Single level current and curvature distributions in mesoscopic systems

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Single level current and curvature distributions in mesoscopic systems

Alex Kamenev, Daniel Braun

To cite this version:

Alex Kamenev, Daniel Braun. Single level current and curvature distributions in mesoscopic systems.

Journal de Physique I, EDP Sciences, 1994, 4 (7), pp.1049-1062. �10.1051/jp1:1994183�. �jpa-00246963�

Classification

Physics Abstracts

05.45 73.35

Sinfle ^{level current and curvature} distributions in mesoscopic

sys

ems

Alex Kamenev

(~)

and Daniel Braun (~)

(~) Department of Nuclear Physics, The Weizmann Institute of Science, ^Rehovot 76100, Israel (~) Laboratoire de Physique des

Solides(*),

Bat 510, Université Paris-Sud, 91405 Orsay, France

(Received

26 October1993, accepted 16 March

1994)

Abstract. Exact analytic results for s1~lgle ^level curre~lt and curvature distribution functions

are denved within the framework of _a 2 _x 2 random matrix model. Current and curvature _are defined _as the first and second derivatives of energy with respect to _a time-reversai symmetry breaking _parameter

(magnetic flux).

Trie apphcability of the obtained distributions for the

spectral statistic of disordered metals is discussed. The most surprising feature of _our results _is the divergence of the second and higher moments of the curvature at _zero flux. _{It is} shown that this divergence also _appears in the general N _x N random matrix model. Furthermore, _we find

an unusual logarithmic behavior of the two point current-current correlation function at small flux.

1. Introduction.

The statistical

properties

of _energy spectra ^of a

single

electron in disordered

mesoscopic

systems have been the

subject

of intensive studies

during

the past ^decade.

Apart

from _a fundamental

interest of its own, the

spectral

statistic is

closely

related to observable characteristics such _as persistent currents, ^anomalous

magnetization,

^and conductance fluctuations. After the seminal studies of Efetov

iii

and Altshuler and Shklovskii _{(2] it} became dear that the

spectral

statistic in the diifusive

regime,

where the _mean free

path

of the electrons is much smaller than trie

sample

_size L, _may be described

by

random matrix

theory (RMT).

Trie

major

feature of trie spectra, ^known

long

_ago from RMT (3,

4],

^is a level

repulsion.

It _means that trie

probability, Pie),

to find _two consecutive levels _a distance

e apart ^tends to _{zero as} e decreases:

Pie) ^Ù

0.

Despite

_{great progress} in trie

theory,

closed

analytical

expressions ^for trie distribution function

(DF), Pie),

_are not available

(except

small _e

behavior).

It _was

demonstrated, however,

in _{a vast} amount of numerical

experiments

_(4], that _a

simple Wigner

surmise, obtained for _a 2 _x2 random Hamiltonian, is _an excellent

approximation

of trie

Pie)

of

large

N _x N matrices.

Originally

(*) Associé _au CNRS.

trie

Wigner

DF _was obtained for three distinct symmetry classes of trie Hamiltonian: Gaussian

orthogonal, unitary

and

symplectic

ensembles

((GOE),(GUE)

and

(GSE), respectively)

_(3].

Subsequently

trie _crossover ensembles from _{one pure} symmetry type to another _were introduced and

investigated

_(Si. In trie present

study

_{we are}

especially

interested in _{a crossover} from GOE to

GUE,

which

corresponds

to _a

graduai breaking

of trie time-reversai symmetry. To this end it _was

proposed

_{(Si to}

study

_a random Harniltonian of trie

following

structure:

Hl'~l(a)

=

Hj~

₊

_ioHi~~,

_{0 §}

a § 1,

(1)

where

Hj~~

and

Hfl

are,

respectively, symmetric

and

antisymmetric

real random N _x N matrices. For _o

= 0 _one bas

GOE,

for _a

= 1 trie GUE-case. It _was shown

recently

_(6] that trie time-reversai symmetry

breaking

parameter a may be

uniquely

connected to _an Aharonov- Bohm

flux,

16 _e

#lbo/21r,

carried

by

_a flux

fine,

which penetrates ^trie system. ^Trie

latter, by

_gouge

invariance,

_may be related to _a

change

of

boundary conditions, imposed

_{on wave}

functions in trie

angular

direction

(around

trie flux

fine):

_lF(21r)

=

lF(0)e~~

[7].

According

to reference _(6] trie relation between _a and

#

bas trie form

a=4#,

(2)

where A is trie _mean level _spacing and Ec is trie Thouless correlation energy of trie system.

Let _{us Dow}

point

out that trie first and second derivatives of trie _energy levels with respect to flux

(berce

with respect to

a)

bave _a clear

physical

interpretation. Trie first derivative ôen

/ôlb

is

exactly

_a

single

level current, carried

by

each _energy level in trie absence of time reversai symmetry

la

>

0).

