Zero modes and commensurate-incommensurate transitions

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Zero modes and commensurate-incommensurate transitions

C. Mantea, A. Corciovei, A. Stepanescu, A. Masoero

To cite this version:

C. Mantea, A. Corciovei, A. Stepanescu, A. Masoero. Zero modes and commensurate-incommensurate

transitions. Journal de Physique I, EDP Sciences, 1992, 2 (1), pp.23-29. �10.1051/jp1:1992121�. �jpa-

00246459�

Classification

Physics

Abstracts

64.60 64.70R

Zero modes and commensurate-incommensurate transitions

C. Mantea

(I),

A. Corciovei

(2),

A.

Stepanescu (3)

and A. Masoero

(4)

(1) Research Institute for Electrical

Engineering

(ICPE), Bucharest, Bd. T. Vladimirescu, 45-47, Romania

(2)

Department

of Fundamental

Physics,

Institute for

Physics

and Nuclear

Engineering, Mhgurele,

Bucharest, MG-6, Romania

(3)

Dipartamento

di Fisica and I.N.F.M.-C.I.S.M., Politecnico di Torino, C. _so Duca

degli

Abruzzi 24, lo129 Torino, Italy

(4) Istituto Elettrotecnico Nazionale Galileo FerJaris and I.N.F.M.-C.I.S.M., C. _so Massimo

D'Azeglio

42, 10125 Torino,

Italy

(Received 22

April

1991, revised 19

August

1991,

accepted

23

September

1991)

AbstracL There _are two different results conceming the functional

relationship

between the chemical

potential

of the lD massive

ThirJing

model and the misfit parameter ^{of the lD} quantum

sine-Gordon model which have been

reported

[4-7]. ^In order to

investigate

this

problem,

the Jordan boson

representation

of fermion field operators ^{is used to obtain the bosonized}

expression

of the ID massive Thimng model. The

equivalence

^{of this} model with the ID quantum ^sine- Gordon model is

proved,

all zero-mode contributions taken into account. The

problem

of

controversial results

previously

reported is solved. Thereafter, by means of Bethe-Ansatz method, _{a new}

expression

for the critical _curve of the commensurate-incommensurate transition in the two-dimensional sine-Gordon model is found.

1. Introduction.

These last years, the commensurate-incommensurate

(C-IC)

transitions have been

thoroughly

studied both

theoretically

and

experimentally.

Such transitions from _a commensurate state,

compared

_{to an} extemal

periodic

structure, ^into an incommensurate _{one, are} manifest in _a

series of two-dimensional

materials,

out of which _we

mention,

as

typical examples

the

monoatomic

layers

adsorbed _on

graphite-substrates

and the

type

^II

superconducting

thin

films.

By applying

an extemal

magnetic

field normal _{to a} thin

superconducting film,

_a

triangular

vortex lattice is

generated

into it. Within this lattice there appears a

competition

between the vortex-vortex

interaction,

that tends _to maintain the vortices in the lattice

sites,

and the interaction with the substrate that tends to modulate the _vortex lattice

according

to its

own

periodicity.

Much attention has been

especially payed

to sine-Gordon

systems

^{which exhibit the} essence

of the C-IC transition. The

thermodynamic properties

of _a two-dimensional

(2D)

classical

24 JOURNAL DE PHYSIQUE I N°

sine-Gordon model

(SGM)

_may be obtained

by investigating

a one-dimensional

(lD)

quantum ^SGM

[1, 2].

The latter model is

commonly mapped

onto the lD massive

Thirring

model

(MTM) [3-7]

and

powerful

methods elaborated for the last model _are then

employed.

There _are two different results

conceming

the functional

relationship

between the chemical

potential

_p of the lD MTM and the misfit parameter

I

_{of the lD}

quantum ^SGM :

p =

~

i _(I.1)

2v~

reported by

Luther _et al.

[5]

and Yamamoto

[7]

and

p =

~ _"

i _(1.2)

VF g

reported by

Schultz

[4]

and Bohr et al.

[6].

It _seems that the

difficulty

and the reason for these different results lie in the

mapping

of the

lD quantum ^SGM onto the ID MTM.

