Transcript A theory for lightly doped Mott insulators

Simulating High Tc Cuprates
T. K. Lee
Institute of Physics, Academia Sinica, Taipei, Taiwan
December 19, 2006, HK Forum, UHK
Collaborators
• Y. C. Chen, Tung Hai University, Taichung, Taiwan
• R. Eder, Forschungszentrum, Karlsruhe, Germany
• C. M. Ho, Tamkang University, Taipei, Taiwan
• C. Y. Mou, National Tsinghua University, Taiwan
• Naoto Nagaosa, University of Tokyo, Japan
• C. T. Shih, Tung Hai University, Taichung, Taiwan
Students:
• Chung Ping Chou, National Tsinghua University, Taiwan
• Wei Cheng Lee, UT Austin
• Hsing Ming Huang, National Tsinghua University, Taiwan
Acknowledgement
Kitaoka and Mukuda, Osaka Univ.
Introduction
To start simulation, need Hamiltonians!
minimal models:
2D Hubbard model –
exact diagonalization (ED) for 18 sites with 2 holes
(Becca et al, PRB 2000)
finite temp. quantum Monte Carlo (QMC) , fermion sign
problem ( Bulut, Adv. in Phys. 2002)
dynamic mean field theory and density matrix
renormalization group, ..
2D t-J type models –
Model proposed by P.W. Anderson in 1987:
t-J model on a two-dimensional square lattice, generalized to
include long range hopping

1 ,
s

spin
2
i
  1

H    tij ci c j  H .C.  J   si  s j  ni n j 
4
 n 
i, j 
i, j 
c c
i
tij = t for nearest neighbor charge hopping,
t’ for 2nd neighbors, t’’ for 3rd
J is for n.n. AF spin-spin interaction
Constraint: For hole-doped systems
two electrons are not allowed
on the same lattice site
Three species: an up spin,
a down spin or an empty site or a
“hole”


i
i
Introduction
• To start simulation, need Hamiltonians!
minimal models:
2D Hubbard model –
exact diagonalization (ED) for 18 sites with 2 holes
(Becca et al, PRB 2000)
finite temp. quantum Monte Carlo (QMC) , fermion sign
problem ( Bulut, Adv. in Phys. 2002).
dynamic mean field theory and density matrix
renormalization group, ..
2D t-J type models –
no finite temp. QMC – sign problem and strong
constraint
ED for 32 sites with 4 holes (Leung, PRB 2006)
Variational Approach!
TrialVariational
wave function:
approach :
1. Construct the trial wave function
2. Treat the constraint exactly by using Monte Carlo method
 tr   a 
 is configuration index
a is Slater determinant

To calc. a quantity
ˆ 
 tr O
tr
 tr  tr


 

,

a a
ˆ 
 O
a


2
 

 a 
ˆ
  p     O
   a 
p 
a

 
2
a
2
Expt. Info.
Phase diagram
electron
Damscelli, Hussain,
hole
and Shen, Rev. Mod. Phys. 2003
electron – hole
asymmetry!
Phase diagram
Only AFM insulator (AFMI)?
How about metal (AFMM),
if no disorder?
Coexistence of
AF and SC?
Related to the
mechanism of SC?
Important info from experiments
• 5 possible phases:
AFMI, AFMM, d-wave SC, and AFM+d-SC,
normal metal.
 e-doped system is different from hole-doped..
Broken symmetries:
• electron and hole symmetry
• Afmm, afm+sc and sc all have different broken
symmetries
From the theory side:
Without the long range hopping terms, t’ and t’’,
the model Hamiltonian, t-J model, has the particlehole symmetry: c   c 

i
i
t’ and t’’ break the
symmetry between doping
electrons and holes!
From hole-doped to electron-doped, just change
t’/t → – t’/t and t”/t → – t”/t
Different Hamiltonians – different phase diagram.


  1

H    tij c c j  H .C.  J   si  s j  ni n j  AFM is natural!
4
 D-wave SC?
i, j 
i, j 

i
In 1987, Anderson pointed out the superexchange term
1
 

H J  J   si  s j  ni n j 
4

i, j 


J
2
 (C
J
2


i ,
C j ,  Ci,C j , )(Ci ,C j ,  Ci ,C j , )
i, j

i, j
i, j
i, j
This provides the pairing mechanism!
It can be easily shown that near half-filling this term
only favors d-wave pairing for 2D Fermi surface!
Spin or charge pairing?
The resonating-valence-bond (RVB) variational wave function
proposed by Anderson ( originally for s-wave and no t’, t’’),


