Complete devil's staircase in the one-dimensional lattice gas

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Submitted on 1 Jan 1983

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Complete devil’s staircase in the one-dimensional lattice

gas

S. Aubry

To cite this version:

S. Aubry. Complete devil’s staircase in the one-dimensional lattice gas. Journal de Physique Lettres,

Edp sciences, 1983, 44 (7), pp.247-250. �10.1051/jphyslet:01983004407024700�. �jpa-00232188�

Complete

devil’s staircase

in the

one-dimensional lattice

_gas

Comment

on «

one-dimensional

lattice

_gas

and

the

universality

of

the devil’s staircase

»

bv S. E.

Burkov

S.

_Aubry

Laboratoire Léon Brillouin, CEN

_Saclay,

91191 Gif sur Yvette Cedex, France

(Re~u le

_{2 fevrier}

1983,

accepte

le

_{15 fevrier}

1983)

Résumé. 2014 Nous calculons

explicitement

la forme exacte de l’escalier du diable

_complet

du modèle

de gaz sur réseau. Nous montrons _{ainsi que le}

comportement

de la transition de

_phase

commensurable-incommensurable

_dépend

essentiellement des

_interactions

à

_longue

distance. Nous mettons en

question

les

propriétés

d’universalité

_suggérées

_{par Burkov.}

Abstract. 2014 We calculate

_explicitly

the exact form of the

_complete

devil’s staircase for _{the lattice gas}

model. Thus, we show that the behaviour at the commensurate-incommensurate

_phase

transition

essentially

depends

on the _long distance interactions. We

_question

the assertion of Burkov about the

universality properties

of the devil’s staircase. Classification

Physics Abstracts

05.20 - 68.20 - 64.90

The devil’s

_staircase,

calculated

_by

Burkov in his

_interesting

_paper

_[1],

can also be

_explicitly

calculated

_by

the same method that we used in the first exact calculation of a

_complete

devil’s

staircase in 1978

[2].

In

addition,

our method has the

_advantage

of

_having

a

_rigorous

mathe-matical foundation

_[3].

Our initial model was a variation on the discrete Frenkel-Kontorova

model where the sine

_periodic

_potential

was

_{replaced by}

a

piece-wise

parabolic periodic

potential.

It was transformed into an

integer

model which had a similar form to _{the lattice gas model} considered

_by

Burkov. The details of these calculations in reference

_[2]

were

_given

in

appendix

A3

of reference

_[4b].

We also noted that the model studied

_by

Bak and Bruinsma

_[5]

could also be transformed into such a lattice gas model

_[6].

In addition

_{by usihg}

a result

given

in reference

_[3c],

which was

_essentially

devoted to

_{obtaining rigorous}

results

_applicable

to

_quite

a more

_general

class of

models,

it was

_pointed

out that our

_proofs

and exact

results

could be extended

_unchanged

to models with _any kind of

_long

_range interactions

_given

some _necessary

convexity

conditions.

The purpose of this comment is not to

_reproduce

_{the substance of papers}

_{already published}

or

about to _appear

_but,

1)

to show how our

method,

which is

_simpler,

can be

_applied

to the model studied

_by

Burkov.

Thus,

we find

again

the

equation

of the devil’s staircase

given by

Burkov,

2)

to

_question

the

_{universality properties}

of the devil’s staircase

_{suggested by}

Burkov.

L-248 JOURNAL DE PHYSIQUE - LETTRES

The one-dimensional lattice _gas model

_[1]

with Hamiltonian

is included in the class of models

(34)

of reference

_[6]

where for each

_Um(x)

there exists a

_positive

constant

_Cm

such that for all x,

Um(x)

>

_em

> 0.

_(Many

_rigorous

results were

_given

for this class of models in reference

_{[3].) (Note}

that if this condition is not

_satisfied,

the devil’s staircase

of (1)

may not

_exist.)

When the atomic mean

_distance,

_I,

which is the inverse of the atomic

concen-tration,

c, i.e.

is

fixed,

the

_ground-state

of model

₍₁₎

is

_{given by}

_(see

ref.

