ARTICLE IN PRESS

Physica B 404 (2009) 3257–3260

Contents lists available at ScienceDirect

Physica B

0921-45

doi:10.1

� Corr

CIT Cam

E-m

(R. Kish

journal homepage: www.elsevier.com/locate/physb

Distinct solutions of infinite U Hubbard model through nested Bethe ansatzand Gutzwiller projection operator approach

A.K. Mishra �, R. Kishore

Insitituto Nacional de Pesquidas Espaciais - INPE, P.O. Box 103, CP 515, S. J. Campos, SP 12245-970, Brazil

a r t i c l e i n f o

PACS:

75.10.Jm

05.30.Pr

71.27.+a

Keywords:

Orthofermions

Infinite U Hubbard model

Strongly correlated electron systems

26/$ - see front matter & 2009 Elsevier B.V. A

016/j.physb.2009.07.116

esponding author. Permanent address: Institu

pus, Chennai 600113, India.

ail addresses: mishra@imsc.res.in (A.K. Mishra

ore).

a b s t r a c t

The exact nested Bethe ansatz solution for the one dimensional (1-D) U infinity Hubbard model show

that the state vectors are a product of spin-less fermion and spin wavefunctions, or an appropriate

superposition of such factorized wavefunctions. The spin-less fermion component of the wavefunctions

ensures no double occupancy at any site. It had been demonstrated that the nested Bethe ansatz

wavefunctions in the U infinity limit obey orthofermi statistics. Gutzwiller projection operator

formalism is the another well known technique employed to handle U infinity Hubbard model. In

general, this approach does not lead to spin-less fermion wavefunctions. Therefore, the nested Bethe

ansatz and Gutzwiller projection operator approach give rise to different kinds of the wavefunctions for

the U infinity limit of 1-D Hubbard Hamiltonian. To compare the consequences of this dissimilarity in

the wavefunctions, we have obtained the ground state energy of a finite system consisting of three

particles on a four site closed chain. It is shown that in the nested Bethe ansatz implemented through

orthofermion algebra, all the permissible 23 spin configurations are degenerate in the ground state. This

eight fold degeneracy of the ground state is absent in the Gutzwiller projection operator approach. This

finding becomes relevant in the context of known exact U infinity results, which require that all the

energy levels are 2N-fold degenerate for an N particle system.

& 2009 Elsevier B.V. All rights reserved.

1. Introduction

In contrast to the permissible double occupancy of an orbitalstate for fermions, only one particle can be present at any sitewhen the intrasite Coulomb repulsion U tends to infinity in theHubbard Hamiltonian. As a consequence, the system satisfies anexclusion principle which is more exclusive than the Pauli’sexclusion principle for fermions wherein both spin up and spindown particles are permitted to occupy a single orbital state. Inorder to implement the ‘no double-occupancy’ constraint, oneneeds to go beyond the conventional fermionic anticommutationrelation, namely,

fiafjb þ fjbfia ¼ 0; ð1Þ

fiaf yjb þ f yjbf yia ¼ dijdab; ð2Þ

where f and f y are particle annihilation and creation operators.The Gutzwiller projection operator (GPO) approach and nested

Bethe ansatz (NBA) are two commonly used methods to imple-

ll rights reserved.

te of Mathematical Sciences,

), kishore@las.inpe.br

ment no double occupancy at any site. In the GPO approach, thedouble occupancy at a site is removed through a projectionoperator [1]. On the other hand, the nested Bethe ansatz providesthe exact solutions for the 1-D Hubbard model [2,3] for any valueof U, from which U-1 result is analytically obtained. Ashighlighted by Fulde [4], a remarkable property of the U infinityNBA wavefunctions is that their spatial part correspond to that ofspin-less fermions for any distribution of particles or holes on thechain. This general feature of the NBA state vectors automaticallyimplements the ‘no double-occupancy’ constraint. Unlike in theGPO approach, no additional projection operators are required.Such wavefunctions have been employed to obtain the momen-tum distribution and spin correlation for the one dimensionallarge U Hubbard model with various electron densities [5,6], andhave been shown to lead to an instability of the Nagaoka statewith more than one hole, again for the 1-D case [7]. It is pertinentto point out that in the presence of holes, the NBA wavefunction isa superposition of spin–charge factorized wavefunctions. Thissuperposition prevents a decoupling between the spin and chargedegrees of freedom. This lack of spin–charge decoupling in theNBA has been pointed out by Zvyagin [8].

