Orbital Maghetism in Two-Dimensional Integrable Systems

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Orbital Maghetism in Two-Dimensional Integrable Systems

E. Gurevich, B. Shapiro

To cite this version:

E. Gurevich, B. Shapiro. Orbital Maghetism in Two-Dimensional Integrable Systems. Journal de

Physique I, EDP Sciences, 1997, 7 (6), pp.807-820. �10.1051/jp1:1997194�. �jpa-00247364�

Orbital Magnetism in Two-Dimensional Integrable Systems

E. Gurevich and B.

Shapiro (*)

Department

of

Physics,

Technion-Israel Institute of

Technology,

Haifa 32000, Israel

(Received

5

January 1997,

received in final form ii

Februajj.

1997,

accepted

19

February1997)

PACS.75.20.-g Diamagnetism

and paramagnetism

PACS.73.20.Dx Electron _states in low-dimensional structures

(superlattices,

quantum ^well structures and

multilayers)

Abstract. We

study

orbital

magnetism

of _a

degenerate

electron _gas in _a number of two- dimensional

integrable

_systems, within linear _response theory. There _are three relevant energy scales:

typical

level

spacing /h,

the energy r, related to the inverse time of

flight

_across the system, and the Fermi energy SF-

Correspondingly,

there _are three distinct temperature

regimes:

microscopic (T

<

hi, mesoscopic (A

« T £

r)

and macroscopic

(r

« T «

eF).

In the first two

regimes

there _are

large

finite-size effects in the

magnetic susceptibility

_x, ~vhereas in the third

regime

_X approaches its macroscopic value. In _some cases, such _{as a} quasi-one-dimensional strip _{or a} harmonic

confining

potential, ^it is possible to obtain

analytic

expressions for x in the

entire temperature _range.

1. Introduction

A

degenerate

electron gas, in the _presence of _a weak

magnetic field,

exhibits weak orbital

magnetism ill (the

Landau

diamagnetism).

For _a two-dimensional _gas the value of the orbital

magnetic susceptibility

is

given by

_{XL =}

^-e~ /12~Mc~,

_where lzI is the electron _mass.

(Double degeneracy

with

respect

to

spin

is

implied

in this

expression,

_as well _as in all

subsequent formulae.)

This result

applies

to _a macroscopic

system.

Whether _a

sample

^of a

given

size L _can be considered _as

macroscopic depends

_on the tempera- ture T.

Namely,

T

(in

_energy

units)

should be

compared

with the characteristic

size-dependent energies

^such as the _mean level

spacing,

A _ci

2~h~/fi£L~,

_or the inverse time of

flight

_across

the

sample,

r _ci AUF

IL

_ci

kF

LA

/2~,

where _vF and

kF

_are the Fermi

velocity

and _wave number.

Generally,

_one should

distinguish

between three temperature

regimes:

ii)

For T _< A

(the "microscopic" regime)

discreteness of the energy levels _comes into

play

and the

sample

_can be viewed _{as a}

giant

atom. The

magnetic

_response in this _{case can} be very

strong

^{and includes such exotic}

possibilities

_as

perfect diamagnetism

and Meissner effect [2].

(ii)

For

higher temperatures,

A _« T

$ r,

the

system enters

_the

_{mesoscopic regime} (for

recent reviews _see Refs.

[3, 4]).

Here the

typical

value of the

magnetic susceptibility

is

(* ^{Author for}

correspondence (e-mail: borisflphysics.technion.ac.il)

©

Les

#ditions

_de

_Physique

₁₉₉₇

of order

(kFL)°

_XL and _can have either

sign.

The

exponent

_a

depends

_on the

sample

_ge-

ometry.

For most two-dimensional

integrable systems

a =

3/2, although

other values _are also

possible

in _some

special

_cases

(see below).

For

completely

chaotic two-dimensional

systems

a = I.

(iii)

For still

higher temperatures,

^{when T} » r

(but

smaller than the Fermi _energy

eF),

the

system

can be considered _as

macroscopic

and its

magnetic susceptibility,

_up to small

corrections,

is

given by

the Landau value _XL

Thus,

at

present

there is _a

good qualitative understanding

of the

phenomenon

of orbital

magnetism

in various

temperature regimes

and for various

geometries. However,

reliable _quan- titative results _{are scarce.} Most of such results refer to the

mesoscopic regime [4-6]

and _are based _{on a} semiclassical

approximation

for the

density

of states. This

approximation

becomes

inadequate

both at very low

temperatures,

^T <

A,

and at

high

temperatures, ^T > r.

It, therefore,

_seems useful to consider _a few

simple systems, starting

with _an exact

expression

^for the

susceptibility

_x, and to observe the behaviour of _x in the entire

temperature

range. This is done in the present paper

by u8ing

_a linear response

expression

^for _x.

In Section 2 _we present ^several

equivalent expressions

for the orbital

magnetic susceptibility

within the linear response

theory.

In Sections

3, 4,

5 _we consider

specific examples

of _a

strip, disc,

and

square-geometries.

Section 6 is devoted to _an electron gas confined

by

_a two- dimensional

parabolic potential.

2. Linear

Response Theory

for the

Magnetic Susceptibility

We consider _an electron gas, confined _{to some} domain in the

xv-plane

and

subjected

to _a weak

magnetic

field B in the z-direction. The

jrand-canonical potential

Q

=

/ dEp(E) In[I

+

efl~"~~)], (l)

fl

where _~t is the chemical

potential, fl

= I

IT

and

p(E)

is the

single-particle density

of states.

Sometimes it is

useful, by integrating by parts twice,

to rewrite

equation (I)

_as:

Q =

/ _{dEQ(E)$ (2)}

where

f(E)

=

(I

+

exp[fl(E ~)])~~

is the Fermi function and the

quantity

Q(E)

=

/~ dE' /~ dE"p(E") ₍₃₎

-c~

-~

has the

meaning

of _a

grand-canonical potential,

for the _same

system,

at zero

temperature

^and with the chemical

potential equal

to E.

The

density

of states _can be written _as

PIE)

=

-)Im

Tr

G(E)

₌

-)Im

Tr

lE

₊

i~ H)~~

,

14)

where

G(E)

is the retarded Green's

function,

at energy E. The full

(single-particle)

Hamilto- nian H is

split

into the

unperturbed part, Ho,

and the

perturbation

~

&Ic~

^{~ ~} _2fifc~ ^~ _' ^~~~

which describes the effect of the

(static) magnetic

field. It is assumed that the vector

potential

A satisfies the condition div A

= 0.

Expanding G(E)

in terms of the

unperturbed

^Green's

functions, Go (E)

=

(E

₊

i~ Ho)~~,

one

obtains,

_up to second

order,

G =

Go

+

GOVGO

+

Go VGOVGO, (6)

which,

on substitution into

equation (4),

leads to the

following expression

^{for the} correction

bp(E)

to the

density

of states:

The first

_and the econd term

describe,

spectively,

the para-

lugging

nto

(I)

^{nd rating} _by

_parts

gives:

IQ =

This

expression

is

proportional

to

B~,

_so that the

susceptibility,

in the B _~ 0

limit,

is x =

-2bQ/B~A,

where A is the

sample

area.

For further calculations it is useful to have _an

expression

for _x in terms of

xo(E)

which is the

susceptibility

of the

system

at T

=

0,

_~

= E

(compare

with

Eq. (2)):

X "

/ _dEXo(E)$

191

Equation (8)

is written in _an abstract operator ^{form. For}

practical

calculations it is useful to use a

particular representation.

For

example,

if _one

computes

^{the traces}

using

_{as a} basis the

eigenstates

of

Ho (with

the

appropriate boundary conditions),

_one

obtains,

after _some

algebra:

AB2

~

bet

^~

~

i

~~~~~~~"

_~

~~

~~'~~ ~~~~

Here the summation is _over all states of the

unperturbed

Harniltonian

Ho,

with

eigenenergies

e~. The

energies _e(

and

el'

denote the first

(I.

_e.

proportional

to

B)

and the second

(proportional

to

B~)

corrections to the

unperturbed energies

_ei. The first correction _can exist

only

for levels which _are

degenerate

in the absence of B.

Another useful

expression

for _{x is} obtained

by writing equation (8)

in the coordinate represen- tation. This results in:

x =

~

~ ^~

Im

/

_dE

_f(E) / _d~r / _d~r'

~AB fi£c

x

(-h2 ^A(ri )Go (r, r'; )j A(r') )Go (r',

_r;

Eij

⁺

^{Miir r'iGo jr, r';} E1421ri), ⁱⁱⁱ⁾

where

Go(r,r'; E)

is the

unperturbed (retarded)

Green's function in the coordinate _represen- tation. Since this

representation

is

usually

the most

appropriate

for

making

various approx-

imations, equation ill)

is _a

good starting point

in many cases. It _was

used,

for

instance,

in the

study

of

mesoscopic

effects in disordered

systems

_[7]

(in

this _case

Go

includes the random

potential

^of

impurities).

It also

helps

to prove in the _most direct way

that,

for T _>

r,

the _sus-

ceptibility approaches

its

macroscopic

value _XL,

independently

of the

sample geometry. Indeed,

the Fermi function

f(E)

has

poles

at values

En

= ~ +

i(~ lpi (2n +1).

Therefore the

integral

over E in

equation (11)

can be

replaced by

a sum

containing Go (r, r'; En).

This function in _an infinite

system decays

with distance _as

exp(-kF (r r'[ ~(2n+1) /fl~).

It is therefore clear that for _a system ^{with size L}

larger

than

fl~/kF,

_i.e. for T _>

r,

the

susceptibility

_ceases to

depend

on

sample

size _{or on} its

geometry. Therefore,

for T _> r linear response

theory

is

applicable

_as

long

_as the

cyclotron

_energy

hwc

is smaller than T.

However,

for T _< r the

susceptibility-

_x

does

depend

on

sample

size and its

geometry,

and the linear response condition

requires

that

hwc

is smaller than the level

spacing

£h

(I.e.

the

magnetic

flux ib

through

the

sample

is smaller

than the flux

quantum

ibo =

2~hcle).

A useful

approximate expression

for _x is obtained _upon

using

the semi-classical

approxima-

tion [8] for the Green's functions in

equation (11).

Let _us

briefly

outline the derivation

(details

are

presented

elsewhere

[9]).

^First, one derives _a semi-classical

approximation

for the function to

(E).

This is done

by rewriting

the Green's functions in _terms of the

propagators K(r, r', t),

approximating

the

propagators by

their semi-classical

expressions

_[8] and

performing

the inte-

grals

within _a

saddle-point approximation.

Then _one substitutes _xo

(El

into

equation (9)

and

integrates

over

E, making

_use of the

approximation

/

dE E"

CDS(F(Eii $

QS /L" CDS

(F(/L)I

l~

i~F'(/LlT) ₍₁₂₁

where

F(E)

is _some smooth function of E

(I.e.

does not

change appreciably

within _an interval of order

T)

and

11(x)

e

xl

sinh _x. The

resulting

semi-classical

expression

for _{x is}

Here I labels families of

primitive periodic

orbits for

integrable systems

or isolated orbits for chaotic _{ones. r} is the

winding number,

_TX is the

period

of _a

primitive periodic

orbit and

Tt "

h/~T.

Factors

d>.r(~)

_are related _to the

oscillating

part ^{of the}

(unperturbed)

semi-

classical

density

of states

[8,10]:

Posc

(Ill

"

~j d>,r(/~). (14)

A,r

For

integrable system (A()

is _an orbit _area

squared

and

averaged

over the

family

~. For _a chaotic

system

it is

simply

the

squared

area of _an isolated orbit.

Averaging

_{over a}

family

amounts to

integration

_{over one} of the

angle

variables [4]

(All

=

) / ^~~ _{A~(9)d9. (15)}

The semi-classical

expression

for x,

equation (13),

coincides with the _one derived in refer-

ence

[4],

^{in the B} ~ 0 limit. This fact demonstrates _that it does not matter which of the two

approximations,

I.e. linear response or semi-classics. is done first.

3.

Strip Geometry

In this section _we consider electrons confined to _a

strip -L~/2

_{< x <}

L~ /2, -Ly/2

< y ^<

Ly/2.

Periodic and _zero

boundary

conditions _are assumed

along

_x and _y directions

respectively,

and

the limit

L~

~ oo is taken. It is convenient to choose the Landau gauge:

A~

=

-By,

Ay

=

Az

= 0.

Stationary

states are labeled

by

two

quantum numbers,

_-cc < k < +cc and

n =

1, 2,.

^The

eigenfunctions

~§n,k

lx, vi

₌

e~k~un,k (VI,

^where

u[,k

satisfies:

1/M )~

^~

_2~f ^~~~

^~

^~~~~~ ~"'~~~~ "~'~~~~'

Treating

the

magnetic

field _{as a}

perturbation,

_one obtains

ill]

that the first order correction

£[~

= 0 and the second order correction

,,

L(

eB ^~ 6

k~L( ^~2

15

1

~"~ 24M _c ~2n2 ^~ ~4 n2 n4 ^~~~~

Thus, only

the first term in

equation (10)

is present, ^{and the} zero-temperature

susceptibility (including spin degeneracy) xo(E)

is

given by

x0(El

=

/~~ /)) ₍₎ ^f _{£i~o(E ~n~i, j17)}

n=1

where _£nk

=

(h~k~/2M)

₊

(h~~~n~/21zIL()

are the

unperturbed

_energy levels.

Next,

_we

integrate

over

k, apply

the Poisson summation formula

ill

to the _{sum over n,} and insert the

resulting expression

^for _xo

(El

into

equation (9).

The

integral

_over E is then

performed, using

the

approximation (12),

which amounts to

neglecting

small terms of order

T/~t.

The final

expression

for _x is:

~

= l ₊

fi~

~°~

~~~(~~

~ ll

~~~)

+

Oil /fi), ₍₁₈₎

xL ~

~~

f

~~

r

where

kF

₌

@,

_r

=

h~kf/fifL~

=

hvf/Ly

and the

function11(x)

has been defined above. In

equation (18)

_we wrote down

only

the

leading oscillating term,

^{of order}

fi,

and the term

I, responsible

for the Landau

diamagnetism.

There exist also small

oscillating corrections,

of order

(kFLy)~~/~

and

smaller,

which _are not written down

explicitly,

_even

though they

_are calculable [9]. ^Let us

only

mention

that,

in addition to

oscillating corrections,

there is also _a small

non-oscillating paramagnetic correction,

9[_XL

(/8kFLy,

to the Landau value xL.

Thus,

the

strip geometry provides

_{a rare}

example

for which it is

possible

to obtain _an

essentially

exact

(up

to small corrections of order

T/~) expression

for the linear

susceptibility

x,

including

all

size-dependent

terms. One _can observe the

change

of _x with T in the entire

temperature

range, from T

= 0 and _up to T _» r when _x becomes

equal

to its

macroscopic

value XL-

Equation (18)

is

quite

similar to the

corresponding expression

for the _case of _a

parabolic

confinement

[12].

^{This fact} demonstrates that the nature of confinement

(I.e.

hard walls _or "soft"

confinement)

is immaterial for the

phenomenon

of

size-dependent

fluctuations.

In

fact,

the

leading oscillating

term in _{x can} be

obtained,

within _a semi-classical

approxi- mation,

for _any

confining potential

and for

arbitrary magnetic

field

[9].

^{For small}

fields,

the oscillations _are

given by

an

expression

similar to

equation (18),

_up to _an overall factor of order

unity

and _an extra

phase

in the

argument

^{of the cosine.}

L~

^{should be understood} as some

effective width of the

confining potential.

12.

~

©~

~

7.

m

~

~

~ ₂

20 22.5 25 27.5 30 32.5 35 37.5

k~R

Fig.

1. Linear

magnetic susceptibility

in _a disc geometry, calculated at T ₌ 0. lb and normalized

to

(XL((kFR)~.

4. Circular

Geometry

The electron gas is confined to a disc of radius R. The

unperturbed,

I.e. zero-field

stationary

states _are

given,

_up to _a normalization

factor, by exp(im9)Jm(~mnr/R),

where

~mn

is the n-th

zero of _a Bessel function

Jm(x).

The

unperturbed energies

_{are £mn}

=

(h~ /2MR~)~$~.

A

pair

of states

[m, n)

and _m,

n)

have the _same energy.

The

perturbation

term in the

Hamiltonian,

due to the

magnetic field,

is:

iheB b

~

e~B~

~2

(19)

V~@@ _8Mc~'

and the first and second-order _energy corrections _are:

~~"

2Mc'~~'

^~~" _8Mc~ ^{~'~~'~~~ ~'~~'~~' ~~~~}

Note that the first-order _term in V does not contribute to the correction

£$~

which is therefore

purely diamagnetic.

It _now follows from

equation (10)

that

1 1 i(mnr~mni/(£mm

+

Ii _m~ Al

(211

We

analyze

first the

low-temperature regime,

T < A _e

2h~ /MR~.

Since

(mn[r~ [mn)

_ci

R~,

the first

(diamagnetic)

term is of order of the total number of

electrons,

N _cf

(kFR)~

₌

4~ IA.

The second

(paramagnetic)

term exhibits

sharp peaks

_every time when the chemical

potential

~ coincides with _an energy level _£mn.

Indeed,

for _~

= £mz~, the function

of /b£mn

is

equal

to

-1/4T,

and it

rapidly

decreases when _~ deviates from _£mz~

by

_a few T. Since the

quantum

number _m, for _{states near ~,} is

typically

of order

kFR,

the

height

of the

paramagnetic peaks

is of order

(kFR)~A/T

_ct

~/T. Thus,

in the

low-temperature regime

the

susceptibility

_x

(normalized

to the Landau value

[xL() Plotted

_{as a} function of _~,

displays

_a

diamagnetic background,

of order

~/A,

with

sharp paramagnetic peaks

of

height ~/T

and width T. An

exact numerical

computation

of the

expression (21)

confirms this

qualitative picture (Fig. Ii.

Next,

_we ^consider

temperatures

^{in the} _range ^A « T

£

^r e AUF

/2R.

For such

temperatures

the

paramagnetic peaks _get

smeared and the

diamagnetic background,

of order

(kFR)~,

cancels

with the

corresponding paramagnetic

term. The net effect is _an

oscillating

_term, of order

(kFR)~/~

The calculation is based _{on a} Poisson summation of the double _sum in

equation (21)

and _{on a} semi-classical

approximation

for the

unperturbed energies,

_or

~mn,

which

satisfy

₁₁

3]:

m2 _m

arccos(m/~mz~)

=

~(n

+

~). ⁽²²⁾

Let _us outline the calculation of the

paramagnetic

term

x~~~/(xL(

"

-(6h~/MR~) ~ m~(bf/b£mn).

As

usual,

it is

simpler

to consider first the

zero-temperature

case and then to _use

equation (9).

At T

=

0,

^b

f/b£

=

-b(~

_El and

xj~~ (E)

₌ 12 XL

~j _m~b(~$~ i~), ₍₂₃₎

m,n

where

i~

=

4~/A.

After

applying

the Poisson summation

formula,

_m and _{n go over} into continuous variables: _{m ~ x,}

(n

₊

_~)

_{~ y.} The

integral

over y is

immediate,

due to the )-function. The

integral

_{over x} is done in the

saddle-point approximation, using

the

large

parameter i and the continuous version of

equation (22).

The

resulting expression

for

xj~~ (E)

is:

x~~~

(E)

= [XL

~

7~

+

i~/~ ~j _{#(K~, K~)}

₊

O(1)

,

~

(24)

K~,Ky

where the _{sum runs over} I <

K~

< cc and 0 <

2K~

<

K~,

and

#(K~, K~)

=

~~

cos~ l~~~) ^sin~/~ _~~~~)

^cos

^21K~

^sin

_~~~~) ^~~K~

⁺

^{j (25)}

/~ _{KY KY KY}

2 4

A similar treatment of the

diamagnetic

term results in _a term

-31~/4 plus oscillating

_correc-

tions of order

@. ^Thus,

the

large non-oscillating

terms cancel and the net result for _xo

(El

is

given by

the second term in

equation (24). Finally, using equations (9, 12),

_we find:

x =

lxL i~/~ ~j _{4(Kz, Ky)}

_7~ ^~

_(~~

_sin

^~ )) ₍₂₆₎

K~,K~ ^Y

Thus,

in the temperature range A « T _<

r,

the

susceptibility

_x

is, typically,

of order

(kFR)~/~

and _can have either

sign.

It

oscillates,

_{as a} function of ~, with ^a

period

of order

(~ A)~/~

_ci r.

In

Figure

2 _we present _x as a function of _{i. as} obtained from

equation (26) (solid line).

For

comparison,

dots

represent

^{the result of} a numerical

computation,

based _on the exact

equation (10),

with £[ and £I'

given by equation (20).

These oscillations reflect the structure of the

density

of states smoothed _{over energy} intervals AE

£ r,

in the _{same way as} the

sharp peaks

of the

low-temperature regime

reflected the exact

(discrete)

_spectrum of the system.

Equation (26)

can be obtained from the

corresponding expression

of reference

[4],

^{in the B} ~ 0 limit. The derivation in reference [4] was based _{on a} semi-classical

approximation

for the

density

of _states

[8,10],

with _a

subsequent

introduction of the

magnetic

field via the classical action.

In

contrast,

we have started with _an exact

expression

^{for the linear}

susceptibility

and used

(in

the

mesoscopic

temperature

regime)

_a semi-classical

approximation

for the energy levels of the

system.

^{In this}

respect

our derivation is similar to that of

Dingle _[13].

He calculated the

orbital

magnetization

of _a thin

cylinder

and _a small

sphere,

and obtained _an

expression

^for _x, which contained

oscillating

terms as well _{as a}

large steady

term. His result for the

oscillating

terms appears to be

correct,

^{while the}

steady

term is due to _{an error.}

Q~ ~

~

_l

%

~ ~

~-l

~ -2

-3

40 50 60 70 80

k~R

Fig.

2. Linear

magnetic susceptibility

in _a disc geometry, calculated at T

= 5zh and normalized to XL

((kFR)~/~. Analytic

result obtained from

equation (26) (solid line)

is

compared

to numerical _one based _on equation

(35) (dots).

5.

Square Geometry

Here _we discuss electrons within _a square of size L. The

unperturbed energies

_are

£nm =

(~~h~/2fi£L~)(n~

₊

m~).

A state

in, m)

is

degenerate

with the state

[m,n) (there

can

be,

in

addition,

accidental

degeneracies

if _a

pair n',

^m' has the _{same sum} of _{squares as} the

pair

_n,

m).

Let _us first consider low temperatures, ^T « A _e

2~h~ /ML~,

and discuss the

paramagnetic peaks

due to the second term in

equation (10).

The first order correction

£[~

is due to

lifting

of the double

degeneracies by

the

magnetic

field.

(We

do not consider accidental

degeneracies, although

the treatment is

readily

extended to include that _{case as}

well.)

The

degeneracy

is lifted

only

if _n and _m have different

parity

and in that _case

~~~ ~~"

~2 &Ic

(n2 m2)3

I

3

~~

d

~

~

~

~

20 30 40 50 60 70

k~L

Fig.

3. Linear

magnetic susceptibility

in _a square geometry, ^{calculated at T} = 0.lA and normalized to

jxLiikfR)~/~

20 40 60 80 100

k~L

Fig.

4. Linear

magnetic susceptibility

in _a square geometry, calculated at T

= 5 A and normalized to (XL_(.

Analytic

result obtained from equation

(28) (solid line)

is compared to numerical _one based

on equation

(35) (dots).

The

largest

corrections _occur when _n

= m + I. In such _cases

£[~

ci

(heB/fi£c)kFL

and the

height

of the

corresponding peak

in

susceptibility

is xmax ^ci XL

(kFL)2A IT, just

_as in the _case of the disc.

Note, however,

that in the square

geometry

^such

large peaks

_are much _{more rare} than in _a disc. For _a disc

large peaks

_were

separated by

_a distance of order A. For _a square such

peaks

_occur,

roughly,

for each

pair (n,

_{n +}

Ii,

I.e. _are

separated by

_a distance of order n/h _ci

kFL/h

_ci r _e AUF

IL.

The difference between _{a square} and _a disc is

clearly

_seen, if _one compares

Figure

3 to

Figure

^I.

Except

for the

large paramagnetic peaks

in

Figure

3 _{one can}

see an

oscillating background.

This

background

_comes form the first _term in

equation (10).

It will be shown below that this term _can be either para- or

diamagnetic

^and its

typical

value is of the order

(kFL)~/~.

In the

mesoscopic temperature regime,

/h _< T _<

r,

the

susceptibility

_x is described

by

the semi-classical

expression (13).

This _{case was} studied in detail in reference [6] and

particularly

in reference _[4] where

expressions

^for the factors

d>,r

^and

(A()

_can be found. The

resulting expression

^for _x is:

X 8

~

~j3

^/2

f _~j

^~~~

^{~~~~ ~~~}

^~

~~~~

(XL

5@

^~

~ ~ ~

~l/2

(q~2 +

q~2)~/~

^q~2q~2

~

/d/

~ ~ ~ ~

2~rfi~T

xll

,

(28)

F

where _u~ and uy are

positive coprime integers,

^{which label}

primitive

orbits.

Only

odd _u~ and

uy ^enter the _sum in

equation (28),

since otherwise the _area enclosed

by

_an orbit is

exactly

zero [4].

As _an

example,

in

Figure

⁴ we

present xl

_{[XL as} calculated from

equation (28) (solid line).

We chose the _same ratio

Tllh

= 5 _as in reference

[4].

^{The result is}

practically indistinguishable

from the numerical _one

(dots),

based _on

equation (10) (Let

us note, ^{that in} reference _{[4] some}

disagreement

between

equation (28)

and numerics _was

observed).

There _{are some}

qualitative

differences between the

mesoscopic

oscillations in the square

geometry (Fig. 4)

^and those for _a circle

(Fig. 2):

in the circle oscillations _are modulated _{on an energy} scale which exceeds r

by

an order of

magnitude.

Comparison

with _an exact numerical

computation

demonstrates that

expression (28)

^is valid down to

temperatures

^T m

/h,

but fails for T _< /h.

Nevertheless,

it _can be used for

estimating

the aforementioned

background,

due to the first term in

equation (10).

The

point

is that this

term _ceases to

change

when

temperature

^is lowered from T _m /h down to T ₌ 0. To obtain

an estimate of

expression (28)

at low

temperatures,

we square it and average ^out the fast oscillations. This

gives,

for the

typical

value of

x~

in the

background,

X)ack

~

X~ i~ _~~~~~~ ~/~~

~~~~~~~~~~~

~~~~

The _{sum over}

repetitions

_r is estimated

by replacing

it with _an

integral,

with _an upper cutoff

provided by

the function 11. This

gives

a

logarithmic factor,

_so that

iXbacki

C~ (XL(

(kFL)~~~@@. ₍₃₀₎

Let _us mention that _a similar

logarithmic

factor _appears in chaotic

billiards,

at low temper-

atures

[5].

^{The behaviour of} _x ^{in that} case is, of _course,

quite

different from the square

geometry:

^the enhancement factor is

kFL,

instead of

(kFL)~/~,

and the

large paramagnetic peaks disappear.

6. Harmonic Confinement

Our last

example

is _a

degenerate

electron _gas confined

by

_a harmonic

potential [14-17].

The

problem

of orbital

magnetism

in this _{case was} considered

previously by

_a number of

authors,

and _some

analytical

and numerical results have been obtained for various temperatures ^and fields. Below _we shall derive _an

analytical expression

for the linear

susceptibility

_{x, as a} function of temperature

[18].

This

expression

demonstrates the _crossover from

strong magnetic

effects

towards the weak Landau

diamagnetism,

under increase of temperature.

We consider _a

slightly anisotropic

harmonic

potential. U(x, y)

=

(M/2) (Hi ^x~

₊

_{Q2 y~} ),

with

Hi

close to

Q2, namely [Hi _Q2(

₊ AR _<

hQ( /~.

In the absence of _a

magnetic

field the spectrum is

given by

_En

m ⁼

hoi (n

₊

))

₊

hQ2 (m

₊

)),

where _{n, m}

=

1, 2,.

^For the

isotropic

case,

Hi

₌

Q2

"

Q,

_energy levels _can be labeled

by

_a

single integer

I

=

1,2,.

,

and the f-th level is t-fold

degenerate.

A small

anisotropy

~Q

splits

the

degenerate

levels into _narrow

"multiplets".

The width of the n-th

multiplet

is of order

than,

which is

~AQ IQ

for

multiplets

near the energy ~. The above formulated

condition,

AR _<

hQ~/~,

is

just

^the

requirement

for

having

well defined

multiplets.

It is clear _on

physical ground,

and is verified

by

the calculation

below,

that such weak

anisotropy

_can affect the

physical properties

^of the system

only

at

temperatures,

T _<

~AQ IQ.

When _a weak

magnetic

field is

applied,

_energy levels

acquire

a second order correction

,,

1 eB ^~

(n

₊

))Qi

-.

(m

₊

))Q2

~"'~

2

~

Mc

Q] -.Q]

^~~~~

Since for _any finite

anisotropy

the first order correction is _zero,

equation (10) gives:

fi

"

~$

Qj Qj ~ ^(~l

₊

_{))fli (nl}

₊

jlfl2j ^{f(£nm). (32)}

n,m

The effective radius R is determined from

MQ~R~/2

= ~, where Q

=

(Hi

₊

Q2)/2

_m

Hi Applying

to the double _sum in

equation (32)

the Poisson summation

formula,

_we obtain for the

zero-temperature susceptibility xo(E):

X°(~)

12

j~

~~~

i/> ,->y

[XL1

7(1 ~2j (~~) dY dx(z ~y)e~~~(lY+k~) ~~~j

I,k=-cu

where ~

=

Q2/Qi

and i =

~t/hQ

=

kFR/2.

Calculating

the

elementary integrals

and

making

_use of

equations (9, 12),

_one can obtain _a final

expression

^for the

susceptibility x(T).

This

expression

is rather cumbersome and will not be

given

here

[19].

For

temperatures

^T »

ihAQ

and

anisotropy

AR « Q

Ii

it

simplifies

to:

~

= -l +

21~ f

cos(2~ri)11 ^~~~

^~

(34)

(XL

~~~

F

where r

=

hvf/2R

=

hQ/2. Except

for the

leading oscillating term,

of order

i~,

there _are smaller

oscillating

terms which _are not written in

equation (34).

This

equation

does not contain

anisotropy,

which demonstrates

that,'for

T much

larger

than the

multiplet

width

ihAQ,

the

anisotropy

does not come into

play (up

to

exponentially

small

corrections).

For T »

r,

the oscillations in

equation (34)

are

negligible

and x = XL For T

£ r,

_{one can}

keep only

the first term in the sum, which results in _an

oscillating

term of order

i~

_ci

(kFR)~.

For

ihAQ

« T _«

r,

many terms contribute to the _sum. The

resulting expression

exhibits

paramagnetic peaks,

of

height (kFR)~r/16T

and width

T,

on a

diamagnetic background

^of order

-(kFR)~.

For T _<

ihAQ, equation (34)

_ceases to be

applicable.

An

analysis

of the full

expression [19]

for

X(T)

shows that it matches the

equation (34)

at T m

ihAQ

and that

only

minor

changes

in

x(T)

_occur when T is lowered below

ihAQ.

This _means that at low

temperatures,

^T <

ihAQ,

the width of the

paramagnetic peaks

becomes

ihAQ

^and their

height

is of order

kFRr/hAQ.

This result is _a manifestation of _a

general

rule. If there is _a cluster of

nearly degenerate

levels about _{some energy £c}

(the

width of the cluster b is much smaller than the

typical

level

spacing /h),

then for T > b the cluster behaves _{as a}

single degenerate

level: it

gives

rise _{to a}

paramagnetic peak

of

height g(kFL)~/h/T,

_{where g} is the number of _non-zero

eigenvalues

of the matrix

(I[iz[j)

in the

subspace

of the cluster. For T _<

b,

the width of the cluster becomes

relevant and the

peak

saturates at a value of order

g(kFL)~/h lb.

This is best _{seen if £( and} £I' in

equation (10)

are written

explicitly using

the

symmetric

_gauge. This leads to:

X

_

~~

~(i)r~)I) f(£1)

₊

~ '~~'~'~~~ /~

^~~

~'~~~

~~~~

~~~l

_~~~~

j~~j

A

~

~

iJ °

When b _< T _< /h the

paramagnetic

contribution of the

cluster,

_as

given by

the second term in

equation (35),

is

approximately -(3~/AM) f'(£c)

tr

L(,

where the trace is within the

subspace

of the cluster.

Estimating

the trace _as

gh~(kFL)~

leads to the

expression g(kFL)~/h IT

for the

paramagnetic peak.

Let us,

finally,

mention that when the

anisotropy

^AR

approaches

the value Q

Ii ii.

_e.

neigh- bouring multiplets

start to

overlap),

the well

pronounced paramagnetic peaks disappear,

al-

though

the oscillations _are still of order

(kFR)~.

7. Conclusions

We have calculated the linear

magnetic susceptibility x(T)

for several two-dimensional inte-

grable systems. Generally,

there _are three distinct energy scales: level

spacing /h,

inverse time of

flight

r and the Fermi energy £F.

For T _<

/h,

the

susceptibility

is sensitive to the detailed structure of the energy

spectrum

as well _as the matrix elements of the

angular

momentum

operator.

In

particular, degeneracies

lead to

large paramagnetic peaks

of order

g(kFL)~/h IT,

_g

being

the level

degeneracy.

At such low temperatures ^the

sample dimensionality

is of _no

importance

and similar

peaks

exist also in three-dimensional systems ^with

degenerate

levels

[20].

^For a

pair

of

nearly degenerate levels,

when the level

separation

b _<

/h,

the

height

of the

corresponding peak

saturates at T _m b.

The

typical

value of _x between the

peaks

is also non-universal: for

systems

with rotational

symmetry

^{it is} of order _XL

(kFL)~

whereas for _a square it is of order [XL

((kFL)~/~ ln(kFL)

and _can be either para- or

diamagnetic.

For the

mesoscopic temperatures,

^/h « T

£ r,

the orbital

magnetic susceptibility

remains

large.

In units of

[xL(,

it is of order

(kFL)"

and

oscillates,

_{as a} function of the chemical

potential

_~t, with _a

period

_{A~t ct} r. For

generic integrable systems

a =

3/2 (compare

to

a = I for chaotic

systems [3]).

For

quasi-one-dimensional

_systems

(a strip)

_a

=

1/2

and for _a

harmonic

confining potential

_{a =} 2.

Thus,

the harmonic

potential

is _{a very}

special

_case even

among the

integrable systems.

^The

point

is

that,

for the

isotropic

_case and at _zero

magnetic field,

the two-dimensional harmonic

potential

exhibits "accidental"

degeneracies,

which _{are not} related to the rotational symmetry

(alike

the "accidental"

degeneracies

ⁱⁿ the

hydrogen atom).

Finally,

for r _« T _« ~t, all

large

orbital

magnetic

^effects

disappear

and _x becomes

equal

to the Landau value _XL-

Thus,

for the

macroscopic

limit to be

achieved,

it is not sufficient to have T »

/h,

_{as one}

might naively expect.

^A much _more

stringent condition,

T _»

r,

is

needed. We close the _paper with the

following

remarks:

(ii Although

_our calculations have been done for the

grand-canonical ensemble,

_{one can}

immediately

infer about the

picture

for the canonical ensemble. For T _«

/h,

the _sus-

ceptibility

_can be very sensitive to the exact number of

particles

in the

system, provided

that the last

occupied

level is

degenerate.

If such _a level is

partially occupied,

the sys- tem is

paramagnetic

with _{x Cf}

c[xL((kFL)~/h/T,

where the factor _c accounts for the

degeneracy

and occupancy of the level. For _a

fully occupied level,

and in the presence of rotational

symmetry,

^the response is

diamagnetic,

with _{x ci}

xL(kFL)~. (The

discus-

sion _can be

generalized

to the _case of _a cluster of

nearly degenerate levels,

_{as was} done above for the

grand-canonical ensemble.)

For

mesoscopic temperatures,

^/h « T

£ r,

the standard transition from the

grand-canonical

ensemble to the canonical _{one ap-}

plies,

I.e.

xcan(N)

=

xgr(~t(N)). Indeed,

the thermal fluctuation IN _ct

fi

in the number of

particles

is much smaller than the

change

in the number of

particles, (AN)

_ci

(A~t)(bN/b~t)

_ci

Tllh, corresponding

to _a

significant change

in _x.

(iii

So far _we have discussed

only

the orbital

magnetic susceptibility.

The Zeeman

splitting,

due to the electron

spin,

also contributes. Within the linear response its contribution _xs is

simply

added to the orbital

part

^{of the}

susceptibility.

This is

clearly

_seen form _equa-

tion

(10),

if the Zeeman

splitting +~tBB

is included into the first order energy correction

£(.

^This

gives

_xi

=

A~~~t[(bN/b~t),

where N

=

£~ f(£i).

At low

temperatures,

T <

/h,

xs exhibits

paramagnetic peaks

_xs ci [XL

/h/T,

which is

just

the Curie

paramagnetism

due to the last

occupied

level. For

temperatures

T »

/h,

_xs is

given by

the Pauli _param-

agnetism,

_xs

"

3[xL(,

_up to small corrections.

Thus,

in this _case, there is _no

appreciable

mesoscopic

effects due to the electron

spin.

(iii)

The

magnetic

^field in this paper was treated _{as a}

given (homogeneous)

external

field, Bext.

For

sufficiently large

_x,

however,

the orbital

magnetic

currents

flowing

in the

sample produce

_a field

Bcurr

which is

comparable

to

Bext.

In three-dimensional

saniplu

this

happens

when the

magnetization

M becomes

comparable

to

Be,t/4~.

_or

ix

C~

1/4~.

In two dimensions the

magnetization

is defined _as the

magnetic

moment per unit _ar(>a

(rather

than per unit

volume),

_so that _x has units of

length. Also,

since the thickness uf the

sample (height

in

z-direction)

is much smaller than its size

L, Bcurr

differs very flinch from the volume

magnetization (demagnetization effect).

So _one has to estimate the field

Bcurr

and compare it to

Bext.

The most

stringent

limitation is set

by

the

requirement,

that the

largest possible

value the local field

Bcurr

_{can assume} should be much smaller than

Bext.

This

yields

_[9]

[x[kF In(kFL)

_« I

, or

[x/xL(

<

In(kFL)/kfre

,

where _re

=

e~/fi£c~

is the classical electron radius.

(Though,

in _some cases, e.g. in _a square

geometry,

^the condition is less _severe,

namely ix /xL(

« L

In(kFL) Ire.)

As _an

example

consider _a disc

geometry,

where at

paramagnetic peaks x/(xL(

Cf

(kFL)~/h/T.

If

lF

_Cf

10~~

_cm and

kFL

_ct

100, then,

_as

long

as T »

10~~/h,

the self-consistent _treatment is not necessary.

If the condition is not

satisfied,

_{one cannot assume anymore a}

given

external field. but should solve the entire

problem self-consistently.

A self-consistent treatment leads to such

possibilities

as Meissner effect and orbital

ferromagnetism,

both in bulk

samples

and

rings [2, 3, 20-23].

An

interesting possibility

_appears to exist in the

strip geometry,

^considered in Section 3. Here the linear

susceptibility, equation (18),

is _{xo Cf}

(xL(@~. _However,

the

differential magnetic susceptibility,

_xd, in _a finite field B

$ &IcvfleLy,

turns out to be of order [XL

[(kFLy)~/~ (see

^Ref.

[12],

^where a

strip

with harmonic confinement _was

analyzed) _[24].

If

[xo[

«

lF /4~ln(kFL)

but

[xd[

>

lF/4~ln(kFL),

then _one encounters a situation similar to the _one which _{can occur} in the de

Haas-van-Alphen

effect and leads to a break _up of the

sample

into

magnetic

domains

[25].

Acknowledgments

This work _was

supported by

the Fund for the Promotion of Research at the Technion.

References

iii

Landau L-D- and Lifshitz

E-M-,

Statistical

Physics,

3rd

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Ginzburg V-L-, Uspekhi

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^Nemeth

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^Azbel

M.,

Solid State Commun. loo

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195.

[18]

^The

analysis

_can be extended to an

arbitrary magnetic

field

(Ref. _[19] ).

[19]

^{Gurevich E. and}

Shapiro B., preprint (1997)

[20]

^See e-g- Buzdin

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Y.E., Phys.

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Kulik

I.O.,

Pisma Zh.

Eksp.

Tear. Fiz. ll

(1970)

407

(JETP

Lett. ll

(1970) 275).

[22]

^Wohlleben

D.,

J. Less Common Metals138

(1988) II;

Wohlleben D. _et

al., Phys.

Rev.

Lett. 66

(1991)

3191.

[23]

^Azbel

M.Ya., Phys.

Rev. 848

(1993)

4592.

[24]

^This can be

explained

in terms of the semi-classical

picture.

In the

strip geometry

the

leading

contribution to xosc comes from the

"bouncing-ball" trajectories (at

B

= 0 these

are

self-retracing ones).

At _zero

magnetic

field _{an area} enclosed

by

the

trajectories

is

exactly

zero. This _area increases

along

with B and becomes of order

L(

at B _~

mcvfleLy,

what

yields

_{Xd ~ )XL}

)(kFL)~/~

at such _a field.

[25]

^Lifshitz

I-M-,

Azbel M.Ya. and

Kaganov M-I-,

Electron

Theory

of Metals

(Consultant

Bureau,

New

York, 1973).