Quantum matter without
quasiparticles:
random fermion models,
black holes, and graphene
Quantum Matter, Spacetime, and Information,
Yukawa International Seminar,
Yukawa Institute for Theoretical Physics, Kyoto
June 15, 2016
Subir Sachdev
Talk online: sachdev.physics.harvard.edu
HARVARDQuantum matter without quasiparticles:
1. Ground states disconnected from independent
electron states: many-particle entanglement
2. No structure
quasiparticles
2. Quasiparticle
of excited states• Superfluid-insulator transition of ultracold bosonic atoms•••

in an optical lattice
Solvable random fermion Sachdev-Ye-Kitaev (SYK) model
Charged black hole horizons in anti-de Sitter space
Graphene
Note: Most states with long-range entanglement, like thefractional quantum Hall states, do have quasiparticles

Quantum matter without quasiparticles:• Superfluid-insulator transition of•••ultracold bosonic atoms in an opticallattice
Solvable random fermion Sachdev-YeKitaev (SYK) model
Charged black hole horizons in anti-deSitter space
Graphene

Superfluid-insulator transition

Ultracold Rb
atoms - bosons
87

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)

T
Quantum
critical
TKT
Insulator

Superfluid0gccg⇠ U/tT“Boltzmann”
theory of NambuGoldstone TKT
phonons andvortices

Quantum
criticalBoltzmann
theory of quasiparticles/holes

Insulator

Superfluid0gccg⇠ U/tT

Conformal field theory(CFT3) at T>0

TKT

Quantum
Quantummattercriticalwithout
quasiparticles

Insulator

Superfluid0gccg⇠ U/tT

Conformal field theory(CFT3) at T>0

Quantum
Shortest possible “phase coherence”
criticalor local thermal
equilibration time
~
⇠
TKT
kB T
Insulator

Superfluid0gccg⇠ U/tA.V. Chubukov, S. Sachdev, and J. Ye, PRB 49, 11919 (1994); K. Damle and S. Sachdev, PRB 56, 8714 (1997);
S. Sachdev, Quantum Phase Transitions, Cambridge (1999)

Local thermal equilibration orphase coherence time, ⌧' :• There is an lower bound on ⌧' in all many-body quantum
systems of order ~/(kB T ),~⌧' > C,kB T
and the lower bound is realized by systems
without quasiparticles.
• In systems with quasiparticles, ⌧' is parametrically larger
at low T ;e.g. in Fermi liquids ⌧' ⇠ 1/T 2 ,and in gapped insulators ⌧' ⇠ e /(kB T ) whereis theenergy gap.
S. Sachdev, Quantum Phase Transitions, Cambridge (1999)

A bound on quantum chaos:• The time over which a many-body quantum system becomes“chaotic” is given by ⌧L = 1/ L , where L is the “Lyapunov
exponent” determining memory of initial conditions.
D(t) = hW (t)V (0)W (t)V (0)i ⇠ c0

✏ c1 eLt,where we make a (system-dependent) choice to arrange ✏ ⌧ 1.
This Lyapunov time is argued to obey the lower bound⌧L

1 ~2⇡ kB T

A. I. Larkin and Y. N. Ovchinnikov, JETP 28, 6 (1969)
J. Maldacena, S. H. Shenker and D. Stanford, arXiv:1503.01409

A bound on quantum chaos:• The time over which a many-body quantum system becomes“chaotic” is given by ⌧L = 1/ L , where L is the “Lyapunov
exponent” determining memory of initial conditions.
D(t) = hW (t)V (0)W (t)V (0)i ⇠ c0

✏ c1 eLt,where we make a (system-dependent) choice to arrange ✏ ⌧ 1.
This Lyapunov time is argued to obey the lower bound⌧L

1 ~2⇡ kB T

A. I. Larkin and Y. N. Ovchinnikov, JETP 28, 6 (1969)

QuantumJ. Maldacena,
matterS. without
H. Shenker andquasiparticles
D. Stanford, arXiv:1503.01409⇡ fastest possible many-body quantum chaosQuantum matter without quasiparticles:• Superfluid-insulator transition of•••ultracold bosonic atoms in an opticallattice
Solvable random fermion Sachdev-YeKitaev (SYK) model
Charged black hole horizons in anti-deSitter space
Graphene

