Screening of an external magnetic field by spin currents: the infinite U Hubbard model

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the infinite U Hubbard model

Jean-Christian Anglès d’Auriac, Benoît Douçot

To cite this version:

Jean-Christian Anglès d’Auriac, Benoît Douçot. Screening of an external magnetic field by spin

currents: the infinite U Hubbard model. Journal de Physique I, EDP Sciences, 1993, 3 (2), pp.501-

513. �10.1051/jp1:1993146�. �jpa-00246737�

Classification

Physics

Abstracts

71.10-75.20

Screening of

_an

external magnetic field by spin currents: the infi-

nite U Hubbard model

J.C.

AnglAs

d'Auriac and B.

Dougot

Cent _re de Recherches _sur (es Trbs Basses

Tempdratures,

Laboratoire associd h l'Universitd

Joseph Fourier,

CNRS, BP 166, 38042 Grenoble Cedex 9, France

(Received

J5

July1992, accepted

in final form 18

August 1992)

R4sum4 Le modble de Hubbard h

r6pulsion

infinie _est 6tud16 _{avec un} seul _trou

en

pr6sence

d'un champ

magndtique

externe orbital _sur des _amas finis de diidrentes

g60mdtries.

Des rdsultats de

diagonalisations

exactes sont prdsentds _pour

l'dnergie,

^le spin ^total et

plusieurs

fonctions de

corrdlation de spin. Ils confirment l'idde que le flux

appliqud

est

compensd

_par le flux fictif provenant d'une

configuration

de spins _non

coplanaires.

Ce phdnombne est aussi bien illustrd par une dtude variationnelle.

Abstract. The infinite U Hubbard model with

a

single

hole in the _presence of _an external orbital

magnetic

field is studied _{on a} finite cluster for several

geometries.

Exact

diagonalization

results for the energy, total spin and several

spin

correlation functions _are

presented. They strongly

_support the idea of the

compensation

of the

applied

flux

by

_a flictitious flux induced

by

a

non-coplanar spin background.

A variational study also shows evidence of this

phenomenon.

The

physics

of

strongly

correlated electronic

systeius,

in

connectivity

with

high temperature superconductivity,

has been the focus of _many recent theoretical

investigations. Among

these

models,

the infinite U Hubbard iuodel is the

simplest~

since it doesn't contain _{any energy} scale and

depends only

_on the electron

density.

Until

recently,

the main result for this

system

_was the

Nagaoka

theorem

iii

which states that the

ground

state for _a

single

hole _{on a}

bipartie

lattice has the maximum total

spin. Iio».ever,

the

stability

of

ferromagnetism

in two dimensions has been further

investigated by

_means of exact

diagonalizations

_on finite clusters

[2-4],

mean field

approximation

_16,

_6],

variational _wave function with _one

spin flip [7-9]

^and

high temperature expansions [10].

^For more than _one

hole,

_some instabilities have been found _on finite lattices

[2-5],

^{but have been}

interpreted

_as finite size effects

ill].

It _seems rather

likely

that the

fully

polarized

state is at least

locally

stable to,vards _a

single spin flip

for _a finite

density

_z of holes up to _{z =} 0.29

[9].

^{But it is} not clear that this remains valid at any finite

temperature [10].

High

temperature

expansions suggest

that _many low

lying

excited states with

arbitrary

small

magnetization

_may dominate the

magnetic properties

at T

#

0.

Part of _our

present understanding

of

strong

correlations borrows from exact results in _one dimension

[12],

^where _a

decoupling

between

charge

and

spin degrees

of freedom has been found.

In the infinite U limit for _a closed

ring, eigenstates

are classified

by

the

crystal

momentum of the associated

squeezed spin

chain and the k values for the holes

[5].

^More

precisely,

_a finite

crystal

momentum in the

spin

chain shifts the allowed k values and thus

plays

the role of _a fictitious

magnetic

flux for the orbital motion of the holes. This idea has been the

starting point

of _{some mean} field

approaches

which _assume that holes _move in _a twisted

spin background [6].

Actually,

_a full

dynamical

treatment _can be formulated in terms of _a gauge

theory [13].

This

analogy

between _a

spin

field and _a fictitious

magnetic

field

suggests

that the

spin configuration

is very sensitive to _an external orbital field.

Intuitively,

the

spins adjust

_so that the fictitious field

they _generate compensates

^the

applied

field. This

phenomenon

has been studied

by

exact

diagonalization [14]

^and

quantum

^{Monte Carlo}

jib]

simulation. The effect of the

spin

background is,

_as

expected

from this

picture,

to reduce the energy

dependence

_{as a} function of the external flux. This result is

quite interesting,

since it

suggests

that fictitious fluxes _can be

generated

without _a

significant

reduction of the

hopping amplitude.

Another motivation to

study

the infinite U Hubbard model for _a

single

hole in _an external field is that in the _presence of _a finite

density

of

holes,

and _zero external

field,

it is

possible

to

represent

^{these holes}

by

hard _core bosons bound to

singular

flux tubes

[16].

In _{a mean} field

type

^of

approximation,

the flux tubes _are

replaced by

_a uniform _average orbital

magnetic

field. Since these holes _are

bosonic,

their behavior is related to the low

lying

states of this elsective _one hole model.

We should stress that for real systems, ^{the external field}

couples

also to the

spins

via the usual Zeeman B-S term. For _a

single hole,

this Zeeman term scales with the

system

^size

by

contrast to the kinetic energy of the

hole,

which remains finite in the

thermodynamical

limit.

As _a

result,

_our model is _not realistic. However the main motivation is _to

study

the

interplay

between

charge

and

spin degrees

of freedom.

This _paper is dedicated to _a numerical studv of t~his

problem, using

both exact

diagonal-

izations _on finite clusters and variational _wave functions. More

specifically,

_we consider the infinite U Hubbard model:

H =

~j ~j _{e'~'? (I}

ni-,

c$cja(I

n

j a)

+ h-c

(I)

(ij) ?

where

c$

is _an electron creation operat~or at site

spin components

c1. The I,

j

suiuiuation _runs

over nearest

neighbor

^{bon(ls and the}

projection

_operator enforces the

non-double-occupancy

constraint. The external field is

iiuplemented by

the

phases Ii

_j, which _are related to the flux

by:

~ _Aij

2~ ^~

(2)

oriented

~°

p>aquett»

,vhere

~a is the flux

through

the

given plaquette

and _{~ao =} is the

elementary

flux quantum.

As

usual,

the

spectrum

^{and all} _gauge invariant obser;ables _are

periodic

in

~a wit-h

period

_~ao.

Two

types

of

geometries

have been

cqnsidered.

First _a 4 _x 4 square lattice with free

boundary

condition and second _a closed

strip

of,vidth 2 and

legnth

8. These

are shown in

figure

I. In the _case of the internal annulus is identical to the flux

~a~ in the external

plaquettes~

and _one for which _{~aj =} 0. For the _square

geometry,

_we have chosen _a uniform flux _~a.

We first discuss the result froiu the exact

diagonalizations.

The

ground

_energy and first two excited states are sho,vn in

figure

2a for the _square cluster. As in

previous studies~

the energy variation is

strongly

reduced in

comparison

to the free

particle

_case. We also observe

a coincidence between the maxima of

dE/d~a

and the minima of level

separation

for these low

-1-1-0

~ ~ ~

Q Q Q Q~ Q~

1 2 1 Q~ Q~

v v v

Q~

0 1 0

~ ~ ~

q~

q

Q~ Q~

a) b)

C)

Fig.

i. Different geometries studied in this work.

a)

The 4 x 4 square lattice with _open boundary conditions. Numbers _on different

plaquettes

label them

according

to the symmetries of the cluster.

They

will be used for

figure

5.

b)

Labels for the different types ^{of sites} for the _square cluster.

They

will be used in

figure

3.

c)

The 4 _x 2

strip,

with the flux _~j for the

unique

internal

plaquette

and _~e for the 8 external

plaquettes.

lying

states. The _same trends _are observed for the two

ring geometries, suggesting

that

they

_are

quite general.

The total energy variation for the

ground

state is of the order of 0.02

by

contrast

to 4

2v5

ci 1.18 for the infinite square lattice and _a

single spinless particle.

This confirms the

picture

where

spins

_screen at least

partially

the

applied

field

by

their induced fictitious field.

This is

strongly supported by

further

analysis

of the

ground

state _wave function. For

instance,

the

probability

to find the hole _{on a}

given

site is

depicted

in

figure

3. As _a consequence of the free

boundary condition,

this

probability

is reduced for smaller coordination

points (ie edges

and

corners)

for the _square cluster. For _a

single spinless particle,

it exhibits

relatively large

fluctuations _{as a} function of the external

magnetic

field and the discontinuities _are due to level

crossings.

These variations _are

again

reduced

by roughly

_a ^factor 10 when

spin degrees

of freedom _are included. We note that the characteristic field scale for the oscillations _are smaller in the

ring geometry

^that in the _square cluster. This is

simply

_a consequence of _a

larger

discretization

step

for the square cluster. The

probability

for _a

spinless particle

_{on a}

strip

to be _on the internal annulus ranges from 0.I to 0.9 when the flux

~a is varied. Moreover this

probability

presents also several discontinuities which _are

again

due to level

crossings.

The

corresponding

_{curves are} not shown in

figures

3c and 3d because

they

_are ols scale.

Another

interesting quaiitity

is the total

spin

_{versus ~a.} This is

plotted

in

figure

4. We observe that it decreases rather

rapidly

with

~a and reaches its minimum value around ~a/~ao = 0.21 _on the _square cluster. This critical ~a/~ao ^{is 0. II for the} ladder with _{~aj =} 0 and 0.10 for _{~aj =} ~ae.

Note however that in the latter _case, the total

spin

is not a monotonous function of _~a. This scale is in

quite good agreement

^{with the results of reference}

[14]

^and

again

_seems not to

depend

much _on the

geometry.

In order to present a more

quantitative description

of the induced fictitious

flux,

_a

good quantity

to _measure is the

expectation

value of

permutation operators

for

spins along given

closed

paths

_on the lattice. This is related to the fact that when the hole _moves

along

such _a closed

path,

it induces _a

cyclic permutation

of the

remaining spins along

this

path.

As

usual,

the

amplitude

for such _{a process} has _an additional Aharanov-Bohm

phase

factor 2~

~apath/~'o

where ~apath ^{is the enclosed flux. Perfect}

screening

_occurs if

spins along

the

path

_are in _an

eigenstate

of the associated

permutation operator

with the

eigenvalue

_exp

(-12~ ~apath/~'o)

4x4 square lattice 3

3.05

3.1

3.15

3.2

3.25

0 0.2 0A 0.6 0.8

Q/Qo (a)

Fig.

2. Ground state arid first excited state

energies

from exact

diagonalization. a) Square

cluster with _a uniform flux _{~ per} plaquette. The dashed _curve is the exact result for _a

spinless particle (I.e.

state with maximal total

spin). b)

2 x 8

strip

with ~j = 0, and _~a

= ~e.

c)

2 _x 8

strip

with _~aj

= ~ae " i2.

Of _course, this cannot be achieved

simultaneously

for all the

paths present

in the

system.

However,

on the average,

spins

_may twist in _a way which tends to reduce the total

phase

accumalated

by

the hole

along

closed

paths. Figure

5 shows the

phase

of the

expectation

value of

cyclic permutation operators.

In

figure 5a,

the _wave function is

projected

_on the

subspace

where the hole

occupies

_one of the sites of _a

given plaquette.

A

cyclic permutation

of the three

spins

_on the

plaquette

is then

performed,

and the

overlap

of the final state with the

ground

state is calculated. The

phase

of this

overlap

is denoted

by 2~~afic/~ao. Figure

^5b

corresponds

to

projecting

the _wave function _on the

subspace

where the hole is absent from the

given plaquette.

In this _{case, we}

apply

_a

corresponding

four

spins cyclic permutation

and calculate the

overlap

with the

ground

state. As

expected,

_~aefr is _zero for the

fully polarized ferromagnetic

state.

When the total

spin

reaches its nfinimal value S

=

1/2,

the

spins

twist in _{a way} which

provides

a rather

good

cancellation of the external flux.

However,

this

matching

is not as

good

when the hole is not _on the

given plaquette

and when ~a/~ao ^is

large.

While the three and four site

paths

are note

exactly

the _same, such _a trend is

sensible,

since _we

expect

the constraint _on the

spin

wave function to be the

strongest

^{in the}

vicinity

of the hole. We should also

emphasize

that these

expectation

values of

spin permutation

_{operators are} related to the

spin chirality

in the

following

_way. Let _us consider _a 4 sites

ring

with

spins Si> 52,

_53>

54,

and denote

by P;j

^the

permutation _operator

for

spins

I and

j.

A three

spin cyclic permutation P3

is

given by

P3= P12P23

and _a four

spin permutation P4 by P4= P12P23P34.

It _can be shown

[17]:

P3-P(

= -4i

Si (52

x

53) (2)

P4 P(

₌ -2i

(Si (52

x

53)

+

52 (53

x

54)

+

53 (54

x

Si

+

54 (Si

_x

52)) (3)

So _a finite

chirality,

defined

by (Si (52

_x

53)) #

0 for instance

implies

that ~afic/~ao is ^not

integer

_nor half

integer.

The results of

figure

⁵ indicate _a finite

spin chirality

in the

ground

state

expected

for ~a/~ao =

1/2

and the

vicinity

of

~a = 0. This is consistent with the fact that

2.75

2x8 periodic strip, _q =o

2.8

2.85

2.9

2.95

3

0 0.2 DA 0.6 0.8

Q/Qo

(b)

2x8 periodic strip. _q,=q~

2.75

2.8

2.85

2.9

2.95

3

0 0.2 0A 0.6 0.8

Q/Qo

icj

Fig.

2.

(continued)

~a/~ao "

1/2

and ~a/~a =

1/2

_are the

only

values for which the Hamiltonian

(I)

does not break time reversal

symmetry. Furthermore,

finite

chirality implies

_a

non-coplanar spin configuration

in

general.

This _non

coplanarity

_{appears very}

naturally

in _{a mean} field semi classical

picture,

where

spins

_are assumed to be

mutually

uncorrelated and

pointing

in different

directions,

thus

defining

_a classical unit vector field

n(I)

_on each site I. With such _an

hypothesis,

the contribution of the

spin background

to the total

phase

_seen

by

the hole

during

its motion around _a closed

path

is half of the solid

angle spanned by

the vector field

n(I)

_on the unit

sphere

when I goes

along

the

path

_[6]. Our exact

diagonalizations suggest

^{that such} a semi- classical _mean field

picture predicts

the

right

variation of the

spin chirality

_{as a} function of the external flux.

However,

it has been shown that _a class of _wave functions leads to _a

strong

JOURNAL DE PHYSIQUE T 3, N' 2, FEBRUARY iW3 _j8

4x4 square lattice 0.06

with spins

- 0.05

j

~

_0.04

f

g

0.03

I

o

0.02 without spin

fl

0.01

0

0 0.2 DA 0.6 0.8

Q/Qo

(a)

4x4 square lattice 0.21

rv

o o-19

j g

=

m

~

g

fl

fl

£

ith spins

O-I1

0

/Qo

(bi

Fig.

3.

Probability

of presence for the hole _on different sites in the

ground

state.

a)

4 _x 4 square lattice for sites of type i _see

Fig. lb).

"With spins" corresponds to the full Hilbert _space and "Without

spin"

to the

maximally polarized

state,

equivalent

to _a

single

spinless

particle. b)

Same _as

previous

case, for sites of type 2.

cl

2 _x 8

strip

with _~a,

= ~ae = i2, for sites _on the internam

ring. d)

2 x 8

strip

with _~aj

= 0 for sites _on the internal. Note that the

magnification

_on the _y axis is much

larger

for _case

c)

and

d)

th;tn for

a)

and

b).

reduction of the modulus of the

hopping amplitude, eventhough

its

phase

_can ^{be tuned} to the

right

value. It would be of _course

interesting

to know if _we

can

reproduce

the

ground

state energy ^at finite

flux, using

_a semi-classical static

spin background.

It is not so

straightforward,

since it

requires

_an

optimization

_over all the associated unit vector fields

n(I)

_on the lattice.

We haven't

attempted

this

yet,

^since we suspect ^{t~hat the}

large

finite twist could lead to _a too

2x8 periodic strip.

q,=0

0.51

'I

c1

a 0,505

bc

E

©

5 _0.5

~

C

i

fl

0.495

p

~

0.49

0 0.2 DA 0.6 0.8

q/q~

icj

2x8 periodic strip. _{q,= q~}

0.51

'Ic1

a 0.505

E

©

E

©

~ 0.5

g

~ 3

fl 0.495

kO

0.49

0 0.2 DA 0.6 0.8 I

q/q~

id)

Fig.

3.

(continued)

siuall absolute value of the

hopping amplitude.

In

principle,

those semi-classical _states

are not

eigenstates

of the total

spin. However,

it is

always possible

t~o

project

them onto the

subspace

with the small value of the total

spin.

It would be

interesting

to

optimize

the

overlap

between such _a state and the actual

ground

state, ^to

get

a better

understanding

of the

physical

nature of this

ground

state.

In this _{paper, we} also

report

some results _{on an a}

priori

different class of variational _wave functions. Rather than

starting

from _a semi-classical

picture,

which is well

adapted

when the

n field varies

only

_{on very}

large length

^scales

(I.e.

when ~a/~ao <

I),

_our trial states

definitely

contain

strong quantuiu

fluctuations. The idea is that the

spin part

of the _wave function should

provide

_{a source} of fictitious flux

acting

_on the

hopping

of the hole. Let _us denote

by

_z the

4x4 square lattice

15/2

13/2

1/2

I

_9/2

(

_7/2

5/2

3/2

1/2

0 0.05 o-1 0.15 0.2 0.25

Q/Qo (a)

2x8 periodic strip.q=0

8

7

6

~

5 I

~ 4

"

°

3

2

0

0 0.2 0 4 0.6 0.8

q/q~

(b)

2x8 periodic strip. q=q~

8

7

6

~

5 I

~ 4

"

o

3

2

0

0 0 2 0 4 0 6 0.8

Q/Qo

(c)

Fig.

4. Total

spin

of the

ground

state _{versus ~a.}

a)

_square lattice.

b)

2 _x 8

strip

for _~aj

= ~e " i2.

c)

2 _x 8 strip for _~j

= 0 and _~ae

" ~.

Plaquefle with hole 0.5

0 _+-

.. I -

DA 2 _+-

k 0.3

m S

0.2

o-1

;.'

_{".._}

0 0.2 DA 0.6 0.8

q/q~

(a)

Plaquefle without hole 0.5

,." .._ 0 +-

," ", -

DA

;.'

_"., 2 +-

I _0.3

)

." ".,

0.2 ;.' ".

oi

0 0.2 DA 0.6 0.8

Q/Qo

16)

Fig.

5. Fictitious flux _per

plaquette

_as defined in text for the _square cluster, for the three types ^of

plaquettes (see Fig. ia). a)

Plaquette with the hole located _on it. b) Plaquette without hole. In both

case

a)

and _case

b)

and for ₌ o-S, _a

degeneracy

of the

ground

stat.elead t-o _a values

slightly

different from o.5. The discrepency has been corrected by hand.

position

of the hole _on the

lattice,

and

by

_{vi, y~,}

,

yn the

positions

of the

n down

spins.

We

investigated

_wave functions of the form:

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

~~'

_Y~

_{~l ~b)(jn} (VI,

, y~ ~~

where @(~,

y)

^is the

polar angle

of site _~ with the

origin

at site _y. This _wave function binds

a fictitious flux tube to each down

spin. _4lsp,n (vi >yn)

^is a

spin

_wave function and

4l)()~

denotes its restriction to the allo,ved states with the hole at site _~, to values of _y,

#

_~. We

assume

4l)j)~

^{to be}

norinalized,

ie:

~

~§(~) ²

Vi <Yj< <y

~~~~' ~~'

'Yn)

_" I.

v,#x ^"

is)

g(~)

is

interpreted

_as the hole _wave function

given

this

spin background.

It

depends

for instance

on the chosen _gauge for the

A(~s.

^{So the} presence of

g(~)

is

required by

_gauge invariance. We chose for

4lspjn

a wave function with _a uniform

spatial density,

with _a tunable amount of

repulsion

between the down

spins.

More

specifically,

_we assumed _a Couloiub

gas-Jastrow

form:

~z

ib[[)n (vi,

,

Yn "

fl

~XP

(io'Yui _{(2 luuil~)} fl

_Ym

Yl l~

/Z~'~(Z). (6)

m=I I<I<m<~z

The

length

£ is chosen to _ensure the

global neutrality

of the

corresponding plasma,

which leads to:

~

£~ =

~'

(7)

2~n

As this

point,

_we stress that this variational

study

has been limited to the _{same square} cluster

case. We chose the

origin

of the coordinate

system

to coincide with the _center of the _square

cluster,

^and used the lattice

spacing

_as the unit

length. z(~)

is _a normalization

factor,

deter- mined in order to

satisfy equation IS).

Unless _p

=

0,

it varies with _~ in

general.

The _wave vectors

Q

have been either

(0, 0)

or

(~, ~)

in _our trivial _wave functions. In all the _{case we} have

studied, Q

₌

(~, ~) gives

_a lower energy. We think this is because each time the hole

hops

towards _a site

occupied by

_a down

spin,

the

phase

factor in

equation (4) acquires

_a

large

additional contribution of

~. This interferes

destruct~ively

with

configurations

where the hole

hops

towards _an up

spin

site. Such _a

negative

interference _no

longer

_occurs if

Q

"

lx, ~).

In

what

follows~

this value of

Q

has been assumed.

For

given

values of

Q,

_{p~ n} and

~a~ ^,ve minimize the

expectation

value of H

by adjusting

the hole _wave function

g(~).

This is

equivalent

to _an effective _one

particle tight binding problem

since

(H)

₌

£tij _{g* (~i)}

_g

_(zj

₊ h-c-

(8)

(iJ) with the normalization constraint:

L

_ig

_(~,

_)i~

= I

19)

In

equation (8), _tip

_a

coiuplex

number

given by:

11

lij

_" e~~~?

~j fl

_eXp

jl _(b (~j,

_ym b

(~i, _Yli ))I _~b[[I/

~

lu'l)~b[[I/llYl) (~°)

Yl< <Yn ill#I Ym#Xj

In

equation (10),

the _{sum runs over}

spin configurations (y) corresponding

to the hole located at site

j.

The

spin configurations (y')

is defined from

(y)

after the hole has moved from

j

to I. From

Cauchy

Schwarz

inequality,

_we have

]I,j

< I.

Quite generally,

_{we can} write:

ii_{j =}

Iii

_j exp I

(A,

j +

a(~)

=

]t;j

_exp

a[) ii1)

a(~

^{is the fictitious vector}

^potential generated by

the

spin background.

We

expect

the total fictitious flux

through

the

system

to be _n~2o in absolute value.

Figure

6 shows the variational energy for _p

=

2,

and several values of _n. The value _p

= 2 _seems to

provide

the best variational

states in the class described

by equation (6).

The main feature is the

multiple crossings

of

the series of _curves. As _a

result,

the variational

ground

state energy

corresponds

to the lower

envelope

of this

family

of _curves. In

general,

for _a

given

value of _n, the _{energy as a} function of ~a/~ao

begins

to

decrease,

_up to the

point corresponding

to _a

vanishing

^{effective flux.} A

consequence of this series of

crossings,

is _a

strong

^reduction of the

amplitude

of variation of the variational energy as a function of the external flux.

However,

the absolute value of the

energy is

quite

above the actual value from exact

diagonalization. IlJrthermore,

the variational method overestimates the total

spin

of the

ground

state

by quite

_a

large

amount. In order to

understand the _nature of these _wave

functions,

it is

helpful

to

study

the effective Hamiltonian defined in

equation (10).

For each

plaquette

_we define _an effective flux froiu the oriented _sum Of

a)f's

around the

plaquette.

The

phase

modulo 2~ is chosen to be in the interval _~,

~].

The _average effective flux (~a) is then obtained

by averaging

over all the oriented

plaquettes

with _an

equal weight.

The result is

plotted

in

figure

7a. As

intuitively expected,

the

optimal

wave function

corresponds

to

approximately miniiuizing

the absolute value of

(~a).

^We note a

systematic

bias towards _a smaller value of _n. This behavior is

simply

related to the results of

figure

7b. This last

figure represents

^the

amplitude it)

which is the average of

]I;j

over all links

on the lattice with

equal weight. Clearly, increasing

the number of

spin flips

reduces

it) quite dramatically,

