Enhancement of Persistent Currents by Hubbard Interactions in Disordered 1D Rings: Avoided Level Crossings Interpretation

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Enhancement of Persistent Currents by Hubbard Interactions in Disordered 1D Rings: Avoided Level

Crossings Interpretation

Michael Kamal, Ziad Musslimani, Assa Auerbach

To cite this version:

Michael Kamal, Ziad Musslimani, Assa Auerbach. Enhancement of Persistent Currents by Hubbard

Interactions in Disordered 1D Rings: Avoided Level Crossings Interpretation. Journal de Physique I,

EDP Sciences, 1995, 5 (11), pp.1487-1499. �10.1051/jp1:1995212�. �jpa-00247152�

Classification

Physics

Abstracts

72.10-d 71.27+a 72.15Rn

Enhancement of Persistent Currents by Hubbard Interactions in Disordered ID Rings: ^Avoided Level Crossings Interpretation

Michael

Kamal(~),

Ziad H.

Musslimani(~)

and Assa

Auerbach(~)

(~) Goldman, Sachs &

Co.,

85 Broad

St.,

New York, NY 10004, USA

(~) Department

of

Physics,

Technion-Israel Institute of

Technology,

Haifa 32000, Israel

(Received

26

January1995, accepted

2

July 1995)

Abstract. We study effects of local electron interactions _on the persistent current of _one dimensional disordered

rings.

For dilferent realizations of disorder _we compute the ensemble average ^current as a function of Aharonov-Bohm flux to zeroth and first orders _in the Hubbard interaction. We find thon the

persistent

current is enhanced

by

_on site interactions.

Using

_an avoided level

crossings approach,

_we derive

analytic

formulas which

explain

the numencal results ai weak disorder. The _same

approach

also

explains

the opposite effect

(suppression)

found for

spinless

fermion models with intersite interactions.

1. Introduction

Recent

experiments

^{have found that small} isolated metallic

rings,

threaded

by magnetic flux,

carry

persistent

currents.

Although

^the existence of this effect had been

anticipated

theoreti-

cally

^for _{many years}

[1-4],

the

magnitude

of the observed current _was found to be much

forger

thon

expected.

Experiments performed

_on

moderately

disordered

rings

with small _transverse width bave measured the

magnetic

_response of _an ensemble of _copper

rings [si,

of isolated

gold loops _[6],

and of _a

dean,

almost

one-dimensional, sample

_[7].

Trie ensemble

averaged persistent

current _was found _to bave _an

amplitude

of order

lo~~evf IL

and

periodicity

4lo

/2, (where

L is the circumference of the

ring,

_vf is the Fermi

velocity,

e is

the electron

charge,

and 4lo " h

le

is the flux

quantum).

The

single loops experiment reported finding

_an

unexpectedly large

current of order _evf

IL,

the free electron

value,

and

displaying

periodicity

in flux with

period 4lo.

Theoretical

approaches

con be classified into

non-interacting

and

interacting

electrons _ap-

proaches.

Non

interacting

theories which take into account the effects of disorder and devia- tions from _a

perfect

ID geometry [8],

predict

values of the average

persistent

current that _are smaller thon

observed,

but still within _{one or two} orders of

magnitude

of trie

experimentally

determined value of trie _average current. More

surprisingly,

the theoretical estimates for the

typical

value of the

persistent

current, measured in the

single loop experiment,

are

perhaps

Q

Les Editions de

Physique

1995

two or three orders of

magnitude

smaller than the

reported

values. It _seems that

agreement

between

theory

and

experiment

_worsens with

increasing

disorder. This

suggests

that

perhaps

interaction corrections _are

quantitatively _important

in the

moderately

disordered

regime.

A first _guess would be that interactions enhance the

impurity scattering

and further _s~tppress the

persistent

_{current, as}

happens

for the conductance of the disordered

luttinger liquid _[9].

However,

this intuitive

analogy

_may be

quite misleading.

