doped antiferromagnets (I)

awfulhihatUrban and Civil

Nov 15, 2013 (3 years and 8 months ago)

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Zheng
-
Yu Weng


Institute for Advanced Study

Tsinghua

University, Beijing



KITPC,
AdS
/CM duality

Nov. 4, 2010

High
-
T
c


superconductivity in
doped
antiferromagnets

(I)

Outline


Introduction: High
-
T
c

experimental phenomenology


pseudogap

phenomenon





High
-
T
c

cuprates

as doped Mott insulators /doped
antiferromagnets


exact sign structure


Pseudogap

state as an RVB state and the slave
-
boson
approach


electron fractionalization and gauge degrees of freedom


Reduced
fermion

signs in doped Mott insulator:


pseudogap

-

emergent mutual Chern
-
Simons gauge fields



Conclusion

Mueller

Bednorz

High
-
T
c

cuprate

superconductors

Megahype

Scientists: dreaming about instant fame

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Business people: getting rich!

over 100000
papers

1990 March meeting: 30

sessions
in
parallel!

Nature of superconducting state?

What is the essential elementary excitation

deciding the superconducting transition?

300
250
200
150
100
50
0
EDC Intensity (A. U.)
-0.4
-0.3
-0.2
-0.1
0.0
0.1
E - E
F
(eV)
ZF3t17O90PP22T0P
Bi2212, Tc=91K
T=17 K
sharp
Bogoliubov

QP peak

(laser ARPES, XJ Zhou, et al.)

BCS
theory for superconductivity



electron pairing by “
glueon
”: phonon, AF fluctuations,



Pb: T
c
=7.19 K
λ
=1.55,
μ
*=0.13,
ω
0
=4.8 meV

Nb
3
Ge: T
c
=21.2 K
λ
=1.73,
μ
*=0.12,
ω
0
=10.7 meV

Strong coupling theory

*

--

Coulomb

pseudopotential

F
E
High
-
T
c

cuprates
:
T
c

~ 160 K
F
eAs

based superconductors:
T
c

~56 K



--

coupling constant

0

--

characteristic energy of the
glueon

Fermi sea

typical energy scales:

*
1
0








e
T
c
K
J
K
D
500
,
1
~

300
~

Phase diagram of
cuprate

superconductors

New state of matter?

(non
-
Fermi
-
liquid)

300
250
200
150
100
50
0
EDC Intensity (A. U.)
-0.4
-0.3
-0.2
-0.1
0.0
0.1
E - E
F
(eV)
ZF3t17O90PP22T0P
Bi2212, Tc=91K
T=17 K
sharp
Bogoliubov

QP peak

(laser ARPES, XJ Zhou, et al.)

,
k

k
Z
Pauli susceptibility

Korringa behavior


Landau paradigm

ARPES

Sommerfeld constant

Fermi degenerate temperature

/
F F B
T E k

Fermi sea

F

typical Fermi liquid behavior:

F
T
T

T
T
const
T
C
s
v



1
/
1
.


K
eV
E
F
000
,
10
1
~

Fermi surface of copper

Paradigm in crisis

Landau’s Fermi
-
liquid: state of interacting electron
system in metals =
Fermi gas of quasiparticles.

Quasiparticle:

Fermion with S =1/2, momentum k, energy E(k)

QP Fermi surface:

La
2
-
x
Sr
x
CuO
4

Spin susceptibility (T
. Nakano, et al. (1994
))

Specific heat (
Loram

et al. 2001)

NMR spin
-
lattice relaxation rate (T
. Imai et al. (1993
))

Pauli susceptibility

Korringa behavior

Sommerfeld constant

typical Fermi liquid behavior:

F
T
T

T
T
const
T
C
s
v



1
/
1
.


T. Nakano, et al. PRB49, 16000(1994)

F

Fermi liquid

Heisenberg model

Uniform spin susceptibility

no indication
of Pauli
susc
.