These

single

level _currents manifest

themselves,

for

example,

in persistent

currents

through

_a

mesoscopic ring.

^As was

conjectured by

Thouless _(8] and

subsequently

discussed

by

Akkermans and Montambaux (9], a

typical single

level curvature

(trie

second derivative with respect to trie

flux)

_may be considered _{as a measure} of trie correlation _energy, hence it is

directly proportional

to trie

dissipative

conductance of trie system. ^Trie Thouless

conjecture

is

usually

written _as (6, 9]

~~

~jé ^{~~~ ~j}

~~~

' ~~~

where trie

angular

brackets

(..

denote _an

averaging

_{over a} random ensemble. It

certainly

makes _sense to ask trie

question,

^what are trie distributions of trie

single

level current and curvature, as a function of

flux,

in trie framework of RMT.

Today

_one knows

only

trie Iowest correlators of these

quantities

Iike _an average

value,

variance

il

0,

iii

_or

twc-point pair

correlator

(see

_e-g- Ref.

(12]).

^This

knowledge

_comes

mainly

from

diagrammatic perturbation theory,

which is

usually

not

applicable

to small values of flux.

Trie _aim of trie present ^{article is} to derive _exact

analytical

results for trie

single

level _current

and curvature

DF'S, using

RMT. We shall also find ail trie moments of these

quantities

and

compare them with available

perturbative

results.

Finally

_we shall calculate trie two point current-current correlation function for _a small flux. As _was done in trie _case of trie level

spacing distribution,

_we shall _{use a}

simple

but

exactly

solvable 2 _x 2 random matrix model.

We prove,

however,

that some

key

features of _our results _are not restricted to this toy model but

apply

to N _x N random matrices _as well.

Trie most

surprising

features

resulting

^from trie present

study

are trie

following:

let _us define

a

single

^level curvature to be

proportional

to trie second flux derivative of _energy

Î

à~fn

~ _'~

£~

ôj2

(numerical

^factors will follow _m trie

body

of trie

article).

Then trie flux

dependent

distribution function of trie _curvature bas trie form

(for

_{(#( <}

#c

_%

A/(2Ec)

_{< 1)}

-1/2,

C < Î,

~-3,

_Î

< C <

~c/Î~Î'

~é~~~ ~' ~~

(4)

~~

)

j /(j(

_«

C$eXp j~

~QC_{#~ c} ^~

This distribution

implies

trie

following

flux

dependence

of trie moments of trie _curvature

~;

m = i,

1)j _~l

+~

El ln14c/1411~~;

m = 2,

(SI

E714c/1411~~~;

_m 2

3,

As is shown in

Appendix A,

this structure of trie _moments is not _an artefact of trie

simple model,

but may be

rigorously

derived from _a

general

N _x N random

Hamiltonian, equation il ).

^Based

on this, one may suggest ^trie

universality

of trie

large

curvature,

(ô~en/à#~(

»

Ec,

behavior of trie

distribution, equation (4).

In contrast, trie small curvature part,

(ô~en/à#~(

<

Ec,

_may be

a

specific

property ^of our toy ^model.

Equation (5) definitely

contradicts trie

frequently

used

expression,

equation

(3),

when eval-

uating

trie _{average over} the disorder _{as an} arithmetical _mean.

Indeed, according

to

equation (5)

trie second and

higher

moments of trie _curvature

diverge

at _zero flux

(or,

_m other

words,

in the GOE

ensemble).

This is _a consequence of the absence of the

exponential

tait in the distribution function _at

exactly

_zero flux. In this _case trie behavior

Po(c)

+w

c~~ _{continues up}

to

infinity.

As _was

already mentioned,

such _a behavior is _{a common} feature of GOE and not _a result of _an

oversimplified

model

(see Appendix A).

Thus when

evaluating

the Thouless _energy,

equation (3),

_one should either _{use a}

geometrical

_{mean, as}

already pointed

out

by

Thouless (8], ^{who assumed} a

simple

Lorentzian distribution of the curvatures, or use another _measure of the

sensivity

of the spectrum to variations of trie

boundary

conditions:

62~

~lc ~4

~

),

ôj2

#=0

The _same

divergence

affects

dramatically

trie current-current correlation function.

Namely,

it will be shown that for

(#(, (#'(

^<

#c,

_one has

jôf«1#) ^ôfn14')

~

_~~ ~ ~,

~~

141

+

li'l

ôj ôj'

^C

j~

Diiferentiating

this result with respect to # and

#'

and trier

putting #'

=

#,

_one returns bock to the just

quoted

second moment of the curvature. This

expression

is

absolutely unexpected

from the

point

^of view of

perturbation theory.

The latter _assumes rather

E)##' (without

logarithm)

for trie above defined correlation function. It would be

definitely interesting

to see if such _a behavior exists _m the framework of

supersymmetric

calculations. In

fact,

these

calculations have

already

been carried _eut in reference (16],

but,

_as for _{as we}

know, only

in

GUE,

where _we do net expect

anything

unusual.

Curvature distributions have

already

been

investigated

in _a number of works (13,

14, 15],

however, ⁱⁿ a very diiferent _context. The considered Hamiltonians had trie structure

H(À)

=

Hi +

ÀH2,

where

H(À) belongs

to trie _same

universality

class in the whole _range of

ils].

Trie curvature is defined _as trie second derivative of _energy with respect to trie pararneter ^À.

Subsequently,

curvature distributions for three _pure symmetry classes _were studied. Since trie above defined parameter does not break trie time-reversai symmetry, it _con not

play

the rote of _a

magnetic

flux.

Thus,

there _{is no sense,} for

example,

in

looking

for _{a current}

(defined

_as

a derivative with respect to À) distribution. For _reasons which become clear in

Appendix A,

our results for _o

= 0 comcide

exactly

with trie GOE results of references

(13-15].

We _recover trie c~3 tait of trie curvature

distribution,

first discovered in reference (13], ^and equation

(28)

of trie present work may be found in reference

ils]. However,

for any a

#

0 _our conclusions

are very diiferent from those of references

(13-15].

^For

example,

in GUE

la

=

1)

_we bave

found _a Gaussian tait of trie _curvature

(with

_{respect to}

flux)

distribution. At trie _same time trie distribution function of trie curvatures, defined with respect to trie parameter À,

decays only

_as trie fourth _power

(see Appendix

A and Ref.

(13]).

Trie present article bas trie

following

structure. In section 2 _we

specify

trie 2 _x 2 model based

on

equation (1)

and re-derive trie known results _concerning trie energy

spacing

distribution.

In sections 3 and 4

single

level current and curvature DF'S _are derived.

Finally

in section 5 _we discuss trie

possible implications

of trie

simple

model to real

physical

_systems. In

Appendix

A

the _moments of trie curvature for _a

general

N _x N random matrix model _are considered. A

summary of the results for _pure ensembles

(GOE

and

GUE)

is

given

in

Appendix

B.

2. Trie mortel and energy

spacing

distribution.

Consider _a model based _on

H(~l(a),

_as defined

by equation il

[xi

^{+ x2 z3 + iax4}

H(2)j

_~~

~

j~)

23 l£X24 Xl 22

where

zj(j

=1.

4)

_are real random

variables,

with _a Gaussian distribution law

1

1~2

