Determinantal quantum theory of d.c. electrical conductivity

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Determinantal quantum theory of d.c. electrical conductivity

A. Fortini

To cite this version:

A. Fortini. Determinantal quantum theory of d.c. electrical conductivity. Journal de Physique I, EDP

Sciences, 1992, 2 (5), pp.625-647. �10.1051/jp1:1992170�. �jpa-00246573�

Classification Physics Abstracts

03.80 72.10B

Determinantal quantum theory ^of d.c. electrical conductivity

A. Fortini

Universitd de Caen-ISMRa, LERMAT (*), 14032 Caen Cedex, France

(Received 22 July J99J, revised J2 December J99J,

accepted

24

January

J992)

Rksumk. Le calcul

quantique

de la conductivitd _en courant contblu d'un

systbme

d'dlectrons _est reconsiddrd h l'aide d'une solution amdlior£e pour la matrice densit6.

L'dquation

d'dvolution utilisde _est d'abord dcrite _{comme un}

systbme

1bldaire dans

l'espace

de Liouville des stats

quantiques

discrets, rdsolue h l'aide de ddterrninants de Cramer et convenablement

adaptde

ensuite _aux

problbmes

de transport. ^{Le calcul de la} moyenne

statistique

de la densit6 de courant,

en

pr6sence

d'un

potentiel

de diffusion, est effectu6 h la limite

therrnodynamique,

en utilisant _une

propr16t6 caract6ristique

des matrices de collision, initialement introduite par van Hove _sous le

nom de

sulgularit6

&. Contrairement _aux m6thodes

pr6cddentes,

bas£es sur les

6quations cindtiques

rdsolues par

perturbations,

la th60rie de Kubo, _ou l'utilisation de fonctions m6moire de type Mod, la rdsolution d'une

6quation int6grale,

ddjh incluse dans les forrnules de Cramer, n'est

plus

n6cessaire. Parmi _ses autres avantages

majeurs,

la m6thode est

capable

de r6soudre la

divergence

bien _connue de la conductivitd lorsque ^la

frdquence

tend _vers zdro. Elle fournit aussi

une

expression explicite

d'un temps ^{de relaxation} effectif, construite

pr6cis6ment

h partir des

s6quences

de transitions

diagonales

responsables ^des effets de

self-dnergie

dans la thdorie de _van Hove. Cette

expression qui

_{ne suppose que} la distribution de

phase

aldatoire des stats

d'6quilibre

initiaux, peut ^{Etre utilisde} pour obtenir des

d6veloppements

maniables _en

puissances

du potentiel de collision. Son domaine de validitd n'est limitd que par l'effet des collisions _sur la

statistique d'6quilibre, qui

se r6vble

ndgligeable

dans les limites du critbre de Peierls h/r « p, off rest le temps de relaxation et p

l'dnergie

de Fermi. Finalement, la

comparaison

_avec la thdorie

616mentaire montre que le temps ^de relaxation est

proche

de

l'expression classique,

mais _ne

pourrait s'y

identifier

complbtement,

h l'ordre le

plus

bas des collisions,

qu'en ignorant

les

divergences apparaissant

au-delh du second orate.

Abstract. The quantum calculation of the d-c- electrical

conductivity

for _an electron system ^is reconsidered

by using

an

improved

form of the

long-time

solution of the

density

matrix. The

starting

time-evolution

equation

which is written _{as a} linear system ^{in the Liouville} space of discrete quantum states, is first solved in terms of Cramer's determinants and next

suitably adapted

to transport

problems.

The calculation of the statistical average of the

many-body

current

density, ^{in the} _presence ^of

scattering

potential, ^{is carried} out in the

thermodynamic

limit,

using

_a

characteristic property ^{of collision matrices introduced earlier} by van Hove _as the so-called 3-

singularity.

In contrast to

previous

methods based _on kinetic

equations

solved

by perturbations,

Kubo's

theory,

_or Mori type memory functions, the resolution of _an

integral equation, blitially

(*) Associd _au C.N.R.S. URA N° 1317.

included in Cramer's formulae, is _no

longer

required.

Among

its

major advantages,

the method succeeds u1

overcoming

the well known _zero

frequency divergent

behaviour of the

conductivity.

It also

yields

_an

explicit

expression ^of an effective relaxation time which is

just

constructed out of those

diagonal

transition sequences

responsible

for

self-energy

effects in _van Hove's

theory.

This

expression

which

only

assumes the randomization of the

phases

of initial states can be used _to get tractable

expansions

in powers of the collision

potential.

Its range of

validity

is limited by ^the effect of collisions _on

equilibrium

statistics, which is shown to be

negligible

within the Peierls criterion, h/r « p, r

being

the relaxation time and _p the Fermi energy.

Finally,

the

comparison

of the present

approach

with the

elementary

transport

theory

reveals that the relaxation time is close to the standard

expression,

even

though

the latter would be

exactly

recovered, in lowest order of

collisions,

only

on

disregarding divergent

terms

arising beyond

second order.

1. Introduction.

The

quantum theory

of transport processes, and

especially

the

problem

of electrical

conduction,

have received _a great ^{deal of attention until} recent years. Since the earliest

Boltzmann

transport equation

which _was solved with the

help

of the old random

phase

approximation (RPA),

various

techniques

have been

investigated.