_[6]

or

[3a,

3b,

3c])

where a is an

_arbitrary

_phase.

The energy per atom with

condition

(2)

is

and can be written

with

The method for

_calculating

_t/lm(l)

is the same as that for

calculating

(A3-11)

in reference

[4b]

or

_{( 13-b)}

in reference

_{[6]. By}

_inspection

of the results in reference

_[4b]

or

_[6],

we

readily

find

For the real model

_{(with 1}

not

_fixed)

the concentration c =

1/1

is determined

by

minimizing

the

«

_{grand potential » (1).}

Thus unlike our models in references

[2, 4

or

_6],

the

_quantity

which has to be minimized is not _{the energy per} atom but the

_{grand potential}

_{per unit}

_length

_(since

the total

number of

_particles

is not

_conserved).

Instead of

_having

a devil’s staircase

~(~)

where 1 is the atomic mean distance

_{and ~}

is a tensile force

_(that

is the

_opposite

of a

_pressure)

the devil’s staircase of model

₍₁₎

describes the concentration c =

1/1

of

particles

_{as a} function of the chemical

potential

~.

The minimization of the

_grand

_{potential (1)}

per unit

length,

which is from

equation (4)

This is the

_{implicit equation}

of the devil’s staircase

_c(P)

of model

₍₁₎

for which we are

_looking.

Using (7)

this

_equation

becomes

which is identical to

_{equation (5)}

as

given by

Burkov in reference

[1].

It is clear that the

_behaviour

ofc(~)

at the commensurate-incommensurate transitions reflects the

_properties

of the interaction

U.(x)

for

_large

x and m. Since for sake of

_brevity

we cannot

_give

the details of the calculations

here,

we

_suggest

tha~

the interested reader should convince himself

_{by checking}

for

_example

the cases

in which for all m :

and

in the

_vicinity

of the

_{registered commensurability}

c = 1. In the first _case,

(11~),

the well known

logarithmic

behaviour

is

_found,

while in the second case

Apparently

this behaviour is not universal and

_only

_depends

on the

_properties

of

_!7~(x)

for

large x

and

_large

m. The

_universality

_properties

claimed

_by

Burkov are not

_clearly

connected to any

physical

observation

and appear

only

as a

_generic

property

of a mathematical nature for the irrational

numlbers.

In our

_opinion,

the hierarchical construction of the devil’s staircase

_proposed

_by

Burkov is an

artefact of his method of calculation.

Indeed,

we have shown above that there exists a

straight

forward calculation of the devil’s staircase

_{of (1)}

which does not _{involve any} details of continued fraction

_expansions

as in reference

_[1].

_Moreover,

the

_generation

of the

_steps

of the devil’s staircase as

_{proposed by}

Burkov does not _appear reasonable

since,

for

_example,

there exist

_steps

_generated

at order two which are

_infinitely

small while at order 3 some of them can be

_large.

It is necessary

to consider the metric

_(that

is the width of the

_steps)

of the devil’s staircase for a

good description

of the

_generation

of the

_steps,

while these

_purely

_topological

consideration

_although

not wrong

mathematically

are

_{physically misleading.}

We

_proposed

in reference

_[7]

another

_hierarchy

which

_corresponds

_to

the

_generation

of

_steps

which

_{approximately}

are in

_decreasing

order of

width. This

_corresponds

to the well known construction of the set of rational numbers. We start

. 0 1 0 1 1

from the two

rational y

and

At first order we

_generate

the

sequence T

and

1.