~~~~ U/~

^~~~

212 (7)

The variance of trie distribution v~ _and the time-reversai symmetry

breaking

parameter a are

trie two free parameters of the model. In the end

they

should be related to

physical

observables such _{as mean} level

spacing

and

magnetic

flux. Let us,

however,

postpone this discussion until

section 5. Trie spectrum ^{of trie}

Hamiltonian, equation (6),

_is

given by

e~ = xi +

(xl

+

xl

₊

_{a~x()~~~, (8)}

and trie _energy

spacing e(a,x) by e(a,x)

_{e e+ e-}

= 2

(xl

+

xl

+

a~x()~~~

^Let us consider

an energy

spacing

DF

Pa(f)

_"

à(f f(O,X)) ^~~ fl _{P(Xj)dXj. (9)}

The calculation of trie

integrals

is

straightforward, finally

_one obtains (17]

~~~~ 4u~à#

_~~~

^~2

_~~~

lÀÎÎu~

_~~~~

In two

limiting

_{cases one} returns

again

to trie familiar distributions: for _o

= 0

FOIE) ₌ ^~ _exp ^~

~

(GOE), Ill)

4u

~

~~~ ~°~ _°

e~ f~

(GUE) (12)

Pi(e)

=

~ exp

-p

'

4/ùu

_U

It is _a well-known fact that in trie _case of

GOE, Po(e)

_{c~ e} for _{e < u,} whereas in trie absence of time-reversai symmetry

(GUE)

trie level

repulsion

is stronger:

Pi (e)

c~ e~;

le

<

u).