The direct resolution of the

density

matrix motion

equation by

means of

perturbation

series has

given

rise to the _so-

called method of

«

quantum

kinetic

equations

_» first

developed by

Kohn and

Luttinger [I],

then

by Argyres

and co-workers

[2, 3a, 3b],

who introduced

appropriate

refinements to

remove the

unsatisfactory repetition

^of the WA _at any time. The Kubo

theory

based _on the

calculation of correlation functions of current

density [4]

and combined with _an extensive _use of Green's

functions, diagrams

and

projection techniques [5-7],

_{represents an} altemative

approach which, surprisingly,

has known _a

widespread

use, in

spite

of its

highly

abstract and fornlal structure. The fundamental work of Mori

[8]

has

generated

_a third kind of

approach

known _as the _« memory function method _»

[9-12],

with much effort to _overcome several

difficulties

remaining

in the

previous

_ones.

The merit of the old Boltzmann

equation mainly

relies _on its remarkable and

meaningful simplicity.

At the lowest order of

scattering

and within the one-electron

approximation,

it is

thought

to

give

the

right

result known _as Drude's

fornlula,

and is often

regarded

as a test for

more elaborated

quantum-mechanical theories,

taken within the _same

validity

conditions.

This is the _reason

why

_a main _concem of

existing

theories is often to

clarify

their connection

with the Boltzmann

theory

result

[1, 13-16].

In

fact,

as stressed

by Argyres [2, 3], beyond

the lowest

order,

all methods

require

the

solution of _an

integral equation. Expressions

of

conductivity

derived from

projection

techniques

^which are claimed to avoid such _a

resolution,

_were shown to be unreliable because of the

remaining

low

frequency divergences. Proper handling

of these

divergences requires

the summation of _an infinite number of tennis

equivalent

to

solving

an

integral equation.

Similar

efforts,

in the memory function scheme, to

by-pass

the

integral equation

resolution

[9, 10]

_were also shown

by Argyres

to become incorrect at low

frequency

and inconsistent with

Drude's fornlula

[3].

Dramatic

improvements

ⁱⁿ the

quantum

transport

theory

_were

accomplished by

_van Hove

in _a famous series of papers

[17a-17e].

As is well

known,

a decisive step ^of van Hove's work

was to

recognize

_a

general

and characteristic

property

^{of collision matrices in} most

applications,

the so-called

3-singularities

in the continuous spectrum, ^I,e. in the ther-

modynamic

^{limit of} _a

large

system. ^This

mainly

stems from the

spatial

extension of the collision

perturbation

in

macroscopic

systems, ^{and leads} to

separate

out _an

overwhelmingly

predominant diagonal _part

in collision transition sets, at any order.

Among important

consequences, this

separation

entails the

preeminence

^of tennis in

(A ~t)~

in the

asymptotic

behaviour

(with

A

denoting

_a diniensionless

parameter

characteristic of the

strength

of the collision

potential),

which in tum entails the elimination of interference effects in the

density

matrix thus

avoiding

the

repeated

_use of WA. It also results in the _occurrence of _a

complete

set of

asymptotically perturbed

states

including self-energy

effects

[17b].

Van Hove's ideas _were next

injected

into various theoretical

approaches

^of transport

problems,

and

particularly

^those derived from Kubo's formalism

[5-7,

18-2

Ii. Recently,

Loss

[7a] pointed

out the interest of

dealing directly

^{with the}

density

matrix in the Liouville space, for

extracting

^from

perturbative expansions,

^the A

~t

_limit

intimately

connected with the 3-

singularities.

^He also

published

_an

application

of his method to the calculation of

conductivity,

in the lowest three orders

(A

_~, A

',

A of collisions.

The determinantal method

previously developed by

the author for

solving

the time-

dependent Schr6dinger equation

of the evolution

operator [22a]

and the

density

matrix

[22b],

is

particularly

well suited to find the

long-time

limit of the response, so

providing

us with _a

drastically

_{new means} for

obtaining

a

steady-state

solution in

quantum transport problems.

In the

present

paper the method will be

applied

to the electrical

conductivity calculation,

in which its

major advantages

^will be

clearly

exhibited. The resolution of an

integral equation

will be

avoided,

convergence will be ensured in any case, and _an

explicit

fornlulation of _a Drude-like relaxation

time,

suited to the

conductivity problem,

will be derived.

Furthernlore,

van Hove's

theory

will tum out to have _a determinant contribution in the derivation of the main

result,

and the connection with Drude's formula to lowest order will be discussed in

detail.

The paper will be

organized

_{so as} to outline the salient features of the

method, leaving

out of consideration irrelevant details such _as discrete

quantum

^{numbers like}

spin indices,

_or

anisotropy

effects.

Further,

since the low

frequency

limit _{was a}

highly

controversial

topic,

it

will be restricted to the d,c, _case, which needs

simpler

mathematics. Extension to the

harmonic _case, where _no

divergence

_occurs, will offer _no additional

difficulty

and _can be

rejected

in

subsequent publications.

To make the paper

self-containing,

the determinantal solution of the

density

matrix will be

bliefly

^{recalled in} section 2. The calculation of

conductivity

will be carried _out in section 3 and

given, first,

in _a

general

^form

capable

of

including many-body effects,

in order _to allow further

developments

and

applications.