While the conductance

depends

on the Fermi surface

properties,

1-e-

velocity

and

scattering

rates, the

persistent

current is

a sum of contributions from ail

occupied

states.

Also, by

Galilean

invariance,

_an

interacting

electron

liquid

without disorder carries the _{same current as} the _non

interacting

_{gas, even}

though

its Fermi

velocity

_may be renormalized. Thus it is

plausible

that in the disordered

system,

interactions could work both _ways, 1-e- either _{suppress or} enhance the

persistent

currents under conditions which need to be

explored.

Exact

diagonalization

studies of

spmless

fermions with intersite interactions _on small

rings [10, iii,

have found that weak interactions red~tce the

persistent

current below its

noninteracting

value. The correlation between the _non

interacting persistent

current and the first order

(in interactions)

corrections have been evaluated

numerically

for the Hubbard model with

spin by Ramin,

Reulet and Bouchiat

(RRB) _[13]

for various disorder realizations. RRB found that the

non

interacting

and interaction correction have the _same

sign,

which agrees with exact results for the disordered Hubbard model found

by

Giamarchi and

Shastry [12]. They

also

explain

this

effect

by estimating

^the

ground

state energy correction. In this _paper

[14]

we shall _{gain more}

insight

into the

puzzle

of interaction corrections

by arriving

at _an

analogous expression

for the interaction correction to the

persistent

current and its

explicit dependence

on disorder

strength.

In Section 2 _we

diagonalize

the

tight binding

Hamiltonian

numerically

for different realizations of

disorder,

and

compute

^the

persistent

current to zeroth and to first order in the Hubbard interactions. We find that interactions enhance the

persistent

current. In Section 3 _we present the Avoided Levels

Crossing (ALC) theory,

which

provides

_an

analytic approximation

to the

numerical results. This _{is an}

expansion,

at weak

disorder,

of

nearly degenerate eigenstates

at fluxes close to

points

of time reversai

symmetry.

^The correlation between the zeroth and first order currents _is

explained by

_{a common}

mechanism,

1e. the avoided level _crossings.

The

opposite (suppression)

effect for

spinless fermions,

is also

explained by

the ALC

theory

_in

Section 4. We conclude

by

_{a summary} and future directions.

2. Numerical Perturbation

Theory

We consider _a

tight-binding

Hamiltoman _{on a}

periodic

chair with

repulsive

on-site Hubbard interaction:

H =

Ho

+ U

~j

niinii,

il

1

Ho

=

~je~~ ⁺ %)cÎ+i~cm

+

h-c-j

+

~ _{eicl~cm. 12)}

m m

where

c)~

^creates an electron at site1 with

spin

_a. 0ur unit of _{energy is} the

hopping

_{energy. fi}

is the dimensionless on-site disorder energy

uniformly

distributed in the interval

[-W/2,W/2].

The chair has L sites,

N~ electrons,

the flux

through

its center is

given by

4l. With these

conventions,

the energy spectrum is

periodic

_in ^{the enclosed} flux with

penod #o,

and the

current is given

by

1(#)

=

~

E(#), (3)

à#

W=0

,--,__ ,,-,,,

"'~

io(~)

^W=4

~/~o

Fig.

l. The ensemble

averaged noninteracting

current (ID) _as a function of the

applied

flux for

several

strengths

of disordered

potential:

W=0,

o-à,

I.o, 2.0 and 4.0. AIT _{curves are} for _a half-filled Iattice of _six sites.

where

Ej4l)

=

Eo14l)

+

Ei14l)

₊

OjU~). j4)

Eo

is the exact

ground

state energy of

Ho,

whose

single

^electron

eigenstates

are determined

numerically. El

is the first order correction in

U,

which is given

by

El

= U

~

niinii

i

nia =

l4~olCÎaCialifo) là)

where

~o

is the Fock

ground

state of

Ho

Thus

by diagonalizing Ho

_{we can}

readily

obtain

f(#)

=

Io(#)

+

Ii (#)

₊

tJ(U~ (6)

The effects of disorder _on the ensemble

averaged

_non

interacting

current

Io (#)

_can be _{seen m}

Figure 1,

where it is shown as a function of flux for _a half filled lattice of six

sites,

for different disorder

strengths

W. Disorder smoothens and reduces trie

magnitude

of ID, and for ^W » it is dominated

by

the first harmonic

sin(2ir#/#o).

