FIELD THEORETICAL RG FOR A 2D FERMI SURFACE

Alvaro FerrazInternacional Centre for Condensed Matter PhysicsUniversity of BrasiliaBrasilia-Brazil

General ScopeGeneral Scope

1-Introduction 2-Lagrangian Model & its 2D Fermi Surface 3-Coupling Function Renormalization at One-

Loop Order 4-Self-Energy Corrections 5-RG at Higher Orders 6-Renormalized Coupling Flows 7- Density Wave & Pairing Susceptibilities 8-Conclusion

1-Introduction1-Introduction

RG proved to be very useful in probing strongly interacting systems.

This is even more so in 2D. We will describe a field theoretical RG calculation of a 2D

electron gas in a presence of nearly flat Fermi surface (FS).

Our results can be related to the cuprate high-Tc superconductors.

A FS identical to ours was observed recently in La2-xSrxCuO4

thin epitaxial film under strain (Abrecht et al, PRL 91,57002( 2003)).

Cuprates are Mott insulators at ½-filling which turn into a spin liquid at very low dopings ( x~0.02! ).

At higher temperatures and concentrations there appears an exotic pseudogap phase and finally at even higher concentration a D-wave high-Tc superconductor.

2-Lagrangian Model and its 2D Fermi Surface2-Lagrangian Model and its 2D Fermi Surface

To describe 2D electrons consider the renormalized lagrangian (A.F.,EPL 61,228(2003)).

Here a= refers to the upper (lower) or right (left) flat patch of our FS.

†( ), ( ),

,

† †2 1 ( ), ( ), ( ), ( ),

, , , , ,

( ) ( , ) ( , )

1( , ) ( , ) ( , ) ( , )

t a a aa

L i p t t

t t t tV

1 2 3

p,

1 2 3 3 2 1p p p

p p

p +p -p p p pg g

Thanks to the flat FS the single-particle dispersion is simply

a F Fv p k p

with and F Fk p k p

In general F Fv v p

The bare couplings represent backward and forward scatterings

Since the non-interacting propagators are

0

0 F F

iiG p

p v p k i

0

0 F F

iiG p

p v p k i

both part-part and part-hole diagrams are IR log divergent

with being a fixed upper energy cut-off2 Fv

3 – Coupling Function Renormalization at One-3 – Coupling Function Renormalization at One-Loop OrderLoop Order

We will now proceed with the calculations of the one-particle irreducible functions

within the framework of the FTRG.

Up to one-loop order the calculation is simple. We set

(4)1 1 2 3

10 20 30 1 1 2 3

( , ; ;

/ 2; / 2) ( , , )R F F F

R

p k p k p k

p p p ig p p p

Using appropriate Feynman rules, we find

1

2

1 1 2 3 1 2 2 12

1 1 1 2 2 1

1( , , ; )

4

2 ln

R R R R RF D

R R R R R R

D

g p p p dk g g g gv

dk g g g g g g

Similarly, using a similar prescription we define

(4)2 1 2 3

10 20 30 2 1 2 3

( , ; ;

/ 2; ) ( , , )R F F F

R

p k p k p k

p p p ig p p p

This gives

Using, again, appropriate Feynman rules it turns out that

Or, equivalently

1

3

2 1 2 3 2 2 1 12

2 1

1( , , ; )

4

ln

R R R R RF D

R R

D

g p p p dk g g g gv

dk g g

The counterterms are in this way continuous functions of the external momenta and the RG scale parameter

Since the bare parameters don’t depend on the RG scale we are naturally led to the RG equations

1 2 3, ,p p p

1 1 1

2 2 2

0

0

R R

R R

d dg g g

d dd d

g g gd d

It follows immediately that the one-loop renormalized coupling flows are then

1

2

1

3

1 1 2 3 1 2 2 12

1 1 1 2 2 1

2 1 2 3 2 2 1 12

2 1

1( , , )

4

2

1( , , )

4

R R R R RF D

R R R R R R

D

R R R R RF D

R R

D

dg p p p dk g g g g

d v

dk g g g g g g

dg p p p dk g g g g

d v

dk g g

4 – Self-Energy Correction4 – Self-Energy Correction

To calculate self-energy corrections we need to estimate at least two-loop contributions

The first four diagrams produce constant shifts in and renormalize .