The subsections

3.4,

3.5 will be _more

especially

concemed with the

important application

of most

important

results to the free electron _case,

leading

to a somewhat

improved

relaxation time

definition, carefully compared

with the standard

theory. Finally,

section 4 will be devoted to a

discussion of the

ability

of the method _{to overcome some} basic

difficulties,

_as

compared

^with

previous

_ones.

2. Determinantal solution of the

density

matrix

equation.

We shall consider _an electron system ⁱⁿ a

periodic

lattice

potential,

described

by

the

unperturbed

Hamiltonian

Ho,

the

eigenstates

of which _are denoted

by b,

_c, with

eigenvalues

_s~

=

hw~,

_s~

=

hw~,

Predominant collision processes are described

by

the

coupling

Hamiltonian

V,

and the

coupling

of electrons

(of charge e)

with _an extemal electric field of

magnitude E, applied along

the

x-direction,

is

represented by eEx,

^{in the} scalar

jauge.

^{The field} is assumed to be switched _on at time t ₌ 0.

Let _us

briefly

recall that the so-called _« determinantal _» method

[22c-22d]

aims at

finding

_a

trace

conserving

the solution to the

Schr6dinger equation

of the

many-body density

matrix

p

(t),

written in the fornl

dp/dt

=

(ih)~ glo

+ V ₊

eEx,

_p

i (p pa)

_s~.

(I)

JOURNAL DE PHYSIQUEI -T 2, N' 5, MAY IW2 25

pa = p

(0)

stands for the initial

equilibrium density

matrix.

Very

slow relaxation _processes which _{are not} included in V _can be

described,

in _a way introduced

by

Lax

[23], through

a

simple phenomenological

relaxation rate s~.

They involve,

in

particular,

the

coupling

with the heat-bath and _ensure the overall coherence of the

theory by allowing dissipation

in the

purely

elastic _case, while

improving

mathematical convergence. In the linear response

appxoximation

we will be restricted to,

hereafter,

_{s~ is} small

enough

for

having

_no

significant

effect _on the result and

thus,

will be

regarded

_{as a}

vanishingly

small

quantity.

Then the

Laplace-transformed density matrix, namely R(v)

_<

p(t),

satisfies the trans- forn1ed

equation

:

(v

+

sr)R(v) +I*-iiHo+

v ₊

eEx, R(v)i

=

po(i

+

e/v) (2)

which will be rewritten in the Liouville space

i~

_{= iJc~ @}

I(~,

^where _iJc~ ^{is the Hilbert} space

spanned by

the

unperturbed eigenstates

of

Ho,

and

I(~,

^{the dual} space

[24]. i~

is sustained

by

a tetradic

representation,

the basis of which is defined

by

all

couples

of

Ho eigenstates (b, c), (bi, ci), Operators

in

I,~

^such as

R(v)

become vectors, written

R(v)

in

i~,

with

components

R~~ =

<cb R(v)>

c

<cip(t)i b>

,

while superoperators ^such as the commutation kemel in

equation (2)

become operators, ^We shall write for any operator ^{Y of}

ix (or

lf in

i~)

lKvlf=ih~~~v,Y]; lK~lf=ih~~[eEx,Y] (3)

with the

following

matrix elements

(cb[ _lKv [ci bi)

= lKv)~ ~' =

ih~

~(V[~ 8)~ _{8j~ V~~)}

,

etc.

(4)

so that

ih-

ICI lv, Yi bl

= ~Kvl~

~ Y~~

b~ ⁼ I*-

(vl~ 8(~

_81~

V(~) Yll

=

ih- ⁱ

(vj~ Yj Yj v(') ₍₅₎

Summation _over

repeated

^{indices will be}

implicit throughout. Diagonal

matrix elements of V will be

ignored

without serious loss of

generality.

^As

previously pointed

out

by

Kohn and

Luttinger ii,

their

only

effect is _to shift the

unperturbed energies by

_a constant. Once

suitably completed by

a

chronological ordering

rule

[22d], expression (4) provides

_us with _a mechanical _means for

calculating

the

repeated

^action of

lKv.

^{Such tetradic}

representations

were

already

introduced in the Liouville space with somewhat different conventions and notation

[7a, 7b, 20].

On the

quantized

state basis of

i~, equation (2)

is

conveniently

rewritten _{as a} linear system :

~~

+ ~<b~

(I~V

^{i ~} +

I~E

^l _~~)

~~

hi ^~ l~^b~ PO

~(l

₊

El" )

,

(6)

where the

d~~'s

stand for the matrix elements of the

following diagonal superopeator

d

=

(v

₊

e~)

ⁱ +

~x~, (7)

denoting

the

identity

superoperator. ^{Then the solution of}

(6)

can be first written in Cramer's

fornl,

~mb R~i

=

fi _{po1, (8)}

expressed

in _tennis of the deternlinant D and the

ml (column)-cb (line)

minors of the matrix in the left-hand member of

(6).

In the

thermodynamic

limit

equation (6)

becomes _an

integral

Fredholm

equation

of the second

kind,

the solution of which is then

given by

the limit of

expression (8) [25].

As

previously

shown in references

[22c, 22d],

the solution

(8)

contains unlinked sets of transitions

(associated

with

unphysical

_sequences which do _not

properly

start from the _« initial state _»

cb)

which _can be eliminated in _a

systematic

_way,

through

the fornlal division of the upper and lower determinants of each fraction of the _sum in

(8), by

the

diagonal

minor

D[( pertaining

to the

starting

state cb

(see

Ref.