Essentially

this

mapping

has been

performed by

means

of Luther-Peschel-Mattis

(LPM)

boson

representation

of fermion fields

[8, 9]

and it is known that this boson

representation

is

plagued

with inconsistencies

[10].

A

major

lack of LPM boson

representation

is the contribution of the zero-modes associated with the

particle

number operators. ^{These zero-modes} were

consistently

taken into account

by

Haldane

[I1, 12].

The Haldane boson

representation [12]

looks very much the _{same as} that encountered in the field-theoretical literature

[13] and,

in

fact,

it _was derived many years ago

by

Jordan

[14-17]

for _a

single

fermion field in _one dimension in his attempt to construct a

neutrinic

theory

of

light.

Haldane

[I1, 12]

used _a normal ordered boson

representation

_so that there is _no need for _a cut-off

parameter

_tx in the bosonized

expression

of fermion fields.

However,

products

of _{two or more} fermion fields _{are to} be calculated and the normal

ordering problem

arises

again.

In order to make the summations _over wavevectors

appearing

in

problems

of this type ^{finite Haldane}

[12] pointed

out _a cut-off

procedure

which is

essentially

the _{same as} that

given by

^{Luther and} Peschel

[8].

Moreover,

there is _a

quantity pointed

out

by

Jordan

[14-17]

and which will be referred _{to as} Jordan's commutator, which has been overlooked _so far

by

all these boson

representations,

Haldane's included. This Jordan's commutator

plays

^the part ^of an additional condition which must be satisfied

by

the boson

representation.

Its

importance

is

directly

connected to the

renormalisation of the

infinitely large density

^of

particles.

A

slight

modification of Haldane's cut-off

procedure [12],

_ensures the _correct

reproduction

of Jordan's commutator.

All the inconsistencies

generated by

_{a wrong} cut-off

procedure

and

by neglecting

_zero-

mode terms have been

recently

removed and _a consistent boson

representation

has been

worked _out

[10, 18, 19].

The purpose of this paper is to

apply

the _new boson

representation

to decide the controversial results mentioned above.

In order to

study

^{the C-IC} transitions in such systems, ^the two-dimensional classical sine- Gordon model is used. The model Hamiltonian may ^be

ivritten [4, 7]

_as

H=

£ (J~[h(I

+

I, j)-h(I, j)+ 3]~+J~[h(I, j

+

I)-h(I, j)]~-2Vcos _{[2 grh(I,} j)])

(,.J)

(1.3)

where

(I, j)

denotes the lattice

sites, J~

and

Jy

are the elastic _constants of the considered monoatomic

layer

and V is the

strength

of the substrate

potential.

The misfit

parameter

^{b is} the difference between the natural distance among the adsorbed _atoms

(in

absence of the substrate

potential)

and the

period

of this substrate. The

thermodynamics

of this 2D model

(with J~»J~ WV)

can be

investigated [1, 2] by studying

the

following

one-dimensional quantum sine-Gordon model

~ a~ ^{2 2} au

H~=- jdx(1I(x)+ (-+3) _{-jcos(gp)j (1.4)}

2 ax g

where

I

= 2

gr3/(gc),

_au

=

(2 gr)~

VI

(J~ c~), g~

₌ ²

gr~T/ fi.

_{T and}

c are the

tempera-

ture and the lattice constant of the 2D classical system

(1.3), respectively.

In the paper, ^at rust the Jordan boson

representation

of fermion field

operators

^{is reviewed}

(Sect. 2). Then,

this

representation

^{is used} to obtain the bosonized

expression

of the ID

massive

Thimng model,

and the

equivalence

of this model with the ID

quantum

sine-Gordon model

(1.4)

is

proved (Sect. 3). Finally,

the quantum sine-Gordon model

(1.3)

is

investigated,

and _some conclusions

regarding

the

problem

studied in the

work,

_are formulated

(Sect. 4).

2. Jordan Boson

representation.

