Exploring quantum matter in the high temperature...

HARVARD

International Center for Theoretical SciencesBangalore

June 20, 2015

Subir Sachdev

Talk online: sachdev.physics.harvard.edu

Exploring quantum matter in the high temperature

superconductors

YBa2Cu3O6+x

High temperature superconductors

CuO2 plane

Cu

O

“Undoped”Anti-

ferromagnet

FilledBand

“Undoped”Anti-

ferromagnet

Anti-ferromagnetwith p holesper square

Anti-ferromagnetwith p holesper square

But relative to the band

insulator, there are 1+ p holesper square

SM

FL

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

SM

Antiferromagnet

FL

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

SM

High temperature Superconductor

FL

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

Conventional metal

Area enclosed by Fermi surface =1+p

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

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

Pseudogap

“Fermi arcs” at

low p

Kyle M. Shen, F. Ronning, D. H. Lu, F. Baumberger, N. J. C. Ingle, W. S. Lee, W. Meevasana,Y. Kohsaka, M. Azuma, M. Takano, H. Takagi, Z.-X. Shen, Science 307, 901 (2005)

SM

FL

Strange metal

Metal(gapless,

compressible state)

without quasi-

particles

SM

FL

1. The pseudogap metal

Fermi liquid co-existening with topological order

2. The strange metal

Metal without quasiparticles

Infinite-range model: dual to extremal charged

black holes and yields

Bekenstein-Hawking entropy

Outline

Pseudogap

“Fermi arcs” at

low p

Kyle M. Shen, F. Ronning, D. H. Lu, F. Baumberger, N. J. C. Ingle, W. S. Lee, W. Meevasana,Y. Kohsaka, M. Azuma, M. Takano, H. Takagi, Z.-X. Shen, Science 307, 901 (2005)

SM

FL

A new metal — a fractionalized

Fermi liquid (FL*) — with electron-like quasiparticles on a Fermi surface of size p coexisting

with topological order

Y. Qi and S. Sachdev, Phys. Rev. B 81, 115129 (2010)M. Punk, A. Allais, and S. Sachdev, arXiv:1501.00978

SM

FL

Pseudogap

Fractionalized Fermi liquid (FL*)

T. Senthil, S. S., M. Vojta Phys. Rev. Lett. 90, 216403 (2003) R. K. Kaul, A. Kolezhuk, M. Levin, S. S., and T. Senthil, Phys. Rev. B 75, 235122 (2007)

E. G. Moon and S. S. Phys. Rev. B 83, 224508 (2011); M. Punk, A. Allais, and S. S., arXiv:1501.00978.

= | ⇥⇤⌅ � | ⇤⇥⌅

Realizes a metal with a

Fermi surface of

area p co-existing

with “topological

order”

Fractionalized Fermi liquid (FL*)

T. Senthil, S. S., M. Vojta Phys. Rev. Lett. 90, 216403 (2003) R. K. Kaul, A. Kolezhuk, M. Levin, S. S., and T. Senthil, Phys. Rev. B 75, 235122 (2007)

E. G. Moon and S. S. Phys. Rev. B 83, 224508 (2011); M. Punk, A. Allais, and S. S., arXiv:1501.00978.

= | ⇥⇤⌅ � | ⇤⇥⌅

Realizes a metal with a

Fermi surface of

area p co-existing

with “topological

order”

Fractionalized Fermi liquid (FL*)

T. Senthil, S. S., M. Vojta Phys. Rev. Lett. 90, 216403 (2003) R. K. Kaul, A. Kolezhuk, M. Levin, S. S., and T. Senthil, Phys. Rev. B 75, 235122 (2007)

E. G. Moon and S. S. Phys. Rev. B 83, 224508 (2011); M. Punk, A. Allais, and S. S., arXiv:1501.00978.

= | ⇥⇤⌅ � | ⇤⇥⌅

Realizes a metal with a

Fermi surface of

area p co-existing

with “topological

order”

T. Senthil, S. S., M. Vojta Phys. Rev. Lett. 90, 216403 (2003) R. K. Kaul, A. Kolezhuk, M. Levin, S. S., and T. Senthil, Phys. Rev. B 75, 235122 (2007)

