Theory of Lattice and Electronic Fluctuations in Weakly Localized Spin-Peierls Systems

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Theory of Lattice and Electronic Fluctuations in Weakly Localized Spin-Peierls Systems

C. Bourbonnais, B. Dumoulin

To cite this version:

C. Bourbonnais, B. Dumoulin. Theory of Lattice and Electronic Fluctuations in Weakly Local- ized Spin-Peierls Systems. Journal de Physique I, EDP Sciences, 1996, 6 (12), pp.1727-1744.

�10.1051/jp1:1996185�. �jpa-00247278�

Theory of Lattice and Electronic Fluctuations in Weakly

Localized Spin.Peierls Systems

C.

Bourbonnais (*)

and B. Dumoulin

Centre de Recherche

en

Physique

du Solide et

Département

de

Physique,

Université de

Sherbrooke, Sherbrooke, Québec,

Canada JIK 2Rl

(Received

18

July

1996, received in final form 26

August

1996,

accepted

29

August 1996)

PACS.71.20.Rv

Polymers

and organic compounds

PACS.64.60.Ak

Renormalization-group, fractal,

and

percolation

studies of

phase

transitions PACS.75.40.-s

Critical-point

effects, specific heats,

short-range

order

Abstract. A theoretical

approach

to the influence of one-dimensional lattice fluctuations

on electronic

properties

_in

weakly

locahzed

spin-Peierls

systems _is

proposed

_using the renormal- ization group and the functional

integral techniques.

The

interplay

between the renormalization group flow of correlated electrons and one-dimensional lattice fluctuations _is taken into account

by

the

ope-dimensional

functional

integral

method in the adiabatic limit. Calculations of _spin- Peierls precursor effects _{on response} functions _are carried eut

explicitly

and the

prediction

for the temperature

dependent magnetic susceptibility

and nuclear relaxation is compared ^with

available

experimental

data for

(TMTTF)2PF6.

1. Introduction

Among

the

possible

structural

phase

transitions in low-dimensional

systems,

^trie

spin-Peierls (SP) instability

bas

recently

received

particular

attention

[1, 2j.

^In contrast with trie Peierls

transition,

whose normal

phase

is

metallic, pronounced insulating properties

and

antiferromag-

netic

spin

correlations _are found far ahead of trie SP

ordering temperature. However, owing

to trie

quasi-one-dimensional

character of these systems, _precursor eifects should _{appear m} both

types

^of transitions. Since

spin

correlations _are

essentially

one-dimensional _m

character,

lattice fluctuations should _{emerge as} diffuse

scattering

ⁱⁿ

X-ray experiments,

and _{as a} so-called

pseudo-gap

eifect in

magnetic properties.

The

pseudo-gap

eflect is well known to be the hall- mark of

strongly amsotropic

Peierls

systems [1,3j,

but

surprisingly,

its observation _{is a} rather

uncommon feature of most SP

systems

^{and trie}

underlying explanation

for this is not known

so far

[2j. Correspondingly,

the mcitement to achieve _a theoretical

description

of fluctuation eifects

m the SP _case did not receive any

experimental justification,

so that most theoretical

efforts _were rather devoted _to the

description

of the ordered state

[2,4, 5j. However,

recent observations of SP fluctuation eifects in the orgamc

systems (TMTTF)2X

and

(BCPTTF)2X

with X

=

PF6, AsF6,

has

prompted

_a renewed interest for this

problem [6-10j.

In both _cases, one-dimensional SP _{precursors are}

clearly

_{seen m} structural and

magnetic

measurements well above the _true SP

ordering

temperature.

(*)Author

for

correspondence le-mail: cbourbonslphysique.usherb.ca)

©

Les

Éditions

_de

Physique

1996

This paper will be

entirely

devoted to the theoretical

description

of one-dimensional lattice and electromc correlations in SP

systems

^{of trie}

(TMTTF)2X

series. Unhke

(BCPTTF)2X,

which _are

already good

insulators at room

temperature, (TMTTF)2X

_are

weakly

localized and still show _a metallic behaviour in this

temperature

range. It is

only

around

Tp

=t 220 K that carriers

undergo

a Mott-Hubbard

locahzation,

below which trie

resistivity

becomes

thermally

activated

Ill].

Albeit trie

magnetic susceptibility

remains unaifected at

Tp,

NMR

spin-lattice

relaxation rate measurements bave revealed trie existence of

strong

^1D

antiferromagnetic spin

correlations below this temperature _range

[î,12].

Below 60

K,

one-dimensional SP lattice soft-

ening

is borne out

by X-rays

and bas _a marked influence _on trie

temperature dependence

of both

susceptibility

and nuclear relaxation

[6, 7].

^{In order to tackle the}

problem

of one-dimensional electromc and lattice fluctuations eifects in SP

systems

we shall combine the renormalization group

technique

for fermions and the

functional-integral

method

[9,10].

A similar

approach

has been

recently applied

with _{some success} to

strongly

localized SP

systems

^{of the}

(BCPTTF)2X

_serres

_[10].

Since the

itineracy

^of carriers is much _more

pronounced

in the

high

temperature

domam of

(TMTTF)2X series,

both spm and

charge degrees

of freedom _must be considered in the renormalization

flow,

which in

practice

makes the extension of the

previous approach slightly

_more

complicated.

2. The Model and the Partition Function

In

setting

_up the essential

ingredients

of the

model,

it is first convenient to

study

the

parti-

tion function of the

electron-acoustic-phonon

interaction

model, namely

when direct electron-

electron interaction is absent. The

corresponding

Hamiltonian _is well known to take the form

[13]:

H =

£ _ep(k)c)

~ ~c~ k _a +

£uJ(Q)(b[bQ

+

p,k,a ^{' '} Q

~

+lL)~~~~ ~j _9(k> Q)C),k+Q,aC-P.k,a(b~

_Q ⁺ bQ)>

Il)

jp,k,Q,ai

where

e~(k)

₌

vF(pk kF)

is the linearized spectrum around the 1D Fermi

points pkf

"

+kF

for free fermions and

uJ(Q)

" UJDÎ

sin(Qa/2)1

is the

dispersion

relation for the

phonons là

=

1).

