Mott Physics, Sign Structure, and
High-Tc Superconductivity
Zheng-Yu Weng
Institute for Advanced Study
Tsinghua University, Beijing
Newton Institute, Cambridge 2013.9.16
Outline
• Introduction to basic experimental phenomenology of high-Tc cuprates
• High-Tc cuprates as doped Mott insulators /doped antiferromagnets
• Basic principles: Mott physics and sign structure
• Nontrivial examples:
(1) one-hole case
(2) finite doping and global phase diagram
(3) ground state wavefunction
• Summary and conclusion
Discovery of high-Tc superconductors
Mueller
Bednorz
1986
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:
Cv  T
Sommerfeld constant
 s  const.
Pauli susceptibility
1 / T1  T
Korringa behavior
La2-xSrxCuO4
Specific heat
Spin susceptibility
(T. Nakano, et al. (1994))
(Loram et al. 2001)
Fermi liquid behavior:
NMR spin-lattice relaxation rate
Cv  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)
uniform susceptibility, resistivity
NMR 1/T1
Optical measurement
Photoemission
Nernst effect
Underdoped phase diagram
T
~ J/kB
Pseudogap:
T0
New quantum state
of matter?
strong AF correlations
A non-Fermi-liquid
Tc  T0
strange metal: maximal
scattering
TN
T*
Tv
lower pseudogap phase
strong SC fluctuations
Tc
0
antiferromagnetic order
FL
QCP
d-wave superconducting order
x
Outline
• Introduction to basic experimental phenomenology of high-Tc
cuprates
• High-Tc cuprates as doped Mott insulators /doped
antiferromagnets
• Basic principles: Mott physics and sign structure
• Nontrivial examples:
(1) one-hole case
(2) finite doping and global phase diagram
(3) ground state wavefunction
• Summary and conclusion
Cuprates = doped Mott Insulator
Anderson, Science 1987
T
~ J/kB
one-band large-U Hubbard model:
T0
TN
T*
Tv
Half-filling:
Mott insulator
Tc
x=0
QCP
x
Mott Insulator/ antiferromagnet
F
Mott insulator
doped Mott insulator
F
F
F
Heisenberg model
t-J model
A minimal model for doped Mott insulators: t-J model
hopping
superexchange
cc 1