E. G. Moon and S. S. Phys. Rev. B 83, 224508 (2011); M. Punk, A. Allais, and S. S., arXiv:1501.00978.

A fermionic “dimer” describing a “bonding” orbital between two sites

Realizes a metal with a

Fermi surface of

area p co-existing

with “topological

order”

Fractionalized Fermi liquid (FL*)

T. Senthil, S. S., M. Vojta Phys. Rev. Lett. 90, 216403 (2003) R. K. Kaul, A. Kolezhuk, M. Levin, S. S., and T. Senthil, Phys. Rev. B 75, 235122 (2007)

E. G. Moon and S. S. Phys. Rev. B 83, 224508 (2011); M. Punk, A. Allais, and S. S., arXiv:1501.00978.

= | ⇥⇤⌅ � | ⇤⇥⌅

Realizes a metal with a

Fermi surface of

area p co-existing

with “topological

order”

Fractionalized Fermi liquid (FL*)

T. Senthil, S. S., M. Vojta Phys. Rev. Lett. 90, 216403 (2003) R. K. Kaul, A. Kolezhuk, M. Levin, S. S., and T. Senthil, Phys. Rev. B 75, 235122 (2007)

E. G. Moon and S. S. Phys. Rev. B 83, 224508 (2011); M. Punk, A. Allais, and S. S., arXiv:1501.00978.

= | ⇥⇤⌅ � | ⇤⇥⌅

Realizes a metal with a

Fermi surface of

area p co-existing

with “topological

order”

Fractionalized Fermi liquid (FL*)

T. Senthil, S. S., M. Vojta Phys. Rev. Lett. 90, 216403 (2003) R. K. Kaul, A. Kolezhuk, M. Levin, S. S., and T. Senthil, Phys. Rev. B 75, 235122 (2007)

E. G. Moon and S. S. Phys. Rev. B 83, 224508 (2011); M. Punk, A. Allais, and S. S., arXiv:1501.00978.

= | ⇥⇤⌅ � | ⇤⇥⌅

Realizes a metal with a

Fermi surface of

area p co-existing

with “topological

order”

Fractionalized Fermi liquid (FL*)

T. Senthil, S. S., M. Vojta Phys. Rev. Lett. 90, 216403 (2003) R. K. Kaul, A. Kolezhuk, M. Levin, S. S., and T. Senthil, Phys. Rev. B 75, 235122 (2007)

E. G. Moon and S. S. Phys. Rev. B 83, 224508 (2011); M. Punk, A. Allais, and S. S., arXiv:1501.00978.

= | ⇥⇤⌅ � | ⇤⇥⌅

Realizes a metal with a

Fermi surface of

area p co-existing

with “topological

order”

Fractionalized Fermi liquid (FL*)

T. Senthil, S. S., M. Vojta Phys. Rev. Lett. 90, 216403 (2003) R. K. Kaul, A. Kolezhuk, M. Levin, S. S., and T. Senthil, Phys. Rev. B 75, 235122 (2007)

E. G. Moon and S. S. Phys. Rev. B 83, 224508 (2011); M. Punk, A. Allais, and S. S., arXiv:1501.00978.

= | ⇥⇤⌅ � | ⇤⇥⌅

Realizes a metal with a

Fermi surface of

area p co-existing

with “topological

order”

Fractionalized Fermi liquid (FL*)

T. Senthil, S. S., M. Vojta Phys. Rev. Lett. 90, 216403 (2003) R. K. Kaul, A. Kolezhuk, M. Levin, S. S., and T. Senthil, Phys. Rev. B 75, 235122 (2007)

E. G. Moon and S. S. Phys. Rev. B 83, 224508 (2011); M. Punk, A. Allais, and S. S., arXiv:1501.00978.

= | ⇥⇤⌅ � | ⇤⇥⌅

Realizes a metal with a

Fermi surface of

area p co-existing

with “topological

order”

Fractionalized Fermi liquid (FL*)

T. Senthil, S. S., M. Vojta Phys. Rev. Lett. 90, 216403 (2003) R. K. Kaul, A. Kolezhuk, M. Levin, S. S., and T. Senthil, Phys. Rev. B 75, 235122 (2007)

E. G. Moon and S. S. Phys. Rev. B 83, 224508 (2011); M. Punk, A. Allais, and S. S., arXiv:1501.00978.

= | ⇥⇤⌅ � | ⇤⇥⌅

Realizes a metal with a

Fermi surface of

area p co-existing

with “topological

order”