[22c])

~m

D7i (Dl()~

~

~fl ~

(~cb)-

^{~0 b}

⁽⁹⁾

cb cb

Note that the form

(9)

cannot

change

the value of the

rigorous

solution

(8),

but

only

the

approximate expressions

of the numerator and the denominator which will be retained in

practice. Indeed,

the deternlinant

quotients arising

in

(9)

can be

given

_a more

explicit

form in tennis of

geometric

series

[22a],

_e,g.,

Dft (D[()~

=

(ml (I

+

Q~~

^{d~ lK}

)~ cb)

,

for

ml

_{~ cb. The occurrence of the}

complementary projector Q~~

^is

simply

connected with

the exclusion of the initial _{state row} cb in the definition of the minor

Dft,

and the

subsequent

division

by

the minor

D[(. Separating

out the

particular

ternl

ml

=

cb, expression (9)

then becomes

~~

8ft (ml

lK

(I

+

Q~~

d~ lK

)~ cb)

~ ~~

d,~

+

(cb[

^lK(1 + 4ll~~ d~ lK

)~ [cb)

^{~° ~'}

with lK

=

lKv

+

lK~,

and

ml

_{# cb in the upper}

angular

bracket.

In transport

theory, expression (10)

is recast into _{a more} suited fornl

[22d], including

_some

simplifications, namely

the

neglect

of natural transition widths

(tennis containing

_lK~ in the

denominator)

with

regard

to collision widths

(tennis

in

lKv alone),

and linearisation. This amounts to

drop

the field kemel lK~ ^{in lK}

everywhere

except ^{in the} first-order-in-E ternl in the

numerator of

(10).

The

leading

result is

given by equations (23)

and

(27)

of reference

[22d] (1),

which will be rewritten in the

following

_compact fornl

nmb (nZb)-

^I

pi

=

~~

~~

^{ih~ eE}

(z[ [x, po][b) (ll)

d~j,

D(D(b)~

The

long-time

limit

v -

0+

(with

_{s~ -}

0)

will

always

be

implied

and assumed to be taken after the

thernlodynamic

limit. On

carrying

out the deternlinant

divisions,

_one obtains _:

