Modulated spiral phases in doped quantum antiferromagnets

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Submitted on 1 Jan 1990

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Modulated spiral phases in doped quantum

antiferromagnets

Thierry Dombre

To cite this version:

847

Modulated

_spiral

_phases

in

_doped

_quantum

_{antiferromagnets}

Thierry

Dombre

Centre de Recherches sur les Très Basses

_{Températures, CNRS,}

BP 166 X, 38042 Grenoble

Cedex, France

(Reçu

le 29 novembre 1989,

accepté

le 15

janvier

1990)

Résumé. 2014 Nous étudions la

phase spirale

découverte par Shraiman et

_Siggia,

dans le

_régime

où

sa constante de raideur devient

_négative.

Nous montrons _que cette instabilité ne

_signale

_pas une

séparation

de

_phase

mais

_plutôt

la formation de murs suivant les directions

_{(1, 1)}

ou

(1,1)

sur un réseau carré, où les trous se localisent. Nous discutons brièvement l’effet d’une

répulsion

coulombienne à

_{longue portée}

dans une telle structure.

Abstract. 2014

We

_study

the

_{spiral phase}

discovered

_by

Shraiman and

_Siggia,

in the

_regime

of

coupling

constant where the effective

_spin

wave stiffness is

_negative.

We show that this

instability

does not

_signal

a

_{phase separation}

but rather the formation of domain walls in the

(1,1)

or

(1,1)

directions on a _{square lattice,} where holes _get

_trapped.

We discuss

_briefly

the

interplay

between

_dipolar

and Coulomb interactions in the

_resulting

structure.

J.

_Phys.

France 51

₍₁₉₉₀₎

847-856 1er MAI 1990,

Classification

Physics

Abstracts 75.10L - 75.25 -

75.30F

1. Introduction.

Recently,

Shraiman and

_Siggia

_[1]

have written within the so-called t - J model a

semiphenomenological

Hamiltonian

_describing

the motion of holes in a

_locally

Néel ordered

antiferromagnet.

Their

_{major finding}

for a

_single

hole was that for wave vectors around the energy minimum

(at

k =

( ±

TT /2,

±

_{TT /2),}

at least for not too

_large

values of

_tIJ),

the vacancy is dressed

_by

a twist of the

_{staggered magnetization taking}

a

_{dipolar configuration}

at

large

scales. This effect comes

_physically

from the

_{coupling through hopping}

of the hole momentum to the

_spin

current carried

_by

the

_magnetic

medium around. The

_approach

followed in

_{[1] (hereafter}

referred as

_I)

was

_essentially

of a semiclassical nature and

_{applies a}

priori

to the limit

_t/JS

1.

_Although

the

_picture

found in 1 is less obvious in the

_{large t}

_limit,

where we know that the hole disorders _very

_strongly

the

_spins

within a core

_region

_[2],

it

seems to be confirmed

_{qualitatively}

in recent numerical studies

_[3].

_{One may therefore think} that the Hamiltonian derived on 1 is still of

_physical

relevance in this limit

_although

with some

renormalized

_coupling

constants.

In a

_subsequent

work

[4] (hereafter

referred as

_II),

Shraiman and

_Siggia

considered the situation of a small but finite

_density

of holes. Thanks to the semiclassical

_analysis

of

_I,

one

JOURNAL DE PHYSIQUE. - T. 51, N° 9, 1CR MAI 1990

can _very

_easily

understand the nature of the interaction between holes at

_large

scales. Indeed thé

_long

_{range tails of the static}

_spin

clouds around each hole mediate a

_dipolar-like

interaction between them. It appears that

beyond

a critical

_{value g, of}

the

_coupling

of the holes to

_spin

waves, holes

prefer

to have their

_dipole

moments

_aligned.

This results in the

formation of a «

_{spiral » phase,}

_{characterized}

_by

a uniform

_planar

rotation of the

_staggered

magnetization

with an

_{incommensurability}

wave number

_scaling

like the

_density

of holes.

