Persistent Current of Interacting Electrons: a Simple Hartree-Fock Picture

(1)

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Persistent Current of Interacting Electrons: a Simple Hartree-Fock Picture

G. Montambaux

To cite this version:

G. Montambaux. Persistent Current of Interacting Electrons: a Simple Hartree-Fock Picture. Journal

de Physique I, EDP Sciences, 1996, 6 (1), pp.1-4. �10.1051/jp1:1996128�. �jpa-00247170�

(2)

Short Communication

Persistent Ctlrrent of Interacting Electrons:

_a

Simple Hartree- Fock Picttlre

G. Montambaux (*)

Laboratoire de

Physique

^des ^Solides

(**),

Université

Paris-Sud,

91405

Orsay,

France

(Received

20

SeptembeT1995, accepted

23 OctobeT

1995)

Abstract. The _average persistent current

(Ii

of diffusive electrons in trie Hartree-Fock _ap-

proximation

_is derived in

a

simple non-diagrammatic picture.

Trie Fourier

expansion directly

reflects trie

wmding

number

decomposition

of trie diffusive motion around trie

ring.

One _recovers trie results of

Ambegaokar

and

Eckern,

and Schmid. Moreover _one finds _an _expression for

(Ii

which is valid

beyond

trie diffusive

regime.

PACS. 05.30Fk Fermion systems and electron gas.

PACS.

72.10Bg

General formulation of transport

theory.

PACS.

71.25Mg

Electron _energy states _m

amorphous

and

glassy

solids.

The

physics

^of

persistent

currents in

mesoscopic

isolated

rings pierced by

_an Aharonov-Bohm flux

#

,

bas attracted _a lot of interest _on tl~e

description

of tl~e

tl~ermodynamic properties

of

mesoscopic metals,

both in tl~e

non-interacting picture

_or in trie presence of electron-electron in-

teractions. Trie

description

of trie interactions is _a

complicated

task:

although

several attempts bave been made to describe trie rote of the interactions in one-dimensional _or few-cl~annel

rings,

with the

help

of

analytical

_arguments _or numerical calculations

[ii,

trie diffusive _nature of trie electronic

motion,

which is

probably

essential in trie

experiments

on metallic

rings

with finite thickness

[2-4],

has been treated

originally

in two series of _papers

by Ambegaokar

and Eckern and Scl~mid

là, 6]. They

calculated tl~e average

persistent

current in trie Hartree-Fock _approx- imation where tl~e interaction is treated

perturbatively

in _a

diagrammatic picture. Althougl~

tl~is calculation

provides

_an _average current smaller that

experimentally

observed

[2],

one can

believe tl~at it contains tl~e essential

pl~ysical ingredient namely weakly interacting

diffusive electrons. Thus it may be

mteresting

to

simplify

_as much _as

possible

trie calculation _in order to

possibly generalize it,

_or to compare it with numerical calculations. In the above _papers, the _average

persistent

current is calculated with _a

diagrammatic technique

where the diffusive motion is described

by

_a

Cooperon pote

with _a _wave vector

quantized by periodic boundary

conditions.

Then, by

Poisson

summation,

this _sum _over diffusive modes is transformed into _a

sum over harmonics with

periodicity #o/2

where

#o

_is trie flux quantum.

In this short note, we propose a

simple

denvation where tl~e _average current is

directly

probability

for _a diffusive

partiale.

Then tl~is return

probability

_is

expanded

according

to tl~e

winding

number of tl~e dilfusive motion around tl~e

ring,

wl~ich _gives

directly (*)

e-mail:

[email protected]

(**)

associé _au CNRS

©

Les

Éditions

_de

Physique

1996

(3)

2 JOURNAL DE

PHYSIQUE

I N°1

access to the harmonics

expansion

of trie current.

Moreover,

_we obtain _a

general expression

which _can be used

beyond

tl~e diffusive

regime.

Consider _a

quasi-one

dimensional

ring

of

perimeter

L and of _transverse section

S,

in wl~icl~

the motion is

supposed

to be diffusive

along

tl~e

perimeter

and uniform

along

tl~e _transverse section.

Tl~e first

step

_is to write tl~e total energy

ET

in tl~e Hartree-Fock

approximation:

ET

=

El

₊

~j U(r r')

_(ifij

(r')(~(ifi~(r)(~drdr' Il)

~~~

Î

£

à«~«~

/ _U(r r')ifij (r')ifij ^jr)

_~§]

(r)ifi~(r')drdr'

~>J

where

E[

is tl~e total _energy in trie absence of interaction. In trie lowest order in trie interaction

parameter,

^the states ifi~ are the _states of tl~e

non-interacting system.

^Tl~e ^summation

£~

~ is over filled _energy levels. _a~ is the

spin

of _a state ifi~.

Considering

that the Coulomb interaction

U(r r')

is screened in trie metallic

regime,

it is

replaced by U(r r')

=

Uô(r r')

where U

=

4ire~/q(~

and _qTF _is trie Thomas-Fermi _wave

vector

[5-7]. Replacing

the interaction _m

equation (1) by

_a à function _is

certainly

correct _as

long

as trie Thomas-Fermi _wave

length

is smaller than tl~e _mean free

patl~ le: qTfle

» 1. Tl~e interaction

being

_now considered _as

à-like,

it is

quite

_easy to _see tl~at the Fock term has the

same structure _as the Hartree term.

Introducing

the local

density n(r)

₌

£~ (ifi~(r)(~,

^the ^total energy can be rewritten:

ET

=

El

₊ _U

/ _n~(r)dr ) / _n~(r)dr ₍₂₎

The Fock term is half the Hartree contribution because of the constraint _on the

spin

and its

sign

is

opposite

because of

exchange

of

partiales.

The local

density n(r)

_can be

expressed

in

terms of the Green function:

n(r)

₌

(-1lin) f)~ ZmGR(r,

_r,

_uJ)dur

_so that trie average ^current

is given

by

_[8]

IIe-e('))

~

(~() (3)

=

$ (

/ / /(G~(r,

^r,

^uJ)G~(r,

^r,

uJ'))duJduJ'dr

where

G~ (G~)

is trie retarded

(advanced)

Green function. Trie

product (G~(r,r)G~(r,r)) simply

_expresses trie

probability

to go from _some

point

_r to itself [9]. More

precisely,

for _a

partiale

at energy E

[10,11]

P(E,w)

=

£iG~(r,

r, E ₊

w/2)G~(r,r,E _w/2)j _(4j

is trie Fourier transform of trie retum

probability P(E,t)

after _a time t _:

P(E, t)

=

P(E,uJ)e~~~~duJ

(5)

2ir

Because of disorder _average, this retum

probability

is

mdependent

of the

position

r. The

current can Dow be

expressed directly

_m terms of

P(E,t). Neglecting

the energy

dependence

(P(E,t)

=

P(t)),

_one

gets:

~ ô

/°° ^P(t,~)

^~~

⁽⁶⁾

(~~-~

_~~~~