Here _VF is the Fermi

velocity

and _UJD is the

Debye frequency,

which coincides with the

phonon

energy ^at

Q

"

+2kF("

+7r

la,

when the band is

half-filled).

As for the

interacting

part, ^the

electron-phonon coupling

constant is

given by:

9~~'Q~

" ~~

àfi ~~~~Q~/~~ ~°~~~~

+

Q~/~~'

_~~~

where II is the molecular _mass and ga =

-dta /dz

> o is the

spatial

modulation of the

longitu-

dinal

hoppmg integral

ta. The

dynamics

of

phonons being purely harmonic,

their

integration

m the

partition

function Z

= Tr

e~P~

can be carried out

exactly, yielding

_an effective _re- tarded interaction among electrons

(fl

=

1/T, kB

_"

1).

The result _can be put in the

following

functional-integral

form

~ ~

P

Z =

[d~~][dfl ^e~e'~

^'~l

exp(- _{dTidT2 ~(Ti T2)), (3)}

over the

anticommuting

Grassman variables

#(*1

The free electromc action _is

given by

S)[i~,*, ~]

=

/~

dT

~j _{~j ~(k, T)[G((k,} T)]~~~~,~(k, T), (4)

o ~

'

where

G((k,T)

=

[-à/ôT ep(k)]~~

is the free electron

propagator.

^{As for} trie retarded interaction

~(Ti T2),

we will

proceed

to _a

"g-ology" decomposition

which

only

retains

2kF phonon exchange

between electrons at

+kF,

with the result

~(Tl

_T2)

"

(~~)

^~

~ _91,ph(T~

_T2)

_~~,ai

_{(~~ +}

_P~~F

_{+ ~,}

_{T~)~~p,a2 (~2 P~kF}

~>T2) (P,k,q,a)

X

~p,~~(k2,T2)~-p,ai(k~,T~)

+

(~~)

^~

~ _93,ph(T~

_T2)

_~Îp,ai

_{(~~ +}

_P~~F

_{~ P~i>}

_{T~)~Îp,a2 (~2 P~kF}

_{+ ~>T2)}

ip,k,q,aj

~ ~P,0E2(k2>

T2)~p,ai(k~,T~), là)

where G

=

4kF

_{is a}

reciprocal

lattice vector. The retarded

backscattering

and

umklapp

_cou-

pling

constants _are

given by

gi,ph(Ti

T2) "