We suppose that the fermion field

operators

1l~~

(x)

₌ L

~~~£

^e'P~ ay

~p )

~

(2.I)

laj~P), al ~P')1

₌

3j.j _3p,p laj~P)> aj,~P')I

=

°

correspond

to a ID two-fermion system ^with

periodic boundary

conditions _{on a} segment ^of

length

L. The index

j

=

1,

2

points

to the fermion

type.

^{The kinetic} Hamiltonian of this ID

system

^is

supposed

to be

Ho

= VF ~X~

($'/ (X) fi$fi(X) $f~ (X) fi$f~(X)) (2.2)

where v~ is the Ferrni

velocity, N,

the norrnal

ordering

operator ^{of the ferrnion field} operators and

fl

=

I alax. The Jordan boson

representation

of the ferrnion field operators ^is

given by

the relation

[10]

~~

(x)

=

q

^{L ~'~}_Sy ^±

e*'~

^"~

^~~'~

^~'~~ _e~

^~/

~~~

e~1"~ _(2.3)

where

Sy

a~<~p) S[

=

3j<j

_~J

(P

^~

~F

~ ~~ ~~'~~

"~~ ~~~~

Bi

⁼ ~*°

^{Z al ~P) ai~P)}

~<°

^{Z ai~P) al ~P)} (2.5)

~2" I ~(~i')~2~i')~ i ~2~l')a(~i')

p«o p»o

Dj(x)

=

~£l £

^'

^{e*'~ «~(+ k). (2.6)}

k,o

The upper

(lower) signs correspond

to

j

₌

(2). Q-numbers

_cy

satisfy

the conditions

[10]

c~

c(

=

cl

c~ ⁼ I

,

(cy, c)

=

(cy,

_c~,) = 0

j

#

j' (2.7)

26 JOURNAL DE PHYSIQUE I N°

The operators

cr~(± ^k)

^from

^(2.6),

^defined

^by

^{the relations}

"i(- k)

=

Eat ~P) ai~P

₊

k), «i(+ k)

=

«/(- k)

~

~ ~

k~0

(2.8)

"2(- k)

=

£

_{a2 ~P} +

k) a2~P), "2(- k)

_{= «2}

(+ k) satisfy

boson-like commutation relations

1"j(S

k

), ml (T k')1

=

)

3~,j 3k<_k

,

i"j (+

k

), «j,(T

k'

)1

= 0

(2.9)

With

(2.1), (2.5)

and

(2.8)

the

following

relation is obtained

"j(x)

=

N[~/ (x) ~~(x)]

₌

)

B~ ⁺

£ _~fj(x)

+

F/ (x)]

~ ~

(2.10) F~(x)

₌

_~ £ ^e~'~ cr~(±k).

k,o

The operator ^{N of norrnal}

ordering

of the ferrnion field operators ^{is not efficient when} a

boson

representation

is used. In order _to obtain

(2.10)

and similar

meaningful

finite results,

owing

to the infinite

filling

of the

non-interacting ground

state at zero temperature, ^{it is}

necessary ^to use Jordan boson

representation (2.3) together

with _a modified Jordan

prescription

for

introducing

the cut-off

parameter

a,

~'/ (x)

_ir~

(y)

=

lim

j

_iir~

(x

_T I

a/2) i~

_$i~~y I la

/2)

(01 i~j (x

_± I

a/2)j+

_{~~ ~y ±}

_«/2) oj j (2. i1)

where

[0)

is the

ground

state at T

= 0 in the absence of the interaction.

The

unitaflty operators

_S~

(2.4)

_may be written

[18]

into the form

S~ ⁼ e"

(2.12)

provided

that the N~ operators

obey

the

following

commutation relations

~J'~'~~~~~J'J' ~~'~~)

3. ID massive

thirring

model bosonization.