The importance of a spin-less fermion based description for theU infinity Hubbard model has been independently shown by Klein[9], Beni et al. [10] and Sokoloff [11]. Using the argument that noreordering of the spins in a linear Hubbard chain is possible in the

ARTICLE IN PRESS

A.K. Mishra, R. Kishore / Physica B 404 (2009) 3257–32603258

U infinity limit, they have shown that the thermodynamicproperties of the system corresponds to that of a spin-lessfermions, albeit with a modified chemical potential which takesinto the account the 2N-fold spin related degeneracies.

Due to the presence of a spin-less fermion component, NBA

wavefunction for the infinite U is not consistent with theantisymmetry associated with the fermions. In general, awavefunction whose spatial component correspond to that of aspin-less fermion cannot be created through the action ofGutzwiller projected fermion creation operators on the vacuumstate [12,13]. In fact one needs to employ orthofermion creationoperators for generating such wavefunctions [12–14]. It may benoted that in the U infinity context, an earlier attempt to constructsuch a spin-less fermion wavefunctions using creation operatorswas unsuccessful [9].

A state vector for particles obeying orthofermion statistics isantisymmetric only in its spatial component; the spin part of itswavefunction exhibits no symmetry relation. States havingdifferent permutations of distinct spin indices s and s are nowindependent orthonormal states. This property is obviouslysatisfied by any spin system. Consequently, the spin degree offreedom belong to a class of infinite statistics proposed byGreenberg [15,16], albeit with two indices only [17]. It had beenshown that the exact NBA wavefunctions in the U-1 limit obeythe orthofermi statistics [12,18]. Using orthofermion algebra, ithas been possible to reproduce the known exact 1-D U infinityresults for the thermodynamics and for the equivalence ofHubbard chain Hamiltonian to that of a spin-less fermionHamiltonian in the absence of magnetic field, in an elegantfashion [13,19–21].

Here we show that NBA implemented through the orthofer-mion algebra and GPO lead to different solutions for infinite U

Hubbard model. Towards this end, a system comprising of threeparticles on a 4-site closed isotropic chain is considered. U is takento be infinity. Consequently, only kinetic energy terms appear inthe Hamiltonian. We also note that when U ¼ 1, all the 2N spinconfigurations corresponding to an N particle system aredegenerate in the NBA approach [4,5].

The commutation relations for orthofermions and Gutzwillerprojected fermions are provided in the following section. TheHamiltonian for a system consisting of four sites with threeparticles and a hole on a closed isotropic chain is presented inSection 3. To compare the consequences of the orthofermion(NBA) and GPO approach for the infinite U case, the ground stateenergy and the associated degeneracies have been determinedand analysed in Section 4, followed by Section 5 on summary andconclusions.

2. Creation and annihilation operators for orthofermions andGutzwiller approach

The orthofermions are characterized by the following commu-tation relations [14]:

aiaajb þ ajaaib ¼ 0; ð3Þ

aiaayjb ¼ dab dij �Xg

ayjgaig

!: ð4Þ

Here Latin and Greek indices, respectively, specify the spatial andspin coordinates. The Gutzwiller projected annihilation (creation)operator g ðgyÞ is expressed in terms of the fermion annihilationand creation operators f and f y

gis ¼ fisð1� f yis fis Þ; gyis ¼ ð1� f y

is fis Þfy

is: ð5Þ

Using the fermion anticommutation relations (1,2) it can beshown that g and gy obey the following commutation relations:

gisgjs þ gjsgis ¼ 0; ð6Þ

gisgjs þ ð1� dijÞgjsgis ¼ 0; ð7Þ

gisgyjs þ ð1� dijÞg

y

jsgis ¼ 0; ð8Þ

gisgyjs þ gyjsgis ¼ dijð1� f yis fis Þ: ð9Þ

Obviously g and gy algebra is not closed. However, employing theconcept of generalized Fock spaces, a closed algebra for g and gy