RVB  Pd  (uk  vk Ck,Ck , ) 0  Pd BCS
 k

The Gutzwiller operator Pd Pd  i (1  nini )
doubly occupied sites for hole-doped systems
vk / uk 
enforces no
Ek   k
,  k   v (cos k x  cos k y )
k
 k  2(cos k x  cos k y )  4t 'v cos k x cos k y  2t ''v (cos 2k x  cos 2k y )  v ,
Ek   k2   k2
four variational parameters, tv’ tv’’ ∆v, and μv
d-RVB = A projected d-wave BCS state!
AFM was not considered.
The simplest way to include AFM:
 tr  exp(h *  (1)i S zi ) RVB
i
Lee and Feng, PRB 1988, for t-J
Use mean field theory to include AFM,
, if i  j  xˆ

, if i  j  yˆ
   ci c j   ci c j 
Chen, et al., PRB42, 1990; Giamarchi and Lhuillier, PRB43, 1991; Lee and Shih,
PRB55, 1997; Himeda and Ogata, PRB60, 1999
Assume AFM order parameters:
staggered magnetization
m  s Az   sBz
And uniform bond order


c
 i c j

Two sublattices and two bands – upper and lower
spin-density-wave (SDW) bands
RVB + AFM for the half-filled ground state
(no t, t’ and t’’)
 /



 
 Ak ak  a k   Bk bk b k  
k


 0  Pd

Ne
2
0
Ne= # of
sites
ak  lower SDW & bk  upper SDW bands
Ek  k
Ak 
k

&
Ek    
2
k
2
k

1
Pd   1  ni ni 
Ek   k
Bk  
k
2
k 

3
4
i
J  cos k x  cos k y   Jm
Variational results
 
 si  s j   0.3324(1)
2
2
2

1
2
“best” results
-0.3344
Liang, Doucot
And Anderson
staggered moment m = 0.367
0.375 ~ 0.3
The wave function for adding holes to the half-filled
RVB+AFM ground state
Lee and Shih, PRB55, 5983(1997); Lee et al., PRL 90
(2003); Lee et al. PRL 91 (2003).
Creating charge excitations to the Mott Insulator “vacuum”.
The state with one hole (two parameters: mv and Δv)
 1h (k , S z  1 / 2)  c k   0
 Pd c

k


/




 Aq aq a q  Bq bqb q 
qk



Ne 1
2
0
A down spin with momentum –k ( & – k + (π, π ) ) is
removed from the half-filled ground state. --- This is
different from all previous wave functions studied.
J/t=0.3
Energy dispersion after one electron
is doped. The minimum is at (π, 0).
t’/t= 0.3, t”/t= - 0.2
Dispersion for a single hole.
t’/t= - 0.3, t”/t= 0.2
1st h+
1st e-
The same wave function is used for both
e-doped and hole-doped cases.
There is no information about t’ and t”
in the wave function used.
□ Kim et. al. , PRL80, 4245 (1998); ○ Wells et. al.. PRL74,
964(1995); ∆ LaRosa et. al. PRB56, R525(1997).
● SCBA for t-t’-t’’-J model, Tohyama and Maekawa, SC Sci. Tech. 13,
R17 (2000)
ARPES for Ca2CuO2Cl2
The lowest
energy at
k  ( / 2,  / 2)
Ronning, Kim and Shen, PRB67 (2003)
Nd2-xCex CuO4 -- with 4% extra electrons
Fermi
surface
around (π,0)
and (0, π)!
Armitage et al., PRL (2002)
Momentum distribution for a single
hole calc. by the quasi-particle wave
function
< nhk↑>
1h (k  ( / 2,  / 2), Sz  1/ 2)
< nhk↓ >
Exact results
for the singlehole ground state
for 32 sites.
Chernyshev et al.
PRB58, 13594(98’)
a QP state kh=(,0) = ks
a SB state ks =(,0), kh=(/2, /2)
Exact 32 site result from P. W. Leung for the lowest energy (π,0) state
t-J
t-t’-t’’-J
Takami TOHYAMA et al.,
J. Phys. Soc. Jpn. Vol 69,
No1, pp. 9-12
J/t=0.4, t’/t-α/3 and
t”/t’=2/3
a
b
c
d
e
(0,0)
QP
(0,0)
SB
(/2,/2)
QP
(,0)
QP
(,0)
SB
0.188
-0.0288
-0.0044
0.123
-0.0313
0.188
-0.0254
-0.0052
0.159
-0.0085
0.202
-0.0302
-0.0005
0.071
-0.002
-0.273
-0.203
-0.2241
-0.353
-0.1921
-0.264
-0.195
-0.2154
-0.279
-0.2115
Our Result: (π, 0)
is QP at t-J model,
but SB for t-t’-t”-J.
(0, 0) is QP for
both
QP:Quasi-Particle state
SB: Spin-Bag state
The state with two holes
 2 h (ktotal  0, S Z  0)  ck  c k
h
h
0


