The magnetic response of chaotic mesoscopic systems

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The magnetic response of chaotic mesoscopic systems

Oded Agam

To cite this version:

Oded Agam. The magnetic response of chaotic mesoscopic systems. Journal de Physique I, EDP

Sciences, 1994, 4 (5), pp.697-730. �10.1051/jp1:1994171�. �jpa-00246942�

Classification Physics Abstracts

05.30F 75.70A 05.45 03.655

The magnetic _response of chaotic mesoscopic systems

Oded

Agam

Departnlent of Physics, Technion, Haifa 32000, Israel

(Received

19 September 1993, accepted 27 January

1994)

Abstract. The magnetic _response of mesoscopic systems with ballistic motion of electrons thon is chaotic _in the classical l1nlit is considered. Semiclassical methods _are applied in order to derive _a formula for the susceptibility which is expressed in terms of _a finite number of classical

periodic orbits. This formula _is used to study the fluctuations of the susceptibility _{in conlpar-}

ison with the fluctuations of random systems. Some of the nlechanisms which lead to these fluctuations _are discussed. At relatively high temperatures ^{the fornlula for the} susceptibility

reduces to _a simple expression _in which only few short periodic orbits dominate the behavior.

An experiment based _on this result is proposed.

1. Introduction.

Mesoscopic systems were

extensively investigated

in recent years

il,

_2]. A mesoscopic system ^is

usually large enough

to contain _a number of electrons that _may be considered _as

thermodynam- ically large,

but still

sufliciently

^small to maintain

phase

coherence _over distances

comparable

to the size of trie

sample.

The elastic

scattering

of the

partiales

from the

impurities

and from the

boundary

generates ^{behavior which is}

usually

chaotic in its dassical limit.

Presently,

there is _a

growing

interest in

understanding

the imprints ^of trie

underlying

dassical

dynamics

_on the quantum ^{behavior of chaotic} systems. Trie

investigations

^of such systems form the field of

"quantum

chaos" [3-8]. ^{In the} nonlocalized

regime

^trie

high

level _wave functions of chaotic

systems

spread

_over ail the system [9,

10], therefore, they

_are sensitive to the

precise

_arrange-

ment of _the

impurities

_as well _as to the

shape

of the

boundary.

The _energy spectrum, _on trie other

hand, displays rigidity

and

long

_range correlations. These correlations _are the

subject

of several recent

investigations.

Many

quantum ^{eifects of} mesoscopic systems are

directly

related to the

spectral

correlation function. For

diffusing electrons,

this function _was calculated

by

Altshuler and Shklovskii

iii]

using

_a

perturbative diagrammatic

method. An alternative derivation [12]

applies

Gutzwiller's trace formula [13, 14] which _expresses the

density

of states in terms of _{a sum over} classical

penodic

orbits.

Following Berry

_[15] and

applying

a

generalization

of

Hannay

and Ozono de Almeida _sum rule [16] one is able to _recover the _saine results obtained

by diagrammatic

expansion.

Recent theoretical

[17-20]

^and

experimental

_[21] studies show also that statistical

properties

of the fluctuations of transport in

irregular mesoscopic

structures, _may be

explained

m the framework of the

periodic

orbit

theory.

This paper focuses _on the orbital

magnetic susceptibility

of

mesoscopic

systems. ^Several authors

investigated

this

problem

for random systems

[22-30].

One of the conclusions which

emerged

from these studies is that the

magnetic

_response in _a

specific sample

is sensitive

to trie exact

configuration

of trie

impunties,

and that the

sample

to

sample

fluctuations _are

proportional

to kil where kF is the Fermi wavenumber and is the elastic _mean free

path.

This result is somewhat counterintuitive since _as the disorder decreases the fluctuations increase. In order to examine this

problem

it is instructive to

study

the

susceptibility

of clean

mesoscopic

systems. The

sample

to

sample

fluctuations of the

magnetic

_response of such systems arise because of trie different

shape

of their boundaries. These _are defined _as

ôx= ((x (x))~),

where _{x is} the

susceptibility

of _some individual

sample,

and (. .) represents ensemble _averagmg associated with the

roughness

^{of the} boundaries. One expects ^that as becomes

sufficiently large compared

to the linear size of the system L, the fluctuations will _{cease to} increase and will remain

proportional

to kFL.

In trie clean

limit,

it is

important

to

distinguish

between two extreme _cases. One _is the situation _m which the classical

dynamics

of _one

partiale

in the

corresponding

system ^{is inte-}

grable,

^while trie other is when it is

completely

chaotic. In between these two _{cases are} the

systems ^with mixed

dynanlics.

It tutus out that trie structure of their classical

phase

_space is

directly

related _to the

magnitude

^{of the} fluctuations of the

susceptibility.

The fluctuations of

a system in which _some of trie

corresponding periodic

orbits _appear in continuons families of d parameters

(e,g, integrable

systems ^{with d}

degrees

of

freedom)

_are in

general proportional

to

(kFL)1(~+d'

_[31]. For

completely

chaotic systems the

periodic

orbits _on the energy shell _are isolated and the fluctuations in trie

susceptibility

_are

proportional

to kFL. One _purpose of this paper is to present a detailed derivation of this result. The

impurities

in the random systems generate a chaotic classical motion.

Therefore,

_{a proper}

comparison

between them and the clean systems _reqmres ^the consideration of systems ^{which in trie classical limit exhibit chaotic}

dynamics.

In _an

early

work _on this

subject

_[32],

Dingle

considered the

magnetic

_response of _a nonin-

teracting

electron gas confined to _move in _a

cylinder

of radius R where _a uniform

magnetic

field is

applied parallel

to the axis of the

cylinder.

He showed that the behavior of this system

depends

_on the ratio of the

cyclotron

^radius at the Fermi _energy,

l~,

to the linear size of the system R. When R _» Rc

("large systems")

there is _a small

steady diamagnetism together

with _some small terms

periodic

in the field which _are

significant only

at low temperature.

The latter

correspond

to de-Haas

van-Alphen

oscillations [33,

34].

^At the other limit R<Rc

("small systems")

he found _a

large steady diamagnetism

and _some

oscillatory

terms which

become

sigmficant

at very low temperature.

(In fact,

the rotational symmetry of the system may account for

perfect diamagnetism [29].)

^A more

general

_case ^of

large

systems where the electrons _are confined _{to move} in _a

cylinder

of

arbitrary

section _was considered

by

Robnik at

high

temperatures [35]. ^{Sivan and} Imry

investigated large

two dimensional systems at very low temperature [36], ^{while Nakamura and Thomas studied}

numerically

billiard systems ^with

both circular and

elliptic shapes

[37].

The model

investigated

in the present _paper consists _{m a}

noninteracting

electron _gas confined to in _a small two dimensional chaotic billiard of

arbitrary shape.

The electronic motion is

assumed _to be

completely ballistic, namely,

the elastic _mean free

path

due _to the

(static) impurities

is infinite. The levels

broadening

because of inelastic processes is

supposed

to be

negligible compared

to the thermal _{energy, i-e-} the inelastic _mean free

path

is also assumed to be infinite. The _spm orbit

couphng

will be

ignored.

The

problem

will be

analyzed using

semiclassical methods. It tums out that this

approach

clarifies the mechanism which leads to trie

large

fluctuations in the

susceptibility.

Recent _re- summation

techniques

which enable to extract from Gutzwiller's _trace formula trie semiclassical information

regarding

indiuidual _energy levels of chaotic systems will be used. This issue and

some other related

topics

are summarized in section 2. In section 3 _we

analyze magnetic

re-

sponse due to a

single

Aharonov-Bohm flux fine. The main result there is _a

general

formula for the

susceptibility expressed

ⁱⁿ terms of _a finite number of

periodic

orbits. At

high

temperatures

it reduces to _a

simple expression

in which

only

trie shortest

periodic

orbits bave substantial contribution. It is dear from trie latter that trie

susceptibility

is _an

oscillating

function of the flux and of the Fermi wavenumber with _a

typical amplitude proportional

to

kFL.

A formula for the low temperature

regime

is obtained

applying

the

Berry-Keating

resummation method

[38].

The fluctuations in this limit _are shown _to

be,

in

general, proportional

to

kFL@@.

Some of trie mechanisms which lead to

anomalously large

fluctuations of the

susceptibility

in this energy

regime

^{will be related} to "scars" and to avoided

crossings.

In section

4,

these results

are

generalized

to the _case of _a uniform

magnetic

field. In

particular

it is shown that for _a wide range of field

strengths,

^trie _response is very similar _to that of _a

magnetic

flux fine.

Finally,

_a

proposed experiment,

and _some related _open

problems

_are discussed in section 5.

2. Semiclassical

quantization.

The semidassical

approximation

_concerns trie

leading

terms of

expansions

in which à

plays

trie rote of _a small parameter,

therefore,

it reflects the

underlying dynamics

of the

correspond- ing

classical system. ^Trie characterization of this

dynamics

is crucial since trie appropriate

semiclassical method of

quantization depends

_on trie nature of this

dynamics.

Integrable

systems ^{of N}

degrees

of freedom contain N

independent

constants of motion _m involution. Their

phase

_space motion is confined _{to a} N dimensional torus, ^and semidassical

quantization

of these systems _may ^{be obtained}

using

Einstein-Brillouin-Keller

(E.B.K) theory [39].

In this _case, each _energy level of the spectrum arises from _{a set} of values of the irreducible

action

integrals

associated with the different

periods

of trie motion.

Chaotic systems, on the other hand, do not contain besides trie energy any constant of _mc- tion. A

typical trajectory

of these systems covers trie whole energy shell

ergodically, therefore,

E-B-K method of

quantization

is

inadequate

for this _case.

Yet,

chaos introduces another sort of

simplicity,

and semidassical

quantization

_may be

formulated,

rather

simply,

in terms of trie

isolated

periodic

orbits of the dassical

dynamics.

Gutzwiller's trace formula

[13, 14],

which expresses the

density

of states as a sum over these

orbits,

_is the main trot in the field of chaos quantization.

Generic systems,

however,

_are neither chaotic _nor

integrable.

Their

phase

_space structure is _a mixture of

regular

and chaotic

domains, consequently,

semidassical quantization of these

systems is _more

complicated.

In _{some cases one may} consider the

regular

and the chaotic parts ^{of the}

phase

_space

separately,

and

apply correspondingly

the E-B-K _or the Gutzwiller's

quantization

^{methods. A remarkable}

example

of this

approach

is the semiclassical

quantization

of the hehum atom [40]. ^{A different route for}

quantizing

such systems is based _on time demain

techniques

which where

recently developed

_[41].

According

to this method the spectrum ^is

obtained from the Fourier transform of the autocorrelation

function,

_<

#(0))#(t)

_>,

expressed

in terms of the Van-Vleck propagator, where

)#(0)

> is _some initial Gaussian _wave

packet.

Trie systems this _paper is

dealing

with _are two dimensional closed chaotic billiards. Trie semiclassical

approximation

for their

density

of states _is obtained

by tracing

trie Green function

using

the

stationary phase

approximation. The result may be written _{as a sum} of _a smooth

background

and

oscillatory corrections,

P(E)

_~

P(E)

+

Posc(E). (2.1)

The first _term

originates

from short

orbits,

1-e- orbits with actions which _are small relative to

&. It is

given approximately by

trie Thomas-Fermi

expression, fl(E)

_~

j / / _dPdqô(E

7i(q>

P))>

(2.2)

where

7i(q, p)

is the Hamiltonian of trie system, and

(q, p)

_are trie coordinates and momenta.

For closed

billiards,

trie

analytic expressions

for trie

higher

corrections of

fl(E)

_are also known [42]. ^Trie

oscillatory contribution, po~c(E),

is related to trie classical

periodic

orbits of trie

system [13]

pose

(E)

₌ Re

£ _Aj e~~~~~J (2.3)

grà

J

The _{sum over} j includes ail trie

penodic orbits, Sj

_is the action of the

j-th orbit, Aj

_is an

amplitude

which characterizes the behavior of trie

trajectories

in the

vicinity

of that

orbit,

and 'tj ^{is trie} Maslov

phase

deterniined

by

the

focusing patins

close to

j.

In the _case where ail the

periodic

orbits

(on

the _energy

shell)

_are

isolated,

this contribution _may be

conveniently expressed

in terms of the

dynaniical

zeta function [43],

(s(E),

_as

pos~(E)

=

Im£ log((s(E)). (2.4)

This

function,

aise called trie

Selberg

zeta

function,

is defined _{as a}

product

_over

primitive periodic

orbits (1.e. ^{orbits which do not} retrace

themselves),

(s(E)

=

fl fl

₁

_{exp()Sp(E) i'tp}

₊

_j)up)

,

(2.5)

P-P-o J

~

where the

subscflpt

_p denotes these

orbits, Sp(E)

is the

corresponding action,

_'tp is the Maslov

phase,

and up is trie

instability

exponent associated with the hneanzed motion _m trie

vicinity

of the

periodic

orbit.

The introduction of

(s(E)

is

extremely

useful. For

example,

the semidassical

quantization

rule for chaotic systems may be formulated _as

(~(E)

= 0.

(2.6)

Unfortunately,

trie

dynamical

zeta function suffers from _severe convergence

problems

due to the

exponential growth

in trie number of

periodic

orbits _as their period increases. In order to

darify

this point, ^{it is} convenient to represent

(~(E)

_{as a sum over}

pseudo-orbits, (~(E)

=

£c~ek~". (2.7)

Each

pseudo-orbit,

labeled here

by

_~t, is a linear combination of primitive orbits where

~1

represents ^trie vector of the repetition numbers of the

primitive

orbits,

(rp)~

=

(ri,

_{r2> r3,}

).

Thus trie

pseudo-orbit

action _is

S~

₌

£ _rpsp, (2.8)

lrpl«

and trie

corresponding period

is

given by 7j

₌

ôS~/ôE.

For

long

orbits and

pseudo-orbits

of

ergodic

systems

7j

and

S~

_are

proportional

and

satisfy

trie relation

~~

~

~lll~

_~~~~

where D is the number of

degrees

of freedom of the system

(two

in _our

case), fl(E)

is the classical

phase

_space ^volunie with energy less than

E,

fl(E)

₌

/ / _{dpdq8[E 7i(q, p)], (2.10)}

and

fl'(E)

is its derivative with respect to the energy. In the above

equation,

8 is the unit step ^function.

Formula

(2.7)

is obtained

by expanding

trie

product (2.5)

and

collecting

trie various terms.

The

resulting amplitudes

_are

given by

_[43]

CH ^"

fl ~~(j~~~~f~le~irp~p

~~~~"

~~

_{~~~ ~ ~~~~}

^(2,Il)

They

decrease

exponentially

with the

period

_{r as}

e~Î~

_{where is the metric entropy associated}

with the _average value of the

instability

exponents. ^{On the other}

hand,

trie number of

pseudo-

orbits

proliferates exponentially

with their

period

_as ^e~T~ where hT is the

topological

entropy.

Since hT > À, ^absolute convergence of the _sum

(2.7)

is obtained

only

for

complex

values of i

là satisfying

_[38]

~

~~~~~

~~&

^~

^~~

₂

2fl(El'

^~~'~~~

Convergence

of trie seniiclassical expansion ^for

complex

values of the energy was discussed

by

Eckhardt and Aurell [44].

To be able to obtain real

eigenenergies

from

(2.6),

an

analytic

continuation of

(s(E)

into the real

1là

axis _is

required.

The rifain conclusion from studies

conceming

this issue is that the _sum

(2.7)

_over

pseudo-orbits

should be

effectively

truncated [38,

45-49]. Generally,

it will include ail

pseudo-orbits

of

period

smaller than trie

Heisenberg

time.

Yet,

_a considerable reduction in

the number of

pseudo-orbits

_may be obtained

using

functional relations that exist within the quantum

theory,

and the

longest

orbit

required (labeled

here

by ji)

bas _a

period equal

to half of trie

Heisenberg time,

6

_"

1> (2'13)

where /h

=1/fl(E)

is trie _mean level _spacing. Trie resummed formula which will be

presented

here _was

recently

obtained

by Berry

and

Keating

[38].

They

considered trie semiclassical

spectral determmant, D(E),

which is related to trie

dynamical

zeta function

by

D(E)

=

e~~"'~~~'(s(E), (2,14)

where

fiÎ(E)

is the smooth level staircase

approximately

_given

by

the classical

phase

_space volume of _energy less than E divided

by h~, namely,

NIE)

_c~

")(~

=

j((. _12.15)

J~~ fl

~(£ ~a)

~

_ d~

~~ _~~À/( ÎÎ

~

~%~ ~'~

0 2000 4000 E

o ()

~~

_

- c

~ uJ

- ~

©

~

6000 8000 E

Fig. l.

(a)

The billiard together with _a typical periodic orbit.

(b)+(c)

the corresponding spectral

determinant for _u= I,à, and L=2m= h=1. It is calculated using

(2,16)

from 17347 penodic orbits.

The tuning parameter is set to K~

= 5 ^~

~~~~~~.

_{The vertical bars along the}

energy axis mark the

à(1/&)2

exact eigenvalues.

Here A is trie _area of the

billiard,

and _m is the

partide

_mass. Like for

(s(E),

the _zeros of the

spectral

determinant

provide

the semiclassical

approximation

for the

eigenenergies.

The

advantage

^of this function _is that,

semiclassically,

it is real for real values of E [43], ^{and unlike}

(s(E)

it is invariant under the

change

_- -& [38].

Exploiting

these

properties, Berry

and

Keating

_were able to continue

analytically

the

spectral

determinant. The

leading

term of the

resulting asymptotic

formula is

~~~~

_'~' ~~

~

_~~~ ~~~~~~~~~~~~~~

Q(ÎÎÎÎÎ~Î _~'

~~ ~~~

where

((~t,

_&,

E)

=

S~

gr

fiÎ(E), (2.17)