pi

=

~~ ~~ ~"'~ ~~~

~ _{~~ ~}

~~~~

~~ ^~~~~

^ih~ eE

(z[ [x, po] [b) (12) d-j,

₊

(zb

IK

(I

₊

Qz~

^d- IK

)~ (zb)

(I)

Due to _{an error} of

sign,

the ₊

sign

must be

changed

in in the numerator of

equation (20)

of

reference [22d], which entails change of

sign

in some subsequent

equations

derived therefrom.

The kemel IK _now stands for

IKv

^{in which the redundant}

subscript

V is

dropped.

The

particular

ternl

ml

= zb for which the numerator of the fraction reduces to I

(see Eq. (

II

))

is

again separated

out

and, therefore,

the index restriction

ml

_# zb

imposed by

the

projector Qz~

holds in the

angular

bracket

of

the _numerator.

It is worth

recalling

too, ^{for further}

discussion,

that if _we carry ^out the division of the minor

by

the deternlinant in

(8),

instead of

keeping

the initial fractional fornl _as done in the above

fornlulae,

_we obtain

R~ I ⁼

Dl't

D~ ^I po

[Id,

~ ⁼

(ml _(I

₊ d- IK

)~ cb)

_po

_[/d~~ (13)

leading strictly

to the

perturbation

series of the

density matrix,

_upon

expansion.

Of _course the

same result could be obtained from

expression (10)

_as

well, showing

that the division of the

numerator

by

the denominator in the deternlinantal fornlulae would

just

have the effect of

leaving

the _numerator alone without the index restriction

imposed by Q~~ [or Qz~

ⁱⁿ

(12)]

therein.

3. Calculation of the electrical

conductivity.

We _now

proceed

to the calculation of the d.c.

conductivity

from the statistical ensemble average 3 of the _current

density,

a

=

Tr

(pi), (14)

where J denotes the current

density

_operator.

Making

_use of _our basic result

(12), equation (14) gives

3~ ieJ~ 8)

8

f

_dji~

ill

lK

(I

+

Qz~

d~ lK

)~ zb)

' E h

d-~

+

(zb

IK

(I

+

Qz~

^{d~ IK}

)~ zb)

^{~~ ~~'}

^{~°~ ~~}

^~~~~

with11

_{~ zb} in the

angular

bracket of the _numerator.

Anisotropic

effects which would add

nothing

new to the

physical

^{features of the}

theory

_are

ignored

and _so «~ is

simply

denoted

by

«.

Here, b,

_{c, z,}

,

I,

m, represent

many-body

states which _can be constructed _out of individual Bloch-states

specified,

_as usual,

by

_a band index _n, the electron momentum k and _a

spin

index. As

already mentioned,

the

spin plays

no

particular

role in the present

study

and will be

ignored

for

simplicity.

We shall also limit ourselves _to the one-band

approximation

(interband

transitions _more

specifically

relate to the

optical

_response which

requires

_a

separate study).

For later _{use, we} shall write out the current

operator

^{J in} tennis of

single- particle velocity

_v

=

p/m

in the band under

consideration,

which is

diagonal

in the Bloch

representation.

In _a second

quantization scheme,

_we have _:

~~

~~i~k~k

^~ _~k' ~~~~

k

~

where flJ denotes the volume of the system ^and

cl,

_{c~ the customary} creation and annihilation

operators.

^The commutator

[x, pal

in

(15)

will also be

given

a second

quantization

fornl in the n, k

representation

IX,

p01

_~

i

IX,

p0fi~'C$k'

Cnk

(17)

n'k'nk

which,

in tum,

requires

the

knowledge

of the related

one-particle

matrix elements, In _our intraband

approximation,

the latter reduce to

[x, po]~'

= i

(V~,

+ V~

)(nk'[po[ nk)

,

(18)

as was

already

mentioned

by

Chester and

Tellung [18].

The demonstration is

briefly

recalled in

Appendix I,

and detailed discussion _can be found in the review paper

by

Blount

[26].

Notice the limit of

equation (18)

when k'- k _:

[x, po]$

= I V~

(nk[po[ nk) (19)

So far the result

(15)

is rather formal. To set up a more tractable

expression

_we rust have _to

remove

divergences occurring therein,

and _to work out _an

explicit

form of the matrix

elements of the commutator

[x, po].

3.I REMOVAL oF DIVERGENCES.- We rust

recognize

in

(15)

the well known

divergent

behaviour due to the factor

dji~ _becoming

_very

_large

in the

long-time

limit. This

primarily

occurs because the

physical quantity,

the statistical _average of which _we have to

calculate,

is

diagonal

in the

unperturbed representation.

A second _reason is that _{we are} in d,c,

regime.

At finite

frequency

_{w, v} would be

replaced by

_v + iw in

dir

and _{so, no}

divergence

would _occur.

However,

the fraction in

(11) (or equivalently

in

(12))

is

necessarily

finite in any case,

irrespective

to the average

quantity

which is to be

calculated,

_or the

frequency

_range of interest. It follows hat this

divergence

should be lifted

by

_means of

L'Hospital's

rule before

going

to the

thermodynamic limit,

after

having singled

out a similar behaviour in

dji'

in the denominator. In

Appendix

II this is done in _a

systematic

_way for both the

dj~'

and

dji' divergences, leading

to

expression (II.4),

written in _a closed form thanks to the

following

definition of the relaxation parameters

T~i

=

iii

IK

(I

+

Qz~ Qii

d~ IK

)~ (zb)

=

iii ~ z

^IK

(- Qz~ Qii

d~

IK)" zb) (20)

i

Using (II.4),

the

conductivity (15)

can

finally

be rewritten in the fornl

j ^b

~b

_j ^f

~fb

" ~

~i fi

(d~ l~r=/)

_~fifi ^~~'

^~°~~

^~~~~

in which the

dfi~ divergences

_{are now} removed.

In the calculation of _« from

(21)

_we shall thus have to deal with the matrix elements of the

T's,

I,e.

expansions

of the fornl

(20).

Let _us write out for definiteness the

expansion

of the ternl in

J~((

to lowest

(second)

order

J,( _{J~~/[x, po][}

=

(ih~ ~)~J~i(V$~ 8(~

8$~

V(~)dj _(~(V)~ 8( _8)~ V(~)[x, po][

=

=

-2j~i( ~'i

_~~~i~~

~°i~

~j vi _{ix, Poi~} vi vi ix,joi~

vi

+

~i

^i~~

°(~ ~'i ~"

dci

_b

dd

₁₆

-1>1

The

general

ternl of any order involves

products exemplified by

V~

V" V~

Vz[x, pot VI ^V~' V~.. V~ Vildqj, d,,n, (22)

The

d~,,,

s,

d,jj,

s... represent ^the energy denominators taken between internlediate states

appearing

somewhere in the sequence.

Unsignificant

indices _are omitted.

To

proceed further,

we have _to take _more

realistically

into _account the

properties

of _sets of collision operators, ^such as those

arising

in

(22)

_or in the

expression

of pa matrix elements below

(see (29)).

As

already mentioned,

this _was

previously

done

by

_van Hove

[17a-17e]

through

the introduction of the characteristic

3-singularity,

valid in the

thernlodynamic

^limit

where the relevant

quantum

^{numbers become}

quasi-continuous variables,

in all _cases of

interest such _as collisions with random scatterers, interaction with

quantized fields,

etc. This fundamental

property

^{holds for}

products

of matrix elements of the

type (22),

_as well.

Here,

the denominators d~,,~ will

only impose

_some condition on the

energies

of certain

couples

of

states, upon

integration,

without

affecting

the essential of Van Hove's arguments

leading

to the

predominence

of

diagonal

contributions

[17b,17d].

In _{a more}

explicit fornl,

the 3-

singularities

entail that _among all

possible

_groups of successive V

operators

^{of the kind shown} in

(22)

and

subgroups therein,

those which _are

diagonal

_are much

larger

than the

off-diagonal

ones

(by

_a factor of the order of the number of

particles

in the

system).

It follows that in

subsequent

summations any

diagonal

set will be at least of the _same order _as the _sum of the