Infinite-range model with quasiparticles
N
X

1
†
H=
t
c
c
+
.
.
.
ij
j
i
(N )1/2 i,j=1††
ci c j + c j c i =ci cj + cj ci = 0 ,1 X †ci ci = QN iij

tij are independent random variables with tij = 0 and |tij |2 = t2

Fermions occupying the eigenstates of a
N x N random matrix

Infinite-range model with quasiparticles
Feynman graph expansion in tij.. , and graph-by-graph average,
yields exact equations in the large N limit:
G(i!) =

1
i! + µ,⌃(⌧ ) = t2 G(⌧ )⌃(i!)
G(⌧ = 0 ) = Q.

G(!) can be determined by solving a quadratic equation.

Im G(!)µ!

Two-body interactions lead to a scattering time of quasiparticle excitations from in(random) single-particle eigenstates which diverges as ⇠ T 2 at the Fermi level.

Infinite-range (SY) model without quasiparticles
N
M
X
X

1
†
†
H=
J
c
c
c
c
ij
i
j↵
i↵
j
1/2
(N M )
i,j=1 ↵, =1†
ci↵ cjci↵ cj + cj ci↵ = 0 ,1 X †ci↵ ci↵ = Q
M ↵

†
+ cj

ci↵ =

ij ↵

2 = J2
Jij are independent random variables with Jij = 0 and Jij
N ! 1 at M = 2 yields spin-glass ground state.
N ! 1 and then M ! 1 yields critical strange metal

S. Sachdev and J.Ye, Phys. Rev. Lett. 70, 3339 (1993)

SYK model without quasiparticles
N
X

1
† †
H=
J
c
c
c
c
ij;k`
i
j
k
`
3/2
(2N )
i,j,k,`=1
,
1 X †
Q=
ci ci
N i
1

3

i

†
†
ci c j + c j c i =ci cj + cj ci = 0

4

ij

2

J4,5,6,11 6

5

J3,5,7,13

X †
µ
ci ci

7

10

8
9

11

J8,9,12,14

12

14
12

13

Jij;k` are independent random variables with Jij;k` = 0 and |Jij;k` |2 = J 2
N ! 1 yields same critical strange metal; simpler to study numericallyA. Kitaev, unpublished; S. Sachdev, PRX 5, 041025 (2015)

SYK model without quasiparticles
N
X

1
† †
H=
J
c
c
c
c
ij;k`
i
j
k
`
3/2
(2N )
i,j,k,`=1
,
1 X †
Q=
ci ci
N i
1

3

i

†
†
ci c j + c j c i =ci cj + cj ci = 0

4

ij

2

J4,5,6,11 6

5

J3,5,7,13

X †
µ
ci ci

7

10

8
9

11

J8,9,12,14

12

14
12

13

A fermion can move only by entangling with another fermion:
the Hamiltonian has “nothing but entanglement”.
A. Kitaev, unpublished; S. Sachdev, PRX 5, 041025 (2015)

SYK model without quasiparticles
Feynman graph expansion in Jij.. , and graph-by-graph average,
yields exact equations in the large N limit:
G(i!) =

1
i! + µ,⌃(⌧ ) =⌃(i!)
G(⌧ = 0 ) = Q.

2

2

J G (⌧ )G( ⌧ )Low frequency analysis shows that the solutions must be gaplessand obey⌃(z) = µ1p
z + ...
A,A
G(z) = pzefor some complex A. Let us also define ⌃(z)
= ⌃(z)µ.

S. Sachdev and J.Ye, Phys. Rev. Lett. 70, 3339 (1993)

SYK model without quasiparticles

Local fermion density of states
⇢
1/2
!
,!>0
⇢(!) = Im G(!) ⇠
2⇡E
1/2
e
|!|
, ! < 0.
E encodes the particle-hole asymmetry
While E determines the low energy spectrum, it is determined by
the total fermion density Q:
1
Q = (34tanh(2⇡E))1
tan 1 e2⇡E .
⇡S. Sachdev and J.Ye, Phys. Rev. Lett. 70, 3339 (1993)
A. Georges, O. Parcollet, and S. Sachdev Phys. Rev. B 63, 134406 (2001)

SYK model without quasiparticles
At non-zero temperature, T , the Green’s function
also fully determined by E.
✓
i✓iCe
GR (!) =(2⇡T )1 2

where= 1/4 and

◆

i!
+ iE
2⇡T
✓
◆
i!
1
+ iE
2⇡T
sin(⇡ + ✓)
2⇡E
e
=
.
sin(⇡
✓)

GR (!)GA (!)