We write the Hamiltonian of the lD massive

Thirdng

model _as

HM~

₌

dX(VF[~/ fl~I ~~ fl~2j

_H

[~/ _~l

₊

~( ~2j)

₊

+

dX(~ll0[~/ ~2

+

~~ ~lj

+ 2 go

~/ ~~ _~2 ~l) (3,I)

where _p is the chemical

potential,

_mo has the dimension of _{a mass} and go ^{is the} interaction

coupling

constant. The bosonized forrn of the kinetic Hamiltonian

(2.2)

has been

previously

obtained

[10]

Ho

" "VF ^L

Z Bj

+ 2 arVF ^L

Z I"/

(- ^k en(-

k)

₊

ml (k) cr2(k)] (3.2)

1 k~o

Using

the Jordan boson

representation (2.3),

bosonized

expressions

of the other _terms of the Hamiltonian

(3.I)

_are

easily

obtained

l~X[~/ ^~l

⁺

^{$'~ $'2j}

^~

^BI

⁺

^{~2 (3.3)}

dx~/ $'I

_4i2

_4'1~

L~

~Bi 82

+ L~

£ ^{[en(- k) cr2(k)}

+

en(k) ml (- k)] (3.4)

k>0

dxj~j _~~

₊

~j ~ ii

=

£

dx _cos

(pw (3.5)

««

where _we have introduced the sine-Gordon field

p

(x)

₌

Ni

+

N~

^{~ "}

(Bi

₊

B~)

_x I ~ _"

£

^'

^e~~~ [«i(k)

₊

r~(k)])

(3.6)

P

L L

k

k

The

expression p

=

(4 wv~)~'~

for the parameter

p

^is obtained

by imposing

the canonical commutation relation

[p (x), H(y )]

=

I

(x

_y

)

to the boson field _p

(x)

and _to its canonical

conjugate

momentum

H(x)

=

~ _"

~~Bi

B~

+

£ ^'e~' [«i(k) r2(k)]~

(3.7)

p

L

~

With

(3.2)-(3.5)

in

(3. I)

the bosonized

expression

of the ID massive

Thirring

Hamiltonian is obtained. With

(3.6)

and

(3.7)

and

by using

^relations

(3.2)-(3.4)

_we obtain

j _{dXi4it (x) in(x)}

+ in

(x)1r2(x)1

=

)

^~~~

dx[ ~ij)~ ^(3.9)

j

_dX

_{4it (x) $it}

(x) $i2(x) $ii(x)

₌

~

j~

dx

[vi ~[)~

^~

_H~(x)j (3.10)

With

(3.5)

and

(3.8)-(3.10)

in

(3.I)

we obtain for the ID massive

Thirring

model

Hamiltonian,

the

expression

~"~

~ ~ ~~ ~

~~j~~

~

~ ~

2

v~ ~~~~~

(3. I1)

+

~~ ^~~~

p dx ^?

~'~~

+

~° dx _cos

[(4 _wv~)~'~

_p

(x)

w ax ax

By operating

in

(1.4)

^{the canonical} transformation _{p -} A _p, H-

H,

A

m

(4 wv~)~'2/g

and

by comparing

the result with

(3.

I

I),

we notice that

H~~

_can be written under the form of the Hamiltonian

(1.4)

of the lD

quantum

sine-Gordon

model,

if the

following

conditions _are

imposed

I 4 _w

g2

_w 4 _w

g2

~~ ~ ~~

~ ~ _" ^~ ~~

~ ~~ ^{~ "}

(3.12)

I

=

~~ ~

»

,

m~ =

j

_«~

2 «

w g

28 JOURNAL DE

PHYSIQUE

I N°

4. C-IC transition in two-dimensional classical sine-Gordon

system.

The relation

p =

~ _"

i _(4.I)

VF g

between the chemical

potential

_p and the misfit parameter

I

we have found is identical with that

previously reported by

Schultz

[4]

and Bohr et al.

[6].

Our

approach disproves

the

relation

(I.I) reported by

Luther _et al.

[5]

and Yamamoto

[7].

The

mapping

of the lD quantum ^SGM

(1.4)

onto the lD MTM

(3.I)

has been

previously performed [3-7] by

_means of LPM boson

representation. Consequently,

the zero-mode

contributions to the bosonized massive

Thirring

Hamiltonian have been overlooked

by

all

these authors. Yamamoto

[7]

has taken into account the zero-mode contributions in the

kinetic Hamiltonian

only. Owing

_to the _use of Jordan boson

representation

_we have taken into account all zero-mode contributions both in the kinetic and in the interaction _terms of the

massive

Thirring

Hamiltonian. The

equations (3.2)-(3.5) presented above,

and

especially

the relation

(3.3)

_prove that in order to establish the

mapping

of the ID SGM and the ID MTM

one onto another _one has _to take into _account the zero-mode contributions to the bosonized Hamiltonian. When

using

the LPM boson

representation

to find the

equivalent

^of the relation

(3.3)

_{one can use no}

rigorous

method and then encounters some inconsistencies. Luther et al.

