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Algebraic Bethe ansatz for gl(2, 1) invariant 36-vertex models
NUCLEAR PHYSICS B [FS] Nuclear Physics B 479 [FS] (1996) 575-593
ELSEVIER
Algebraic Bethe ansatz for gl(2, 1) invariant 36-vertex models Markus E Pfannmtiller 1, Holger Frahm 2 Institut fiir Theoretische Physik, Universitgit Hannover, D-30167 Hannover, Germany Received 27 March 1996; revised 5 July 1996; accepted 7 August 1996
Abstract
Four-dimensional irreducible representations of the superalgebra gl(2, 1) carry a free parameter. We compute the spectra of the corresponding transfer matrices by means of the nested algebraic Bethe ansatz together with a generalized fusion procedure. PACS: 71.27.+a; 75.10.Lp; 05.70.Jk Keywords: Algebraic Bethe ansatz; gl(2, 1); Vertex models
1. Introduction
Recently, vertex models built from representations of the superalgebra g/(2, 1) or q-deformations thereof have attracted increasing attention [ 1-4]. One reason is their relation to integrable models of interacting electrons in one spatial dimension: for example, the supersymmetric t-J model [5-7] is obtained in the hamiltonian limit of the transfer matrix for the vertex model based on the three-dimensional fundamental representation [½]+ of gl(2, 1) [3,4]. A special feature of this algebra is the existence of a family of four-dimensional representations [b, l ] parametrized by a parameter b 4: -4-½. From the corresponding vertex model a one-parametric integrable model of interacting electrons can be derived [ 1 ]. This system of electrons with correlated hopping has been solved recently by means of the coordinate Bethe ansatz [8,9]. The relation to integrable vertex models provides an embedding of these models into the framework of the Quantum Inverse Scattering Method. However, up to now a direct J E-maih [email protected]. 2 E-mail: frahm @itp.uni-hannover.de 0550-3213/96/$15.00 Copyright (~) 1996 Published by Elsevier Science B.V. All rights reserved PII S 0 5 5 0 - 3 2 1 3 ( 9 6 ) 0 0 4 2 5 - 7
M.P Pfannmaller, H. Frahm/Nuclear Physics B 479 [FS] (1996) 575-593
576
solution by means of the nested algebraic Bethe ansatz has been obtained only for the supersymmetric t - J model [3,4]. The fusion method used to solve vertex models corresponding to higher-dimensional representations of ordinary Lie algebras such as sl, is not applicable due to the peculiarities of the representation theory of a superalgebra. Only a single member of the family of four-dimensional representations [b, ½], namely the one with b = ~, can be obtained as the irreducible component of a tensor product of two representations of dimension less than 4. Thus only in the special case of b = 3 the fusion procedure can be applied. Knowing the eigenvalues of the transfer matrix for this case Maassarani [2] has made a conjecture for the general case which is in agreement with the spectrum of the hamiltonian obtained from the coordinate Bethe ansatz. In this paper we extend the approach used by Maassarani to vertex models built from R-matrices defined on tensor products of two different four-dimensional representations [bl, ½] ® [b2, ½] to compute the eigenvalues of the corresponding transfer matrix z t''t'2 (/,): In the following section we give a short overview over the three- and four-dimensional representations of gl(2, 1) that are used together with the R-matrices acting on tensor products of these. In Section 3 and 4 we obtain two sets of Bethe ansatz equations for this model corresponding to a different choice of the reference state. Then we use the fusion procedure to find the eigenvalues of the transfer matrix for bl = 3. Then, using the set of Yang-Baxter equations for the intertwiners between the different representations in addition with the known analytic properties of the eigenvalues of the transfer matrix we can determine the eigenvalues of 7b~b2(#) up to an overall factor which is fixed by studying the model built from two vertices only.
2. R-matrices for [½]+ and [b, ½] representations In this section we present the R-matrices acting on tensorproducts of three-dimensional [ ½] + and four-dimensional [ b, ½] representations of gl(2, 1) along with the corresponding Yang-Baxter equations. Before we discuss the particular form of the representations we introduce the notation [x] for the grading of an object x:
[x]
~" 0 if x is bosonic (even), L 1 if x is fermionic (odd).
(2.1)
The multiplication rule in graded tensor products differs from the ordinary one by the appearance of additional signs. For homogeneous elements B and v we get (A•B)(v®w)
=(-1)[8]b:l(Av)®(Bw).
(2.2)
Using homogeneous bases in the two vectorspaces this equation can be written in components:
(A ® B)i2,.i2il ,A = ( _ 1) ( [i2]+ [.]2[) [jl] AiJ ,.h Bi2,j2
.
(2.3)
The even part of the superalgebra gl(2, 1) consist of a the direct sum of an su(2) and a u( 1 ) Lie algebra. Thus the basis vectors of the irreducible representation can be labeled
M.P Pfannmaller, H. Frahm/Nuclear Physics B 479 [FS] (1996) 575-593
577
by the eigenvalue B of the u(1) operator, the total spin and the z-component of the spin: IB, S, Sz) [10,11]. The three-dimensional representation [ 1 ] + contains a doublet with B = ~1 and a singlet with B = 1. We arrange the basis in the following order: 13, 3' ½)'
13, ½ - ½ ) ,
I1,0,0) •
(2.4)
The first two states are considered as bosonic (grading 0), the last one as fermionic (grading 1). For the supersymmetric t - J model they are identified with the electronic states with a spin up or a spin down electron and an empty site. In this case one should choose the opposite grading [ 1,1,0]. In the four-dimensional [b, 3] representation we find a doublet with B = b and two singlets with B = b i 3, respectively. The basis is ordered according to Ib, 71 ½),
Ib, g1 , - g 1 ) ,
ib
g1 , 0 , 0 ) ,
t b + 7 ,1 0 , 0 ) .
(2.5)
Here the grading of the basis vectors is [1, 1,0,0]. For the model with correlated hopping they correspond to states with a single spin up or spin down electron, an empty site and a doubly occupied site. The bosonic ones may be exchanged leading to an equivalent model. On the tensor product of two [ ½] + representations we have an R-matrix r 33: r 3 3 ( A ) = a ( / ) i d 9 + b(,~)H33 •
(2.6)
Here id9 denotes the 9 x 9 identity matrix and H33 is the graded permutation operator .,ii..jj = ( --1"1[it][i216' • The functions a and b are given with matrix elements ( H 33)i2,j2 •J tl,J2• 6.t2,./l' by a(a) -
A h+l'
b(A) -
1 A+I
(2.7)
This R-matrix is a solution of the Yang-Baxter equation 33 rl2('~ - / z ) r ~ ( A ) r33 23 ( /x ) = r 5333 ( t x ) r 1333 ( A ) r 1 233( , ~ - # ) .
(2.8)
Here the lower indices denote the spaces in which the R-matrix acts. The R-matrix R 3b on the tensor product [ 3 ] + ® [b, ½] can be constructed from the elementary intertwiners on the irreducible components of the tensor product [12]. The result is 4/x - 2b - 3 1 , R3b(#) = Ih + 4/1. 4. 2 b ~ - 3 t2
(2.9)
1 1 ] subwhere It1 and It2 are the operators intertwining the eight-dimensional [b 4- g, representation and the four-dimensional [ b + 1, 3] subrepresentation, respectively. Their matrix representation can be found in Appendix A. The tensorproduct [bl, 3] ® [b2, ½] contains three irreducible components, namely [bl + b 2 , 1 ] ( D = 8 ) , [bl + b 2 + 1,½] (D = 4 ) and [bl 4 . b 2 - 1,½] (D = 4 ) . The R-matrix is given by the following combination of intertwiners:
578
M.P. Pfannmfiller. H. Frahm/Nuclear Physics B 479 [FS] (1996) 575-593
R&b2(l~) = 2 # -- b~ - b2+__ }Itl + 2/z - bt - b2 q-- 1 2/z - bl - b2 - l i t 2 + I t 3 ,
2 # + b~ ~ b2
2/z q- bl q- b2 - 1 2 # + bl + b2 q- 1 (2.10)
where Ih, It2 and It3 are the intertwiners for the eight and the two tbur-dimensional irreducible components, respectively. They are presented in Appendix B. A peculiar feature of the intertwiners and thus of the R-matrix R b'b2 is that they are not symmetric under the exchange of b~ and b2. This is a direct consequence of the the rule (2.2) for the graded tensor product. Further it should be noted that there are matrix elements which are proportional to b~ - b2 and thus vanish only if the representation in the two spaces are identical. These entries are the reason that R-matrix R b'b2 has a shift point (i.e. becomes proportional to the graded permutation operator for some value of the spectral paramter) only if the two representations coincide. For the R-matrices defined in Eqs. (2.9) and (2.10) the following Yang-Baxter equations hold: 3b
3b
3b
3b
33
r33(A -- /z) RI3(A) R23(/z) = R23(Iz ) R13 ( A ) r12( A - t z ) 12 3t,1 -~3b2 nblb2 r , l~blb2(l.t,) R~b2( k ) l~3bl RI2 ( A - / . £ ) / 4 1 3 ( a ) / x 2 3 ~,/./,)="23 "'12 ( a - ~ ) , Rb~b2 12 ( k _ ~,, ' ~" Ob~b3 1 3 (A)R2b~b3(/z) = Ob263 . , o b ~ b ~ , O b "'23 (/*)rt13 ~ , / t 1 'l~b' ,1 2 " ( ~ ' - - ].Z) .
(2.11) (2.12) (2.13)
Writing the Yang-Baxter equations in components one has to include additional signs due to the grading • ~il,.il R rAx.h,kl R t . ,~j2,k2g 1)[.jz]([.jll+lkll) r 1 2 ( ~ -- t.~)i2,J2 13~, 1i3,j3 23klc~)j3,k31,--I
i2,j2 ~',il,Jl r r s z ) k ,~' ( [.j2]([il]+ljt]) = R23(Iz)i3.h R13(alj3,k~ 12~A -- -- j2,k2" -- 1 )
(2.14)
From the R-matrices for the different representations we can construct monodromy matrices by taking matrix products in one component of the tensorproduct - the auxiliary or matrix space, T c
"~a,t~l,...,°'L
R ~ ,aa,
R~" ,a,,a,_,
.
R " '"~'a2RZ" ,a2.h ~ ,)E,:2([,~,l+[~il)ES=I'l,~,l(2.15)
Again the grading gives rise to additional signs. As a consequence of Eqs. (2.8) and (2.11 ) - (2.13) the monodromy matrices satisfy the following Yang-Baxter equations: 33
33
33
33
33
33
r12(a -- ~ ) T i 3 ( a ) T 2 3 ( l * ) =g~3 ( ~ ) g i 3 ( a ) rl2(A - ~,) , r,~(a
-
u)r?~(a)r33b(F*)= r ~ 3 b ( u ) r ? ~ ( , ~ ) r 3 3 ( a
- t,),
(2.16) (2.17)
R ~ ' ( A - tz)T3b2(,~ )Tb~e'(iz) = rb~b2(iz)T3~2( a ) R~b'( A -- I X ) ,
(2.18)
ab~b2(t~ __ [,g) TI3bib,.(,,~) T~3b2b3(t.L) =123,~b2b3/(/,Z)l13\,a,-blb3(l~)R}?2(/~ -- /{.g) .