R

ReG (!)ImGR (!)!
S. Sachdev and J.Ye, Phys. Rev. Lett. 70, 3339 (1993)
A. Georges and O. Parcollet PRB 59, 5341 (1999)
A. Georges, O. Parcollet, and S. Sachdev Phys. Rev. B 63, 134406 (2001)

SYK model without quasiparticles

The entropy per site, S, has a non-zero limit as T ! 0,
and its limiting value obeys
✓
◆
✓
◆
@S
@µ
=
= 2⇡E
@Q T
@T Q
Note that S and E involve low-lying states, while Q depends upon all states, and details of the UV structure
S(Q, T ! 0) = 2⇡

Z f 1 (Q)1dx xf 0 (x) ,f (x) =

(3tanh(2⇡x)
4

tan 1 (e2⇡x )⇡O. Parcollet, A. Georges, G. Kotliar, and A. Sengupta Phys. Rev. B 58, 3794 (1998)
A. Georges, O. Parcollet, and S. Sachdev Phys. Rev. B 63, 134406 (2001)

Infinite-range (SYK) model without quasiparticles
After integrating the fermions, the partition function can be written as a path integralZ
Z = DG(⌧1 , ⌧2 )D⌃(⌧1 , ⌧2 ) exp( N S)
S = ln det [ (⌧1 ⌧2 )(@⌧1 + µ) ⌃(⌧1 , ⌧2 )]
Z
⇥
⇤
2
2
2
+ d⌧1 d⌧2 ⌃(⌧1 , ⌧2 ) G(⌧2 , ⌧1 ) + (J /2)G (⌧2 , ⌧1 )G (⌧1 , ⌧2 )At frequencies ⌧ J, the time derivative in the determinant is lessimportant, and without it the path integral is invariant under the
reparametrization and gauge transformations

A. Georges and O. Parcollet
PRB 59, 5341 (1999)
A. Kitaev, unpublished
S. Sachdev, PRX 5, 041025 (2015)⌧ = f( )00G(⌧1 , ⌧2 ) = [f ( 1 )f ( 2 )]1/4 g( 1 )G( 1 ,g( 2 )
3/4 g( 1 )00⌃(⌧1 , ⌧2 ) = [f ( 1 )f ( 2 )]⌃( 1 ,g( 2 )

where f ( ) and g( ) are arbitrary functions.

2)2)

Infinite-range (SYK) model without quasiparticles
Reparametrization zero mode
Expand about the saddle point solutions, Gs (⌧1 ⌧2 ) and ⌃s (⌧1 ⌧2 ) by
writing G(⌧1 , ⌧2 ) = [f 0 (⌧1 )f 0 (⌧2 )]1/4 Gs (f (⌧1 ) f (⌧2 )) (and similarly for⌃) and obtain an e↵ective action for f (⌧ ). This action does not vanish
because of the time derivative in the determinant which is not reparameterization invariant. However it must vanish for the SL(2,R) transformationa⌧ + bf (⌧ ) =, ad bc = 1
c⌧ + dbecause Gs , ⌃s are invariant under it. In this manner we obtain thee↵ective action as a Schwarzian
N Se↵ =

N4⇡ 2

where the co-efficientZd⌧ {f, ⌧ } ,000f{f, ⌧ } ⌘ 0
f

3
2

✓

00

f
f0

◆2

,

determines the specific heat, C
@S
C=T
=N T
@T

J. Maldacena and D. Stanford, arXiv:1604.07818
See also A. Kitaev, unpublished, and J. Polchinski and V. Rosenhaus, arXiv:1601.06768

Infinite-range (SYK) model without quasiparticles
The Schwarzian describes fluctuations of the energy operator with scaling
dimension h = 2. There are an infinite number of other scalar operatorswith irrational scaling dimensions given by the roots of✓◆⇡(2h 1)
1 2h
tan
=
4
3
)

h = 3.77354 . . . , 5.67946 . . . , 7.63197 . . . , 9.60396 . . . , . . .

