High-Tc superconductivity in
doped antiferromagnets (I)
Zheng-Yu Weng
Institute for Advanced Study
Tsinghua University, Beijing
KITPC, AdS/CM duality
Nov. 4, 2010
Outline
• Introduction: High-Tc experimental phenomenology
pseudogap phenomenon
• High-Tc 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
High-Tc cuprate superconductors
Mueller
Bednorz
Megahype
Scientists: dreaming about instant fame
1990 March meeting: 30
sessions in parallel!
over 100000 papers
Woodstock of physics
Business people: getting rich!
Nature of superconducting state?
300
ZF3t17O90PP22T0P
Bi2212, Tc=91K
T=17 K
EDC Intensity (A. U.)
250
200
What is the essential elementary excitation
deciding the superconducting transition?
150
100
50
0
-0.4
sharp Bogoliubov QP peak
-0.3
-0.2
-0.1
E - EF (eV)
0.0
0.1
(laser ARPES, XJ Zhou, et al.)
BCS theory for superconductivity
electron pairing by “glueon”: phonon, AF fluctuations, …
Tc  0e

EF
1
  *
 -- coupling constant
Fermi sea
 * -- Coulomb
pseudopotential
0 -- characteristic energy of the glueon
Strong coupling theory
Pb:
Nb3Ge:
Tc=7.19 K
Tc=21.2 K
λ=1.55, μ*=0.13,
λ=1.73, μ*=0.12,
High-Tc cuprates: Tc ~ 160 K
typical energy scales:
ω0=4.8 meV
ω0=10.7 meV
FeAs based superconductors: Tc ~56 K
D ~ 300K
J ~ 1,500K
Phase diagram of cuprate superconductors
300
ZF3t17O90PP22T0P
Bi2212, Tc=91K
T=17 K
EDC Intensity (A. U.)
250
200
New state of matter?
(non-Fermi-liquid)
150
100
50
0
-0.4
-0.3
-0.2
-0.1
E - EF (eV)
0.0
0.1
sharp Bogoliubov QP peak
(laser ARPES, XJ Zhou, et al.)
Landau paradigm
F
ARPES
 k , Zk
Fermi sea
Fermi surface of copper
Fermi degenerate temperature
TF  EF / kB
EF ~ 1eV  10,000K
T  TF
typical Fermi liquid behavior:
C v  T
Sommerfeld constant
 s  const .
Pauli susceptibility
1 / T1  T
Korringa behavior
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:
La2-xSrxCuO4
Specific heat
Spin susceptibility
(T. Nakano, et al. (1994))
(Loram et al. 2001)
T  TF
NMR spin-lattice relaxation rate
typical Fermi liquid behavior:
C v  T
Sommerfeld constant
 s  const .
Pauli susceptibility
1 / T1  T
Korringa behavior
(T. Imai et al. (1993))
Uniform spin susceptibility
Fermi liquid
Heisenberg model
F
no indication of Pauli susc.
J
T. Nakano, et al. PRB49, 16000(1994)
Resistivity measurement
T. Nakano, et al. PRB49,
16000(1994)
T. Shibauchi, et al.
(2001)
T. Imai et al., PRL 70 (1993)
Guo-qing Zheng et al. PRL (2005)
Kawasaki, et al. PRL (2010)
NMR 1/T1
L. Taillefer- arXiv
1003.2972
Optical measurement
Photoemission
Y.S. Lee et al. PRB 72, (2005)
Nernst effect
Vortex Nernst effect and diamagnetism in the pseudogap regime
B
v
-T
Xu et al., Nature (2000),
Wang et al., PRB (2001).
Uemura’s Plot: BEC?

nodal quasiparticle excitations
s (T )  s (0)  aT
P.A. Lee and X.G. Wen (1997)
ns
 s
m*
Phase fluctuations
L
1
2
2
d
r





s
2
Emery & Kivelson,
(1995)
“resonant mode” in neutron exp.
P.C. Dai et al, 2007
“resonant mode” in neutron scattering
Raman scattering experiment
Sacuto& Bourges’ Group, 2002
Raman scattering in A1g
channel
Two sets of experiments
Eg
k BTc
6
Pseudogap phase
T
~ J/kB
strange metal: maximal scattering
Pseudogap:
T0
New quantum state
of matter
strong AF correlations
A non-Fermi-liquid
Tc  T0
upper pseudogap phase
TN
T*
Tv
lower pseudogap phase
strong SC fluctuations
Tc
x
antiferromagnetic order
QCP
d-wave superconducting order
Cuprates = doped Mott Insulator
Anderson, Science 1987
T
~ J/kB
T0
TN
T*
Tv
Half-filling:
Mott insulator
Tc
x=0
QCP
x
Half-filling: Mott Insulator/Heisenberg
antiferromagnet
Mott insulator
F
F
Heisenberg model
F
on-site Coulomb repulsion U causes a Mott insulator
H=J
 Si · Sj
Half-filling: Low-energy physics is described by Heisenberg model
Pure CuO2 plane
H=J
Si · S j

nn
large J = 135 meV
quantum spin S =1/2
neutron scattering
Raman scattering
Spin flip breaks 6
bonds, costs 3J.
J ～ 135 meV
Antiferromagnetism at x=0 is well described by the
Heisenberg model
inverse spin-spin correlation length
Chakaravarty, Halperin, Nelson
PRL (1988)
Heisenberg model
1

H J  J  S i  S j  
ij  
4
J: superexchange coupling
n





n
Z  Tr  e  H   Tr  

H
 J  
 n n!