J

T. Nakano, et al. PRB49,

16000(1994)

Resistivity measurement

T.
Shibauchi
,
et al.

(2001)

Guo
-
qing

Zheng et al. PRL (2005)

T. Imai et al., PRL 70 (1993)

Kawasaki,
et al.
PRL (2010)

1003.2972

L.
Taillefer
-

arXiv


Photoemission

Y.S. Lee et al. PRB 72, (2005)

Optical measurement

NMR 1/T
1

Nernst effect

Xu et al., Nature (2000),


Wang et al., PRB (2001).

B

v

-

T

Vortex Nernst
effect and diamagnetism in the
pseudogap

regime


Uemura’s

Plot: BEC?



2
2
1
2
s
L d r
 
 

Emery &
Kivelson
,
(1995)

P.A. Lee and X.G.
Wen

(1997)

Phase fluctuations

( ) (0)
s s
T aT
 
 
*
s
s
n
m

 
nodal
quasiparticle

excitations

“resonant mode” in neutron exp.

“resonant mode” in neutron scattering

P.C. Dai et al, 2007

Raman scattering experiment

Sacuto
& Bourges’ Group, 2002

Raman scattering in A
1g
channel



Two sets of experiments

6

c
B
g
T
k
E
d
-
wave
superconducting order

T

T
0

x

antiferromagnetic

order

~
J/
k
B

strong SC fluctuations

strong AF correlations

upper
pseudogap

phase



lower
pseudogap

phase



Pseudogap

phase

strange
metal: maximal scattering

T
*

T
N

T
v

T
c

QCP

Pseudogap
:


New quantum state

of matter




A non
-
Fermi
-
liquid

0
T
T
c

T

T
0

x

~
J/
k
B

T
*

T
N

T
v

T
c

QCP

Half
-
filling:

Mott insulator

x=0

Anderson, Science 1987


Cuprates

= doped Mott Insulator

Half
-
filling: Mott Insulator/Heisenberg


antiferromagnet


Mott insulator

Heisenberg model

F

F

F


H
= J



S
i



S
j

on
-
site Coulomb repulsion U causes a Mott insulator

Pure CuO
2

plane


H
= J



S
i



S
j

nn

large
J

= 135
meV


quantum
spin
S

=1/2

Half
-
filling: Low
-
energy physics is described by Heisenberg model

neutron scattering

Raman scattering

Spin flip breaks 6
bonds, costs 3J.

J



135
meV

Chakaravarty, Halperin, Nelson


PRL (1988)

inverse spin
-
spin correlation length

Antiferromagnetism

at x=0 is well described by the


Heisenberg model

1
4
J i j
ij
H J
 
 
  

 
 
S S
Heisenberg model







Tr Tr 1/
!
n
n
H
J B
n
Z e H k T
n




 
  
 
 
 





high
-
T expansion



1
RVB
2
3
RVB
4
i j i j
ij
i j i j
ij
      
 
S S S S

J:
superexchange

coupling

Mott insulator

J


Resonating Valence Bond (RVB)


+

+










2
1
P. W. Anderson, Science,
235
, 1196 (1987)

RVB pair



)
(
)
)(
(
)
(
RVB
1
1
,
1
1
n
n
B
j
A
i
n
n
j
i
j
i
j
i
W
j
i
W












Bosonic

RVB

wavefunction

Liang
,
Doucot
, Anderson, PRL (1988)



1
( )
2
i i
j j
ij
     
31
.
0

3346
0

numerics
exact
30
.
0

3344
0
energy

l
variationa




m

.
-

E
m

.
-

E
G
G
Ground state at half
-
filling

A spin singlet pair

Good understanding of the Mott

antiferromagnet
/paramagnet at half
-
filling!

