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Analytical Results for the Grand Canonical Partition Function for Unidimensional Hubbard Model, Up to Order β 5 . I.C. Charret1 , Departamento de F´ısica Instituto de Ciˆencias Exatas Universidade Federal de Minas Gerais Campus da Pampulha Belo Horizonte, M.G., 31270–901 BRAZIL E.V. Corrˆea Silva2 Centro Brasileiro de Pesquisas F´ısicas R. Dr. Xavier Sigaud n.o 150 Rio de Janeiro, R.J., 22290-180 BRAZIL S.M. de Souza3 , Departamento de Ciˆencias Exatas Universidade Federal de Lavras C.P.: 37 Lavras, M.G., 37200–000 BRAZIL M.T. Thomaz4 Instituto de F´ısica Universidade Federal Fluminense Av. Gal. Milton Tavares de Souza s/n.o Campus da Praia Vermelha Niter´ oi, R.J., 24210–310 BRAZIL Abstract We calculate the exact analytical coefficients of the β–expansion of the grand canonical partition function of the unidimensional Hubbard model up to order β 5 , using an alternative method, based on properties of the Grassmann algebra. The results derived are non–perturbative and no restrictions on the set of parameters that characterize the model are required. By applying this method we obtain analytical results for the themodynamical quantities, in the high–temperature limits, for arbitrary density of electrons in the unidimensional chain. 1

E–mail: [email protected] E–mail: [email protected] 3 E–mail: [email protected] 4 Corresponding author: Dr. Maria Teresa Thomaz; R. Domingos S´avio Nogueira Saad n.o 120 apto 404, Niter´oi, R.J., 24210–340, BRAZIL –Phone/Fax: (21) 620–6735; E–mail: [email protected] 2

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1. Introduction Working with fermionic variables seems discouraging, for their non–commuting nature, in a certain way, compells one to a higher degree of care than that required by commutative variables. Nevertheless, Grassmann algebra properties justify their use in many circumstances [1]. With that in mind, we have recently described an alternative method to calculate the terms of the β– expansion of the grand canonical partition function of periodic unidimensional self–interacting fermionic models, in which Grassmann algebra properties play a central role. No auxiliary fields are needed, and we express our results in terms of matrices with commuting elements. The general approach for a periodic unidimensional fermionic model has been applied to the unidimensional Hubbard model up to order β 3 [2]. An important point about the method developed in reference [2] is that even though the unidimensional Hubbard model has exact solutions [3], the analytical expressions are only known in the half– filled case. This drawback hinders the analytical evaluation of the partition function for the model from the knowledge of its energy spectrum when we are not considering the half–filling case. Takahashi[4] derived a closed expression for the grand canonical partition function of the unidimensional Hubbard model, but besides requiring the formulation of some additional hypotesis, a simple closed expression for the grand canonical partition function is only obtained in the strong coupling limit. The literature offers many examples of high temperature expansions of the grand canonical partition function for the Hubbard model in different space dimensions, some of them up to order β 9 [5]. However, all these works referer to either some approximation in which one of the characteristic constants of the model has to be much bigger than the other, or some consideration based on important numerical analysis. Our results are analytical and do not rest upon any additional hypothesis on the constants that characterize the model. We should point out that we do not perform a perturbative expansion valid in the high temperature limit [5], but a β–expansion of the grand canonical partition function [6] where we calculate the exact analytical coefficients of the terms up to order β 5 for any density of electrons in the unidimensional chain. In the present paper, we will apply the approach developed in reference [2] to get the exact terms at orders β 4 and β 5 of the grand canonical partition function of the unidimensional Hubbard model. In section 2 we present a review of the results of reference [2]. In section 3, 1

we write down the unidimensional Hubbard model and the Grassmann functions necessary in the calculations that follow. In section 4 we obtain the coefficients at orders β 4 and β 5 of the β–expansion of the grand canonical partition function of the unidimensional Hubbard model. We use the grand potencial derived up to order β 4 to calculate some physical quantities. A certain property of the multivariable Grassmann integrals — namely, its factorization — opens the way to extending the method of reference [2] to orders higher than β 3 . This factorization property is presented in Appendix A through one example. In Appendix B we introduce a graphical notation that greatly simplifies the calculations. In Appendix C, we present a table of the necessary multivariable Grassmann integrals. Section 5 is dedicated to our conclusions.

2. Review of Previous Results [2] The grand canonical partition function of any system is given by: Z(β; µ) = Tr(e−βK ), where β =

1 kT

(2.1)

, k is the Boltzmann constant, T is the absolute temperature, and

K = H − µN,

(2.1a)

H is the hamiltonian of the system, µ is the chemical potential and N is the total number of particles operator. In the high temperature limit, β ≪ 1, Z(β; µ) has the expansion ∞ X (−β)n Z(β, µ) = Tr[1I − βK] + Tr[Kn ], n! n=2

(2.2)

that we call the β–expansion of the grand canonical partition function. For any self–interacting fermionic quantum system, in reference [2] we used the Grassmann algebra to show that 2nN

n

Tr[K ] =

Z 2nN Y

dηI d¯ ηI e

P

I,J =1

I=1

n

η ¯I AIJ ηJ

×

n n × K (¯ η , η; ν = 0) K (¯ η, η; ν = 1) · · · K (¯ η , η; ν = n − 1),

2

(2.3)

where the matrix A is given by,

A=

A↑↑ O l

O l A

(2.3a)

,

↓↓

and

A↑↑ = A↓↓

1lN×N

O lN×N = . .. 1lN×N

−1lN×N

O lN×N

1lN×N

−1lN×N

O lN×N

O lN×N

··· O lN×N

··· O lN×N . .. .

(2.3b)

· · · 1lN×N

Each matrix Aσσ has dimension nN × nN , 1lN×N been the identity matrix in dimension N × N and O lN×N the null matrix in this dimension. N is the number of space sites and n is the power of the β term. The non–null elements of Aσσ , σ =↑ and σ =↓, are

Aσσ IJ

a = 1, I = 1, 2, · · · , nN II I = 1, 2, · · · , (n − 2)N = aI,I+N = −1, a(n−1)N+I,I = 1, I = 1, 2, · · · , N.

(2.3c)

n The Grassmann function K (¯ η , η) is the kernel of the fermionic operator K in the normal

order [7]. In writing down eq.(2.3), we have used a particular mapping for the Grassmann generators [2], that greatly simplify our calculations: η↑ (xl , τν ) ≡ ηνN+l

(2.4a)

η↓ (xl , τν ) ≡ η(n+ν)N+l ,

(2.4b)

and

where l = 1, 2, · · · , N , and ν = 0, 1, · · · , n − 1. The mappings (2.4a–b) can be summarized as [8]:

ησ (xl , τν ) ≡ η[ (1−σ) n+ν]N+l . 2

The generators η¯σ (xl , τν ) have an equivalent mapping. 3

(2.4c)

In reference [9] we showed that the Grassmann integrals (2.3) can be written as co–factors of the matrix Aσσ . As we have stated before, the calculation of the co–factors of matrix Aσσ gets simpler when we diagonalize it through a similarity transformation P−1 Aσσ P = D,

(2.5)

where the matrix D is, λ 1l 1 N×N O lN×N D= .. .

O lN×N λ2 1lN×N

··· ···

O lN×N O lN×N .. .