However,

it was

_pointed

out in II that

if g

exceeds

ô

_gC, the

_spin

wave stiffness as

renormalized

_{by particle}

hole fluctuations becomes

_negative

in the same _way as it does in an

homogeneous

Néel

_{phase for g}

> _g,. It is the main purpose of this paper to elucidate the nature of the

_phase

which should _{appear above this second threshold. We} find that this second

instability

corresponds

to the formation of domain walls in the

_spin

orientation,

along

the

(1, 1)

or

(1,1 )

directions on a _{square lattice,} where holes

_get

_{self-consistently trapped.}

Once

they

are

_formed,

it is

_{energetically}

favourable for these domain walls to accomodate as _many

holes as

_they

can. A _proper

_description

of the final structure

_consisting

of

_sharply

localized soliton lines would

_require

more

_knowledge

on the short range

physics

in a

_doped

antiferromagnet.

This matter is

_{clearly beyond}

_{the scope of} our

_long

wave

length approach

which nevertheless is able to _{prove the formation of} a « modulated »

_{spiral phase right}

above

g =

à

_gc. Since there is _an accumulation of

charges

_on these domain

walls,

it is worth

investigating

how their formation is affected

_by

a

_long

_{range Coulomb}

_repulsion.

If the

associated Bohr radius ao is of the order of the lattice

spacing

a, then the

spiral phase

is

restabilized

_by

Coulomb effects which are however

_likely

to

_promote

in this case the

formation of a

_{Wigner crystal}

at low

_density.

If on the

_contrary

_{ao is much}

greater

than a

_(a

somewhat unrealistic

_hypothesis)

we find that domain walls could still appear

provided

that the

_density

of holes is

_{high enough.}

The paper is

organized

as follows. In section 2 we recall some basic features of the

Shraiman-Siggia

theory

and introduce a

_simple

version of their Hamiltonian which contains all the instabilities we are interested in. In section

_3,

we _{calculate the energy of} a

_single

domain wall and show that it becomes

_negative

_{at g} =

h

gC,

independently

of the

density

of

holes within the wall. We _{argue that the interaction between domain walls is}

_{repulsive. Finally}

Coulomb

_long

_{range effects} are discussed in section 4.

2.

_Simple

considérations on the

_{spiral phase.}

We first recall some basic features of the

_{Shraiman-Siggia}

_theory.

Holes are

_represented

by

spinless

fermions,

spin degrees

of freedom

_{by Schwinger}

bosons or in the

_{large S}

limit a unit

vector à

whose

_dynamics

is controlled

_by

a non-linear u model. Local Néel order halves the

Brillouin zone and it is convenient to introduce two

_components

for the hole annihilation

operator,

.p A

and

_.pB

_{corresponding}

to the two sublattices. These two

_components

form a

spinor,

hereafter noted , One

complementary important assumption

of this work is that the energy minimum for a

_single

hole is at

_k1

=

( + TT /2,

₊

7r/2)

_or

k2

=

(

+

TT /2, -

_{TT /2). Although}

this fact is confirmed

_by

various numerical works at values of

tIJ

smaller than

4,

finite size effects make it more difficult to check for

_larger

tIJ

[3, 5].

As

will be clear in the

_following,

the

_physics

to be discussed _{in this paper} is _very

sensitive to the

_position

_{of the energy minimum.}

849

the

_long

wave

_length

_components

of

_.pi.

It is

_{straightforward}

to take the continuum limit of the

Shraiman-Siggia

Hamiltonian for the fields Xi. It reads

In this

_equation

the last term is

_purely

_{magnetic :}

it is the non-linear a model

_governing

the

spin

fluctuations at

_large

scales. 0 and cp are the Euler

_{angles parameterising}

the local

direction of the

_staggered

order

_parameter

à.

m is the local

_{magnetization}

_operator

which is

conjugate

to

03A9. X ~ J-1

is the

_{magnetic susceptibility}

and p ~

JS2

the

spin

wave stiffness.

The first two terms take their

_origin

from the

_hopping

_processus. The first one

_gives

the

band structure of the holes within each

_valley.

It builds _up

_{predominantly}

from the

_coupling

of holes to

_high

_{energy virtual}

_spin

fluctuations. The band structure is known to be very

anisotropic

and we have to

_specify

here our choice of

_spatial

coordinates. We take the

_{(1, 1)}

and

_(1,

_{1 )}

directions as

_principal

axes i.e. we define and. . In such

coordinates the effective inverse mass tensor is

_diagonal

and

Both

_{perpendicular}

and

_parallel

masses scale in the same _{way with}

(t/f): U -1

_{goes like}

l 2 a2

at small t and becomes of order

Ja 2

at

_{large t [6].}

It is worth

_noting

that this first term

represents

hopping

events where the hole remains on the same sublattice and as such should

be affected

_by

Aharonov-Bohm effects discussed

_by

_{many authors}

_[7],

which

_originate

from

the

_{complex overlap}

between

_{neighbouring spins.}

However,

the

_phases

we are

_going

to

discuss do not _{involve the creation of any fictitious flux because}

_they

have no

_skyrmions,

in

contrast for instance to the double

_{spiral phases}

discovered in

_[8].

Therefore for our

_present

purpose, we can

_forget

about this

_{complication.}

The second term is the central

_part

of

_{Shraiman-Siggia’s theory}

since it deals with the

problem

of coherent

_hopping

from one sublattice to the

_other,

as is made clear

_{formally by}

the

presence of the u x or

_o-Y

matrices : the unit vectors

_pi

are directed

_along

the momenta

ki

within each

_valley

to

_leading

order in n. As said in the introduction these vectors _appear like

_dipole

moments which

_couple

to the « electric fields » sin 0

_Vq

and V 0. In the small t

limit,

this term is obtained

_by

_writing

the

_hopping

_part

of the hamiltonian in terms of

Schwinger

bosons and

_spinless

holes

operators

and

_expanding

the

_resulting

_expression

to lowest order in

_gradients.

One

_{gets g}

= 2

à St

(note

that _our definition

of g

differs

by

_a

factor

.J2

from the one used in

_I).

It was

_argued

in 1 that in the

large t

limit g

should saturate to some value of order J which seems

_{unfortunately}

_{very difficult} to calculate

_{microscopically.}

Note that in

₍₁₎

we discarded the

_coupling

of the hole

_density

to the local

_{magnetization}

present

also in 1 because it is

_higher

order in n for wave vectors concentrated near

k1

or

_k2.

Equation (1)

constitutes the

startinf

point

of our

_analysis.

We can

_simplify

_things

further

_by

rotating

the

_spinors

_{like X}

=

e - i (7T /4 u y X

(as

was also done in

_II)

in such a _{way that the x}

It is observed that the first term in

₍₂₎

is

_diagonal

in

_spin

_{space in this} new

_{representation.}

It

is

now _easy to understand the nature of the

spiral

_phase.

We

_imagine

a

_phase

where the

_spins

stay

in a

_plane

which can be

_conveniently

chosen to

_correspond

to 0 =

7r/2.

Each

valley splits

into two subbands with an effective

_spin

index i

or 1

and the conditions for

_reaching

a

stationary

energy minimum are

_{given by}

The

_{spiral phase corresponds}

to _{the appearance of} a finite but uniform value of

d1’P

or

_a2~.

It is

_simple

exercise to _{calculate the energy of} a

_{polarized phase}

_{(ni t #}

_Ni

_).

One obtains for each

_valley

_{and per unit} area

Thus,

provided

that ;

it is

_advantageous

to

_{fully polarize}

each

valley

i. e. ni =

ni f, ni = 0. This indeed

maximises

1 ni t

-

ni

at a

_given

value of the

density ni

carried

_by

each

_valley.

The _energy of a

_{fully polarized valley}

is

where the coefficient of

_nl

in

_{equation (5)}

is

_positive.

This means in

particular

that at a

_{given density n}

=

n, + _{n2, it is}

_{energetically}

favourable to have ni =

n2, i.e. an

_equal

distribution of the holes between the two

_valleys.

_Tuming

back to

equation

(3)

we see that this result

_{implies 1 dl Cf’ 1 =}

_{a2cp .}

In other

_words,

the uniform twist of the

_{staggered magnetization}

characteristic of the

_{spiral phase}

has to be directed

_along

the

(1, 0)

or

_{(0, 1)}

_directions,

as obtained in II. On the other

hand,

when A > 1 the

compressibility

of each

_valley

is

_negative

and the

_{spiral phase}

cannot be

_{thermodynamically}

stable.

A more elaborate calculation was

_given

in II where the renormalization of the

_spin

wave

propagator

by

particle-hole

fluctuations was considered. One finds the renormalized static stiffness

_{p (q ) = p - g Z a 2 X d (q, 0 )}

where

_{X d (q, 0 )}

is the static

_{susceptibility}

of

_2D-free

electrons. To

_leading

order in n, a

_{given valley}

contributes

_only

to the renormalization of the

sin 8

_aIq

or

_{aI o}

terms where i is the index of the

_valley.

At zero wave vector

X d =

(u,

Ir 7ril

)1l2

and one recovers the same criterion for the

_instability

of the Néel

_phase.

It

is

_interesting

to

_repeat

the same calculation in the

_{spiral phase.}

The

_spin

fluctuations in the

plane

of the

_spiral,

as described

_by

_’Vcp,

do not mix the bands

_{1’ and 1 according}

to

₍₂₎

and

_they

are renormalized as in the Néel state

_except

that _Xd has to be divided

_by

two because we are now

_dealing

with a

_{fully polarized}

state. Therefore these fluctuations

_acquire

a

_negative

static

stiffness

_{for g}

>

.J2

_gc. In _{contrast, the}

spin

fluctuations out of the

plane

of the

spiral

induce

interband transitions and

_they

will be renormalized

_by

the hole’s

_{bubble-diagrams}

in a

851

As

before,

only

the

_valley

_carrying

the index i contributes to the renormalization of the

propagator

of

_{Õi 8.}

_By

_{definition, the gap between the}

two subbands of a same

_valley

in the

spiral phase

is

_{2 ài}

= 2

ga

1 ai -r

+ 2 g2 a2

ni.

Carrying

back this result in

(6),

one sees that

P

P0

vanishes at zero wave _vector,

_regardless

of the value of the

_{coupling,constant}

g. In

fact,

it

can be shown that

_{p ~ (q )}

as defined above is a

_{positive quantity}

for all values of q,

behaving

like

_q2

at small wave vectors. Since the same

_property

should hold at finite

_frequency,

we

conclude that the transverse fluctuations of the order

parameter à

are

_completely

stable in

the presence of a

_spiral

distortion

of Û,

even for a > 1.

These considerations lead us to concentrate our search for structures

_appearing

above a = 1 on

_{purely planar phases}

i.e.

_{involving only}

variations of an azimuthal

_angle

cp at a fixed

value of 0. To motivate the next

_section,

let us

_point

out that the t - J model in the small t limit does indeed

_yield

a

_coupling

_{constant g}

which exceeds

.J2

_g,,. In this limit all

quantities

can be calculated

_exactly.

We

_already

_quoted

in the text p =

JS2 and g

= 2

.J2

tS. The

effective masses _{IL 1.}

_{and U}

_Il can be obtained from standard second order

_{perturbation theory}

in

_t/J

_[9]

or

_numerically

from exact

_{diagonalization}

methods on small clusters

_[3].

Both

approaches

agree to

_give

_{IL 1. --}

1.6 lt- 2 a- 2,

ml! ~ 4.2 lt- 2 a- 2.

_Combining

these

_expressions,

we find a - 3.3. Even if the t - J model is not

very

physical

in the small

t-limit,

this remark

means that a numerical simulation of this

_problem

for a sufficient

_large

size of the lattice

would not

_give

a

_{spiral phase}

but a different structure which we would like now to understand

(see

however the end of section 3 for a brief discussion of other

possible

instabilities in the

t - J

model).

In the

_{large t}

limit,

« must take on

_{general grounds}

a value of order 1

independent

of

_t/J

but we cannot _{say whether it is}

_bigger

than 1

_(or

_bigger

than 1/2

corresponding

to the first

_{instability).}

3. Modulated

_{spiral phases.}

From the discussion of the

_preceding

_section,

it should be clear that in order to restabilize the

spin

waves one must find a _way to lower the hole

_{susceptibility.}

This can be achieved

_by

opening

gaps in the hole

dispersion

relations,

or in other words

_creating

localized holes states. We thus shall consider

_{inhomogeneous}

distributions of the

_gradients

of cp :

by choosing

cp

(x, y)

of the form

F (x1 )

+

_{F (X2)}

we

_decouple

the two

_valleys

_(to

the order of

_{approximation}

of Hamiltonian

_(1)).

Since we want the two

_valleys

still to be

_{fully polarized,}

we take F to be a

continuously increasing

function of xi or _x2.

_According

to

_(1),

in that case,

only

the lower subbands will be

_populated

at zero

_temperature

within each

_valley.

The

_{spiral phase}

can be described

_by

_{F (x )}

=

Qx.

For the

valley

1,

_we have _to find the

eigenstates

of the Hamiltonian

with the condition that

_axl

_{where the average is taken} over all

_occupied

states

_(of

_negative

_energy).

We first

_study

the case of one

_single

localized structure or domain wall across the

xl direction, uniform

_{along x2}

and

_carrying

an

_arbitrary

number of holes

_Nh.

This number

divided

_by

the

_{total length}

in

_{the x2}

direction defines a linear

_density

of hole within the defect

which we _{call np . The hole}

_eigenstates

are still

_plane

waves in the x2 direction but we assume

that

_they

share the same wave function

_{perpendicular}

to the domain wall.

_According

to

energy solution normalized to

_unity

of the

_stationary

non-linear

_{Schrôdinger equation}

Rescaling

the wave function and the coordinate as 1 with the

_{length f}

given by

we obtain the

_following

_equation

_{for §}

_(with

the condition

that /l

be of unit

_norm)

The solution of this

_problem

is well known : to obtain it

_simply,

we

_identify

u =

1 /

I É I 03C8

and T =

I É I

X _as

respectively

_a

position

and _a time and _treat

equation (9)

as the classical motion of zero _{energy of} a unit mass

_point

in the

_potential

_{V (u) =}

In this way we

_get

the localized solution whence follows the

value of _{the energy}

_However,

to avoid

_counting

twice the

interaction energy, we have to add to this result so that the

global

energy per hole

coming

from the motion

perpendicular

to the soliton line is - 1/3 in reduced units.

_Reexpressing

_{this result in real energy units and}

_combining

it with the kinetic energy in the X2 direction, one _{finds the total energy of this} structure, divided

_by

the number of holes to be

_{given by}

Thus

_EW

becomes

_negative

at

_exactly

the same threshold as the

_spin

wave stiffness.

A

_periodic

_{array of such soliton lines with} an inverse

_spacing

_nl

= n

is characterized

_by

an

nI

energy per unit area

_nEW,

if we assume that the solitons are so

_widely

spaced

that

they

negligibly

interact. A

_comparison

of

_nEW

to _{the energy} of the

_spiral

_{phase (Eq.}

₍₅₎₎

shows at

once that the modulated

_phase

_{is of lower energy,} as soon as It is

also clear that the lowest energy state _{among these modulated}

_phases

is reached when np takes its maximal value

l la

corresponding

to one _{hole per site}

_along

the soliton lines. Note

that even if we were

_assuming

that the Hamiltonian

₍₁₎

is correct down to the scale of the lattice

_spacing,

there should be corrections to

₍₁₀₎

when f

becomes

_comparable

to a. In

particular

we do not

_expect

_EW

to scale like

a 2 in

the

_large

a limit but rather like a.

_Evidently,

we cannot trust our initial Hamiltonian well before

_reaching

such short

length

scales. But the

nature of the

_instability

is

_undoubtedly

_correctly

understood

_by

our

_purely

_large

scale

approach

since all the calculations are consistent for

_{na2, nI! a 1}

and a -1.

_{Up to now, we}

restricted our discussion to one

_valley

but it

_applies

without

difficulty

to the case of two

853

overall

_{physical picture}

is a

_periodic

_pattern

of _square cells

along

the

diagonal

directions of the

square lattice, with an area

_scaling

like

n - 2

and an almost uniform orientation of the Néel order

_parameter

within each cell.

_By

each

_crossing

of a soliton

_marking

the

boundary

between

cells,

the

_angle

cp of the

_spins

in the

_plane

x, y

rotatès

by

a finite

amount G

_{an// . This}

quantity

p

becomes of order 1

_{when nI ~ 1/a.}

The

_tendency

of holes to form lines can be

_simply

understood from the

_following

classical

argument :

we have seen in section 2 that each hole can be

assigned

a

dipole

moment

Pi, depending

on the

_valley

it

_{occupies. Spin}

waves mediate between two holes a static

interaction which takes the form of the 2D electrostatic

_dipolar

interaction with a minus

sign

i.e. The

_change

of

_sign

comes from the fact that in the

energy enters, besides the

_magnetic

term which

_corresponds

to the usual electrostatic _energy,

the interaction term between the

_dipole

moments of the holes and the

_spin

current. From the above

_expression,

it is clear that two holes

_carrying

the same

_dipole

moment

_p

attract when their relative

_position

_rij

is

_{perpendicular}

_{to p}

and

_repell

when

_they

lie

_parallel

to

p.

It follows that holes

_belonging

to the same

_valley

_gain

_energy

by aligning

themselves

perpendicularly

to

_p.

We wish now to discuss the

_question

of the interaction between soliton lines. The formation of walls screens

_completely

the

_dipolar

interaction on a classical level

(in

electrostatics,

the

electric field outside an infinite

_capacitor

vanishes).

This is no more true on a

_quantum

mechanical level where we

_expect

a weak

_repulsion

due to the small

_overlap

between

neighbouring self-trapped

states. If we trust Hamiltonian

₍₁₎

down to the lowest

_length

scales,

then the

_problem

of

_finding

one-dimensional

_{stationary periodic}

solutions of

_{equation (7)}

is

equivalent

to the Peierls-Frôhlich

_problem,

for which exact solutions are known

_{[10, 11].}

In the small

_density

limit

_{(n a -2),}

_{the energy of} an individual soliton inside a

_periodic

_{array of}

self-trapped

states, is

_{given by}

_[11]

where L is the distance between successive

_solitons,

which scales like

_{(an )-1}

and

f

is their width

_(of

order

_a).

This result shows that the domain wall structure has an

exponentially

weak but

_positive

_{compressibility.}

On the other

hand,

distortions of each individual line are controlled

_{by energies}

of order

_J(nll

_{a )2 _ j}

_{for nll a ’"}

1. Therefore the modulated

_{spiral phase}

should be seen as a _gas

_{of weakly interacting}

almost

_rigid

soliton lines. We shall not

_expand

further on the

_physics

of this

_phase

since we lack a

_microscopic

understanding

of its short scale features. It is worth

_pointing

out that such

_charged

solitons have

_already

been found in the Hubbard model from different

_approaches.

In the weak

coupling

limit,

Schulz

_[12]

found incommensurate

_{antiferromagnetic}

_phases

close to half

filling :

in that case solitons form in the

_{(1, 0)}

or

(0, 1)

directions. In the

large

U

limit,

motivated

_by

a

_previous

work on the two-band Hubbard model

_[13],

Poilblanc and Rice

_[14]

also found

_numerically

soliton like solutions within Hartree-Fock

_theory,

this time

_along

the

diagonal

directions like in the

_present

work. In both

_approaches,

soliton lines

separate

domains of

_opposite

direction of the Néel order

_parameter.

In the

present

theory,

the

_jump

àç

in the orientation of the

_spins

is a

_priori

not

_quantized.

_However,

it is conceivable that in the

_limiting

case nll a ~ 1, our soliton lines become in fact

_equivalent

to the

_previous

ones or in

other words that

_{they correspond}

to

_{a jump}

_A~

= 03C0r .

Besides the

_instability

towards domain wall

_formation,

the

_long

_range

_dipolar

interaction

the

_{pairs strongly overlap.}

In the moderate

_coupling

limit

_{(a - 1 )}

and small

_density

of

concern to us in this paper, it is more

_adequate

to look

_directly

at the

_pairing

between two isolated holes. In the case of an

_isotropic

mass tensor, the

problem

can be solved

_analytically

since it becomes

_separable

in the variables

_log

r and 8

_(where

r, 8

are the relative

_polar

coordinates of the two

_holes).

