state becomes

degenerate due to gauge fluxes

enclosed by cycles of the

torus

Place insulator

on a torus:



D.J. Thouless, PRB 36, 7187 (1987)S.A. Kivelson, D.S. Rokhsar and J.P. Sethna, Europhys. Lett. 6, 353 (1988)

obtain “topological” states nearly

degenerate with the ground state:

number of dimers crossing

red line is conserved modulo 2

Place insulator

on a torus:

= (|"#i |#"i) /p2





Place insulator

on a torus:



= (|"#i |#"i) /p2

4





Place insulator

on a torus:



= (|"#i |#"i) /p2

2





Place insulator

on a torus:



= (|"#i |#"i) /p2

0





Place insulator

on a torus:



= (|"#i |#"i) /p2

2

Topology and the Fermi surface size

Lx

Ly

Φ

We take N particles, each with charge Q, on a Lx

Ly

lattice on a torus.

We pierce flux = hc/Q through a hole of the torus.

An exact computation shows that the change in crystal momentum of the

many-body state due to flux piercing is

Pxf

Pxi

=

2N

Lx

(mod 2) = 2Ly

(mod 2)

where = N/(Lx

Ly

) is the density.

M. Oshikawa, PRL 84, 3370 (2000)A. Paramekanti and A. Vishwanath,

PRB 70, 245118 (2004)

Proof of

Pxf

Pxi

=

2N

Lx

(mod 2) = 2Ly

(mod 2).

The initial and final Hamiltonians are related by a gauge transformation

UG

Hf

U1G

= Hi

, UG

= exp

i2

Lx

X

i

xi

ni

!.

while the wavefunction evolves from | i

i to UT

| i

i, where UT

is the time evolutionoperator. We want to work in a fixed gauge in which the initial and final Hamilto-nians are the same: in this gauge, the final state is |

f

i = UG

UT

| i

i. Let Tx

bethe lattice translation operator. Then we can establish the above result using thedefinitions

Tx

| i

i = eiP

xi | i

i , Tx

| f

i = eiP

xf | f

i ,

and the easily established properties

Tx

UT

= UT

Tx

, Tx

UG

= exp

i2

N

Lx

UG

Tx


Px

= 2Ly

(mod 2) , Py

= 2Lx

(mod 2)

Now we compute the momentum balance assuming that the only low energy exci-

tations are quasiparticles near the Fermi surface, and these react like free particles

to a suciently slow flux insertion. So each quasiparticle picks up a momentum

~

p (2/L

x

, 0), and then we can write (with n

p

the quasiparticle density excited

by the flux insertion)

P

x

=

X

p

n

p

p

x

.

Now n

p

= ±1 on a shell of thickness

~

p · d~Sp

on the Fermi surface (where

~

S

p

is an

area element on the Fermi surface). So we can write the above as a surface integral

P

x

=

I

FSp

x

L

x

L

y

4

2

~

p · d~Sp

= (

~

p · x)Z

FV

L

x

L

y

4

2

dV

by the divergence theorem. So

P

x

=

2

L

x

L

x

L

y

4

2VFS , P

y

=

2

L

y

L

x

L

y

4

2VFS

where VFS is the volume of the Fermi surface. So, although the quasiparticles

are only defined near the Fermi surface, by using Gauss’s Law on the momentum

acquired by quasiparticles near the Fermi surface, we have converted the answer to

an integral over the volume enclosed by the Fermi surface.

Now we equate these values to those obtained above, and obtain

N L

x

L

y

VFS

4

2= L

x

m

x

, N L

x

L

y

VFS

4

2= L

y

m

y

for some integers m

x

, m

y

. By choosing L

x

, L

y

mutually prime integers we can now

show

=

N

L

x

L

y

=

VFS

4

2+m

for some integer m: this is Luttinger’s theorem.


~p

Px

= 2Ly

(mod 2) , Py

= 2Lx

(mod 2)

Now we compute the momentum balance assuming that the only low energy exci-

tations are quasiparticles near the Fermi surface, and these react like free particles

to a suciently slow flux insertion. So each quasiparticle picks up a momentum

~

p (2/L

x

, 0), and then we can write (with n

p

the quasiparticle density excited

by the flux insertion)

P

x

=

X

p

n

p

p

x

.

Now n

p

= ±1 on a shell of thickness

~

p · d~Sp

on the Fermi surface (where

~

S

p

is an

area element on the Fermi surface). So we can write the above as a surface integral

P

x

=

I

FSp

x

L

x

L

y

4

2

~

p · d~Sp

= (

~

p · x)Z

FV

L

x

L

y

4

2

dV

by the divergence theorem. So

P

x

=

2

L

x

L

x

L

y

4

2VFS , P

y

=

2

L

y

L

x

L

y

4

2VFS

where VFS is the volume of the Fermi surface. So, although the quasiparticles

are only defined near the Fermi surface, by using Gauss’s Law on the momentum

acquired by quasiparticles near the Fermi surface, we have converted the answer to

an integral over the volume enclosed by the Fermi surface.