g(-PkF, p2kF)g(pkf, -P2kF)D° (2kF,

_{Ti T2)}

#

~~~ D~ (2kF,

_{T~ T2}

), MàJD

93,ph(T~

T2) "

~~Î9~,ph(T~

T2)>

where

D°(2kF,T~

_T2)

=

-[e~~D'~i~~2'

+

2(eP"D -1)~~cosh(U~D(T~ T2))j

is the

phonon propagator

at

2kF. Here,

_~/ < 1 is _a

positive

constant that reflects the half-filled character of trie band. For _a

single

half-filled

band,

_~/

= 1 while for

systems

hke

(TMTTF)2X

with _a

quarter-filled

band and _a small dimerization gap, one can take _~/ < 1

[14j.

^From

above,

one

observes that for

exchange

of

2kF

acoustic

phonons,

_g3,ph and g~,ph ^are

opposite

m sign. Trie retarded interaction _can then be written

umquely

_m terms of

composite

fields:

~(Tl

_T2)

"

/

_dz

_{~j g$(Tl T2)°~*(X, T~)°~(Z,}

T2), (6)

~

where _we have introduced the

composite

field

°~* (~, T)

_#

(°*(Z, T)

+

àΰ(T, T)),

2 with

°(~'~)

"

~j _{~5,a(~> T)~+,OE(~,} Tl'

a

These

corresponds

to "site"

(M

=

+)

and "bond"

(M

=

-) charge-density-wave (CDW)

correlations at

half-filling.

The related combinations of

couphngs

_{are given}

by g$(T~

T2) "

9~,ph(T~

T2) +

~Îg3,ph(T~ T2).

In order to

analyze

these

correlations,

we

apply

_an Hubbard-Stratonovich transformation _on the retarded

part

^{of the} interaction

z "

/ld~~lldùlld4~l~~~i~~'~'

x

exp(- dzdT~dT2 ~ 4~(z,

_T~)

_gf(T~

_{T2) Î~~}

4~(z,T2))

M=+

x

exp(-

^d~dT

~j ÀMO~(z,T)çi~(z,T)), ⁽⁷⁾

M=+

where

çi*

_are real

auxiliary

fields for site and bond fluctuations

respectively,

and

À+

₌

2,

À-

=

21. It is worth

noting that,

in the absence of

umklapp scattering, gj~

=

g)~

^{and there is no}

diiference between bond and site

density

_wave fluctuations _so that bath

auxiliary

fields will combine to

produce amplitude

and

phase

fluctuations of the

complex

field _çi

=

çi+

+

içi~

=

(çi(e~~, as found in incommensurate Peierls

systems.

Unretarded

repulsive

electron-electron interaction at

half-filling

will

promote

bond with _re-

spect

to site electronic correlations. These essential

ingredients

of trie SP

instability

_can be added to trie model

by following

trie

"g-ology" prescription [15j,

in which trie direct interaction among

right-

and

left-moving

carriers is

decomposed

in terms of backward

(g~),

^forward

(g2)

and

(g3) umklapp scattering coupling

constants. Trie full

partition

^{function of} trie model in trie Fourier-Matsubara _space then becomes

Zlh~l

"

"

/ld~*1ldill ld4~l

_exP

SI _{fil*, ill}

₊

S°14~l

₊

Si W*, ~, 4'~I

₊

Si fil*, 4

₊

ShW*, ~, h'~I

+

/ldt~*lldt~lld4~l

^exp

~ IGÎ(1)l~~ #],a(1)~/,p,a(1) ~14~(4)l~lg$(uJm)l~~

p,1,a 4

+

/~jÀMO'~(4)4'~14)

4

+

~ _~

g~t~],~~ li~

₊

Q)t~lp,~~ll~ Q)t~-p,ai ll~)t~p,a~lÎ2)

lP,1,Q,al

+~ ~ _~j

92il],ai ^lÎ~

+

4)ilLp,a~ lÎ2 _4)ilp,ai (Î~)il-p,a~(12)

jp,à,<,aj

+

) _~

g3t~],~~ lt~

₊

_Qp)t~],~~ ll~ _Qp

₊

p©)~-~,~~ (l~ )~-~,~~ (i~)

lP,1,Q,al

+

~j (°f(Q)hf*14)

+

H-C) ), 18)

4,v

where k

=

(k,uJn

=

(2n +1)7rT)), Qp

₌

(pQ,

_mm

=

27rmT),

G

=

(4kF,o), G((k)

= [iuJn

ep(k)]~~,

^and

gf(uJm)

=

8g((MUJD)~~(1- M~/)D°(2kF, mm).

Trie

integration

_{measures are}

[d~l*][d~l]

=

fl

_j

~

d~l(,~(k)d~lp,~(k)

and

[dçi'~]

₌

fl~ _Q~~ gf(uJm) (~~dçi'~(Q)dçi'~(-Q)

for fermion and

alxiliary fields, respectively.

Here ^'

D°(2kF, mm)

₌

~

)~~

,

(9)

uJ~ uJ~

corresponds

to trie bare

phonon propagator. Finally, Sh

is _an additional term which

couples

an infinitesimal _source field

hf

_to site bond

(M

=

+,

_{~t =}

o)

CDW and site

(AI

= +,

~t =

1, 2,3)

SDW correlations. These

correspond

to composite fields defined

by Of*(i~)

₌

Î~~Î(Q)

+

ÀΰH(Q)),

Wl~ll

°~=0,~,2,3(Q)

"

(~/~)~~~ ~Î(,~,pl~~,a(~

^~

Q)~~~l~-,fl(~),

_~l,2,3

and _ao

being

the Pauli and the

identity

matrices

respectively.

In the

following

the source-field term will be useful for the calculation of relevant response functions for the SP

instability.

3. Renormalization

Group

Results

The

problem

of

low-energy

electronic and lattice correlations reduces to the

study

^of

interacting

electrons

coupled

to the

fluctuating

field

çi'~,

which _can be done

by

first

applying

the _renor- malization group

approach developed

in references

[10,16]

for the fermion

degrees

of freedom.

It consists of first

integrating high-energy

fermion states,

namely

_we write

~l(*)

-

~l(*) +1(*~,

where

~l(*)

describes

degrees

of freedom to be

integrated

_over in the outer energy shell of thick-

ness

Eo(f)df

_on both sides of the Fermi level at

+kF

and for all _mn. Here

Eo(f)

=

Eoe~~

is 2

the band _energy cut-off at the

step

f and

Eo

+

2EF

is the initial band

width,

which is twice the Fermi energy.

Keeping

trie

çi'~'s fixed,

this is

formally

written _as

Z[h'~]

_c~

/ [d~l*][d~lj[dçi~j e~'~*'~"" ~"lt~P~'~"t / _{[dj~j Idi] e~Î'~*~l}

(e~~+~I+~h

~ _{~_~}

"

/ ldi~~lldi~lld4~l ~~'~~'~'~~"~~'i'~~~'~~"'~~P (~j _j(IÉÀ

+

ÉI

+

Éh)~)o.s)

<

~

~ç

Î jd~j*jjd~/,jjdjmj ~s[~*,~,#~,h~]e+dt-8~Î4"lt+di~ jio)

which leads to _a recursion relation for electromc parameters ^{of trie action}

mcluding

those related

to _source fields for the calculation of _response functions

(see below>

and for

7[çi~]1,

which is

a

free-energy

functional which collects all the contributions in the

auxiliary

field

çi'~.

At finite temperature, ^the

partial integration

is conducted down to

éT

=

In(EF/T).

In the

present

renormahzation group

procedure,

electronic

degrees

of freedom _are treated in the continuum limit _so that the discreteness of the

underlymg

lattice will be

neglected

in the

following _[10].

3.1. ELECTRONIC PART

3.1.1.

Couplings

and One-Particle

Self-Energy.

It is useful for the

following

discussion to recall the well-known

two-loop

RG results for the

purely

electronic

part (çi'~

=

o)

in _zero field

(h"

=

o).

The eifect of the lattice fluctuations and the calculation of relevant _response func- tions will be considered afterwards. In this

purely

electronic

limit,

the

smgle-particle

fermion

propagator

^transforms

according

to trie recursion relation _z~

(é+dé)[G(]

^~~ _{= z~}

(é)z~ (dé) _[G(]

^~~

At trie

two-loop

level trie outer-shell correction to z~

(dé)

_comes from trie _n

= 2

(S/)~_~

terms of

(10),

which

yields

trie flow

equation:

~

()~

=

-jl12i~(é) ^{i~(é)l~ fil(f)}

+

31iil

+

3iilll, iii)

where trie

j~ (é)

e

g~(é) (7ruF )~~

are trie normalized

couplmg

constants at f. Trie latter transform

as

J(é

₊

dé)

=

j~(é)zp~(dé)z2,3,4(dé), which, together

with _zi, are obtained from

one-loop

(É/)~

~

and

two-loop (É/)~_~

outer shell corrections to

two-partiale four-point

vertex functions

r~,2,3(é

+

df)

=

z2,3,4(dé)r~,2,3(é).

This _is known to

yield

the flow

equations [15-17]

d9~

~2

1~3

$

^~~

2~~'

~~~~(j

^{~~~ = ii~11}

^{~121~ J~ )i,}

()

=

i~(21~ i~)11 121~

ii

_)i

(ii, i12)

For

(TMTTF)2X compounds, j~

_=t

j2

_> o _are

repulsive

at é

=

o,

whereas _a finite dimerization of the TMTTF stacks _can be

parametrized by

_a small and

positive j3

<

j~ [9,14].

