Simple representation of the eigenstates of the U↦∞ one dimensional Hubbard model

(1)

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Simple representation of the eigenstates of the U� ∞ one dimensional Hubbard model

Ricardo Dias, J. M. B. Lopes dos Santos

To cite this version:

Ricardo Dias, J. M. B. Lopes dos Santos. Simple representation of the eigenstates of the U� ∞ one dimensional Hubbard model. Journal de Physique I, EDP Sciences, 1992, 2 (10), pp.1889-1897.

�10.1051/jp1:1992252�. �jpa-00246669�

(2)

J.

Fhys.

I France 2 (1992) 1889-1897 OCTOBER PAGE

Classification

Physics

Abstracts

71 30 75.10

Simple representation of the eigenstates of the U-

_aJ _one

dimensional Hubbard model

Ricardo G. Dias

(*)

and J. M. B.

Lopes

dos Santos

Labora16rio _e Cemro de Fisica da Universidade do Porto,

Praga

Games Teixeira, 4000 Porto, Portugal

(Received 21 April 1992,

accepted

15 June 1992)

Abstract. We have constructed _an _exact map of the U-cc

one dimensional Hubbard Hamiltonian _onto _a system ^of ^free fermions related _to the charge degrees of freedom (holons). The

spin configuration only

influences the energy in _so far _as the fermions _move in _a loop ^which ^is threaded

by

a fictitious

magnetic

flux

proportional

to the

spin configuration's

total momentum. A rather

simple

and

physically appealing interpretation

of _some of the features of the Bethe Ansatz

solution is

possible

in this

representation.

We illustrated its _use with _a calculation of the t/U corrections to the energy.

I. Introduction.

The

discovery

of

high

_T~

superconductivity

and Anderson's

subsequent suggestions [II originated

a renewed interest in the Hubbard Model. It has become clear that this model poses

some very

interesting

unanswered

questions regarding strongly interacting

fermion systems.

Despite

Lieb and Wu's _exact solution

[2]

the one-dimensional version has _not been

spared

the recent attention of many theorists. Some

important physical quantities

had not been calculated

[3, 4]

and _an elucidation of the nature of the solution of Lieb and Wu has often been used to

suggest

^ideas ^which may

apply

in wider circumstances

[5, 6, 7].

In this paper we look at the U

- co limit of the Hubbard model in ID with

periodic boundary conditions,

_an

arbitrary

number of holes in _a half filled band and construct a very

simple representation

^of the

eigenstates

of the

system. Contrary

to other authors

[3, 6, 8]

we do not

start from the Bethe Ansatz solution

(henceforth

abbreviated _as

BA)

and take the limit

U _- co. It has

recently

been shown

[9]

that the BA

eigenstates

are not a

complete

set. Of all the states of _a

given spin multiplet only

the _ones of maximum and minimum S~

projection

_are in the BA basis. We construct

directly

_an exact map of the

original

Hamiltonian into _a

simple tight binding

Hamiltonian of

noninteracting

^fermions

(holons).

The

background spins only

influence the energy levels in _so far _as

they provide

a fictitious

magnetic

flux

threading

the

loop

in which the holons _move. There is of _course _a

large spin degeneracy

which,

however,

is not

complete

_except in the

thermodynamic

limit. The existence of this

representation

has been

(*) Supported by _a Grant from Instituto Nacional de Investigagio Cientifica.

(3)

1890 JOURNAL DE PHYSIQUE I N° 10

hinted _at

by

^several authors

[3, 6, 10],

and _more

recently

and

explicitely by

Schofield _et al.

[I II.

^Our formulation is in _terms of

empty

^sites ^and

spins which,

as we will _see,

considerably simplifies

the

interpretation

of the results for the energy. We illustrate its usefulness

by calculating

^the ^t/U corrections to the energy. Like

Ogata

and Shiba

[3]

_we find this

problem

to be

equivalent

to

finding

the

eigenstates

of _a

spin

1/2

Heisenberg antiferromagnetic

chain, with

a well defined overall momentum.