Fractionalized Fermi liquid (FL*)

T. Senthil, S. S., M. Vojta Phys. Rev. Lett. 90, 216403 (2003) R. K. Kaul, A. Kolezhuk, M. Levin, S. S., and T. Senthil, Phys. Rev. B 75, 235122 (2007)

E. G. Moon and S. S. Phys. Rev. B 83, 224508 (2011); M. Punk, A. Allais, and S. S., arXiv:1501.00978.

= | ⇥⇤⌅ � | ⇤⇥⌅

Realizes a metal with a

Fermi surface of

area p co-existing

with “topological

order”

Place pseudogap metal on a

torus;

Topological order

Place pseudogap metal on a

torus;obtain

“topological” states nearly degenerate

with the ground state: change sign of every dimer

across red line

Topological order

= | ⇥⇤⌅ � | ⇤⇥⌅

Topological order Place

pseudogap metal on a

torus;obtain

“topological” states nearly degenerate

with the ground state: change sign of every dimer

across red line

= | ⇥⇤⌅ � | ⇤⇥⌅

-1 -1-1-1

Topological order Place

pseudogap metal on a

torus;obtain

“topological” states nearly degenerate

with the ground state: change sign of every dimer

across red line

= | ⇥⇤⌅ � | ⇤⇥⌅

-1-1

Topological order Place

pseudogap metal on a

torus;obtain

“topological” states nearly degenerate

with the ground state: change sign of every dimer

across red line

= | ⇥⇤⌅ � | ⇤⇥⌅

-1 -1-1-1

Topological order Place

pseudogap metal on a

torus;obtain

“topological” states nearly degenerate

with the ground state: change sign of every dimer

across red line

A new metal — a fractionalized

Fermi liquid (FL*) — with electron-like quasiparticles on a Fermi surface of size p coexisting

with topological order

Y. Qi and S. Sachdev, Phys. Rev. B 81, 115129 (2010)M. Punk, A. Allais, and S. Sachdev, arXiv:1501.00978

SM

FL

Pseudogap

1. The pseudogap metal

Fermi liquid co-existening with topological order

2. The strange metal

Metal without quasiparticles

Infinite-range model: dual to extremal charged

black holes and yields

Bekenstein-Hawking entropy

Outline

Strange metal

Metal(gapless,

compressible state)

without quasi-

particles

SM

FL

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

cicj + cjci = 0

cic†j + c†jci = �ij

Jij;k` independent

random numbers

An infinite-range model of a strange metal

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

S. Sachdev, arXiv:1506.05111

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

Local fermion density of states

⇢(!) ⇠⇢

!�1/2 , ! > 0

e�2⇡E |!|�1/2, ! < 0.

cicj + cjci = 0

cic†j + c†jci = �ij

Jij;k` independent

random numbers

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

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

Local fermion density of states

⇢(!) ⇠⇢

!�1/2 , ! > 0

e�2⇡E |!|�1/2, ! < 0.

cicj + cjci = 0

cic†j + c†jci = �ij

Jij;k` independent

random numbers

A. Georges, O. Parcollet, and S. Sachdev Phys. Rev. B 63, 134406 (2001)

Q =1

4(3� tanh(2⇡E))� 1

⇡tan�1

�e2⇡E

�

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

Microscopic zero temperature

entropy density, S, obeys@S@Q = 2⇡E

Local fermion density of states

⇢(!) ⇠⇢

!�1/2 , ! > 0

e�2⇡E |!|�1/2, ! < 0.

cicj + cjci = 0

cic†j + c†jci = �ij

Jij;k` independent

random numbers

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)

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

Microscopic zero temperature

entropy density, S, obeys@S@Q = 2⇡E

Local fermion density of states

⇢(!) ⇠⇢

!�1/2 , ! > 0

e�2⇡E |!|�1/2, ! < 0.

Einstein-Maxwell theory

+ cosmological constant

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

Microscopic zero temperature

entropy density, S, obeys@S@Q = 2⇡E

Local fermion density of states

⇢(!) ⇠⇢

!�1/2 , ! > 0

e�2⇡E |!|�1/2, ! < 0.

Einstein-Maxwell theory

+ cosmological constant

Boundary

area Ab;

charge

density Q

⇣~x

Horizon area Ah;AdS2 ⇥R

d

ds

2 = (d⇣2 � dt

2)/⇣2 + d~x

2

Gauge field: A = (E/⇣)dt

⇣ = 1

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

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

Microscopic zero temperature

entropy density, S, obeys@S@Q = 2⇡E

Local fermion density of states

⇢(!) ⇠⇢

!�1/2 , ! > 0

e�2⇡E |!|�1/2, ! < 0.

Einstein-Maxwell theory

+ cosmological constant

Boundary

area Ab;

charge

density Q

⇣~x

Horizon area Ah;AdS2 ⇥R

d

ds

2 = (d⇣2 � dt

2)/⇣2 + d~x

2

Gauge field: A = (E/⇣)dt

⇣ = 1

L = �↵D↵ +m

Local fermion density of states

⇢(!) ⇠⇢

!�1/2 , ! > 0

e�2⇡E |!|�1/2, ! < 0.

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

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

Microscopic zero temperature

entropy density, S, obeys@S@Q = 2⇡E

Local fermion density of states

⇢(!) ⇠⇢

!�1/2 , ! > 0

e�2⇡E |!|�1/2, ! < 0.

Einstein-Maxwell theory

+ cosmological constant

Boundary

area Ab;

charge

density Q

⇣~x

Horizon area Ah;AdS2 ⇥R

d

ds

2 = (d⇣2 � dt

2)/⇣2 + d~x

2

Gauge field: A = (E/⇣)dt

⇣ = 1

L = �↵D↵ +m

Local fermion density of states

⇢(!) ⇠⇢

!�1/2 , ! > 0

e�2⇡E |!|�1/2, ! < 0.

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

of spacetime far from the AdS2

S. Sachdev, arXiv:1506.05111

Eliminate r0 between

Q =rd�10

p2d [(d� 1)R2 + (d+ 1)r20]

2gF

E =gF r0

p2d [(d� 1)R2 + (d+ 1)r20]

2 [(d� 1)2R2 + d(d+ 1)r20]

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

Microscopic zero temperature

entropy density, S, obeys@S@Q = 2⇡E

Local fermion density of states

⇢(!) ⇠⇢

!�1/2 , ! > 0

e�2⇡E |!|�1/2, ! < 0.

Einstein-Maxwell theory

+ cosmological constant

Boundary

area Ab;

charge

density Q

⇣~x

Horizon area Ah;AdS2 ⇥R

d

ds

2 = (d⇣2 � dt

2)/⇣2 + d~x

2

Gauge field: A = (E/⇣)dt

⇣ = 1

L = �↵D↵ +m

Local fermion density of states

⇢(!) ⇠⇢

!�1/2 , ! > 0

e�2⇡E |!|�1/2, ! < 0.

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

of spacetime far from the AdS2

S. Sachdev, arXiv:1506.05111

Black hole thermodynamics

(classical GR) yields

1

Ab

@Ah

@Q = 8⇡GNE

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

Microscopic zero temperature

entropy density, S, obeys@S@Q = 2⇡E

Local fermion density of states

⇢(!) ⇠⇢

!�1/2 , ! > 0

e�2⇡E |!|�1/2, ! < 0.

Einstein-Maxwell theory

+ cosmological constant

Boundary

area Ab;

charge

density Q

⇣~x

Horizon area Ah;AdS2 ⇥R

d

ds

2 = (d⇣2 � dt

2)/⇣2 + d~x

2

Gauge field: A = (E/⇣)dt

⇣ = 1

L = �↵D↵ +m

Local fermion density of states

⇢(!) ⇠⇢

!�1/2 , ! > 0

e�2⇡E |!|�1/2, ! < 0.

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

of spacetime far from the AdS2

S. Sachdev, arXiv:1506.05111

Black hole thermodynamics

(classical GR) yields

1

Ab

@Ah

@Q = 8⇡GNE

Combination:

S =

Ah

4GNAb

SM

FL

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

Pseudogap Strange metal

1. The pseudogap metal

Fermi liquid co-existening with topological order

2. The strange metal

Metal without quasiparticles

Infinite-range model: dual to extremal charged

black holes and yields

Bekenstein-Hawking entropy