On the lattice corrections to the free energy of kink-bearing nonlinear one-dimensional scalar systems

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kink-bearing nonlinear one-dimensional scalar systems

D. Grecu, A. Visinescu

To cite this version:

D. Grecu, A. Visinescu. On the lattice corrections to the free energy of kink-bearing nonlinear one-dimensional scalar systems. Journal de Physique I, EDP Sciences, 1993, 3 (7), pp.1541-1550.

�10.1051/jp1:1993199�. �jpa-00246815�

J.

Phys.

I France 3

(1993)

1541-1550 _JULY 1993, PAGE 1541

Classification

Physics

^Abstracts

05.90

On the lattice corrections _to the free _energy of kink-bearing

nonlinear one-dimensional scalar systems

D. Grecu and A. Visinescu

Department

of Theoretical

Physics,

Institute of Atomic

Physics,

P-O- Box

MG-6, Magurele, Bucharest,

Romania

(Received

5

October1992,

revised 25

February1993, accepted

26 March

1993)

Abstract The effective potential ^obtained by

Trullinger

and Sasaki is

discused.Using

_asymp-

totic methods from the theory of differential equations

depending

_{on a}

large

parameter, ^the lattice corrections to the kink and kink-kink contributions to the free energy are calculated.

The results _are in

complete

agreement with _a first order correction to the energy of the static kink.

1 Introduction.

The influence of lattice discreteness _on the

properties

of nonlinear

systems having

kink solutions

was

investigated by

several authors

[1-10].

^{These studies have}

pointed

out _a

large variety

of

effects, including

modification of kink

velocity

and its form and

leading

sometimes to the

pinning

of the kink _on the lattice. But _on the other side numerical

investigations

_[9] have revealed the existence of _narrow

solitary

_waves

propagating

in certain discrete lattices without energy loss. There is also _a close connection of these

problems

with the

complete integrability

of

specific

models in nonlinear continuum and discrete

systems [11-13].

Since the

pioneering

_paper of Krumhansl and Schrieffer

[14]

^{the role}

played by

the

elementary

excitations of kink

type

ⁱⁿ the

thermodynamics

of one-dimensional nonlinear

systems

_was

investigated by

_many authors

([15-20]

^{and the} references

therein).

As _concerns the influence of lattice discreteness _on these

thermodynamic properties

this has received _a smaller attention

[20-23].

The class of i-D

systems mostly

discussed is described

by

the Hamiltonian

(in

the notations of CKBT

[15])

H =

fliA()#)

+

)I#;+1-#;)~

+

Wlvl#,)) ^Ii)

where the

nonlinearity

enters

only througb

the

potential V(#),

assumed to have at least two

degenerate

minima with

nonvanishing

curvature. It is the merit of

Trullinger

and Sasaki

[21]

to have shown that the first lattice corrections _are taken into account if the

potential V(#)

is

replaced by

_an effective

potential

where d ₌ ^~° is the _mean width of the static kink. In the

present

_paper we intend to

complete

wo

their results and determine also the lattice corrections to the multi-kink contributions to the free energy.

The

partition

function and the free energy are

usually

calculated

using

the transfer

integral operator (TIO)

method. In the

thermodynamic

limit the lowest

eigenvalues

of TIO

play

the most

important

role. In the

displacive/continuum limit,

_«

_1,

it is

possible

to transform the Fredholm

integral equation representing

the TIO into _a differential

equation.

This is

easily

done when the

two-body

interaction in

(1)

has _an harmonic character (~

[17, 19].

One obtains

exPl-7Vl#))

_exP

()D~) _Aim#)

=

exP1-7in) All#). 13)

Here ₇

=

fllAw],

m*

=

(flAcowo)~,

_Am ^{is the}

eigenvector

of TIO

corresponding

to the

eigen-

value _en and

in

_{= en} + ln

~'~. (3a)

7 7

The left hand side of

(3)

contains the

product ^{e~ e~}

with A and B two

noncommuting

_opera-

tors,

and in order to obtain the claimed differential

equation

_we have _to

put

it _as

e~,

where the

operator

^C can be written in terms of A and B

using

the Baker-Hausdorff formula

[20-22].