Curiously,

when the number of electrons has the form

N

=

4 _n

(n integer)

this momentum differs

by

_ar from the _momentum of the

ground

state of the chain itself. The results of Des Cloiseaux and Pearson

[12]

_on the excitations of the

chain,

then lead to the

amusing

conclusion that the

ground

state of the Hubbard

model,

in the

large

U

limit, is,

in _some _cases, _a

triplet.

2. Basic concept.

The interaction term in the Hubbard model

JC

= t

I

_{C)~ Cj~} + hC ₊ u

jj

_n, _i _n, _j

(i

<>J>

is U times the number of

doubly occupied

sites. If _we let U

- co, the _states with

doubly

occupied

sites _are

pushed

to infinite energy and may be

projected

out. For this purpose it is useful _to

decompose

^the ^electron

^operators

^as

^c,~=c~~(I -n,_~)+c,~n,_~

and

c)~

₌

(l

_n,

~ c,~ + n,

~

c)~.

The

projection

of the Hubbard Hamiltonian onto the space of zero,

doubly occupied

sites

becomes,

for _a

ring

of L

sites,

L

~ ~

~

~~

~

_~~

~~<+l-~)C<+la~ia(~ -~<-a)+~C (~)

,=l

which is also _one of the _terms of the famous t J model.

The basic idea of this paper ^can be

simply explained by considering

a state with _a

single

empty ^site. ^When the Hamiltonian of

equation (2)

acts on such a state it

simply exchanges

the empty ^site ^with a

spin

with _an

amplitude

t. One

might

therefore think that, for each

spin configuration,

one could map the system onto a

tight binding

Hamiltonian with _a

single particle.

However _one

easily

sees that if the empty ^site ^is ^taken

through

a closed

loop by

successive

application

of the

Hamiltonian,

the

spin configuration

is not restored to the

original

one but, rather, to a circular

permutation

of it. The hole motion

effectively

mixes the

spin configurations.

Consider _now what

happens

in _a

single loop.

Label the sites1,.

,

L in _a clockwise fashion. Let

~-i

Ii

«j,.

,

«L-i)

=

fl cl «1°) (3)

j~i

denote _a state with site I empty,

I-e-,

_b~

=

j

for

j

_< and

b~ ⁼

j

+ I for

j

_m I. It is clear

that, 3C[I,

_«j,.

,

«~_j)

=

-t[I

-I, _«j,.

,

«~_j) -t[I

+1, _«j,.

,

«~_j) (4)

unless I

= or ₌ L. In

fact, 3C[1,

_«j,.,

«~~j)

=

-t(-)~-~ _[L,

«~_j, «j,.,

«~_~) (5)

t[2,

_«j,.

,

«~_j) 3C(L,

_al,

, "L-

II

"

t(- t~~ _11,

_«2, _"3,

...,

"1) (6)

-t(L-

i, _al,

,

"L-1)

(4)

- cc

There is _an _extra

phase

factor due to the

reordering

of the electron creation operators but, more

importantly,

the

spin configuration

is altered when the

empty

^site

hops

between I and L. If _we

construct the Bloch _states for the

spin configuration

_«j,

, «~_ _j

Ii

_; _«,

q) =) _z _e'o~li

; «i-m,

,

«~-i-m) (?)

~~=~

where q ₌

n(2 ar/r~)

with _n

=

0,

,

r~

I,

and _a labels the classes of

spin configurations

which _are circular

permutations

of each other

(r~

is the

period

of each

class).

In view of

equations (4)

to

(6),

the action of the Hamiltonian in this

basis, gives Xii

_; _a,

q)

=

t([I

₊ I _; _a,

q)

₊

Ii

I _a,

q) ) (8)

(I #I,L)

3C

(1

; a,

q)

=

t(- f~~

_e~'~

_[L

a,

q)

_t

[2

_a,

q) (9)

3C[L;

_a,

q)

=

-t(- f~~e'~[I

_;

a,q) -t[L-

I _;

a,q). (10)

It is _now obvious that the Hamiltonian is identical _to _a

tight-binding

_one with _an added

spin degeneracy

and

enclosing

a fictitious

magnetic

flux related _to the

spin configuration's

momentum. We _can restore the translational invariance with _a gauge transformation. A similar argument ^has ^been

presented by

Schofield _et al.