Since in our calculation both and SF are kept fixed we may ignore those contributions altogether.

In contrast, the remaining diagrams give us

FvFk

Fv

4

0 1 1 2 24 2

01 2 2 1

0

00

0

( ) 2 264

( )ln

( )

( )ln ; ( ) ...

( )

F F R R R RF D

F FR R R R

F F

F FF F

F F

ip v p k dk dq g g g g

v

v p k p ig g g g

v p k p i

v p k p ii Z p p v p k

v p k p i

; ; 1Z p Z p

where Z is the quasiparticle weight which relates the bare and renormalized fields:

( )

1/ 2( ) ( ) ( ; ) ( )

a

Ra Z p

p p

Defining the renormalized one-particle irreducible function

such that(2)( ) ( ; )R p

(2)( ) 0( , )R Fp p k p = = ,

Here

it follows immediately that

; ( ) lnZ p p

where the anomalous dimension is given by

4

1 1 2 24 2

1 2 2 1

( ) 2 232 R R R R

F D

R R R R

ip dk dq g g g g

v

g g g g

It is now straightforward to estabilish the RG equation for the quasiparticle weight Z:

( ; )( ) ( ; )

Z pp Z p

We point out that our renormalized Lagrangian can now be put in a more convenient form:

†( ), ( ),

,

14 2

2 1, , , , , 1

† †( ), 1 2 3 ( ), 3 ( ), 2 ( ), 1

( ) ( ) ( )

1( ; )

( - ) ( ) ( ) ( )

1 2 3

p,

p p p

R Ra t F F a

a

ii

R R R R

L Z p i v p k p

Z PV

p p p p p p

g g

where

1/ 24

1

( ; ) ; ;i i iR iRi

g Z p g p g p

5 – RG at Higher Orders5 – RG at Higher Orders

To calculate corrections for and we need to take into account the higher order diagrams which are also

1Rg2Rg

ln

These non-parquet diagrams in two-loop order for both backscattering and forward scattering channels are

Taking again into account the RG condition

the RG flow equations for the renormalized coupling functions in two-loops become

/ 0idg d

4

1 2 31 2 3

1

1 2 3

( , , ) 1( , , )

2

( , , )

iRi iR

i

iR

dg p p pp g p p p

d

g p p p

6 – Renormalized Coupling Flows6 – Renormalized Coupling Flows

It is impossible to solve our RG equations analytically.

We need therefore numerical methods to estabilish the flow of the renormalized coupling functions (H. Freire, E. Corrêa, A. F., PRB 71,165113 (2005)).

To do this we discretize the FS replacing by a finite set of points.

For convenience we take where is our fixed upper energy cutoff and l our RG step.

Notice that max l is limited by the fact that cannot be shorter than the distance between neighboring points in our discretization procedure.

p

exp( )l

Initially we depict the one-loop results for different choices of external momenta.

They reproduce previous one-loop results.

We show next the quasiparticle weight Z

If we take initially , Z is mildly reduced from unity.1 2 1R Rg g

We show next the RG flows for and

in two-loop order1 2 31 ( , , )Rg p p p

1 2 32 ( , , )Rg p p p

In contrast with one-loop results the renormalized couplings approach plateau values in a fixed point like regime.

Many renormalized couplings now approach zero continuously as a result of the suppression of Z.

This is a strong indicative that there is no symmetry breaking and no onset of long range order in the physical system.

To test the leading instabilities in this new regime we need to calculate the charge and spin susceptibilities.

7 – Density Wave and Pairing Susceptibilities7 – Density Wave and Pairing Susceptibilities

Since the susceptibilities are essentially mean values of composite operators we add to our original Lagrangian the contributions (E. Corrêa, H. Freire and A. Ferraz (2005)).

† †( ), ( ),

,

( , ) ( , ) ( , ) ( , ) . .SCSC SCL h q t k q k q t k t h c

q,k,

and

†( ), ( ),

,

( , ) ( , ) ( , ) ( , ) . .DWDW DWL h q t k q k q t k t h c

q,k,

The addition of composite operators generate new divergencies which must be regularized in their own right.

As a result we must have

1/ 2 1/ 2( , ) ( ) ( )

( , ) ( , )

SC

SC SCR R

k q Z k q Z k

k q k q

1/ 2 1/ 2( , ) ( ) ( )

( , ) ( , )

DW

DW DWR R

k q Z k q Z k

k q k q

and

The density wave renormalized vertex should be symmetrized with respect to the spin to give

DWR

CDW DW DWR R R

andSDW DW DWR R R

Similarly, associated with the ’s we define the singlet and triplet pairing vertices

SCR

SSC SC SCR R R

and TSC SC SCR R R

Diagrammatically the ’s and ’s are directly related with the one-particle irreducible vertex function

In one-loop order we get

DWR SC

R

(2,1)R

For the DW channel we use the prescription(2,1)

0 0( , , ; , 2 ; 0) ( , )DWR F F Rp p k p q q k q i p q

We use a similar condition for the SC channels

Taking into account the RG condition for the bare vertices we arrive at the RG equations

/ 0d d

1( )

2R R

i Ri exterallegs

d dp

d d

with or ., ,CDW SDW SSC TSC

Due to the particular shape of our flat FS the renormalized couplings must be symmetrical with respect to the exchange of + and – particles and change of sign of the external ’s :p

1 2 3 4 2 1 4 3

1 2 3 4 3 2 1

1 2 3 4 1 2 3 4

( , ; , ) ( , ; , )

( , ; , ) ( , ; , )

( , ; , ) ( , ; , )

iR iR

iR iR

iR iR

g p p p p g p p p p

g p p p p g p p p p

g p p p p g p p p p

In view of that it turns out that the RG equations for the renormalized vertices are symmetrical with respect to the sign reversal of in for a fixed (A. Zheleznyak et al PRB 55, 3200 (1997)).

We therefore define two irreducible representations of this symmetry which never mix

p ( , )R p q q

( , ) ( , ) ( , )aSC aSC aSCR R Rp q p q p q

with a = S,T.

is associated with s-wave symmetry whereas shows d-wave character.

SSCR

SSCR

Instead, for density wave symmetries

( , ) ( , ) ( , )bDW bDW bDWR R Rp q p q p q

With b = S,C.

Here, the antisymmetrical ones are associated with the so-called flux phases.

Once the are found the related susceptibilities ,

associated with the related , follow immediately.

R R

(0,2)R

or, equivalently

3

4

*

2

*

2

1( ; ) ( , ) ( , )

4

1( ; ) ( , ) ( , )

4

R R RF D

R R RF D

dq d p p q p q

d v

dq d p p q p q

d v

,CDW SDW and ,SSC TSC

Following the same numerical procedure as before we can estimate if there is any symmetry breaking and of what kind.

Here we take a Hubbard like initial condition together with

1 2 10R Rg g

and

1

22

CDW SDW SSC TSCR R R R

CDW SDW SSC TSCR R R R

pSin

We show the corresponding one-loop and two-loops

contributions for the various symm and

antisymm renormalizedR

8 - Conclusions8 - Conclusions

Although the one-loop ’s seem to announce symmetry breaking and, in particular, the predominance of the SDW+ instability the two-loop ’s seem to approach plateau values characteristic of short-range ordered states only.

Among them the SSC- (d-wave like) & SDW- (spin flux phase) appear closely together and slightly above the remaining susceptibilities.

The exception to that is the SDW+ which appears to grow indefinitely. However as opposed to the one-loop result, this growth seems spurious since it is slowered down considerably when we consider high-order effects.

To check this conclusion we calculated the associated spin and charge uniform susceptibilities (H. Freire, E. Corrêa and A. Ferraz, cond-mat/0506682) and we find indeed no sign of long-range order since both

0unifSDW and 0unif

CDW