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IL NUOVO CIMENTO VOL. 112 B, N. 6 Giugno 1997 NOTE BREVI

Multi-particle fermionic realization of the su( 2 ) Lie algebra

C. F. LO

Department of Physics, The Chinese University of Hong Kong Shatin, New Territories, Hong Kong

(ricevuto il 22 Gennaio 1997; approvato il 25 Marzo 1997)

Summary. — We have constructed the multi-particle fermionic realization of the

su( 2 ) Lie algebra. It is observed that the unitary displacement operator can be identified as the operator which induces the generalized multi-particle fermionic Bogoliubov transformation. We have also shown that the h-pairing algebra of the Hubbard model on a bipartite lattice is, in fact, a special case of this multi-particle fermionic representation.

PACS 03.65.Fd – Algebraic methods. PACS 02.20 – Group theory.

It is well known that the Lie algebra su( 2 ) has applications in many areas of physics and group theory [1-3]. For instance, the su( 2 ) Lie algebra has been used by many researchers in the study of the nonclassical properties of light in quantum-optical systems. They have found the so-called SU( 2 ) squeezing for the su( 2 ) generators, associated with the SU( 2 ) generalized coherent states in the (two-mode) Schwinger bosonic representation of the su( 2 ) Lie algebra, in the study of interferometers as well as other applications in quantum optics [4, 5]. Furthermore, the bosonic realization of the su( 2 ) Lie algebra has been receiving extensive attention in nuclear physics recently [6-8]. The su( 2 ) Lie algebra consists of three generators K0, K1 and K2

satisfying the commutation relations [2, 4]

[K0, K6] 46K6, [K1, K2] 42K0.

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The corresponding Casimir operator C is given by

C 4K021

1

2(K1K21 K2K1) , (2)

which satisfies [C , K₆_{] 4 [C, K}0] 40. The discrete representation of the su(2) Lie

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C.F.LO 938 algebra is characterized by [2]

.

/

´

CNm, jb 4j( j11)Nm, jb , K0Nm , jb 4 mNm , jb , K₆_{Nm , jb 4}

k

( j Z m)( j 6m11)Nm61, jb , (3)

where K₂_{N 2 j , jb 4 K}₁_{Nj , jb 4 0. In this case j 4 0 , 1 /2 , 1 , 3 /2 , 2 , R , and m 4 2j ,} 2j 1 1 , R , j 2 1 , j. The set of states ]Nm , jb: m 4 2j , 2j 1 1 , R , j 2 1 , j ; j 4 const( forms a complete orthonormal basis:

a j , mNn, jb 4dm , n,

!

j

m 42jNm , jba j , mN 4 1 . (4)

Following Perelomov [2], the SU( 2 ) generalized coherent states Nu, fb are defined as Nu , fb 4 exp [aK12 a * K2] N2j, jb ,

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where a 4 (u/2) exp [2if], 0 GuGp, and 0 GfG2p. The ladder operators K6select

the vacuum state Nvacb from the states Nm, jb in the usual way, namely K2Nvacb 4

K₂_{N 2 j , jb 4 0.}

In addition to the bosonic representation, there also exists a fermionic representa-tion, in which the generators are given by

K₁_{4 c}1†c2†, K24 c2c14 2c1c2, K04 1 2(c1 †_c 11 c2†c22 1 ) , (6)

where the fermionic operators satisfy the anticommutation relations: ]ck, cl†( 4 dkl and ]ck, cl( 4 ]ck†, cl†( 4 0. In this representation the index j can be either 1 /2 or 0 only. For j 41/2, the vacuum state Nvacb is just the state with no fermion, i.e. Nvacb 4 N0 b1N0 b2, while for j 40, the vacuum state is doubly degenerate, namely Nvacb 4

N1 b1N0 b2 or N0b1N1 b2. The corresponding unitary displacement operator D(a) 4

exp [aK₁_{2 a * K}₂] induces the following transformation of the annihilation and creation operators:

.

`

/

`

´

D(a)†_c

1D(a) 4c1cos (NaN)1c2†

a

NaNsin (NaN) ,

D(a)†_c†

1D(a) 4c1†cos (NaN)1c2

a *

NaNsin (NaN) ,

D(a)†_c

2D(a) 4c2cos (NaN)2c1†

a

NaNsin (NaN) ,

D(a)†_c†

2D(a) 4c2†cos (NaN)2c1

a *

NaNsin (NaN) , (7)

which is the well-known fermionic Bogoliubov tranformation. In the present work we shall generalize the two-particle fermionic realization of the su( 2 ) Lie algebra to the multi-particle case and discuss its applications.

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MULTI-PARTICLE FERMIONIC REALIZATION ETC. 939 generators as follows:

.

/

´

K₁₄

!

k , l bklck H† cl I† , K24 K † 14 2

!

k , l b *klck Hcl I, K04 1 2

k

k , l

!

(bb†₎ kl(ck H† cl H1 cl I† ck I) 2

!

k (bb†₎ kk

l

. (8)

In order to satisfy the commutation relations in eq. (1), we must requirebto be unitary,

i.e.bb†₄b†b_{4 1. As a result, the operator K}0can be rewritten as

K04 1 2

!

k (c† k Hck H1 ck I† ck I2 1 ) 4 1 2(N 2M) , (9)

where N 4

!

k(ck H† ck H1 ck I† ck I) represents the total number of fermions and

M 4

!

k1 the total number of different species of fermions (characterized by the index k). In the case of only one species, we shall recover a special case of the usual two-particle fermionic realization. In this multi-particle fermionic representation

j 40, 1/2, 1, 3/2, 2, R, M/2. For j4M/2, the vacuum state is uniquely given by

Nvacb 4

»

kN0 bk, whilst for other values of j, the vacuum state is highly degen-erate. Similar to the two-particle case, the unitary displacement operator

D(a)4exp [aK12a * K2] transforms the annihilation and creation operators as follows:

.