can be obtained [22].A main difference between the orthofermion algebra and

Gutzwiller projection operator algebra is that the former isinvariant under a unitary transfer, and hence it is representationinvariant. In contrast, the Gutzwiller operator algebra depends onthe representation as it is not invariant under a unitarytransformation [12]. But this algebra is invariant under the phasetransformation

hia ¼ eifiagia ð10Þ

which implies particle number operator Nia exists for this case.The orthofermion algebra is not invariant under the phasetransformation eifia . Thus Nia does not exist for orthofermions.This algebra is invariant only with respect to following phasetransformations:

bia ¼ eifi aia; dia ¼ eifaaia: ð11Þ

Therefore only separate number operators Ni for orbital popula-tion and Na for spin number counting exist. This feature is aconsequence of the factorization of charge and spin componentsin the wavefunction in orthofermions and nested Bethe ansatzsolution for U-1.

3. System Hamiltonian

When only the kinetic energy term is present in the U ¼ 1

limit, the Hamiltonian for three particles distributed on the foursites of a closed isotropic chain is written as

H ¼ tX/ijS

Xs

cyiscjs: ð12Þ

The annihilation and creation operators c and cy take the value a

and ay respectively in the NBA or orthofermion approach, and areequal to g and gy in the Gutzwiller approach. The range of i and j

are 1–4, and /ijS implies a summation over the nearestneighbour sites only. The above system admits 32 configurationscorresponding to the four different locations of the hole on thechain times 23 spin configurations. The Hamiltonian does notcontain a term leading to a spin flip process. Therefore theproblem of finding the spectrum of H gets reduced to diagonaliz-ing it in sectors corresponding to Sz equal to 73

2 and 712. A basis

set for the Sz ¼32 sector is

jF3=2S ¼ fj1s2s3sS; j1s2s4sS; j1s3s4sS; j2s3s4sSg: ð13Þ

Both for the orthofermions and Gutzwiller projected fermions, theHamiltonian in Eq. (12) admits following representation in thisbasis set:

/F3=2jHjF3=2S ¼

0 t 0 t

t 0 t 0

0 t 0 t

t 0 t 0

0BBB@

1CCCA: ð14Þ

ARTICLE IN PRESS

A.K. Mishra, R. Kishore / Physica B 404 (2009) 3257–3260 3259

The symmetry of the problem ensures that the above representa-tion of the Hamiltonian is also valid for the Sz ¼ �

32 sector.

In the Sz ¼12 sector, a basis set having 12 elements is

jF1=2S ¼ fj1s2s3sS; j1s2s4sS; j1s3s4sS; j1s2s3sS; j1s2s4sS;

j1s3s4sS; j1s2s3sS; j1s2s4sS; j1s3s4sS; j2s3s4sS;

j2s3s4sS; j2s3s4sSg: ð15Þ

The Hamiltonian (12) now admits distinct representations fororthofermions and Gutzwiller projected fermions. In the case offormer, the Hamiltonian matrix for the Sz ¼71

2 sectors is given as

/F1=2jHjF1=2S ¼

0 t 0 0 0 0 0 0 0 t 0 0

t 0 t 0 0 0 0 0 0 0 0 0

0 t 0 0 0 0 0 0 0 t 0 0

0 0 0 0 t 0 0 0 0 0 t 0

0 0 0 t 0 t 0 0 0 0 0 0

0 0 0 0 t 0 0 0 0 0 t 0

0 0 0 0 0 0 0 t 0 0 0 t

0 0 0 0 0 0 t 0 t 0 0 0

0 0 0 0 0 0 0 t 0 0 0 t

t 0 t 0 0 0 0 0 0 0 0 0

0 0 0 t 0 t 0 0 0 0 0 0

0 0 0 0 0 0 t 0 t 0 0 0

0BBBBBBBBBBBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCCCCCCCCCCA

:

ð16Þ

For Gutzwiller projected fermions, the matrix for the Hamiltonian(12) in the Sz ¼71