(2.19)
M.P Pfannmgiller, H. Frahm/Nuclear Physics B 479 [FS] (1996) 575-593
579
From the monodromy matrix the transfer matrix is obtained by taking the supertrace in the auxiliary space: r(/z) = }-'~',~(-1)[alT(#)aa. The Yang-Baxter equations (2.16) and (2.17)-(2.19) imply that transfer matrices acting in the same quantum space commute:
[7"33(a), T33(/£)] = 0,
[~3b(a), ~.3b(~)] = o, [~.362(a), ~b,b: (;~)] = 0, [~b~b, (a), ~b2b, (;~)] = 0. These equations imply that T3b, "1"bIb and "t"b2b share a system of common eigenvectors and thus can be diagonalized simultaneously.
3. Nested algebraic Bethe ansatz for ~ b
The transfer matrix r 3b can be diagonalized directly by means of a nested algebraic Bethe ansatz. The calculations can be performed in close analogy to the NABA for the supersymmetric t-J model [ 3 ]. Thus we omit the details here and present only the most relevant steps. We can represent T 3b as a 3 × 3 matrix in the auxiliary space with entries beeing operators in the L-fold tensorproduct of four-dimensional quantum spaces { A l l ( A ) AIZ(A) B 1 ( A ) ) T3b(A) = [ A z l ( A ) A22(A) B2(A) . \ C1(,~) C2(,~) D(A)
(3.1)
From the Yang-Baxter equation (2.17) we can derive commutation relations for these quantum operators. The ones needed in the sequel are
D( A)Bi(tz) -
1
a ( t z - A)
Bi(tz)D( A) +
b(a
-
~)
a( A - tz)
Bi( A)D(tz) ,
1
x fi,ll B . iz) r(A - tz)i:,t: ~1(A)Al_,k2(~) b(a - ~)
Ai2k2 (t z) Be1(A) - a ( a -
Bi:(tz)Ailk2 (,~),
a(A - / z )
Bi: (i,) Bi~ (a) =
(3.2)
1
b ( a - ~ ) - a ( a - ~z)
r( A - t*: .'~il'll i2,12B,,1 (A) Bt2 (/z)
(3.3) (3.4)
Here r(/z) is the R-matrix of the rational six vertex model, both states being bosonic: r(/z) = a(/z)id4 + b(/z)Hz2.
(3.5)
We choose the state 10) = ® L i b - ½, O, O)
(3.6)
Ml! Pfannmiille,: H. Frahm/Nuclear
580
as reference
Physics B 479 [FS] (1996) 575-593
state. The action of the monodromy
P(/.L)10)
= i
1
0
O
l
0
0
hence it is an eigenstate make the following IA,,...,
matrix on this state is
(3.7)
of T3’(p)
= -D(p)
+ Al 1(,u) +A22(,4).
Starting from IO) we
ansatz for the Bethe vectors:
A,) = B,,, (AI ) . . B, (A,,) 10)F”Jr-.“’,
where summation
(3.8)
over repeated indices is implied and the amplitudes Pt...“’ are funcht , . . . , A,. In order to calculate the action of the transfer
tions of the spectral parameters
matrix T~~(,LL) on such a state, we use relations
(3.2)
and (3.3)
to commute
the oper-
ators D and A through all B’s until they hit the vacuum,
Here ~(‘1 is the transfer matrix of an inhomogeneous is constructed from the R-matrix (3.5).
The amplitudes
PI,...*“’ can now be identified
six vertex model with II sites that
with the components
of a vector F in the
state space of this n-site model. As can be seen from Eqs. (3.9) and (3.10) a sufficient condition for IAt, . . , A,) to be an eigenvector of r(p) is that the unwanted terms & and Ak cancel and that the vector F is an eigenvector of the nested transfer matrix 7(1)(/L). The condition that the unwanted terms & and Ak ought to cancel leads to a set of equations for the Spectral parameterS hk:
4Ak- 2b+ 5> L
F”
4Ak + 2b + 3
,,.....
I”
=
7 (1)
( hk),h:J:;~;
fT’““.-Jl
,
k= I,...,n.
(3.12)
581
M.P. Pfannmiiller, H. Frahm/Nuclear Physics B 479 [FS] (1996) 575-593
In a second step (nesting) we have to diagonalize the transfer matrix r(l~. This goal is achieved by another Bethe ansatz which gives the well kown results for the inhomogeneous, rational six vertex model. The amplitudes F a.......a, are the components of the corresponding eigenvectors. The eigenvalues are found to be n A(~) (,al & . . . . .
& lp l . . . . .
"1
1
"1
v n , ) = "[-[ ~. a ( A i - At ) ~. ] - [x a ( ~,j _ At i=1
1
+ ]--[Jf.~
j=l
"=
a(At
-
z,j)
"
(3.13) The spectral parameters ~,j are subject to the following set of Bethe equations:
f
a(,~ i
--
12.j
=
i=1
i
I-I a(u_____~Z ~'J__2 kq=j a(~'./ - ~'k) '
j = 1 .... '
(3.14)
nl .
If we insert the elgenvalues (3.13) of the nested transfer matrix into Eq. (3.12) we obtain the first set of Bethe equations: nl
(4A~-
2b+5)
-4-~ ; - - ~ +
1
L = H
.j=l
k=l ....
a ( A ~ - ~,j ) '
n.
(3.15)
"
From Eqs. (3.9) and (3.10) we can read of the eigenvalues of r3b(At) as now the eigenvalues of ~-(1)(At) are known: A3b ( A t l & . . . . .
.....
v.,
=-
4--~T2b+3 n,
+ n .j= 1 a ( u / -•
1
i:~ a(aiLAt) nl
I-t) + . =
1 a(At
n v i•)
=
1 a(Ai--
At)
•
(3.16) The eigenvalues and Bethe equations should be compared to the rational limit of the corresponding equations in Ref. [2]. We find complete agreement.
4. A second Bethe ansatz
In general the specific form of the eigenvalues and the Bethe equations depends on the particular choice of the reference state from which the Bethe vectors are built. For the transfer matrix ~.3b there exist a second possibility besides (3.6), namely the state
IO) = ®Lib+
½, 0, 0 ) .
The action of the monodromy matrix on this state is
(4.1)
M.P Pfannmfiller, H. Frahm/Nuclear Physics B 479 [FS] (1996) 575-593
582
T3b(/2) 10) =
(4/.* + 2b k.4/2 + 2b T
0
0
10).
(4tx-Zb-
(4.2)
\4/2 + 2b+3]
For the Bethe ansatz we need the following commutation relations between these operators, which are derived from the Yang-Baxter equation (2.17). D(/2)C~(A) -
1
a(/2 - 1)
Ai~k~ ( A)C~2(tx) - a ( A 1- / 2 )
Ci(A)D(tz)
b(A-/2)
+
a(a - ~)
(4.3)
Ci(tz)D(a),
kl ,/l r ( A - ix)k2,t2Ct2(/2)Ai~h(A)
b( A - ix)
a(a-
tz) Ck, (A)Ai,k2 (/z) ,
(4.4) k1,11
(4.5)
Ck, ( A ) C k 2 ( # ) = b( A - # ) -1 a ( A - tx ) r ( A - / 2 ) ~ 2 , 1 2 C l 2 ( / 2 ) B h ( A ) .
Here r(/z) is again the R-matrix (3.5) of the rational six vertex state model, with two bosonic states. Now we use the operators C~ to build the Bethe vectors starting from the new reference state:
IAl . . . . . & ) = Ca, ( & ) . . .
Ca,, ( 1 . ) 1 0 ) F ........
1
(4.6)
.
The action of ~.3t, on such a Bethe state is given by D(/z),A,,...,An)=(4-~/2-2b-3)L
=nilI a ( / 2 - A 1 i)
b~ b F a ...aIB -t-E(Ak)a,:::::a:: .... bk(~)
k=l
. . . . ,An)
fI Bb,(A.i)lO)
(4.7)
j4: k
[All ( # ) + A22(/x) ] IA1. . . . . An) 4/X + 2b + ~
L ,
1
a ( / 2 -- Ai) r
(1) . ~bl,...,b,,Fa .......al ........
"=
l ~B lb(jAO )j, /
j=l
K-"r A k),,i...',a;, xbl ... b,Fa ..... + Z_.,t ' ' al B b kr. t ~ J~ k=l
]-[
(4.8)
Bb;(A.i)[O) j4.k
Here r (l) is the transfer matrix of an inhomogeneous n-site model that is constructed from the R-matrix (3.5), xbl,...,b, dd. - d,,,d,,- l ~d3,d2r(,, t/2)al ........ = r ( / 2 - An)b;,,a,r(/2 -- A,-l)b,,_,a,,_l " . r ( / 2 -- ~A2)b2,a2 ~t~
./.( I ) r
_
"~d2,d AlJbl,al "
(4.9)
M.P. Pfannmidler; H. Frahm/Nuclear
The condition
that the unwanted
Physics B 479 [FS] (1996) 575-593
terms ought to cancel
583
leads to the following
set of
equations: 4/ik - 2b-3
4Ak + 26 - 1 >
LFPnv..#l
=
7(1)(Ak)ln)::::::‘~:F”.....,“1
,
As before, the nested transfer matrix r (I) is diagonalized eigenvalues are found to be
k
=
1,.
. . ,H.
(4.10)
by a second Bethe ansatz. The
corresponding
where V,i are solutions
of the following
set of Bethe equations:
j=
If we insert
the eigenvalues
l,...,n,
A(‘) mto Eqs.
(4.12)
(4.10)
we obtain
the first level Bethe
equations: 4Ak - 2b-
3
L
(4.13)
4hk + 2b - 1 > From Eqs. (4.7) and (4.8) we can now determine
the eigenvalues
of the transfer matrix
PQ):
n3b(plA,,.. .,Anlv1,.. .,vn,) =-
This completes between
4p-2b-3 4,u+fbf3
1 L ’ rI i=, a(~ - Ai)
the the second Bethe ansatz. We postpone
the discussion
of the relation
the two Bethe ansatze to Section 6.