In

Figure 2,

the average value of

Ii (#)

is

plotted.

There _are several features of the first order interaction correction that deserve comment.

First,

the most

important

observation is that it

generaiiy

enhances the _non

interacting

current for ail values of disorder

strength

and flux. That is to say: there is _a

positive

correlation between the _non

interacting

current and the first order

correction,

jiojù)iijù))~,~ à

o.

ii)

which is in

agreement

with the numerical results obtained

by

RRB for the _same mortel

[13].

This result is fourra to holà for ail reahzations of disorder which _we have used.

Second,

in the limit of weak

disorder, Ii(#)

becomes

singular

at

discontinuity points

of

Io(#),

which _are at

W=1

Iii~)

w=~

~/~o

Fig.

2. The first order ensemble

averaged interacting

current

(Ii)

_{as a} function of the

apphed

flux for disordered

potential

values: W= I.o, 2.0 and 4.0. As before, ail _{curves are} for _a half-filled Iattice of six sites.

fluxes

(m

₊

) )#o, (m#o

for _even

(odd)

number of filled

orbitals,

where _{m are}

integers. Finally,

we see the first order _current also becomes dominated

by

its first harmonic _at

large

^disorder.

To

study

the

scaling properties

of the current with

system

size and disorder

strength,

it is convenient to characterize the

strength

of both

Io

and

fi by

their

amplitude

at #

=

#o/4.

And while it does Dot capture ^the

singular

nature of

fi (#)

_near the

regime

of very weak

disorder,

this characterization _is still useful to

study

the

scaling properties

of the first order current.

Previous studies have found

[4, loi

both

numerically

and

analytically,

that the

amplitude

of the

noninteracting

current,

averaged

_over

disorder,

behaves like:

iloi*o/4)1

_"

)) _eXPi-L/1).

18) By fitting

to this functional form _we extracted the localization

length (

for different

system

sizes and disorder values.

Figure

3 shows the values of the localization

length

obtained for sizes

L=6,10,14,20

and

25,

at

half-filling, averaged

_{over many}

(up

to four

thousand) impurity configurations

for each value of W. We find

good _agreement

between _Dur mferred value of the localization

length

and the known

asymptotic

form _m the weak disorder limit:

(

₌

105/W~,

valid when W _< 2ir for

large systems

at

half-filling.

There is less

agreement

in the

strongly

locahzed limit where the localization

length

^is known to behave like:

(

=

(In(W/2) 1)~~.

We find _a better fit

taking (

=

Il In(W°.~/2.5)

for the L ₌ 25 data. This

discrepancy

could be due to the small _sizes

considered,

_{or a} breakdown of

equation (8)

when

(

is of order

unity.

We have calculated the

amplitude fi(#

₌

#o/4)

as

function of

strength

of

disorder,

characterized

by

the

scaling

parameter

L/(,

for system sizes

of

L=6,10,14,20,25.

At weak disorder

(large (),

the

amplitude

_increases with the

strength

of disorder

achieving

its _maximum value at some intermediate

strength

of disorder

(

_m L

).

^In this

weakly

disordered

regime,

_a

single scaling

function could be used _to descnbe the results:

)~

~

AL/()/L. 19)

8.0

60

Înf

_4.0

2.0

o-o

~ ~ ~ ~

W

Fig.

3. The

Iogarithm

of the Iocalization

Iength (In(), implied by equation (8),

_is

plotted

_{as a} function of

strength

of disorder

(W)

for half-filled _rings with N=6, 10, 14, 20 and 25 sites. Data for the different system sizes

collapse

weII onto _a

surgie

_curve.

This behavior _is similar to that of the

amplitude

of the

noninteracting

current

given by (8).