  J
high-T expansion
Si  S j RVB
ij
 Si  S j

Mott insulator
＝1/ k BT

1

2
3
RVB
4
ij
i
 －
j
i

j

Resonating Valence Bond (RVB)
+
+
RVB pair
≡

1
    
2
…

P. W. Anderson, Science, 235, 1196 (1987)
Ground state at half-filling
A spin singlet pair
1
2
(ij ) 
Bosonic RVB wavefunction
RVB 
W (i
iA, jB
1

i

j
 i 
j


Liang, Doucot, Anderson, PRL (1988)
 j1 )    W (in  j n )(i1 j1 )    (in j n )
variationa l energy
EG  -0.3344
m  0.30
exact numerics
EG  -0.3346
m  0.31
Good understanding of the Mott
antiferromagnet/paramagnet at half-filling!
Cuprates = doped Mott Insulator
T
~ J/kB
T0
TN
T*
Tv
Half-filling:
Mott insulator
Tc
x=0
QCP
x
Doping the Mott Insulator/ antiferromagnet
F
Mott insulator
doped Mott insulator
F
F
F
Heisenberg model
t-J model
The cuprates are doped Mott insulators
Single band Hubbard model, or its strong
coupling limit, the t-J model
Pure CuO2 plane
Dope
holes
t

H=J
Si · Sj

nn
J
no double occupancy constraint PG :
t3J
cc 1


†
i
i
A minimal model for doped Mott insulators: t-J model
hopping
superexchange
cc 1



i
i
Mottness and intrinsic guage invariance
cc 1



i
Conservations of spin and charge separately:
Spin-charge separation and emergent gauge fields
in low-energy action !
i
Fermion signs
Fermi sea
F
Antisymmetry of wave function
ARPES
Fermi surface of copper
Landau-Fermi liquid behavior
k ,
1/  k
Zk
k  F
2
C v  T
Sommerfeld coefficient
 ~ const.
Pauli susceptibility
1 / T1  T
Korringa behavior
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

Zt  J  Tr e  H J

A complete basis
   n

n
 Tr  
  H J  
 n n!


such that
Marshall sign rule
Heisenberg model
1

H J  J  S i  S j  
ij  
4
c  1
Total disapperance of fermion signs!
Ground state at half-filling
A spin singlet pair
1
2
(ij ) 
Bosonic RVB wavefunction
RVB 
W (i
iA, jB
1

i

j
 i 
j


Liang, Doucot, Anderson, PRL (1988)
 j1 )    W (in  j n )(i1 j1 )    (in j n )
variationa l energy
EG  -0.3344
m  0.30
exact numerics
EG  -0.3346
m  0.31
Disappearance of the fermion signs at half-filling
Reduced fermion signs in doped case:
single hole case

(1) Nc  (1)  (1)  (1) 
 c
+
-
-
Phase String Effect
-
+
+
+
+
-
+
-
-
loop c
+
+
 c  (1) N

h (c)
D. N. Sheng, et al. PRL (1996)
K.Wu, ZYW, J. Zaanen (2008)
Exact phase
string
effect in the t-J
Exact
sign
structure
ofmodel
the t-J model

Z  Tr e
 H t  J
   W (C )
arbitrary doping, temperature
dimenions
C
C
 C  (1)
2t 2t 2t ( J / 2) n
W (C )   ... 
 M h  M  ,n  0
J 
J 
J n
n!

h
Nh  N ex
M h (C )
M h (C )
= total steps of hole hoppings
M  (C ) = total number of spin exchange processes
h
N ex ( C )  N h  K ( C ) - number of hole loops
For a given path C:
+
+
-
+
-
-
+
+
-
-
-
+
+
+
+
+
+
+
-
-
+
-
Single-particle propagator
－
＋
＋
＋
＋
－
－
－
＋
－
－
Phase string factor
＋
－
－
Goldstone theorem for the ground state energy
phase string factor
Cuprates as doped Mott insulators
Overdoping: Recovering
more fermion signs
Mott insulator: No fermion signs
Doped Mott insulator:
Reduced fermion signs
Charge-spin entanglement induced by phase string
-+
+
+
-
+
-
+
+
AFM state
+
+
+
+
+
+
Nagaoka state
an extreme case ignoring
the superexchange energy
+
-
+
+
-
+
+
+
+
-
+
-
+
+
+
-
+
+
RVB/Pseudogap
-
+
-
-
minimizing the total exchange
and kinetic energy
Summary
• Pseudogap state is firmly established by experiment
as one of the most exotic phases in the cuprates
which is closely related to high-Tc 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