5. Fusion
The tensor product of two [ i ] + representations contains a four-dimensional [b = 2, i] and five-dimensional [ 1 ] + representation. As mentioned in the introduction [b = i, 31 is the only menber of the family of four-dimensional representations [b, i] which
584
M.P. PfannmMler, H. Frahm/Nuclear Physics B 479 [FS] (1996) 575-593
can be obtained by taking tensor products of lower-dimensional irreducible representations. We construct the fused R-matrix for the "sacttering" of two composite "[ ~]+ particles" with a third "[b, ½] particle" from R,2,3(A) = R,3(A3b+ Ao)R~3b(A - Ao) •
(5.1)
3 71 ] state In general [ 23-,½] and [ 1 ] + states are mixed by the scattering. However, the [ ~, will not be destroyed if the triangularity condition holds: 3
p52RI2,3Pl~2 = 0.
(5.2)
3
Here P52 and Pl~ denote the projectors onto the five-dimensional representation [1]+ and the four-dimensional [3, ½] representation, respectively, r 33 becomes proportional to a projector onto the five-dimensional representation for the special value 1 of the spectral parameter 33 ri5 (1) ~ Pis .
(5.3)
From the Yang-Baxter equation (2.1 1) we obtain l 3b 3a Z~p3bcx I)p5 . Pi~2R~4(A + 7)R23 (• - ½) = R2b(A - 2 ' " , 3 ''~ + 7 ,2"
(5.4)
This allows us to prove the triangularity condition (5.2) for R12,3 for h0 = ½. Thus the matrix 2 ! R( 12),3 = Pi22Rlz,3P122 (5.5) 1 indeed describes the "scattering" of a [ 3, 7] particle with a [b, 1 ] particle. A YangBaxter equation holds for this R-matrix:
R(12),3(/~ -/~)R(12),4(~.)R~b(#) = R~b(#)R(12),4(/~)R(12),3(,~ - # ) .
(5.6)
The triangularity condition (5.2) states that R,2,3 becomes an upper block-triangular matrix if we change the basis in the tensor product 1 ® 2 such that the first four vectors form a basis for [ 3, ½] and the next five a basis for [ 1 ] +. Let B be the corresponding transformation matrix. Then we have
B-~R12,3B =
,)
R5 b
.
(5.7)
We can replicate this R-matrix to a chain of L sites (the auxiliary space is the tensorproduct 1 ® 2!). Taking the supertrace over spaces 1 and 2 leads to the following relation between the transfer matrices:
T3b( A _}_ 1) T3b(,~ _ 1) = T~b( a) _]_T5b(/~) .
(5.8)
The eigenvalues are thus related by
A3b(A + 1) A3b(A _ 1) = A3b(A) + A5b(A).
(5.9)
M . P Pfannmffller, H. F r a h m / N u c l e a r
585
Physics B 4 7 9 [FS] (1996) 5 7 5 - 5 9 3
The reference state 10) (3.6) is an eigenvector of each diagonal part of the monodromy matrices T 3b and T sb on its own. The eigenvalues (vacuum eigenvalues) corresponding to T} b are
T(1 (IX)I0> =T2=2 (IX)I0> =
2IX+ b +
10>,
~b
T~ (ix)[o) = 11o), Y4~/(IX) I0) = For
+b+~ 5 2ix+~+
(5.10)
Io).
find
T 54 w e
T~(IX) Io) = T ~ (IX)Io) = ~ ( i x ) I o ) - 11o),
--
+b+
(5.11)
10).
As usual the eigenvalues of the transfer matrices r~ b and r 5b corresponding to an arbitrary Bethe vector (3.8) can be written as a sum of these vacuum eigenvalues dressed by factors depending on the spectral parameters of the Bethe state. The 1.h.s. of Eq. (5.9) can be evaluated with the help of Eq. (3.16): A 3b (tx - 1) A3b (IX + ½)
( 2IX
-
2IX +
+
b +
b
+
_
3n
tx -
~ti -
2
IX -
IX
/~i
1 j=l Ix -
1
PJ
2
/~J "~ 2
Ix - as - 5 _ ~ - vJ + ~ 1 l i=l # A i + g j=l # - v j + g 2/x - b + 3 2 # - b + g
+
7b+½ 2ix-b+
7b+~ +
2ix~b+
+1
Ix --
+1-I
n
Ix -
i=l Ix
2
i=, I X - A i + ~
IX- a i - ~
5
1
_ Ix-aiT~
1
Ix-v,-
,=, I x - v , - ~
~ 1
Ix-vi+~
7,72'-
{hi_ IX--
~
~
Ix-&-~
,=,ix
#-Ai 5
~t i -
Ix "i 3
_~
/~i -- ~1
hi -
1 ~I
~
n.j=l
IX -
, ~ i 7 T j=l ]£
1
IX - - P j -- ~ +
' a
Ix
Pj 7
vj -
~ IX -
vj +
P.J
½ ]£
12.j -~- 1
3
~1
IX - - A i -- ~ a i -~- ~1 .
i=1 Ix
3}
IX -- P j + "~ IX
P.j
3 21
(5.12)
-~
'
M.P Pfannmgiller,H. Frahm/NuclearPhysicsB 479 [FS] (1996)575-593
586
Comparing to Eqs. (5.10) and (5.1 1) the first two terms are identified as contributions to A3b(/z) and the third one as part of ksb(/,). The knowledge of the vacuum eigenvalues alone is not sufficient to separate the third part of Eq. (5.12) into contributions to A~b(,~) and A-%(/l) because the vacuum eigenvalue 1 is found in Eq. (5.10) as well as in (5.11). As second argument we use the analyticity of the eigenvalues [13]: The Bethe equations (3.14) and (3.15) are known to guarantee that the residues of the eigenvalues at the poles of (5.12) vanish. The parts that have been identified above as contributions to A3b(/z) have poles at l /-£ ---- /~i -- ~I and ~ = ~'.i + ½. The Bethe equations (3.14) and (3.15) /,.I, = /~i -[- ~, lead to vanishing residues at the last two of these. In order to have a zero residue at =- /~i "~ 1 as well the contribution corresponding to the vacuum eigenvalue 1 has to be n
--
I~i=l ( # - ai 3)/(# be expressed as
_ ,~i
3b
__
½). Collecting the results the eigenvalues of r~ (/z) can
A~b(tz)=--(2~--b+3)L{
"iHl
+b+½
i=I /.Z
}
+
,
ki 7 ~ .j=l /'£
2 # - b + 3 2/z +b+
3I- I
_ /*-ai
-
~2/z+b+
v.i+ 71
I'L -- l~i
23 --~
+1
P.J +
b q -7
1
'
½,=l/z
L
~
n
.= /-£
1
/~i -~- ~
i~g _ ,~i _
i=1 /--/,
/~i
~ 1
(5.13) As was pointed out in the introduction b = 23-is the only member of the whole family of four-dimensional representations which can be obtained as a component of the tensor product of lower-dimensional representations. Thus r~ b2 ( # ) is the only representative of transfer matrices rb'b2(Iz) whoose eigenvalues can be calculated by means of the fusion procedure. This situation is of course not satisfactory as one wishes to diagonalize the transfer matrix for an arbitrary [bl, ½] representation in the auxilliary space. In particular, we cannot handle the situation of [b, ½] ® [b, ½] with arbitrary b which gives rise to models of correlated electrons with an additional free parameter. To extend our results to the complete family of four-dimensional representations in the auxilliary space, we again make use of the analytic properties of the eigenvalues and Bethe ansatz equations [13]. From the fact that 7"~b2(/z) and rblb2(a) commute and thus share a system of common eigenvectors, it is clear that the Bethe ansatz equations (3.14) and (3.15) must be preserved. The eigenvalue of T ~b2 on the pseudo vacuum (3.6) is known to be
A&b2t_.~ 0
'~, =
2 ( 2tz + bl - b2 ) L 2/z+bl+ff2
( 2t,+b,-l,2
~ 1
2t,+I,,-b2+2
+ \2/7-7- bl 7-b2 - Z 1 2tz + b, 7 b 2 + l J
L
+ 1.
(5.14)
M.P PfannmiiUer,H. Frahrn/NuclearPhysics B 479 [FS] (1996) 575-593
587
In order to generalize Eq. (5.13) we replace the vacuum eigenvalues of the diagonal elements of T 362 (5.10) by those of T b'b2 given in Eq. (5.14). The next step is to determine the necessary changes in the "dressing factors". The structure of the eigenvalues must be the same as in Eq. (5.13). Therefore we make the ansatz
Ab'b2(#) =
_(
\ 2 # + bl + b2 - 1
)L
X{ niH._l#-- l~i'~-Oll f i ]d'- pj ~-~1 _
+
(
A i + /31 j=l # -
#
I-~
# - - Ai-'~O/l
/'~i -~ 81 + i=1 #
2# +_b, 2# + bl + b2 - 1 2 # + bl ~ b2 +
# -- P._j/+ Y2 "[
"~i -'1-/32
# - vj +8!
J
# - - Ai-{- ~ 1 i=1 # - / I i +/32
l/
(5.15) i:1 ~ - - - a i +/31 This expression has poles at # = Ai -/31, # = Ai -/32 and # = vj the first two of these poles vanish if
--
81. The residues at
nl
=H#-VJ 2 # + bl + b2 - 1 (2_~_~-+-bl-b2+~)
+ bl 7 b2 +
/x=ai-,Si
L
j=l /..L
+& /"j -~ ")/I
nl
= H ~ -- pj -~- ')/2 /"J -'~ 81 /x=ai-,82 j=l # ]z=~.i --/~2
(5.16)
Now comparing the l.h.s, of these equations to the 1.h.s. of the first Bethe equation (3.15) we can determine 131 and/32: bt /31- 2
5 4'
bl /32- 2
1 4
(5.17)
Inserting these values for/31 and/32 into the RHS of Eqs. (5.16) and comparing to the RHS of the Bethe equation (3.15) leads to bl Yl - 2
5 4'
bl 3 ")/2 = ~- + ~ ,
bl 1 81 = ~- - ~ .
(5.18)
The parameter oq is the only one that remains unkown. Since it appears in an overall factor only it cannot be determined using the above arguments. We use the first nontrivial solution of the BAeq's for a two-site model, namely n = 1, nl = 0, AI = - 1 and determine the corresponding eigenvalue by operating with the transfer matrix on the corresponding Bethe vector. Evaluating the result at # = (b2 - b l ) / 2 isolates the last term. This leads to the final result for the eigenvalues of rb'b2:
(2#+bi-b2)L{ Ab'b2(#) = -- \ 2 ~ 7 b l +if2 7 1
"in l 2# - 2ai - bl - 3 f i 2# - 2vj + bt _ 21,
2 A i 7 bl
~ j=l 2/1,
2vj
-{-
bl
2
51 2
588
M.P. Pfannmfiller, H. Frahm/Nuclear Physics B 479 [FS] (1996) 575-593
2# - 2,~i - bj - ~ 2# - 2uj + b14-_ 5 + i:l 2/z 2~i ~-bl 2p.j 4- b! 1 2I j=l 2# (
2/z+b~-b2
l
+ Hn 2 # -
2,~i - bl -
i:l 2 #
2,h.i'~
c 1LI 2/z
2#+bl-b24-_2~
+ \2 -T bl
Tb + l J
bl
32
i=l 2/Z -- 2Ai 4- bl
2
-
2,~ i -
1
3
(5.19) ~'
It should be noted that bl and b2 enter this expression in an asymmetric manner. This asymmetry cannot be removed by a shift in the spectral parameter. It has its origin in the non-commutativity of the graded tensor product which was already mentioned in Section 2. We have checked our results by completely diagonalizing the transfer matrix for L = 2 and arbitrary values of bl, b2 and /z. For bl = b2 the equations are equivalent to the rational limit of the conjectures in Ref. [2]. Taking the logarithmic derivative of the eigenvalues we find the energies that where calculated in Ref. [9] by means of coordinate Bethe ansatz. We can proceed with results from the second Bethe ansatz in a similar manner. To keep the presentation short we will only give the corresponding results for the more general case of inhomogeneous chains in Section 6.
6. Inhomogeneous chains A straightforward generalisation is now to construct an inhomogeneous model. We build the monodromy matrix from R bb' matrices with a representation [b, ½] in the auxiliary space and [bi, ½] in the quantum space at site i. This model is also integrable by construction because of the Yang-Baxter equation (2.13). Modifying the vacuum expectation values, the eigenvalues and Bethe ansatz equations can be derived from the ones obtained in the previous section. It is convenient to rescale and shift the spectral parameters according to ,~k ~ -i,~k - 1, u i - + -i~,j + ½: L
AJ'{I'~} (tz) = - H k=l n
2/,t+b~l
_ 2#+2~/+b---
½ j=l 2 / z + 2 i ~ ' j + b + i 2
2# + 2iAi - b + ½ I-I 21z + 2i~') + b + ~ } b + ~3 .i=~ 2tz + 2i~,j + b + 7
+II 2# + 2iAi ~ i=1 L
(
_2/z+ b ~_b5
e / z + b - bk +
+II \2a+b+bk--12 k=l 17
+II i=l
2tz+2iAi-b+
7
+bTb,+ 1
2/z + 2iAi - b + ~ 2 # + 2iAi -+ b 2I
(6.1) "
589
M.P. Pfannmiiller, H. Frahm/Nuclear Physics B 479 [FS] (1996) 575-593
The Bethe equations are
~_i a t _ i ( 1 ~ b k - ~ 1) = ~ A l - - v j + T ~. t k=l Al+i(½bk ¼) Al--vi--½i' r I "~t--l"iq-½i-1=1 /~l
--
½i
/~i
"
n~IPJ--Piq-i,-.j=l pj
l = l . . . . . n,
(6.2)
i = 1. . . . • nl.
(6.3)
Pi
In a similar fashion we can proceed with the results from the second Bethe ansatz. Finally we obtain
L ( 21,z-b+bk 2 / z - b - b k + l ) Ab{b'}(Iz)=--l--[ 2 / ~ T - b ~ k : 1 2 / z + b + b k + l k=l
21,z + 2iAi --k b +__½ n 2t.z + 2i~,j - b - g3 2tz+2iAs b 7 j=~ 2# + 2it,j b+ 1
× _
+ H" i=1
2 # + 2i,~i + b + ~l ~ - i 2 # + 2 i v i - b + ~ 2/z + 2iAi
- - b --~ -~ 3 j=l
L (2#-b+bk
2tz-b+bk+2)~-[2tz+2i,~i+b+½
+I'I
2/~-b~-~-k----12/zTb+bk+
+ I-[
2# - b - b~ +__l 2tz - b - bk Tb~bk 1 2#~b-+b~T
k=l
k=l
}
2# + 2i~'j - - - - ~ J
i=1
2/z+2iA/
b + 53
i)fx
21z + 2ihi +__b +__5 ' i=1 2 / z + 2 i a i b 1. 2 (6.4)
The Bethe equations are
at-i(½bk+¼) k=l a t ~ i ( l b k + ¼ ) = I-I
,
at-Pi--2t /=1 A,-vj+½i ..... 17
,n,
(6.5)
I
'~j - "-~ +- i t=l ,~, At - ,,i Pi +_ ½, ½ i - _ 1-I j=l pj Pi - i '
i= 1..... nl.
(6.6)
This second set of Bethe equations can be obtained from the first set (6.2), (6.3) by replacing bk by -bk. If in addition b is replaced by - b the eigenvalues (6.1) and (6.4) are found to be equal up to an overall factor. This factor can be removed by a suitable normalisation of the R-matrix (2.10). This behaviour is a consequence of the existence of an automorphism of the superalgebra gl(2, 1 ) which maps the u( 1 ) operator B onto --B.
590
M.P Pfimnmiiller, 1f. Frahm/Nuclear Physics B 479 [FS] (1996) 575-593
7. Discussion In this paper we have computed the spectrum of vertex models invariant under the action of the superalgebra gl(2, 1) by means of the Bethe ansatz. Depending on the choice of the reference state in the four-dimensional quantum space of the local vertices two different Bethe ansiitze are possible which are related to the automorphism of the superalgebra which maps the corresponding states onto each other. The solutions start from one of the bosonic highest weigth states. This is different from the situation in the three state model corresponding to the supersymmetric t - J model for which three Bethe Ans~itze corresponding to the various possibilities of ordering of the basis (2.4)--namely FFB, FBF and BFF--can be constructed [3,4]. The fact that there exists a family of four-dimensional representations for this superalgebra allows to introduce a new type of inhomogenous four-state vertex models by allowing the parameter b to take different values in different quantum spaces. Studying the Hamiltonian limit of this class of inhomogenous vertex models leads to systems of electrons with correlated hopping with a spatially varying parameter. It should be noted however, that the R-matrix Rb'b2(#) becomes proportional to a (graded) permutation operator for some value o f / z only" if bj = b2 = b. The existence of such a shift point is necessary for the construction of a local Hamiltonian from the transfer matrix. Thus to limit the range of interaction one should consider a model with a sufficient number of sites carrying the same representation as the auxiliary space of the monodromy matrix. A possible example is a single b' "impurity" in a chain built from R bb otherwise. We shall study the effect of such an impurity on the thermodynamic properties of an correlated electronic system in a forthcoming paper.
Note added After completion of this work we received a preprint by EB. Ramos and M.J. Martins [14] who obtain the spectrum (5.19) of the transfer matrix rbh(#) by applying the algebraic Bethe ansatz to the 4 x 4 monodromy matrix Tb~(/x) directly. Their results concide with ours; it should be noted though that their discussion of the unwanted terms arising in this procedure is not complete.
Acknowledgments This work has been supported in part by the Deutsche Forschungsgemeinschaft under grant Fr 737/2-1.
M.P Pfannm&'ller, H. Frahm/Nuclear Physics B 479 [FS] (1996) 575-593
591
Appendix A. Intertwiners for the tensor product -z~[:ll+® [b, =11 2J In this appendix we explicitly present the elementary intertwiners for the tensor product [ 1 ] + ® [ b, 71 ] = [ b + 3' 1 1 ] ® [ b + 1, ½]. For the details on the construction of R-matrices from elementary intertwiners we refer to the literature [ 12]. The basis in the tensor product is chosen to be the tensor product basis constructed from (2.4) and (2.5). In the following we give only the diagonal matrix elements and the non-zero off-diagonal entries. Ih corresponds to the [b + ½, 1 ] subrepresentation (D = 8 ) , It2 to [ b + 1,½] ( D = 4 ) .
1
2b+ 1 2b+3
2b 2b+3
r_
2b+l 2b+3 2 2b+3
r+ 2b+l 2b+3
Itl =
--r_
1
1
(A.1) 2b+ 1 2b+3
r+ 2 2b+3
r+ r+ r_
--r_
2 2b+3
4
2b+3
,
where the elements r_ and r+ are given by v'~-2 2b+3 m
r_-
'
r+-
v/~+2 2b+3
For the second intertwiner It2 we have the simple representation It2 = id12 - Itt.
(A.2)
Appendix B. Intertwiners for the tensor product [bl, ½] ® [b2, 1] Here we give the matrix representations of the elementary intertwiners for the tensorproduct [bl, ½] ® [b2, ½] = [bl + b2, 1] ® [b~ + b2 + 1, ½] @ [bl + b2 - 1, ½]. The basis in the tensor product is chosen to be the tensor product basis constructed from (2.5). As before only the diagonal matrix elements and the non-zero off-diagonal entries are written down. It is usefull to define the quantities fit = 2bl + 1,
~ = 2bl - 1,
(B.1)
2bz + 1,
~2 = 2b2 - 1.
(B.2)
/~2 =
The intertwiner for the [ bl + b2 4- 1,½ ] representation is given by
592
M.P.. Pfannmiiller, H. Frahm/Nuclear Physics B 479 [FS] (1996) 575-593
AI,I
AI,2
0
AI,3
Bl,1
ALl
A1,4 ) B1,2
A2,2
A2,3
0
A2,4
0 Bl,1
It2 =
Bi,2 0
A3,1 B2,1 A4,1
0 0
A3,2
A3,3
A3,4 B2,2
B2,1
A4,2
(B.3)
A4,3
B2,2
A4,4 1
The non-vanishing entries are
A=
(/~1 + ~2)(~1 q- ~2)
~
--V~l~l
~'
/~. -
and
B -
(BI + /32)
For the [bl +
b2 - 1, ½] subrepresentation we have the intertwiner
0
C1,2
Cl,1
D2,1
Dj,1
C2,1
It3 =
C2,2
0
D2,2 D2A
C3,1
C3,2
C1,4
Ca,3
C2,4
O1,2
Dl,1
D2,1
C1,3
(B.4)
D2,2
0
C3,4
C3,3 0
C4,1
C4,2
C4,3
0
C4,4
M.P Pfannmfiller, H. Frahm/Nuclear Physics B 479 [FS] (1996) 575-593
593
where the non-vanishing entries are given by
( 8 1 ÷ 1~2)(~1 ÷ ~ 2 )
(/31 + & ) Finally the intertwiner Itl for the [bl + b2, 1 ] subrepresentation can be represented as It1 = id16 - It2 - It3.