Upon increasing

the

impunty scattering further,

the first order current decreases with disorder.

When the

single partiale

wavefunctions _are

sufficiently locahzed, fi

_cari be descnbed

by

_a

different

scaling

form:

Î~

'~

~~~~~~'

_~~~~

where g is a

decreasing

function of its

argument. Equation (10) suggests

that in the localized regime

fi

dominates

fo

for

large

system

sizei.

However for localized

doubly occupied single

electron states

l~fiai~fiai lu ~niiniil ~fiai~fiai)

~

(ii)

~

Thus _even for weak

interactions,

the interaction corrections may be

larger

than the _non inter-

acting

level

spacings

which go as

1/L.

This invahdates

pert~trbation theory

in U in the iocaiized

regime.

3. Avoided Level

Crossings Theory

Here _we will discuss how the numerical results of the

previous

section _can be understood in

terms of avoided level

crossings (ALC)

at weak disorder.

First,

_we denve the ALC

approxima-

tion for the

noninteracting

current

fo.

3.1. ÀLC _{THEORY FOR}

Îo.

In

Figure

4 _{we can see a}

typical

spectrum for _a

tight-binding

JOURNAL DBFHYSIQUBL -T. 5,N°ii NOWMBBR 1995 59

z-o

i o

e~(§j)

o-o

-i o

-z,o

~

i.5 -1 o -o 5 O.O o 5 o 5

ié/4Ùo

Fig.

4.

Energy

Ievel spectrum of the

tight-binding Hamiltoman(12)

_{as a} function of

apphed

flux for _a N=6

ring

_m the absence of disorder

(solid Iines)

and for weak disorder

(dotted Iines).

Hamiltonian of _a 6 site

ring,

_{as a} function of the

apphed

flux. In the absence of disorder

(W

=

0)

the

eigenenergies

_are

en(#)

= -2 _cos

() _in

+

)))

,

(12)

o

with

period #o

" 2ir. At

points

of time reversai

symmetry,

i-e- where # is _an

integer multiple

of

4lo/2,

level

crossings

_occurs between states of

opposite angular

momenta. The

noninteracting persistent

current is _{a sum over} the currents carried

by

ail

occupied levels,

fo(4l)

=

(

=

~j (en(4l), ⁽¹³⁾

where _{n,s are} the orbital and

spin index, respectively.

The

nonmteracting

current

fo(4l)

will be _a smooth function of 4l away from the

points

^of level

crossings. By symmetry

^of # _-

-4l,

any

pair

^{of levels} _cross with

opposite slopes,

and thus if

they

_are both

fully occupied

their contribution to the total current cancels. The

only nonvamshing

contribution _comes from _a

topmost

^level which is not

compensated by

its

partner.

^{Then the} current

changes

_sign

abruptly

as the

occupation

_moves from _one branch to another. Thus for _an odd number of

fully occupied

orbitals the current

discontinuity

_occur at #

=

(m +1/2)#o,

otherwise the discontinuities will

occur at #

=

m#o.

In between the discontinuities the current _varies

hnearly

with the

flux,

which

explains

the

penodic

sawtooth

shape

for

fo

_{as seen in}

Figure

1.

Introducing

_a small

amount of disorder lifts the

degeneracy

at the _crossings

by

_opening small _gaps. This reduces

the

persistent

current

fo

_smce the

occupied

levels have smaller

slopes

_near the former _crossing points. ^This

explains

the behavior shown _m

Figure

1: weak

impunty scattenng

softens the

discontinuities and leads _{to an} overall reduction of the

magnitude

of current. We _cari

quantify

this observation

by examining pairs

of levels with momenta eh which _cross at #

= 0. We

specialize

to the _case of _{an eveii} number of filled levels with

equal occupations

for bath _spin directions. The

unperturbed

_{energies are given}

by

f+k(4l)

=

-2cos1+k

^{+ ~~}_L

_#o

^~

⁽¹⁴⁾

with k

=

2irm/L,

for _a

positive integer

m. Consider the effect of _a weak random

potential

_fi which lifts the

degeneracy

in the 2 _x 2

subspace

of

k, -k,

Ho

-

~Îl ~lji~

lis)

~~~~~

j~

_~ =

( exp(-12kzi)ei. l~~~

L

i=1

From _{now on we} will omit the

subcripts k,

^-k off

Ù.