In trie

intermediate

region

0 _{< a} < 1 _one has

ej2

~

~a

1 ~2 2U

fia

^{' ~}

Ùfi'

~~~~~

~

~~~ ~~~

e 1

~

~a

~~~~

2~

2fi'

^~

fi'

The level

repulsion

is

quadratic

for small energy

intervals,

and becomes linear for

Iarger

_ones.

The moments of the distribution _are

given by

j_ijn ^~3n-1/2

ôn

Ii

^~2

w wm w

^{~~~ ~~~~~~}

Î/ "=i

_{'~ "} ~" _~~

(e'~)a

₌ ~'~

(14)

i-11"2~"+~ $ _lt~ia~t

+ i

a~)~/~l

^{i<=i m} = 2».

In

particular

for trie first moment _one has (17]

(e)~

= ~

~

a

+

Il a~)~~/~

arctan

/ _ils)

É ~

This is _a smooth monotonous function of _o, which varies from

(e)o

"

~/j[

up ^to

(e)1

"

~jj2.

_The fact that it is almost constant will be useful in section

5,

where _we shall try to

give

_a

physical interpretation

of trie results.

3.

Single

level current distribution function.

We _now define trie

single

level _currents in _a 2 _x 2 model _as

~~1°'~l

_~

ÎÎ

"

~eÎÎÎÎj

^l~~~

We will look for _a DF,

Pa(1)

of1 _e 1+ ^{> 0.}

Obviously

the distribution of1- _is the _same

(up

to the minus

sign

of the

argument).

If _{one is} interested in the DF

of1,

_{which may have}

either

sign,

_one

simply

bas

Éa(1)

₌

Pa((Î()/2 (trie

coefficient

1/2

takes _cane of trie correct

normalization).

To evaluate

P~(1)

let _us first calculate trie

joint (current

and _energy

space) distribution, Pa(1,e).

Besides technical

advantages,

this _way of calculation

provides

_some

additional information.

Namely,

_one will be able _to

identify

those energy

spacings

_e, that _are

mainly responsible

for _a

given

current, ^{1. Trie}

joint

DF is defined _as

oo 4

Pa(1,e)

=

Îm

à(1

-1(a, x))à(e e(a,x)) fl _{P(xj)dxj. (17)}

~_~

After _some calculations _one gets

Paii f)

_-

~if 2°1)1i~à~~ G

exP

(-

^~~°~

^+liili

^°~~~

i18)

Here

Ù(x)

_{is a} usual step function

(remember

that bath 1,e >

0).

To get trie current distribution, Fa

(1),

_one

should, according

to equation

(17), integrate

trie last

expression

over e.

Pa(1)

=

~

(20i)~ /~(l

₊

t)~/~

_exp

(-

^~~

(t~a~

+

t(1

₊

a~)

₊

))

dt

(19)

2

(~/~)3

o 2i'~ ^'

where _a variable t was introduced _{as e}

=

2ai(t +1).

Trie last

integral

is net known in

special functions,

_except in trie two

limiting

_{cases a}

= 0 and _o

= 1

(see below).

However _{one can} work out its

asymptotic

behavior m various

regions

Î

~

₃

Fa

)

~

~ ⁱ ~

The values of _energy

spacings

_e, which

provide

the main contribution in each _{case are}

designated

in brackets.

Equation (20)

shows that the

single

level current DF bas _an

integrable

_square root

singularity

at small currents. Realizations with _{energy spacings} of trie order of _{an average}

one are

mostly responsible

for this

singularity.

In trie intermediate

region,

^which exists

only

if _{o «} 1, ^trie current DF decreases _as i~3

Finally,

for

large

currents trie distribution bas _a

Gaussian tait. For _a < 1 trie last two

regions

_anse due to realizations with

extremely

small energy

spacings.

For trie two pure cases one can calculate trie distributions

analytically:

for _a

= 0

Po(1)

₌ 26

(1)

,

(GOE) (21)

i' i'

which is evident without _any

calculations,

and for _a

=

~~~~ àfi~ Î'

~à~~

'

~~~~~

_~~~~

where

Fia, x)

is _an

incomplete

_gamma function. Trie

asymptotic

behavior of

Pi

₍₁₎ is

given by

the first and the third fines of

equation (20) (with

_a

=

1).

^The moments of the current

distribution _are given

by

fia

m = 1,

jjmj

_{~_} ~m ~3£Y~ in_{£Y ill}

" 2,

&<1 '- j~~~

~~Î~ ^°~~ ~~ Î

^~

m 2 ^{' ~}

~ ~'

for _{a <} in

leading

order in _a, and

jjmj

~~ ^~_ ~m

2m/2

~

(m

^{+ 3 i j~~~}

"~

@

2 _m

+1/2'

for _{a m} 1 in

leading

order in

il o~).

Let _us mention

particularly

that trie second moment of _a

single

level current is

given by (i~)~«i

_G3 -3~~a~lna. This _can be _seen

directly

from

equation

(20).

Trie

leading

term _comes from _the mtermediate

region

of the currents

(~o

<

<

~).

Trie main contribution to

(i~)a«i

arises from realizations with very small

spacings

e m a~~ _<

(fi. Starting

from trie third _one, ail trie moments _are determined

by

_a Gaussian tait of trie distribution (~ <

ii,

whereas trie average ^current

(1)~«i

_comes from trie

opposite

region1<

_~.

4. Curvature distribution function.

Following

the _same scheme _{as m} the _previous

section,

consider _now trie

single

^level curvature DF,

~~~°'~~

ôo2 ^~

e(a,x) e3(a,x)

^~~~~

Again

_we consider _a distribution of _a positive defined

quantity

_{c e c+} > 0

(it

is mdeed positive

as e > 2ax4,

according

to

equation (8)).