^This

~

formula turns to be _a mixed

expansion

in the small

parameters j

_and ~

=

(m*)

² and

consequently

is _very convenient for the

proposed

_purposes. The first

latZce co$ection

_{is ob-}

tained if _one retaines

only

the _terms

proportional

with ^~~

Contrary

the

opinion

of

Trullinger

d2

and Sasaki

[21]

^the same result is found if _one work with tbe

symmetrised

_or

non-symmetric

form of TIO. It is

possible

to show

through

a set of transformations and

neglecting system- atically higher

_powers in

i2 j,

^{that the}

eigenvalues

of

TIO,

with the first lattice corrections

included,

_can be found from the

equation _{[21] (~)}

~~fi

+

(in l~a(#))tvn

= 0.

(4)

2m*

d#

The result is _very

general

and valid for _any kind of

potential

function

V(#).

In the _next section the

leading

terms _{as an}

asymptotic expansion

of the lowest

eigenvalues

of

(4)

will be found in the low

temperature limit,

m* _» 1. The method

used, briefly

sketched in the

Appendix,

allows _us to make _a clear distinction between the contributions of

phonons

(~)

^As was shown

by Guyer

and Miller [17] even for _more

complicated

interactions between nearest

neighbours

it is

possible

to write _a formal

expression

for the

partition

function and the free energy

usinj

^{a cumulant}

^expansion.

The

discrepancy

between the results of [21] and [20] comes from _an

inadequate

treatment of _a certain

equation

obtained in _an intermediate step of the

proof

in _our

previous

_paper [20].

N°7 LATTICE CORRECTIONS TO KINK SYSTEMS ₁₅₄₃

and of the various kink _sectors.

Explicit expressions

for the lattice corrections _to the kink and kink-kink contributions _to the free _energy will be

given.

The last section is devoted to _a discussion of the results in the

spirit

of the CKBT

[15] phenomenology

of

independent phonons

and renormalized kinks. As it is known

[15, 16,

_19] this

phenomenology

is exact in the low

temperature

limit of the continuum models and the results of

Trullinger

and Sasaki

[21]

^and those of the

present

_paper

strongly support

^the

conjecture

that it remains valid _even when the first lattice corrections _are taken into account.

2. Lattice corrections to the free energy.

Using (3a)

the free _{energy per} unit

length

becomes

where _To is the lowest

eigenvalue

of

equation (4).

In the low

temperature

limit

fl

»

1,(m*

_»

₁₎

there _are several _ways to find

approximate

solutions of this

equation,

all included in so-called class of the

improved

WKB methods

[15-20, 23, _24] (and

the references

therein).

We shall _use the method

developed by

_us in _a series of papers

[20, 23, 24]

^{which has the}

advantage

to make _a clear distinction between the various

contributions _to the free energy:

phonons, i-kink,

2-kink and _so

on sectors. The method is

based _on well known results from the

theory

of

asymptotic

solutions of second order differential

equations depending

on a

large parameter [25]

an is

presented

in the

Appendix.

As _{we are} interested in the

leading

term in m* the exact

equation (4)

is

approximated by

the

comparison equation (A.9).

With

Vea replacing

the

potential

V in the definition

(A.8)

of the constant _a the first term in the

asymptotic expansion

of the lowest

eigenvalue

of the isolated

potential

well is

given by

~° 2~$ ^{~ ~~2~}

~

~~/*~'

~~~

Then the

corresponding

contribution to the free _energy

density ₁₆₎

writes

' l _~~~~~°°~~

~

~d ~ ~d~~

~~~

and is

easily

identified with the first terms in the series

expansion

in powers of

l~/d~

of the exact free energy of _an

independent

_gas of

phonons

with the

dispersion

relation

w~(k)

₌

w( 1+4~ ~ sin~

^~~

(8)

2

Higher

order terms in

(m*)~~/~, neglected

in

(6),

will

correspond

to anharmonic interactions between the

phonons [23].

The presence of the other

degenerate

minima determines _a

symmetrical splitting

of each

eigenvalue

ofthe isolated

potential

^well. In the _case ofthe

#~

model _we obtain

Eo

_-

io

_"

Eo+

to where

~ 12

~°

@

~ 4d~~

~ ~~~

with _c

= I. As is

explained

in the

Appendix

the value of _u is determined from the

boundary

conditions

(A,ll) imposed

on the

eigenfunctions

of the

comparison equation.