For holes in the same

_valley

and with the same

_spin

_index,

the

orbital wave function must be odd and we find a finite threshold for the formation of a bound

state

a c - 1 >-

1. On the

_contrary

for holes

_belonging

to different

_valleys

or

_carrying

a

V2

different

_spin

index,

the orbital wave function can be an s-wave state

_(with

an admixture of

higher

even

_angular

momentum

_components)

and there is no threshold _{anymore. The}

_dipolar

interaction

_by

itself cannot form bound states but

_only

«

collapsing

» states which

_continuously

decrease their energy

by shrinking

their size to zero. This is because the

_dipolar

interaction

scales

_exactly

like the kinetic _{energy in}

_{1 /r2.}

To

_regularize

the

_problem

one has to restore a

short range cut-off and the energy of the bound states will _{be very}

_dependent

on the

_physics

at short scales.

So,

in view of all these uncertainties _{it is very difficult} to make a

_general

statement on the relative

_stability

of a

_{superconducting phase}

or an

_insulating

one at

« > 1. We leave this

question

for further work.

4. Discussion of

long range Coulomb effects.

In the

_previous

section we have identified the nature of the

_instability

which affects the

_spiral

phase

at « > 1. The troublesome feature of our result is that it

_unavoidably

leads to short scale structures with

_highly

non-uniform distribution of

_charges.

In this section we would like

to see how a

_long-range

Coulomb

_repulsion

can

_fight

this

_tendency

and whether it restabilizes

the

_spiral

_phase.

We write the Coulomb interaction like and define The

dipolar

interaction between holes takes the form with

of the same order as The

_length

_ao =

a(ld/lc)

determines the scale

beyond

which the Coulomb

_repulsion

dominates both

_dipolar

_{interaction and kinetic energy which} scale in the same _{way and} are of the same order of

magnitude.

Within the

_present

approach,

the most

_{plausible assumption}

is

_{JC > Jd}

since

_Jd

is

_typically

of order J whereas

Te

can

_easily

be some sizeable fraction of U. If

_consequently

_ao is of the order of a or

_less,

then

a

_{Wigner crystal}

will form at low

_density

and the

_question

we ask looses its sense. We shall rather assume that

_{Jd > Je}

and that a

_{spiral phase}

has formed.

Let us reconsider the renormalization of the

_spin

wave stiffness. We take the twist in the

(1, 0)

direction and look first at the

_spin

fluctuations in the

_plane.

The field

_aycp

_couples

to the

holes like the difference of

_density

between the two

_valleys

_(see

_Eq.

_(3)).

Therefore its inverse

propagator

is not affected

_by

Coulomb effects and remains therefore

_negative

at

« > 1. On the

_contrary,

_{AXq couples}

to the

_density

_{and may be screened.}

_Performing

the usual RPA

calculation,

one finds for its inverse

_propagator

To be

_complete,

we also

_analyzed

the transverse

_spin

fluctuations. Since

_they

are

855

screened in the

_{long wavelength}

limit

by

the Coulomb interaction which conserves the

_spin

indices

_(they

are corrected

_{only by}

ladder

_diagrams).

However,

we have

_argued

in section 2

that these transverse fluctuations are stable in the

spiral phase.

So

they

do not _{raise any}

trouble.

The

_instability

with

_respect

to fluctuations of

_ayq

can be cured

_by

_emptying

_completely

one

of the two

_valleys.

This also amounts to rotate the twist of the

_spiral

from the

_{(1, 0)}

to the

(1, 1)

direction for

_instance,

as

_previously

noted in II.

Then,

_only

_{the ax1p}

_component

still

couples

to the holes and it is

_strictly

_equivalent

to a

_density

fluctuation : so its stiffness is

_given

by

formula

₍₁₂₎

with a factor 2

_missing

in the denominator. If

_{n- 1/2 > ao > a,}

then for

q kF,

_{DXl xl (q, 0 )}

simplifies

into

_{p [1 -}

_{1 q 1 ao ], a}

_{clearly positive quantity.}

Under these

conditions the

_{spiral phase}

is stable unless the

_density

is so low that a

_{Wigner crystal}

forms. If

on the

_contrary

_{ao > n- 1/2 » a,}

we see from

equation (12)

that

_spin

waves are stable at scales

larger

than ao but unstable at _{intermediate scales between ao and}

n-1I2.