Now we equate these values to those obtained above, and obtain

N L

x

L

y

VFS

4

2= L

x

m

x

, N L

x

L

y

VFS

4

2= L

y

m

y

for some integers m

x

, m

y

. Now choose L

x

, L

y

mutually prime integers; then

m

x

L

x

= m

y

L

y

implies that m

x

L

x

= m

y

L

y

= pL

x

L

y

for some integer p. Then

we obtain

=

N

L

x

L

y

=

VFS

4

2+ p.

This is Luttinger’s theorem.


Lx

Ly

Φ

Topology and the Fermi surface size in FL*

T. Senthil, M. Vojta, and S. Sachdev, Phys. Rev. B 69, 035111 (2004)

To obtain a di↵erent Fermi surface size, we need low energy

excitations on a torus which are not composites of quasiparticles

around the Fermi surface. The degenerate ground states of a

Z2 spin liquid can provide the needed excitation, and lead to

a Z2-FL* state with a Fermi surface size of p, rather than the

Luttinger size of 1 + p.

Lx

Ly

Φ

Topology and the Fermi surface size in FL*


The exact momentum transfers Px

= 2(1 + p)Ly

(mod2)and P

y

= 2(1+p)Lx

(mod2) due to flux piercing arise from

• A contribution 2pLx,y

from the small Fermi surface of

quasiparticles of size p.

• The remainder is made up by the topological sector: flux

insertion creates a “vison” in the hole of the torus.

Start with a spin liquid and then remove

electrons

= (|"#i |#"i) /p2

= (|"#i |#"i) /p2

Start with a spin liquid and then remove

electrons

= (|"#i |#"i) /p2

A mobile charge +e, but

carrying no spin

= (|"#i |#"i) /p2

S.A. Kivelson, D.S. Rokhsar and J.P. Sethna, PRB 35, 8865 (1987)

N. Read and B. Chakraborty, PRB 40, 7133 (1989)

Spin liquidwith density p of spinless, charge +e “holons”.

These can form a Fermi surface of size p, but

not of electrons

S.A. Kivelson, D.S. Rokhsar and J.P. Sethna, PRB 35, 8865 (1987)

N. Read and B. Chakraborty, PRB 40, 7133 (1989)

= (|"#i |#"i) /p2

Spin liquidwith density p of spinless, charge +e “holons”.

These can form a Fermi surface of size p, but

not of electrons

= (|"#i |#"i) /p2

FL*

Mobile S=1/2, charge +e fermionic dimers: form

a Fermi surface of size p of electrons

R. K. Kaul, A. Kolezhuk, M. Levin, S. Sachdev, and T. Senthil, PRB 75, 235122 (2007)S. Sachdev PRB 49, 6770 (1994); X.-G. Wen and P. A. Lee PRL 76, 503 (1996)

M. Punk, A. Allais, and S. Sachdev, PNAS 112, 9552 (2015)

= (|"#i |#"i) /p2 = (|" i+ | "i) /

p2

R. K. Kaul, A. Kolezhuk, M. Levin, S. Sachdev, and T. Senthil, PRB 75, 235122 (2007)S. Sachdev PRB 49, 6770 (1994); X.-G. Wen and P. A. Lee PRL 76, 503 (1996)

M. Punk, A. Allais, and S. Sachdev, PNAS 112, 9552 (2015)

= (|"#i |#"i) /p2 = (|" i+ | "i) /

p2

FL*

Mobile S=1/2, charge +e fermionic dimers: form

a Fermi surface of size p of electrons

Place FL* on a torus:


degenerate with quasiparticle

states: number of dimers

crossing red line is conserved

modulo 2


= (|"#i |#"i) /p2 = (|" i+ | "i) /

p2

2

FL*






modulo 2


= (|"#i |#"i) /p2 = (|" i+ | "i) /

p2

0

FL*






modulo 2


= (|"#i |#"i) /p2 = (|" i+ | "i) /

p2

2

FL*

SM

FL


SM

FL


FL*

FL

SM

FL


FL*

FL

Strange Metal

S. Sachdev, M. A. Metlitski, Y. Qi, and C. Xu, Phys. Rev. B 80, 155129 (2009)D. Chowdhury and S. Sachdev, PRB 91, 115123 (2015)

• A SU(2) gauge boson.

• Fermion , transforming as a gauge SU(2) fundamental,with dispersion "k from the band structure, at a non-zero chemical potential: has a “large” Fermi surface, andcarries electromagnetic charge

• A complex Higgs field, H, transforming as a gauge SU(2)adjoint, carrying non-zero lattice momentum.

• A SU(2) fundamental scalar z↵, carrying electron-spin andelectromagnetically neutral.

SU(2) gauge theory for transition between Z2-FL* and FL

S. Sachdev, M. A. Metlitski, Y. Qi, and C. Xu, Phys. Rev. B 80, 155129 (2009)D. Chowdhury and S. Sachdev, PRB 91, 115123 (2015)

• The ‘Higgs’ phase with hHi has a deconfined Z2 gaugefield, and realizes a Z2-FL* with Ising-nematic order.

• The ‘confining’ phase is a FL or a superconductor, and noIsing-nematic order.