For the

#~

^model

we

get

~-;xv

-1 =

2j%

_exp 21J +

u(1

+

ln2) -(1+ 2u)In(1+ 2u) (10)

r(-u)

2

This

expression

is valid in

leading

order in

(m*)~~/~,

^but in any order in _u, and is the

starting point

to find multi-kink contributions to the free energy. Here J is _an

integral

defined

by

J =

/~'/fid# ₍₁₁₎

o

with pi the left

turning point

of the isolated

potential

well. As _{we are} interested in the

ground

state the

quantity

I from

(11)

is connected to _u

by

~

2~ ^~ l)

2

_Ii

+

2uc)

=

fi (i

_24d2^l~

⁽¹²⁾

where

io

is _now

independent

_on the lattice

spacing

I. It is _seen that J

depends

_on

l~/d~

both

through

the

integrand

and the upper limit of

integration. Using (12)

and the

expression

of

l~a

the

turning points

of the

#~ model,

calculated in the order

O(l~/d~),

_are

given by

12 14

p],~

= i ~

fi

₊

~fi (~j

+

fi)

+

O(~). ⁽¹³⁾

In the _same order

J ₌

b _{£"' Ii} &<~l ^{/(Pi <2)(Pi} <2) d<

₊

O(i~/d~). (14) Writing

~

~

~°

~

~d2

~~ ~~~~

where _now

Jo

and

Ji

does _not

depend

_{any more on}

l~/d~,

_we

_get

These

_ntegrals

can

be alculated in

terms of complete

elliptic ntegrals of

modulus in _the

2ViJo

t

io(1

+

In$)

Co

2ViJi

t

~

+sic. (16)

N°7 LATTICE CORRECTIONS TO KINK SYSTEMS 1545

Introducing

these results into

(10)

we

get

@e ~~~~~~ ~~e ~

=

~~~

~ ~ ~

e~'~"

-u In(12flE)°~)

"@

(~~) ~~

r(-u)

^{~ ~ ~~~}

where

E)°~

₌

( Acowo

is the _energy of the static kink.

In the sine-Gordon _case

(periodic potential)

each

eigenvalue

of the isolated

potential

well is

symmetrically splitted

in _an allowed band of width

2to

where

to

is

given by (9)

with _c

=

2,

and

u results from

boundary

conditions

imposed

_on the

eigenfunction

at the

point #

= ir.

Finally

we

get

~;xv rj1

_~

_~)

_{~ ~~}

l =

2@

^~ _exp 21J +

2u(1

+

ln2) In(1

+

4u)

,

(10a)

~(~U) r( I)

2

where the

integral

J is defined

by

J

=

/ ^~ _{fi~d#. (11a)}

12

The

integral

J is calculated in the _same linear

approximation

in

j

^{and the}

^leading

^{terms in}

io

_{is given}

by

~

~~ ~~ ~~2

~

~

~~

~~2 ~

~~~~~

where

io

is defined in

(12) being independent

_on lattice corrections.

Introducing (16a)

into

(10a)

^and

using

^the

expression E)°~

=

8Acowo

for the _energy of the static kink _one obtains

e~~~~~~~~

~~e

41~?

= l~

~;xu yji

_~

_~)

~~~ ~,

=

2

e-~"

'~~fl~k

e~"W II?)

SG

~(-U) ~())

In order _to find various kink contributions _we have to

expand

the

right

^hand side of

equations (17)

in _powers of _u and to write also

u = uk + _ukk +..

(18)

The

single

kink term uk comes from the

expansion

of

~

r(-u) ^~~~~~

(7

= 0.577 Euler's

constant)

and _one obtains

easily

bk =

@~-pE)°)j~_

^{,2 ~~~}

~

~~~~

~

48d2 (19)

_~i4

~

@~

p~io~~~ ^,, i~

lr

~ ~~'

ml _jig)

sG

This result is in

complete agreement

with that found

by Trullinger

and Sasaki

[21].

In

finding

the kink-kink contribution _{ukk we} have to take into account that the

expansion (18)

is _a formal

expansion

in _powers of

e~fl~i~

_and

consequently

_ukk is of the _same order _as

VI.

^{Then it is}

easily

found that

ukk "

u( (7

⁺

In(12flE)°~) ^~~~

+

ir) (20) #~

24d

One _sees that _ukk

gets

an

imaginary part satisfying

the _very

simple

relation

Imukk

_"

iru( (21)

This relation has been obtained

by

Zinn-Justin

[26]

^{in his}

analysis

of the multi-instanton contributions in

quantum

^{mechanics and is}

tightly

related to the non-Borel

summability

of the

Rayleigh-Schr6dinger perturbation expansion

when tbe

potential

has

degenerate

minima.