If _{one wants} to include also _{c-, one} should

again

make trie distribution

symmetrical

with _{a proper} normalization. Trie

joint (curvature,

_energy

spacing) DF,

Fa

(c, e),

_may be found in

elementary

functions _m the closed form

~"~~'~~

_~~~

~°~~~Î (~à)3 ^Î~

^~~~

ÎÎ2 ~~~~)

xB~~

(wmexp ($

(~ B)

⁺

^wmexp (- ^~~ )~ B)j

_,

^j26j

~ 2a 8~ 2a

where B

= 1-

802cle.

One _now integrates ^trie last expression _{over e} and obtains trie

following asymptotics

for trie _curvature DF:

Î~~~Îl'

_~~~'

~~~~~

p~j~~

~s

3

Ill

^~

~ « C «

1, ^if

~

a~C)

~

~ ~

Î~

~~P

Î ^"~

ÎÎ

^~

^~~~

~

~~,

^~~~ ~~ ~

°~~

ail 02)

(27)

where _ga

=

(7a~ ^+18a~

₊

7) fil

+

a~).

Trie first two fines in

equation (27)

look _very similar to those of trie current distribution

(cf. Eq. (20)).

Trie _{reason is} that for _{c <}

~la

trie last

term _m trie

expression

for the curvature,

equation (25),

_may be

omitted,

thus _one bas _a trivial

relationship

between current and curvature, = oc. Trie

large

curvature

(c

»

~/o)

tait of trie

distribution _is aifected

by

trie last term of

equation (25), resulting

in _a

complicated

form of trie tait of trie

DF,

equation

(27).

For _a

= 0

(GOE)

trie tait

disappears completely,

in this _case

one con get _an exact result for trie curvature distribution

(see

aise Ref.

ils]

where

Da(x)

is a Whittaker

parabolic cylinder

function. Trie asymptotic behavior of this DF

is given

by

trie first two fines of trie

general expression, equation (27).

Due to trie absence of _an

exponential

tait in _a GOE curvature

distribution, equation (28),

ail trie moments of it,

starting

from trie second _one,

diverge. Indeed,

_a direct evaluation of trie _{moments m}

leading

order in

o results in

fi;

m = 1,

-31n _{o; m =} 2,

(c'~)~«i

_" ~'~

(29)

~

(-i)P

^~

~~Î~ °~

_~~

~~ Î ^~ (

m 2

/2p

'

~ ~ ~'

P=

for _o <

(cf.

with

Eq. (23)),

and

~mj ~ ~m

2~/~

~

(~

^{+ 3}

f

^~~

^IF

~"~ ~

@

2 _m j~~~

~/21p'

P=0

for _{o m}

(cf.

with

Eq. (24)

);

l~

is _a binomial coefficient. As in trie _case of trie moments P

of trie current, ail trie moments of trie curvature,

starting

from trie second _one, arise from realizations with small energy

spacings

_e <

(fi (if

_{a <}

1).

At _o

= 0

only

trie first moment

exists, whereas ail

higher

moments

diverge.

This

unexpected

fact will be discussed in _more detail in trie next section.

5. Discussion of trie results.

As

already

mentioned in section 1, trie

Wigner

surmise obtained for 2 _x 2 matrices works

extremely

well aise for _a

large

N. TO establish this connection _one should relate the

phe- nomenological

_pararneter ^~~ the variance of the distribution _{to an average} level

spacing

A.

One

simply

demands that

jej~

₌ à.

Strictly speaking

^the average

spacing (e)~

is _a function of _a

(magnetic flux), although

trie _mean level

spacing

A is

presumably

_a constant,

independent

of externat parameters.

However,

_{as we}

noticed after

equation ils),

^trie

dependence

_{on a is very} ^weak

(especially

^for small

a). Using

this fact _we shall

disregard

its

a-dependence

and

just

admit

(e)o " à

=

~/~, ₍₃₁₎

where _we bave used

equation Ils).

TO be

honest,

_one should

re-identify

trie parameters ^for each value of _o

separately.

This

procedure, although trivial,

is net transparent

enough

for _Dur

illustrative purposes.

Having

an energy

spacing

DF _{as an}

example,

_{one may}

hope

that the

single

level current

and curvature

DF'S,

^{derived for} _a 2 _x 2 system, _may be suitable for

forger

systems _as well. TO

support ^{the last} statement let _us put forward trie

following

_arguments. As _we bave _seen in

previous sections,

ail trie _moments of _a

single

level _current and curvature

DF'S, starting

^from

trie second _one, arise

mainly

^from realizations of _a random Hamiltonian with _very small _gaps.

For these realizations trie 2 _x 2 ansatz is

supposed

to be

essentially

correct, ^because for trie close pair ^of levels

only

their mutual interaction appears ^to be important. In

Appendix

A _we prove that trie small flux behavior of trie _{moments is} indeed observed in trie

general

N _x N model _as well. The first moment,

however,

is determined

by

realizations with _{an energy gap} of the order of the _{average one.} In this _case the 2 _x 2 scheme need not be _precise. Thus

one should not trust trie value of trie first moment, but rather connect it

phenomenologically

with the microscopic characteristics of _a system.

Followmg

Thouless, _{one may} relate _a

typical

second derivative

(not

_{r-m-s !)} with respect to flux at zero flux to a correlation energy

~~

62~

ôjl

~' ~~~~

#=0

cf. with equation

(3).

On the other

hand,

_we had

(see Eqs. (25), (29))

~l~~Î ^_~~~ ~É'

Using

the definition of _{a mean} level

spacing, equation (31),

_one obtains

a =