off-diagonal

_ones. For

example,

the matrix elements of the

product

V...

A~VAI V,

where A i,

A~,

are

diagonal,

_can be

split

_as follows into _a

singular

part

involving

_a 3-function and _a

regular part

^{which does} not

(b'(V. _A~ VAT V[ b)

=

F~(b)

⁸

(b'- b)

₊

F~(b', b). (23)

Upon integration,

this becomes

ldb'(b'[V. ^{A~ VAT Vi b)}

=

F~(b)

+

db'f~ (b', b),

where the first _{ternl on} the

right-hand

site is at least of the _same order _as the second _one.

Now,

due to the very

3-singularity property,

a new kind of

divergence

arises every time _a denominator like

d~~,

ⁱⁿ

(22), vanishes,

I,e, if _n

= m.

Indeed,

in such _{a case} the

3-singularity

of the related _mm

diagonal

subset

represents

a very

large (macroscopic)

number of tennis with the _same

vanishing

denominator

d~~

in the

long-time

limit. In

fact,

these

divergent

tennis tum out to cancel. Let _us consider the

general (n

+

I)

th order ternl of fornl

(22)

in

expansion (20).

On

starting

from the initial zb element and

going

to the last _one,

it,

the _occurrence of the successive

divergent diagonal

sets _can be

pictured by writing

_{a segment up} to _some qq

state, in the

following simplified

form

(see Appendix III)

~ ~- ^l j~qp ~ ^l

j~pa'~a

qq~ qq Pq PP ap' a',

where

X$,

represents the matrix elements of the undetailed

subproduct

in the

right-hand side, including

the

starting

element

[x, pa[ (the

Q's which

simply impose

that qq,

pp'#

zb _are

omitted).

It is shown in

Appendix

m that S~~ ^{cancels term}

by

term with S~~, except ^when qq

represents

^the final

it

_{state of the}

(n

₊ I

)

th order set under

consideration,

since then the

I- dependent

^factor

J(

comes into

play.

The related

dji' divergence, however,

has

already

been

lifted

by

_means of

L'Hospital's rule, leading

to

equation (21).

It may be concluded that the

general

term of fornl

(22)

in

expansion (20)

_{represents a} well defined finite

quantity

: the

singular part

^of its

diagonal

subsets associated with

divergent

denominators will be

eliminated,

whilst in the

thermodynamic

limit the

regular

_part cannot entail

divergence

but

only

a

Cauchy-type integration [25],

which is handled

by

_means of the well-known rule

- w8

(w~~, )

^I

,

(24)

f

+

I(1~

₊

w~~,)

Wqq,

P

where _v

=

f +17~

(7~ m

0),

the limit

being

taken for

f,

7~ - 0. The

subscript

P indicates the

principal

value.

3.2 MATRIX _ELEMENTS oF THE COMMUTATOR

[x, po].

The

equilibrium density

matrix elements po

[

will be written in the usual

grand

canonical scheme

po = exp

i- p (Ho

₊ V _qn

)i/Tr

_exp

i- p (Ho

₊ V _qn

)1 (25)

where

p

=

I/kB

T is the inverse of the thermal energy. The Fermi energy ~ is fixed whereas the number of

panicles

is not, this is _more convenient because all electron dishibutions _on the

available states _are then allowed without any restriction. In the second

quantization

representation,

Ho

_~n

=

£ (e~

_~

) c(

_c~

(26)

k

The calculation of matrix elements of po between the

eigenstates

of

Ho

will be facilitated if

we first recast

(25)

in _{a more} convenient form. To this end _use will be made of the

exponential

of the _sum of _non

commuting

_operators

given

in Bourbaki

[27] (see Appendix IV), yielding

exp

i- p (Ho

_qn + V

)i

_{= exp}

ip (Ho

_qn

)i

_x

lm ^L~

^V

x exp

p (Ho

_l~n

)

₊

z (- p

_)~ ⁺ _{j exp}

i- p (Ho

_l~n

)1

,

(27)

n ° ~'

where L stands for the customary ^Liouville

operator

defined

by Lx~

^V = @Io,

Vi.

We shall

simply

denote

by

S the first two

exponential

factors in

(27), describing

the

perturbative

effect of collisions _on the statistics S

= exp

ip (Ho

_l~n

)i

exp

i- p (Ho

_qn

) p

_exp

(- pLxo) Vi (28)

The

equilibrium density

matrix _{can now} be rewritten iri the form S _exp

i- p (Ho

_l~n

)1

~° ~~

Tr

(S expi- p (Ho

_l~n

)1)

^'

where the

unperturbed part

^is well

separated

out. The derivation of the second

quantization expression

of the

unperturbed

Boltzmann factor from

(26)

^is

straightforward

^and leads to the usual

expression

of po, in the absence of collision

(S =1) [28]

exp

p £ (Ho ~n) c(

_c~

po(V

~

=

0)

=

Tr exp

p £ (Ho

~n

) c(

_c~

~

~

fl

^~~k

^~~

^{+ ~~P ~~}

^~

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

=

fl a~ (30)

~

l ₊ exp

[- p (8k

_l~

)I

k

where

&~

=

ia(e~)

denotes the

occupation _operator

~°k "

(1 fk )

_{CkC~ +}

fk C(

_Ck

(31)

in terms of the

single-state

Fermi function

f~

=

I/(exp §3 (e~

_~

)]

₊ l

).

In

(30)

use has been

made of the property ^that

Tr

leXp(-P £(H0~$~n)C( kjj

~

fl (I +eXp[-P(8k~$~)1).

k k

Now,

the collision factor S

splits,

_upon

expansion,

into sequences of transitions

generated by increasing

_powers ^of the

arguments

in

(28),

which will

again

exhibit

singular diagonal

parts. ^Due to the

property (23),

these contribute _to the _same order of all

nondiagonal

ones.

But,

_{as was} discussed in detail

by

_van Hove

[17d], nondiagonal

elements of initial _states entail interference effects which cancel out for _an

overwhelming majority

of _cases and _so,

they

_can be

ignored

without loss of accuracy. In another way, this amounts to _assume the RPA

restricted _to the initial states of the

equilibrium density

matrix po.

The effect of collisions will thus be described

by

_means of the

diagonal

_part of S, the

expansion

of which in lowest order is calculated in

Appendix

IV

S~

₌ I +

£

_S~

(k'k) cl

_c~,

cl,

_c~

(32)

k'k

with

S~(k' k)

~~P(- _p

_e~,

~

)

i ~

El,

~

~

$ IVl'

^~

~'k

(33)

(e~,~

_{= e~,-}

e~).

From

(17), (19)

and the above

expressions

from

(29)

to

(33),

the

commutator matrix elements of interest in

(21)

will in _tum reduce to

diagonal

_ones of the

form

[x, po]

= I

£ c(

_c~ V~

fl

_&~, +

O( IV ~), ₍₃₄₎

k

~

k'

where the

leading

term on the

right-hand

side is of zeroth order in collisions. The relative contribution of the correction in

IV

^~ associated with the S factor

(32-33)

will be first

ignored

in what

follows,

and discussed later _on, in Section

3.5,

in connection with the Peierls criterion.

3.3 RELAXATiON PARAMETERS. The f~ matrix elements _as

given by (20)

_{represent many-}

body

relaxation

parameters

associated with V

collisions,

in _a

general

and

rigorous form,

where all electrons _are assumed to be concemed at _once in collision processes. In lowest

order,

I.e. the second _one since V has _no

diagonal elements, J~((

for instance is written _as

J~ll

₌

ibb11~Q~~

^{d- 1~}

bb)

₊

= h-

~(vt, al'- at, VI') di [, (vi' 3t, al'vi, )

₊

V$ Vi _{V~ V~'}

=

h~~

^' ₊ ^' ₊

(35)

Iic,b Iibb,

with ci

bi,

# bb.

Obviously, expression (35)

is real. More

generally,

_any

J~((

^defined in

(20)

is _a real

quantity.

This

simply

results from the

conjugation

relations

(j~C,b2~~ ~b,C2

_{~~ ~}

c2b, b~c, > cj hi hic,

(which straightforwardly

derive from the definitions

(3)

and

(7))

when

applied

to each matrix element of d~

~li

_iri

expansion (20),

at any order.

Assuming

a

single-particle

^collision

potential,

and

using

the second

quantization expression

V

=

£

_V~~

cl

_c~,

(35)

takes _on the _more detailed form

k~k,

' ~ '

f~bb

~

£ ^{fi-2 vk, vkJ}

~ ~.

(36)

bb k~ k, ~ ~