For the sine-Gordon model _one obtains in

a similar _way

~~

~~( (7

+

ln4pE)o)

^{12 i~}

~~~~ ~

(~~~

^~~

Both

(19) #~

and

(20)-

SG in the limit 12

j ^- 0 _are in

complete agreement

^with

previous

d

calculations of the kink-kink _sector contributions

[26-28].

3

Concluding

remarks.

According

to CKBT

phenomenology

the

thermodynamic properties

of the systems described

by

the Hamiltonians of the form

(1)

_are influenced

by

the existence of the static kinks

[15-19].

A

complete agreement

between this

phenomenology

and the exact results of TIO in the low

temperature

^{limit is found if} one takes into account the

scattering

of

phonons

_on the static

kink, leading

to _a renormalization of the kink energy.

The results of

Trullinger

and Sasaki and of the

present

_paper are

showing

that lattice corrections _are

easily

included into the

thermodynamics

of this

systems. Although

_a

complete proof

of _a similar CKBT

phenomenology

does not exist at

present,

the

existing

results

strongly support

the idea that their

phenomenology

is still valid at low

temperatures.

This _comes both from the

phonon part, which,

as mentioned

above, reproduces exactly

the first terms in the

12

series

expansion

in

j

^{of the free energy of a}

^pbonon

^{lattice gas, and from the kink} contribution.

As is _seen from

(19)

in the kink contribution _{appear a} lattice corrected kink energy

Ek

₌

E)°~ (

E)~~ (21)

where

E)°~

^{is the known}

unperturbed

static kink energy and the correction

E)~~

^{for the SG} and

#~

^model are

given by

N°7 LATTICE CORRECTIONS _TO KINK SYSTEMS 1547

E)~~

=

E)°~, E)°~

=

~A0woco (22) #~

These corrected values have been obtained also

by

^other authors

[5, 6],

^and as will be shown

below,

result from _a very

simple perturbation theory.

Indeed

expanding

the field variable

#(z

+

I)

ⁱⁿ a

Taylor

series the discrete

Euler-Lagrange equation

transforms into _a fourth order differential

equation

~~~

~

~~2 ~~

^~

~~ ^~

^~~~~

where _y

= z

Id

and

Vj

₌

(~'. ^Looking

^for a

perturbed

kink solution

#

4k(Y)

"

4[°~lY)

+

)#[~~lY)1 ¹²⁴⁾

where

#)°~

^{is the kink solution of the continuum}

limit,

the linearized

equation determining

~(l)

i~

k

~~~i~~

~~~j~(0)~ ~(l) _d~~~~

dy~

^{~ ~}

dy~ j~$)

The

homogeneous

_part of

(25)

has

appeared

_{some years ago} in the

study

of the kink

dy-

namics in the _presence of

perturbations

and has _as solution the "translation mode"

(Goldstone mode) [15, 16, 29].

^Then

equation (25)

_can be solved

by

the method of variation of constants

and the obtained results _are in

complete _agreement

with those

already existing

in literature

[4-7, _10]. Using

the _same

Taylor expansion

for the field variable _{one can} write down _{a cor-}

rected

integral

of energy.

Introducing

in it the _new

expansion

of

#k (24)

the

integration

is

easily performed

^and the result is

just

that

presented

ⁱⁿ

(22).

Still unsolved remains the

prob-

lem of calculation of renormalization of kinks due _to the

scattering

^{of lattice}

^phonons

on the

perturbed kinks,

renormalization which determines the factor

multiplying

the

exponentials

in

l19).

Acknowledgement.

The authors would like to thank Dr. Marianne Croitoru for _a fruitful collaboration

during

many years.

Appendix.

The lowest

eigenvalues

of

(4)

_can be

found,

in the low

temperature

^{limit m*} »

I, using asymptotic

methods known from the

theory

of second order differential

equation depending

_on

a

large parameter [25].

^The

equation

under

investigation

is of the

type

where the

potential V(#)

has at least two

degenerate

minima and

nonvanishing

curvature at the minima.

The

problem

will be solved in two distinct steps. ^{In the}

first,

based _on

Langer's

transforma-

tion,

_one looks for _an uniform valid

expansion

of the solution _near the minimum. Due to the

peculiarities

mentioned

above,

this transformation is chosen in such _{a way} to

give

the harmonic oscillator behaviour _{as a}

leading

term. In the second

step,

the existence of the other minima of

V(#)

_are ^{taken into} account

using symmetry properties

of the _wave functions. Each

eigenvalue

of _an isolated

potential

^well is

slightely splitted.

These

tunneling

terms _are

directly

related to the kink contribution to the free _energy.

Let _pi and _p2 be the two

turning points

of _an isolated

potential

well. One _passes from the variables tV and

#

to the _{new ones} R and _z.