(B.5)
References [ 1] [2] [31 [41 [51 [61 [71 [81 [91 [101 [ 111 1121 1131 1141
A.J. Bracken, M.D. Gould, J.R. Links and Y.-Z. Zhang, Phys. Rev. Lett. 74 (1995) 2768. Z. Maassarani, J. Phys. A 28 (1995) 1305. EH.L. Essler and V.E. Korepin, Phys. Rev. B 46 (1992) 9147. A. Foerster and M. Karowski, Nucl. Phys. B 396 (1993) 611. C.K. Lai, J. Math. Phys. 15 (1974) 1675. B. Sutherland, Phys. Rev. B 12 (1975) 3795. P. Schlottmann, Phys. Rev. B 36 (1987) 5177. I.N. Karnaukhov, Phys. Rev. Lett. 73 (1994) 1130. G. BediJrftig and H. Frahm, J. Phys. A 28 (1995) 4453. M. Scheunert, W. Nahm and V. Rittenberg, J. Math. Phys. 18 (1977) 155. M. Marcu, J. Math. Phys. 21 (1980) 1277. G.W. Delius, M.D. Gould, J.R. Links and Y.-Z. Zhang, Int. J. Mod. Phys. A 10 (1995) 3259. N.Yu. Reshetikhin, Lett. Math. Phys. 7 (1983) 205. P.B. Ramos and M.J. Martins, One parameter family of an integrable spl(2] 1) vertex model: Algebraic Bethe ansatz and ground state structure, preprint UFSCARF-TH-96-03, (1996).
ELSEVIER
Algebraic Bethe ansatz for gl(2, 1) invariant 36-vertex models Markus E Pfannmtiller 1, Holger Frahm 2 Institut fiir Theoretische Physik, Universitgit Hannover, D-30167 Hannover, Germany Received 27 March 1996; revised 5 July 1996; accepted 7 August 1996
Abstract
Four-dimensional irreducible representations of the superalgebra gl(2, 1) carry a free parameter. We compute the spectra of the corresponding transfer matrices by means of the nested algebraic Bethe ansatz together with a generalized fusion procedure. PACS: 71.27.+a; 75.10.Lp; 05.70.Jk Keywords: Algebraic Bethe ansatz; gl(2, 1); Vertex models
1. Introduction
Recently, vertex models built from representations of the superalgebra g/(2, 1) or q-deformations thereof have attracted increasing attention [ 1-4]. One reason is their relation to integrable models of interacting electrons in one spatial dimension: for example, the supersymmetric t-J model [5-7] is obtained in the hamiltonian limit of the transfer matrix for the vertex model based on the three-dimensional fundamental representation [½]+ of gl(2, 1) [3,4]. A special feature of this algebra is the existence of a family of four-dimensional representations [b, l ] parametrized by a parameter b 4: -4-½. From the corresponding vertex model a one-parametric integrable model of interacting electrons can be derived [ 1 ]. This system of electrons with correlated hopping has been solved recently by means of the coordinate Bethe ansatz [8,9]. The relation to integrable vertex models provides an embedding of these models into the framework of the Quantum Inverse Scattering Method. However, up to now a direct J E-maih [email protected]. 2 E-mail: frahm @itp.uni-hannover.de 0550-3213/96/$15.00 Copyright (~) 1996 Published by Elsevier Science B.V. All rights reserved PII S 0 5 5 0 - 3 2 1 3 ( 9 6 ) 0 0 4 2 5 - 7
M.P Pfannmaller, H. Frahm/Nuclear Physics B 479 [FS] (1996) 575-593
576
solution by means of the nested algebraic Bethe ansatz has been obtained only for the supersymmetric t - J model [3,4]. The fusion method used to solve vertex models corresponding to higher-dimensional representations of ordinary Lie algebras such as sl, is not applicable due to the peculiarities of the representation theory of a superalgebra. Only a single member of the family of four-dimensional representations [b, ½], namely the one with b = ~, can be obtained as the irreducible component of a tensor product of two representations of dimension less than 4. Thus only in the special case of b = 3 the fusion procedure can be applied. Knowing the eigenvalues of the transfer matrix for this case Maassarani [2] has made a conjecture for the general case which is in agreement with the spectrum of the hamiltonian obtained from the coordinate Bethe ansatz. In this paper we extend the approach used by Maassarani to vertex models built from R-matrices defined on tensor products of two different four-dimensional representations [bl, ½] ® [b2, ½] to compute the eigenvalues of the corresponding transfer matrix z t''t'2 (/,): In the following section we give a short overview over the three- and four-dimensional representations of gl(2, 1) that are used together with the R-matrices acting on tensor products of these. In Section 3 and 4 we obtain two sets of Bethe ansatz equations for this model corresponding to a different choice of the reference state. Then we use the fusion procedure to find the eigenvalues of the transfer matrix for bl = 3. Then, using the set of Yang-Baxter equations for the intertwiners between the different representations in addition with the known analytic properties of the eigenvalues of the transfer matrix we can determine the eigenvalues of 7b~b2(#) up to an overall factor which is fixed by studying the model built from two vertices only.
2. R-matrices for [½]+ and [b, ½] representations In this section we present the R-matrices acting on tensorproducts of three-dimensional [ ½] + and four-dimensional [ b, ½] representations of gl(2, 1) along with the corresponding Yang-Baxter equations. Before we discuss the particular form of the representations we introduce the notation [x] for the grading of an object x:
[x]
~" 0 if x is bosonic (even), L 1 if x is fermionic (odd).
(2.1)
The multiplication rule in graded tensor products differs from the ordinary one by the appearance of additional signs. For homogeneous elements B and v we get (A•B)(v®w)
=(-1)[8]b:l(Av)®(Bw).
(2.2)
Using homogeneous bases in the two vectorspaces this equation can be written in components:
(A ® B)i2,.i2il ,A = ( _ 1) ( [i2]+ [.]2[) [jl] AiJ ,.h Bi2,j2
.
(2.3)
The even part of the superalgebra gl(2, 1) consist of a the direct sum of an su(2) and a u( 1 ) Lie algebra. Thus the basis vectors of the irreducible representation can be labeled
M.P Pfannmaller, H. Frahm/Nuclear Physics B 479 [FS] (1996) 575-593
577
by the eigenvalue B of the u(1) operator, the total spin and the z-component of the spin: IB, S, Sz) [10,11]. The three-dimensional representation [ 1 ] + contains a doublet with B = ~1 and a singlet with B = 1. We arrange the basis in the following order: 13, 3' ½)'
13, ½ - ½ ) ,
I1,0,0) •
(2.4)
The first two states are considered as bosonic (grading 0), the last one as fermionic (grading 1). For the supersymmetric t - J model they are identified with the electronic states with a spin up or a spin down electron and an empty site. In this case one should choose the opposite grading [ 1,1,0]. In the four-dimensional [b, 3] representation we find a doublet with B = b and two singlets with B = b i 3, respectively. The basis is ordered according to Ib, 71 ½),
Ib, g1 , - g 1 ) ,
ib
g1 , 0 , 0 ) ,
t b + 7 ,1 0 , 0 ) .
(2.5)
Here the grading of the basis vectors is [1, 1,0,0]. For the model with correlated hopping they correspond to states with a single spin up or spin down electron, an empty site and a doubly occupied site. The bosonic ones may be exchanged leading to an equivalent model. On the tensor product of two [ ½] + representations we have an R-matrix r 33: r 3 3 ( A ) = a ( / ) i d 9 + b(,~)H33 •
(2.6)
Here id9 denotes the 9 x 9 identity matrix and H33 is the graded permutation operator .,ii..jj = ( --1"1[it][i216' • The functions a and b are given with matrix elements ( H 33)i2,j2 •J tl,J2• 6.t2,./l' by a(a) -
A h+l'
b(A) -
1 A+I
(2.7)
This R-matrix is a solution of the Yang-Baxter equation 33 rl2('~ - / z ) r ~ ( A ) r33 23 ( /x ) = r 5333 ( t x ) r 1333 ( A ) r 1 233( , ~ - # ) .
(2.8)
Here the lower indices denote the spaces in which the R-matrix acts. The R-matrix R 3b on the tensor product [ 3 ] + ® [b, ½] can be constructed from the elementary intertwiners on the irreducible components of the tensor product [12]. The result is 4/x - 2b - 3 1 , R3b(#) = Ih + 4/1. 4. 2 b ~ - 3 t2
(2.9)
1 1 ] subwhere It1 and It2 are the operators intertwining the eight-dimensional [b 4- g, representation and the four-dimensional [ b + 1, 3] subrepresentation, respectively. Their matrix representation can be found in Appendix A. The tensorproduct [bl, 3] ® [b2, ½] contains three irreducible components, namely [bl + b 2 , 1 ] ( D = 8 ) , [bl + b 2 + 1,½] (D = 4 ) and [bl 4 . b 2 - 1,½] (D = 4 ) . The R-matrix is given by the following combination of intertwiners:
578
M.P. Pfannmfiller. H. Frahm/Nuclear Physics B 479 [FS] (1996) 575-593
R&b2(l~) = 2 # -- b~ - b2+__ }Itl + 2/z - bt - b2 q-- 1 2/z - bl - b2 - l i t 2 + I t 3 ,
2 # + b~ ~ b2
2/z q- bl q- b2 - 1 2 # + bl + b2 q- 1 (2.10)
where Ih, It2 and It3 are the intertwiners for the eight and the two tbur-dimensional irreducible components, respectively. They are presented in Appendix B. A peculiar feature of the intertwiners and thus of the R-matrix R b'b2 is that they are not symmetric under the exchange of b~ and b2. This is a direct consequence of the the rule (2.2) for the graded tensor product. Further it should be noted that there are matrix elements which are proportional to b~ - b2 and thus vanish only if the representation in the two spaces are identical. These entries are the reason that R-matrix R b'b2 has a shift point (i.e. becomes proportional to the graded permutation operator for some value of the spectral paramter) only if the two representations coincide. For the R-matrices defined in Eqs. (2.9) and (2.10) the following Yang-Baxter equations hold: 3b
3b
3b
3b
33
r33(A -- /z) RI3(A) R23(/z) = R23(Iz ) R13 ( A ) r12( A - t z ) 12 3t,1 -~3b2 nblb2 r , l~blb2(l.t,) R~b2( k ) l~3bl RI2 ( A - / . £ ) / 4 1 3 ( a ) / x 2 3 ~,/./,)="23 "'12 ( a - ~ ) , Rb~b2 12 ( k _ ~,, ' ~" Ob~b3 1 3 (A)R2b~b3(/z) = Ob263 . , o b ~ b ~ , O b "'23 (/*)rt13 ~ , / t 1 'l~b' ,1 2 " ( ~ ' - - ].Z) .
(2.11) (2.12) (2.13)
Writing the Yang-Baxter equations in components one has to include additional signs due to the grading • ~il,.il R rAx.h,kl R t . ,~j2,k2g 1)[.jz]([.jll+lkll) r 1 2 ( ~ -- t.~)i2,J2 13~, 1i3,j3 23klc~)j3,k31,--I
i2,j2 ~',il,Jl r r s z ) k ,~' ( [.j2]([il]+ljt]) = R23(Iz)i3.h R13(alj3,k~ 12~A -- -- j2,k2" -- 1 )
(2.14)
From the R-matrices for the different representations we can construct monodromy matrices by taking matrix products in one component of the tensorproduct - the auxiliary or matrix space, T c
"~a,t~l,...,°'L
R ~ ,aa,
R~" ,a,,a,_,
.
R " '"~'a2RZ" ,a2.h ~ ,)E,:2([,~,l+[~il)ES=I'l,~,l(2.15)
Again the grading gives rise to additional signs. As a consequence of Eqs. (2.8) and (2.11 ) - (2.13) the monodromy matrices satisfy the following Yang-Baxter equations: 33
33
33
33
33
33
r12(a -- ~ ) T i 3 ( a ) T 2 3 ( l * ) =g~3 ( ~ ) g i 3 ( a ) rl2(A - ~,) , r,~(a
-
u)r?~(a)r33b(F*)= r ~ 3 b ( u ) r ? ~ ( , ~ ) r 3 3 ( a
- t,),
(2.16) (2.17)
R ~ ' ( A - tz)T3b2(,~ )Tb~e'(iz) = rb~b2(iz)T3~2( a ) R~b'( A -- I X ) ,
(2.18)
ab~b2(t~ __ [,g) TI3bib,.(,,~) T~3b2b3(t.L) =123,~b2b3/(/,Z)l13\,a,-blb3(l~)R}?2(/~ -- /{.g) .
(2.19)
M.P Pfannmgiller, H. Frahm/Nuclear Physics B 479 [FS] (1996) 575-593
579
From the monodromy matrix the transfer matrix is obtained by taking the supertrace in the auxiliary space: r(/z) = }-'~',~(-1)[alT(#)aa. The Yang-Baxter equations (2.16) and (2.17)-(2.19) imply that transfer matrices acting in the same quantum space commute:
[7"33(a), T33(/£)] = 0,
[~3b(a), ~.3b(~)] = o, [~.362(a), ~b,b: (;~)] = 0, [~b~b, (a), ~b2b, (;~)] = 0. These equations imply that T3b, "1"bIb and "t"b2b share a system of common eigenvectors and thus can be diagonalized simultaneously.
3. Nested algebraic Bethe ansatz for ~ b
The transfer matrix r 3b can be diagonalized directly by means of a nested algebraic Bethe ansatz. The calculations can be performed in close analogy to the NABA for the supersymmetric t-J model [ 3 ]. Thus we omit the details here and present only the most relevant steps. We can represent T 3b as a 3 × 3 matrix in the auxiliary space with entries beeing operators in the L-fold tensorproduct of four-dimensional quantum spaces { A l l ( A ) AIZ(A) B 1 ( A ) ) T3b(A) = [ A z l ( A ) A22(A) B2(A) . \ C1(,~) C2(,~) D(A)
(3.1)
From the Yang-Baxter equation (2.17) we can derive commutation relations for these quantum operators. The ones needed in the sequel are
D( A)Bi(tz) -
1
a ( t z - A)
Bi(tz)D( A) +
b(a
-
~)
a( A - tz)
Bi( A)D(tz) ,
1
x fi,ll B . iz) r(A - tz)i:,t: ~1(A)Al_,k2(~) b(a - ~)
Ai2k2 (t z) Be1(A) - a ( a -
Bi:(tz)Ailk2 (,~),
a(A - / z )
Bi: (i,) Bi~ (a) =
(3.2)
1
b ( a - ~ ) - a ( a - ~z)
r( A - t*: .'~il'll i2,12B,,1 (A) Bt2 (/z)
(3.3) (3.4)
Here r(/z) is the R-matrix of the rational six vertex model, both states being bosonic: r(/z) = a(/z)id4 + b(/z)Hz2.
(3.5)
We choose the state 10) = ® L i b - ½, O, O)
(3.6)
Ml! Pfannmiille,: H. Frahm/Nuclear
580
as reference
Physics B 479 [FS] (1996) 575-593
state. The action of the monodromy
P(/.L)10)
= i
1
0
O
l
0
0
hence it is an eigenstate make the following IA,,...,
matrix on this state is
(3.7)
of T3’(p)
= -D(p)
+ Al 1(,u) +A22(,4).
Starting from IO) we
ansatz for the Bethe vectors:
A,) = B,,, (AI ) . . B, (A,,) 10)F”Jr-.“’,
where summation
(3.8)
over repeated indices is implied and the amplitudes Pt...“’ are funcht , . . . , A,. In order to calculate the action of the transfer
tions of the spectral parameters
matrix T~~(,LL) on such a state, we use relations
(3.2)
and (3.3)
to commute
the oper-
ators D and A through all B’s until they hit the vacuum,
Here ~(‘1 is the transfer matrix of an inhomogeneous is constructed from the R-matrix (3.5).
The amplitudes
PI,...*“’ can now be identified
six vertex model with II sites that
with the components
of a vector F in the
state space of this n-site model. As can be seen from Eqs. (3.9) and (3.10) a sufficient condition for IAt, . . , A,) to be an eigenvector of r(p) is that the unwanted terms & and Ak cancel and that the vector F is an eigenvector of the nested transfer matrix 7(1)(/L). The condition that the unwanted terms & and Ak ought to cancel leads to a set of equations for the Spectral parameterS hk:
4Ak- 2b+ 5> L
F”
4Ak + 2b + 3
,,.....
I”
=
7 (1)
( hk),h:J:;~;
fT’““.-Jl
,
k= I,...,n.
(3.12)
581
M.P. Pfannmiiller, H. Frahm/Nuclear Physics B 479 [FS] (1996) 575-593
In a second step (nesting) we have to diagonalize the transfer matrix r(l~. This goal is achieved by another Bethe ansatz which gives the well kown results for the inhomogeneous, rational six vertex model. The amplitudes F a.......a, are the components of the corresponding eigenvectors. The eigenvalues are found to be n A(~) (,al & . . . . .
& lp l . . . . .
"1
1
"1
v n , ) = "[-[ ~. a ( A i - At ) ~. ] - [x a ( ~,j _ At i=1
1
+ ]--[Jf.~
j=l
"=
a(At
-
z,j)
"
(3.13) The spectral parameters ~,j are subject to the following set of Bethe equations:
f
a(,~ i
--
12.j
=
i=1
i
I-I a(u_____~Z ~'J__2 kq=j a(~'./ - ~'k) '
j = 1 .... '
(3.14)
nl .
If we insert the elgenvalues (3.13) of the nested transfer matrix into Eq. (3.12) we obtain the first set of Bethe equations: nl
(4A~-
2b+5)
-4-~ ; - - ~ +
1
L = H
.j=l
k=l ....
a ( A ~ - ~,j ) '
n.
(3.15)
"
From Eqs. (3.9) and (3.10) we can read of the eigenvalues of r3b(At) as now the eigenvalues of ~-(1)(At) are known: A3b ( A t l & . . . . .
.....
v.,
=-
4--~T2b+3 n,
+ n .j= 1 a ( u / -•
1
i:~ a(aiLAt) nl
I-t) + . =
1 a(At
n v i•)
=
1 a(Ai--
At)
•
(3.16) The eigenvalues and Bethe equations should be compared to the rational limit of the corresponding equations in Ref. [2]. We find complete agreement.
4. A second Bethe ansatz
In general the specific form of the eigenvalues and the Bethe equations depends on the particular choice of the reference state from which the Bethe vectors are built. For the transfer matrix ~.3b there exist a second possibility besides (3.6), namely the state
IO) = ®Lib+
½, 0, 0 ) .
The action of the monodromy matrix on this state is
(4.1)
M.P Pfannmfiller, H. Frahm/Nuclear Physics B 479 [FS] (1996) 575-593
582
T3b(/2) 10) =
(4/.* + 2b k.4/2 + 2b T
0
0
10).
(4tx-Zb-
(4.2)
\4/2 + 2b+3]
For the Bethe ansatz we need the following commutation relations between these operators, which are derived from the Yang-Baxter equation (2.17). D(/2)C~(A) -
1
a(/2 - 1)
Ai~k~ ( A)C~2(tx) - a ( A 1- / 2 )
Ci(A)D(tz)
b(A-/2)
+
a(a - ~)
(4.3)
Ci(tz)D(a),
kl ,/l r ( A - ix)k2,t2Ct2(/2)Ai~h(A)
b( A - ix)
a(a-
tz) Ck, (A)Ai,k2 (/z) ,
(4.4) k1,11
(4.5)
Ck, ( A ) C k 2 ( # ) = b( A - # ) -1 a ( A - tx ) r ( A - / 2 ) ~ 2 , 1 2 C l 2 ( / 2 ) B h ( A ) .
Here r(/z) is again the R-matrix (3.5) of the rational six vertex state model, with two bosonic states. Now we use the operators C~ to build the Bethe vectors starting from the new reference state:
IAl . . . . . & ) = Ca, ( & ) . . .
Ca,, ( 1 . ) 1 0 ) F ........
1
(4.6)
.
The action of ~.3t, on such a Bethe state is given by D(/z),A,,...,An)=(4-~/2-2b-3)L
=nilI a ( / 2 - A 1 i)
b~ b F a ...aIB -t-E(Ak)a,:::::a:: .... bk(~)
k=l
. . . . ,An)
fI Bb,(A.i)lO)
(4.7)
j4: k
[All ( # ) + A22(/x) ] IA1. . . . . An) 4/X + 2b + ~
L ,
1
a ( / 2 -- Ai) r
(1) . ~bl,...,b,,Fa .......al ........
"=
l ~B lb(jAO )j, /
j=l
K-"r A k),,i...',a;, xbl ... b,Fa ..... + Z_.,t ' ' al B b kr. t ~ J~ k=l
]-[
(4.8)
Bb;(A.i)[O) j4.k
Here r (l) is the transfer matrix of an inhomogeneous n-site model that is constructed from the R-matrix (3.5), xbl,...,b, dd. - d,,,d,,- l ~d3,d2r(,, t/2)al ........ = r ( / 2 - An)b;,,a,r(/2 -- A,-l)b,,_,a,,_l " . r ( / 2 -- ~A2)b2,a2 ~t~
./.( I ) r
_
"~d2,d AlJbl,al "
(4.9)
M.P. Pfannmidler; H. Frahm/Nuclear
The condition
that the unwanted
Physics B 479 [FS] (1996) 575-593
terms ought to cancel
583
leads to the following
set of
equations: 4/ik - 2b-3
4Ak + 26 - 1 >
LFPnv..#l
=
7(1)(Ak)ln)::::::‘~:F”.....,“1
,
As before, the nested transfer matrix r (I) is diagonalized eigenvalues are found to be
k
=
1,.
. . ,H.