~~~#, ₍₃₃₎

This should be

compared

with the

conjecture

of

Dupuis

and Montambaux [6],

equation (2) (with

N

=

2).

As _{one sees,} trie agreement is

extremely good,

trie

slight discrepancy

_may be attributed to trie fact that equation

(2)

_was obtained for trie

large

N limit. We conjecture thus that, with trie identifications, equations

(31), (33),

trie tails of trie distributions obtained for _a 2 _x 2 model _are

applicable

for

larger

systems as well. Let us discuss trie further consequences of this rather strong

assumption.

First of ail _one notices that trie _a

=

(GUE)

_case

corresponds

to trie value of _a flux

#~ ₌

à/(2Ec).

This value is well-known _{as a} correlation flux.

Up

to this flux _a

typical

level may

change parabolically,

without

crossing

other levels. At # ₌ #c the first

avoiding crossing

event

usually happens,

and trie

simple

2 x2 scheme

obviously

breaks down. It _was demonstrated

numerically

_(6], that at

#

m

#c

trie _crossover to GUE is indeed

practically completed.

This shows that trie

applicability

of _a 2 _x 2 model for 0 <

#

< #~

(0

< _a <

ii

is

quite reasonable,

as well _as trie identification of trie

#

₌ #~

point

with GUE.

Consider _now trie second moment of trie

single

level current in GUE

(or,

trie _{same, at}

#

=

#c).

Using

equation

(24),

_one obtains

((i~)~=i)~~~

"

~/fl,

or in

physical

_parameters

(using Eqs (31), (33))