[5],

_e-g- have used the sine-Gordon field

W(x)

=

I

hi')e~~~~"'~"~j«i(k)

+

«2(kjj (4.2)

to find

I jdx(?()~~) =[izi«i(k)+«~(k)I _jdxe-'k~

"

[~l (°)

+

"2(°)j (4.3)

" GF H

dX[~/ ~l

+

~~ ~2j

We have notice here that

(4.3)

contains the zero-mode

(k

=

0)

_terms while in

(4.2)

these

terms have to be absent in order that _p

(x)

makes _sense. In _our

approach

the zero-mode _terms

are

properly

taken into account and

consequently

it is free of all such inconsistencies.

By using

the relation

(4.I)

and the _same Bethe-Ansatz _as Yamamoto

[7]

_we find for the critical _curve

&~ =

&~(T)

of the C-IC transition in the 2D classical sine-Gordon system, the relation

&~(T)

T.

/(T) (4.4)

where

&/(T)

is the critical _curve found

by

Yamamoto

(Eq. (15)

in

[7]).

So, the contributions of zero-mode

modify,

in _a

slight

but nontrivial _manner, the critical

curve of the considered C-IC transition. The

slope

of the _curve

(4.4)

around

g~

₌ ⁴ _vr

point

is

twice _as much _as

reported by

Yamamoto

[7].

When

deriving

the relation

(4.4)

we have restricted ourselves _to the

region

vr/3

~

A ~ w, A

= cot~

(go/2)

where _no

stings

_appear ⁱⁿ the

ground

state. We have restricted

our discussion _to the _case

~ 0. For the _case

~

0,

from the

symmetry

^{of the}

original

sine- Gordon system one expects

[7]

to find the _{same curve} except ^{the difference of the}

sign.

We

point

out that the

present

Bethe-Ansatz

approach

and therefore the _curve

&~(T)

_we have found _are

only

valid _near

g~

=

4 _w. One may expect to obtain

&~(T)

down to T

=

0 _{as a} classical limit when

using

the Bethe-Ansatz

approach developed by

Haldane

[20]

and

Okwamoto

[21]

to

investigate

the C-IC transition at

general

temperatures.

References

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[2] STOECKLY B. and SCALAPINO D. J.,

Phys.

Rev. 811 (1975) 205.

[3] LUTHER A.,

Phys.

Rev. 815 (1977) 403.

[4] SCHULTZ H. J.,

Phys.

Rev. B 22

(1980)

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[5] LUTHER A., TtMONEN J. and POKROVSKY V., in Phase Transitions in Surface Films, J. G. Dash and J. Ruvalds Eds. (Plenum, New York, 1980).

[6] BOHR T., POKROVSKY V. L, and TALAPOV A. L., L-D- Landau Institute

Preprint

1982-3.

[7] YAMAMOTO H., Frog. ^Theor. Phys. 91(1983) 1281.

[8] LUTHER A. and PESCHEL I., Phys. Rev. B 9 (1974) 2911.

[9] MATTIS D. C., J. Math. Phys.

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[10] APOSTOL M., J.

Phys.

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[I ii HALDANE F. D. M., J.

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Phys.

C14

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[13] HEtDENREICH R., SCHROER B., SEtLER R. and UHLENBROCK D., Phys. Leit. 54A (1975) 119.

[14] JORDAN P., Z.

Phys.

93 (1935) 464.

[15] JORDAN P., Z. Phys. 99 (1936) 109.

[16] JORDAN P., Z.

Phys.

102 (1936) 243.

[17] JORDAN P., Z.

Phys.

105 (1937) 114, 229.

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Phys.

C 20 (1987) 3 II1.

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Phys.

Scr. 39 (1989) 294.

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Proofs

not corrected

by

the authors

JOURNAL DE PHYSIQUEI T 2, N' I, JANVJER >W2 2