thus

favoring

the state with smaller _n at _a fixed value of ](~a)]. ^{We conclude this} section

by stressing

that _{p =} 2 _seems to be _a rather

sharp

minimum for the trial _wave function energy. If _p is not close to this

value,

the _curves

corresponding

to

figure

6 do _{not even cross,} due to _a

stronger

reduction of

iii

_{as a} function of _n.

.8

~~-,, "..._ n=1

~ ".._ n=2

',, ".._ n=3

", ".,

,:

..'" n=4

$

2.2

'',., _'".,

_{".._}

_:" ,,~'

_."

~I~

( ',,

"=:_"'

,,,"

_{_:"} n=7

fl

-2.4

..

'',,

_{",, :.'} ,?~~ .;." _:.'

_g ... ..

',,

_I",., ,I' _{_."} ^"..

fi ".._ _{',_ _:"}

,,?,

,,.[

)

^-2.6

,~~

".._ '"..,

_2'j~,,j'

^_:.'

2.8

""

3

0 o-1 0.2 0.3 DA 0.5

Q/Qo

Fig.

6. Variational

energy for the best trial states defined in text on a 4 x 4 cluster for various

values nd of the number of down

spins.

The thick _curve is the lower

enveloppe.

Note that the

fully polarized

state

(nd

=

0)

has not been included. It is

optimal

state at small

fi.

To conclude _our

report,

^{all the} calculations

st.rongly support

the

picture

that the fictitious fields induced

by

the

spin

currents

(I.e.

_non

coplanar spin configurations)

^cancel _an external

n=1

0.75 n=2

n=3

0.5 ⁿ⁼⁴

n=5

0.25 ,,,'

~"~

fi ^~~

[

0 ... ;,

v ,,,'

_., _:...

0.25 _...."'

,,''',

""

~...."

_:;.."

,, _,,"

__...."'

0.5 k~

),.,,')j (_:..."

0.75 '~) ).).~.."

l

0 o-1 0.2 0.3 0.4 0 5

Q/Qo

iaj

o.95

o-g

0.85

0.8

0.75

0 2 3 4 5 6 7

nd

16)

Fig.

7. Effective

Hefr

for the hole for the variational _wave functions

(see Eq. (10)). a) Average

effective flux

(see Eq. iii)).

The thick line represents ^{the actual} variational

ground

state.

b) Average hopping amplitude.

For each value of _~, nd is chosen _so that the variational _energy is minimal.

orbital

magnetic

^field,

However,

it does not _seem to be _easy to

reproduce

the rather

large

effective bandwidth in the _presence of _a

large

twist of the

spins by simple

variational

schemes,

such _as the mean-field semi-classical states of reference [6] or the Jastrow

type

wave functions

discussed here. It _seems that the

systems

take

advantage

of the

exponentially large

dimension-

ality

of the

subspace corresponding

to the minimal value of the total

spin.

Those who would derive _a

simple picture

for these

ground

states should concentrate on

iuaxiiuizing

the value of

(t),

which _appears to be _more difficult than

simply adjusting

^{the effective} flux.

Another

important

^{issue is} to understand the finite

temperature

^{behavior. Some}

preliminary

results of recent Monte Carlo siiuulations

suggest

^{that the fictitious flux} ~afic exhibits _{a very}

singular

behavior at any finite T in the low

~a

regime, specifically _[5],

when T is much smaller than the

bandwidth, f~f)

~

~. Extrapolated

to T

= 0 this would lead to _a finite value

~=o

of _~afic in the _presence of _an infinitesimal

positive

value of _~a. This _{may seem} at first

glance

in contradiction with the results of

figure

5.

However, taking

the T ₌ 0 limit in the Monte-Carlo simulation of reference

[15]

^{amounts to}

taking

the

thermodynamic

limit. We expect then that the excitation energy of the minimal

spin subspace

above the

Nagoaka

state for _a

single

hole at ~a ⁼ 0 to vanish in that limit.

Furthermore,

these low

spin

states may dominate the

entropy

at any finite

temperature.

^One of the

striking

results of reference

[14]

^{is the} observation of

a finite

chirality

of ~afic in the limit of _{~2 -} 0 for the minimal value of the total

spin.

Of course, these states have _a finite excitation energy for _a finite size

system.

If

they

survive in the

thermodynamic limit, they

_may be the _way to reconcile finite cluster

diagonalizations

and finite

temperature

studies. The idea that low

spin

states with

quite

different

properties compared

to the

ferromagnetic

state may dominate

physical

observables at finite T is also

suggested

in the

high

T series

expansions

of reference

[10] where,

in the absence of external

flux,

_no evidence of

ferromagnetism

_appears at any

density

of holes.

Acknowledgements.

The work _was initiated while R. Rammal _was still among us. He

provided

the

impetus

and _many ideas. His

suggestion

to

study

several

geometries certainly

reflects his taste for _a

systematic approach

in the face of _{a new}

problem.

We

imagine

that he would have

appreciated figure

61

(see

Ref.

[18])

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