Il

~~~

~-~c

~Î ÎÎ _/~.

This result is aise well-known from perturbation

theory (up

to trie numerical

coefficient) Ill, loi.

Being

thus convinced that trie obtained results lead to reasonable predictions for real

physical

systems, let _us discuss trie most

surprising

^feature of trie considered DF'S: at _zero flux ail trie

moments of trie curvature,

starting

from trie second _one,

diverge. Thus,

when

calculating

trie correlation _energy from the curvatures, one lias to _use another _measure for their

typical

value than

just

trie root _mean square, like

equation (32)

_or trie

geometrical

_mean

proposed by

Thouless (8].

Trie consequences of trie discussed

divergence

_are,

however, deeper

than

just

trie

necessity

of _{a more} careful definition of trie correlation energy. One aise should reconsider trie _univer- sal

relationship

between

dissipative

and correlation

conductances,

derived

by

^Akkermans and

Montambaux (9].

Mathematically

this relation _was

expressed

_{as (9]}

~ ~

~~" 2

~

~j

^{~ ~"}

jl/2

64 64~ ~=0

~

where bar denotes

integration

with respect to

flux,

and _{a is a} universal numerical factor. Ac-

cording

to trie present ^{results this relation} cannot

hold,

when trie

typical

curvature is calculated

as an arithmetical _mean. Indeed,

using equations (19), (31), (33),

_one obtains for the 1h-s- of trie last _expression

fis (up

to a coefficient of trie order of

unity),

whereas trie r-h-s-

diverges.

To understand trie _reason for this

phenomenon

let _us consider trie _two point current-current correlation function

C(°,°')

₊

(%(°)%(°')). (34)

A very similar

object

_was

recently

considered in reference (12]. ^One con

explicitly perform

trie

averaging

in _a 2 _x 2 model

by integrating

over dx with trie

corresponding weight, precisely

_as

one did in the previous sections. The

general

_answer is

cumbersome,

but _one needs

only

the behavior for small flux. In this _{case one}

easily

gets

C(a, a')

_m

-3~~aa'ln(a

₊

o'),

0 < o, a' _< 1.

(35) Putting

here _a

=

a',

_one returns agoni ^to trie _expression for trie second moment of trie current

(the

second fine in

equation (23)).

On the other

bond, diiferentiating

equation

(35)

with respect to a and a' and then

putting

_a

=

a',

_{one recognizes} the second fine of

equation (29).

In

physical

parameters

equation (35)

_may be rewritten _as

~j ~ ~,j

~

jôf~14) ^ôf~14')

_~

^_~~-i~~ ~ ~,

~~

141+14'l

~j

~,j

~

~ j~~j

'

ai ail

^~

j~

^{' ' ~'}

This should be

compared

with the

corresponding

result of the

perturbative

calculations

Ôi<, <'i

_m

127r-~El«' ()) _{<, <'}

_«

_{<c, 137)}

where ~ is a cut off in

perturbation theory,

which is

usually supposed

to be of trie order of à (6]. ^Trie

discrepancy

between trie two results is rather dramatic. Whereas equation

(37)

leads to a finite second moment of trie curvature

(m E)), _{equation (36)}

results in _a

divergent

second

moment. Let _us also point out that the

perturbative

result

obviously

_may be

expressed

in _a

form

Ô(#, #')

=

f(#

+

#') fil #'),

which may be traced bock _to Diifuson and

Cooperon

channels in the

diagrammatic

_expansion.

Equation (36)

does not allow such _a

decomposition.

This

might

be _a

point

where the present scheme contradicts the derivation of reference (9].

Indeed,

it _was assumed

explicitly

_(9], that the

Diifuson-Cooperon decomposition (which

is

certainly

correct for _a

large

^flux

#

»

Ici

is also valid in trie

vicinity

of _zero flux.

According

to the present consideration this _is not the _case.

It is not clear _at the moment whether the discussed

divergence

has _a real

physical meaning,

but if _so, it

might

cause difficulties in numerical calculations of correlation functions _m GOE.

We conclude that further

analytical (both

RMT-like and

supersymmetric)

calculations and numerical work _are necessary to

clarify

this

unexpectedly

controversial issue.

Acknowledgements.

We _are very

grateful

to Yuval

Gefen,

Gilles

Montambaux,

Eric Akkermans and

Uzy Smilansky

for _numerous and

helpful

discussions. We want to

acknowlege

trie

hospitality

of trie Institute

for Scientific

Interchange (ISI), Torino,

where this work _was

completed.

One of _us

IA-K-j

was

supported by

trie German-Israel Foundation

(GIF)

and trie U.S.-Israel Binational Science Foundation

(BSF).

Note added in

prooE

After this article _was

accepted

for

publication

_we received _an

unpublished preprint

^of N.

Taniguchi,

A.

Hashimoto,

B-D-

Simons,

and B-L- Altshuler where trie

logarithmic

low-flux behavior of trie correlation function _was obtained

using

trie supersymmetry method.

Appendix

A.

Moments of trie curvature in _an N _x N mortel.

Let _{us now} show that trie flux

dependence

of trie moments of curvature, derived for _a 2 _x 2 model, equation

(29),

and summarized in

equation (Si,

_may be obtained from a

general

N _x N model. Consider _an N _x N random matrix

Hamiltonian, given by

equation

il).

Without loss of

generality

_{one may} assume that its spectrum is not

degenerate. Then, using

^{second order}

perturbation theory,

one obtains the

following

ezact

relationship

p~j ²

62~ j~yj ^{N < CY,}

k(H~

_{(Yl, a} >

~~~

^~

~

~"~~~ ~~~~~

~~

_~~

where _eh

(a)

and (k, a > are

eigenvalues

and

eigenfunctions

of the full

Hamiltonian, Hl'~l(a).

As it well-known from RMT (4], ^statistics of

eigenvalues

and statistics of

eigenfunctions

_are

completely independent

of each other.

Let _us first consider the _case of

exactly

_zero flux

la

= 0,

GOE).

In this _case the energies

m the denominator _on the r-h-s of equation

(A.l)

_are

eigenvalues

of

Hj~l,

whereas in trie numerator _one bas matrix elements of

Hfl _Hence,

_matrix

_elements,

(<

^0,

k(Hi~~(n,

0

(~,

and

eigenvalues,

_eh + eh

(0),

_may be considered _as

independent

random variables. Trie statistic of trie

eigenvalues

is

given by

a

Wigner-Dyson (GOE)

distribution _(4]

PN(el,

, EN " const x exp

- ( ^(j _fl

(en

ekl, (A.2)

~

k=1 1<k<n<N

whereas _an exact form of trie matrix element distribution is not

important

for _our purposes.

O _ne is _now in _a position to consider trie moments of trie random variable _c

= (ô~en

la /ôa~

_(~=o

Doing this,

_one will be interested

only

in trie _manner of

divergence, omitting

alI trie

prefactors

(.

as weII _as less

divergent

terms.