One then

gets

^the

inequality j~ 2j2

_<

j3 indicating

that

umklapp scattering

^{is relevant. The flow of}

charge couplings 2j2 -à~

and

j3

then scales towards

strong coupling

while the

spin coupling j~

is found to be

margmally

irrelevant with the fixed

point

values

fi (é

_-

cc)

_-

2, fi (é

_-

cc)

_-

1,

and

fi

_- o. The

strong coupling

sector of the RG flow is reached when

g3(é

e

ép)

m

1,

where ép =

In(EF/Tp)

defines the

temperature

scale for the _presence of _a

charge (Mott-Hubbard)

gap

~lp

e

2Tp. Neglecting

transients between weak and

strong coupling regimes, one-partiale self-energy

corrections

resulting

from the solution of

equation (11)

_can be

put

ⁱⁿ the

followmg scaling

form

zi~(é)

_m

zi~(ép) (Eo(é)/Ap)°Î, (13)

which

depicts

the reduction of the

quasi-partiale weight

at the Fermi level with

9*(g(, g], g()

=

3/4 being

evaluated at trie fixed

point.

3.1.2.

Response

Fbnctions. Trie

pertinent

electronic _response functions involved in trie de-

scription

of the SP

instability

_{m one} dimension _are those related to site SDW (~t

# o,

^II ₌

+)

and bond CDW (~t =

o,M

=

-)

correlations [9].

They

_can be

readily

calculated via the renormalization of the _source field _term

Sh

_116]

which,

at scale

é,

reads

shl#~ #. h~li

"

~j (~f(é)hf~(4)°f (4)

_{+ H'~'}

xfl~kf> é)hf~(4)hf14)), _l14)

lv,M,ôl

where

zf _lé)

_is the renormalization factor for the

pair

vertex part, ^while

xÎ(2kF,é)

=

-(7ruF)~~ Î~ Ù$(é')dé', (15)

is the

2kF

_response function and

ii

"

(zf)~

is the

auxiliary susceptibility

_m the

(~t,M)

channel considered. The _one- and

two-loop

corrections to

zf

_come from the outer shell _averages

(ÉhÉI)o.s

and

(ÉhÉ/)~_s

^for n = 2 and _{n =} 3

respectively.

This leads to the flow

equation

~

~()"

=

if(é) (àÎ(é)

₊

iiÎ(é) à~(é)12(é)

₊

)iiÎ(é)1, ⁽¹⁶⁾

where

j* _lé)

=

j2(é)

_~

j3 If) 2j~ lé)

for the CDW channel and

j+~~(é)

=

j2(é)

+

j3(é)

for the SDW channel.

Followmg

the

example

of

(13)

for the

one-particl$self-energy renormalization,

if

_can be

expressed

in the

followmg scaling

form

xi jt)

_~s

ii _{(t~) jEo(t) /à~) ~~i'~, (17)}

where

if _(ép)

is the weak

couphng

contribution below the

charge

_gap. The exponents

~[(

~ ⁼

~*~

₌

3/2

obtained

by evaluating

the r-h-s of

(16)

at the fixed

point

indicate that

onli'site

SDW and bond CDW correlations _are

singular

at

half-filling.

It should be

mentioned, however,

the

g]

^obtained at

two-loop

level _are well known to overestimate the values of the

exponents.

Higher

order contributions _are

expected

to

bring

them doser to the exact value

~~Î,3

"

~*~

₊

~* "1, (18)

which is known to result from _more elaborate calculations at é » ép

[18j.

Before

closing

this section, it is useful for

applications (see

Sects. 3.2.5 and

4.2)

to compute the

imaginary part

of the retarded _response function at low

(real) frequency.

From reference

[16],

^{it is}

directly

connected _to

if

_as follows:

Im

,yf(2kF

_{+ q,}uJ) =

if _(T)

_Im

_y°(q

₊

_{2kF, uJ),}

uJ - o

(19)

for _ufq < T, where

Im

x° (q

+

2kF,

_uJ)

=

~~Im ~j _{Gf (k, uJn)G( (k}

₊

_2kF

₊

q, mn + uJ +

io+)

~ k,w~

=

~Î

uJ -

o, (20)

Tufcosh (flufq/4)

is the

imagmary part

^{of the}

low-frequency non-interacting

_response function _near

2kF (see Appendix A.2).

3.2. INFLUENCE _OF LATTICE FLUCTUATIONS

3.2.1.

Ginzburg-Landau

Functional. When the lattice

auxiliary

field

çi'~

is taken into _ac- count, the

partial integration (8) generates

a series of terms _{m n} > 2 powers of the

çi'~'s

_ma the closed fermion

loops (É[) In!.

This will not

only

lead to corrections for

S° [çi'~]1

^for n =

2,

but it

yields

_a recursion relation for the

quantum Ginzburg-Landau free-energy

functional

7[çi'~]1,

when combined to the _{n >} 2 terms

[10,16j.

Since

only 2kF

bond correlations _are

smgular

in the CDW channel for

repulsive couplings, only

the

dependence

_on the bond field

çi~

needs to be

retained,

with the result

flJ~i<~ii+di

=

flJ~i<~ii ~ ~ _Bnidé) <-141) <-14n)ôz _~~a. 121)

n22

o

The

quadratic (n

₌

2)

and

quartic (n

₌

4)

outer shell contributions lead to

82(dé)

=

(7ruF)~~[z~(é)]~ ^dé,

~~

_~~~~

Î~ÎÎÎEOÎÎÎ]~

~~'

_~~~~

where

((3)

_ci 1.2... For the

problem

at

hand,

_{one can assume} that

adiabaticity

^between electrons and the lattice _is

sufliciently _strong

that

quantum

^lattice corrections _can be

neglected, thereby allowing

the static limit.

Up

to the

quartic

term, one obtains the

followmg Ginzburg-

Landau free energy functional

lin

real

space)

71<~ii~

₌

/

dz

aiT)(<~ ix))

^~ _{+ c}

()

+

b14~ ix))

^~

l, ₍₂₃₎

~

at

temperature T,

where

a(T),

_c and b _are trie

Ginzburg-Landau parameters.

Trie coefficients

a(T)

and _c of the

quadratic

term _are

respectively given by

~(~)

_"

_Î9ph(°)1 ~+X (~~F>l~)

m

a'(T/T)p-1)

c m

~a'(uF/7rT)p)~, (24)

2

where a'

=

(7ruF)~~i~(T)p).

Here the bond CDW response function

X~(2kF,T)

in equa- tion

(15), together

with

(17),

bave been used in trie linearization of

a(T)

around trie SP

mean-field

temperature

T)p

m

Tp((jj~((~(Tp))~~~~ ⁽²⁵⁾

for

Tp

»

T)p.

It is worth

noting

that _smce

(jj~(

e

(7ruF)~~Î9jhÎ

^c~

g(,

then

T)p

_c~

g(

for

~*

=

1,

which turns out to be trie _same power

dependence

_on trie

electron-phonon coupling

constant that _was found in _previous mean-field calculations _{on more} localized

systems

_{[4]. As}

for trie

rigidity parameter

_c, it is

readily

obtained in Fourier space from trie

Q

"

2kF

+ q

expansion of

x~(Q, T)p) _using

trie

approximate expression

é(T(p, 2kF

+

q)

₌ In

~

[1 ⁺

(ufq/7rT(p)~]~i

TSP

,

as

boundary

condition of trie

logarithmic integration

_m

(15)

near

T)p.

This form

essentially

comcides with trie

Q dependence

of trie

elementary

Peierls bubble _near

2kF. Finally,

trie flow of trie mode-mode

coupling

m

(22)

is

stopped

at

éTo

^{and reads}

"

~~~~~

^~

16(1

Î~ÎÎ7rT)p)2

~~

~~~~~~~'

~~~~

for

Tp

»

T)p.

Fluctuation eifects in

çi~

below

T)p

_are then

governed by

trie classical functional

integral

Z =

J [dçi~j exp(-fl7[ô~j ),

which _can be carried out

exactly using

trie transfer matrix method

[19].

^An

important quantity

to compute is trie static 1D SP _response function _~sP. From trie

results of reference

[19],

one finds

xsp(2kF

_{+ q,}

T)

=

j /

d~

((4~ (z)4~10)))

e~~~~+~~~~~

2fl (( (4~)~ )) (SP

l +

q~fip

^' ~~~~

where

(( (. ))

=

z-~ /id<-i ^(. exp(-flJ~i<-1),

denotes _a statistical average over

ô~

Here

(sP

is trie correlation

length

of trie real order field

çi~,

which grows

exponentially

below

T)p according

to

(sP

"

~ _~

eP/~, (28)

p (a(

where

~~

~ =

(29)

~~~~~~~~~~~

~

~'~~~P'

for T <

T)p.

The

temperature profile

of both

(sP

and the _{mean square} fluctuations

(((çi~ )~))

are

plotted

in

Figure

1.

3.5 1.6

z

n __

n

«

~w ^°'~

?~Î

v -

"

0.4

O.o

0.4 O.fi

T/T~~

o

Fig.

l.

Temperature

variation of the _{mean square} of the SP order parameter

(left scale)

and the

inverse of the correlation

length (right scale)

normalized

by

the lattice constant.

~+

~

~fi

Fig.

2.

Leading one-partiale self-energy

correction due to SP fluctuations for electrons at +kF. The

wiggly (dashed)

fine

corresponds

to the lattice field

(electronic -kF)

propagator _xsP.

3.2.2.

Pseudo-Gap

Eisect. The influence of the static SP lattice fluctuations

governed by (23)

_on

one-partiale

electronic

properties

at the

step éT

^{of the RG}

procedure

will be

analyzed by considering

to

leading

order the one-electron

self-energy

contribution of

Figure

2 [3], ^which

can be obtained from

(8)

after _an

integration

_over the

çi~

field. The result is

Lj(k,(çi~))

=

-TL~~zp~zp ~j ^G° _~(k p2kF q,iuJn)xsP(2kF

_{+ q,}

T). (30)

q

The bare

propagator

appearing in

S)

then becomes

iGpii,14~i)i~~

=

z~iGjii)i~~ Li(1,14~i)

~

~~

_{~ ~~~~}

zi~(zi)~ ((14~i~))

" ~ ~~~~

iu~~ + ep

(k)

₊

ivf(il (T)

A relevant

quantity

to

compute

^is the reduction of the

density

of states per

spin resulting

from SP fluctuations

(pseudo-gap eifect).

After

analytic

continuation of

[Gp]~~

^{to real}

frequencies,

this is found _to be

~~~'~~

~~

iÎ~(~~~~~~'~'~~

^~~

p,

~

i

a121d

+

4~/~

~~~~

7ruF

12(d

+

~) a2]d'

where trie presence of trie renormalization

factpr

_{z~ ensures} that the

pseudo-gap

^{eifect is} the result of lattice fluctuations.

Followmg

trie notation of reference

[3j,

we bave

~ ~

~-l((jfl~-j2)j-1/2

F SP

b)

, /

Fig.

3. Dressed electron-hole bubbles _in the

a)

Landau and

b)

Peierls

(or Cooper)

channels. Thick and thin continuous

(dashed)

fines

correspond

to dressed and bore electron propagators,

respectively

at +kF

(-kF).

Fig.

4.

Two-trop a) one-partiale self-energy

and

b)

four points vertex

singular diagrams

involved

in the RG flow

including

the

leading

order corrections due to lattice fluctuations.

j

~((jfl~-j2))-~/2

~ = 1+

~o~-&i~

4

d =

(K~+i~a~)~/~, ₍₃₃₎

where _we bave defined

(4l~

_{(~ =}

zp~(z~)~

_(çi~

(~

Therefore,

_as trie

temperature

decreases below trie characteristic

T)p,

lattice fluctuations _grow and will

progressively

freeze electromc

degrees

of freedom. In

tutu,

this reduction will affect trie RG flow of various

quantities

considered in Section 3. The inclusion of lattice fluctuation eifects _m trie RG flow will be considered _to

lowest order where vertex corrections due to

exchange

of lattice fluctuations _are

neglected.

For

example,

at

one-loop

level of trie

RG,

this amounts to

substituting G(k, (çi~))

for _one of trie

propagators

_appearing in trie formal

expression

of trie

Landau, Peierls,

^and

Cooper elementary susceptibilities (Fig.

3 and

Appendix A).

At trie

two-loop level,

all trie relevant

next-to-leading singular diagrams

shown _m

Figure

4 involve

absorption

and emission of _an electron-hole _pair at small

fin

trie intermediate state, which _can be dressed

by

lattice fluctuations

(Appendix B).

From the results of

Appendices

A and B, this amounts to

replacing

the outer shell

logarithmic

contribution of all

diagrams by

^the

followmg expression

dé - 7ruF

D[Eo(é)/2, (çi~)jdé, (34)

where

D[Eo(f) /2, (çi~)j

is

density

of states

(32)

_m the presence of _a

pseudo-gap.

Substituting

this RG generator m the flow

equation

for the renormahzation factor

zp~

in

(11)

will slow down the electronic contribution to the

decay

of the

quasi-partiale weight

below

T)p.

Following

the

example

of the

purely

electronic _case, the

approximate "scaling" (transient-free)

form

(13)

then becomes

~i~lT)

"

~i~ITÎP)(T/TÎP)~~~~i> 135)

where

zp~(T)p)

is

given by (11)

and

9j(T)

₌

7ruFD[T,(çi~ )]9j

^becomes a

temperature depen-

dent

exponent

^below

T)p.

3.2.3.

Staggered Response

Fbnctions. The

pseudo-gap

will affect electronic correlations of the (~t,

AI)

channel _as well.

Indeed, substituting (34)

in

(16) yields

the

following approximate

"scaling"

form

1$(T,i<-i)

m

t$(Tip)(T/Tipl~~*~°~~, ⁽³⁶⁾

where

if (T)p)

is

given by (1î)

and

~*(T)

₌

7ruFD[T, (çi~)]~*.

As for the

imagmary part (19),

the above results and

Appendix

A.2 lead to Im

xi _iq

₊

_2k~,

u~,

jçi-j)

=

-iijT, _jçi-j)Im xiq

₊

2kF, W,14~i)

~

~~fi~'i~~i~ ÎÎÎÎUÎÎ/Î/ÎÎ

~' ~ ~ ° ~~~~

which is further reduced

by

lattice fluctuations.

3.2.4.

Magnetic Susceptibility.

The calculation of the

spin

_response ^function _ys at small

(q,

uJ) ^{is known} in the

purely

electronic situation

[21j.

^Its

generalization

when lattice fluctua- tions _are

present

^is

straightforward

and

yields

xs

(q,

_W,

14~1)

=

(~~~~'~' ^{i~ i~}

,

138)

1

jgi(T)x(q, ^W,14~i)

where in the above scheme of

approximation ~(q,uJ, (çi~))

is the

elementary

electron-hole bubble dressed

by

lattice fluctuations

(Fig.

3a and

Appendix A.1),

and g~

(T)

is given

by (12)

at

éT. According

to the results of

Appendix A,

_one finds

~~

~~~'~' ~~

_~~

Î~ ~~~" ~~

_~~

ÎÎ'~ ^{~~' ~°} pÎÎ~

uJ'

p

IIU

Xl~,

_~>

14~1)

"

/ ~ ~l~" 14~ Il (~ )) _{d~' ~j} P~~F~ô(PUF~ ~) (39)

p

for the real and

imaginary

parts ^{of the dressed} electron-hole bubble. It follows that in the static

lu

_-

o)

and uniform

(q

_-

o) hmit,

the

temperature

^{variation of the}

spin susceptibil- ity ~s(T, (çi~))

will be

depressed

in the presence of lattice fluctuations that grow up below

T)p (Fig. 5a).

In the very

low-temperature domam,

the SP correlation

length

becomes _ex-

ponentially large

and the

magnetic susceptibility

becomes

thermally

^{activated. When these}

temperature

^conditions

prevail,

the

system

is almost

long-range

ordered.

3.2.5. Nuclear Relaxation Rate. From the above

results,

the

temperature dependence

of the nuclear

spin-lattice

relaxation rate

Tp~

_can be

readily

calculated. The basic expression for

Tp~

in one dimension _is well known to be

Tp~

₌

(-Î(~T /dQ~~ ^{~~~'~~, (40)}

uJ

4.0 a)

3.0 j

) ~°

Z-o

~

l-o

o-o

Î

b) 40

j

î zo

~~

0

O.o 0.5 1-o 1.5 Z-o 2.5 3.0

T/T~~

o

Fig.

5. Calculated low temperature variation of

a)

the uniform

magnetic susceptibility

and

b)

_nu-

clear relaxation rate in the _presence

(continuous fine)

and absence

(dashed fine)

of lattice fluctuations.

which _can be taken in the limit _{uJ -} o. Here Im _x is the imagmary part of the retarded _spm response function and

À

_is

a constant

proportional

to the

hyperfine couphng [20j.

It is well

established that uniform

(Q

+~

o)

and AF

(Q

'~

2kF)

_spm fluctuations _grue the essential _con- tributions to the relaxation in _one dimension

[21], allowing

in tutu to make the

decomposition

t~

"

t~lo

'~ ol +

Ti~lo

'~

~kfj. (41)

For the

staggered

_part

Tp~ [Q

'~

2kFÎ,

we have _seen from

(37)

that Im

x(2kF

_{+q, uJ)} whose defi- nition coincides with -Im

x$~o(2kF

+q,

uJ))

is

peaked

when _q lies in the interval

[-T/uF, T/UFÎ,

which leads to

Tp~[Q

'~

2kFÎ

Cf

CITD[T, (çi~)]$((T, (çi~)), (42)

where C~ =

7rvj~(À(~tanh(1/4).

As for the uniform

contribution,

the _use of

(38-39) immediately

leads to

~ ~

DjT, ii-11

,

TP lQ

_'~

°'

_"

~° (43)

ji jg~ (T)DjT, ii- iii

^~

where

Go

_"

47r(uF)~~ _ÎÀ(~.

As

expected,

the reduction of _spm fluctuations

by

short _range lattice correlations below

T(p

will decrease the

amplitude

of the relaxation via the reduction of the

density

of states. At

sufliciently

low

temperature, namely

when the correlation

length (SP

becomes

exponentially large,

_one finds

Ti~

'~ xs '~

e~P/~.

_{In the}

high-temperature

_regime, lattice fluctuations _are small and the uniform component

Tp~

r-

COTXÎ eventually

dommates the relaxation.

Using

the

scaling

form

(36)

_m

(42)

and

(43),

the

temperature dependence

of

Tp~

is summarized in

Figure

5b where it is

compared

to the _case without fluctuations.

(TMTTFJ~PF~ ^{iP i Bar)}

_ o

~ ~sP

~

Q «

CD

j/

Tsp +

~t+#

+++

'

~

/

~~

₊

O.O

0 20 40 60 80

TeniperaLure (K)

Fig.

6.

Comparison

between calculated

(continuous fine)

and observed

(crosses)

temperature _pro- files of

magnetic susceptibility

for

(TMTTF)2PF6.

The data _are taken from reference [7].

4.

Application

to the SP

System (TMTTF)2PF6

By

_way of

application

^{of the}

present theory

to the

sulphur

based

organic compound (TMTTF)2 PF6.

Previous

analysis

of this

system

^{in the normal}

phase using

the one-dimensional electron gas mortel _can be used for the determmation of the

input

^bare

parameters

^{of the}

purely

elec-

tronic

part

of the model. NMR

analysis

of Wzietek et ai.

[12]

^{have shown that}

j~

_ct

j2

Cf

0.9,

with

EF

_Cf 1600

K, give

a rather

good description

of the

temperature dependent magnetic susceptibility

in the

high temperature

domain of this material. As for the observed character- istic

temperature

scale

Tp

re 220 K

(below

which the

system presents insulating properties),

it can be used _to

identify, together

with

(12),

trie

low-temperature

domam of

strong umklapp scattering,

which then allows _one to take

f3

_~S o.2 [9]. As for trie

input parameters

^{for trie} lattice

component,

one will fix trie value of

T)p

at 60 Il which is trie characteristic

temperature

scale for trie onset of

strong

^lattice fluctuations in

X-ray experiments [6j.

^{From trie above set} of

figures

ail

quantities

of interest _can be calculated.

4.1. MAGNETIC SUSCEPTIBILITY. Trie

temperature-dependent

EPR

spin susceptibility

(TMTTF)2PF6

measured

by

Creuzet et ai.

I?i

^is

reported

in trie

low-temperature

^domam in

Figure

6.

xs(T)

decreases

monotonously

from trie

high temperature

domain and becomes

weakly temperature dependent

_near 80

K, y7hich

is

typical

of all members of trie

sulphur

_serres

m the normal state.

However,

m the low

temperature

^{domain below 60}

K,

trie spm

suscepti- bility

decreases

by roughly

40$io down to the true SP transition at

TSF

_~S 19

K,

below which it

becomes

thermally

activated.

Using

the above set of

parameters

^{for the}

model,

the theoretical

prediction

for

ys(T. (çi~ ))

is illustrated in

Figure

6 and

gives

_a

fairly good description

^{of the}

temperature

variation of xs m the SP

pseudo-gap regime.

At very low

temperature,

^the

present theory predicts

_a

thermally

activated behaviour when

(sP

_grows

exponentially,

which is found to mimic the actual

temperature dependence

below the true transition

temperature

at 19 K.

However,

a more realistic

description

of the SP system m this low temperature

region,

^would _require the inclusion of the interchain

coupling.

60

/

l'i&iTTF) PF

Ù 40

~

~

~

aj

~u sp

0 40

Temperature (K)

Fig.

7.

Comparison

between calculated

(continuous fine)

and observed

(crosses)

temperature _pro- files of nuclear relaxation rate for

(TMTTF)2PF6.

_{The data}

are taken from reference [7].

4.$.

NUCLEAR RELAXATION. The

temperature profile

of

Tp~

for

(TMTTF)2PF6,

_measured

by

Creuzet et ai.

I?i,

is

given

in

Figure

7. From the

analysis

of nuclear relaxation in the

high temperature

domam

T)p

< T _<

Tp,

where SP fluctuations _are

weak,

the contribution to

Tp~

is well known to be

purely

electronic in character. A

quantitative description

of the relaxation rate in this

regime

_can be obtained from

(42)

and

(43) neglecting

the

dependence

on

çi~ [12].

^Below

T)p,

the relaxation rate shows _a 30$io decrease between

T)p

and

TSF

due to one-dimensional lattice fluctuations.

According

to

(42)

and

(43)

both trie

staggered

and uniform parts of trie relaxation _are aflected below

T)p.

Thus, _usmg these

expressions

for trie above set of parameters, one obtains trie

Tp~ temperature profile

shown in

Figure

7. Trie theoretical _curve is obtained from trie

expressions (43), (42)

and

(36)

in which trie values of trie

constants

Co(7ruF)~~

_Cf 12.1 and

C~(7ruF)~~Tp(((Tp)

ci 9.6 results from trie

analysis

of

Tp~

data made

by

Wzietek et ai.

[12],

ⁱⁿ trie

high temperature

domam T »

T)p.

Therefore trie above low temperature results for the nuclear relaxation rate

complete

the

previous analysis

made in reference

[12].

Acknowledgments

The authors thank J.-P.

Pouget

and L. G. Caron for _numerous discussions. We would also like to thank D. Sénéchal for useful comments about the manuscript. Financial support ^from the Natural Sciences and

Engineering

Research Council of Canada

(NSERC),

le Fonds _pour la

Formation de Chercheurs _et l'Aide à la Recherche du Gouvernement du

Québec (FCAR)

and Canadian Institute for Advanced Research

(DAR)

_is

gratefully acknowledged.

Appendix

A

Elementary Susceptibilities

in Presence of Phonons

In this

appendix,

we

proceed

to the calculation of the

elementary

electron-hole bubbles dressed

by

lattice fluctuations _m the

Landau,

Peierls and

Cooper channels, respectively.

SUSCEPTIBILITY AT SMALL q ^AND u~. The

expression corresponding

ta the electron-hale bubble of

Figure

^3a is

xii,14~i)

=

-) _~ Gpii,14~i)Giii

₊

_@, (A.l) p,1

for _one

spm~rientation. _Using

_the

_spectral representation

^of

Gp(k, (çi~))

and

performmg

the fermion

frequency

_{sum, one} finds

~~~'~~

_~~

_~~~~

~~j Î_~ ~~'~~~~~~'~"~~ ~~iÎÎ~ÎÎ(Î~ÎqÎ~~ÎÎ' ^~~'~~

,

~~ _'

Since most of the

spectral weight

_appears in the

region

^uJ' re

ep(k),

_{one can}

replace ep(k) by

uJ'

m the ratio of the above

integral,

which _can be cut off at

+Eo/2.

