Transcript A new possibility for the metal

A new
scenario
for the metalCritical
behavior
near a two dimensional
Mott insulator
Mott insulator transition in 2D
Why 2D is so special ?
S. Sorella
Coll. F. Becca, M. Capello, S. Yunoki
Sherbrook 8 July 2005
1
The still unexplained phase diagram
A huge non-Fermi liquid region close to a Mott insulator
?
2
The variational Jastrow-Slater
Min G 
T hetask:
correlation
G H G
G G
mean field
 G  exp( v q nq nq ) Fermigas
q
nq  1 / L  e nR
iqR
R
vq  g
Gutzwiller wavefunction
3
The Gaskell-RPA solution
2 v q  1 / N  (1 / N )  2U q /( q / 2m)
0
q
0 2
q
2 2
N  FermiGas nqnq FermiGas | q |
0
q
Within the same RPA the structure factor is:
Nq  N /(1  2vq N ) | q |
0
q
0
q
4
In a lattice model with short range interaction?
Uq  U
T he Hubbard U
h q / 2m  2t (cos(q)  1)
2 2
(?)
And the f-sum rule?
q | Kin Energy| (1  cos(q)) / Nq
Thus no way to get an insulator with Jastrow-Slater?
See e.g. Millis-Coppersmith PRB 43, (1991).
The 1d numerical solution U/t=4 L=82
Insulator  v q ~ 1 / q
Metal
 vq ~ 1 / q
2
M. Capello et al. PRL 2005
J  exp( v qnqnq )
q
5
A long range Jastrow correlation can
drive a metallic Fermi sea to a Mott insulator!!
vq  g
vq ~ 1/ q2
vq  g
vq ~ 1/ q2
For an insulator :
No charge stiffness
Incompressible fluid
6
What in higher dimension?
Brinkmann-Rice:
 Z  ( Jump of nk )  0 ~ (U c  U )

*
-1
at Uc : 
m   ~ (U c  U )

-1



~
(
U

U
)
s
c

7
Infinite dimension (DMFT)
U /D 1
U /D 2
U / D  2.5
U /D 3
U /D 4
The insulator is more realistic
n n  0
8
Now we can do the same in 2D
(obviously we neglect AF as in DMFT or in BR)
Jastrow factor q-space
25
98
20
2
q vq
15
10
162 242 #Sites/U
7
8
8.5
9
10
KT
5
0
0.0
0.5
1.0
q // (1,1)
KT means Kosterlitz-Thouless transition point, explained later…
9
A clear transition is found
1.00
Zk
F
<D>
0.2
0.1
0.75
0.0
5
0.50
10
15
20
U/t
Z  (U c  U )
0.25
0.00
# Sites
98
162
242
98
162
242
 1
4
8
12
16
20
24
Uc / t U/t
8.75  0.25
10
Feynmann never lies (assumed)
Excitation energy q induced by nq  0
where  0 is the exact ground state of a physical Hamiltonian
 q  const.(2  cos(qx )  cos(q y )) / N q
 q  Hubbard gap (~ U) in theinsulator
The reason is simple
for q  0 nq   e nR  N (# particles)
iqR
R
T hus
H , n   0 and n
q
q
0 
Exact eigenstate
11
Now let us start from the insulator
Doblon
holon Singly occupied
q 1
q  1

q0 q0


T ake  ( x )  exp(V ( x) / T )
x 
2
0
eff
holonand doblon positions
12
Mapping to a classical model
Quantum
 0 nqnq  0
Nq 
0 0
Classical
exp(V ( x ) / T ) n ( x )n



exp(V ( x ) / T )


eff
q
q
( x)
x
eff
x
Notice:
nq ( x)  1 / L
q e
iqxi
i
Position charges
For large U/t we are in the very dilute regime
13
Nq  q
Now ask how can we satisfy
2
No way out, for any insulator U>>t (any D):
V ( x )   v q nq ( x ) n q ( x )
q
vq  1 / q
2
In 2D a singular v between holon and doblon
14
Exact mapping to the 2D CG model
 ( x )  exp(V ( x ) / T )
2
0
eff
V ( x )   qi q j log(| ri  rj |)  Less singular
i j
We can classify all 2D insulators in terms of
and what for T
eff
T
eff
 TKosterlitz-Thouless ?
true 2D Mott Insulator
(no broken translation symmetry)
15
A KT transition is found
0.5
2D Fermi-Coulomb gas
Confined phase
1/
1.0
578
1250
2450
Plasma phase
0.0
0.2
0.3
0.4
0.5
0.6
0.7
0.8
eff
T
  1  2 /(q T ) Nq
1
2
eff
16
In the “plasma phase” , similar to Luttinger liquid:
Fermi surface but no Fermi jump
1.0
98
162
242
338
1250
T eff  0.75
k
0.6
0.4
Jump n
nk
0.8
0.2
0.5
nk  (k  k F )
  0.14
0.4
100
0.0
0.0

1000
# Sites
0.2
0.4
0.6
0.8
1.0
q //(1,1)
Similar conclusions in Wen & Bares PRB (1993)
17
Instead in the confined phase T
T eff  0.25
k
Jump n
0.4
 TKT
50
98
162
242
338
nk
0.6
eff
0.01
0.001
100 200 300
# Sites
0.0
0.2
0.4
0.6
0.8
1.0
k //(1,1)
The density matrix appears to decay exponentially
i.e. the momentum distribution is analytic in k
18
Anomalous exponents for Z in 2D
t-J (projected wf)
 BCS ~ 1 / 2
 FG ~ 3 / 4
Prediction HTc:
Z ~ doping
Hubbard
1/ T
eff
(non trivialexponent)
19
From 2D Coulomb gas (see P. Minnhagen RPM ’87) :
The charge correlation decays as power law > 4 T eff  TKT
2
N ( q)  A | q |  B | q |
2
 n0nR  R
A>0

 2
because
... i.e :
( 4  )
there is a gap at q0 according to Feynmann
correlations are decaying as power laws
A gap with power laws !!!
20
n.b. This implies that any band insulator  plasma phase T eff  TKT
Holon-Doblon
N(qmin)/qmin
2
0.003
10
-3
10
-4
10
-5
10
-6
10
-7
10
-8
1250
338
1
Distance
0.002
0.00
0.02
0.04
0.06
1/#Sites
21
It looks consistent, though it is impossible to prove numerically
No Friedel oscillations (Mott insulator)
Clearly quadratic
22
New scenario T=0 D=2
(compatible with VMC on Hubbard)
The Hubbard gap:
q0  K  q2 / N (q)  0
Consistent with DMFT D  
U /t
Fermi liquid
Non Fermi liquid MotteffInsulator
T  TKT
critical point
23
Even more new scenario T=0 D=2
(long range interactions?)
A charge gap opens up continuously
U /t
T eff  TKT
T eff  TKT
Fermi liquid Non Fermi liquid Mott Insulator
(or d-wave BCS) incompressible
(with preformed pairs)
24
In the plasma phase for T eff  0.4 we have:
1) Z0 Non Fermi liquid, singular at kF ~ ( / 2, / 2)
2) No d-wave ODLRO (preformed pairs at T=0)
pseudogap T=0 phase ( BCS ~ (pseudogap)  0 )
T
eff
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 Z0 in the insulating/metallic phase
• Power law correlations in the insulator with gap
Non Fermi liquid phase possible in 2D?
26
Finite doping ?
1
D-wave SC
Non Fermi liquid Mott Insulator
with preformed pairs