At second order

0 1 1 ~ 1 . 0 l~l 2 1 3 3 1 .

we

generate T’ ’3 ’ ’2 ’ "3 ’ T ’ at

thud order

T’4’~5’2’5’4’T

and so on. The next sequence is obtained from the

_previous

one

_{by inserting}

between two consecutive rationals of the _sequence

~~+1/~+1 ~

new

_{rational r"}

+

_{r. - , Is.}

+

_{~Sn 1 ~}

Let us also note that Burkov considers it obvious that the Cantor set called

_M1

in _{his paper is}

self-similar. This is a metric

_property

from which he concludes that

_M1

has measure zero which

L-250 JOURNAL DE _{PHYSIQUE -} LETTRES

could be

_incomplete

_[2]

and then this set

_M,

has finite measure. Of course in this case the Cantor

set

_M,

cannot be self-similar. It is not self-similar even when

_M,

has zero measure, at least in the

case when the

_long

_{range interactions decrease fast}

_enough.

In a

_forthcoming

_paper we will

present

the

_proof

for the existence of an

_incomplete

devil’s staircase in the Frenkel-Kontorova

model for a small

_{periodic potential [8]}

which confirms our

_{early predictions [2].}

Thus a

_rigorous

proof

for the

_completeness

of the devil’s staircase in the case of the lattice _gas model is needed. This

_proof,

which is in fact _very

_simple,

can be found in reference

_[6].

Finally, although

we do not believe the

_{physical interpretation}

of the hierarchical construction

in the form

_proposed

_by

Burkov

_{by using}

the continued fraction

_expansion

of irrational

numbers,

this method of

_expansion

turns out to be _very useful for

_{understanding}

the « transition

_{by breaking}

of

_analyticity

»

_[2]

which is

_closely

related to the

_{stochasticity}

threshold

_[9]

but such a transition does not exist in model

(1).

The

_{physical meaning}

of this

_expansion

_{becomes very}

_transparent

_by

considering

the renormalization group

approaches

used to

_investigate

this kind of transition

_[10].

(I apologize

to Prof. Burkov and to _{the reader that my}

_early

works on the devil’s staircase are

not

_easily

available in the literature because

they

are

incompletely published

in

journal

which

have

_only

a limited

_readership.

This was due

_largely

to serious difficulties with their

_{publication.)}

References

[1] BURKOV, S. E., One-dimensional lattice gas and the universality of the devil’s staircase, J. _Physique Lett. 44 ₍₁₉₈₃₎ L-179.

[2]

AUBRY, S., Soliton in Condensed Matter, Solid State Sci. 8 (1978) 264.

[3]

a) AUBRY, S., On modulated

_{crystallographic}

structures. Exact results on the classical _ground-state

of a one-dimensional model,

preprint unpublished (rejected for

publication)

(1978).

b) AUBRY, S. and ANDRÉ, G., Ann. Israel. _Phys. Soc. 3 (1980) 133.

c) AUBRY, S. and LE _DAERON, P. _Y.,

_preprint

_(1982). The discrete Frenkel-Kontorova model and its extension. I. Exact results for the

_{ground-state,}

submitted to

_Physica

D.

d) AUBRY, S., The twist map, the extended Frenkel-Kontorova model and the devil’s _staircase, to

appear in

Physica

D _(1983).

[4] a)

AUBRY, S., Ferroelectrics 24 (1980) 53.

b) AUBRY, S., The devil’s staircase _{transformation} in commensurate lattices, in _Proceeding of IHES and

Columbia 1979-1980 _(Eds. D. and G. _Chudnovsky), Lectures Notes in Mathematics _{925, 221.}

[5]

BAK, P. and BRUINSMA, R., Phys. Rev. Lett. 49 ₍₁₉₈₂₎ 249.

[6]

AUBRY, S., Exact models with a

_complete

devil’s staircase, to _{appear in J.} Phvs. C

(1983).

[7]

AXEL, F. and AUBRY, S., J.

_Phys.

C 14

₍₁₉₈₁₎

5433 (third

section).

[8]

DE SEZE, L. and AUBRY, S., in

_preparation.

[9] GREENE,

J., J. Math.

_Phys.

20

₍₁₉₇₈₎

1183.

[10]

PEYRARD, M. and _{AUBRY, S.,} Critical behavior at the transition

_by

_breaking

of

_analyticity,

to appear in J.

_Phys.

C

₍₁₉₈₃₎

and references therein.