Trie

eigenvalues

and normalized

eigenfunctions

of

(là)

_ai-e:

e+iù)

_.

_ifi+

=

fi _(A+eik~

₊

B+e-ik~)

e-iù) ifi-

=

fi _(A-eik~

+

B-e-ik~) (ii)

~vhere:

~~

4(Ù(~

~~~

~ _{~ ~}

fk

~-k +

IIe-k _~k)~

₊

_~ÎÎ~Î~

^{~ '}

~y ~

~~~~ A

,

(18)

f-k fk +

IIe-k _~k)~

₊

_4(V(~

and,

ek + e-k +

~/(ek e-k)~

+

4(Ù(2

f+ "

,

(19)

2 Whicii

SatisfY.

~~j~)

+

~~iù)

=

e~i#)

+

e-ki*).

_~~°~

We _see that the contribution of the

occupied

orbitals to

Io

is

unchanged by

weak disorder since:

~"~

"

~~~il ^~~

"

~~~~ù~~~ ^l~~~

Let _{us now use}

(13)

to calculate

Io(#)

for free electrons with weak disorder

by summing

over

occupied

levels. We tilt ail levels _up to

kf

=

2irmf IL, plus

two electrons at

kf

_so that

an even number of orbital levels _is filled.

Although

~ve restrict _our calculations to _{an even} number of filled

orbitals,

the ALC

theory

_cari be

easily

extended to ail other

fillings

_as follows:

we note that the _non

interacting

wavefunctions have two-fold _spm

degeneracy.

An

unpaired

spin

in _an open orbital will net

produce

_any interaction correction to the

persistent

current since its

density

interacts with _an

approximately

uniform _core

density.

For odd number of

doubly occupied orbitals,

the

important

level1 _{crossmgs occur} at # _i

+)#o,

rather than _near

# ₁ 0. The total

noninteracting

current is

given by

the contribution from the filled level _pairs,

together

with the contribution of lowest _m

= 0

(nondegenerate)

level and the

topmost

orbital level at mf:

10

(#)

_"

~j _If

₊

If"°

₊

_I~~

m

" ~~

~~

~~

S

^~~~

Z ~~

^~

Î~ _{~W l~~~}

Neglecting

corrections of order

1/L~,

_we obtain the expression:

~ ~

8irsin(ki) ~

# #

/

^{# ~}

L(V(

^~

°~

~~

~~

L#o #o~ _{#o #o}

^~

_4irsin(kf)

_'

for # _E

[-#o/2, #o/2].

This _expression for _{ID is} valid _as

long

as the energy scale of the disorder

is much smaller than the level

spacing

at the Fermi level:

iii

_{« 47r}

_{sinjkf) /L. j24)}

Using

^{the relation between}

Vi

and the _mean free

path

_l~i defined

by

the _one dimensional Born

approximation,

i~i e

~~ ~~~~ ~~

(25)

(V(~L According

to

(24),

the ALC

approximation

_is vahd for

8iri~i/L _{/ 1,} (26)

which is the

ballistic,

or delocalized

regime(~

In terms of l~i one can write

I01*)

^~~

)jl~f) _~

*

~

#

/

4~ ² ~

o

#~ $ -)

~

~

*o 8irl~i (27)

In order to compare

(27)

to the numerical result for

equation (2),

_one needs to determme the

parameter

l~i Since fluctuations _{in ID} for different disorder realizations _are

large,

_we determine l~i

by fitting (27)

to the numerical ensemble

averaged

_ID Once l~i is determined for _a

particular

disorder

realization,

_{we use} it to evaluate

fi

_as shown below.

3.2. ÀLC THEORY FOR

Ii.