~~~~

~

~~

_~~

(4)

Q

= LS is the volume. Since the relation

(4)

is exact, ^the

expression (6)

is

quite general

and is valid

beyond

the diffusive

regime.

In the classical

approximation

for the diffusive

regime (le

<

L),

the diffusion

probability

is the solution of _a classical diffusion

equation

D/lP ₌

ôP/ôt

where D is the diffusion coefficient taken here _at the Fermi

level,

D

=

une/3.

It is

given by

~~

_~~'~"

~~

4irjÎt)d/2

~ ~~ ~'~~~~~~

~~~

In the

geometry

^of a

quasi-one-dimensional ring,

the return

probability

_can thus be

expanded according

to the

winding

number of the diffusive motion:

P(t, ~)

=

~j

_e~

^Ù

[1 ⁺

co8(~~rmp)j (8)

S@$

~

where _ç7

=

#/#o.

The second term, of

importance here,

results from the

phase

interference between time-reversed

paths

in the semi-classical

approximation,

each

path accumulating

_a

phase ~2irmç7 [10,12].

In _zero

flux,

the return

probability

^is twice the classical _one. This

expansion according

to the

winding

number _grues

directly

the Fourier

decomposition

of the current:

ii-e (#))

=

£ _Im sin(4irmç7) (9)

m

with

__

i~

₌

8mUPo / ^~-i

^dt

_~ioj

#cris

o

t5/2

We have introduced the Thouless energy

E~

=

hD/L~. _Defining

a dimensionless

winding

number _w

by ^w2

₌

m~/4E~t, Im

_can be

simply

rewritten _as

~ ~

~~

~P0 ~c /" _~2~-w~

_~~

~

fi #o

m2

~

~~~~ ~

~~~~

which is the result of

Ambegaokar

and Eckem _[5] and Schmid [6].

We finish with the calculation of the current at finite

temperature

^T ^where two Fermi factors

f(e)

have to be introduced in

equation (3). Dring

the standard substitution

f f(e)g(e)de

=

2iirT£~ g(iuJn)

where _mn ₌

(2n +1)irT,

the average ^current ^at finite T is

straightforwardly given by:"

IIe-e)

=

-QUAD

^~

l _47r~T~ _~j _Pii~J» _i~Ji)

~

~~~° Ii Î si~lh

ÎTt)2 ^~'~~

^~~~~

By introducing

the _same dimensionless

wmding

number _w _as

above,

the harmonics

Im

_are

given by

~

~ ~~

_~~~

/Î Î _~~~~~

~~

ÎÎÎ

~~ smh

21fi

^~~~

where

Tm

₌

E~/m~

is the effective

temperature

associated with the

winding

number _m and Ùm =

T/Tm. Although

the

integrand

is

quite different,

this

temperature dependence

is identical

(5)

JOURNAL DE

PHYSIQUE

I N°1

to the _one found

by Ambegaokar

and Eckern [5]. ^It

directly

_expresses the current in _terms of

a

temperature

square average of _a

winding

number.

Equation (ii

shows that the

persistent

current is

proportional

to the interaction parameter.

It is known that this current is smaller than

experimentally

observed [2].

Moreover, Cooper

channel renormalization reduces further trie

amplitude

of the estimated current

[15].

^Aise ^the

importance

^{of trie}

self-consistency

_m trie Hartree-Fock

approximation

in still under numerical

investigation _[13].

A different

approach,

based _on

Density

Functional

Theory,

also

gives

similar

results,

smaller than the

experimental

_one

iii.

In

conclusion,

_we have calculated the average

persistent

current in the first order of the Hartree-Fock

approximation. By writing

the current

directly

_m terms of the

winding

number

decomposition

of the return

probability,

_we have avoided the _use of the

diagrammatic

calculation and

directly

found the harmonic

expansion

^of the current.

Although

this calculation

is reminiscent of trie semidassical

description

of the

spectral

correlations and of the average current of

non-interacting partides

_m the canonical ensemble

[14],

^it ^does not use any semidassical _sum

rule,

since here the correlation function of interest can be

exactly

written in terms of

the retum

probability

without _any

approximation. Moreover,

_we have written _an

expression

for the average ^current

(Eqs. (6)

and

(12)

which is valid

beyond

trie diffusive _regime and may be used for

example

_m trie ballistic regime where

interesting magnetic

_response have also been

observed

[16].

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Lewenkopf C.H., Europhys.

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