T

T
0

x

~
J/
k
B

T
*

T
N

T
v

T
c

QCP

Half
-
filling:

Mott insulator

x=0


Cuprates

= doped Mott Insulator

Doping the Mott Insulator/
antiferromagnet

Mott insulator


doped Mott insulator

Heisenberg model

t
-
J model

F

F

F

F

The
cuprates

are doped Mott insulators

Pure CuO
2

plane

Single band Hubbard model, or its strong


coupling limit, the t
-
J model





Dope

holes

t

J

t



3
J


H
= J



S
i



S
j

nn

no double occupancy constraint


: 1
G i i
P c c
 



hopping

superexchange

A minimal model for doped Mott insulators: t
-
J model

1






i
i
c
c
1






i
i
c
c
Mottness

and intrinsic
guage

invariance

Conservations of spin and charge separately:




Spin
-
charge separation and emergent gauge fields

in low
-
energy action !

Fermion signs

Antisymmetry of wave function

Fermi sea

ARPES

,
k

k
Z
2
1/
k k F
  

T
T
T
C
v


1
/
1
const.

~


Pauli susceptibility

Korringa behavior

Sommerfeld coefficient

F

Landau
-
Fermi

liquid

behavior

Fermi surface of copper

Fermion signs in Feynman‘s path
-
integral

Imaginary time path
-
integral formulation of
partition function:

Fermion

signs

Absence of fermion signs at half
-
filling

Mott insulator

Heisenberg model

1
4
J i j
ij
H J
 
 
  

 
 
S S






Tr Tr
!
J
n
n
H
t J J
n
Z e H
n




 
  
 
 
 

A complete basis

such that

1
c


Total
disapperance

of
fermion

signs!

Marshall sign rule

)
(
)
)(
(
)
(
RVB
1
1
,
1
1
n
n
B
j
A
i
n
n
j
i
j
i
j
i
W
j
i
W












Bosonic

RVB

wavefunction

Liang
,
Doucot
, Anderson, PRL (1988)



1
( )
2
i i
j j
ij
     
31
.
0

3346
0

numerics
exact
30
.
0

3344
0
energy

l
variationa




m

.
-

E
m

.
-

E
G
G
Ground state at half
-
filling

A spin singlet pair

Disappearance of the
fermion

signs at half
-
filling

Reduced
fermion

signs in doped case:
single hole case

+

-

-

Phase String Effect

-

-

+

+

-

-

-

+

+

+

+

+

( )
( 1)
h
N c
c


 
( 1) ( 1) ( 1) ( 1)

c
N
c


       

Phase String Effect

D. N. Sheng, et al. PRL (1996)


K.Wu, ZYW, J. Zaanen (2008)

loop c

( 1)
h h
ex
N N
C



 
0
!
)
2
/
(
2
...
2
2
)
(
,
)
(






n
n
M
M
n
h
C
h
M
n
J
J
t
J
t
J
t
C
W







+

-


+

+


-

+

+

+

+

+

+

+

+

+

-

-

-

-

-

-

-

-

-

-

+

= total steps of hole hoppings

)
(
C
M

= total number of spin exchange processes

)
(
C
M
h
For a given path C:

)
(
)
(
C
K
N
C
N
h
h
ex


-

number of hole loops

Exact phase string effect in the t
-
J model

Exact sign structure of the t
-
J model








C
C
H
C
W
e
Z
J
t
)
(
Tr


arbitrary doping, temperature

dimenions


Phase string factor





























Single
-
particle propagator

Goldstone theorem for the ground state energy

phase string factor

Cuprates as doped Mott insulators

Mott insulator: No fermion signs

Doped Mott insulator:

Reduced fermion signs

Overdoping: Recovering

more fermion signs

-

-

+

+

-

-

-

+

+

+

-

+


Nagaoka state

+

+

+

+

+

+

+

+

+

+

+

+

an extreme case ignoring

the superexchange energy

-

-

+

+

-

+

-

-

+

+

-

+

-

+


RVB/
Pseudogap


minimizing the total exchange

and kinetic energy

Charge
-
spin entanglement induced by phase string


AFM state

-

+

Summary



Pseudogap

state is firmly established by experiment


as one of the most exotic phases in the
cuprates


which is closely related to high
-
T
c

superconductivity




Doped Mott insulator/
antiferromagnet

provides a suitable


microscopic model to understand the
pseudogap

physics




Mott constraint leads to a new sign structure greatly


reduced from the
fermion

signs at low doping