Two-Dimensional Fermi Gas at Half Filling is a Luttinger Liquid

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Two-Dimensional Fermi Gas at Half Filling is a Luttinger Liquid

E. Batkilin

To cite this version:

E. Batkilin. Two-Dimensional Fermi Gas at Half Filling is a Luttinger Liquid. Journal de Physique I,

EDP Sciences, 1995, 5 (11), pp.1481-1486. �10.1051/jp1:1995211�. �jpa-00247151�

Classification

Physics

Abstracts

05.30Fk 71.10+x

Two-Dimensional Fermi Gas at Half Filling is

_a

Luttinger Liquid

E. Batkilin

Department

of

Physics,

Technion Israel Institute of

Technology,

32000

Haifa,

Israel

(Received

27

April

1995,

accepted

6

July1995)

Abstract. It is shown that in

a mortel with

a

regular long

_range interaction, _a 2D fermion gas at half

filling

of the band is _a

Luttinger Liquid.

^It is found that under certain conditions the

jump

in the momentum distribution of such

a gas ^at the Fermi surface

deperlds

_on the

filling

factor _v in trie form

+~

((2v -1)/A(~°,

where A~~

is the range of trie interaction and _{a is a}

positive

function of the position k _on the Fermi surface. The jump vanishes when _v

=

1/2.

The

question why

the fermion _gas in the normal state of

high temperature superconductors

is not a Fermi

Liquid (FL)

has been of

great

^{interest in the last several} years. Anderson

iii

has

suggested

that _at half

filling

_a 2D Fermi _{gas in} trie

conducting planes

of

high-Tc cuprates

can be

regarded

_{as a}

Luttinger Liquid ILL).

This LL has _no

jump

in the momentum distribution and its

parameters

_are

dependent

_on the

position

_on the Fermi surface.

According

to Anderson

iii

this

happens

due to the

singular

interaction which appears in

systems

close to the Mott transition.

It is known that the ID Fermi gas is _a LL

[2]. Recently

it has been

suggested [3,4]

that there exists _an effective

lowering

of the

dimensionality

which is

produced by

the

anisotropic

momentum

dependence

of the

quasipartide

_energy. Such _an effect _can lead to the LL behavior

[3].

^{The nature of the normal state of the 2D fermion}

system

^{has been}

investigated

in detail

mainly

in the _case of the 2D

homogeneous

Fermi _gas

là, 6].

^{It has been found that} a dilute Fermi gas with _a short _range

repulsive

interaction _{is a} FL

[si.

^The 2D

homogeneous

fermion gas with _a

regular long

_range fermion~fermion interaction is also known to be a FL

[6].

^{It has}

been shown [6] that in the

homogeneous

fermion _gas with _a

regular long

_range interaction the non~fL behavior

persists

^{for D<} 2. It has been

pointed

out [6] that this reveals _a

possibility

of _a non-FL behavior in the 2D fermion _gas close to half

filling

due _to the

singular

interaction 11,

7].

The

anisotropic crystalline

electron _gas with _a short _range

(Hubbard)

interaction has been shown to

display

_a non-FL behavior close _to half

filling

_{[8] due} to effects of _momentum transfer with

large

values of momentum

connecting opposite

sides of the nested Fermi surface.

Recently,

it has been

suggested

_[3] that the

screening length

in the

high-Tc cuprates

is rather

large

and therefore the effective fermion-fermion interaction is

long

_range.

In this paper we

study

the behavior close to half

filling iv

₌

1/2)

of the 2D fermion gas with _a

regular long

_range interaction. In contrast to the short _range interaction model

[8],

the

present long

_range interaction mortel restricts the momentum transfer _to small _momentum

@

Les Editions de

Physique

1995

1482 JOURNAL DE

PHYSIQUE

I N°11

values,

therefore effects connected with

nesting

_{[8] are} not essential. We find the

jump

ⁱⁿ the

momentum distribution _{as a} function of the

filling

factor _v. For this _{purpose we} calculate

the Green function of the fermion _gas

by

summation of the contributions of all orders of

perturbation theory [6,9].

We show here that _at half

filling

there exists _a condition for which the

one~particle

Green function has _no

jump

ⁱⁿ the momentum distribution. These results

support

the

suggestion iii

that _{at v}

=

1/2

there _{exists a} LL fixed

point.

This _{is in}

agreement

with recent conclusions of Luther

[10],

who used trie bozonization

approach,

which is not

surprising

since it has been shown

iii]

that the Ward identities

approach [6,9]

is

equivalent

to the bozonization

approach.

We

analyze

in the

present

^{work the} hamiltonian of fermions with _a

long

_range interaction:

H

=

fl elP)aΫap«

₊

fl ~Î ~""'lq)aΫaÎ,«,ap,+q«,ap-q« li)

POE a,a' P,P',q

(2)

Where

ejpaj

=

-w(coslPz)

+

C°SIPY)).

Here _p is the

quasimomentum,

_a is the

spin, (2W(

is the

bandwidth, ~""'(q)

_{is the}

interaction,

and the lattice _spacing is chosen to be

unity.

^As in

[9],

we

investigate

the _case in which

~""'(q)

=

~("' #

0 in _{a narrow}

quasimomentum region

_q < A <

2x,

where 2x is the

period

of trie unit cell in

reciprocal

_space. This _{is a} mortel of _a dense fermion _gas where trie range of trie interaction

A~~

is much

greater

than trie average

inter-particle

distance. Trie interaction

is assumed _to be small _as

compared

to trie bandwidth:

(~("'/W(

< i.

To determine whether the

system

is a

LL,

_one should first show that in the second order per- turbation

theory

the

self-energy

nez

(p, w)

for small p and w behaves _as

(w-e(p) )

In _{w e}

(p)

In the

present

case there _are two relevant

diagrammatic contributions, namely,

zero-sound di- agrams and

spin-wave diagrams.

The zero-sound

diagrams

_are

responsible

for the forward

scattenng

^of

particles

with almost

antiparallel

momenta, which in tum leads to the LL behav- ior of ID

systems

^with

long

_range interaction.

Therefore,

_we first consider

systems

^of

spinless

fermions and later _we shall consider

spin-wave

contributions in the

self-energy.

In the

one-loop (the

second order in the

interaction) approximation

the self _energy of the fermion _is

given by

L(p, w)

₌

1~

^~

(~((q)H°(w ^e(p

+

q), q)sgn (e(p

₊

q)

_eF

(3)

where _~o

= ~"" and

~~~~'~~ Î

Î~Î Î _{ÎÎ ~~~~}

~ ~°" ~ ~

~~~~~~°" ~~

_~~~

Here, G°(w, p)

#

(w (e(p) eF)

+

iôsgn(w))~~

is the Green function of the free fermion and à is _a small

positive

number. As

usual,

_we take _{into account} the Hartree~fock contributions

by

the renormalization of the Fermi energy.

It is _easy to see that

n°ifl _~)

-

/

n

~ip, lllilillipll~

n~ip~i là)

where _nF

(e(p))

is the Fermi distribution and

AP, q)

=

elP

₊

q) elP). 16)

B D

c

Fig.

1. The Fermi surface _m the

representation

of the coordinates p+ and _p-.

All fermions which contribute to

H° Ill, q)

for _{any q <} A _are situated in _{a narrow} momentum

strip

^close to the Fermi surface. In this _narrow

strip

with _a width of trie order

A,

_one has

e(p)

_< W.

Introducing

_new

variables,

P+ #

lPz

+

PY)/2>

P- #

lPz Py)/2 ₁₇₎

we can represent our

dispersion

law _as

e(p)

=

-2Wcos(p+) cos(p-) (8)

Due to the

symmetry

^{of the}

problem

it is suflicient to restrict ourselves to calculation in the first Brillouin _zone. At half

filling iv

₌

1/2)

at the Fermi

surface,

in the first

quadrant

of the

Brillouin zone, one has p+ =

x/2.

Therefore in the

vicinity

of the Fermi surface _we obtain

e(p)

=

2Wcos(p-)p[, ((q, p)

=

2Wcos(p-)q+ (9)

here

p[

= p+

x/2,

_{q+ =}

1/2(q~

₊

_qy)

and _q-

=

1/2(q~ qy).

Thus the _energy of the

partide~hole

excitation

((q, pi

is q-

independent.

Calculating ZmH°(fl,q+, q-j (Eq. (5))

we obtain in this _case the _sum of the two terms

H+ Ill, q+)

and H-

Ill, q-),

where

H+

is _a contribution of the

partide~hole pairs

from

regions (Ai

and

(Ci

in

Figure

i and H- from

regions (B)

and

(D)

_on the _same

figure.

We obtain

zmnlin, q+1=

^{~~, uniess}

⁽ⁿ⁽

^<

^12Wq+1

°~

^~ _otherwise

^~~°l

H- _can be obtained from

H+ by

the substitution _{q+ - q-.} Since in the first

quadrant e(p

₊

q)

= 2W

cos(p-)(p[q+)

^and thus q-

mdependent,

the

self-energy

has two

contributions,

LL contribution

[2,

6] which is due to the interaction with

regions

A and C

(Fig. i)

and FL contribution

là,12]

^from the interaction with

regions

B and D in the _same

figure.

The

LL contribution to the

self-energy LAC(w,p+, p-)

behaves _as

a(p-)wIn

_(w

e(p+, p-)/WA(,

where

°~~ ~~~ÎÎÎIÎ~Î~~

^~~~~

Close to the point _{p- =} ⁰ we have additional

divergence

of the

self-energy. However,

as we

will show below

(see Eq. (16)),

this is _an artifact of the second~order calculations. Close to

1484 JOURNAL DE

PHYSIQUE

I N°11

Fig.

2.

Graphical

representation ofthe effective interaction. The dashed line represents the effective

interaction,

the dotted fine represents ^{the bare} interaction and the

loop

represents _a

partide-hole

_pair.

the _corners of the Fermi surface the LL contribution vanishes with

a(p-)

+~ cosp- - 0. This result is consistent with _our

assumption

that the

approximation

in

equation (9)

is

only

reliable

far from the _corners.

In the Random Phase

Approximation (RPA)

_{we sum up} all bubble

diagrams

_in the so-called effective interaction

D(fl, q) (see Fig. 2).

^One obtains

~~~'~~

l ~o

(H$lQ,qÎ)

+

Hflo, q-Il

^~~~~

The function D

in, q)

has two collective

poles

^for

given

_q+ and _q-. One

pole

_is associated with

divergence

of the

H(

and another

pole

is associated with

divergence

of the

H° Using D(fl, q)

we calculate the

self-energy

~~~'~°~

~

Î _ÎÎ Î ^ÎxÎ2 _~~~°

~°"

~~~~

_{~~°" ~ ~~ ~~~~}

We _{can see}

that,

_{as m} the second-order

approximation, possible divergences

arise from the fact that the intemal fermion line

(see Eq. (13)) depends

_{on q+ or q-} alone and the bare bubble

diagram

_can be

decomposed

into the _sum of

parts

which

depend

_on either _{q+ or q-.} The structure of

H+ Ill,

_q+ and H-

Ill,

_{q- gives} rise to the

pole

structure of the effective interaction.

For

(Q(, W(q+(

<

W(q-(

or

(Q(, W(q+(

»

W(q-(

_we obtain

~(H° Ill, q-)(

+~

~o/W

< i. For

these limits _we obtain

~(H° Ill,

_{q- )( +~}

~o/W

_< i.

Neglecting

the values of order

~o/W

in the effective interaction and

retaining only

the most

divergent

terms _as in

[7,13],

_we obtain for this

region

~~~~~+~

'~ l

V0i(fl,q+)

(14)

The

resulting

effective interaction

D(fl,q+)

in this hmit is

highly anisotropic similarly

to

[3].

^This can be

interpreted

_as effective

lowering

of the

dimensionality

_[4]. The function vo

Ill voH( Ill,

_q+

ii

^has

poles

^for _every ^value of _q+. Such

poles

exist if

vo/W

>

0, namely

vo, W > 0 _{or vo,} W _< 0. In the _{case vo}

/W

<

0,

the

poles

_appear close to _zero energy and the effective

lowenng

of the

dimensionality

does not occur. In this _case the 2D FL behavior

is

preserved,

_as it was assumed in [3].

Assuming

_v

=

1/2

and _vo

/W

_>

0, by integrating

^with

logarithmic

_accuracy

[13]

we obtain the LL contributions from the

poles

in the

self-energy

L for (w

e(p)(

< WA:

ReL(w, p)

=

a(p-)

_w In ^~° ~~~~

(15)

WA

where

ivo/2Wj2A

~ ~~ ~

(27rj4(~~

+

cosjp- jj2 i16)

,--- ',

"

' '

.

/

_~ ~

^{i ,}

i i

(a)

j~=j+~6~+...

(b)

=»....

+...i~j....

+...

(c)

Fig.

3.

Graphical representation

of the effective summation of trie

diagrammatic

contributions of all

orders; ai Self~energy, b)

vertex

correction, ci

effective interaction. The

large

dot represents ^the

renormalized vertex.

where

u]

=

4W~

₊

(vo/2x2)2.

Note that close to p- = 0 the

self-energy

has _no additional

divergence

in contradiction with second order calculations

(see Eq. iii )).

To go

beyond

the RPA _{we must sum}

diagrams

^of all orders of

perturbation theory.

For such

a summation _{we cari use} the Ward

identity

for the vertex

r(p,

_{p +}

q)

for the small momentum transfer _q, derived _{in [9]} for the ID _case and extended _to the D > i in

[6,14].

These identities

are reliable for the

long

_range ^forces since in this _case the momentum transfer is restricted for small momenta.

They

connect the vertex

r(p,p

+

q) (Fig. 3)

with the

one-partide

Green

function

G(p)

=

G(po,P)°

FTP,

_P +

qj

₌ ^~

lP) ^G~~

_IV +

q)

qo

E(P

+

q)

₊

ejpj

¹¹

^7j

It is

easily

_seen that this vertex

depends only

_{on q+ or q-.}

Using

this vertex one can obtain _an

equation

for the Green function _[9]

(vo EIP))GIP)

= +

fi / _dqo / _d~q

~~

~j/1(~

~ ~

j~~ GIP q) i18)

In trie effective interaction

D(q),

trie

self-energy

and trie vertex corrections cancel each other.

Therefore

D(q)

_can be found in trie RPR

by equation (12).

Solving equation (18) iteratively

and

integrating

over

ImG(p),

_one obtains trie

following

momentum distribution

, 2njp-)

nlPl

"

j

^1-

~) ^{sgnjp+j j19j}

where

a(p-

is

given by equation (16).

This

implies

that there is _no

jump

at the Fermi surface.

Considering

the _same

problem

in the

system

with _a

filling

which is

slightly

different from

half,

_we define _r

=

(2v -1)/8~vi,

where _{r <} A. For _q > r the _previous considerations _are

1486 JOURNAL DE

PHYSIQUE

I N°11

true. Thus _{r is a} natural infrared cut-off

parameter

^of the

theory.

The momentum distribution

in this _case is

11(P)

_'~

Il ^~~j~~'~~

~~~~

sgn(P+)j _(2°)

Thus the

jump

is

jr /A(~"~~~~

=

((2v -1) /8xv5A(~~~~~~.

Now _{we are} able to indude the

spin

into _our consideration. Let _us introduce two interaction parameters,

namely,

_vc for the interaction of the

partiales

with

parallel spins

and _vs for the

partiales

with

antiparallel spins. Proper analysis

shows that if _vc and _vs have the _same

sign,

the results of the

previous

consideration remain

essentially unchanged.

When _vc and _vs have

opposite sign,

the Fermi gas is a LL whenever _any of the values _vc

/W

_{or vs}

/W

is

positive.

^In the

present

^work we have discussed the

s-type

bond. In the

d-type

bond _case, which is

thought

to be _a

good

^{candidate for} the

explanation

of the

high-Tc phenomena,

the situation is much

more

complicated

and therefore it demands additional

study.

In

conclusion,

_we have calculated the

anisotropic

contribution in the fermion

self-energy.

We have shown that there exists _a

quasi

ID contribution in the fermion-fermion effective interaction. We have shown that such _a contribution _con lead to the LL behavior at half

filling.

The

dependence

of the

properties

of the 2D Fermi gas on the

sign

of the _vo

/W

leads to the

inconsistency

of the

theory

of the normal state of the 2D Fermi gas ^at half

filling

from both the ID LL [9] and the 2D

homogeneous

Fermi gas

[5, fil.

^{The effective}

lowering

of the

dimensionality depends essentially

_on the

parameter

vo

/W.

For _vo

/W

_>

0,

the normal state of the

system

is

quasi

ID. For _vo

/W

<

0,

the 2D features in the effective interaction

prevail

and _a Fermi _{gas con} be considered _{as a} FL.

Acknowledgments

The author wishes to thank L. P.

Pitaevskii,

J.

Felsteiner,

and Yu.

Nepomnyaschii

for _many frmtful and critical discussions.

References

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