2 sectors becomes

/F1=2jHjF1=2S ¼

0 t 0 0 0 0 0 0 0 0 0 t

t 0 t 0 0 0 0 0 0 0 0 0

0 t 0 0 0 0 0 0 0 t 0 0

0 0 0 0 t 0 0 0 0 t 0 0

0 0 0 t 0 t 0 0 0 0 0 0

0 0 0 0 t 0 0 0 0 0 t 0

0 0 0 0 0 0 0 t 0 0 t 0

0 0 0 0 0 0 t 0 t 0 0 0

0 0 0 0 0 0 0 t 0 0 0 t

0 0 t t 0 0 0 0 0 0 0 0

0 0 0 0 0 t t 0 0 0 0 0

t 0 0 0 0 0 0 0 t 0 0 0

0BBBBBBBBBBBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCCCCCCCCCCA

:

ð17Þ

The eigenvalues of H can be obtained as the solutions of thesecular equations for matrices in Eq. (14) and Eqs. (16) and (17).Next as H, S and Sz mutually commute, these operators havecommon eigenfunctions. As the exact nested Bethe ansatz showsthat the ground state is 23-fold degenerate in the U infinity limit,we provide here only the ground state energy and the associateddegeneracy, both for the orthofermions and GPO formalism. The t

is taken to be positive.In case the system has a 23 fold energy degeneracy, one would

expect that all these three matrices have common eigenvalues.We find that this simplification results only for the orthofermionsapproach and not in the GPO formalism.

4. Ground state energy

Using the orthofermion algebra (cf. Eqs. (3) and (4)), one getsthe energy eigenvalues for the ground state as Eo ¼ �2t. This stateis eight-fold degenerate. The associated wavefunctions are alsoeigenfunctions of the total spin operators S and Sz. The four spinstates having eigenvalue 3

2 for S and Sz taking the four values32 ;

12 ; �

12, and �3

2 have no degeneracy in the ground state. On the

other hand, both the spin states with S ¼ 12 and Sz ¼71

2 aretwo-fold degenerate in the ground state. This result for ortho-fermions ground states is similar to that of an N particle NBA

ground state which is 2N-fold degenerate even in the presence ofholes [4,5].

When the Gutzwiller projection operator algebra (cf. Eqs.(6)–(9)) is employed, the ground state energy is again �2t [23].But unlike the 23 fold degeneracy in orthofermions or NBA case,the ground state is now only four fold degenerate whichcorresponds to high spin state S ¼ 3

2, with Sz taking all the fourvalues 3

2 ;12 ; �

12 ; �

32. Thus the solution to the infinite U Hubbard

model are different in orthofermion ðNBAÞ and GPO approaches.Alternatively stated, the removal of double occupancy at any sitethrough Gutzwiller projector and through spin-less fermions leadto different solutions.

5. Summary and conclusion

The ground state properties of a system of three particles on afour site closed isotropic chain in the limit of U infinity have beenobtained employing the orthofermion algebra which is consistentwith the exact NBA solution in the U-1 limit, and Gutzwillerprojection operator formalism. It has been demonstrated thatthese two well known approaches for the U infinity HubbardHamiltonian lead to different results, and the Gutzwiller projec-tion operator formalism does not correctly reproduce the groundstate degeneracy. The reason for this discrepancy is that thewavefunctions employed to enforce the ‘no double-occupancy’ atany site are different in two approaches. In the former case, it isimplemented through spin-less fermions and in the later case, aprojection operator is employed to remove double occupancy. Inspite of this important distinction in these two approaches, oftena Hamiltonian employing Gutzwiller projected creation andannihilation operators have been erroneously employed in thecontext of nested Bethe ansatz [5,7]. Our analysis brings out thelimitation of the Gutzwiller projection operator technique and atthe same time highlights the equivalence between the nestedBethe ansatz formalism and orthofermions for the U-1 limit ofthe Hubbard Hamiltonian. The results also establish the relevanceof orthofermi statistics for the systems for which U-1 limit isapplicable.

Acknowledgement

A.K. Mishra acknowledges the financial support from CAPES,Brazil.

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