i
i
Half-filling: Low-energy physics is described by Heisenberg model
Pure CuO2 plane
H=J
å Si · S j
<ij>
large J = 135 meV
quantum spin S =1/2
charge localization
at low doping
ARPES result: A broad peak at x=0
Ando et al, PRL 87, 017001 (2001)
K. M. Shen et al, PRL 93, 267002 (2004)
Doping the Mott Insulator/ antiferromagnet
La-Sr-Cu-O
La-Bi2201
Peng, et al., arXiv:1302.3017
(2013)
Sebastian, et al., Reports on
progress in physics 75, 102501
(2012)
Doping the Mott Insulator/ antiferromagnet
La-Sr-Cu-O
charge localization
La-Bi2201
Peng, et al., arXiv:1302.3017
(2013)
Sebastian, et al., Reports on
progress in physics 75, 102501
(2012)
Questions
• If charge localization is intrinsic in a doped Mott insulator
with AFLRO?
• If charge delocalization (superconductivity) arises by
destroying the AFLRO?
• Is localization-delocalization the underlying driving force
or the T=0 phase diagram of the underdoped cuprates?
Outline
• Introduction to basic experimental phenomenology of highTc cuprates and high-Tc cuprates as doped Mott insulators
/doped antiferromagnets
• Basic principles: Mott physics and sign structure
• Nontrivial examples:
(1) one-hole case
(2) finite doping and global phase diagram
(3) ground state wavefunction
• Summary and conclusion
Statistical sign structure for Fermion systems
Fermion signs
y (x1, x2,...) = -y (x2, x1,...)
Landau Fermi Liquid
(
) å (-1)
Z FG º Tr e- b H =
loop c
N ex (c)
W(c) , W(c) ³ 0
interacting fermions: fractal nodes F. Kruger and J. Zaanen, (2008)
y (x1, x2,...) = -y (x2, x1,...)
nodal hypersurface
Pauli hypersurface
Nodal hypersurface
d=2
Test particle
(1) Fermi liquid: Fermion signs
Z FG =
N ex (c)
(-1)
W (c) W(c) ³ 0
å
loop c
(2) Off Diagonal Long Rang Order (ODLRO):
compensating the Fermion signs
Bose condensation
y (x1, x2,...) = +y (x2, x1,...)
Cooper pairing in SC state
CDW (“exciton” condensation)
SDW (weak coupling)
normal state: Fermi liquid
Antiferromagnetic order (strong coupling)
Z Heisenberg =
å W (c)
loop c
(b J / 2)n
W (c) = å
d M-¯ +MQ ,n ³ 0
n!
n
Complete disappearance of Fermion signs!
(3) Single-hole doped Heiserberg model:
Z1-hole =
å t W (c)
c
loop c
t c º (+1) ´ (-1) ´ (-1) ´ ......
= ( -1)
N h¯ (c)
2t 2t 2t (b J / 2)n
W (c) = × ... å
d M h +M-¯ ,n ³ 0
J J J n
n!
M h (C )
＋
－
Phase string effect
D.N. Sheng, Y.C. Chen, ZYW, PRL (1996)
(4) Exact sign structure of the t-J model
at arbitrary doping, dimensions, temperature
M h (C )
= total steps of hole hoppings
M  (C ) = total number of spin exchange processes
M Q (C )
= total number of opposite spin encounters
Wu, Weng, Zaanen, PRB (2008)
For a given path c:
+
-
-
+
-
-
+
+
-
+
+
+ +
+
-
(-)
+
(-)3
+
-
+
-
-
+
K. Wu, ZYW, J. Zaanen, PRB (2008)
Emergent gauge force in doped Mott insulators!
Z=
å t W (c)
c
W(c) ³ 0
loop c
(+1) ´ (-1) ´ (-1) ´...... º t c
Mutual Chern-Simons gauge theory
Kou, Qi, ZYW PRB (2005);
ZYW et al (1997) (1998)
Ye, Tian, Qi, ZYW, PRL (2011); Nucl. Phys. B (2012)
Nonintegrable phase factor:
B
Pe
e
i
Am dx m
c
ò
A
“An intrinsic and complete description of electromagnetism” A
“Gauge symmetry dictates the form of the fundamental forces in nature”
C. N. Yang (1974) , Wu and Yang (1975)
B
“smooth” paths good for
mean-field treatment
singular quantum phase
interference
New guiding principles:
•
Mott physics = phase string sign structure replacing the Fermion signs
•
Strong correlations = charge and spin are long-range entangled
•
Sign structure + restricted Hilbert space = unique fractionalization
Outline
• Introduction to basic experimental phenomenology of highTc cuprates and high-Tc cuprates as doped Mott insulators
/doped antiferromagnets
• Basic principles: Sign structure and Mott physics
• Nontrivial examples:
(1) one-hole case
(2) finite doping and global phase diagram
(3) ground state wavefunction
• Summary and conclusion
DMRG numerical study
Z. Zhu, H-C Jiang, Y. Qi, C.S. Tian, ZYW,
Scientific Report 3, 2586 (2013）
t-J ladder systems
Effect of phase string effect
no phase string effect
t^ = 0
Self-localization of the hole!
σ
Removing the phase string:
A sign-free model
σ
Z=
å t W (c)
c
loop c
tc º1
no phase string effect!
2t 2t 2t (b J / 2)n
W (c) = × ... å
d M h +M-¯ ,n ³ 0
J J J n
n!
M h (C )
Momentum distribution
Quasiparticle picture restored!
without phase string effect
localization-delocalization transition
t’
t
Theoretical understading of self-localization of
the one-hole in 2D
-+
+
-
-
-
+
-
+
+
+
+
-
destructive quantum phase interference
leads to self-localization
D.N. Sheng, et al. PRL (1996);
Holon localization at low doping:
ZYW, et al. PRB (2001)
S.P. Kou, ZYW, PRL (2003)
T.-P. Choy and Philip Phillips, PRL (2005)
P. Ye and Q.R. Wang, Nucl. Phys. B (2013)
Outline
• Introduction to basic experimental phenomenology of highTc cuprates and high-Tc cuprates as doped Mott insulators
/doped antiferromagnets
• Basic principles: Sign structure and Mott physics
• Nontrivial examples:
(1) one-hole case
(2) finite doping and global phase diagram
(3) ground state wavefunction
• Summary and conclusion
Example II: Delocalization and superconductivity
-+
+
-
-+
+
-
-
-
-
+
+
-
+
-
+
+
localization/AFLRO
AF
-
+
+
+
+
-
-
delocalization/SC
spin liquid/RVB!
spin liquid
doping
localization
SC
Non-BCS elementary excitation in SC state
-+
+
-
-+
+
-
-
+
-
+
+
+
-
-
+
+
-
+
-
-
spin-roton
-+
+
Superconducting transition
-
+
-
spinon confinement-deconfinement transition
+
+
spinon-vortex
-
Global phase diagram
T
charge-spin long-range entanglement by phase string effect
“strange metal”
T0
pseudogap
AF
SC
FL
δ
localization
AF = long-range
RVB


