Exploring quantum matter in the high temperature superconductorsqpt.physics.harvard.edu/talks/icts15.pdf · 2015-06-21 · Exploring quantum matter in the high temperature superconductors

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


SM

High temperature Superconductor

FL


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


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”




= | ⇥⇤⌅ � | ⇤⇥⌅


Fermi surface of

area p co-existing

with “topological

order”




= | ⇥⇤⌅ � | ⇤⇥⌅


Fermi surface of

area p co-existing

with “topological

order”



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


Fermi surface of

area p co-existing

with “topological

order”




= | ⇥⇤⌅ � | ⇤⇥⌅


Fermi surface of

area p co-existing

with “topological

order”




= | ⇥⇤⌅ � | ⇤⇥⌅


Fermi surface of

area p co-existing

with “topological

order”




= | ⇥⇤⌅ � | ⇤⇥⌅


Fermi surface of

area p co-existing

with “topological

order”




= | ⇥⇤⌅ � | ⇤⇥⌅


Fermi surface of

area p co-existing

with “topological

order”




= | ⇥⇤⌅ � | ⇤⇥⌅


Fermi surface of

area p co-existing

with “topological

order”




= | ⇥⇤⌅ � | ⇤⇥⌅


Fermi surface of

area p co-existing

with “topological

order”




= | ⇥⇤⌅ � | ⇤⇥⌅


Fermi surface of

area p co-existing

with “topological

order”




= | ⇥⇤⌅ � | ⇤⇥⌅


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



across red line

= | ⇥⇤⌅ � | ⇤⇥⌅

-1 -1-1-1



torus;obtain



across red line

= | ⇥⇤⌅ � | ⇤⇥⌅

-1-1



torus;obtain



across red line

= | ⇥⇤⌅ � | ⇤⇥⌅

-1 -1-1-1



torus;obtain



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








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


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


⇢(!) ⇠⇢

!�1/2 , ! > 0

e�2⇡E |!|�1/2, ! < 0.

cicj + cjci = 0


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



Microscopic zero temperature

entropy density, S, obeys@S@Q = 2⇡E


⇢(!) ⇠⇢

!�1/2 , ! > 0

e�2⇡E |!|�1/2, ! < 0.

cicj + cjci = 0


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






⇢(!) ⇠⇢

!�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






⇢(!) ⇠⇢

!�1/2 , ! > 0

e�2⇡E |!|�1/2, ! < 0.



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






⇢(!) ⇠⇢

!�1/2 , ! > 0

e�2⇡E |!|�1/2, ! < 0.



Boundary

area Ab;

charge

density Q

⇣~x


d

ds

2 = (d⇣2 � dt

2)/⇣2 + d~x

2


⇣ = 1

L = �↵D↵ +m


⇢(!) ⇠⇢

!�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






⇢(!) ⇠⇢

!�1/2 , ! > 0

e�2⇡E |!|�1/2, ! < 0.



Boundary

area Ab;

charge

density Q

⇣~x


d

ds

2 = (d⇣2 � dt

2)/⇣2 + d~x

2


⇣ = 1

L = �↵D↵ +m


⇢(!) ⇠⇢

!�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


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






⇢(!) ⇠⇢

!�1/2 , ! > 0

e�2⇡E |!|�1/2, ! < 0.



Boundary

area Ab;

charge

density Q

⇣~x


d

ds

2 = (d⇣2 � dt

2)/⇣2 + d~x

2


⇣ = 1

L = �↵D↵ +m


⇢(!) ⇠⇢

!�1/2 , ! > 0

e�2⇡E |!|�1/2, ! < 0.




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






⇢(!) ⇠⇢

!�1/2 , ! > 0

e�2⇡E |!|�1/2, ! < 0.



Boundary

area Ab;

charge

density Q

⇣~x


d

ds

2 = (d⇣2 � dt

2)/⇣2 + d~x

2


⇣ = 1

L = �↵D↵ +m


⇢(!) ⇠⇢

!�1/2 , ! > 0

e�2⇡E |!|�1/2, ! < 0.




Black hole thermodynamics

(classical GR) yields

1

Ab

@Ah

@Q = 8⇡GNE

Combination:

S =

Ah

4GNAb

SM

FL


Pseudogap Strange metal








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Exploring quantum matter in the high temperature superconductorsqpt.physics.harvard.edu/talks/icts15.pdf · 2015-06-21 · Exploring quantum matter in the high temperature superconductors