J. Maldacena and D. Stanford, arXiv:1604.07818
See also A. Kitaev, unpublished, and J. Polchinski and V. Rosenhaus, arXiv:1601.06768

SYK and AdS2
AdS2 ⇥ R2
ds2 = (d⇣ 2 dt2 )/⇣ 2 + d~x2
Gauge field: A = (E/⇣)dt⇣=1

151602 (2010)

chargedensity Q⇣

Selected for a Viewpoint in Physics
PHYSICAL REVIEW LETTERS~xwee
105, 151602 (2010)
8 OCT

Holographic Metals and the Fractionalized Fermi Liquid
Subir Sachdev
Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA(Received 23 June 2010; published 4 October 2010)
We show that there is a close correspondence between the physical properties of holographic metals
near charged black holes in anti–de Sitter (AdS) space, and the fractionalized Fermi liquid phase of thelattice Anderson model. The latter phase has a ‘‘small’’ Fermi surface of conduction electrons, along witha spin liquid of local moments. This correspondence implies that certain mean-field gapless spin liquids
are states of matter at nonzero density realizing the near-horizon, AdS2 ! R2 physics of ReissnerNordström black holes.

SYK and AdS2
AdS2 ⇥ R2
ds2 = (d⇣ 2 dt2 )/⇣ 2 + d~x2
Gauge field: A = (E/⇣)dt⇣=1⇣

chargedensity Q~x• The non-zero T ! 0 entropy density, S, matches the BekensteinHawking-Wald entropy density of extremal AdS2 horizons, and thedependence of the fermion Green’s function on !, T , and E, matches
that of a Dirac fermion in AdS2 (as computed by T. Faulkner, Hong
Liu, J. McGreevy, and D. Vegh, PRD 83, 125002 (2011)).
S. Sachdev, PRL 105, 151602 (2010)• More recently, it was noted that the relation (@S/@Q)T = 2⇡E also
matches between SYK and gravity, where E, the electric field on thehorizon, also determines the spectral asymmetry of the Dirac fermion.
S. Sachdev, PRX 5, 041025 (2015)

SYK and AdS2
AdS2 ⇥ R2
ds2 = (d⇣ 2 dt2 )/⇣ 2 + d~x2
Gauge field: A = (E/⇣)dt⇣=1⇣

chargedensity Q~x

The same Schwarzian e↵ective action describes low energy
fluctuations on the boundary theory of gravity theories with
AdS2 near-horizon geometries (including the AdS-ReissnerNordstrom solution of Einstein-Maxwell theory in 4 spacetime dimensions). And the co-efficient of the Schwarzian,N /4⇡ 2 , determines the specific heat C = N T .
A. Kitaev, unpublished; A. Almheiri and J. Polchinski, JHEP 1511 (2015) 014; J. Polchinski and V. Rosenhaus,
arXiv: 1601.06768; J. Maldacena and D. Stanford, arXiv:1604.07818; K. Jensen, arXiv:1605.06098; J. Engelsoy,
T.G. Mertens, and H.Verlinde, arXiv:1606.03438; A. Almheiri and B. Kang, arXiv: 1606.04108

SYK and AdS2
AdS2 ⇥ R2
ds2 = (d⇣ 2 dt2 )/⇣ 2 + d~x2
Gauge field: A = (E/⇣)dt⇣=1⇣

chargedensity Q~x

The Schwarzian e↵ective action implies that theSYK model and the AdS2 theories saturate thelower bound on the Lyapunov time
1 ~⌧L =2⇡ kB T
A. Kitaev, unpublished; A. Almheiri and J. Polchinski, JHEP 1511 (2015) 014; J. Polchinski and V. Rosenhaus,
arXiv: 1601.06768; J. Maldacena and D. Stanford, arXiv:1604.07818; K. Jensen, arXiv:1605.06098; J. Engelsoy,
T.G. Mertens, and H.Verlinde, arXiv:1606.03438; A. Almheiri and B. Kang, arXiv: 1606.04108

Quantum matter without quasiparticles:• Superfluid-insulator transition of•••ultracold bosonic atoms in an opticallattice
Solvable random fermion Sachdev-YeKitaev (SYK) model
Charged black hole horizons in anti-deSitter space
GrapheneJesse Crossno
Philip Kim

