and the phases of matter - Harvard Universityqpt.physics.harvard.edu/talks/unc16.pdfQuantum entanglement and the phases of matter University of North Carolina, Chapel Hill April 18,

Quantum entanglement and the

phases of matter

University of North Carolina, Chapel HillApril 18, 2016

Subir Sachdev

HARVARD

Talk online: sachdev.physics.harvard.edu

Foundations of quantum many body theory:

1. Ground states connected adiabatically toindependent electron states

2. Quasiparticle structure of excited states

E

MetalMetal

carryinga current

InsulatorSuperconductor

k

E

MetalMetal

carryinga current


k

Metals

E

MetalMetal

carryinga current


k

E

MetalMetal

carryinga current


k

Metals



2. Boltzmann-Landau theory of quasiparticles

E

MetalMetal

carryinga current


k

E

MetalMetal

carryinga current


k

Metals




A Guide to Feynman Diagrams in the Many-body Problem, R.D. Mattuck, Dover (1992)

Quantum entanglement:

EPR pair: Non-local correlations between quantum measurements due to superposition

between many-electron states

_

Hydrogen molecule




_




_

E

MetalMetal

carryinga current


k

E

MetalMetal

carryinga current


k

Metals




A Guide to Feynman Diagrams in the Many-body Problem, R.D. Mattuck, Dover (1992)

Modern phases of quantum matter:

1. Ground states disconnected from independentelectron states: many-particle entanglement


Famous example:

The fractional quantum Hall effect of electrons in two dimensions (e.g. in graphene) in the presence of a

strong magnetic field. The ground state is described by Laughlin’s wavefunction, and the excitations are

quasiparticles which carry fractional charge.

Modern phases of quantum matter:


2. Quasiparticle structure of excited states2. No quasiparticles

Quantum matter without quasiparticles:


2. Quasiparticle structure of excited states2. No quasiparticles

• Superfluid-insulator transition of ultracold bosonic atoms in an optical lattice

• Graphene

• Strange metals in high temperature superconductors

• Quark-gluon plasma

• Charged black hole horizons in anti-de Sitter space



• Graphene




M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415, 39 (2002). Xibo Zhang, Chen-Lung Hung, Shih-Kuang Tung, and Cheng Chin, Science 335, 1070 (2012)

Ultracold 87Rbatoms - bosons

Superfluid-insulator transition

U t

|Ground statei =Y

i

b†i |0i

Insulator (the vacuum) at large repulsion between bosons

On-site repulsion between bosons = UTunneling amplitude between sites = t

Excitations of the insulator:

U t



U t



U t



U t



U t



U t


|Ground statei ="X

i

b†i

#N

|0i

U t

Superfluid at small repulsion between bosons


g

T

gc

0

InsulatorSuperfluid

Quantumcritical

TKT

c

h i 6= 0 h i = 0

U/t

g

T

gc

0

InsulatorSuperfluid

Quantumcritical

TKT

c

h i 6= 0 h i = 0

! a complex field representing the

Bose-Einstein condensate of the superfluid

U/t

g

T

gc

0

InsulatorSuperfluid

Quantumcritical

TKT

c

h i 6= 0 h i = 0

S =

Zd2rdt

|@t |2 c2|rr |2 V ( )

V ( ) = ( c)| |2 + u| |2

2

U/t

Long-range quantum entanglement

and no quasiparticles:

A conformal field theory

in 2+1 spacetime dimensions:

a CFT3

g

T

gc

0

InsulatorSuperfluid

Quantumcritical

TKT

c U/t

g

T

gc

0

InsulatorSuperfluid

Quantumcritical

TKT

c

“Boltzmann” theory of Nambu-

Goldstone phonons and

vortices

Boltzmann theory of

particles/holes

U/t

g

T

gc

0

InsulatorSuperfluid

Quantumcritical

TKT

c

CFT3 at T>0

U/t

g

T

gc

0

InsulatorSuperfluid

Quantumcritical

TKT

c

CFT3 at T>0

Quantum matterwithout

quasiparticles

U/t

g

T

gc

0

InsulatorSuperfluid

Quantumcritical

TKT

c

CFT3 at T>0

U/t

Shortest possible

local thermal equilibration

time ~kBT

K. Damle and S. Sachdev, PRB 56, 8714 (1997); S. Sachdev, Quantum Phase Transitions, Cambridge (1999)



• Graphene




Andrew Lucas

Philip KimJesse Crossno

Kin Chung Fong

kx

ky

Graphene

Same “Hubbard” model as for ultracold atoms, but for electrons on

the honeycomb lattice

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

D. E. Sheehy and J. Schmalian, PRL 99, 226803 (2007)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

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

Predictedstrange metal

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

scattering events

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

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

S. A. Hartnoll, P. K. Kovtun, M. Müller, and S. Sachdev, PRB 76, 144502 (2007)M. Müller and S. Sachdev, PRB 78, 115419 (2008)

Transport in Strange Metals For a strange metalwith a “relativistic” Hamiltonian,hydrodynamic, holographic,and memory function methods yieldLorentz ratio L = /(T)

=v2FHimp

T 2Q

1

(1 + e2v2FQ2imp/(HQ))2

Q ! electron density; H ! enthalpy densityQ ! quantum critical conductivityimp ! momentum relaxation time from impurities.Note that for a clean system (imp ! 1 first),the Lorentz ratio diverges L 1/Q4,as we approach “zero” electron density at the Dirac point.

-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



-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



-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



Impurity scattering dominates

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 R

versus 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

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

Red dots: data

Blue line: value for L = L0

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







Red dots: data


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







Red dots: data


-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



Impurity scattering dominates


Wiedemann-Franz Law Violations in Experiment

0

4

8

12

16

20L

/ L0

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]

Wiedemann-Franz obeyed

Strange metal in grapheneJ. Crossno et al., Science 351, 1058 (2016)


Wiedemann-Franz Law Violations in Experiment

0

4

8

12

16

20L

/ L0

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]

Wiedemann-Franz violated !

Strange metal in grapheneJ. Crossno et al., Science 351, 1058 (2016)

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




[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


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

Strange metal in graphene

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

Strange metal in graphene

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”


• Graphene





SM

FL

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

YBa2Cu3O6+x

SM

FLProposed a SU(2) gauge theory for

transtion for quantum-criticality at optimal doping, as the

origin of strange metal (SM) behavior

at higher T

Quark-gluon plasma


• Graphene





depth ofentanglement

D-dimensionalspace

Tensor network representation of entanglement at quantum critical point d

G. Vidal, Phys. Rev. Lett. 99, 220405 (2007)


D-dimensionalspace

Tensor network representation of entanglement at quantum critical point

Emergent direction of an equivalent theory of quantum gravity Brian Swingle, arXiv:0905.1317

d

String theory near a D-brane


D-dimensionalspace

Emergent directionof AdS4

d

Brian Swingle, arXiv:0905.1317

Infinite-range (SYK) model of a strange metal

H =1

(2N)3/2

NX

i,j,k,`=1

Jij;k` c†i c

†jckc` µ

X

i

c†i ci

cicj + cjci = 0 , cic†j + c†jci = ij

Q =1

N

X

i

c†i ci

1 2

3 45 6

78

9

10

1112

12 13

14

J3,5,7,13J4,5,6,11

J8,9,12,14

A. Kitaev, unpublished; S. Sachdev, PRX 5, 041025 (2015)

S. Sachdev and J. Ye, PRL 70, 3339 (1993)

Jij;k` are independent random variables with Jij;k` = 0 and |Jij;k`|2 = J2

N ! 1 yields critical strange metal.

Infinite-range (SYK) model of a strange metal

H =1

(2N)3/2

NX

i,j,k,`=1

Jij;k` c†i c

†jckc` µ

X

i

c†i ci

cicj + cjci = 0 , cic†j + c†jci = ij

Q =1

N

X

i

c†i ci

1 2

3 45 6

78

9

10

1112

12 13

14

J3,5,7,13J4,5,6,11

J8,9,12,14

A. Kitaev, unpublished; S. Sachdev, PRX 5, 041025 (2015)

S. Sachdev and J. Ye, PRL 70, 3339 (1993)

A fermion can move only by entangling with another fermion:the Hamiltonian has “nothing but entanglement”.