,

(2.5a)

O lN×N O lN×N · · · λn 1lN×N λi , i = 1, 2, · · · , n, are the eigenvalues of matrices Aσσ , σ =↑, ↓. At the same time, we transform of the anti–commuting variables, η ′ = P−1 η

and

η¯′ = η¯ P.

(2.5b)

The matrices P and P−1 have a block–structure, and in reference [2] we got the elements of matrix P and its inverse for the particular cases n = 2 and n = 3, besides the eigenvalues of matrices Aσσ , σ =↑, ↓. However, for any value of n, we have that 1 iπ ′ (n) pνν ′ = √ e n (2ν +1)(ν+1) , n

(2.6a)

′ iπ 1 (n) qν ′ ν = √ e− n (2ν +1)(ν+1) , n

(2.6b)

and

with ν, ν ′ = 0, 1, · · · , n − 1, and

and

(n)

p00 1lN×N .. P= . (n) pn−1,0 1lN×N

P−1

(n)

q00 1lN×N .. = . (n) qn−1,0 1lN×N

(n) p0,n−1 1lN×N .. . (n) · · · pn−1,n−1 1lN×N ···

(n) q0,n−1 1lN×N .. . . (n) · · · qn−1,n−1 1lN×N ···

4

(2.6c)

(2.6d)

The diagonal elements of matrix D are: iπ

n (2ν+1) , λ(n) ν =1−e

ν = 0, 1, · · · , n − 1.

(2.6e)

where the eigenvalues are N–fold degenerated, N being the number of space sites. Due to (n)

(n)

space translation symmetry, we should note that the elements pνν ′ and qνν ′ do not carry any space site index. Once the matrix A has a block–structure as depicted in eq.(2.3), the integrals (2.3) are equal to the product of the integral in the sector σσ =↑↑ times the integral in the sector σσ =↓↓. The Grassmann integrals in the sector ↑↑ have the form:

M (L, K) =

Z Y nN

i=1

nN P

η ¯i A↑↑ η ij j

dηi d¯ ηi η¯l1 ηk1 · · · η¯lm ηkm ei,j=1

(2.7)

,

with L = {l1 , · · · , lm } and K = {k1 , · · · , km }. The products η¯η are ordered in such a way that l1 < l2 < · · · < lm and k1 < k2 < · · · < km . From reference [9], the result of this type of integrals is equal to: M (L, K) = (−1)(l1 +l2 +···+lm )+(k1 +k2 +···+km ) A(L, K),

(2.7a)

where A(L, K) is the determinant of the matrix obtained from matrix A by deleting the lines {l1 , · · · , lm } and the columns: {k1 , · · · , km }. The Grassmann integrals to be calculated in sector ↓↓ are of the same type as that of eq.(2.7). After the similarity transformation (2.5) and the change of variables (2.5b), in a schematic way, the integral (2.7) becomes

M (L, K) =

Z Y nN

i=1

−1

dηi d¯ ηi (¯ ηP

−1

)l1 (Pη)k1 · · · (¯ ηP

)lm (Pη)km e

nN P

i,j=1

η ¯i Dij ηj

, (2.8)

where D is a diagonal matrix whose entries are given by eq.(2.6e). We should point out that the relations (2.6a)–(2.6e) are valid for any unidimensional self–interacting fermionic model with space translation symmetry. 5

We have a large number of integrals that contribute to eq.(2.3). For a discussion on some useful symmetries and their use in the reduction of the number of contributing integrals, the reader is referred to [2].

3. Unidimensional Hubbard Model The hamiltonian that describes the Hubbard model in one space dimension is [10]:

H=

N X X i=1 j=1

tij a†iσ ajσ

+U

σ=−1,1

N X

a†i↑ ai↑ a†i↓ ai↓

+ λB

N X X

σa†iσ aiσ

(3.1)

i=1 σ=−1,1

i=1

where a†iσ is the creation operator of an electron in site i with spin σ and aiσ is the destruction operator of an electron in site i with spin σ. The first term on the r.h.s. of eq.(3.1) is the kinetic energy operator. All diagonal elements of tij are equal, tii = E0 , the only non–null off–diagonal terms are ti,i−1 = ti,i+1 = t, where i = 1, 2, . . . , N, and they contribute to the hopping term. U is the strength of the interaction between the electrons in the same site but with different spins. We have defined λB = − 12 gµB B, where g is the Land´e’s factor, µB is the Bohr’s magneton and B is the constant external magnetic field in the zˆ direction. The periodic boundary condition in space is implemented by imposing that a0σ ≡ aNσ

and aN+1,σ ≡ a1σ . Therefore, the hopping terms t10 a†1σ a0σ and tN,N+1 a†Nσ aN+1,σ become

t1N a†1σ aNσ and tN,1 a†Nσ a1σ respectively. We point out that the hamiltonian (3.1) is already

in normal order. The kernel of the operator K (eq.(2.1a)) for the unidimensional Hubbard model on a lattice with N space sites, written in terms of the generators η¯I and ηJ , is equal to

n

K (¯ η , η; ν)) = +

N X X

l=1 σ=±1

+

N X

N X X

l=1 σ=±1

(E0 + σλB − µ) η¯[ (1−σ) n+ν]N+l η[ (1−σ) n+ν]N+l + 2

2

t[¯ η[ (1−σ) n+ν]N+l η[ (1−σ) n+ν]N+l+1 + η¯[ (1−σ) n+ν]N+l η[ (1−σ) n+ν]N+l−1 ]+ 2

2

2

U η¯(n+ν)N+l η(n+ν)N+l η¯νN+l ηνN+l ,

l=1

with the periodic spatial boundary condition: 6

2

(3.2a)

t η¯[ (1−σ) n+ν]N+N η[ (1−σ) n+ν]N+N+1 ≡ t η¯[ (1−σ) n+ν]N+N η[ (1−σ) n+ν]N+1 2

2

2

(3.2b)

2

and t η¯[ (1−σ) n+ν]N+1 η[ (1−σ) n+ν]N ≡ t η¯[ (1−σ) n+ν]N+1 η[ (1−σ) n+ν]N+N , 2

2

2

(3.2c)

2

and the anti–periodic boundary condition in ν:

η[ (1−σ) n+n]N+l = −η[ (1−σ) n]N+l , 2

(3.2d)

2

for l = 1, 2, · · · , N , and σ =↑, ↓. We have used the mapping (2.4c) to write the previous expressions. In order to write down the terms that contribute to T r[K4 ] and T r[K5 ] in a simplified way , we define:

E(¯ η, η; ν; σ) ≡

N X

η¯[ (1−σ) n+ν]N+l η[ (1−σ) n+ν]N+l ;

(3.3a)

T − (¯ η , η; ν; σ) ≡

N X

η¯[ (1−σ) n+ν]N+l η[ (1−σ) n+ν]N+l+1 ;

(3.3b)

N X

η¯[ (1−σ) n+ν]N+l η[ (1−σ) n+ν]N+l−1 .