Kin Chung Fong

Andrew Lucas

GraphenekySame “Hubbard” model as forultracold atoms, but for electrons on
the honeycomb latticekxGraphene
T (K)

600

Quantum critical

500

Dirac liquid

Predicted“strange metal”
without quasiparticles

400∼ √1n (1 + λ ln √Λn )

300
200
100

Hole
Fermi liquid
-1

-0.5

ElectronFermi liquid00.5

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

1n1012 /m2

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” aroundimpurities

Wiedemann-Franz;<

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 toimpurities. Supression of the electronic contribution
to thermal conductivity and hence the separation of
the lattice and electronic parts of conductivity can bedone by application of a transverse magnetic field andhence the Lorenz number can be evaluated. The deviation of the Lorenz number in some degenerate semi8
conductors is attributed to phonon drag. In somec Fluidin Graphene
Thermaland electrical conductivity

with quasiparticles
10 ~

I

I
101

I

0o
Wiedemann-Franz
Law

.I

10 2

Relative electrical conductivity

FigureI l Relative

thermal conductivities, A,
measuredby Fermi liquid:
Wiedemann-Franzlawina
Wiedemann and Franz (AAg assumed to be = 100) and relativeelectrical conductivities, ~, measured by ( 9 Riess, (A) Becquerel,and (V) Lorenz. C~Agassumed to be - 100. After Wiedemann andFranz [1].


L0 = T1017

1 0 16

I

~

'

I

I

1

~

30
c
Nc- 2.5

SiBe3

Se

i

" 2.0
I

I

r

10 3

\
SiGe1
I

t

;

I

H~

Y

AI

/.

/

W1

Cs2

Fe Ir

Coz Li2 AUz

Yb ~o1
t

10 4

1 024

f

I

[

10 6

Mg

Nb~b.,~o/Cu Fi 9 i e

Au3

Ag

Ga

AI2

All

\\\!!

, ,L T, o y t w , p, t

I

Au

As

\."

SiG%

O

I

1 0 23

Cs

Sb
SiGez
~oe
\

t

1 0 22

SiGe2

Bi

~

W·⌦
.
K2

Carrier c o n c e n t r a t i o n (cm 3)
1 019
1 0 2~
1 0 21

1018~'v3"0t
2.5
O 2.0

10 2

2
2
⇡ kB
⇡
⇡
2.45
⇥
10
3e2

Feq z,~/
Cr ''
/
/
2 As
t31~ Bi2
BiI

Er

I

I

[

I

10 e

Electrical c o n d u c t i v i t y

i

I

10 7

K Pt,\\ X~.,
Sb ~

t

Rh

Cq
I

Ni2

I

10 8

I

I

uA
~"

I

10 9

I

!

g
I

101o

(~,~-1 cm-1)

Figure 2 Experimental Lorenz number of elemental metals in the low-temperature residual resistance regime, see Table I. Also shown are ourG.(Table
S. Kumar,
G. Prasad,
R.O.
Pohl,conductivity
J. Mat. Sci.and
28,also4261(1993)
own data points on a doped, degenerate semiconductor
III). Data
are plottedandversuselectricalversuscarrierTransport in Strange Metals
For a strange metal
with a “relativistic” Hamiltonian,hydrodynamic, holographic,
and memory function methods yield
Lorentz ratio L = /(T )vF2 H⌧imp
1=
T 2 Q (1 + e2 vF2 Q2 ⌧imp /(H Q ))2

Q ! electron density; H ! enthalpy density
Q ! quantum critical conductivity⌧imp ! 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.
S. A. Hartnoll, P. K. Kovtun, M. Müller, and S. Sachdev, PRB 76, 144502 (2007)
M. Müller and S. Sachdev, PRB 78, 115419 (2008)

Graphene
T (K)

600

Quantum critical

500

Dirac liquid

Predicted
strange metal400∼ √1n (1 + λ ln √Λn )

300
200
100

Hole
Fermi liquid
-1

-0.5

ElectronFermi liquid00.5

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

1n1012 /m2

Graphene
T (K)

600

Quantum critical

500

Dirac liquid

Predicted
strange metal400∼ √1n (1 + λ ln √Λn )

300
200
100

Hole
Fermi liquid
-1

-0.5

ElectronFermi liquid00.5

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

1n1012 /m2

Graphene
T (K)