[I Il.

In the

following

_we show that this argument can be

generalized

for any number of

empty

^sites.

3. Formal

development.

Hubbard

[13] (see

also Zou and Anderson

[14]) pointed

out that the _states of the Hubbard model _can be

mapped

onto a

subspace

of those of _a system ^of ^four types ^of

particles,

two

bosons and _two

fermions,

with second

quantized

operators

S,~, S)~,

_e,,

e)

and

d,,

d).

_An

_empty

_site _contains

an

e-particle (holon),

a

doubly occupied

_one a

d-particle

and _a

singly occupied

site _a

S-particle (a spinon),

either up or down. The

subspace

of this

larger

Hilbert space which

corresponds

to that of the

original

electron system ^is characterized

by

the constraint

e)

_e, +

d) d,

+

z s)~ s,~

= i ( ii

,,«

The electron operators

correspond

to

c)~

=

S)~

_e, + «

d) S,~.

In the U

- co limit

d-particles

do _not appear and _we _can

simply

make the

replacements (I-n;_~)c)~-S)~e,,

c,~

(l

_n,

~

)

_-

e) S,~.

Note that Zou and Anderson

[14]

take the holons to be bosons and the

spinons

to be fermions. In

fact, given

the constraint of

equation (11),

_to

reproduce

the anticommutation relations of the

physical

electron

operators

one of these

species

must be _a fermion the other _a

boson,

but _one _can choose which is which. It will prove convenient _to take the holons _as fermions. The Hamiltonian becomes

L

JC=-t

£ S)+i«e,+ie) S,«+hC. (12)

,=1

We will denote kets in the electron

representation

_as

[.. _)~

and in the

spinon-holon

representation

_as

[.. _)~~.

If

b,,

I

=

I,

,

N denotes the sites with _one electron _we can define

N

~~l,

,

hN, (~l,

,

"N) _)~

"

fl

^~~

a~

'~) (13)

j=1

(5)

1892 JOURNAL DE

PHYSIQUE

I N° 10

L

The

physical

_vacuum in the

spinon representation

is

[0)~

=

fl e( _[0)~~. Expressing

the

k=I

electron operators ⁱⁿ terms of the

spinon

and holon _ones, and

keeping

the b sites ordered clockwise I _w

hi,

,

bN

< L

N M

hi,

,

bN

_;

jai,

~

«N)

=

(- )~l~l fl S)

~

fl e( _[0)~~ (14)

~

j=1

~~,=i

^'

N

where

3(b)

m

jj (b, I)

and the a's _are the empty ^sites. ^This suggests ^the

following

i =i

notation for _our basis

N M

(aj,

,

aJ~) («j,

,

«N)

~ ⁼

(- )~l~l fl S)

~

fl _{e) [0)~~.} (15)

~

j =1

~ i

'

With this definition it is clear that these states can be put ⁱⁿ a one to one

correspondance

with the

product

states of _a set of M free

spinless

fermions in _a

ring

of L sites

jai,

,

aJ~)

~~

and _a chain of N localized

spins

_«

j, _,

«N)

~.

If _we define

jai,

,

a[)

~~ ^m e,

~ j

e) jai,

,

aJ~)

~~, it is a

simple

matter to prove that

Ii _~~+l

« ~< _+l ~~

<a) (~l,

,

~M) (~l,

,

~N))sh

_~

~

(- ₎₍ (a(,

,

all ("i,

,

"N)

)~~ ^for ^I ^{# L}

(- )~~ (al,

,

al)

_;

(«~,

_ml,

, "N

ill

~

for I

=

L.

~~~~

Proceeding exactly

as in

equation (7)

_we define the Bloch _states for the

spin

system

a ~z,

,«-<

~~~~ fi

~Z

^~'~~

jai

_;

j«~

°

~'' ' "N- _i

-ml)~~ ^~i~~

with the

q's given

as before.