zl  z jd
 h   h
 hd | zlh  z jd
   (l , l ,..., l
|

h
1
2
Nh
)
Outline
• Introduction to basic experimental phenomenology of highTc cuprates and high-Tc cuprates as doped Mott insulators
/doped antiferromagnets
• Basic principles: Sign structure and Mott physics
• Nontrivial examples:
(1) one-hole case
(2) finite doping and global phase diagram
(3) ground state wavefunction
• Summary and conclusion
Example III : “Parent” ground state
ZYW, New J. Phys. (2011)


zlh  z jd
 h  
 hd | zlh  z jd
   (l , l ,..., l
|

h
1
2
Nh
)
lh
AFM state:
h (l1, l2 ,..., lN )  constant
h
Superconducting state:
emergent (ghost) spin
liquid
gij : BCS-like pairing
short-ranged
iu
jd
Summary and Conclusion
• Cuprates are doped Mott insulators with strong Coulomb interaction
• New organizing principles of Mott physics:
An altered fermion sign structure due to large-U
• Consequences:
(1) Intrinsic charge localization in a lightly doped antiferromagnet
(2) Charge delocalization (superconductivity) arises by destroying
the AFLRO
(3) Localization-delocalization is the underlying driving force for the
T=0 phase diagram of the underdoped cuprates
• Non-BCS-like ground state wavefunction
Fermionic RVB theories
P. W. Anderson: Resonating valence bond (RVB) theory (1987)
Slave-boson mean-field theory: Baskaran, Zou, Anderson (1988)
Kotliar, Liu (1988) …
Gauge theory description: U(1) P.A. Lee, N. Nagaosa, A. Larkin, …
SU(2) X.G. Wen, P. A. Lee, …
Z2
Sentil, Fisher
……..
Variational wave function: Gros, Anderson, Lee, Randeria, Rice, Trivedi,
Zhang; T.K. Lee; Tao Li, …
Anderson, et al., J. Phys.:
Condens. Mater (2004)
Lee, Nagaosa, Wen, RMP (2006)
(5) Hubbard model on bipartite lattices: A general sign structure
(Long Zhang & ZYW, 2013 )
Hilbert space:
spinons
holon (h) doublon (d)
Basic hopping processes in the Hubbard model
Partition function：
Z=
å t W (c)
t
W(c) ³ 0
c
U
loop c
t c = ( -1)
N h¯ (c)+N d¯ (c)
( -1)
J
h
d
N ex
(c)+Nex
(c)
+
+
-
+ +
half-filling:
t c Þ1 U / t ® ¥
tc Þ
+
+
-
Fermi signs
U/t®0
（-）
intermediate U / t
t c Þ phase string effect
- +
Spin-charge separation
three-leg ladder: