Excitations and finite-size properties of an integrable supersymmetric electronic model

Excitations and finite-size properties of an integrable supersymmetric electronic model

A. L. Malvezzi

Departamento de Física, Faculdade de Ciências, UNESP - Universidade Estadual Paulista, CP 473, 17033-360 Bauru, SP, Brazil M. J. Martins

Departamento de Física, Universidade Federal de São Carlos, CP 676, 13565-905 São Carlos, SP, Brazil 共Received 13 October 2009; revised manuscript received 30 November 2009; published 27 January 2010兲

We investigate the bulk and finite-size properties of an integrable extension of the Hubbard model with a free parameter␥related to the quantum deformation of the superalgebra sl共2兩2兲共2兲_{. The Bethe ansatz solution} is used to determine the ground state and the nature of the spin and charge excitations. As␥→0 the charge dispersion relation for low momenta crossover from linear to parabolic foreseeing a change from antiferro-magnetic to ferroantiferro-magnetic behavior. The study of the finite-size corrections to the spectrum reveals us that the underlying conformal theory has central chargec= −1 and a related effective central chargec_ef= 2 as well as critical exponents depending on the parameter␥. We note that exact results at the isotropic point␥= 0 can be established without recourse to the Bethe ansatz solution.

DOI:10.1103/PhysRevB.81.035120 PACS number共s兲: 71.10.Fd, 75.10.Jm, 71.10.Pm, 64.60.an

I. INTRODUCTION

The study of electron correlation effects in one-dimensional systems has by now attracted the attention of theorists for more than half-century. The physical behavior of one-dimensional correlated electron models are expected to be drastically different from that of free electrons.1 _The corresponding elementary excitations have a collective char-acter and nonperturbative techniques are essential to unveil the physical behavior. In this sense, electronic lattice systems solvable by the Bethe ansatz, such as the Hubbard model2 and quantum lattice gases,3 _{have provided relevant insights} in the understanding of correlation effects in low dimensions.4

Of particular interest are integrable extensions of Hubbard model containing extra correlation mechanisms besides the standard charge-charge interactions among electrons. These models can be derived exploring solutions of the graded Yang-Baxter equation5 _{with two fermionic and two bosonic} degrees of freedom. The respective Hamiltonian is built up from the logarithmic derivative of the Yang-Baxter solution based on a given Lie superalgebra. The Hamiltonian can then be made invariant by the underlying Lie superalgebra and in the scope of condensed matter such invariance is often re-ferred as supersymmetries.6,7 _{This invariance does not} con-tain the Lorentz group and therefore should not be confused with spacetime supersymmetry of field theories. We remark however that fermionic lattice models with underlying space-time supersymmetry have also been proposed and investi-gated in the literature.8_{Their Hamiltonians are derived from} the anticommutator of supersymmetric generators providing for instance a number of exact results for the spectrum.9_For a review of the properties of this class of models and further references see Ref.10.

To what concerns extended Hubbard models arising from supergroups, representative examples are the models based on the four dimensional representations of the sl共2兩2兲 and gl共2兩1兲Lie superalgebras.6,7,11,12_{We remark that further} gen-eralizations can be constructed by means of the quantum deformations of such algebras13–15 _{as well as on the central}

extension of sl共2兩2兲.17_{The purpose of this paper is to} inves-tigate the critical properties of an extended Hubbard model based on the quantum deformation of the twisted sl共2兩2兲共2兲

algebra.14We recall here that this model appears to provide a lattice regularization of an interesting integrable

共1 + 1兲-dimensional quantum field of two coupled massive Dirac fermions.16 _{Though the respective Bethe ansatz} solu-tion is known15_{it has not yet been explored to extract} infor-mation about the physical properties of such lattice elec-tronic model. Following15 _{the model Hamiltonian can be} rewritten as

H=

兺

i=1

L

兺

␴=⫾

关c_i†_,␴ci+1,␴+ H.c.兴关1 −X␴ni,−␴−X¯␴ni+1,−␴兴

+U

兺

i=1

L

ni,+ni,−+V

兺

i=1

L

关ni,+ni+1,−+ni,−ni+1,+兴

+Y

兺

i=1

L

关c_i†_,+c†_i,−ci+1,−ci+1,++ H.c.兴

+J

兺

i=1

L

关c_i†_,+c_i†_+1,−ci,−ci+1,++ H.c.兴−␮

兺

i=1

L

关ni,++ni,−兴

共1兲

where c_i†_,␴ and ci,␴ are fermionic creation and annihilation operators with spin index␴=⫾acting on a chain of lengthL. The operatorni,␴=ci,␴

†

ci,␴represents the number of electrons with spin␴on the ith site.

Apart from the standard kinetic hopping amplitude and the on-site Coulomb term U we see that Hamiltonian 共1兲

X_␴= 1 +␴sin共␥兲, X¯_␴= 1 −␴sin共␥兲,

U

2 =V=J=Y= cos共␥兲, 共2兲 where the anisotropy ␥ is related to the q deformation of sl共2兩2兲共2兲_{byq= exp关i␥兴.}

The potential ␮ is in principle arbitrary since the model conserves the total number of electrons with spin ␴=⫾. However, the invariance of Hamiltonian共1兲by the superal-gebra Uq关sl共2兩2兲共2兲兴 fixes a relation between ␮ and ␥, namely,14,15

␮= 2 cos共␥兲. 共3兲

Considering parametrizations共2兲and 共3兲 one can relate the spectra of Hamiltonian Eq.共1兲at the points ␥ and␲−␥. In fact, by performing a combination of particle-hole ci,␴

→ci,␴

†

and the parity ci,␴→共−1兲ici,␴ transformations one is able to find the following relation:

H共␥兲= −H共␲−␥兲. 共4兲

Due to property共4兲the analysis of the physical properties of Hamiltonian共1兲subjected to the constraints共2,3兲can be re-stricted to the antiferromagnetic interval 0ⱕ␥ⱕ␲/_{2. In this}

work we shall argue that the low-energy behavior of this model in the regime 0⬍␥ⱕ␲/_{2 is that of a conformally}

invariant theory with central charge c= −1 and underlying effective central charge cef= 2. The point ␥= 0 is special since the model reduces to the supersymmetric isotropic sl共2兩2兲extended Hubbard model.6In this case it was argued that though the excitations are gapless the dispersion rela-tions have a nonrelativistic branch.18,19_{In fact, we found that} for the electronic model 共1兲–共3兲 the speed of sound of the underlying low-lying excitations is proportional to sin共␥兲

which vanishes in the␥→0 limit.

We have organized this paper as follows. In next section we shall explore the Bethe ansatz solution to determine the

ground state and the nature of the excitations of the elec-tronic model 共1兲–共3兲. A particular characteristic is that the dispersion relation of charge excitations combines both the behavior of massless and massive degrees of freedom. We note that as ␥_→0 the charge dispersion for low energies changes its form from linear to quadratic dependence on the momenta and a corresponding change on the ground state behavior. In Sec.IIIwe study that finite-size properties of the spectrum of Hamiltonian共1兲–共3兲by both analytical and nu-merical approach. We argue that the critical properties are described by a critical line with central charge c= −1. Our conclusions are summarized in Sec.IV.

II. THERMODYNAMIC LIMIT

Here we will determine the ground state and the nature of the elementary excitations of the electronic model of Sec.I. These properties can be investigated by exploring the diago-nalization of Hamiltonians 共1兲–共3兲 by the Bethe ansatz method. It was found that the corresponding spectrum is pa-rameterized by the following nested Bethe equations:15

冋

sinh共␭j/2 −i␥/2兲 sinh共␭j/_{2 +}i␥/₂兲

册

L =

兿

k=1

N+

sinh共␭j−␮k−i␥兲 sinh共␭j−␮k+i␥兲 ,

j= 1, . . . ,N++N− 共5兲

and

兿

k=1

N₊+N₋

sinh共␮j−␭k−i␥兲 sinh共␮j−␭k+i␥兲

= −

兿

k=1

N₊

sinh共␮j−␮k− 2i␥兲 sinh共␮j−␮k+ 2i␥兲 ,

j= 1, . . . ,N+, 共6兲

where the integers N␴ denote the total number of electrons with spin␴=⫾.

The eigenvaluesE共L,␥兲of Hamiltonian共1兲–共3兲are given in terms of the variables␭jby

-1.6 -1.2 -0.8 -0.4 0.0 0.4 0.8 1.2 1.6

Re[λ,µ]

0.4 0.8 1.2

iπ/2

Im[λ,µ]

λ_j µ_j

-1.2 -0.8 -0.4 0.0 0.4 0.8 1.2

Re[λ,µ]

0.4 0.8 1.2

iπ/2

Im[λ,µ] (b)

(a)

FIG. 1. The ground state roots␭j共crosses兲and␮j共circles兲for

␥=␲/5 and L= 12 in sectors 共a兲 N+=L/2 , N−=L/2 − 1 and 共b兲

N+=L/2 − 1 , N−=L/2. We note that the roots␭jare the same for both sectors.

-1.6 -1.2 -0.8 -0.4 0.0 0.4 0.8 1.2 1.6

Re[λ,µ] 0.8

2.4 3.6

iπ/2

iπ

Im[λ,µ]

λ_j µ_j

-1.2 -0.8 -0.4 0.0 0.4 0.8 1.2

Re[λ,µ] 0.4

0.8 1.2

iπ/2

iπ/5

-iπ/5 -0.4 Im[λ,µ]

(b) (a)

FIG. 2. The first excited state roots␭_j共crosses兲and␮j共circles兲 for ␥=␲/5 and L= 12. Note that 共b兲 has two roots ␭j fixed at

E共L,␥兲=

兺

j=1

N₊+N₋

2 sin2共␥兲

cos共␥兲− cosh共␭j兲

. 共7兲

To make further progress it is important to identify the distribution of roots 兵␭j,␮k其 on the complex plane which reproduce the low-lying energies of Hamiltonian 共1兲–共3兲. This task is performed by first determining the particle num-ber sectors of the low-lying eigenvalues. This is done by means of brute force diagonalization of the Hamiltonian for small chainsLⱕ12 and a few values of the parameter␥. We then compare these eigenvalues with the results coming from the numerical analysis of the solutions of the Bethe ansatz equations 共5兲–共7兲. By performing this analysis we find that the ground state in the regime 0⬍␥ⱕ␲/_{2 for L even sits,}

for periodic boundary conditions, in sectors N+

=L/_{2⫾1 ,} N−=L/2 orN+=L/2 , N−=L/2⫾1 and therefore

it is fourfold degenerated. Due to the particle-hole symmetry it is sufficient to determine the respective pattern of the Be-the roots 兵␭j,␮k其for the sector with the minimum possible number of roots. In Fig.1we exhibit the ground state Bethe roots for L= 12 in sectors N+=L/2 , N−=L/2 − 1 and N+

=L/_{2 − 1 ,} N₋=L/_{2. We clearly see that the roots␭jare real}

while ␮k have a fixed imaginary part ati␲/2. The first ex-cited state is double degenerated and lies in sector N+=N−

=L/_{2. In Fig. 2 we show the corresponding Bethe roots} 兵␭j,␮k其 for L= 12. By performing this analysis for the low-energy excitations we find that they can be described mostly in terms of real variables when the second Bethe roots␮kis shifted by the complex number i␲/_{2. Considering this}

dis-cussion we find convenient to introduce the following variables:

␭j=␭j

共1兲

, ␮j=␭j

共2兲

+i␲

2, 共8兲

where␭共_ja兲苸Rfor a= 1 , 2.

Now by substituting Eq.共8兲in the Bethe ansatz equations

共5兲and共6兲and afterward by taking their logarithms we find that the resulting relations for␭共_ja兲 are

L⌽

冉

␭j

共1兲

2 ,

␥

2

冊

= 2␲Qj

共1兲

−

兺

k=1

N₊

⌽

冉

␭_j共1兲−␭_k共2兲,␲ 2 −␥

冊

,

j= 1, . . . ,N₊+N₋ 共9兲

and

−

兺

k=1

k⫽j N₊

⌽共␭共_j2兲−␭_k共2兲,2␥兲+ 2␲Q_j共2兲

=

兺

k=1

N₊+N₋ ⌽

冉

␭j

共2兲

−␭k

共1兲

,␲ 2 −␥

冊

,

j= 1, . . . ,N+ 共10兲

where function ⌽共␭,␥兲= 2 arctan关cot共␥兲tanh共␭兲兴.

The numbers Q共_ja兲 define the many possible logarithm branches and in general are integers or half-integers. Consid-ering our previous numerical analysis we find that the low-lying spectrum is well described by the following sequence of Q共_ja兲 numbers:

Q共_j1兲= −1

2关L−n+−n−− 1兴+j− 1, j= 1, . . . ,L−n+−n−,

共11兲

Q_j共2兲= −1 2

冋

L

2 −n+− 1

册

+j− 1, j= 1, . . . ,

L

2 −n+,

共12兲

where n_⫾ are integers labeling the sector with N_⫾=L/₂

−n_⫾particles with spin␴=⫾.

For largeLthe number of roots tend toward a continuous distribution on the real axis whose density can be defined in terms of the counting function Z共␭j

共a兲_兲

=Q_j共a兲/L by the expression

␳共a兲_共␭共a兲_兲=dZ共␭j

共a兲_兲

d␭j

共a兲 , a= 1,2. 共13兲

In the thermodynamicL_→⬁limit the Bethe equations共9兲

and 共10兲 turn into coupled linear integral relations for the densities␳共a兲_共␭共a兲_{兲which can be solved by the Fourier}

trans-form method. The final results for the densities are

␳共1兲_共␭共1兲_兲= 2

␲

sin共␥兲cosh共␭共1兲_兲 关cosh共2␭共1兲_兲

− cos共2␥兲兴,

␳共2兲_共␭共2兲_兲

= 1

2␲cosh共␭共2兲_兲. 共14兲

Now from the expressions for the density ␳共1兲_共␭共1兲_{兲 and}

Eq. 共7兲 we can compute the ground state energy per site

e_⬁共␥兲= limL_→⬁E0共L,␥兲/L. By writing the infinite volume

limit of Eq.共7兲in terms of its Fourier transform we find

e_⬁共␥兲= − 4 sin共␥兲

冕

0

⬁

d␻cosh关␻共␲/2 −␥兲兴sinh关␻共␲−␥兲兴

cosh关␻␲/₂兴_sinh关␻␲兴 for 0⬍␥ⱕ

␲

Let us consider the behavior of the low-lying excited states about the ground state. As usual these states are ob-tained from the Bethe equations 共9兲and共10兲by making al-ternative choices of numbersQ共_ja兲 over the ground state con-figuration. This procedure is nowadays familiar to models solved by Bethe ansatz and for technical details see, for instance.3,20 _{It turns out that the expressions for the energy} ␧共a兲_共␭共a兲_{兲 and the momenta p}共a兲_共␭共a兲_{兲, measured from the}

ground state, of a hole excitation on theath branch is given by

␧共a兲共␭共a兲兲= 2␲␳共a兲_共␭共a兲_{兲, p}共a兲_共␭共a兲_兲=

冕

␭共a兲 ⬁

␧共a兲共x兲dx.

共16兲

To compute the dispersion relation␧共a兲_共p共a兲_{兲one has to}

elimi-nate the auxiliary variable ␭共a兲 _{which connects energy and}

momentum. This is done by first computing the integrals in Eq. 共16兲 with the help of the roots densities关Eq. 共14兲兴. We then are able to eliminate the rapidity␭共a兲_from␧共a兲_共␭共a兲_兲and

the final results for the dispersion relations are

␧共1兲共p共1兲兲= 4 cos共␥兲sin

冉

p

共1兲

2

冊

冑

sin

2

冉

p

共1兲

2

冊

+ tan

2_共

␥兲,

共17兲

␧共2兲共p共2兲兲= 2 sin共␥兲sin共p共2兲兲. 共18兲

Note that the dispersion relation associated to particle num-ber excitations ␧共1兲_共p共1兲_{兲 has the interesting feature of being}

factorized in terms of two physically distinct types of disper-sions. The first part has a massless behavior while the second one has a massive character with a mass term proportional to tan共␥兲. We see that near the point␥= 0 the mass gap becomes small and the terms depending on the momenta dominate Eq.

共17兲. This means that at low energies the dispersion relation can crossover from a linear to a parabolic dependence on the momenta as␥ approaches zero. This is indeed the situation since we shall see below that the expression 共17兲 remains valid down to␥= 0. By way of contrast the dispersion related to the spin branch ␧共2兲_共p共2兲_{兲 is similar to the spin-waves of}

the antiferromagnetic Heisenberg XXZ model. For low mo-menta and as long as␥⫽_{0 both charge and spin excitations}

behave linearly with the respective momenta. Their disper-sion relations have in fact a common slope at p共a兲_{= 0,}

namely,

␧共a兲_共p共a兲_{兲 ⬃}

2 sin共␥兲p共a兲 for 0⬍␥ⱕ␲

2 共19兲

and therefore they travel with the same speed of sound v_s

= 2 sin␥.

Let us turn our attention to the physical properties of the model at special point ␥= 0. In this case, the Hamiltonian

共1兲–共3兲 commutes also with the number of local electrons pairs18 _{and it is proportional to the graded permutator,}

H共␥= 0兲=

兺

j=1

L

兺

a,b=1 4

共− 1兲p_ap_be ab

共j兲_丢e

ba

共j+1兲

−L 共20兲

wheree_ab共j兲 denotes 4⫻4 Weyl matrices acting on the jth site and the Grassmann parities are given by p1= 0, p2= 1, p3

= 1, and p4= 0.

The diagonalization of the Hamiltonian共20兲by the Bethe ansatz was discussed in the literature since long ago.3,5 _We remark that the respective Bethe equations do not follow immediately from Eqs. 共5兲 and 共6兲 when ␥→0 due to the

peculiar pattern of the Bethe roots 兵␮j其. We find, however, that certain properties of the model at ␥= 0 can be inferred without the need of using its Bethe ansatz solution. This is done by first investigating the pattern of the ground state degeneracies of Hamiltonian 共20兲by means of exact diago-nalization up to L= 12. This study has revealed that the ground state sits in many different sectors whose total num-ber of particles is either Lor L⫾1. This tells us the ground state for a givenLis 4L-fold degenerated and that its energy and low-lying excitations can be computed from the particu-lar simple sectors N+=L, N−= 0 orN+= 0 , N−=L. Because

these are typical ferromagnetic states the calculations are rather direct. Denoting by p the momentum of an excitation with spin ␴= − over the state N₊=L, N₋= 0 one finds that the corresponding energy is

E共p兲= − 2L+ 4 sin2

冉

p

2

冊

, 共21兲

where for a finite Lthe momenta p=2_L␲K, K= 0 , . . . ,L− 1. From Eq.共21兲we conclude that the ground state per site is e_⬁共␥= 0兲= −2 and that for low momenta p the excitation energy are proportional to p2. Therefore, the system has a nonrelativistic behavior in accordance with previous works in the literature.18,19_{Interesting enough, we observe that such} results can also be derived from Eqs.共15兲and共17兲by taking the limit␥_→0. To obtain the ground state energy from Eq.

共15兲 we first perform the change of variable ␻_→␻/_{␥ and}

afterward take the␥_→0 limit. On the other hand, the disper-sion relation␧共p兲= 4 sin2_{共p/2兲follows directly from Eq.共17兲}

by substituting␥= 0.

We have now the basic ingredients to investigate in next section the finite-size effects in the spectrum of the electronic models 共1兲–共3兲for 0⬍␥ⱕ␲/_2.

III. CRITICAL PROPERTIES

The results of the previous section suggests to us that the generalized Hubbard model 共1兲–共3兲 in the regime 0⬍␥

ⱕ␲/_{2 is conformally invariant. This means that the}

corre-sponding critical properties can be evaluated investigating the eigenspectrum finite-size corrections.21,22 For periodic boundary conditions, the ground stateE0共L,␥兲 are expected

to scale as

E0共L,␥兲

L =e⬁−

␲v_s共␥兲c

6L2 +O共L

−2_兲

, 共22兲

From the excited statesE␣共L,␥兲we are able to determine the dimensions X_␣共␥兲 of the respective primary operators, namely,

E_␣共L,␥兲

L −

E0共L,␥兲

L =

2␲v_s共␥兲X_␣共␥兲 L2 +O共L

−2_兲

. 共23兲

A first insight on the structure of the finite-size corrections can be obtained by applying the so-called density root method.23–25 _{This approach explores the Bethe ansatz} solu-tion and it makes possible to compute theO共L−2兲corrections to the densities of roots ␳共a兲_共␭共a兲_{兲. This method is however}

only suitable for systems whose ground state and low-lying excitations are described by real roots. Fortunately, this is exactly the situation we have found in Sec.IIonce the sec-ond root is shifted byi␲/_{2. Considering this subtlety on the}

root density approach we find that the leading finite-size be-havior of the eigenenergies is

E共L,␥兲

L =e⬁共␥兲+

2␲

L2vs共␥兲

冋

−

1 6+Xn+,n−

m,m₋

共␥兲

册

+O共L−2兲,

共24兲

where the dependence of the scaling dimensionsX_n

+,n− m,m_−共

␥兲on the anisotropy ␥is

X_n

+,n− m,m_−共

␥兲=1 4

冋

n+

2

+n− 2

+ 2

冉

1 −2␥

␲

冊

n+n−

册

+ ␲2

4␥共␲−␥兲

⫻

冋

m2+m₋2− 2

冉

1 −2␥

␲

冊

mm−

册

. 共25兲

As before the integersn_⫾parameterizes the numbers of elec-trons N_⫾=L/_{2 −}n_⫾ with spin ␴=⫾. The indices m=m+

+m₋ andm₊ characterize the presence of holes in theQ_j共1兲

and Q共_j2兲 distributions and in principle can be integers or half-integers. This approach is however not able to predict either the possible values for the vortex numbers mandm+

as well possible constraints with the corresponding spin-wave integersn andn+. To shed some light on this problem

we shall first study the finite-size effects at the particular point␥=␲/_{2. For}␥=␲/_{2 we see that all the interactions in}

Hamiltonian 共1兲–共3兲 cancel out and we remain with two coupled free fermion models. In this case standard Fourier technique is able to provide us the exact expressions for the low-lying energies in the case of arbitraryL. Due to periodic boundary conditions, the respective calculations depend on the total number of electrons on the lattice L. We find that when n=n++n− is odd that the expression for the lowest

energy in this sector is given by

Eodd

冉

L,

␲

2

冊

= − 2

冋

cos

冉

␲n+

L

冊

+ cos

冉

␲n−

L

冊

册

sin

冉

␲

L

冊

. 共26兲

Considering the asymptotic expansion of Eq.共26兲for largeL

one finds,

Eodd

冉

L,

␲

2

冊

L =e⬁

冉

␲

2

冊

+ 2␲

L2 v_s

冉

␲

2

冊

冋

− 1 6 +

n₊2+n₋2

4

册

+O共L−2兲. 共27兲

By way of contrast when n=n++n− is an even number the

lowest energy is

Eeven

冉

L,

␲

2

冊

= −␴

兺

=⫾

cos

冋

␲共n␴+ 1兲

L

册

sin

冉

␲

L

冊

+

cos

冋

␲共n␴− 1兲

L

册

sin

冉

␲

L

冊

,

共28兲

whose expansion for largeL is

Eeven

冉

L,

␲

2

冊

L =e⬁

冉

␲

2

冊

+ 2␲

L2vs

冉

␲

2

冊

冋

− 1 6+

n₊2+n₋2

4 +

1 2

册

+O共L−2兲. 共29兲

Taking into account Eqs. 共27兲 and 共29兲 we see that the expected finite-size corrections depend whether the index n

is an odd or even integer. In addition, by comparing Eqs.

共27兲 and 共29兲 with the general results 共24兲 and 共25兲 at ␥

=␲/_{2 we clearly see that for} n odd the numbers mandm−

appear to start from zero while forneven the lowest allowed value formandm₋is in fact one-half. This analysis strongly suggests that possible values for the vortex numbersm and

m+ should satisfy the following rule:

共1兲 for n odd→m,m−= 0,⫾1,⫾2, . . . ,

共2兲 for n even→m,m−= ⫾

1 2,⫾

3 2,⫾

5

2, . . . . 共30兲 Let us now check if the above proposal remains valid for other values of the parameter␥. This is done mostly by solv-ing numerically the original Bethe equations 共5兲 and共6兲 up toL= 32. For the excited states whose respective Bethe roots are unstable already for moderate values of Lwe have used the data obtained from the numerical diagonalization through the Lanczos method. This numerical work enables us to com-pute for eachL the following sequence:

X共L兲=

冉

E共L,␥兲

L −e⬁共␥兲

冊

L2

2␲v_s共␥兲

+1

6. 共31兲 By extrapolatingX共L兲for several values ofLwe are able to verify expression共25兲forX_n

+,n− m,m₋

共␥兲and constraints共30兲. In TablesI–IIIwe exhibit the finite-size sequence关Eq.共31兲兴for six lowest dimensions on the even sector to make an exten-sive check of the less unusual part of rule 共30兲. For sake of completeness we also present three conformal dimensions corresponding to thenodd sector. All those numerical results confirm conjectures 共25兲and共30兲 for the finite-size proper-ties of the generalized Hubbard models共1兲–共3兲.

sits in the sectorsn+=⫾1 andn−= 0 orn+= 0 andn−=⫾1.

Considering rule 共30兲 the corresponding vortex numbers have the lowest possible valuesm=m+= 0 and from Eqs.共24兲

and共25兲we derive the following finite-size behavior:

E0共L,␥兲

L =e⬁+

␲v_s共␥兲

6L2 +O共L

−2_兲

. 共32兲

Direct comparison between Eqs.共22兲and共32兲leads us to conclude that the central charge of the underlying conformal theory is

c= − 1 for 0⬍␥ⱕ␲

2. 共33兲

Recall here that central charge关Eq.共33兲兴has been derived from the finite-size corrections with strict periodic boundary conditions. In the operator content of conformal theories with cⱕ0 it is however expected the presence of negative conformal weights. This means that ifX_⬍is the lowest such scaling dimension the true ground state is then governed by the effective central cef=c− 12X⬍. This type of sector in models with underlying superalgebra can be obtained by considering antiperiodic boundary conditions for the

fermi-onic degrees of freedom, see for instance.26_{The antiperiodic} twist prompts the sector with maximum number of Bethe rootsn_⫾= 0 which is now compatible with the vortex quan-tum numbers m=m−= 0. Denoting by E⬍共L,␥兲 the energy corresponding to this state we derive from the Eqs.共24兲and

共25兲the following scaling behavior:

E_⬍共L,␥兲

L −

E₀共L,␥兲

L =

2␲v_s共␥兲 L2

冉

−

1 4

冊

+O共L

−2_{兲. 共34兲}

From Eq.共34兲we see that the negative scaling dimension isX⬍= −

1

4 and the respective effective central charge is

there-fore cef= −1 +

12

4= 2. It is expected that the behavior of the

heat capacity at low-temperatureT be governed by such ef-fective central charge.21_{This means that the leading term of} the heat capacity per length of such electronic model should be ₃2v␲_s共T␥兲 for 0⬍␥ⱕ␲/2.

Let us now conclude by discussing the conformal dimen-sions on the periodic sector. The scaling dimendimen-sions of the primary operators X¯_n,n

+ m,m_+共

␥兲 depend on the anisotropy␥ and they should be measured from the ground state E0共L,␥兲.

Considering Eqs. 共24兲 and 共25兲 together with Eq. 共32兲 we find that they are given by

TABLE II. Finite-size sequences关Eq.共31兲兴of the anomalous dimensions for␥=␲/_{5 ,␲}/_{3 from the Bethe}

ansatz. The exact conformal dimensions areX_1,00,0共␥兲=1₄,X0,0_2,−1共␥兲=₄1+2_␲␥, andX

2,−2 1 2,

1

2_共␥兲=4_␲␥+_4共1−1_␥/␲兲.

L X_1,00,0共␲

5兲 X2,−1

0,0 _共␲

5兲 X2,−2

1 2,

1 2共␲

5兲 X1,0

0,0_共␲

3兲 X2,−1

0,0_共␲

3兲 X2,−2

1 2,

1 2共␲

3兲

8 0.251098 0.642630 1.000395 0.252587 0.902529 1.541194

12 0.250523 0.646574 1.047807 0.251149 0.910140 1.622978

16 0.250301 0.648003 1.068757 0.250646 0.912924 1.655748

20 0.250195 0.648689 1.080196 0.250413 0.914244 1.672266

24 0.250136 0.649072 1.087265 0.250287 0.914971 1.681827

28 0.250101 0.649307 1.092001 0.250211 0.915414 1.687896

32 0.250077 0.649463 1.095375 0.250161 0.915704 1.692009

Extrap. 0.250003⫾1 0.65003⫾1 1.1124⫾1 0.250004⫾2 0.9167⫾2 1.70825⫾1

Exact 0.25 0.65 1.1125 0.25 0.91666¯ 1.70833¯

TABLE I. Finite-size sequences关Eq.共31兲兴of the anomalous dimensions for␥=␲/5 ,␲/3 from the Bethe

ansatz. The expected exact conformal dimensions areX_0,0

1 2,

1

2共␥兲=_4共1−1_␥/_␲兲, X_0,0 1 2,−

1

2共␥兲=_4共␥1/_␲兲, andX_1,−1 1 2,

1 2共␥兲=_␲␥ +_4共1−1_␥/␲兲.