Using equation (16)

and the _one _to _one

correspondence

of the sh- basis states and those of free fermions and localized

spins

it becomes clear that

lla'l

;

tY', q' z S)+

i _« e,

+ i

e) S,« lal

_; _tY,

~

=

~

sh

(-)(a'[eje)[a)(a',q'[a,q)~

^for ^I#L

(-

_)~ e~'Q

(a'[e~

~j

e) a) (a', q'icy, q)~

^for I

=

L.

~~~~

s

Given the

orthogonality

of

spin

states with different a's _or

q's

this

implies

that the Hamiltonian

can be

diagonaiized

within each

subspace

of _a

given

_a and q, inside

which,

it is

equivalent

to

L

3C(q)

=

£

_{t, e,}

~ j

e)

₊ hc

(19)

1=1

where

t~ = -t for I#L

t~ ₌

(- t-

_e- IQ t ₌

e-'i~-'~L) (-1). ~~°~

(6)

N° 10 EIGENSTATES OF THE U

- cc HUBBARD

A

simple

_gauge transformation e~ - e~

e~'~~~~~

~~~ transforms the

hopping amplitudes

_as

_,(£_~)

~~ ~ ~~ ^~ ^~ ~°~ _'~ _{~ ~}

<(f-r)L-<(~-r) (21)

t~ _-

t~

e

~ for _m

=

L

and makes the Hamiltonian

translationaily

^invariant

L

3C'(q)

= t

e~'Q'~ £ e)

_e,

~ j + hc

(22)

=1

This Hamiltonian describes a set of free ferrnions

moving

in _a

ring

threaded

by

_a fictitious

magnetic

^flux _~b~ ₌

q~bolar

^where _q ^varies ^between ⁰ ^and 2 _ar. The band is

anti-bonding (the

top ^{of the} ^band ^is at k

=

0)

which is _no

surprise

since it describes the motion of the holes in the

original problem.

4. Ground state energy.

The

illuminating

discussion of the energy of the

ground

state. If

kj,

,

kJ~ are the Bioch wavevectors of the

occupied

holon states, the total energy is

M q

(23)

E(kj,

,

kM)

_" ^~

i

_~°~

~'

L

It is convenient _to choose the unit cell of the

reciprocal

lattice between 0 and 2 _ar,

k~ =

n(2

^«IL with _n

=

0, 1,

,

L I.

In

figure

I _we show the energy of the k _states for _an _even numbered

ring,

L

=

8,

both for q = 0 and q = ar, A _non _zero value of q shifts the cosine _curve of the

dispersion

relation

by

o

» »

1 3 5

-0

-1

Fig.

I. The energy of the hole states in _a

eight

site

ring

when the

spin

chain has momentum 0 (circles) and _ar

(squares).

(7)

1894 JOURNAL DE

PHYSIQUE

I N° 10

0.

1 2 3 4 5 6

-0

@@

-1

Fig.

2. The energy of the hole _states in _a nine site

ring

when the

spin

chain has momentum 0 (circles) and _ar

(squares),

q/L,

^which means that the

energies

are the same

again

_{for q}

=

2 ar, Because there is _a state at k = ar, for _one hole the

ground

state occurs for q = 0. If there _are _two

holes, however,

the

ground

state will _occur for q = ar. A trivial calculation confirms that the

ground

state is _at q = 0 for _an odd number of

holes,

and q = ar for _an _even _one, There is in either _case _a

large spin degeneracy,

but note that when q #

0,

the

ferromagnetic configuration

is excluded from

the

ground

state, Another way of

stating

these results is _to say

that,

in the

ground

state, the

spin

chain has momentum q = ar if

N,

the number of electrons is _even, and q = 0 if N is odd, Note that when N is _even and not all the

spins

_are _up _or

down,

there _are

always configurations

with

even

periodicity

_r~

and, therefore,

states with momentum w, Stated in this way these results _are also valid for _an odd numbered

ring (Fig. 2).

There is _no k

= ar state and therefore the

ground

state occurs for _q

= ar for _an odd number of holes and for q = 0 for _an _even _one, In this _case the one hole

ground

state is _not

ferromagnetic,

In all _cases the

ground

state energy is

easily

shown to be

~~~

=

~~~ ~

~~

(24)

2 _t

ar

sin L

In

comparing

_our results with the _ones obtained with the BA

analysis

_one should bear in mind that the latter _are

usually expressed

in _terms of number of

electrons,

N and the number of down

spins Nj.

It has

recently

been shown

[9]

that _a BA

eigenstate

with _a

given

N and

Nj

is part ^of a

spin multiplet

with S

=

jN

N _j

.

For instance, if

Nj

=

N/2 the state is _a

singlet,

If this is taken into account we find _our results in total agreement ^with ^the ones

previously reported by

^other ^authors

^[3, 8]

both with

regard

to the energy and the _momentum of the

spin

chain.

(8)

N° 10 OF

- cc

Another

interesting

feature which is

easily

understood in this scheme _concerns the response to _an extemal

magnetic

field. If _a Peierls

coupled

extemal

magnetic

field is

applied,

its flux _~b

through

the

loop

will

simply

add to ~b~,

changing

_{q in}

equation (22)

to q~ = q

ar~b/40,

The value of q~ in the

ground

state will be _as close _as

possible

to 0

(odd N)

or ar

(even fi§,

The energy will be

periodic

in the extemal flux with _a

period

2

~bo/r~,

^which

depends

_on the

spin configuration.

This

implies

that the

spin configuration

_may

change

with _~b, In the ther-

modynamic limit, though,

the energy will be

independent

^of extemal flux, These facts have been noted

by

Kusmartsev

[8]

and Schofield _et al.

5. Finite U corrections to the energy.

In the t/U« I limit _a canonical transfonnation _to eliminate

doubly occupied

sites in the Hubbard model generates, ⁱⁿ ^the ^lowest

order,

_two _extra _terms in addition _to the _one of

equation (2) [15].

In the

spinon

holon

representation they

can be written _as

(J

= 2

it [~/U)

3C^~~~

= 2 J

£

_S~

~ j

S,

n,

~ j n~

(25)

, i

~

~ ^~~~

~

i ^~l

_TWW'

(

~WW~ ~~

+ l _W ~'

+ l ~~ _l

~'

l W' + hC

(~~)

,_a_ a

where

S,

₌

£ ^(1/2)S)~ S,~

_r~~,, _n,

=

£S)~ S~~,

^and _r are the Pauli matrices. When the

««. _«

operator

S)~, S,~

acts on a state of the basis defined in

equation (15)

it either

gives

zero or

changes

some

spin

in the list

(«j,

,

«N)

from _« _to «'.

Keeping

this in mind _one

readily

shows the

following

result

s, iai

_;

I.

,

am,

ii

=

r««~(i el e,)i iai I.,

_«,

ii (27)

where _m will in

general depend

_on the a's. The factor

(I e)

_e, _ensures the

equality

when the site I is empty. ^Then it follows that the matrix elements of 3C^~~~ can be factorized _as

products

of

matrix elements defined in the space of

spinless

fermions and of the

spin

chain

(la'l ^ltr'l ^S<

+i S<

(n,

+

in<) ^lal

^;

inn)~~

⁼

~

l~'l

_~~

~l

+ i e<

+ i

)(I ej

_e,

al

~~

~

("') ^Sm

+ i Sm tT

(28)

s

We have _seen in the

subspace

of the

ground

state is characterized

by

a

given

value of q, and under _a

spin

translation the _states _are then

multiplied simply by

_a

phase

factor e'Q This _means, of _course that the matrix element

(a', ⁾ _(S~~i ^S~

₄ a,

q)

does not

depend

_on _m and _can therefore be

replaced by

the _sum _over all sites in the

spin chain,

divided N. Hence

1lk'l

_;

«',

_q13C~~~l

ikl

_; _«,

q)

=

lk'l ij _(I _el+

i e;

+ i

)(i el

_e,

_lkl

x

~ ~

a',

_q

£ _(S~

~ j

S~

_a,

ql, ⁽²⁹⁾

N

~1

~

(9)

1896 JOURNAL DE PHYSIQUE I N° 10

A similar discussion _can be made

regarding

the 3C^~~~ term. There is _a

subtlety

involved

though,

When the m~~

spin

is

destroyed

at site I

I,

and created at site I ₊ I, ^it

effectively

becomes the

(m

+ )~~ one,

exchanging position,

in the

spin

chain with the

spin

which in the real lattice is _at site I. This

permutation

of _two

spins 1/2,

is well known _to be

equivalent

to an

exchange

interaction and _one ends up with the

following

result

lla'l;tY',ql3C~~~llal;tY,q)= IL a'zei+i(i-e)e,)e)-i+hc ⁾

x

, sf

x

~

la',

^q

^£ ^S~

~ j

S~)

^a, ^q

⁽³⁰⁾

N

~1

~

s

The

problem

of

diagonalizing

3C~~~ + 3C~~~ in the

ground

state

subspace

of the Hamiltonian of

equation (12)

has been reduced to that of _an

antiferromagnetic Heisenberg

Hamiltonian in _a

one dimensional chain of N

spins,

with _a well defined total momentum q. However this

momentum, when N/2 is _even, is _not that of the

ground

state of the

Heinsenberg spin chain,

q = 0

[12].

It differs

exactly by

_ar.

According

to the results of Des Cloiseaux and Pearson

[12],

this

implies

that the

ground

state will be S

=

I. In the

thermodynamic

limit

though,

the excitation energy of _a

ar

spin

_wave vanishes and the

ground

state energy correction

(GS

3C^~~~ ₊ 3C

~~~[

GS),

is

just

the average of 2 J

(S~

~ i

S~

in the

ground

state of the

Heisenberg

chain

multiplied by

the

following spinless

fermion correlation function

[3]

1( ⁽¹

e,

~ j e;

~ j

)(I e)

_e,

(e)

~ j

(I e)

_e,

) e)

i +

c)~

=

(n~

ⁿ ^~~~

~ _"~

L

, =1

~ ~ _'~

(31)

where _n

=

I M/L is the real electron concentration and the average is calculated in the fermion

ground

state. This result agrees with the exact one to this order in t/U

[16].

6. Conclusions.

We have introduced _a

representation

of the

eigenstates

of the U

- co Hubbard chain which _we believe sheds _some

light

on several results

normally

^obtained

by looking

at the Bethe Ansatz in this limit, We find _a

separation

of

charge

and

spin degrees

of freedom in the calculation of several

ground

state averages. However this

separation

has tumed _out _to be rather subtle, as was evident in the calculation of the finite U corrections to the energy. It remains _to be _seen

whether this formulation _can be useful in wider circumstances,

References

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Rev. Lett. 20 (1965) 1445, [3] OGATA M,, SHIBA H,,

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Rev. B 41(1990) 2326, [4] PAROLA A,, SORELLA S., Fhys. Rev. Lett. 64 (1990) 1831, [5] ANDERSON P, W.. Fhys. Rep. t84 (1989) 195,

[6] DOUqOT B., WEN X, G,, Fhys. Rev. B 40 (1989) 2719,

[7] ^CARMELO J,, OVCHINNIKOV A, A,, J.

Phys.

Cond. Matter 3 (1988) 757, [8] KUSMARTSEV F, V., J.

Fhys.

Cond. Matter 3

(1991)

3199,

[9] ESSLER F, H, L,, KOREPIN V., SCHOUTENS K,,

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(10)

- cc

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[14] ZOU Z., ANDERSON P. W., Fhys. Rev. B 37 (1988) 627.

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