Infinite-range strange metals

E encodes the particle-hole asymmetry

S. Sachdev and J. Ye, Phys. Rev. Lett. 70, 3339 (1993)A. Georges, O. Parcollet, and S. Sachdev Phys. Rev. B 63, 134406 (2001)

Local fermion density of states

(!) = ImG(!)

!1/2 , ! > 0

e2E |!|1/2, ! < 0.

While E determines the low energy spectrum, it is determined by

the total fermion density Q:

Q =

1

4

(3 tanh(2E)) 1

tan

1e2E

.

Analog of the relationship between Q and kF in a Fermi liquid.

Q =1

N

X

i

Dc†i ci

E.

H =1

(2N)3/2

NX

i,j,k,`=1

Jij;k` c†i c

†jckc`

1 23 4

5 6

78

9

10

11 12

12 1314

J3,5,7,13J4,5,6,11

J8,9,12,14

Known ‘equation of state’

determines E as a function of Q


(!)

!1/2 , ! > 0

e2E |!|1/2, ! < 0.

Einstein-Maxwell theory

+ cosmological constant

Boundary

area Ab;

charge

density Q

~x

Horizon area Ah;AdS2 R

d

ds

2 = (d2 dt

2)/2 + d~x

2

Gauge field: A = (E/)dt

= 1

L = ↵D↵ +m


(!)

!1/2 , ! > 0

e2E |!|1/2, ! < 0.

‘Equation of state’ relating Eand Q depends upon the geometry

of spacetime far from the AdS2

A. Sen, arXiv:hep-th/0506177; S. Sachdev, arXiv:1506.05111

Black hole thermodynamics

(classical general relativity) yields

@SBH

@Q = 2E

Q =1

N

X

i

Dc†i ci

E.

H =1

(2N)3/2

NX

i,j,k,`=1

Jij;k` c†i c

†jckc`

1 23 4

5 6

78

9

10

11 12

12 1314

J3,5,7,13J4,5,6,11

J8,9,12,14



Microscopic zero temperature

entropy density, S, obeys@S@Q = 2E


(!)

!1/2 , ! > 0

e2E |!|1/2, ! < 0.



Boundary

area Ab;

charge

density Q

~x


d

ds

2 = (d2 dt

2)/2 + d~x

2


= 1

L = ↵D↵ +m


(!)

!1/2 , ! > 0

e2E |!|1/2, ! < 0.






@SBH

@Q = 2E

Holographic gravity theoryStart with simplest theory of Einstein gravity and Maxwell electromagnetism

SEM =

Zd

d+2x

pg

1

22

R+

d(d+ 1)

L

2 R

2

g

2F

F

2

Holographic gravity theoryStart with simplest theory of Einstein gravity and Maxwell electromagnetism

SEM =

Zd

d+2x

pg

1

22

R+

d(d+ 1)

L

2 R

2

g

2F

F

2

Solve equations of motion in the the presence of a

d-dimensional flat boundary with charge density Q

+

++

++

+Electric flux

hQi6= 0

rAdS-Reissner-Nordstrom Quantummatter on

the boundary with

a variable charge

densityQ of a global

U(1) symmetry.

A. Chamblin, R. Emparan, C. V. Johnson, and R. C. Myers, PRD 60, 064018 (1999)

Charged black branes

Realizes a strange metal: a state with an unbroken global U(1)

symmetry with a continuously variable charge density, Q, at

T = 0 which does not have any quasiparticle excitations.

+

++

++

+Electric flux

hQi6= 0

rAdS-Reissner-Nordstrom Quantummatter on

the boundary with

a variable charge

densityQ of a global

U(1) symmetry.

Quantum fields on charged black branes

~x = 1

AdS2 R

d

ds

2 = (d2 dt

2)/2 + d~x

2


chargedensity Q

E encodes the particle-hole asymmetry

T. Faulkner, Hong Liu, J. McGreevy, and D. Vegh, PRD 83, 125002 (2011)

AdS2 boundary Green’s function of at T = 0

ImG(!)

!(12) , ! > 0

e2E |!|(12), ! < 0.

where the fermion scaling dimension is a function of m

Dirac fermion of mass m

Q =1

N

X

i

Dc†i ci

E.

H =1

(2N)3/2

NX

i,j,k,`=1

Jij;k` c†i c

†jckc`

1 23 4

5 6

78

9

10

11 12

12 1314

J3,5,7,13J4,5,6,11

J8,9,12,14






(!)

!1/2 , ! > 0

e2E |!|1/2, ! < 0.



Boundary

area Ab;

charge

density Q

~x


d

ds

2 = (d2 dt

2)/2 + d~x

2


= 1

L = ↵D↵ +m


(!)

!1/2 , ! > 0

e2E |!|1/2, ! < 0.






@SBH

@Q = 2E

Q =1

N

X

i

Dc†i ci

E.

H =1

(2N)3/2

NX

i,j,k,`=1

Jij;k` c†i c

†jckc`

1 23 4

5 6

78

9

10

11 12

12 1314

J3,5,7,13J4,5,6,11

J8,9,12,14






(!)

!1/2 , ! > 0

e2E |!|1/2, ! < 0.



Boundary

area Ab;

charge

density Q

~x


d

ds

2 = (d2 dt

2)/2 + d~x

2


= 1

L = ↵D↵ +m


(!)

!1/2 , ! > 0

e2E |!|1/2, ! < 0.



Q =1

N

X

i

Dc†i ci

E.

H =1

(2N)3/2

NX

i,j,k,`=1

Jij;k` c†i c

†jckc`

1 23 4

5 6

78

9

10

11 12

12 1314

J3,5,7,13J4,5,6,11

J8,9,12,14






(!)

!1/2 , ! > 0

e2E |!|1/2, ! < 0.



Boundary

area Ab;

charge

density Q

~x


d

ds

2 = (d2 dt

2)/2 + d~x

2


= 1

L = ↵D↵ +m


(!)

!1/2 , ! > 0

e2E |!|1/2, ! < 0.



A. Sen hep-th/0506177; S. Sachdev PRX 5, 041025 (2015)



@SBH

@Q = 2E

Q =1

N

X

i

Dc†i ci

E.

H =1

(2N)3/2

NX

i,j,k,`=1

Jij;k` c†i c

†jckc`

1 23 4

5 6

78

9

10

11 12

12 1314

J3,5,7,13J4,5,6,11

J8,9,12,14




(!)

!1/2 , ! > 0

e2E |!|1/2, ! < 0.



Boundary

area Ab;

charge

density Q

~x


d

ds

2 = (d2 dt

2)/2 + d~x

2


= 1

L = ↵D↵ +m


(!)

!1/2 , ! > 0

e2E |!|1/2, ! < 0.



S. Sachdev, PRL 105, 151602 (2010); PRX 5, 041025 (2015)



@SBH

@Q = 2E



Evidence for

AdS2 gravity

dual of H

S. H. Shenker and D. Stanford, arXiv:1306.0622

J. Maldacena, S. H. Shenker and D. Stanford, arXiv:1503.01409

A. Kitaev, unpublished J. Polchinski and V. Rosenhaus, arXiv:1601.06768

A bound on quantum chaos:

• The “Lyapunov exponent” for chaos, L, is given

by out-of-time-order correlators, and for quan-

tum systems near equilibrium, it obeys the bound

L 2kBT/~.

• This bound is saturated by holographic theories

with Einstein gravity. This makes it similar to

the /s > 1/(4) bound.

• The bound is also saturated by the SYK model








L 2kBT/~.












L 2kBT/~.






• Graphene




Entangled quantum matter without quasiparticles

Entangled quantum matter without quasiparticles

• No quasiparticle excitations

• Shortest possible “collision time”, or more precisely, fastest

possible local equilibration time ~kBT

• Continuously variable density, Q(conformal field theories are usually at fixed density, Q = 0)

• Theory built from hydrodynamics/holography

/memory-functions/strong-coupled-field-theory

• Exciting experimental realization in graphene.

• Future work: detection of hydrodynamic flow in other strange

metals . . . . . .

Documents

and the phases of matter - Harvard Universityqpt.physics.harvard.edu/talks/unc16.pdfQuantum entanglement and the phases of matter University of North Carolina, Chapel Hill April 18,