• When the Higgs potential is critical, we obtain a non-Fermi liquid of a Fermi surface coupled to Landau-damped gauge bosons, and critical Landau-damped Higgsfield. This is a candidate for describing the strange metal.

SU(2) gauge theory for transition between Z2-FL* and FL

Fermi liquids: quasiparticles moving ballistically between impurity (red circles)

scattering events

Strange metals: electrons scatter frequently off each other, so there is no regime of ballistic quasiparticle motion.

The electron “liquid” then “flows” around impurities

kx

ky

Graphene

Electron Fermi surface

Graphene

Hole Fermi surface

Electron Fermi surface

Graphene

-1 -0.5 0 0.5 1

100

200

300

400

500

600

-1 -0.5 0 0.5 1

100

200

300

400

500

600

∼ 1√n(1 + λ ln Λ√

n)

n1012/m2

T (K)

Dirac liquid

ElectronFermi liquid

HoleFermi liquid

Quantum critical

Graphene

M. Müller, L. Fritz, and S. Sachdev, PRB 78, 115406 (2008)M. Müller and S. Sachdev, PRB 78, 115419 (2008)

-1 -0.5 0 0.5 1

100

200

300

400

500

600

-1 -0.5 0 0.5 1

100

200

300

400

500

600

∼ 1√n(1 + λ ln Λ√

n)

n1012/m2

T (K)

Dirac liquid


HoleFermi liquid

Quantum critical

Graphene


Predictedstrange metal

Dirac Fluid in Graphene 26

Wiedemann-Franz Law

I Wiedemann-Franz law in a Fermi liquid:

T 2k2

B

3e2 2.45 108 W ·

K2.

I in hydrodynamics one finds

T=

Lhydro

(1 + (Q/Q0)2)2 , Lhydro 1.

hence the Lorenz ratio, L, departs from the Sommer- feld value, L o

L - efT (4)

The important scattering processes in thermal and electrical conduction are: (i) elastic scattering by solute atoms, impurities and lattice defects, (ii) scattering of the electrons by phonons, and (iii) electron-electron interactions. In the elastic scattering region, i.e. at very low temperature, IE = IT and hence L = L 0. At higher temperatures, electron-electron scattering and electron-phonon scattering dominate and the collisions are inelastic. Then IE#l T and hence L deviates from L o.

Deviations from the Sommerfeld value of the Lorenz number are due to various reasons. In metals,

1 0 2 A g /e

"73 t - O C~

"~I01

0) >

rr"

10 ~

Cu o ~ ,", /

Brass O V , "

Sn o z~ /v Fe GerP~n Pt Q~ z~/~z~

silv U

Wiedemann-Franz ;< 0 o

Figure l Relative

I I I .I 101 10 2

Re la t i ve e lec t r ica l c o n d u c t i v i t y

thermal conductivities, A, measured by Wiedemann and Franz (AAg assumed to be = 100) and relative electrical conductivities, ~, measured by ( 9 Riess, (A) Becquerel, and (V) Lorenz. C~Ag assumed to be - 100. After Wiedemann and Franz [1].

at low temperatures the deviations are due to the inelastic nature of electron-phonon interactions. In some cases, a higher Lorenz number is due to the presence of impurities. The phonon contribution to thermal conductivity sometimes increases the Lorenz number, and this contribution, when phonon Umklapp scattering is present, is inversely propor- tional to the temperature. The deviations in Lorenz number can also be due to the changes in band structure. In magnetic materials, the presence of mag- nons also can change the Lorenz number at low temperatures. In the presence of a magnetic field, the Lorenz number varies directly with magnetic field. Changes in Lorenz number are sometimes due to structural phase transitions. In recent years, the Lorenz number has also been investigated at higher temperatures and has been found to deviate from the Sommerfeld value [14-20] and it is sometimes attributed to the incomplete degeneracy (Fermi smearing) [21] of electron gas. The Lorenz number has also been found to vary with pressure [-22, 23].

In alloys, the thermal conductivity and hence the Lorenz number have contributions from the electronic and lattice parts at low temperatures. The apparent Lorenz ratio (L/Lo) for many alloys has a peak at low temperatures. At higher temperatures the apparent Lorenz ratio is constant for each sample and approaches Lo as the percentage of alloying, x, increases. In certain alloys at high temperatures, the ordering causes a peak in L/L o.

The Lorenz number of degenerate semiconductors also shows a similar deviation to that observed in metals and alloys. Up to a certain temperature, inelastic scattering determines the Lorenz number value, and below this the scattering is elastic which is due to impurities. Supression of the electronic contribution to thermal conductivity and hence the separation of the lattice and electronic parts of conductivity can be done by application of a transverse magnetic field and hence the Lorenz number can be evaluated. The deviation of the Lorenz number in some degenerate semiconductors is attributed to phonon drag. In some

1 0 16

~'v 3"0 t ~ 2.5 O 2.0

1017 1018 I ~ ' I

Bi

Carr ier concen t ra t i on (cm 3) 1 019 1 0 2~ 1 0 21 1 0 22 1 0 23

I 1 ~ t I t ; I

SiGe 2 Cs AI Au Mg

SiBe3 /. / Nb~b.,~o/Cu F i i e 9

Sb As Ag Ga

1 024

30 c N 2.5 c-

O " 2.0

SiGe z ~oe \ SiG%

i \ SiGe 1

I I r I I t f

1 0 2 1 0 3 1 0 4

Se H~ Y W1 Cs2 Fe Ir Co z Li2 AUz Au3 AI2 All

\." \ \ \ ! ! , ,L T, o y t w , p, t Feq z , ~ / Cr '' \ X~., K Pt , \ Ni2 u A ~ "

Yb ~o 1 Er / / 2 As Sb ~ Rh g t31~ Bi 2 Bi I Cq

I [ I I [ I i I t I I I I I I

1 0 6 1 0 e 1 0 7 1 0 8 1 0 9 Electr ical conduc t i v i t y (~,~-1 cm-1)

! I

101o

Figure 2 Experimental Lorenz number of elemental metals in the low-temperature residual resistance regime, see Table I. Also shown are our own data points on a doped, degenerate semiconductor (Table III). Data are plotted versus electrical conductivity and also versus carrier concentration, taken from Ashcroft and Mermin [24] except for the semiconductors.

4262

G. S. Kumar, G. Prasad, and R.O. Pohl, J. Mat. Sci. 28, 4261 (1993)

L0 =

Thermal and electrical conductivity with quasiparticles

-1 -0.5 0 0.5 1

100

200

300

400

500

600

-1 -0.5 0 0.5 1

100

200

300

400

500

600

∼ 1√n(1 + λ ln Λ√

n)

n1012/m2

T (K)

Dirac liquid


HoleFermi liquid

Quantum critical

Graphene


Predictedstrange metal


Wiedemann-Franz Law Violations in Experiment

0

4

8

12

16

20

L / L

0

10

20

30

40

50

60

70

80

90

100T

ba

th (

K)

−10−15 15−5 0 5 10

n (109 cm-2)

disorder-limited

phonon-limited

[Crossno et al, submitted]

Strange metal in graphene

J. Crossno, Jing K. Shi, Ke Wang, Xiaomeng Liu, A. Harzheim, A. Lucas, S. Sachdev, Philip Kim, Takashi Taniguchi, Kenji Watanabe, T. A. Ohki, and Kin Chung Fong, Science 351, 1058 (2016)

L =

T

L0 =2k2B3e2



0

4

8

12

16

20

L / L

0

10

20

30

40

50

60

70

80

90

100T

ba

th (

K)

−10−15 15−5 0 5 10

n (109 cm-2)

disorder-limited

phonon-limited



L =

T

L0 =2k2B3e2

Wiedemann-Franz obeyed




0

4

8

12

16

20

L / L

0

10

20

30

40

50

60

70

80

90

100T

ba

th (

K)

−10−15 15−5 0 5 10

n (109 cm-2)

disorder-limited

phonon-limited



L =

T

L0 =2k2B3e2

Wiedemann-Franz violated !


0

4

8

12

16

20

L / L

0

10

20

30

40

50

60

70

80

90

100

Tb

ath

(K

)

−10−15 15−5 0 5 10

n (109 cm-2)2

−10 −5 0 5 10

0

0.6

1.2

Vg (V)

R (

kΩ)

250 K150 K

100 K

50 K4 K

A

10

-1012 -1011 -1010 1010 1011 1012

100

n (cm-2)

Ele

c. C

on

du

ctiv

ity

(4 e

2/h

)

Tb

ath (K)

e-h+

B

∆Vg (V) Tbath (K)-0.5 0.50 0 50 100 150

Th

erm

al C

on

du

ctiv

ity

(n

W/K

)

0

2

4

6

8

0

2

4

6

0

2

4

6

8

0

01

1

10 mm

20 K

-0.5 V

0 V

40 K

75 K

C D

E

Tel-ph

Tdis

κe

σTL0

FIG. 1. Temperature and density dependent electrical and thermal conductivity. (A) Resistance versus gate voltageat various temperatures. (B) Electrical conductivity (blue) as a function of the charge density set by the back gate for di↵erentbath temperatures. The residual carrier density at the neutrality point (green) is estimated by the intersection of the minimumconductivity with a linear fit to log() away from neutrality (dashed grey lines). Curves have been o↵set vertically such thatthe minimum density (green) aligns with the temperature axis to the right. Solid black lines correspond to 4e2/h. At lowtemperature, the minimum density is limited by disorder (charge puddles). However, above Tdis 40 K, a crossover markedin the half-tone background, thermal excitations begin to dominate and the sample enters the non-degenerate regime nearthe neutrality point. (C-D) Thermal conductivity (red points) as a function of (C) gate voltage and (D) bath temperaturecompared to the Wiedemann-Franz law, TL0 (blue lines). At low temperature and/or high doping (|µ| kBT ), we find theWF law to hold. This is a non-trivial check on the quality of our measurement. In the non-degenerate regime (|µ| < kBT )the thermal conductivity is enhanced and the WF law is violated. Above Telph 80 K, electron-phonon coupling becomesappreciable and begins to dominate thermal transport at all measured gate voltages. All data from this figure is taken fromsample S2 (inset 1E).