We shall _now

proceed

to use the _same

approximation

to

explain

the behavior of

fi

_(~

(~)There

_{is no} intermediate "diffusive

regime"

between the locahzed and ballistic regimes m one

dimension [15].

(~ This

approach

is similar to the discussion of Aharonov-Bohm oscillations of the

participation

ratio

m disordered rings [16].

According

to

(5),

the first order energy is _{a sum} of the

density squared

at aII sites,

Ei(4l)

= U

~j

_n~in~i

(28)

where for the

filling

^of _{an even} ^{number of orbitals} _per

spin,

the

single

_spm

density

_is,

n~s(«)

=

lino(1)i~

+

iJfim,,-(i)i~

+

~É~ ~ iifim,«(1)i~- (29)

We wiII _now show that the first order _energy,

Ei(4l)

= U

~j

_n~in~i

₍₃₀₎

is enhanced _near Ievel

crossings.

For _a

pair

of

fully occupied

Ievels _one

has, by unitary

lifim,+(1)l~

+

lifim,-(1)l~

₌

lifik(1)l~

+

titi-k(1)l~

₌

). ₍₃₁₎

Consequently,

for weak

disorder,

aII

occupied Ievels,

except ^{for the Iast} one, contribute constant values to the total

density:

n4 =

~~~

+

(ifi-(1)(~ (32)

L

or m terms of the coefficients

A,

B

(17)

n~ =

~ [ ^~2k~,

L~

~

~

~/

+

~~~5~ ^~~~~' L The first order energy is thus

given by (32),

Ei(4l)/U

=

~~~~

+

~~~~~~~~~~~

(34) Using (17)

and

(25)

_{we con} write

El (4l) /U

_m

^~~~

+ l +

~~~~'

~

,

(35)

L

~

2L L 4lo

~ ^~

Dilferentiating (35)

with respect to flux

yields

Ii(«)/U

-

ll~il Ill

₊

_l~l~l ^{Ill j~} ₍₃₆₎

In

Figure 5, (36)

is

compared

to the numencal ensemble

averaged

result for

Ii

for _vanous values of disorder. For each disorder

realization,

_{l~j is} determined

by fitting

the numerical and ALC results for the average

Io.

We _see that for weak disorder there _{is a}

satisfying agreement

between the ALC

approximation

^{and the numencal} results. In

Figure 5c,

the disorder is _too

large,

and the ALC

approximation

^faits

badly.

Equation (36) explams

both the

positive

correlation between

Io

and

Ii

of

(7).

Since _{m one} dimension the _mean free

path _(l~j)

and the Iocahzation

Iength (~)

ailler

by

_a

proportionality

factor of order unity

[15], (36)

_agrees with the

empirical scahng

form

(9) iii Î/U

+w

f(L/l~j) IL

which _was found

numerically

at weak disorder.

W=025

ii(~)/U

~)

^w~o

w ₌ o-s

Ii(~)/u

b)

^~/~o

W ₌ 1

~~ ~mo

Fig.

5.

Comparison

between

numerically

determined interactions correction Ii

(dashed fines)

and the

analytic

ALC result

(36) (sohd fines). a) c)

show results for three values of disorder

strength

W.

loi(W)

_are determined by

fitting

equation

(27)

to the numerical disorder

averaged Io(4l).

Error bars depict fluctuations of numerical Ii for different disorder reahzations.

4. Difference of Hubbard Model and

Spinless

Fermions

In recent papers

[10, iii,

small disordered

rings

of

spinless

electrons have been

exactly diago-

nalized. The

persistent

currents have been

computed

_{as a} function of disorder and interaction

strength.

In _contrast to _our result for the Hubbard

mortel,

interactions have been _seen to reduce the

persistent

current at weak disorder. Here _we

apply

the ALC

approach

to

explain

this apparent ^{dilference between the} mortels.

The

spinless

Hamiltonian H~ is given

by

H~ =

Hi +~jU~n~nj,

~

Hi

=

-~j[e~~+(~c)~~c~+h.c.]+ _~je~c)c~. (37)

~ ~

The _non

interacting

current,

Ii,

_is

given by

half the value of

equation (13).

In the ALC

approximation,

it is

given by

half the value of

equation (27).

Following

the

analogous

derivation of

Ii,

we use

(33)

to obtain:

E((4l)

=

4mf~U(0)

₊

2(A- (~(B- (~U(2kf), (38)

where

Ù(k)

=

j ~exp(-ikxj)U~, (39)

J#,

which

implies

E((4l)

₌

4m)U(0)

₊

Û(2kf)

₊