/




 Pd   Aq aq a q  Bq bqb q 
q  kh



Ne 1
2
0
Ground state is
kh=(π/2, π/2)
for two 0e holes;
kh=(π,0) for
two 2e holes.
Similar construction for more holes and more electrons.
The Mott insulator at half-filling is considered as the vacuum state.
Thus hole- and electron-doped states are considered as the
negative and positive charge excitations .
Fermi surface becomes pockets in the k-space!
Lee et al., PRL 90 (2003); Lee et al. PRL 91 (2003).
The rotation symmetry of the wave function
alternates between s and d symmetry for 2h,4h, 6h,…!
d-wave pairing correlation function
2 holes
in 144 sites
The new wave function has AFM but negligible pairing.
An AFM metallic (AFMM) phase but no SC!
Increase doping, pockets are connected to form a
Fermi surface:
Cooper pairs formed by SDW quasiparticles
three new variational parameters: μv, t’v and t”v
AFMM shows more hole-hole
repulsive correlation than
AFMM+SC.
The pair-pair correlation of
AFMM is much smaller than
AFMM+SC.
Difference between AFMM and AFMM+SC
The doping dependence of AFMM is quite
different from AFMM+SC.
The pair-pair correlation of AFMM is much lower than AFMM+SC.
12X12
16X16
12X12
16X16
AFMM shows more hole-hole repulsive correlation than AFMM+SC.
mv and Δv
mv and Δv
μv, t’v and t”v
Δv, μv, t’v and
t”v
Variational Energy
Phase diagram
   triplet    (cos kx  cos ky )ck  c k  Q 
k
Possible Phase Diagrams for the t-J model
CT Shih et al., LTP (’05) and PRB (‘04)
t’/t=0 t’’/t=0
AFMM+SC
t’/t=-0.2
t’’/t=0.1
AFMM
0.1
0.2
0.3
x
0.4
0.5
0.6
t’/t=-0.3, t’’/t=0.2
H. Mukuda et al., PRL 96, 087001 (2006).
Phase diagram for hole-doped systems
The “ideal” Cu-O plane
Extended t-J model, t’/t=-0.2, t’’/t=0.1
H Mukuda et al., PRL (’06)
QP excitation
Ne  1, k
 PdCk, ( aqCq,Cq, ) Ne / 2 0
q
Excitation energies are fitted with
Ek  ( k2  2k )
Example:
t’/t, t’’/t, and /t are
renormalized and “Fermi
surface” is determined by
Setting =0 in the
excitation energy.
spectral weight:
STM Conductance is related to the spectral weight
A(k ,  )   m Ck , 0    Em  E0    m C
2
m

k ,
m
0    Em  E0 
2
Quasi-particle contribution to the conductance ratio
d-BCS
E=0.3t for 12X12
E=0.2t for 20X20
d-RVB (t’=-0.3, t’’=0.2)
Beyond VMC approach – the Power-Lanczos method
For a given trial wave function, |   , we approach the ground state
in two steps:
1. Lanczos iteration
|  (1)  |   
|
( 2)
C1
H| 
N
We denote this state as |PL1>
C1
C2 2
 |    H |    2 H |  
N
N
C1 and C2 are taken as variational parameters
2. Power Method
|  (nl )  (W  H) n |  (l ) 
| GS 
lim
n 
|  (nl )  (W  H) n |  ( l ) 
|PL2>
Controversies about pairing
Only t-J (without t’) is sufficient to explain high Tc?
No! --- Shih, Chen, Lin and Lee, PRL 81, 1294 (1997)
64 sites, J=0.4, PL0=VMC, PL1-1st order Lanczos, PL2-2nd order
P. W. Leung, PRB 2006
Summary
• Variational approach has provided a
number of semi-quantitative successes in
understanding high Tc cuprates : ground
state phase diagram and excitaiytons,
spectral weight, STM conductance
asymmetry, etc..
• Detailed questions about pseudogap,
effects of disorder and electron-phonon
interactions, etc.. are still to be resolved.
• Finite temp?
Thank you for your attention!b