Realization of the Dirac fluid in graphene requires thatthe thermal energy be larger than the local chemical po-tential µ(r), defined at position r: kBT & |µ(r)|. Impu-rities cause spatial variations in the local chemical po-tential, and even when the sample is globally neutral, itis locally doped to form electron-hole puddles with finiteµ(r) [25–28]. Formation of the DF is further complicatedby phonon scattering at high temperature which can re-lax momentum by creating additional inelastic scatteringchannels. This high temperature limit occurs when theelectron-phonon scattering rate becomes comparable tothe electron-electron scattering rate. These two temper-atures set the experimental window in which the DF andthe breakdown of the WF law can be observed.

To minimize disorder, the monolayer graphene samplesused in this report are encapsulated in hexagonal boronnitride (hBN) [29]. All devices used in this study aretwo-terminal to keep a well-defined temperature profile

[30] with contacts fabricated using the one-dimensionaledge technique [31] in order to minimize contact resis-tance. We employ a back gate voltage Vg applied tothe silicon substrate to tune the charge carrier densityn = ne nh, where ne and nh are the electron and holedensity, respectively (see supplementary materials (SM)).All measurements are performed in a cryostat controllingthe temperature Tbath. Fig. 1A shows the resistance Rversus Vg measured at various fixed temperatures for arepresentative device (see SM for all samples). From this,we estimate the electrical conductivity (Fig. 1B) usingthe known sample dimensions. At the CNP, the residualcharge carrier density nmin can be estimated by extrap-olating a linear fit of log() as a function of log(n) outto the minimum conductivity [32]. At the lowest tem-peratures we find nmin saturates to 8109 cm2. Wenote that the extraction of nmin by this method overesti-mates the charge puddle energy, consistent with previous

4

0 100 2000

5

10

15

20

25

Temperature (K)

L /

L 0

B

1 10 100 1000109

1010

1011

Temperature (K)

nm

in (c

m-2

)

DisorderLimited

ThermallyLimited

S3S2S1

A

−6 −4 −2 0 2 4 60

4

8

12

16

20

n (1010 cm−2)

L/L 0

C

40 60 80 1000

2

4

6

8

10

T (K)

H

(eV

/µm

2)

CHe

h

-V+Ve

h

∆Vg = 0

FIG. 3. Disorder in the Dirac fluid. (A) Minimum car-rier density as a function of temperature for all three sam-ples. At low temperature each sample is limited by disorder.At high temperature all samples become limited by thermalexcitations. Dashed lines are a guide to the eye. (B) TheLorentz ratio of all three samples as a function of bath tem-perature. The largest WF violation is seen in the cleanestsample. (C) The gate dependence of the Lorentz ratio is wellfit to hydrodynamic theory of Ref. [5, 6]. Fits of all threesamples are shown at 60 K. All samples return to the Fermiliquid value (black dashed line) at high density. Inset showsthe fitted enthalpy density as a function of temperature andthe theoretical value in clean graphene (black dashed line).Schematic inset illustrates the di↵erence between heat andcharge current in the neutral Dirac plasma.

more pronounced peak but also a narrower density de-pendence, as predicted [5, 6].

More quantitative analysis of L(n) in our experimentcan be done by employing a quasi-relativistic hydrody-namic theory of the DF incorporating the e↵ects of weakimpurity scattering [5, 6, 39].

L =LDF

(1 + (n/n0)2)2 (2)

where

LDF =HvFlmT 2min

and n20 =

Hmin

e2vFlm. (3)

Here vF is the Fermi velocity in graphene, min is the elec-trical conductivity at the CNP, H is the fluid enthalpydensity, and lm is the momentum relaxation length from

impurities. Two parameters in Eqn. 2 are undeterminedfor any given sample: lm and H. For simplicity, we as-sume we are well within the DF limit where lm and Hare approximately independent of n. We fit the experi-mentally measured L(n) to Eqn. (2) for all temperaturesand densities in the Dirac fluid regime to obtain lm andH for each sample. Fig 3C shows three representative fitsto Eqn. (2) taken at 60 K. lm is estimated to be 1.5, 0.6,and 0.034 µm for samples S1, S2, and S3, respectively.For the system to be well described by hydrodynamics,lm should be long compared to the electron-electron scat-tering length of 0.1 µm expected for the Dirac fluid at60 K [18]. This is consistent with the pronounced sig-natures of hydrodynamics in S1 and S2, but not in S3,where only a glimpse of the DF appears in this moredisordered sample. Our analysis also allows us to es-timate the thermodynamic quantity H(T ) for the DF.The Fig. 3C inset shows the fitted enthalpy density asa function of temperature compared to that expected inclean graphene (dashed line) [18], excluding renormal-ization of the Fermi velocity. In the cleanest sample Hvaries from 1.1-2.3 eV/µm2 for Tdis < T < Telph. Thisenthalpy density corresponds to 20 meV or 4kBTper charge carrier — about a factor of 2 larger than themodel calculation without disorder [18].

In a hydrodynamic system, the ratio of shear viscosity to entropy density s is an indicator of the strength ofthe interactions between constituent particles. It is sug-gested that the DF can behave as a nearly perfect fluid[18]: /s approaches a “universal” lower bound conjec-ture by Kovtun-Son-Starinets, (/s)/(~/kB) 1/4 fora strongly interacting system [40]. Though we cannotdirectly measure , we comment on the implications ofour measurement for its value. Within relativistic hy-drodynamics, we can estimate the shear viscosity of theelectron-hole plasma in graphene from the enthalpy den-sity as Hee [40], where ee is the electron-electronscattering time. Increasing the strength of interactionsdecreases ee, which in turn decreases and /s. Employ-ing the expected Heisenberg limited inter-particle scat-tering time, ee ~/kBT [5, 6], we find a shear viscosityof 1020 kg/s in two-dimensional units, correspondingto 1010 Pa · s. The value of ee used here is consistentwith recent optical experiments on graphene [14, 16, 17].Using the theoretical entropy density for clean graphene(SM), we estimate (/s)/(~/kB) 3. This is comparableto 0.7 found in liquid helium at the Lambda-point [41],0.3 measured in cold atoms [3], and 0.4 for quark-gluon plasmas [4].

To fully incorporate the e↵ects of disorder, a hydrody-namic theory treating inhomogeneity non-perturbativelymay be needed [42]. The enthalpy densities reported hereare larger than the theoretical estimation obtained fordisorder free graphene; consistent with the picture thatchemical potential fluctuations prevent the sample fromreaching the Dirac point. While we find thermal conduc-

S. A. Hartnoll, P. K. Kovtun, M. Müller, and S. Sachdev, PRB 76, 144502 (2007)

Lorentz ratio L = /(T)

=

v2FHimp

T 2Q

1

(1 + e2v2FQ2imp/(HQ))2

Q ! electron density; H ! enthalpy density

Q ! quantum critical conductivity

imp ! momentum relaxation time from impurities

J. Crossno et al., Science 351, 1058 (2016)

2

FIG. 1: Current streamlines and potential map for vis-cous and ohmic flows. White lines show current stream-lines, colors show electrical potential, arrows show the direc-tion of current. Panel a) presents the mechanism of a negativeelectrical response: Viscous shear flow generates vorticity anda back flow on the side of the main current path, which leadsto charge buildup of the sign opposing the flow and results ina negative nonlocal voltage. Streamlines and electrical poten-tial are obtained from Eq.(5) and Eq.(6). The resulting po-tential profile exhibits multiple sign changes and ±45o nodallines, see Eq.(7). This provides directly measurable signaturesof shear flows and vorticity. Panel b) shows that, in contrast,ohmic currents flow down the potential gradient, producing anonlocal voltage in the flow direction.

The quantum-critical behavior is predicted to be par-ticularly prominent in graphene.[10–12] Electron inter-actions in graphene are strengthened near charge neu-trality (CN) due to the lack of screening at low car-rier densities.[12, 24] As a result, carrier collisions areexpected to dominate transport in pristine graphene ina wide range of temperatures and dopings.[25] Further-more, estimates of electronic viscosity near CN yield oneof the lowest known viscosity-to-entropy ratios which ap-proaches the universal AdS/CFT bound.[5]

Despite the general agreement that graphene holds thekey to electron viscosity, experimental progress has beenhampered by the lack of easily discernible signatures inmacroscopic transport. Several striking e↵ects have beenpredicted, such as vortex shedding in the preturbulentregime induced by strong current[6], as well as nonsta-tionary flow in a ‘viscometer’ comprised of an AC-drivenCorbino disc.[9] These proposals, however, rely on fairlycomplex AC phenomena originating from high-frequencydynamics in the electron system. In each of these cases,as well as in those of Refs.[8, 20], a model-dependentanalysis was required to delineate the e↵ects of viscosityfrom ‘extraneous’ contributions. In contrast, the nonlo-cal DC response considered here is a direct manifestationof the collective momentum transport mode which under-pins viscous flow, therefore providing an unambiguous,

FIG. 2: Nonlocal response for di↵erent resistivity-to-viscosity ratios /. Plotted is voltage V (x) at a distance xfrom current leads obtained from Eq.(12) for the setup shownin the inset. The voltage is positive in the ohmic-dominatedregion at large |x| and negative in the viscosity-dominatedregion closer to the leads (positive values at even smaller |x|reflect the finite contact size a 0.05w used in simulation).Viscous flow dominates up to fairly large resistivity values,resulting in negative response persisting up to values as largeas (new)2/ 120. Nodal points, marked by arrows, aresensitive to the / value, which provides a way to directlymeasure viscosity (see text).

almost textbook, diagnostic of the viscous regime.Nonlocal electrical response mediated by chargeless

modes was found recently to be uniquely sensitive to thequantities which are not directly accessible in electricaltransport measurements, in particular spin currents andvalley currents.[21–23] In a similar manner, the nonlo-cal response discussed here gives a diagnostic of viscoustransport, which is more direct and powerful than anyapproaches based on local transport.There are several aspects of the electron system in

graphene that are particularly well suited for studyingelectronic viscosity. First, the momentum-nonconservingUmklapp processes are forbidden in two-body collisionsbecause of graphene crystal structure and symmetry.This ensures the prominence of momentum conservationand associated collective transport. Second, while carrierscattering is weak away from charge neutrality, it can beenhanced by several orders of magnitude by tuning thecarrier density to the neutrality point. This allows tocover the regimes of high and low viscosity, respectively,in a single sample. Lastly, the two-dimensional structureand atomic thickness makes electronic states in graphenefully exposed and amenable to sensitive electric probes.To show that the timescales are favorable for the hy-

drodynamical regime, we will use parameter values esti-mated for pristine graphene samples which are almostdefect free, such as free-standing graphene.[28] Kine-matic viscosity can be estimated as the momentum dif-

L. Levitov and G. Falkovich, arXiv:1508.00836, Nature Physics online


1

Negative local resistance due to viscous electron backflow in graphene

D. A. Bandurin1, I. Torre2,3, R. Krishna Kumar1,4, M. Ben Shalom1,5, A. Tomadin6, A. Principi7, G. H. Auton5, E. Khestanova1,5, K. S. NovoseIov5, I. V. Grigorieva1, L. A. Ponomarenko1,4, A. K. Geim1, M. Polini3,6

1School of Physics & Astronomy, University of Manchester, Oxford Road, Manchester M13 9PL, United Kingdom 2National Enterprise for nanoScience and nanoTechnology, Scuola Normale Superiore, I‐56126 Pisa, Italy

3Istituto Italiano di Tecnologia, Graphene labs, Via Morego 30 I‐16163 Genova (Italy) 4Physics Department, Lancaster University, Lancaster LA14YB, United Kingdom

5National Graphene Institute, University of Manchester, Manchester M13 9PL, United Kingdom 6National Enterprise for nanoScience and nanoTechnology, Istituto Nanoscienze‐Consiglio Nazionale delle Ricerche

and Scuola Normale Superiore, I‐56126 Pisa, Italy 7Radboud University, Institute for Molecules and Materials, NL‐6525 AJ Nijmegen, The Netherlands

Graphene hosts a unique electron system that due to weak electron‐phonon scattering allows micrometer‐scale ballistic transport even at room temperature whereas the local equilibrium is provided by frequent electron‐electron collisions. Under these conditions, electrons can behave as a viscous liquid and exhibit hydrodynamic phenomena similar to classical liquids. Here we report unambiguous evidence for this long‐sought transport regime. In particular, doped graphene exhibits an anomalous (negative) voltage drop near current injection contacts, which is attributed to the formation of submicrometer‐size whirlpools in the electron flow. The viscosity of graphene’s electron liquid is found to be an order of magnitude larger than that of honey, in quantitative agreement with many‐body theory. Our work shows a possibility to study electron hydrodynamics using high quality graphene.

Collective behavior of many‐particle systems that undergo frequent inter‐particle collisions has been studied for more than two centuries and is routinely described by the theory of hydrodynamics [1,2]. The theory relies only on the conservation of mass, momentum and energy and is highly successful in explaining the response of classical gases and liquids to external perturbations varying slowly in space and time. More recently, it has been shown that hydrodynamics can also be applied to strongly interacting quantum systems including ultra‐hot nuclear matter and ultra‐cold atomic Fermi gases in the unitarity limit [3‐6].

In principle, the hydrodynamic approach can also be employed to describe many‐electron phenomena in condensed matter physics [7‐13]. The theory becomes applicable if electron‐electron scattering provides the shortest spatial scale in the problem such that ℓee ≪ , ℓ where ℓee is the electron‐electron scattering length, the sample size, ℓ ≡ the mean free path, the Fermi velocity, and the mean free time with respect to momentum‐non‐conserving collisions such as those involving impurities, phonons, etc. The above inequalities are difficult to meet experimentally. Indeed, at low temperatures ( ) ℓee varies approximately as ∝ reaching a micrometer scale below a few K [14], which

1

Negative local resistance due to viscous electron backflow in graphene

D. A. Bandurin1, I. Torre2,3, R. Krishna Kumar1,4, M. Ben Shalom1,5, A. Tomadin6, A. Principi7, G. H. Auton5, E. Khestanova1,5, K. S. NovoseIov5, I. V. Grigorieva1, L. A. Ponomarenko1,4, A. K. Geim1, M. Polini3,6

1School of Physics & Astronomy, University of Manchester, Oxford Road, Manchester M13 9PL, United Kingdom 2National Enterprise for nanoScience and nanoTechnology, Scuola Normale Superiore, I‐56126 Pisa, Italy

3Istituto Italiano di Tecnologia, Graphene labs, Via Morego 30 I‐16163 Genova (Italy) 4Physics Department, Lancaster University, Lancaster LA14YB, United Kingdom