~~~, I,I, k,

k~

where the

k;

and ky summations _{now run over} all

occupied

dud

empty single-states

in

b,

respectively.

3.4 CONDUCTIVITY AND RELAXATiON TIME. The

foregoing expressions

of the

[x, po]

and

J~'s matrix elements enable _one to calculate the

conductivity

from

(21). Assuming

the WA of initial states as discussed

above,

and

substituting J~

=

euJiJ,

we have

i~2

~ ~+[j

jx, p~j(

"

~

^{~Xb ~Xf}

@

_J~hh ^~~~~

ii _fib

When

only

_a few electrons _are correlated to one

another,

the form

(35)

_or

(36)

of the J~'scan be fiirther

simplified.

To work _{out an} illustrative test, ^and for the sake of

comparison

with the standard

theory, below,

_we shall _assume

completely independent

and

isotropic

electrons in what follows.

Then,

V

coupling only

_occurs between

many-body

states which differ from _one another

by

_a

single-state occupation,

all other states

being regarded

_as

quenched

at

equilibrium.

The set of b states _can then be

approximately split

into

uncoupled

subsets of this kind and their related contribution to the determinants D and

Df(

in

(11)

_are factorized into

independent

blocks which

separate

_upon

simplification,

the

D((

minor

being

next redefined

accordingly

in each block. All of the above determinantal

procedures

remain in each

subspace.

In

fact, only

blocks in which

occupations

_are close _to statistical

equilibrium

are relevant.

Doing

_so,

J~()

becomes _a

unique

function of the

single

state k under

consideration,

which _we shall

simply

denote

by

r

(k),

and

J~()

a function r

(k', k)

^{of the}

single

states k' and k. Of course

dynamical exchange

effects _are somewhat

oversimplified by

^this

way.