L X

0,0 1 2,

1 2_共␲

5兲 X0,0

1 2,−

1 2_共␲

5兲 X1,−1

1 2,

1 2_共␲

5兲 X0,0

1 2,

1 2_共␲

3兲 X0,0

1 2,−

1 2_共␲

3兲 X1,−1

1 2,

1 2_共␲

3兲

8 0.313380 1.227954 0.488672 0.380231 0.752958 0.681073

12 0.312980 1.239120 0.498637 0.377310 0.751250 0.694030

16 0.312787 1.243526 0.502997 0.376297 0.750687 0.699292

20 0.312689 1.245526 0.505395 0.375829 0.750434 0.701996

24 0.312633 1.247012 0.506892 0.375575 0.750299 0.703591

28 0.312595 1.247954 0.507906 0.375423 0.750218 0.704619

32 0.312576 1.248063 0.508633 0.375323 0.750166 0.705327

Extrap. 0.31249⫾1 1.2504⫾2 0.51219⫾1 0.37498⫾1 0.74999⫾1 0.708336⫾1

X ¯

n,n₊ m,m₊

共␥兲=X_n,n

+ m,m₊

共␥兲−1

4 for 0⬍␥ⱕ

␲

2. 共35兲 To our knowledge, models exhibiting this kind of univer-sality class have so far been found in a not self-adjoint theory based on the deformedosp共2兩2兲symmetry.27_{Therefore, the} correlated electron system 共1兲–共3兲appears to be the first ex-ample of a Hermitian Hamiltonian whose continuum limit is described by a field theory with c= −1 with continuously varying anomalous dimensions. The fact that a line of critical exponents with c⬍0 can be realized in terms of Hermitian models could be of importance for practical applications in condensed matter such as in the physics of disordered systems.

IV. CONCLUSIONS

We have studied the thermodynamic limit and the finite-size behavior of an exactly solvable generalization of the Hubbard model with free parameter␥related to the quantum

Uq关SU共2兩2兴兲 superalgebra whereq= exp共i␥兲. We have deter-mined the nature of the ground state and the behavior of the elementary excitations. One interesting feature of the model is that as ␥ approaches zero the spectrum associated with several distinct sectors degenerate to that of the ferromag-netic sector N_⫾=L. Interesting enough, this behavior is somehow encoded on the structure of the dispersion relation for the charge sector. It has the peculiar feature of being given in terms of the product of massless and massive energy-momenta relations. When␥_→0 the mass gap goes to zero and the dependence on the momenta prevail in the

dis-persion relation. As a result, we clearly observe that charge dispersion relation changes its dependence on low momenta from linear to parabolic. The latter being a characteristic of ferromagnetic behavior.

In the regime 0⬍␥ⱕ␲/_{2 the low-lying excitations have}

relativistic behavior and they travel with the same speed of soundv_s共␥兲= 2 sin共␥兲. The underlying conformal theory has

central chargec= −1 and a line of continuously varying ex-ponents. We stress that conformal theories with cⱕ0 are expected to have negative conformal weights and the true ground state is then governed by the effective central charge. In our case we found that such effective central charge is

cef= 2 predicting that the leading behavior of the heat capac-ity should be ₃2v␲_s共T␥兲 for 0⬍␥ⱕ␲/2.

For the particular point ␥= 0 we have argued that basic properties can be obtained without recourse to the Bethe an-satz solution. We expect that this observation remains valid for all integrable models based on the Lie superalgebra sl共p兩q兲for arbitrary finite number of bosonic共p兲and fermi-onic 共q兲 degrees of freedom. This suggests that the models based on the deformed sl共p兩q兲symmetry may also have ex-citation modes with dispersion relation exhibiting both mass-less and massive behaviors. We hope to investigate this in-teresting possibility as well as its consequences in a future publication.

ACKNOWLEDGMENTS

This work was supported by the Brazilian Research Agen-cies CNPq and FAPESP.

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