`

/

`

´

D(a)†c_{k H}_{D(a) 4c}_{k H}_{cos (NaN)1} a

NaNsin (NaN)

!

l

bklcl I

†

D(a)†_c†

k HD(a) 4ck H† cos (NaN)1

a *

NaNsin(NaN)

!

l

bklcl I,

D(a)†_c

k ID(a) 4ck Icos (NaN)2

a NaNsin(NaN)

!

l blkcl H† , D(a)†_c† k ID(a) 4ck I† cos(NaN)2 a * NaNsin(NaN)

!

l blkcl H. (10)

Clearly, this transformation is a multi-particle generalization of the fermionic Bogoliubov transformation in the two-particle case. Furthermore, the corresponding

SU( 2 ) generalized coherent state Nu, fb 4D(a)

»

kN0 bk will be very useful in variational approaches (both time-dependent and time-independent) to the many-body fermionic systems [9].

In what follows we study the application of the multi-particle fermionic realization of the su( 2 ) Lie algebra in the Hubbard model on a bipartite lattice with M sites. The Hamiltonian of this model is given by

H 42 t

!

ar , r 8b (c_{r H}† c_{r 8H}_{1 c}_{r I}† c_{r 8I}_{) 1U}

!

r cr H † c_{r H}c_{r I}† c_{r I}, (11)

where

!

_{ar , r 8b} denotes the summation over nearest-neighboring lattice sites. It is straightforward to show that both the Casimir operator C and the generator K0

commute with the Hamiltonian H : [C , H] 4 [K0, H] 40, and that the generators K6

form the following commutation relations with H: [H , K₁_{] 42[H, K}₂]† 4 2t

!

ar , r 8b

!

r 9 (b_{r 8 r 9}c† r Hcr 9I† 1 br 9 r 8cr 9H† cr I† ) 1 (12) 1U

!

r , r 8 b_{rr 8}(c† r Hcr 8I† cr I† cr I1 cr 8H† cr 8Hcr H† cr 8I† ) .

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C.F.LO

940

If the unitary matrixbis diagonal, i.e. b_{rr 8}_{4 d}_{rr 8}exp [ifr] with frbeing real, then [H , K₁_{] 4UK}₁_{2 t}

!

ar , r 8b

[

exp [ifr] 1exp [ifr 8]

]

c

r H†

c

r 8I†

.

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The second term will vanish provided that exp [ifr] 42exp [ifr 8]. Since r and r 8 are nearest-neighboring lattice sites on a bipartite lattice, a unique choice of such a phase factor is given by exp [ifr] 4 (21)r. With this choice ofb, we obtain

[K₆_{, H] 4ZUK}₆. (14)

Accordingly, we have derived the h-pairing operators introduced by Yang for the Hubbard model on a bipartite lattice [10, 11]:

h₁₄

!

r(21) r_c† r Hcr I† , h24 h1 † _, _J z4 1 2(N 2M) . (15)

In other words, the h-pairing algebra is a special case of our multi-particle fermionic realization of the su( 2 ) Lie algebra.

In summary, we have constructed the multi-particle fermionic realization of the

su( 2 ) Lie algebra. It is observed that the corresponding unitary displacement operator

can be identified as the operator which induces the generalized multi-particle fermionic Bogoliubov transformation. We have also shown that the h-pairing algebra introduced by Yang for the Hubbard model on a bipartite lattice is indeed a special case of this multi-particle fermionic representation. Furthermore, since the SU( 2 ) group is very useful in many branches of physics, and according to the Levi theorem [3], the su( 2 ) Lie algebra is one of the essential building blocks of every Lie algebra—this means that we can deal with a generic Lie algebra by decomposing it into its fundamental blocks, we believe that the results obtained in the present work should have valuable potential applications. For instance, the SU( 2 ) generalized coherent state can be used in variational approaches to the many-body fermionic systems [9].

* * *

This work is partially supported by the Direct Grant for Research from the Research Grants Council of the Hong Kong Government. The author would like to thank Dr. K. L. LIUfor many helpful discussions.

R E F E R E N C E S

[1] KLAUDER J. R. and SKARGERSTAM B. S., Coherent States (World Scientific, Singapore) 1985.

[2] PERELOMOVA. M., Generalized Coherent State and its Applications (Springer-Verlag, New York) 1986.

[3] WYBOUNEB. G., Classical Groups for Physicists (Wiley, New York) 1974. [4] WO´DKIEWICZK. and EBERLYJ. H., J. Opt. Soc. Am. B, 2 (1985) 458. [5] YURKEB., MCCALLS. L. and KLAUDERJ. R., Phys. Rev. A, 33 (1986) 4033.

[6] TSUEY., FUJIWARAY., KURIYAMAA. and YAMAMURAM., Prog. Theor. Phys., 85 (1991) 693. [7] YAMAMURAM. and KURIYAMAA., Prog. Theor. Phys., 88 (1991) 711.

[8] YAMAMURAM., KURIYAMAA. and TSUEY., Prog. Theor. Phys., 88 (1991) 719. [9] ZHANGW. M., FENGD. H. and GILMORER., Rev. Mod. Phys., 62 (1990) 867. [10] YANGC. N., Phys. Rev. Lett., 63 (1989) 2144.