600

Quantum critical

500

Dirac liquid

Predicted
strange metal400∼ √1n (1 + λ ln √Λn )

300
200
100

Hole
Fermi liquid
-1

-0.5

ElectronFermi liquid
0

0.5

1

n
1012 /m2

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

)

10

8

100 K
150 K
250 K

100

10
1011 1012

Thermal Conductivity (nW/K)

5

Tbath (K)10

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

6

Tdis

C

D

8
6

75 K

κe

4
2

Tel-ph4

Red dots:
data
2
L
0
T
σ
Blue line: value
L = L00 V for0E

01 40 K0
1 20 K
0-0.5
0 0.5 0∆Vg (V)

6
10 mm

4
2
-0.5 V050
100 150
Tbath (K)

)

10

8

100 K
150 K
250 K

100

10
1011 1012

Thermal Conductivity (nW/K)

5

Tbath (K)10

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

6

Tdis

C

D

8
6

75 K

κe

4
2

Tel-ph4

Red dots:
data
2
L
0
T
σ
Blue line: value
L = L00 V for0E

01 40 K0
1 20 K
0-0.5
0 0.5 0∆Vg (V)

6
10 mm

4
2
-0.5 V050
100 150
Tbath (K)

)

10

8

100 K
150 K
250 K

100

10
1011 1012

Thermal Conductivity (nW/K)

5

Tbath (K)10

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

6

Tdis

C

D

8
6

75 K

κe

4
2

Tel-ph4

Red dots:
data
2
L
0
T
σ
Blue line: value
L = L00 V for0E

01 40 K0
1 20 K
0-0.5
0 0.5 0∆Vg (V)

6
10 mm

4
2
-0.5 V050
100 150
Tbath (K)Graphene
T (K)

600

Quantum critical

500

Dirac liquid

Predicted
strange metal400∼ √1n (1 + λ ln √Λn )

300
200
100

Hole
Fermi liquid
-1

-0.5

ElectronFermi liquid
0

0.5

1

n
1012 /m2

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

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

Dirac Fluid in Graphene

28

Strange metal in graphene

Wiedemann-Franz Law Violations in Experiment

100

20

phonon-limited

90

16

80

12

60
50

L / L0Tbath (K)

70

8

40
30

disorder-limited

4

20
10
−15

0
−10

[Crossno et al, submitted]−505n (109 cm-2)

10

15

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

Dirac Fluid in Graphene

28

Strange metal in graphene

Wiedemann-Franz Law Violations in Experiment

100

20

phonon-limited

90

16

80

12

60
50

L / L0Tbath (K)

70

8

40
30

disorder-limited

4

20
10
−15

0
−10

[Crossno et al, submitted]−505n (109 cm-2)

10

15

Wiedemann-Franz
violated !

20

20

90
16

80

16

V)

50

8

40
30

4

20
10
−15

0
−10

−5

0
5
n (109 cm-2)

102

15

L/L0

12

60

L / L0Tbath (K)70

Temperature (K)

C

10
8

H

e

H (eV/µm2)

100Temperature (K)

C

h

12
8

+V

e
h

-V

6
4
2
0
40

4
0

−6

−4

−2

0

2

60 80 100
T (K)

4

6

n (1010 cm−2)
Tdis

Tel-ph

8

D

6
κe

4
2

σTL 0

0V 0

E

6
10 mm

4
2
-0.5 V0.5 0

050
100 150
Tbath (K)

FIG. 3.LDisorder
Lorentz ratio= /(Tin )the Dirac fluid. (A) Minimum carrier
2 density as a function of temperature for all three samvples.
impAt
low temperature each 1sample is limited by disorder.
F H⌧= At hightemperature2all2samples
become limited 2by thermal
2
2
T
Q (1
+ e linesvF Q/(H)) (B) The
Q eye.
excitations.
Dashedare ⌧aimp
guideto theLorentz ratio of all three samples as a function of bath temQ ! electron
density;H!
enthalpy
densityperature.
The largest
WF violationis seenin the cleanest
sample. (C)The gateconductivity
dependence of the Lorentz ratio is wellcriticalQ ! quantum
fit to hydrodynamic theory of Ref. [5, 6]. Fits of all three⌧imp ! momentum
relaxationfromimpurities
samples are shownat 60 K. Alltime
samplesreturnto the Fermi
liquid value (black dashed line) at high density. Inset shows
S. A. Hartnoll, P. K. Kovtun, M. Müller, and S. Sachdev, PRB 76, 144502 (2007)
the 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 andJ. current
Crossnoal., Science
351, 1058 (2016)
chargein theetneutral
Dirac plasma.