~~~~'

^~

L 4lo

I((4l)

=

~~j)~ ^U(2kf)

¹ +

~~~~' ())

^{~ ~}

₍₄₀₎

o L

o

For the

spinless

nearest

neighbor

case studied in reference

il Ii El (4)

" u

~j

il~n~+1>

(41)

~

the

corresponding

Fourier coefficient is

U(2kf)

= 2U

cos(2kf). (42)

Above _a

quarter filling, kf

> _x

/4,

U is

negative. Consequently Ii

has the

opposite sign

to that of

Ii

in

equation (36),

and to

Ii.

That is to say, trie persistent ^current of the

spinless

fermion mortel is

suppressed by

the interactions which is _m

agreement

^{with the results obtained}

by

reference

Ill, _13].

The difference between the mortels _con be attributed to the different effects of interstte interactions

(in

the

spmless mortel),

_versus local interactions

(in

the Hubbard

mortel).

5. Remarks and Conclusions

We have

investigated

the first order effect of Hubbard interactions _on the

persistent

current in

one dimensional disordered

rings.

The

findings

_con be summarized _as follows.

Ii

_was round ta correlate _in

sign

with

Io,

_a fact which _goes

contrary

to the _naïve intuition that interactions suppress ^currents

(e.g.

_as

they

do for the

conductivity

of the

Luttinger

mortel with _an

impurity [9]).

We _con understand this result

by observing

that

Ii depends

_on

charge

fluctuations which vary

strongly

with flux _neon the

degeneracy points,

which also determine the

sign

and

magnitude

of

Io. Using

the avoided Ievel

crossings theory,

we obtain

analytical expressions

which fit the numerical results for both

Io

and

Ii

We _{con use} the _same

theory

to

explain why spinless

fermions with intersite interactions exhibit

suppression

^of currents rather thon enhancement.

We have found

numerically

that

Ii (L If)

scales

differently

with the Iocalization

Iength

thon

Io(L If),

it _seems that effects of electron-electron interactions grow as disorder is increased.

However,

first order

perturbation theory

in the

weakly

disordered _{case is}

expected

to holà _as

long

as the interaction matrix elements do not exceed the

single particle

Ievel

spacings.

Thus _we

are restricted to the

regime

U < 1. We _con draw _on the exact

diagonalization

results

[10, iii

where for weak

disorder,

the numerical currents are found to vary

hnearly

with interaction

strength

in _a sizeable regime. Thus _we believe that first order

perturbation theory

should be valid for

physically interesting

interaction

parameters.

It ,vould be _very

satisfying

lia similar

analysis

coula be

applied

to the

experimentally

relevant

case of three dimensional

rings. Preliminary

results

yield positive

correlations between

Io

and

Ii,

but where _a

simple

mmded

application

of the ALC

approach

cannot descnbe the diffusive

regime of _{l~j <} L. It would also be

important

to understand the effects of true

long

_range Coulomb interactions with the screening and

exchange

^{elfects which} are absent in the Hubbard mortel.

Acknowledgments

We thank E.

Akkermans,

R. Berkovits and Y. Gefen for useful discussions. The

support

^of the Us~lsrael Binational Science

Foundation,

the Israeli

Academy

^{of Sciences} and the Fund for Promotion of Research at Technion is

gratefully acknowledged.

Part of this work has been

supported by

grant ^{from US}

Department

of

Energy,

DE-FG02-91ER45441. M.K. thanks the

Institute for Theoretical

Physics

at Technion for its

hospitahty.

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