ôi

and

~

Q(l~>

à>

l~)~ l~~ +1) _~/l

)2

~~~~'

_~~'~~~

h

Inspection

of this formula reveals that the

complementary

_errer function term cuts

smoothly

the infinite _{sum over}

pseudc-orbits.

The center of the smoothed cutoif

(2.13)

_may be found from

iii, à, E)

₌

0,

and the width of the

smoothing region

is determined

by

K which is _a free fine

tuning

parameter. ^This width is minimal when K~

=

~ ~~

fiÎ(E).

à

à(1/à)2

The

applicability

of formula

(2,16)

is demonstrated in

figure

1. The system consist in _a

partiale

of _{mass m moving}

freely

inside a two dimensional chaotic

billiard,

which is _a closed version of the _open

hyperbola

^{billiard studied}

by

^Sieber and Steiner [50].

figure

la shows the billiard

together

with _a

typical periodic

orbit. The

corresponding spectral determinant,

calculated in the semiclassical

approximation

from 17347

periodic

orbits

(the

shortest _among those which bounce the

hyperbola boundary

segment _up to 13

times),

is drawn _m

figures

16 and lc. Trie exact

eigenvalues

^of the systeni, obtained

by solving

^trie

Shrôdinger equation

numerically,

_are ^indicated

by

vertical bars

along

trie _{energy axis,} and trie _zeros of

D(E) give

the

corresponding

semiclassical

approximation.

In this

figure,

122 of them _are shown and found in

good

agreenient ^{with the} exact

eigenenergies.

Even after the resummation, the number of

periodic

orbits

required

for the calculation of the

spectral

determinant

proliferates exponentially

with the _energy. This

proliferation

is the

major

obstacle in

calculating

the

spectral

deterniinant at

high energies. Using

arguments

related to the

pruning

of

pseudo,orbits, BogomoIny proposed

_{a way} for

reducing

their number for _a

given

_accuracy [51].

However,

it is still not clear whether this

approach, indeed, improves

the accuracy ^to work ratio [52].

Part of the

analysis presented

in the

following

section, ^{is formulated in} terms of the

dynamical

zeta function rather than in terms of the

spectral

determinant. A resummed formula for this function _may be

easily

obtained from

(2.14)

and

(2.16).

A

simplified

expression, ⁱⁿ which the

complementary

_error function is

replaced by

_a

sharp cutoif,

is

given by (s(E)

t

£

(cpeÉ~"

⁺

Îe~~'~~~~~~k~"Î (2.19)

n<y

3. The

magnetic

_response of Aharonov-Bohm billiards.

In this

section,

_we

investigate

^the

magnetic

response of _a two dimensional

noninteracting

electron _gas, due to _a

single

Aharonov,Bohm flux hne. The

confining potential

of the electrons

is a chaotic billiard of _area A

lying

in the _~y

plane.

The

magnetic

^flux

#,

located at some

point (~o, vo)

within the

bilhard,

is related to the vector

potential

A

satisfying

1#à(~ ~o)à(y

_{vo "} V _x A.

(3.1)

Its components

expressed

in

polar

coordinates

(r,Ù,z)

where

(~o, go)

is chosen _as the

origin

are Ar =Az =0, and Ao

=#/2grr.

The

thermodynamic properties

of the system are determmed

by

the

corresponding

one par, tiffe

density

of states

p(E),

and _may be derived from the

grand potential

_[53]

fl = -T

/dE ^p(E)

^In

₍₁

+

e~l)

,

(3.2)

where the temperature ^{units chosen here and henceforth} are such that the Boltzman _constant kB is umty, ^and il is the chemical

potential.

For two dimensional systems at low temperature

(T<i~)

here considered it may be

approximated by

the Fermi energy,

ilmEF. Equation (3.2)

is _correct for

spinless partides. Nevertheless,

in Aharonov-Bohm

billiards,

the effect of the

spin

amounts to

multiply by

_a factor of 2 the

integral

in

(3.2).

In what follows this factor will be

ignored.

Consider first the _case of _zero temperature. To the

ground

state of the system with

##

0

corresponds

_nonzero

planar

current

density j~(r).

It generates _a

magnetic

moment M _per-

pendicular

to the

plane

of the

billiard, given by

Biot-Savart law M

=

/d~rr

x

j~.

For 2c

noninteracting

electrons this moment is built up front moments associated with the

single

elec-

tron states

#~

of _energy En below the Fermi _energy.

Namely,

M

=

£~~~~~

Mn with Mn

=

~

<

#n)rvo)#n

>, where _r and _are

polar

coordinates with the

origm

located at the flux 2c

hne,

and _vo is the

tangential

component of the

velocity

operator v. The

thermodynamic

current which flows around the flux fine is In ₌

-côEn lai.

Let _us define the

quantity: ùn

=

-A~~"

ai

=

-~

<

#~)

^~ _vo

)#~

>. In

general

M~ and

ùn

_are

_{different, however,}

_{for the}

case of _a uni-

c 2grr

form magnetic field where # is taken _as the total flux

through

the two dimensional

sample

they

coincide. In

analogy

with the standard definition of the

susceptibility

_as the denvative of the

magnetic

moment with respect to the

field,

_we define the _response due to _a flux hne

by

x=

£ _{£ ù~.}

_For

nonzero temperature it is

~

E~<E~

x = -A

([il

~~,~

(3.3)

The

starting

point ^{for the} semidassical

analysis

_is the Gutzwiller

approximation

for

p(E) expressed

in terms of the dassical

periodic

orbits of the system [13].

Substituting (2.1)

in

(3.2)

and

integrating by

parts

yields

the

grand potential

_{as a sum} of smooth and

oscillatory

terms

analogous

to the

density

of _states:

fl _ci ù _{+ flo~~.}

(3.4)

The smooth part

corresponding

to the contribution of short orbits is

ù

=

/

_dE

^~~ _$~~

,

(3.5)

1 _{+ e T}

where

fiÎ(E)

is the smooth level staircase

(2.15).

The

oscillatory

term commg from the

periodic

orbits is

expressed

in terms of the

dynamical

zeta function

(2.5)

_as

flosc = Im

/

_dE ^~°~

_{~(~Î~ (3. 6)}

« 1 + e T

The semidassical

approach requires

the exact

dependence

of various dassical quantities in the flux. Here it is

simple,

since the dassical

dynamics

_is not affected

by

the flux fine. In

particular,

^the

phase

_space volume is

independent

of the flux. This leads _to the invariance of smooth level staircase,

fiÎ(E,#)

=

fiÎ(E)

_[54].

Consequently,

^the

only

contribution to the

susceptibility,

in the semidassical

limit,

_comes from the dassical

periodic

orbits of the system.

Substituting (3.6)

in

(3.3), diiferentiating

with respect to the

flux,

and

converting

the

integral

into _a Matsubara _sum

(by integrating along

the _contour shown in

Fig. 2),

we obtain _a

general

formula for the

susceptibility

of chaotic billiards:

x c~

i2xoiRe( 1[~llii~j

~

~llii~j

~_~~~~~~

(3.7)

Here _q~

=

<I<D

denotes the reduced flux 1-e- the

magnetic

flux measured in units of the flux quantum

#o

_"

~~,

_Ko

"

~~

~

is the Landau

susceptibility

in _two

dimensions,

/h

=

~~~~

e 12grmc mA

is the _mean level

spacing,

and

(Î~~(E;q~)

denotes the

j-th

denvative of the

dynamical

zeta function with respect _{to q~.} ^The Matsubara

frequencies

_{are En}

là

with

En =

(2n

₊

_1)grT. (3.8)

The _use of

Cauchy

theorem for the _conversion of the

integral (3.6)

into _{a sum over} the residues

(3.7)

is

justified by

the

following properties

of the

dynamical

zeta function and its derivatives:

(1) lim~-m (s(ee~~;q~) ^{=1 for}

gr/2

> > 0;

(ii)

in the _same hmit the derivatives (Î~~(ee~~;q~)

decay exponentially; (iii)

the contribution from the

integral along

the

imagmary

axis _may be

1>nE

Matsubara

frequencies

+ + + + + + +

2nT(

^~

~~ j

^{Re E}

zeros of trie dynani (cal zeta function

Fig. 2. The contour of trie integral which leads to equation

(3.7).

shown

(integrating by parts)

to be of trie order

e~~F/~, therefore,

it may be

neglected

when T<EF.

We _now

investigate

the

dependence

of

(~(E;

_{q~) on} the flux. For this _purpose it is sufficient to examine the

dependence

for an

arbitrary periodic

orbit _as the

magnetic

flux

changes.

Consider first trie _zero flux liniit

q~ ⁼ 0. In this _case the system ^{exhibits time} reversai

invariance,

and this synimetry is reflected in trie

periodic

orbits structure

by

the fact that there are two types of orbits. There _are orbits that trace themselves back

along

the _same

path.

These exhibit

a time reversai invanance. On the other hard orbits which do not have this symmetry _may be traced in two

opposite directions,

and therefore to every such orbit

corresponds

_a time reversed counterpart. ^The time reversai symmetry presents ^itself in the Gutzwiller formula

by

the property ^that such

pairs

of orbits _are characterized

by exactly

the _same dassical quantities.

When _q~

#

0 this

degeneracy

is lifted. Trie

magnetic

flux fine is introduced in trie equations

by replacing

the

original

momentum p=mv

by

_p=mv-

~A,

_{where A is the vector}

_potential

c

which satisfies

(3.1).

This

replacement

leads _{to a} flux

dependence

of the

periodic

orbits actions of the form [55]

Sp(q~) =

jÎ ^pdq

⁼

^{Sp hwpq~, (3.9)}

where

Sp

^is

independent

of _q~ and

corresponds

to the _zero flux

value,

and Wp ^{is trie}

winding

number of trie orbit around trie flux fine at (~o>

go)-

Note that trie other relevant classical

quantities, namely,

the

instability

exponent ^{and the Maslov}

phase

_are not aifected

by

the flux fine.

The

change

_m the time reversai

properties

of trie system is related to the

possible change

in the actions of pairs ^of time reversed orbits. The opposite

signs

of their

winding

numbers lead _{to a} diiference _m their actions.

Yet,

whenever

q~ is _an

integer

(1.e

#

is _an

integer multiple

of

#o),

there will be _a

phase

coherence between the orbit and its time reversed counterpart.

Therefore,

the _various

thermodynamic properties

^{of this} system are

penodic

functions of

q~

[56].

Inserting

expression

(3.9)

for the action into the approximate formula

(2.19)

and

taking

the

derivatives with respect to _q~

yields

((J)j

E.

~)

r~~

£ ~(J>~#s~(~)

~

~u)*~~2«fit(E)-js~(~)j j~

~~)

~

~ <m

~ ~

where the

pseudo-orbits actions, S~(q~),

are

given by

^formula

(2.8)

with

Sp replaced by Sp(q~),

and the

amplitudes

_are

c))

_{= c~}

(-2griW~)?

,

(3.Il)

where c~ ^is

given by (2.Il),

and

W~

^{is the}

pseudo-orbit winding number, W~

=

£ _{rpwp. (3.12)}

lrpl«

Note that at _q~ = 0,

(Î~~(E;

_q~) vanishes. It follows from the linear

dependence

of the

amplitudes cÎ~

_on the

winding numbers,

which

implies

that the contributions from _a pair ^of time reversed

pseudo-orbits

add _up

destructively.

At q~=0 these

exactly

cancel each other.

Forniula

(3.7)

for the

susceptibility

of chaotic billiards will _now be

investigated.

We _assume

a

large enough

Fermi energy (1.e. kFL »1 where kF is the Fernii _wave

number),

much

larger

than the thermal excitations

(EF

»

T).

Under this

assumption,

the

pseudo-orbits

actions evaluated _at EF + iEn _may be

expanded

to first order in _En

/EF

_as

S~(EF

+

iEn)

m

S~(EF) +17j(2n

+

1)grT, (3.13)

where

7j

₌

ôS~/ôE

is the

pseudo-orbit period.

This _expansion holds

only

for the low Mat- subara

frequencies

for which _n <

$.

It tums out that contributions from

large

_n terms are

negligibly

small _as will be

argued

whengr the contribution of the varions Matsubara

frequencies

will be calculated

(see

discussion

following (3.29)).

The

imaginary

part ^of

(3.13)

results in _an

exponential damping

of the

corres/onding pseudo-orbit

contribution. For the lowest

frequency (n

=

0)

it is

proportional

to

e~MG,

while for

higher frequencies

it is much stronger. This

damping imphes

that at

high

temperature

only

the shortest

pseudo-orbits

contribute

signifi- cantly

to the

susceptibility.

One _may define _a threshold temperature,

Tth, according

to the

requirement

that the Upper bound _on these

damping

factors

(which corresponds

to the short- est

periodic orbit)

is of order

e~~.

Since the shortest

penodic

orbit

length

is of order 2L this

condition leads to the definition of the threshold temperature:

~~

(Î)2~~~~'

^~~'~~~

where

fl

=

/À/L

is _a

geometrical

factor of order

unity.

There _are,

therefore,

three relevant energy scales in the

problem:

₍₁₎ the _niean level

spacing /h; (ii)

the threshold teniperature

Tth

associated with the tinte it takes for _a partide at the Fernii energy ^to cross the

sample; (iii)

the Fermi energy EF_fQS ^~

(kFL)~.

When the Fermi _energy is

sulficiently high,

_so that kFL » (2gr)~,

4gr

these scales _are well

separated, namely, /h«Tth

<EF.

At

high

temperatures

(T>Tth)>

the thermal fluctuations wash ont structures in the energy scale of the _mean level _spacing, and

only

^short

periodic

orbits contribute

eifectively

to the

susceptibility.

We then expect _a

simple dependence

_on the parameters. ^As the temperature decreases below Tth

longer periodic

orbits become _more important, ^and at very low tempera- tures

(T

_<

/h)

ail orbits

(of period

smaller than half of the

Heisenberg time)

have

significant

contributions.

Fig. 3. A schematic illustration of _a billiard with rough boundary

(thick

fine), and _some of the shortest penodic orbits

(thin

fines), which almost _cover the _energy shell uniformly. L is the linear dimension of the billiard, A is its area, and LR _is the scale associated with the roughness of the boundary. The latter is assumed to be large conlpared to the Fermi wavelength.

In what

follows,

_an estimation for the

magnitude

of the

susceptibility,

in two ranges of temperature, /h > T and Tth < T, is calculated. For this purpose, some statistical

properties

of the

periodic

orbits _are

required (cf. Appendix A),

and the

following

discussion will be

confined to billiards for which the

boundary

is

complicated enough

so that _{one may} consider the

corresponding

set of the shortest

periodic

orbits

(e.g.

the orbits which bounce the

boundary twice)

_as

covering

the energy shell almost

uniformly

in _an ensemble of billiards. An

example

of such billiard

together

with _some of the shortest

periodic

orbits _is illustrated in

figure

3. In

addition,

it will be assumed that the scale associated with the

roughness

of the

boundary,

LR>

is

large compared

with the Fermi

wavelength.

We shall also introduce _two types of ensemble

averaging,

_(. and

((. )).

The first represents an

averaging

_{over some} externat parameter related to the

roughness

of the

boundary

and is _an average over an ensemble of

billiards,

while the second indudes also _an additional averaging over the flux.

3.1 THE HIGH TEMPERATURE REGIME T > Tth. It is clear tram the above discussion

that at temperatures

sufficiently high

_so that T > Th _is

satisfied,

the shortest

periodic

orbits dominate the behavior. In this _{case one may}

exploit

the natural cutoif of the

periodic

orbit

sum

(2.3)

due to the temperature in order to derive _a

simpler

formula for the

susceptibility

which does not require the resummed _expression of the zeta function.

Substituting (2.3)

in

(3.2)

and

using (3.3)

leads to

x cf

12xo )Re £ £ Îo _(? (2«Wj )~e~+~~~J, _(3.15)

j ?

~ ~

where the _sum is _over ail the

pefiodic

orbits

(primitive

and

repeated),

and

Aj, Tj

and

Wj

are the

j-th periodic

orbit

amplitude, penod,

and

winding

number

respectively.

In the range of teniperature ^here considered, the rifain contribution _comes from the lowest Matsubara fre-

quency n =0, and the

higher frequencies

_may be

neglected.

Then

using

first order _expansion of the actions similar _to

(3.13)

_we obtain:

x t

12xo )Re £ ^? (2grWj)~e~ÎÎ+fi~~~~J

_{Tth 1 T « EF}

_(3.16)

~ J

A

simpler

form of this

formula,

where the

dependence

_on the varions parameters is

dearer,

is obtained

by substituting

the

explicit

form of the actions

(at

_zero

flux) Sj(EF)

"

PFLj

where

oD

CQ

*S~

à

'

, , , /

- ' / ' /

', /"' / ' /", ,~

o

-0.4 -0 2 0 0.2 0 4

#

Fig. 4. The fluctuations of the susceptibility ôx> _in units of x*

"

Îxo)kFL,

_as function of the reduced

flux _~=

#/#o

for various values of the temperature. ^{The sohd fine} corresponds to T=Tth, the dashed Iine to T=2Tth, and the dotted fine to T=3Tth. The threshold temperature, Tth, _{is given} by

(3,14).

pF =

àkf

_is the Fermi momentum and

Lj

is the

length

of the

orbit,

_as well _as the flux

dependent

part

given by (3.9).

Then _one finds

x m

24xo) £' (? e~$(2grWj)~ cos(kFLj -'tj) cos(2grWjq~), (3.17)

j J

where here

£(

denotes _{a sum over}

pairs

of time reversed orbits. This formula shows

that,

in

general,

_{x is an}

oscillatory

function of the Fermi wavenumber and of the flux. For the shortest orbits

Aj/Tj

is of order

unity

and

Lj

_m 2L,

therefore,

the

amplitude

of these oscillations at T =Tth is

proportional

to kFL and it decreases

exponentially

as

e~~/~th

_{for T > Tth. In order}

to estimate the

magnitude

of the

sample

to

sample

fluctuations _we shall consider

ôX _"

À~. _(3.18)

Its calculation

(given

in

Appendix A)

is based _on two results: _one is the

Hannay

and Ozorio de Almeida _sum rule [16] ^which expresses the

density

of the periodic orbits

weighted by

their

amplitudes

_{as a}

simple

function of their

periods,

and the other is _a result of

Berry

and Robnik [55]

according

to which the

winding numbers, Wj,

of

periodic

orbits of chaotic billiards

satisfy

a discrete Gaussian distribution. The

corresponding

_vanance ^a~ is found _to be

proportional

to the orbits

period

t,

namely

~~

~ÎÎ~'

~~'~~~

where _{VF is} the Fermi

velocity,

and C is _a constant

approximately give

_j Cm 0.215. The average over the

roughness

of the fluctuation

function, ôx,

takes the form

~~

m

@

^~

_I(T,

q~) Tth < T < EF

(3.20)

Xo ~

where

I(T,

_{q~) is a}

periodic

function of

q~ for which _{an exact} formula is

given

in

Appendix

A

(Eq.

(A.14)).

It is

presented

in

figure

4. Note that ôX _{is a}

periodic

function of

q~ with

periodicity 1/2.

^This

periodicity

_{is a} consequence of the coherent interference of the orbits and their time reversed counterparts ^{which takes}

place

not

only

at

integer

^{values of} _q~ ^{but also} at half

integers

which

correspond

to false time

breaking

symmetry [55]. ^{In order to estimate the}

magnitude

of

the fluctuations it is instructive to consider

ôy

which is the _same function defined in

(3.18)

but with

(.

.)

replaced by ((. .)).

It is

straightforward

to show that

ôy

^satisfies _a similar formula but with

I(T,

_q~)

replaced by

_a

simpler

function

Î(T) _given by (A.17).

In

particular,

_as shown at the end of

Appendix A,

at T ₌

Tth

it

yields

~~

=

3@e~~CkF/À

m

2kFL. (3.21)

ÎXO

Note that kFL is of the order of the _square root of the number of electrons in the system, thus at the threshold temperature the fluctuations in the

susceptibility

_are in

general

_very

large compared

to )xo_Î. ^The

asymptotic

behavior for _{T »}

Tth

_may be also deduced from

(A.17),

and in this

regime

the _average of the fluctuations decreases

exponentially

_as

)

m

4.5kF/À

^~

e~Î

_Th

< T < EF

(3.22)

XOÎ Tth

The temperature ^{for which the} fluctuations _are of order of the Landau

susceptibility

_xo is

approximately

TmTth

In(4.5kFL)

1-e-

typically larger

than Th.

3.2 THE _Low TEMPERATURE REGIME T _< /h. Consider _now the low temperature

regime

T < A. It is

iniportant

to note that unlike

high

temperatures ^where the flux

dependence

of the chemical

potential

is very weak

(and

therefore _was

ignored),

here it is very strong. The

reason is that in order _to maintain the nuniber of

partides

constant, ^{the chemical}

potential

niust lie in the middle between two energy

levels,

therefore it must follow the variations of the energy levels due to

changes

in the flux. At

high

temperatures, _on the other

hand,

the chemical

potential

is aifected

by

the occupation of _many levels in the energy interval of size T. The flux does not affect the _mean

density

of _states and therefore it does not affect the chemical

potential

as well. Note also that

fixing

the number of electrons while

considering

the

grand potential (3.2)

is

justified

in the limit of

large

number of

partides

where the diiferences between the

thermodynamic properties

derived from the canonical and from the

grand

canonical ensembles

are small.

In order to obtain _a dosed formula for the

susceptibility

in the low temperature _regime,

we shall

apply Berry

^and

Keating

resummation method [38]. ^{Since this method} was

already

described

(see

also Ref.

[57]),

here it will

only

be outlined

briefly.

Let

ù

_{be defined}

as

T E -E

fl =

/

_dE

_{R(E) In(1}

+ e T

), (3.23)

«

where

R(E)

is the resolvent given

by

~~~~ ~

E +

iÎ _E~'

~~'~~~

and

E~

_are the

eigenenergies

^{of the} system _(E is an infinitesimal

positive number).

The _imag- mary part ^of

ù

_{is the}

grand potential

defined in

(3.2).

An important ^{relation satisfied}

by ù

_is

the exact functional

equation:

ù(-à)

=

fl*(&). (3.25)

It is due _to the fact that the

eigenvalues

En _are invanant while _E

changes

_sign under à reversai

[38, 57].

^Let

1

be related _to fl _as in

(3.3), namely

ô2~j

~~ ~

ôj2

^{' ~~'~~~}

EF,T

JOURNAL DE PHYSIQUE T 4 N' ~ MAY 1994