From the definition of the

density

of states

(32),

_one

finds,

after

analytic

continuation to real

frequencies,

~~

~~~'~'~~

_~~

Î~~Î~ ^{~~~" ~~}

^~~

ÎÎ'~ ^~~'~° pÎÎ~

uJ p

Im

x(q,

_uJ.

(çi~ ))

=

~~~~~

D[uJ', (ô~)] (- ^Î~ du'~jp7ruFqô(pufq uJ). (A.3)

2 ~~/~ ôàJ

P

Here

D(uJ, (çi~ ))

is the

density

of states per spm in the presence of the

spin-Peierls pseudo-gap.

The calculation of

y(f, (çi~))

at the

step

^{é of} the renormahzation group

procedure gives

the

same expression, except ^for

Eo

which is

replaced by Eo(é).

PEIERLS AND COOPER SUSCEPTIBILITIES. The calculation of the Peierls electron-hale bub- ble of

Figure

3b starts with the

following expression

x(2kF

+ q, mm,

(çi~))

=

~~

~j G-(k,

_mn,

(çi~))G((k

₊

2kF

+ q, uJn +

mm), (AA)

~ k,w~

for both

spin

orientations. At _zero external variables and after _a summation _over the fermion

frequencies

and the _use of the

spectral representation

for

G-(k, (çi~)),

_one

_gets

x(2kF, (4~))

"

-~ _{~j /~~}

_{du~' Im}

_G-(k,

u~,

(ô~)) ~~~), /) ^j~~~~, _(A.5)

k ^-"

which

actually

coincides with the real

part

^{of the Peierls bubble.}

Assummg

that Im

G-(k, uJ',

(çi-))

is

peaked

in the region uJ' _re

e-(k),

from which _one

replaces e-(k) by

uJ' _m the ratio appearing m the

integral,

we find

"

x(2kF,14~i)

=

~°~~

DIW', ~ildW'~~~~~jÉ'~~~

IA-fi)

As for the electron-electron

(Cooper) elementary

bubble at _zero external variables

correspond-

mg ^to trie expression

x(lP~i)

=

( _~ G-(1,14~i)Gi(-1), _(A.7)

1

trie

property G( (-k,

_-mn

=

-G( (k+2kF,

_mn leads to trie relation

x(2kF, (4~ ))

_"

xl (4~ ))

Within trie renormalization group scheme at

é,

trie _same expressions become

dx(2k~,jçi-j)

₌

-DjEo(é)/2,jçi-jjdé

=

-dx(14~i), (A.8)

when ekaluated in trie outer energy shell.

Finally,

after

analytic

continuation to

rell frequencies

of

(A4),

_{we can carry over} trie _same

type

^of calculation for the

imaginary part

of the Peierls bubble at T in the hmit of small

frequency

and _we find

Im

~~2kF

_{+ ~}

i lil/Î[[~,lj~,,

j< ii ini~ii niuJl

₊

vf~ii

à

i~'

₊

_{~~ ~~~~/~~}

~ 2 -Eo/2

(~'~~

=

~

)~~~~~~~ ^~~

^~

Appendix

B

One-Particle

Self-Energy

and Four-Point Vertex Part in Presence of Phonons

ONE-PARTICLE SELF-ENERGY. The

expression

^for the _one

particle self-energy diagram

of

Figure

4a reads

L+ (k, 14~1)

=

-2g~ )~

^~

~

_G-

(1',14~1)Gf (1'

₊

@GÎ Ii

i>,1

ci

2g~ ~~ ~j ~j /~~ ^duJ'lm G(k', uJ', (çi~ ))

XL

~, ^_~

~

[iuJn

e+(k) -ÎÎÎÎ

+

ÎÎÎÎ ÎÎÎ'Î ^ÎÎÎ~~ _e-(k')

+

uFqÎ' ^~~'~~

where the second fine results from _a fermion

frequency

summation and the _use of the

spectral weight representation.

The

approximation

scheme of

Appendix

A then allows to put

e-(k')

_m

uJ' _m the ratio appearmg m r-h-s of this last

expression

and to cut off the

mtegral

_over ^uJ' at

+Eo/2.

In trie RG

procedure,

trie outer energy shell evaluation of

L+

at

é,

which is obtained after trie

frequency

_{sum over mm} leads to

dL+

₌

dL(

₊

_dLj,

where

dLl(1,14~i)

=

-(DiEo(é)/2,i4~iidEo(é)

~

j ~~ ml+Eo(é)/2 vfql n1+Eo(é)/21) lnB1-vfql

+

nlvfq

+

e+(k)1) 2v~q

_{+ iu~n e+}

jk)

Cf

( iG°ii)i~~ DiEoié)/2, 14~ ii dé, iB.2)

to

leading

order _m

[G°(k)]~~

₌

iuJn e+(k),

for

flEo(é)

_»

_1,

and where _nB

(xi

=

(eP~ 1)~~

Here the

integration

_over trie momentum transfer _{q is} found to contribute

only

_m the interval

2uFqo

>

2uF

_(q( >

Eo(é),

where _qo is _a momentum transfer cut-off.

VERTEX PART. The evaluation of the

two-Îoop

vertex

part

ⁱⁿ the presence of Îattice fluctu- ations

proceeds along

similar lines. The

diagrams

of

Figure

^4b

corresponds

_to

ni

+

r)~~

^and

to the

generic expression

r[~)((çi~ ))

j~2

=

-g~

~

~j G~(k', çi~)G((k' -1)G( (k~,2 +1)G$(k~,2 +1 _@. (B.3)

~ [,

)

In trie RG _{sense one can}

drop

the

dependence

_on the external variables

(ki, k2, iÎl Performing

frequency

_{sums on mn>} and _mm> and usmg trie above

approximation scheme,

_one

gets

dr[11

t g~

~)~j DlEo (é)/2, 14 Il dEo (é)

x

/ j _iniEo(é)/2

+

v~q'i

_n

iEoié)/211 inB i-v~q'i

_{+ n}

_iv~q'ii

₊

Eoié)

_-

-Eo if))

t g~

jDlEoié)/2, ^{Ii- Il dé, (B.4)}

for

2uFqo

>

2uFÎq'Î

>

Eo lé), flufq'

_»

1,

and

flEo(é)

_> 1.

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