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.

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 still unexplained phase diagram

A huge non-Fermi liquid region close to a Mott insulator

? 2

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

Infinite dimension (DMFT)1/ DU

2/ DU

5.2/ DU

3/ DU

4/ DU

The insulator is more realistic 0nn 8

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

qq

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

Clearly quadratic

No Friedel oscillations (Mott insulator)

22

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

1

D-wave SC Non Fermi liquid Mott Insulator with preformed pairs

Finite doping ?