a new scenario for the metal- mott insulator transition in 2d why 2d is so special ? s. sorella...
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A new scenario for the metal-Mott insulator transition in 2D
Why 2D is so special ?
S. Sorella Coll. F. Becca, M. Capello, S. Yunoki
Sherbrook 8 July 2005 1
Critical behavior near a two dimensional Mott insulator
The variational Jastrow-Slater
RR
iqRq
qqq
qG
neLn
nn
/1
gas Fermi)vexp(
correlation mean field
ion wavefunctGutzwiller vq g3
GG
GGG
HMin
: taskThe
The Gaskell-RPA solution
||Gas FermiGas Fermi
)2//(2)/1(/1v2
0
22200
qnnN
mqUNN
qqq
qqqq
||)v21/( 00 qNNN qqqq
Within the same RPA the structure factor is:
4
In a lattice model with short range interaction?
(?) )1)(cos(22/
U HubbardThe 22
qtmqh
UUq
And the f-sum rule?
qq Nq /))cos(1( | EnergyKin|
Thus no way to get an insulator with Jastrow-Slater?See e.g. Millis-Coppersmith PRB 43, (1991).
q
q
q
q
/1~ v Metal
/1~ v Insulator 2
M. Capello et al. PRL 2005
)vexp( qqq
q nnJ
The 1d numerical solution U/t=4 L=82
5
A long range Jastrow correlation candrive a metallic Fermi sea to a Mott insulator!!
6
gqv
2q /1~v q
2q /1~v q
gqv
For an insulator :
No charge stiffness
Incompressible fluid
What in higher dimension?
Brinkmann-Rice:
) (~
) (~
)(~0) of Jump(
:at U1-
s
1-*c
UU
UUm
UUnZ
c
c
ck
7
Now we can do the same in 2D(obviously we neglect AF as in DMFT or in BR)
0.0 0.5 1.00
5
10
15
20
25
KT
Jastrow factor q-space
q
2 v q
q // (1,1)
98 162 242 #Sites/U 7 8 8.5 9 10
KT means Kosterlitz-Thouless transition point, explained later…
9
4 8 12 16 20 240.00
0.25
0.50
0.75
1.00
Zk F
U/t
# Sites 98 162 242
1
)(
UUZ c
25.075.8/ tUc
A clear transition is found
5 10 15 200.0
0.1
0.2
<D
>
U/t
98 162 242
10
Feynmann never lies (assumed)
insulator thein U)(~ gap Hubbard
/))cos()cos(2.(
q
qyxq Nqqconst
Excitation energy induced by where is the exact ground state of a physical Hamiltonian
q 0qn0
The reason is simple
0 and 0, Thus
particles)(# 0for
RR
iqRq
nnH
Nnenq
Exact eigenstate11
Now let us start from the insulator
1q
1q Doblon holon Singly occupied
0q
0q
positions doblon and holon x
)/)(exp()( Take 20
effTxVx
12
x
eff
qqx
eff
qqq TxV
xnxnTxVnn
N)/)(exp(
)()( )/)(exp(
00
00
Quantum Classical
charges Position
q /1)(
:Notice
iiqxieqLxn
For large U/t we are in the very dilute regime
Mapping to a classical model
13
Now ask how can we satisfy 2qNq
No way out, for any insulator U>>t (any D):
2q /1 v
)()(v)(
q
xnxnxV qqq
q
In 2D a singular v between holon and doblon14
Exact mapping to the 2D CG model
singular Less|)log(|)(
)/)(exp()( 20
jijji
i
eff
rrqqxV
TxVx
We can classify all 2D insulators in terms of effT?for what and Thouless-KosterlitzTT eff
true 2D Mott Insulator (no broken translation symmetry) 15
A KT transition is found
0.2 0.3 0.4 0.5 0.6 0.7 0.80.0
0.5
1.0
Con
fine
d ph
ase
Plasma phase
2D Fermi-Coulomb gas
1/
Teff
578 1250 2450
qeff NTq )/(21 21
16
In the “plasma phase” , similar to Luttinger liquid: Fermi surface but no Fermi jump
Similar conclusions in Wen & Bares PRB (1993)
0.14
)(
Fk kkn
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
100 1000
0.4
0.5
Jum
p n
k
# Sites
nk
q //(1,1)
98 162 242 338 1250
17
75.0effT
Instead in the confined phase KTeff TT
0.0 0.2 0.4 0.6 0.8 1.0
0.4
0.6
100 200 300
0.001
0.01
Ju
mp
nk
# Sites
nk
k //(1,1)
50 98 162 242 338
The density matrix appears to decay exponentiallyi.e. the momentum distribution is analytic in k 18
25.0effT
Anomalous exponents for Z in 2D
t-J (projected wf) Hubbard
effT/119
4/3~
2/1~
FG
BCS
Prediction HTc: exponent) trivial(non doping~Z
From 2D Coulomb gas (see P. Minnhagen RPM ’87) :
The charge correlation decays as power law > 4 because
)4(0
22
:i.e ...||||)(
Rnn
qBqAqN
R
A>0 there is a gap at q0 according to Feynmann
correlations are decaying as power laws2
A gap with power laws !!!20
n.b. This implies that any band insulator plasma phase KT
eff TT
KTeff TT
0.00 0.02 0.04 0.06
0.002
0.003
110 -8
10 -7
10 -6
10 -5
10 -4
10 -3
Ho
lon
-Do
blo
n
D istance
1250 338
N(q
min)/
qm
in
2
1/#Sites
It looks consistent, though it is impossible to prove numerically
21
tU /Fermi liquid Non Fermi liquid Mott Insulator critical point
New scenario T=0 D=2 (compatible with VMC on Hubbard)
KTeff TT
The Hubbard gap:
Consistent with DMFT
0)(/20 qNqKq
D
23
tU /
Fermi liquid Non Fermi liquid Mott Insulator(or d-wave BCS) incompressible (with preformed pairs)
KTeff TT KT
eff TT
Even more new scenario T=0 D=2 (long range interactions?)
A charge gap opens up continuously
24
effT
In the plasma phase for we have: 1) Z0 Non Fermi liquid, singular at 2) No d-wave ODLRO (preformed pairs at T=0) pseudogap T=0 phase ( )
4.0effT
0)pseudogap(~ BCS
)2/,2/(~ Fk
25
Conclusions
• A Mott transition is found in 2D Hubbard (VMC)
• Mapping to 2D Coulomb gas
confined phase= Mott insulator
plasma phase=Non Fermi liquid metal
• Critical Z0 in the insulating/metallic phase
• Power law correlations in the insulator with gap
Non Fermi liquid phase possible in 2D?26