(4.10)
by a second Bethe ansatz. The
corresponding
where V,i are solutions
of the following
set of Bethe equations:
j=
If we insert
the eigenvalues
l,...,n,
A(‘) mto Eqs.
(4.12)
(4.10)
we obtain
the first level Bethe
equations: 4Ak - 2b-
3
L
(4.13)
4hk + 2b - 1 > From Eqs. (4.7) and (4.8) we can now determine
the eigenvalues
of the transfer matrix
PQ):
n3b(plA,,.. .,Anlv1,.. .,vn,) =-
This completes between
4p-2b-3 4,u+fbf3
1 L ’ rI i=, a(~ - Ai)
the the second Bethe ansatz. We postpone
the discussion
of the relation
the two Bethe ansatze to Section 6.
5. Fusion
The tensor product of two [ i ] + representations contains a four-dimensional [b = 2, i] and five-dimensional [ 1 ] + representation. As mentioned in the introduction [b = i, 31 is the only menber of the family of four-dimensional representations [b, i] which
584
M.P. PfannmMler, H. Frahm/Nuclear Physics B 479 [FS] (1996) 575-593
can be obtained by taking tensor products of lower-dimensional irreducible representations. We construct the fused R-matrix for the "sacttering" of two composite "[ ~]+ particles" with a third "[b, ½] particle" from R,2,3(A) = R,3(A3b+ Ao)R~3b(A - Ao) •
(5.1)
3 71 ] state In general [ 23-,½] and [ 1 ] + states are mixed by the scattering. However, the [ ~, will not be destroyed if the triangularity condition holds: 3
p52RI2,3Pl~2 = 0.
(5.2)
3
Here P52 and Pl~ denote the projectors onto the five-dimensional representation [1]+ and the four-dimensional [3, ½] representation, respectively, r 33 becomes proportional to a projector onto the five-dimensional representation for the special value 1 of the spectral parameter 33 ri5 (1) ~ Pis .
(5.3)
From the Yang-Baxter equation (2.1 1) we obtain l 3b 3a Z~p3bcx I)p5 . Pi~2R~4(A + 7)R23 (• - ½) = R2b(A - 2 ' " , 3 ''~ + 7 ,2"
(5.4)
This allows us to prove the triangularity condition (5.2) for R12,3 for h0 = ½. Thus the matrix 2 ! R( 12),3 = Pi22Rlz,3P122 (5.5) 1 indeed describes the "scattering" of a [ 3, 7] particle with a [b, 1 ] particle. A YangBaxter equation holds for this R-matrix:
R(12),3(/~ -/~)R(12),4(~.)R~b(#) = R~b(#)R(12),4(/~)R(12),3(,~ - # ) .
(5.6)
The triangularity condition (5.2) states that R,2,3 becomes an upper block-triangular matrix if we change the basis in the tensor product 1 ® 2 such that the first four vectors form a basis for [ 3, ½] and the next five a basis for [ 1 ] +. Let B be the corresponding transformation matrix. Then we have
B-~R12,3B =
,)
R5 b
.
(5.7)
We can replicate this R-matrix to a chain of L sites (the auxiliary space is the tensorproduct 1 ® 2!). Taking the supertrace over spaces 1 and 2 leads to the following relation between the transfer matrices:
T3b( A _}_ 1) T3b(,~ _ 1) = T~b( a) _]_T5b(/~) .
(5.8)
The eigenvalues are thus related by
A3b(A + 1) A3b(A _ 1) = A3b(A) + A5b(A).
(5.9)
M . P Pfannmffller, H. F r a h m / N u c l e a r
585
Physics B 4 7 9 [FS] (1996) 5 7 5 - 5 9 3
The reference state 10) (3.6) is an eigenvector of each diagonal part of the monodromy matrices T 3b and T sb on its own. The eigenvalues (vacuum eigenvalues) corresponding to T} b are
T(1 (IX)I0> =T2=2 (IX)I0> =
2IX+ b +
10>,
~b
T~ (ix)[o) = 11o), Y4~/(IX) I0) = For
+b+~ 5 2ix+~+
(5.10)
Io).
find
T 54 w e
T~(IX) Io) = T ~ (IX)Io) = ~ ( i x ) I o ) - 11o),
--
+b+
(5.11)
10).
As usual the eigenvalues of the transfer matrices r~ b and r 5b corresponding to an arbitrary Bethe vector (3.8) can be written as a sum of these vacuum eigenvalues dressed by factors depending on the spectral parameters of the Bethe state. The 1.h.s. of Eq. (5.9) can be evaluated with the help of Eq. (3.16): A 3b (tx - 1) A3b (IX + ½)
( 2IX
-
2IX +
+
b +
b
+
_
3n
tx -
~ti -
2
IX -
IX
/~i
1 j=l Ix -
1
PJ
2
/~J "~ 2
Ix - as - 5 _ ~ - vJ + ~ 1 l i=l # A i + g j=l # - v j + g 2/x - b + 3 2 # - b + g
+
7b+½ 2ix-b+
7b+~ +
2ix~b+
+1
Ix --
+1-I
n
Ix -
i=l Ix
2
i=, I X - A i + ~
IX- a i - ~
5
1
_ Ix-aiT~
1
Ix-v,-
,=, I x - v , - ~
~ 1
Ix-vi+~
7,72'-
{hi_ IX--
~
~
Ix-&-~
,=,ix
#-Ai 5
~t i -
Ix "i 3
_~
/~i -- ~1
hi -
1 ~I
~
n.j=l
IX -
, ~ i 7 T j=l ]£
1
IX - - P j -- ~ +
' a
Ix
Pj 7
vj -
~ IX -
vj +
P.J
½ ]£
12.j -~- 1
3
~1
IX - - A i -- ~ a i -~- ~1 .
i=1 Ix
3}
IX -- P j + "~ IX
P.j
3 21
(5.12)
-~
'
M.P Pfannmgiller,H. Frahm/NuclearPhysicsB 479 [FS] (1996)575-593
586
Comparing to Eqs. (5.10) and (5.1 1) the first two terms are identified as contributions to A3b(/z) and the third one as part of ksb(/,). The knowledge of the vacuum eigenvalues alone is not sufficient to separate the third part of Eq. (5.12) into contributions to A~b(,~) and A-%(/l) because the vacuum eigenvalue 1 is found in Eq. (5.10) as well as in (5.11). As second argument we use the analyticity of the eigenvalues [13]: The Bethe equations (3.14) and (3.15) are known to guarantee that the residues of the eigenvalues at the poles of (5.12) vanish. The parts that have been identified above as contributions to A3b(/z) have poles at l /-£ ---- /~i -- ~I and ~ = ~'.i + ½. The Bethe equations (3.14) and (3.15) /,.I, = /~i -[- ~, lead to vanishing residues at the last two of these. In order to have a zero residue at =- /~i "~ 1 as well the contribution corresponding to the vacuum eigenvalue 1 has to be n
--
I~i=l ( # - ai 3)/(# be expressed as
_ ,~i
3b
__
½). Collecting the results the eigenvalues of r~ (/z) can
A~b(tz)=--(2~--b+3)L{
"iHl
+b+½
i=I /.Z
}
+
,
ki 7 ~ .j=l /'£
2 # - b + 3 2/z +b+
3I- I
_ /*-ai
-
~2/z+b+
v.i+ 71
I'L -- l~i
23 --~
+1
P.J +
b q -7
1
'
½,=l/z
L
~
n
.= /-£
1
/~i -~- ~
i~g _ ,~i _
i=1 /--/,
/~i
~ 1
(5.13) As was pointed out in the introduction b = 23-is the only member of the whole family of four-dimensional representations which can be obtained as a component of the tensor product of lower-dimensional representations. Thus r~ b2 ( # ) is the only representative of transfer matrices rb'b2(Iz) whoose eigenvalues can be calculated by means of the fusion procedure. This situation is of course not satisfactory as one wishes to diagonalize the transfer matrix for an arbitrary [bl, ½] representation in the auxilliary space. In particular, we cannot handle the situation of [b, ½] ® [b, ½] with arbitrary b which gives rise to models of correlated electrons with an additional free parameter. To extend our results to the complete family of four-dimensional representations in the auxilliary space, we again make use of the analytic properties of the eigenvalues and Bethe ansatz equations [13]. From the fact that 7"~b2(/z) and rblb2(a) commute and thus share a system of common eigenvectors, it is clear that the Bethe ansatz equations (3.14) and (3.15) must be preserved. The eigenvalue of T ~b2 on the pseudo vacuum (3.6) is known to be
A&b2t_.~ 0
'~, =
2 ( 2tz + bl - b2 ) L 2/z+bl+ff2
( 2t,+b,-l,2
~ 1
2t,+I,,-b2+2
+ \2/7-7- bl 7-b2 - Z 1 2tz + b, 7 b 2 + l J
L
+ 1.
(5.14)
M.P PfannmiiUer,H. Frahrn/NuclearPhysics B 479 [FS] (1996) 575-593
587
In order to generalize Eq. (5.13) we replace the vacuum eigenvalues of the diagonal elements of T 362 (5.10) by those of T b'b2 given in Eq. (5.14). The next step is to determine the necessary changes in the "dressing factors". The structure of the eigenvalues must be the same as in Eq. (5.13). Therefore we make the ansatz
Ab'b2(#) =
_(
\ 2 # + bl + b2 - 1
)L
X{ niH._l#-- l~i'~-Oll f i ]d'- pj ~-~1 _
+
(
A i + /31 j=l # -
#
I-~
# - - Ai-'~O/l
/'~i -~ 81 + i=1 #
2# +_b, 2# + bl + b2 - 1 2 # + bl ~ b2 +
# -- P._j/+ Y2 "[
"~i -'1-/32
# - vj +8!
J
# - - Ai-{- ~ 1 i=1 # - / I i +/32
l/
(5.15) i:1 ~ - - - a i +/31 This expression has poles at # = Ai -/31, # = Ai -/32 and # = vj the first two of these poles vanish if
--
81. The residues at
nl
=H#-VJ 2 # + bl + b2 - 1 (2_~_~-+-bl-b2+~)
+ bl 7 b2 +
/x=ai-,Si
L
j=l /..L
+& /"j -~ ")/I
nl
= H ~ -- pj -~- ')/2 /"J -'~ 81 /x=ai-,82 j=l # ]z=~.i --/~2
(5.16)
Now comparing the l.h.s, of these equations to the 1.h.s. of the first Bethe equation (3.15) we can determine 131 and/32: bt /31- 2
5 4'
bl /32- 2
1 4
(5.17)
Inserting these values for/31 and/32 into the RHS of Eqs. (5.16) and comparing to the RHS of the Bethe equation (3.15) leads to bl Yl - 2
5 4'
bl 3 ")/2 = ~- + ~ ,
bl 1 81 = ~- - ~ .