Raising equation (A.l)

to the m~~ _{power and}

averaging,

_one obtains

(c'~)

+~

N~~ II

^~~~~~'~~~

_deide2

+

,

(A.3)

(El

f21~

p~j ²

where

averaging

_over matrix

elements, (< 0,k(H~ (n,0 _>(

,

leads to _some omitted constant

prefactor,

and " ." denotes less

divergent

terms,

arismg

from the

non-diagonal contributions,

like

II/ ^~~

^~~~'~~'~~ _{dei de2de3}

(El

f21~

^~_{Îf2 f31} ,

etc. Here

R~(ei,

., fi is _an 1-point correlation function (4], ^for

example

~~~~~'~~~

(ÎÎÎ

1) ÎÎÎ ÎÎ _{~~~~~' '~~~~~~ ~~~'}

~~'~~

In the limit of

large

N, the correlation functions Ri

depend only

_on diiferences of the

eigen-

values. Then the

integral

over fi + e2 in

equation (A.3)

leads to _some constant of the order of N, and _one

finally

obtains

(c'~)

~ +w

/ _{~~(~ de, (A.5)}

o ^e

where _e

=

(fi

_e21.

Taking

into account the well-known GOE result,

R2(e)

+w e

if

_<

à),

_one

notices that all the moments of the curvature,

starting

from the second _one,

diverge.

This is in exact agreement with trie result for _a 2 x 2

model,

but _{Dow one} bas demonstrated trie

validity

of this _statement for _a

general

N _x N model.

Trie above

developed

scheme _is

applicable

without _any

changes

for trie _case where _a is not _a time-reversal symmetry

breaking

parameter, hence

Hl'~l(o) _belongs

_to trie _same

universality

class in trie whole range of _o

(13,

15].

Indeed,

in this _case trie matrix elements of trie pertur- bation

(numerator

_on r-h-s of Eq.

(A.l))

and trie

eigenenergies (denominator)

cari

always

be

considered _as

independent

random variables. One _can, for

example,

repeat

exactly

trie _same arguments ^{for GUE with} a

single

modification:

R2(e)

+w

e~. Then _one obtains trie

divergence

of ail trie moments of trie _curvature

starting

from trie third _one. This

perfectly

_agrees with trie

result for trie tail of trie _curvature

distribution, proved

in references

(13,

15]:

Pic)

+~

c~~. In trie _{same way} for

GSE, using R2(e)

+~ e~, _one obtains trie

divergence

^of trie _moments

starting

from trie

fijth

_one, in agreement with

Pic)

+w

c~~

_(13,

15].

^This is trie _reason

why

trie result for

trie GOE curvature

distribution, equation (28),

coincides

exactly

with that of reference

ils].

However,

when _a

plays

trie role of _a

magnetic ^flux,

it does

change

trie symmetry ^{class of trie} Hamiltonian. Then trie

energies

eh

(a)

_are

eigenvalues

of trie _sum

Hj~~

+

iaHfl,

_{and thus}

no

longer statistically independent

of the matrix

elements,

_{< a,}

k(Hfl(n,

a >. We _are

going

to show that in this _case, for _a

# 0,

ail the _{moments converge.} This in tutu _means

that,

_contrary

to trie _case of references (13,

15],

^trie curvature DF bas _an

exponential

tail. For small _a and small

(fi

_{e21 one can} show that

(El

(°)

_{~2(£X)1 ~ ((~1}

f2)~

_{+ £X~l~~)~~~,}

where trie

quantity

X

depends only

_on trie matrix elements

(and

not _on trie

energies),

hence it is

statistically independent

of trie zero-flux

energies,

_eh

Repeating again

trie steps,

leading

to

equation (A.5)

_one obtains

OnSt(

^{Rl # Î,}

~~~~ '~

/ _{Î~~~~~ /~}

j~2 +

à~2 jm/2

^{~~ '~ ~~ ~ ~' ~ ~'}

^~~'~~

~2-m,

y~ > 3

' '

where

f(X)

denotes trie

(unspecified)

distribution of matrix elements. In

equation (A.6)

_we

used

again R2(e)

_+~ e. Thus _we bave shown that trie small flux behavior of trie moments of trie curvature,

initially

derived for _a 2 _x 2

model, equation (5),

is valid for

large

N _as well.

Appendix

B.

List of distributions for _pure ensembles.

GOE;a

= 0

Spacing

distribution:

~ ~2

~~~~

4~2 ^~~~ 8~2 ^'

~~'~~

(e'~)o

_"

~'~2~'~/~r 6'~

~ ~

(B.2)

2

Current distribution

Po(1)=~26

^~

,

(B.3)

i'

i'~

li~lo

₌ o.

(B.4)

Curvature distribution