Strange metal in graphene
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. Polini3,61School of Physics & Astronomy, University of Manchester, Oxford Road, Manchester M13 9PL, United Kingdom
2
National Enterprise for nanoScience and nanoTechnology, Scuola Normale Superiore, I‐56126 Pisa, Italy
3
Istituto Italiano di Tecnologia, Graphene labs, Via Morego 30 I‐16163 Genova (Italy)
4
Physics Department, Lancaster University, Lancaster LA14YB, United Kingdom
5
National Graphene Institute, University of Manchester, Manchester M13 9PL, United Kingdom
6
National Enterprise for nanoScience and nanoTechnology, Istituto Nanoscienze‐Consiglio Nazionale delle Ricercheand Scuola Normale Superiore, I‐56126 Pisa, Italy
7
Radboud 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 isprovided 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 reportunambiguous evidence for this long‐sought transport regime. In particular, doped graphene exhibits
L. Levitov
and G.
Falkovich,
NaturePhysics
onlineFIG.
1: Current
streamlines
and injection
potential
mapwhichforvisFIG.to2:theNoan anomalous(negative)
voltagedrop arXiv:1508.00836,
near current
contacts,
is attributedcous
and ohmic whirlpools
flows. Whitelines show
current
streamviscosity
formation ofsubmicrometer‐sizein the electron
flow. The
viscosityof graphene’s
electron rStrange metal in graphene

Science 351, 1055 (2016)

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. Polini3,61School of Physics & Astronomy, University of Manchester, Oxford Road, Manchester M13 9PL, United Kingdom
2
National Enterprise for nanoScience and nanoTechnology, Scuola Normale Superiore, I‐56126 Pisa, Italy
3
Istituto Italiano di Tecnologia, Graphene labs, Via Morego 30 I‐16163 Genova (Italy)
4
Physics Department, Lancaster University, Lancaster LA14YB, United Kingdom
5
National Graphene Institute, University of Manchester, Manchester M13 9PL, United Kingdom
6
National Enterprise for nanoScience and nanoTechnology, Istituto Nanoscienze‐Consiglio Nazionale delle Ricercheand Scuola Normale Superiore, I‐56126 Pisa, Italy
7
Radboud 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 isprovided by frequent electron‐electron collisions. Under these conditions, electrons can behave as a
Figure 1. Viscous backflow in doped graphene. (a,b) Steady‐state distribution of current injected throughliquids.
we report
viscousexhibithydrodynamic
phenomena
to classicala narrowliquidslit for anda classicalconducting
medium withzero(a)similar
and a viscous
Fermi liquid(b). Here(c) Opticalunambiguous
evidencelong‐soughttransport
regime.
In particular,geometry
doped graphene
micrograph of oneof ourfor
SLGthisdevices.
The schematic
explains
the measurement
for vicinityexhibits
resistance.
(d,e) (negative)
Longitudinalvoltage
conductivity
forinjection
this device
as a function
induced by to theananomalous
drop nearandcurrent
contacts,
whichof is attributed
applying gate
0.3 PA; whirlpools
1 Pm. Forinmore
detail, seeflow.
Supplementary
Information.
formationof voltage.
submicrometer‐size
the electron
The viscosityof graphene’s electron

Entangled quantum matter without quasiparticles• No quasiparticle excitations• Shortest possible “phase coherence” time, fastest possible local
equilibration time, or fastest possible Lyapunov time towards~
quantum chaos, all of orderkB T• Realization in solvable SYK model, which saturates the lower
bound on the Lyapunov time. Its properties have some similarities to non-rational, large central charge CFT2s.
• Remarkable match between SYK and quantum gravity of black
holes with AdS2 horizons, including a SL(2,R)-invariant Schwarziane↵ective action for thermal energy fluctuations.
• Experiments on graphene agree well with predictions of a theoryof a nearly relativistic quantum liquid without quasiparticles.