~ =

x~iR _(A.2)

((z)

₌

f~ ^{W@ d< (A.3)}

x =

[j~[. ^jA.4)

Here

1'

=

~~,

and _z is still

an undefined function of

#.

Then

IA-i)

transforms into dz

d2~

j +

l~R

= 6R

IA-b)

where 6

=

-x~1~~~~ l'~

and

l~

= 2m*. The function _x is

subjected

to the

requirement

to be

d#

regular

and not to vanish in the interval of interest. In order _to obtain the harmonic oscillator behaviour _as the

leading term,

_one ^takes

jl')~

=

4a~(1 z~). jA.6)

This is

a function with two

simple

_zeroes and _z

= -1 is associated with the

turning point

#

= pi and

z = +1 to

#

_{= p2.} Now it is

possible

to determine the relation between the _new variable

z and the old _one

#. Integrating _IA-G)

_we

get

a(ir ^cos~~

_{z +}

z/~)

=

/~

V

d# IA.?)

p,

where the constant _a is

given by

P~

a =

Vd#. (A.8)

lr

/,

It _can be

easily proved

that for all the

domain,

even in the

asymptotic region,

6

-w

~,

and I

taking

into account that

l~

= 2m* _»

1,

^{the first order}

approximation

_can be found from the

following comparison equation _[25]

~

+

4a~l~(1 z~)R

=

0, (A.9)

whose solutions _are the functions of the

parabolic cylinder.

For the

#~ model,

the

only

_conve-

nient solution

(decreasing exponentially

when _{z -}

cc)

is Whittaker's function

Dv(y),

where y =

2fi

z and _u

= al

). (A.10)

N°7 LATTICE CORRECTIONS TO KINK SYSTEMS ₁₅₄₉

The

eigenvalues

of the isolated

potential

well _are obtained for _u

= n an

integer

^number. In

leading

order in I _one finds

j(°)

g~ +

n)

₊

o(j2) jA ₁₁₎

"

@

2

Now,

due to existence of the other

degenerate

minima of the

potential V(#)

each

eigenvalue

of the isolated well is

splitted

into two very near levels.

They

_can be found

using

the

boundary

conditions for the _wave function and its derivative

The solution

WI))

of

IA.1)

is related to the solution

R(z)

of

(A.9) by Wi#)

₌

I(l~l Rlz)

,

and the

explicit

connection formula between the old variable

#

and the _new variable _z is

(asymptotic region

of

~)

z~

_m

/~~ ^{Vi d#}

^{+ +}

^ln2[z[

⁺

^{O(z~~). (A.13)}

a o 2

In

applying

the

boundary

conditions

(A.

ⁱⁱ we have to _use the

asymptotic expansion

of

Dv(y)

1301

~~

/~

i /

D"lY)

_~'

Y~~~~

~j_~~

^~"'~

p

^{~ ~}

^lA.14)

The value of _u is found from _a

matching

between the dominant _e /₄ and the subdominant term /

e~ 4 Then from

(A.12-A.14)

_one obtains

~-;xv

Pi

-1 ₌

2j%

_exp 21

W/ _d#

₊

u(1

+ In

2) -(1

+

2u) In(1

+

2u)

r(-u)

2

(A.15)

The sine-Gordon _{case can} be treated in _a very similar _way, the solution for the

comparison equation (A.9) being,

in that _case,

expressed

more convenient in terms of Kummer's functions

having

a definite

parity.

Each

eigenvalue

of _an isolated

potential

well is

splitted

into _{a narrow} allowed band. If the

potential V(#)

has _a

2ir-periodicity

the index _u

corresponding

to the lower and _upper

boundary

of the lowest band follows from similar conditions _as

(A.12)

calculated in the

point #

= ir.

Finally

_{a very} similar relation with

(A. lb)

is

found, namely

~-ixu rj1

_~

~)

~~

~~

rj_ I) /jl)

~

2 2

x exp

21 /" W/ _d#

₊

2u(1

_{+ In}

2) (l

₊

4u) In(1

+

4u)) ^(A.15a)

j ²

Here

by +j

we have denoted the

turning points

of the isolated

potential well,

centred around the minimum

#

= 0. Also the definition of _u is

slightly

modified and insted of

(A.10)

_we have 2u ₌ al

j.

To conclude _we have to indicate the connection relation between the

symmetric

splitting

of each

eigenvalue

of the isolated

potential well, Eo

~

io

"

Eo

+ to and the

quantity

u. We

get

to ₌

flu _IAiG)

where _c

= I _or 2 for the

#~

_or the SG model

respectively.

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