(3.3c)

l=1

l=1

2

2

2

2

and +

T (¯ η , η; ν; σ) ≡

l=1

2

2

We also define

E(¯ η, η; ν) ≡ T − (¯ η, η; ν) ≡ T + (¯ η, η; ν) ≡

X

E(σ)E(¯ η, η; ν; σ),

(3.4a)

X

t T − (¯ η , η; ν; σ),

(3.4b)

X

t T + (¯ η , η; ν; σ),

(3.4c)

σ=±1

σ=±1

σ=±1

and U(¯ η , η; ν) ≡

N X

η¯(n+ν)N+l η(n+ν)N+l η¯νN+l ηνN+l ,

l=1

7

(3.4d)

where E(σ) ≡ E0 − σλB − µ. The term E(¯ η , η; ν) represents the diagonal part of the kinetic energy , T − (¯ η , η; ν) and T + (¯ η , η; ν) are the hopping terms and U(¯ η , η; ν) is the fermionic

interaction term. n For the unidimensional Hubbard model, the Grassmannian function K (¯ η , η; ν) is written

as

n K (¯ η, η; ν) = E(¯ η, η; ν) + T − (¯ η , η; ν) + T + (¯ η , η; ν) + U(¯ η , η; ν).

(3.5)

4. The Exact Coefficients of the β–Expansion of the Grand Canonical Partition Function for the Unidimensional Hubbard Model In eq.(2.2) we have the β–expansion of the grand canonical partition function for any quantum system. For the particular case of self–interacting unidimensional fermionic models, the expression of T r[Kn ] is given by eq.(2.3). In reference [2], we calculated the exact coefficients of the terms β 2 and β 3 of the expression (2.2) for the unidimensional Hubbard model for arbitrary values of the constants E0 , t, U and µ, that characterize the model, and for any value of the constant external magnetic field B. The evaluation of integrals has been performed by a number of procedures (computer programs) developed by the authors in the symbolic language Maple V.3, that consist in the computational implementation of the method described in reference [2]. We have called this package of procedures GINT. The method applied in [2] greatly simplifies the calculations made in reference [11], but memory utilization problems have appeared as we tried to go beyond n > 3. Luckily, the factorization property of multivariable integrals of type (2.8) allowed us to optimize the performance of the package. In Appendix A we consider one typical Grassmann integral to exemplify the factorization of the sub–graphs. The procedure perm is one of the procedures contained in the package, being a useful tool to calculate the independent non–null terms [12] that contribute to T r[K4 ] and T r[K5 ]. In this procedure are implemented the symmetries discussed in reference [2]. The procedure 8

gint has been used to calculate the multivariable Grassmann integrals, taking into account the factorization into sub–graphs. The package can be downloaded through ftp from the site http:/www.if.uff.br.

4.1. Calculation of Tr[K4 ] For the case n = 4, we get from eq.(2.3) that

4

Tr[K ] =

Z Y 8N

dηI d¯ ηI e

8N P

η¯I AIJ ηJ

×

I,J =1

I=1

n n n n × K (¯ η , η; ν = 0) K (¯ η , η; ν = 1) K (¯ η , η; ν = 2) K (¯ η, η; ν = 3). (4)

(4)

(4)

The expressions of Aσσ , pνν ′ , qνν ′ and λν

(4.1.1)

are obtained from eqs.(2.3b), (2.6a), (2.6b) and

(2.6e). From eq.(3.5), for the unidimensional Hubbard model, we have that the Grassmann n function K (¯ η , η; ν) is equal to

n K (¯ η , η; ν) = E(¯ η, η; ν) + T − (¯ η , η; ν) + T + (¯ η, η; ν) + U(¯ η , η; ν).

(4.1.1a)

We defined a simplified notation,

2nN

< O1 (ν1 ) · · · Om (νm ) > ≡

Z 2nN Y

dηI d¯ ηI e

P

η ¯I AIJ ηJ

×

I,J =1

I=1

× O1 (¯ η , η; ν1 ) · · · Om (¯ η , η; νm )

(4.1.2a)

and 2nN

< O1 (σ, ν1 ) · · · Om (σ, νm ) > ≡

Z

(3−σ)nN

Y

dηI d¯ ηI e

P

η ¯(1−σ)nN +I AIJ η(1−σ)nN +J

I,J =1

I=(1−σ)nN+1

× O1 (¯ η , η; ν1 ) · · · Om (¯ η , η; νm ), where Oj (¯ η , η; νj ) are Grassmann functions. The independent terms that contribute to T r[K4 ] are: 9

× (4.1.2b)

T r[K4 ] =< E0 , E0 , E0 , E0 > +4 < U, E0 , E0 , E0 > +2 < U, E0 , U, E0 > +8 < U, T − , T + , E0 > + + 4 < U, U, E0 , E0 > +4 < U, U, U, E0 > + < U, U, U, U > +4 < T − , E0 , T + , E0 > + + 8 < T − , U, T + , E0 > +4 < T − , U, T + , U > +8 < T − , T + , E0 , E0 > + + 8 < T − , T + , U, E0 > +8 < T − , T + , U, U > +2 < T − , T + , T − , T + > + + 4 < T −, T −, T +, T + > .

(4.1.3)

In order to calculate the terms on the r.h.s. of (4.1.3), we need the result of a set of Grassmann multivariable integrals that are presented on Appendix C. We used the procedure gint to calculate them. Before using the procedure gint to calculate the terms in eq.(4.1.3) that include E0 , T − and T + , we need to explicit the contributions coming from the sectors σ =↑ and σ =↓. For example, in the term < E(0), E(1), E(2), E(3) > we have 16 terms when we explicitly write down the spin–sectors. However, the number of terms is diminished when we use the symmetries discussed in reference [2] and the fact that A↑↑ = A↓↓ . The application of these symmetries is simplified by using the graphic notation explained in Appendix B. By taking into account the results of integrals in Appendix C and their contributions to the sum over the space indices in each term of T r[K4 ] (eq.(4.1.3)), we finally obtain 1 3 1 4 3 2 2 4 3 U + U ∆E + ∆E + U ∆E + N 3∆E 2 λ2B + 3∆E 4 + T r[K ] = N U ∆E + 256 8 16 9 3 9 4 45 2 + 3t2 U ∆E + U ∆E 3 + U 2 λ2B + 6t2 ∆E 2 + U + U ∆E 2 + 2 16 128 16 3 3 3 3 51 3 ∆E 4 + U 2 λ2B + 3U ∆E 3 + U 2 ∆E 2 + + t2 U 2 + U 3 ∆E + U ∆Eλ2B + N 2 8 4 2 4 16 16 3 3 4 9 2 2 21 3 51 4 2 2 2 2 2 4 2 2 + 3t U ∆E + ∆E λB + 3t λB + λB + t U + 3t + 3t ∆E + U ∆E + U + 2 4 8 16 256 1 5 3 1 3 − N 3t2 λ2B + t4 + 3t2 ∆E 2 + U ∆E 3 + t2 U 2 + U 2 λ2B + ∆E 4 + 2 2 4 8 4 3 2 1 4 1 3 3 3 4 3 2 2 2 2 2 U . + 3t U ∆E + ∆E λB + U ∆E + λB + U ∆E + U ∆EλB + 2 8 4 8 2 128 (4.1.4) 4

4

3

We use the short notation: ∆E ≡ E0 − µ. 10

4.2. Calculation of Tr[K5 ] For n = 5, the expression of T r[K5 ] obtained from the procedure perm, where each term is multiplied by the respective constant, is: T r[K5 ] = 5 < U, E0 , E0 , E0 , E0 > +5 < U, E0 , U, E0 , E0 > +5 < U, U, E0 , E0 , E0 > + + 5 < U, U, E0 , U, E0 > +10 < U, U, T − , T + , E0 > +5 < U, U, U, E0 , E0 > + + 5 < U, U, U, U, E0 > + < U, U, U, U, U > + < E0 , E0 , E0 , E0 , E0 > + + 10 < T − , T + , T − , T + , E0 > +10 < T − , T + , T − , T + , U > +10 < T − , U, U, T + , E0 > + + 10 < T − , T + , E0 , U, E0 > +10 < T − , T + , E0 , E0 , E0 > +10 < T − , T + , U, E0 , E0 > + + 10 < T − , T + , U, U, E0 > +10 < T − , T + , U, U, U > +10 < T − , E0 , T + , U, E0 > + + 10 < T − , E0 , T + , E0 , E0 > +10 < T − , U, T + , E0 , E0 > +10 < T − , U, T + , U, E0 > + + 10 < T − , U, T + , U, U > +10 < T + , T − , T − , T + , E0 > +10 < T + , T − , T − , T + , U > + + 10 < U, T − , U, T + , E0 > +10 < U, T − , E0 , T + , E0 > +10 < U, T − , T + , E0 , E0 > + + 10 < U, T − , T + , U, E0 > +10 < T − , T − , T + , T + , E0 > +10 < T − , T − , T + , T + , U >, (4.2.1) where we are using the convention (4.1.2a) to write down each term. In n = 5 we have seven new type of integrals that do not have an equivalent one for n < 5. We table those integrals in section C.2 of Appendix C. After a long but convergent calculation, we get 5 4 5 5 3 5 2 1 5 4 2 5 3 5 5 U + U ∆E + U ∆E + U ∆E + ∆E + U ∆E + T r[K ] = N 1024 256 4 32 8 15 5 35 105 2 U ∆E 3 + U 2 ∆Eλ2B + U 3 t2 + 10t2 ∆E 3 + 5∆E 3 λ2B + 5∆E 5 + U ∆E 4 + + N4 16 16 32 4 15 15 15 5 5 155 3 55 4 + U ∆E 2 λ2B + U t2 ∆E 2 + U + U 3 λ2B + U ∆E 2 + U ∆E+ 4 512 64 64 128 2 195 45 15 U ∆E 4 + U ∆E 2 λ2B + + U 2 t2 ∆E + N 3 15t2 ∆E 3 + 15t2 ∆Eλ2B + 8 16 8 15 15 15 15 15 225 2 45 + U λ4B + ∆E 3 λ2B + ∆E 5 + ∆Eλ4B + U t4 + U ∆E 3 + U 2 ∆Eλ2B + 16 2 4 4 4 16 16 15 75 495 4 15 15 + U t2 λ2B + U t2 ∆E 2 + 15t4 ∆E + U ∆E + U 3 λ2B + U 3 ∆E 2 + 4 4 32 2 256 75 2 2 25 195 5 45 3 2 5 + U t ∆E + U + U t + N 2 − U 2 t2 ∆E + U 2 ∆E 3 − 8 1024 32 4 4 11

75 5 15 75 35 15 15 2 U ∆Eλ2B + U − U ∆E 4 − U ∆E 2 λ2B − U λ4B − t4 ∆E− 4 512 16 8 16 2 115 15 5 5 15 U 4 ∆E − ∆E 3 λ2B − ∆E 5 − ∆Eλ4B − 15t2 ∆E 3 − − 15U t2 ∆E 2 − U t2 λ2B + 2 128 2 4 4 45 125 3 15 5 − 15t2 ∆Eλ2B − U 3 λ2B + U ∆E 2 − U t4 − U 3 t2 + 64 64 8 8 15 15 15 25 65 5 + N − U 2 ∆E 3 − U t2 ∆E 2 + U t2 λ2B − U 2 t2 ∆E − U 4 ∆E − U 3 ∆E 2 + 2 2 2 2 32 32 15 5 15 3 2 5 5 5 U − U t − U ∆E 4 + U λ4B . + U 3 λ2B − 32 128 8 4 4 (4.2.2)

−

We continue to use the notation: ∆E ≡ E0 − µ. Certainly, the most subtle part in calculating expression (4.2.2) comes from the product of Grassmann integrals for different σ–sectors when we have Grassmann generators at the same space indice. In this case, we have to suit the conditions satisfied by both integrals and calculate the contribution of the product to the sum over space indices.

4.3. The β–Expansion of the Grand Potential Up to Order β 4 and Physical Quantities The relation between the grand potential W(β; µ) and the grand canonical partition function Z(β; µ) is W(β; µ) = −

1 ln Z(β; µ). β

(4.3.1)

The β–expansion of Z(β; µ) up to order β 5 is (see eq.(2.2)): Z(β, µ) ≈ Tr[1I − βK] +

β2 β3 β4 β5 Tr[K2 ] − Tr[K3 ] + Tr[K4 ] − Tr[K5 ]. 2! 3! 4! 5!

(4.3.2)

The first term on the r.h.s. of (4.3.2) was calculated in reference[11], the second and third terms were calculated in reference [2] and in its last two terms we substitute the results of eqs.(4.1.4) and (4.2.2). From eqs.(4.3.1) and (4.3.2), we get the grand potential up to order β 4 , that is, 12

1 1 1 1 13 3 U t2 λ2B + U 2 t2 ∆E + U 5 + U t2 ∆E 2 + U ∆E 2 − β 16 16 1024 16 768 1 1 5 4 1 2 1 3 2 4 1 3 2 4 4 3 U λB − U λB + U ∆E + U ∆E + U ∆E + U t β − − 768 96 96 768 48 64 1 2 1 1 1 1 1 1 − t U ∆E + U ∆Eλ2B + ∆E 4 + t4 + λ4B + U 4 + U ∆E 3 + 8 16 96 16 96 1024 48 1 1 3 5 1 1 1 + t2 ∆E 2 + U ∆E + t2 U 2 + U 2 λ2B + U 2 ∆E 2 + ∆E 2 λ2B + 8 96 64 64 16 192 3 U 1 1 1 1 2 2 + U λ2B − ∆E 2 U − ∆EU 2 β 2 + + t λB β 3 + − 8 64 16 16 16 o 1 U 1 1 t2 3 + + O(β 5 ) . ∆E 2 + λ2B + ∆E U + + U 2 β − ∆E + 4 4 4 2 32 4

W(β; µ) = −N

n2

ln 2 + −

(4.3.3) It is important to stress out that the coefficients of the β–expansion of function W(β; µ) are exact for any set of constants: E0 , t, U , µ and B for the unidimensional Hubbard model. From expression (4.3.3) we can get the strong limit approximation by taking U ≫ t, as well the atomic limit approximation when U ≪ t. From expression (4.3.3), we can derive any physical quantity for the model at thermal equilibrium at high temperature. As examples, we consider the following quantities: i) specific heat at constant length and constant number of fermions: CL (β). ∂ 2 ∂W(β; µ) β , CL (β) = −kβ ∂β ∂β

(4.3.4)

where k is the Boltzmann constant. From eq.(4.3.3), we get,

5 5 5 5 65 3 5 5 3 2 U − U λ4B + U ∆E 4 + U ∆E 2 + U 2 ∆E 3 − U λB − 256 24 24 192 12 192 3 4 5 2 5 2 2 25 4 5 3 2 5 5 2 2 2 U ∆E + U t β + − U − − U t λB + U t ∆E + U t ∆E + 4 4 4 192 16 256 3 1 3 1 5 1 3 3 − t2 ∆E 2 − λ4B − t4 − ∆E 4 − t2 U 2 − U 3 ∆E − U ∆Eλ2B − ∆E 2 λ2B − 2 8 4 8 8 16 4 4 3 2 2 3 2 3 2 2 3 2 1 − t λB − t U ∆E − U λB − U ∆E 2 − U ∆E 3 β 4 + 2 2 16 16 4

CL (β) = N k

13

3 3 3 + − U 2 ∆E − U 3 + U λ2B − 8 32 8 3 1 1 U ∆E + t2 + U 2 + λ2B + + 2 16 2

3 2 U ∆E β 3 + 8 1 2 6 2 ∆E β + O(β ) . 2

(4.3.4a)

ii) average energy per site: < h > (β). The simplest way to derive the average energy per site from the grand potential, is to scale the constants: (E0 , t, U, λB ) → (αE0 , αt, αU, αλB ) and substitute in eq.(3.2a) to obtain W(β; µ; α). From the scaled grand potential, we have < h > (β) = From eq.(4.3.3), we get that

1 ∂W(β; µ; α) . N ∂α α=1

(4.3.5)

3 2 1 1 1 2 1 1 2 U 2 < h > (β) = E0 + + −t − U − E0 U + U µ − E0 + E0 µ − λB β+ 4 16 2 4 2 2 2 3 1 2 3 3 1 1 3 3 2 2 2 2 2 + E0 U − U µ − U λB + U E0 − U E0 µ + U µ + U β + 16 8 16 16 4 16 64 1 4 1 4 1 1 5 1 3 U + t + λ4B + U 2 λ2B + t2 U 2 + t2 λ2B + + − U E02 µ + 16 256 4 24 16 24 2 1 1 1 1 1 1 1 + U 2 E02 + U 2 µ2 + U E03 − U µ3 + U 3 E0 − U 3 µ + E04 − 16 32 12 48 48 64 24 1 1 3 1 1 3 2 − U E0 µ + U E0 µ2 + U t2 E0 − U t2 µ − E03 µ + E02 µ2 − 32 8 2 8 8 8 1 1 1 3 1 3 − E0 µ3 + t2 E02 + t2 µ2 − t2 E0 µ + U E0 λ2B − λ2B E0 µ− 24 2 4 4 4 8 1 1 3 − U λ2B µ + λ2B E02 + λ2B µ2 β 3 + 16 4 8 5 25 4 5 5 5 4 5 + − U5 + − U E0 − U 3 t2 + U λ4B + U µ − U 2 E03 + 1024 768 64 96 192 48 5 1 65 3 2 1 1 5 3 2 U λB − U E04 − U µ4 − U E0 + U 2 µ3 + U E03 µ− + 768 96 96 768 24 6 13 3 1 3 1 5 + U 3 E0 µ − U E02 µ2 + U 2 E02 µ − U 2 E0 µ2 + U E0 µ3 − U 2 t2 E0 + 96 16 4 16 12 16 5 5 3 1 1 13 + U t2 λ2B − U t2 E02 − U t2 µ2 + t2 U 2 µ + U t2 E0 µ − U 3 µ2 β 4 + 16 16 16 4 2 256 + O(β 5 ).

(4.3.5a) 14

iii) difference between average numbers of spin up and spin down particles per site: < n↑ > − < n↓ >. From the definition of the grand potential (eq.(4.3.1)), we have that

< n↑ > (β)− < n↓ > (β) =

1 ∂W(β; µ) . N ∂λB

(4.3.6)

Up to order β 4 , we get from eq.(4.3.3) that, 1 2 1 2 λB 2 2 − U β + 4 β + ∆E + U ∆E + λB + U + 2t β 3 + < n↑ > (β)− < n↓ > (β) = − 8 3 4 1 1 2 (4.3.6a) U λB + U t2 + U 3 β 4 + O(β 5 ) . + 3 48

iv) average of the square of the magnetization per site: < m2z > (β). λB < (ni↑ − ni↓ )2 > B2 2 1 h ∂W(β; µ) 1 ∂W(β; µ) i , = − gµB +2 2 N ∂µ ∂U

< m2z > (β) =

(4.3.7)

where B is the external magnetic field. From eq.(4.3.3), we obtain that

<

m2z

1 2 2 1 1 1 1 1 1 1 > (β) = g µB + U β + − U E0 + µU − U 2 + λ2B − E02 + 4 2 8 8 8 32 8 8 1 1 1 3 3 1 U t2 + U β + + E 0 µ − µ2 β 2 − 4 8 12 384 15 2 2 15 2 2 1 1 5 U4 + U E0 + U µ + U E03 − U µ3 + + 1536 384 384 24 24 7 3 7 3 1 3 2 2 1 1 1 + U E0 − U µ + t2 U 2 − U λB − t2 λ2B − λ4B − U E02 µ+ 384 384 32 384 8 48 8 1 1 1 1 1 1 1 + U E0 µ2 + U t2 E0 − U t2 µ + E 4 + µ4 − E03 µ + E02 µ2 − 8 8 8 48 48 12 8 1 2 2 1 2 2 1 2 15 2 1 3 4 5 U E0 µ β + O(β ) , (4.3.7a) − E0 µ + t E0 + t µ − t E0 µ − 12 8 8 4 192

where g is the Land´e’s factor and µB is the Bohr’s magneton. v) magnetic susceptibility: χ(β). 15

χ(β) = −

2 1 ∂ 2 W(β; µ) 1 . gµB 2 N ∂λ2B

(4.3.8)

From eq. (4.3.3), we obtain that

2 1 2 1 2 1 2 1 2 1 3 4 1 1 U t + U λB + U β + t + λB + U ∆E+ χ(β) = − gµB 2 8 8 384 4 8 8 1 1 1 1 + ∆E 2 + U 2 β 3 − U β 2 − β + O(β 5 ) . (4.3.8a) 8 32 8 2

5. Conclusions With the implementation of the factorization into sub–graphs of the Grassmann multivariable integrals, we can certainly go beyond the calculation of the term β 4 of the β–expansion of the grand potential of the unidimensional Hubbrad model. Even though the physics for U > 0 and U < 0 are different, the results of section 4.3 apply equally well for both cases. Recently, dos Santos and Thomaz have applied the results of reference [2] to calculate the β–expansion of the grand canonical partition function of the extended unidimensional Hubbard model up to orde β 3 [16]. But the important point is that the present approach opens the possibility to calculate the first terms of the β–expansion of the grand canonical partition function of the Hubbard model in two space dimensions, as well as of unidimensional models with impurities. We believe that improvements on the present approach will render a valuable tool for tackling with such problems.

16

Appendix A Factorization of Grassmannian Sub–Graphs For calculating the co–factors of matrices Aσσ , σ =↑, ↓, it helps to have the value of their determinant. For arbitrary n, the determinant of these matrices is equal to h n−1 Y

det Aσσ =

λ(n) ν

ν=0

(n)

where N is the number of space sites and λν

iN

(A.1)

,

are the N–fold degenerated eigenvalue of Aσσ .

From eq.(2.6e) we have that iπ

λ(n) = 1 − e n (2ν+1) . ν (n) ∗

(n)

We should notice that λ(n−1)−ν = λν

(A.2)

.

We define:

P (n) ≡

n−1 Y

λ(n) ν .

(A.3)

ν=0

For calculating P (n) we need to consider the cases n even and n odd separately. For n even, expression (A.3) can be rewritten as, n−2 2

P

(n)

=

Y

ν=0

π 2 − 2 cos (2ν + 1) n

= 2,

(A.4)

π (2ν + 1) . sin 2n

(A.5)

where the last equality is already known [13]. For n odd, expression (A.3) can be rewritten as,

P

(n)

=2×2

n−1

n−1 Y ν=0

From reference [13], we have that [14]:

2

n−1

n−1 Y ν=0

π (2ν + 1) = 1, sin 2n

that substituted in eq. (A.5) gives 17

(A.6)

P (n) = 2.

(A.7)

From the results (A.4) and (A.7), for any n, we get that

det A

σσ

=

h n−1 Y

λ(n) ν

ν=0 N

iN

=2 .

(A.8)

To present the factorization of the Grassmannian integrals, we consider an example and use the graphic notation explained in Appendix B. Let us consider the integral for fixed space indices l1 and l3 , that contributes to < E0 (↑), E1 (↑), E2 (↑) >, I(l1 , l1 , l3 ) =

Z Y 4N

dηI d¯ ηI e

4N P

η η ¯I A↑↑ IJ J

η¯l1 ηl1 η¯N+l1 ηN+l1 η¯2N+l3 η2N+l3

I,J =1

(A.9)

I=1

where l1 6= l3 . Under the similarity transformation (2.5), eq.(A.9) becomes Z Y 4N 3 h X i ′ ′ I(l1 , l1 , l3 ) = d¯ ηI dηI qν1 0 p0τ1 qν2 1 p1τ2 η¯ν′ 1 N+l1 ητ′ 1 N+l1 η¯ν′ 2 N+l1 ητ′ 2 N+l1 × ν1 ,ν2 =0 τ1 ,τ2 =0

I=1

× h = 2N

3 X

ν1 ,ν2 =0 τ1 ,τ2 =0

3 hX

ν3 =0

4N P η ¯I i qν3 2 p2ν3 η¯ν′ 3 N+l3 ην′ 3 N+l3 eI,J =1

qν1 0 p0τ1 qν2 1 p1τ2 i (4) (4)

λν1 λν2

×

3 hX

ν3 =0

qν3 2 p2ν3 i , (4) λν3

DIJ ηJ

(A.10)

where to write the second equality on the r.h.s. of eq.(A.10), we used the result (A.8). By explicitly writting down the expressions, we see that,

2

N

3 X

ν1 ,ν2 =0 τ1 ,τ2 =0

qν1 0 p0τ1 qν2 1 p1τ2 (4) (4)

λν1 λν1

=

Z Y 4N

4N P

dηI d¯ ηI eI,J =1

η¯I A↑↑ η IJ J

η¯l1 ηl1 η¯N+l1 ηN+l1 ,

I=1

(A.10a) and 18

3 X

qν3 2 p2ν3

ν3 =0

(4)

λν3

4N P Z Y 4N η ¯I 1 = N dηI d¯ ηI eI,J =1 2

A↑↑ η IJ J

η¯2N+l3 η2N+l3 .

I=1

(A.10b) (A.11)

Using the graphic representation of Appendix B, we write result (A.10) as, The factorization (A.11) comes directly from the fact that the matrix D is diagonal (see eq.(2.5a)) and the result (2.7a). Once the presence of Grassmann generators in the integrand of integrals (2.8) correspond to cutting lines and columns of matrix D,then only for cutts at the same space index and any ν–indices we get co–factors of matrix D that are non–zero. In summary, the factorization of the type (A.11) always happens when two or more space indices are different. In a similar way and by the reasons discussed before, it is simple to show that I(l1 ,(A.12) l2 , l3 ), where all the space indices li , i = 1, 2, 3, are distinct, is easily written as:

19

Appendix B Graphic Notation of the Multivariable Grassmann Integrals To exemplify our graphic notation for the graphs that contribute to eq. (2.3), for fixed value of n, we consider some terms of T r[K4 ]. This graphic notation is very helpful when we apply the symmetries discussed in reference [2] to identify equivalent terms in T r[Kn ]. We present the graphic notation through examples and its application to the identification of equivalent integrals. 1) < E(↑, 0), E(↓, 1), E(↑, 2), E(↑, 3) >= E(↑)3 E(↓)× 2) The constants E(↑) and E(↓) are defined just below eq.(3.4d). In order to show that terms E(↓, 0)E(↑, 1)E(↑, 3) >= E(↑)3 E(↓)× (B.1) and<(B.2) are equal, we2)E(↑, use the invariance of the integrals under a cyclic translation in the temperature parameter ν in each sector σ =↑ and σ =↓, separately. Therefore,

< E(↑, 0)E(↓, 1)E(↑, 2)E(↑, 3) >=< E(↓, 0), E(↑, 1), E(↑, 2), E(↑, 3) >

20

(B.3)

3) Due to the presence of the term U(0), the integrals in the two σ–sectors have one space index < U(0)T − (↑, 1)T + (↑, 2)E(↑, 3) >= U t2 E(↑)× l in common.

4)

A↑↑

The terms (B.4) and (B.5) are equal, up to a multiplicative factor, due to the fact that < U(0)T − (↓, 1)T + (↓, 2)E(↓, 3) >= U t2 E(↓)× = A↓↓ . The graphic notation was used along all the calculations and permitted us to considerably

reduce the number of terms that contribute to the expressions of T r[K4 ] and T r[K5 ].

21

Appendix C Useful Multivariable Grassmann Integrals at n = 4 and n = 5 C.1. Useful integrals for n=4 We need the result of twelve integrals only, to calculate the terms that contribute to (4.1.3). In this Appendix, we present the value of these integrals according to the conditions satisfied by the space indices [15]. 1) (4) I1 (l)

≡

Z Y 4N

4N P

dηI d¯ ηI e

η ¯I A↑↑ η IJ J

I,J =1

η¯l ηl = 2N−1 ,

(C.1.1)

I=1

for l = 1, 2, · · · , N . 2)

(4) I2 (l1 , l2 )

≡

Z Y 4N

=

4N P

η ¯I A↑↑ η IJ J

η¯l1 ηl1 η¯N+l2 ηN+l2

dηI d¯ ηI eI,J =1

I=1

2N−1 , l1 = l2 , 2N−2 , l2 = 6 l1 ,

l1 = 1, 2, · · · , N l1 , l2 = 1, 2, · · · , N.

(C.1.2)

3)

(4) I3 (l1 , l2 )

≡

Z Y 4N

4N P

η ¯I A↑↑ η IJ J

η¯l1 ηl1 +1 η¯N+l2 ηN+l2 −1

dηI d¯ ηI eI,J =1

I=1

= 2N−2 , l2 = l1 + 1,

l1 = 1, 2, · · · , N.

(C.1.3)

4)

(4) I4 (l1 , l2 , l3 )

≡

Z Y 4N

=

4N P

η ¯I A↑↑ η IJ J

η¯l1 ηl1 η¯N+l2 ηN+l2 +1 η¯2N+l3 η2N+l3 −1

dηI d¯ ηI eI,J =1

I=1

2N−3 , l3 = l2 + 1 2N−2 , l1 = l3 = l2 − 1. 22

(C.1.4)

5)

(4) I5 (l1 , l2 , l3 )

≡

Z Y 4N

4N P

(4) I6 (l1 , l2 , l3 )

η¯l1 ηl1 η¯N+l2 ηN+l2 η¯2N+l3 η2N+l3

dηI d¯ ηI eI,J =1

I=1

2N−3 , = 2N−2 , N−1 2 ,

6)

η¯I A↑↑ η IJ J

≡

Z Y 4N

=

≡

Z Y 4N

=

l1 = 6 l2 6= l3 l1 = l2 , or, l1 = l3 , or, l2 = l3 l1 = l2 = l3 .

4N P

(C.1.5)

η η ¯I A↑↑ IJ J

η¯l1 ηl1 +1 η¯N+l2 ηN+l2 η¯2N+l3 η2N+l3 −1

dηI d¯ ηI eI,J =1

I=1

2N−3 , l3 = l1 + 1 2N−2 , l2 = l1 and l3 = l1 + 1.

(C.1.6)

7)

(4) I7 (l1 , l2 , l3 )

4N P

η ¯I A↑↑ η IJ J

η¯l1 ηl1 η¯N+l2 ηN+l2 +1 η¯2N+l3 η2N+l3 −1

dηI d¯ ηI eI,J =1

I=1

2N−3 , l2 = l3 − 1 2N−2 , l1 = l3 = l2 + 1.

(C.1.7)

8)

(4) I8 (l1 , l2 , l3 , l4 )

≡

Z Y 4N

4N P

dηI d¯ ηI eI,J =1

η¯I A↑↑ η IJ J

η¯l1 ηl1 η¯N+l2 ηN+l2 η¯2N+l3 η2N+l3 η¯3N+l4 η3N+l4

I=1

N−4 2 , 2N−3 , = 2N−2 , 2N−1 ,

l1 l1 l1 l1 l1

6= l2 6= l3 6= l4 = l2 , or, · · · , or, l3 = l4 (C.1.8) = l2 and l3 = l4 , or, all permutations 2 by 2 = l2 = l3 , or, all permutations with 3 equal space indices = l2 = l3 = l4 .

23

9) (4) I9 (l1 , l2 , l3 , l4 )

≡

Z Y 4N

dηI d¯ ηI e

4N P

η ¯I A↑↑ η IJ J

η¯l1 ηl1 +1 η¯N+l2 ηN+l2 −1 η¯2N+l3 η2N+l3 +1 ×

I,J =1

I=1

× η3N+l4 η3N+l4 −1 =

2N−4 , 2N−2 ,

l2 = l1 + 1 and l4 = l3 + 1, or, l2 = l3 + 1 and l4 = l1 + 1 l2 = l3 + 1 and l1 = l3 and l4 = l3 + 1.

(C.1.9)

10) (4) I10 (l1 , l2 , l3 , l4 )

≡

Z Y 4N

4N P

η ¯I A↑↑ η IJ J

η¯l1 ηl1 −1 η¯N+l2 ηN+l2 −1 η¯2N+l3 η2N+l3 +1 ×

dηI d¯ ηI eI,J =1

I=1

× η3N+l4 η3N+l4 +1

11)

2N−4 , = 2N−3 ,

(4) I11 (l1 , l2 , l3 , l4 )

≡

Z Y 4N

l2 = l3 + 1 and l1 = l4 + 1, or, l4 = l2 − 1 and l1 = l3 + 1 (C.1.10) l2 = l3 + 1 and l1 = l3 + 2 and l4 = l3 + 1 l2 = l3 + 1 and l1 = l3 and l4 = l3 − 1. 4N P

η ¯I A↑↑ η IJ J

η¯l1 ηl1 η¯N+l2 ηN+l2 η¯2N+l3 η2N+l3 +1 ×

dηI d¯ ηI eI,J =1

I=1

× η¯3N+l4 η3N+l4 −1

12)

N−4 2 , N−3 2 , = 2N−2 ,

(4) I12 (l1 , l2 , l3 , l4 )

≡

Z Y 4N

l4 = l3 + 1 l2 = l3 + 1 and l4 = l3 + 1, or, l1 = l2 and l4 = l3 + 1, or, l1 = l3 + 1 and l4 = l3 + 1 l1 = l3 + 1 and l2 = l3 + 1 and l4 = l3 + 1.

dηI d¯ ηI e

4N P

(C.1.11)

η ¯I A↑↑ η IJ J

η¯l1 ηl1 η¯N+l2 ηN+l2 +1 η¯2N+l3 η2N+l3 ×

I,J =1

I=1

× η¯3N+l4 η3N+l4 −1 N−4 2 , N−3 2 , = 2N−2 ,

l4 = l2 + 1 l1 = l3 and l4 = l2 + 1, or, l1 = l2 + 1 and l4 = l2 + 1, or, l2 = l3 and l4 = l2 + 1 l1 = l2 + 1 and l3 = l2 and l4 = l2 + 1. 24

(C.1.12)

C.2. Useful integrals for n=5 We present here the seven integrals that have no equivalent ones for n < 5; i.e., integrals that cannot be factorized into any of the integrals for n < 5 for all the conditions satisfied by the apace indices. In some graphs, we do not have generators in the integrand of integrals of type (2.7) at a given value of ν, as we can see for example in the graphs presented in Appendix B. Those rings in the integrals of type (2.7) that have no associated Grassmann generators in the integrand, we call empty rings. For example, in (B.1) we have one empty ring at σ =↑ (ν = 1), and three empty rings at σ =↓ (ν = 0, 1, and 3). The integrals for n = 5 with empty rings give the same results to the equivalent integrals for n = 4. We have not demonstrated this property in general form for any n, but we have detected it by evaluating these integrals through the procedure gint. The seven integrals for n = 5 and the conditions satisfied by the space indices [15] are: 1)

(5) G1 (l1 , l2 , l3 , l4 , l5 )

≡

Z Y 5N

dηI d¯ ηI e

5N P

η ¯I A↑↑ η IJ J

η¯l1 ηl1 η¯N+l2 ηN+l2 η¯2N+l3 η2N+l3 ×

I,J =1

I=1

× η¯3N+l4 η3N+l4 η¯4N+l5 η4N+l5 N−5 2 , N−4 2 , N−3 2 ,

l1 6= l2 6= l3 6= l4 6= l5 l1 = l2 , or, all permutations with 2 equal space indices l1 = l2 = l3 , or, all permutations with 3 equal space indices l1 = l2 and l3 = l4 , or, all permutations 2 by 2 = (C.2.1) N−2 2 , l1 = l2 = l3 = l5 , or, all permutations with 4 equal space indices l1 = l2 = l3 and l4 = l5 , or, all permutations with 2 or 3 equal space indices 2N−1 , l1 = l2 = l3 = l4 = l5 . 2)

(5) G2 (l1 , l2 , l3 , l4 , l5 )

≡

Z Y 5N

5N P

η ¯I A↑↑ η IJ J

η¯l1 ηl1 η¯N+l2 ηN+l2 +1 η¯2N+l3 η2N+l3 ×

dηI d¯ ηI eI,J =1

I=1

× η¯3N+l4 η3N+l4 −1 η¯4N+l5 η4N+l5

25

N−5 2 , l4 = l2 + 1 N−4 2 , l4 = l2 + 1 and two other space indices are equal 2N−3 , l4 = l2 + 1 and l1 = l4 and l3 = l5 , = or, all permutations 2 by 2 l4 = l2 + 1 and l1 = l4 = l5 , or, all permutations with 3 equal space indices N−2 2 , l2 = l5 − 1 = l3 and l1 = l5 = l4 .

(C.2.2)

3)

(5) G3 (l1 , l2 , l3 , l4 , l5 )

≡

Z Y 5N

dηI d¯ ηI e

5N P

η ¯I A↑↑ η IJ J

η¯l1 ηl1 η¯N+l2 ηN+l2 +1 η¯2N+l3 η2N+l3 −1 ×

I,J =1

I=1

× η¯3N+l4 η3N+l4 η¯4N+l5 η4N+l5 N−5 2 , l3 = l2 + 1 N−4 2 , l3 = l2 + 1 and two other space indices are equal 2N−3 , l3 = l2 + 1 and l1 = l4 and l3 = l5 , or, = all permutations 2 by 2 l3 = l2 + 1 and l1 = l3 = l4 , or, all permutations with 3 equal space indices N−2 2 , l3 = l2 + 1 and l1 = l3 = l4 = l5 .

(C.2.3)

4)

(5) G4 (l1 , l2 , l3 , l4 , l5 )

≡

Z Y 5N

dηI d¯ ηI e

5N P

η¯I A↑↑ η IJ J

η¯l1 ηl1 −1 η¯N+l2 ηN+l2 +1 η¯2N+l3 η2N+l3 +1 ×

I,J =1

I=1

× η¯3N+l4 η3N+l4 −1 η¯4N+l5 η4N+l5 N−5 2 , 2N−4 ,

l1 l1 l1 l2 l2 = l1 l3 l 1 N−3 2 , l 1 l1 l1

= l2 + 1 and l4 = l3 + 1, or, = l3 + 1 and l4 = l2 + 1 = l2 + 1 and l5 = l3 + 1 and l4 = l5 , or, = l4 = l1 − 1 = l3 + 1, or, = l5 = l1 − 1 and l3 = l4 − 1, or, (C.2.4) = l3 + 1 and l4 = l5 = l2 + 1, or, = l5 = l1 − 1 and l2 = l4 − 1, or, = l3 = l4 − 1 and l4 = l2 + 2 = l3 = l5 + 1 and l2 = l5 and l4 = l2 + 2, or, = l5 + 1 and l2 = l5 and l3 = l5 − 1 and l4 = l5 , or, = l3 = l5 − 1 and l2 = l5 − 2 and l4 = l5 . 26

5)

(5) G5 (l1 , l2 , l3 , l4 , l5 )

≡

Z Y 5N

dηI d¯ ηI e

5N P

η ¯I A↑↑ η IJ J

η¯l1 ηl1 +1 η¯N+l2 ηN+l2 η¯2N+l3 η2N+l3 ×

I,J =1

I=1

× η¯3N+l4 η3N+l4 −1 η¯4N+l5 η4N+l5 N−5 2 , l4 = l1 + 1 N−4 2 , l4 = l1 + 1 and two other space indices are equal 2N−3 , l4 = l1 + 1 and l1 = l2 = l3 , or, = all permutations with 3 equal space indices l4 = l1 + 1 and l1 = l2 and l3 = l5 , or, all permutations 2 by 2 N−2 2 , l4 = l1 + 1 and l1 = l2 = l3 = l5 − 1.

(C.2.5)

6)

(5) G6 (l1 , l2 , l3 , l4 , l5 )

≡

Z Y 5N

dηI d¯ ηI e

5N P

η¯I A↑↑ η IJ J

η¯l1 ηl1 +1 η¯N+l2 ηN+l2 −1 η¯2N+l3 η2N+l3 +1 ×

I,J =1

I=1

× η¯3N+l4 η3N+l4 −1 η¯4N+l5 η4N+l5 N−5 2 , l2 = l1 + 1 and l4 = l3 + 1, or, l4 = l1 + 1 and l2 = l3 + 1 N−4 2 , l 2 = l1 + 1 and l4 = l5 = l3 + 1 l2 = l5 = l1 + 1 and l4 = l3 + 1 = l 4 = l5 = l1 + 1 and l2 = l3 + 1 l1 = l4 − 1 and l3 = l5 = l2 − 1 N−3 , l1 = l3 = l4 − 1 and l2 = l4 2 2N−2 , l1 = l3 = l5 − 1 and l2 = l4 = l5

(C.2.6)

7)

(5) G7 (l1 , l2 , l3 , l4 , l5 )

≡

Z Y 5N

5N P

η¯I A↑↑ η IJ J

η¯l1 ηl1 +1 η¯N+l2 ηN+l2 +1 η¯2N+l3 η2N+l3 −1 ×

dηI d¯ ηI eI,J =1

I=1

× η¯3N+l4 η3N+l4 −1 η¯4N+l5 η4N+l5

27

N−5 2 , 2N−4 ,

l3 l4 l3 l3 l3 = l 4 l 2 2N−3 , l2 l1 l2

= l1 + 1 and l4 = l2 + 1, or, = l1 + 1 and l3 = l2 + 1 = l1 + 1 and l4 = l5 = l2 + 1, or, = l5 = l1 + 1 and l4 = l2 + 1, or, = l5 = l2 + 1 and l4 = l1 + 1, or, = l5 = l1 + 1 and l2 = l3 − 1, or, = l4 = l3 − 1 = l1 + 1, or l1 = l3 = l4 − 1 = l2 + 1 = l4 = l5 and l1 = l5 − 1 and l3 = l5 + 1, or, = l3 = l5 − 1 and l4 = l5 and l2 = l5 − 2, or, = l4 = l5 − 1 and l3 = l5 and l1 = l5 − 2.

(C.2.7)

Acknowledgements The authors thank J. Florencio Jr. for interesting discutions and A.T. Costa Jr. for making the figures. I.C.C thanks FAPMG and E.V.C.S. thanks CNPq for financial support. M.T.T. thanks CNPq and FINEP for partial financial support.

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E.H. Lieb and F.Y. Wu, Phys. Rev. Letters 20 (1968) 1445; A.A. Ovchinnikov, Sov. Phys. JETP 30 (1970) 1160;

4. M. Takahashi, Prog. Theoret. Phys. (Kyoto) 43 (1970) 1619; 5. K. Kubo and M. Tada, Prog. Theoret. Phys. 69 (1983) 1345; C.J. Thompson, Y.S. Young, A.J. Guttmann and M.F. Sykes, J. Phys. A: Math. Gen. 24 (1991) 1261; J.A. Henderson, J. Oitmaa and M.C.B. Ashley, Phys. Rev. B46 (1992) 6328; 6. H.E. Stanley, Introduction to Phase Transitions and Critical Phenomena, Oxford Univ. Press (1971); 28

7. C. Itzykson and J.–B. Zuber, Quantum Field Theory, McGraw–Hill (1980); U. Wolf, Nucl. Phys. B225 [FS9] (1983) 391; 8. We are using the convention: σ = 1 =↑ and σ = −1 =↓; 9. I.C. Charret, S.M. de Souza and M.T. Thomaz, Braz. Jour. of Phys. 26 (1996) 720; 10. J. Hubbard, Proc. Roy. Soc. A277 (1963) 237; A281 (1964) 401; M. Gutzwiller, Phys. Rev. Lett. 10 (1963) 159; Phys. Rev. A137 (1965) 1726; 11. I.C. Charret, E.V. Corrˆea Silva, S.M. de Souza and M.T. Thomaz, J. Math. Phys. 36 (1995) 4100; 12. See section 4.1 of reference [2] for a discussion on the vanishing Grassmann integrals; 13. I.S. Gradshteyn and I.M. Ryzhik; Table of Integrals, Series and Products, 4th edition, Academic Press (1965); expression: 1.396.4; 14. Reference [11], expression: 1.392.2; 15. When the space index does not appear among the conditions, we mean that it is distinct from any other space index in the graph; 16. O.R. dos Santos and M.T. Thomaz, private communication.

29