5National Graphene Institute, University of Manchester, Manchester M13 9PL, United Kingdom 6National Enterprise for nanoScience and nanoTechnology, Istituto Nanoscienze‐Consiglio Nazionale delle Ricerche

and Scuola Normale Superiore, I‐56126 Pisa, Italy 7Radboud University, Institute for Molecules and Materials, NL‐6525 AJ Nijmegen, The Netherlands

Graphene hosts a unique electron system that due to weak electron‐phonon scattering allows micrometer‐scale ballistic transport even at room temperature whereas the local equilibrium is provided by frequent electron‐electron collisions. Under these conditions, electrons can behave as a viscous liquid and exhibit hydrodynamic phenomena similar to classical liquids. Here we report unambiguous evidence for this long‐sought transport regime. In particular, doped graphene exhibits an anomalous (negative) voltage drop near current injection contacts, which is attributed to the formation of submicrometer‐size whirlpools in the electron flow. The viscosity of graphene’s electron liquid is found to be an order of magnitude larger than that of honey, in quantitative agreement with many‐body theory. Our work shows a possibility to study electron hydrodynamics using high quality graphene.

Collective behavior of many‐particle systems that undergo frequent inter‐particle collisions has been studied for more than two centuries and is routinely described by the theory of hydrodynamics [1,2]. The theory relies only on the conservation of mass, momentum and energy and is highly successful in explaining the response of classical gases and liquids to external perturbations varying slowly in space and time. More recently, it has been shown that hydrodynamics can also be applied to strongly interacting quantum systems including ultra‐hot nuclear matter and ultra‐cold atomic Fermi gases in the unitarity limit [3‐6].

In principle, the hydrodynamic approach can also be employed to describe many‐electron phenomena in condensed matter physics [7‐13]. The theory becomes applicable if electron‐electron scattering provides the shortest spatial scale in the problem such that ℓee ≪ , ℓ where ℓee is the electron‐electron scattering length, the sample size, ℓ ≡ the mean free path, the Fermi velocity, and the mean free time with respect to momentum‐non‐conserving collisions such as those involving impurities, phonons, etc. The above inequalities are difficult to meet experimentally. Indeed, at low temperatures ( ) ℓee varies approximately as ∝ reaching a micrometer scale below a few K [14], which


3

Figure 1. Viscous backflow in doped graphene. (a,b) Steady‐state distribution of current injected through a narrow slit for a classical conducting medium with zero (a) and a viscous Fermi liquid (b). (c) Optical micrograph of one of our SLG devices. The schematic explains the measurement geometry for vicinity resistance. (d,e) Longitudinal conductivity and for this device as a function of induced by applying gate voltage. 0.3PA; 1Pm. For more detail, see Supplementary Information.

To elucidate hydrodynamics effects, we employed the geometry shown in Fig. 1c. In this case, is injected through a narrow constriction into the graphene bulk, and the voltage drop is measured at the nearby side contacts located only at the distance ~1 Pm away from the injection point. This can be considered as nonlocal measurements, although stray currents are not exponentially small [24]. To distinguish from the proper nonlocal geometry [24], we refer to the linear‐response signal measured in our geometry as “vicinity resistance”, / . The idea is that, in the case of a viscous flow, whirlpools emerge as shown in Fig. 1b, and their appearance can then be detected as sign reversal of

, which is positive for the conventional current flow (Fig. 1a) and negative for viscous backflow (Fig. 1b). Fig. 1e shows examples of for the same SLG device as in Fig. 1d, and other devices exhibited similar behavior (Supplementary Fig. 1). One can see that, away from the charge neutrality point (CNP),

is indeed negative over a large range of and that the conventional behavior is recovered at room .

Figure 2 details our observations further by showing maps , for SLG and BLG. Near room , SLG devices exhibited positive that evolved qualitatively similar to longitudinal resistivity 1/ as expected for a classical electron system. In contrast, was found negative over a large range of

250 K and for away from the CNP. The behavior was slightly different for BLG (Fig. 2b) reflecting the different electronic spectrum but again was negative over a large range of and . Two more

maps are provided in Supplementary Figure 9. In total, six multiterminal devices were investigated showing the vicinity behavior that was highly reproducible for both different contacts on a same device and different devices, although we note that the electron backflow was more pronounced for devices with highest and lowest charge inhomogeneity.

Science 351, 1055 (2016)

Graphene: “a metal that behaves like water”

SM

FL


Strange Metal

Similar memory-function/

hydrodynamic/holographic analyses for strange metal in

cuprates…..

SM

FL


Pseudogap metal matches properties of Z2-FL* phase

We have described a metal with:

A Fermi surface of electrons enclosing volume p, and not the Luttinger volume of 1+p Additional low energy quantum states on a torus not associated with quasiparticle excitations i.e. emergent gauge fields

FL*

We have described a metal with:

A Fermi surface of electrons enclosing volume p, and not the Luttinger volume of 1+p Additional low energy quantum states on a torus not associated with quasiparticle excitations i.e. emergent gauge fields

There is a general and fundamental relationship between these two characteristics. Promising indications that such a metal describes the pseudogap of the cuprate supercondutors

FL*

emergent gauge fields and the high temperature...

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