Assuming

for

simplicity

a

weakly occupied

band the intermediate

occupation

factors _can be

ignored,

_so

that,

_once reduced to one-electron

transitions, expression (36)

becomes in the

two lowest orders

l~(k)

=

2 Re

h~

2

~~'

~~~

ih-3 ^{~~2 ~~~}

~~'

&,~~

d,

,

d,

~

~

,

(k2, ki

# ^k

). (38)

Similarly,

_one would find from

(20) r(k', k)

=

~ ~~

fi-2 ^{~~' ~~'}

~ ~- ³

~k' vk, _~k

~ l

~ l

~ ~~~~

lij k

~' ~ ~'

~iI I

I~II'

I~I _I I~I _k

Iii

j

Iii

k

Notice that in the

integrations

_over intermediate states

arising

in

expansions (38)

_or

(39),

the energy denominators

obviously

_{may pass}

through

zero in the

integration

_range. From the discussion of section 3.I it follows that the

divergent

terms due to the

3-singularities

of the related

diagonal

subsets cancel and _can thus be

ignored.

This _means that

only

^the

regular

parts

^{of these}

diagonal

subsets _are to be

considered, leading

to a finite result

through

Cauchy's type integration (24).

Of _course, all

diagonal

subsets _are not

necessarily

associated

witli

vanishing denominators,

but _{one can} be _sure that _no furtlier

divergence

of tliis kind may

occur. In all _cases

using

_van Hove's

techniques

_may

help

in

finding

the

leading contributions,

such _as saw-tooth _or

symmetrical

transition sets, as was done in

previous

similar calculations

[7, 18, 19].

In the

grand

canonical

scheme,

the summation _over b in

(37)

_{runs over} all

possible

^electron

occupations

of individual states.

Any given

term in the

expansion (39)

contributes the

conductivity (37) through

_a definite set of electron states, _e.g.

k, ki,

k' for the above third order _term. It is

readily

seen that all other electron states which _are not involved will be washed out. Let indeed b' denotes such _a restricted

many-body

state

(I.e. excluding k, ki,

k' in the

preceding example).

From

(34),

their contribution to

(37)

will be

clearly

factorized in the form

I ~l (1 fk~ ) fl fk,

~

ll (1 fk"

+

fk" )

~

l

(40)

b' k~ k, k"Eb'

Next,

introducing

in

(37)

the

single

electron velocities from

(16),

and

making

_use of

(34) (the

effect of collisions in the commutator

[x, po]

is

postponed

for later

discussion),

_we obtain

"£j[~jtiiii~j~(£~

~~~~

This

expression immediately _suggests

the

following

definition of _an effective

conductivity

relaxation time

by

~~~~ _-

&

Ii 16 _[[,j~ l~~~~

~~~~~~~

~~

~ =

~~

£ T(k) V~(Vk/k)

~

~~

l~ ^~~~~ ~~ ^~~~

~~~~

fi%

~

~

k

~~~

which has the form of the result

given by

the standard transport

theory,

For the sake of _{a more} detailed

comparison

with

previous theories,

_{we are}

going

to work out an

explicit expression

of

r~k),

in second

order,

in _our

isotropic

one-electron

model, assuming purely

elastic collisions. Then v~ and k are

parallel

vectors which _can be

specified by

their

magnitude

and their

angular polar coordinates, 0,

_p, with

respect

to any

given

direction.

From

(38),

we first have

q~

(V~~(~d~ki

r(k)

= 2 Re

~ ~

(v

_-

0+) (2 w)

^h

(v

₊

iw~ ~)

q~

(V~'(~k)dki

~

~2

_fi2 dw

~

~ ~~°k>k~ ~~°k> ^~~~

~l ~~l ~~

l

0

~, pi are the

angular

coordinate of

ki

with respect to k, ^and use has been made of

(24)

in the limit _v -0+ The 3-function

imposes ki

=

k,

and the

integration

_{over k~} is carried out

through

the introduction of the

density

of states in the

band, p(w~)

=

~lJk~dk/2 w~dw~,

leading

to

r(k)

= wh~ ^~ p

(w~,) (V~'(~

^sin 0~

d01. (44)

r is _a function of the

only magnitude

of k.

Similarly,

the second-order term in

(39)

is

easily

transformed into

r~k~, k)

₌ 2 Re

vl'(

~/h2

d;

~ ⁼ 2 grin- ²

vl'(

^~ a

(w~,

~

(45)

The calculation of _r

(k)

^from

(42), (44)

and

(45)

is

straightforward.

We _use the

subscript

E when the

angular

coordinate _are referred _to the field direction

(x-axis)

and denote

by 0', p'

the

angular

coordinates of k' with

respect

to the k direction

(Fig. 1). (42) gives

q~ u~, cos

0)

2 _w

Vl'

^~

_k,2

_dk,

~~~ ~ ~~ ~

(2

_W)~ Vk ^CDS

0E

fi~

l~(k')

^dw _k'

~

~~~'~

~~

~' ~~ ~' ~~'

~'~

k

o

Fig.

I.

-Angular

coordinates of the electron momentum k, ^{k' witli} respect to each otlier (9') and to tile direction of the

applied

field

E(e~, 9i).

Energy

conservation

imposes

_v~,

= v~ ^{and the}

angular

coordinates _are linked

by

the well- known

trigonometric

relation

cos

0)

₌ sin

0)

sin 0 _cos

(p~ p')

_{+ cos}

0~

cos 0'

Upon

substitution into the above

equation,

the term in _cos

(p~ p')

vanishes

through

integration. Making

_use

again

of the

density

^of states p

(w~, ),

and

taking

into _account that

r(k')

=

r(k)

if k'

=

k,

_{we are} left with

l~

r

(k )

= r

~(k )

I ₊ wh~ ^~ r~

~(k ) VI'

^~ _p

(w

~,

)

_cos 0' sin 0' do'

(46)

o

This

expression

does _not

markedly

differ from the usual _one. Since the ratio

£cos0'r(k',k)/r(k)

is

noticeably

smaller than I because of the

integration

in

k'

sin 0'cos

0',

one can rewrite

(41) approximately

_as

« >

£ _z

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