In that case, there is

place

for the formation of domain walls.

Although

it is not clear to us which kind of structure realizes

_the

best

_compromise

between the Coulomb

_repulsion

and the

_dipolar

interaction we

_adopt

the

_premise

that a

_regular

_array

of soliton lines can still form. Our

_arguments

in favour of this choice are two-fold. First we

have estimated the total cost in Coulomb _energy

_accompanying

the accumulation of holes

along

lines and have found that it is

_generally

lower than the

_gain

in

_dipolar

energy. For this purpose, we borrow from

_[16]

the

_expression

of the classical interaction

_energy

between holes

occupying

sites of a 2-D Bravais

_Crystal

where dt and _{a, is the} area of an

_elementary

cell. The summation is to be carried over all lattice sites

_{x (1 )}

different from the

_{origin. Viewing}

the domain wall

structure as a

_rectangular

Bravais lattice with a

_spacing

(na )-1

in one direction and a

in

the

other,

one

_easily

_gets

in this

_{strongly anisotropic}

situation the

_singular

contribution

e2

El ~ -

e 2 log

na 2.

The Coulomb energy cost _per hole is

_equal

to

_EI/2.

On the other

_hand,

ira

the

_gain

in

_dipolar

_{energy per hole is of the form}

_{/3 Jd (with (3 =}

_{TT 2/6)}

in a

_strictly

classical

model or

_generally

a constant of order

_1).

If the hole

_density

satisfies

na 2 »

exp -

{3ao/a

(a

less

_stringent

condition in fact than

JÀ

_{ao >}

1 for values of

_j3

around

_1),

it is

_{energetically}

favourable to form a modulated

_{spiral phase.}

Within the same

_{simple-minded}

classical

model,

we can also

_study

the local

_stability

of the

modulated

_phase

and this leads to

_exactly

the same conclusions. We looked in

_particular

at

the transversal

_phonon

modes of such an _{array of}

_dipole

lines for wave vectors

_parallel

to the chains. One

_single

chain is found to destabilize

_against

transversal distortions at scales of the order a

_{exp /3ao/a}

=

f o.

Physically,

beyond

this characteristic

length,

the _cost in Coulomb

energy to add a

_{supplementary}

hole to a

_preexisting

_segment

of

_fola

holes overcomes the

_gain

in

_dipolar

energy. However, for a

_regular

_array of

_lines,

the

_singular

Coulomb contribution

-

Jc (qa )2

log qa is

cut-off at _wave

vectors q

na like

JC (qa )2

_{log na 2.}

The

_dipolar

interaction

provides

a

_restoring

_{energy -}

{3J d (qa)2

and one recovers the

_stability

of the 2-D structure

under

the same condition as before.

_So,

in the

_interesting

_(but

somewhat

_unlikely)

limit

modulated

spiral phase

is a metastable state of lower energy than the

homogeneous spiral

phase,

provided

the hole

_density

is not too low.

5. Conclusion.

In this

work,

we have shown the

_possibility

of domain wall formation within Shraiman

_Siggia

phenomenological

Hamiltonian,

for suitable

_strength

of the effective

_dipolar

interaction

between holes.

_Clearly,

it would be

_interesting

to know the real value of « in the

_strong

coupling

limit t >

_J,

to see whether this

_instability

can occur.

_{Also, quantum}

_spin

fluctuations

around the various « mean field » solutions found up to now, remain to be

worked

out and may

seriously modify

our views.

Acknowledgments.

1 would like to thank Eric

_Siggia

for

_suggesting

to me this

_problem

and him and Boris

Shraiman _{for very}

_stimulating

discussions. 1 am

_grateful

to S. Brasovskii for

_pointing

out the

existence of references

_[10]

and

_[11].

_{The final version of this paper} was

_{completed during}

a

most

_pleasant

and

_interesting

_stay

at the Institute for Scientific

_Interchange

in Torino

_(Italy).

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