(5.18)
The parameter oq is the only one that remains unkown. Since it appears in an overall factor only it cannot be determined using the above arguments. We use the first nontrivial solution of the BAeq's for a two-site model, namely n = 1, nl = 0, AI = - 1 and determine the corresponding eigenvalue by operating with the transfer matrix on the corresponding Bethe vector. Evaluating the result at # = (b2 - b l ) / 2 isolates the last term. This leads to the final result for the eigenvalues of rb'b2:
(2#+bi-b2)L{ Ab'b2(#) = -- \ 2 ~ 7 b l +if2 7 1
"in l 2# - 2ai - bl - 3 f i 2# - 2vj + bt _ 21,
2 A i 7 bl
~ j=l 2/1,
2vj
-{-
bl
2
51 2
588
M.P. Pfannmfiller, H. Frahm/Nuclear Physics B 479 [FS] (1996) 575-593
2# - 2,~i - bj - ~ 2# - 2uj + b14-_ 5 + i:l 2/z 2~i ~-bl 2p.j 4- b! 1 2I j=l 2# (
2/z+b~-b2
l
+ Hn 2 # -
2,~i - bl -
i:l 2 #
2,h.i'~
c 1LI 2/z
2#+bl-b24-_2~
+ \2 -T bl
Tb + l J
bl
32
i=l 2/Z -- 2Ai 4- bl
2
-
2,~ i -
1
3
(5.19) ~'
It should be noted that bl and b2 enter this expression in an asymmetric manner. This asymmetry cannot be removed by a shift in the spectral parameter. It has its origin in the non-commutativity of the graded tensor product which was already mentioned in Section 2. We have checked our results by completely diagonalizing the transfer matrix for L = 2 and arbitrary values of bl, b2 and /z. For bl = b2 the equations are equivalent to the rational limit of the conjectures in Ref. [2]. Taking the logarithmic derivative of the eigenvalues we find the energies that where calculated in Ref. [9] by means of coordinate Bethe ansatz. We can proceed with results from the second Bethe ansatz in a similar manner. To keep the presentation short we will only give the corresponding results for the more general case of inhomogeneous chains in Section 6.
6. Inhomogeneous chains A straightforward generalisation is now to construct an inhomogeneous model. We build the monodromy matrix from R bb' matrices with a representation [b, ½] in the auxiliary space and [bi, ½] in the quantum space at site i. This model is also integrable by construction because of the Yang-Baxter equation (2.13). Modifying the vacuum expectation values, the eigenvalues and Bethe ansatz equations can be derived from the ones obtained in the previous section. It is convenient to rescale and shift the spectral parameters according to ,~k ~ -i,~k - 1, u i - + -i~,j + ½: L
AJ'{I'~} (tz) = - H k=l n
2/,t+b~l
_ 2#+2~/+b---
½ j=l 2 / z + 2 i ~ ' j + b + i 2
2# + 2iAi - b + ½ I-I 21z + 2i~') + b + ~ } b + ~3 .i=~ 2tz + 2i~,j + b + 7
+II 2# + 2iAi ~ i=1 L
(
_2/z+ b ~_b5
e / z + b - bk +
+II \2a+b+bk--12 k=l 17
+II i=l
2tz+2iAi-b+
7
+bTb,+ 1
2/z + 2iAi - b + ~ 2 # + 2iAi -+ b 2I
(6.1) "
589
M.P. Pfannmiiller, H. Frahm/Nuclear Physics B 479 [FS] (1996) 575-593
The Bethe equations are
~_i a t _ i ( 1 ~ b k - ~ 1) = ~ A l - - v j + T ~. t k=l Al+i(½bk ¼) Al--vi--½i' r I "~t--l"iq-½i-1=1 /~l
--
½i
/~i
"
n~IPJ--Piq-i,-.j=l pj
l = l . . . . . n,
(6.2)
i = 1. . . . • nl.
(6.3)
Pi
In a similar fashion we can proceed with the results from the second Bethe ansatz. Finally we obtain
L ( 21,z-b+bk 2 / z - b - b k + l ) Ab{b'}(Iz)=--l--[ 2 / ~ T - b ~ k : 1 2 / z + b + b k + l k=l
21,z + 2iAi --k b +__½ n 2t.z + 2i~,j - b - g3 2tz+2iAs b 7 j=~ 2# + 2it,j b+ 1
× _
+ H" i=1
2 # + 2i,~i + b + ~l ~ - i 2 # + 2 i v i - b + ~ 2/z + 2iAi
- - b --~ -~ 3 j=l
L (2#-b+bk
2tz-b+bk+2)~-[2tz+2i,~i+b+½
+I'I
2/~-b~-~-k----12/zTb+bk+
+ I-[
2# - b - b~ +__l 2tz - b - bk Tb~bk 1 2#~b-+b~T
k=l
k=l
}
2# + 2i~'j - - - - ~ J
i=1
2/z+2iA/
b + 53
i)fx
21z + 2ihi +__b +__5 ' i=1 2 / z + 2 i a i b 1. 2 (6.4)
The Bethe equations are
at-i(½bk+¼) k=l a t ~ i ( l b k + ¼ ) = I-I
,
at-Pi--2t /=1 A,-vj+½i ..... 17
,n,
(6.5)
I
'~j - "-~ +- i t=l ,~, At - ,,i Pi +_ ½, ½ i - _ 1-I j=l pj Pi - i '
i= 1..... nl.
(6.6)
This second set of Bethe equations can be obtained from the first set (6.2), (6.3) by replacing bk by -bk. If in addition b is replaced by - b the eigenvalues (6.1) and (6.4) are found to be equal up to an overall factor. This factor can be removed by a suitable normalisation of the R-matrix (2.10). This behaviour is a consequence of the existence of an automorphism of the superalgebra gl(2, 1 ) which maps the u( 1 ) operator B onto --B.
590
M.P Pfimnmiiller, 1f. Frahm/Nuclear Physics B 479 [FS] (1996) 575-593
7. Discussion In this paper we have computed the spectrum of vertex models invariant under the action of the superalgebra gl(2, 1) by means of the Bethe ansatz. Depending on the choice of the reference state in the four-dimensional quantum space of the local vertices two different Bethe ansiitze are possible which are related to the automorphism of the superalgebra which maps the corresponding states onto each other. The solutions start from one of the bosonic highest weigth states. This is different from the situation in the three state model corresponding to the supersymmetric t - J model for which three Bethe Ans~itze corresponding to the various possibilities of ordering of the basis (2.4)--namely FFB, FBF and BFF--can be constructed [3,4]. The fact that there exists a family of four-dimensional representations for this superalgebra allows to introduce a new type of inhomogenous four-state vertex models by allowing the parameter b to take different values in different quantum spaces. Studying the Hamiltonian limit of this class of inhomogenous vertex models leads to systems of electrons with correlated hopping with a spatially varying parameter. It should be noted however, that the R-matrix Rb'b2(#) becomes proportional to a (graded) permutation operator for some value o f / z only" if bj = b2 = b. The existence of such a shift point is necessary for the construction of a local Hamiltonian from the transfer matrix. Thus to limit the range of interaction one should consider a model with a sufficient number of sites carrying the same representation as the auxiliary space of the monodromy matrix. A possible example is a single b' "impurity" in a chain built from R bb otherwise. We shall study the effect of such an impurity on the thermodynamic properties of an correlated electronic system in a forthcoming paper.
Note added After completion of this work we received a preprint by EB. Ramos and M.J. Martins [14] who obtain the spectrum (5.19) of the transfer matrix rbh(#) by applying the algebraic Bethe ansatz to the 4 x 4 monodromy matrix Tb~(/x) directly. Their results concide with ours; it should be noted though that their discussion of the unwanted terms arising in this procedure is not complete.
Acknowledgments This work has been supported in part by the Deutsche Forschungsgemeinschaft under grant Fr 737/2-1.
M.P Pfannm&'ller, H. Frahm/Nuclear Physics B 479 [FS] (1996) 575-593
591
Appendix A. Intertwiners for the tensor product -z~[:ll+® [b, =11 2J In this appendix we explicitly present the elementary intertwiners for the tensor product [ 1 ] + ® [ b, 71 ] = [ b + 3' 1 1 ] ® [ b + 1, ½]. For the details on the construction of R-matrices from elementary intertwiners we refer to the literature [ 12]. The basis in the tensor product is chosen to be the tensor product basis constructed from (2.4) and (2.5). In the following we give only the diagonal matrix elements and the non-zero off-diagonal entries. Ih corresponds to the [b + ½, 1 ] subrepresentation (D = 8 ) , It2 to [ b + 1,½] ( D = 4 ) .
1
2b+ 1 2b+3
2b 2b+3
r_
2b+l 2b+3 2 2b+3
r+ 2b+l 2b+3
Itl =
--r_
1
1
(A.1) 2b+ 1 2b+3
r+ 2 2b+3
r+ r+ r_
--r_
2 2b+3
4
2b+3
,
where the elements r_ and r+ are given by v'~-2 2b+3 m
r_-
'
r+-
v/~+2 2b+3
For the second intertwiner It2 we have the simple representation It2 = id12 - Itt.
(A.2)
Appendix B. Intertwiners for the tensor product [bl, ½] ® [b2, 1] Here we give the matrix representations of the elementary intertwiners for the tensorproduct [bl, ½] ® [b2, ½] = [bl + b2, 1] ® [b~ + b2 + 1, ½] @ [bl + b2 - 1, ½]. The basis in the tensor product is chosen to be the tensor product basis constructed from (2.5). As before only the diagonal matrix elements and the non-zero off-diagonal entries are written down. It is usefull to define the quantities fit = 2bl + 1,
~ = 2bl - 1,
(B.1)
2bz + 1,
~2 = 2b2 - 1.
(B.2)
/~2 =
The intertwiner for the [ bl + b2 4- 1,½ ] representation is given by
592
M.P.. Pfannmiiller, H. Frahm/Nuclear Physics B 479 [FS] (1996) 575-593
AI,I
AI,2
0
AI,3
Bl,1
ALl
A1,4 ) B1,2
A2,2
A2,3
0
A2,4
0 Bl,1
It2 =
Bi,2 0
A3,1 B2,1 A4,1
0 0
A3,2
A3,3
A3,4 B2,2
B2,1
A4,2
(B.3)
A4,3
B2,2
A4,4 1
The non-vanishing entries are
A=
(/~1 + ~2)(~1 q- ~2)
~
--V~l~l
~'
/~. -
and
B -
(BI + /32)
For the [bl +
b2 - 1, ½] subrepresentation we have the intertwiner
0
C1,2
Cl,1
D2,1
Dj,1
C2,1
It3 =
C2,2
0
D2,2 D2A
C3,1
C3,2
C1,4
Ca,3
C2,4
O1,2
Dl,1
D2,1
C1,3
(B.4)
D2,2
0
C3,4
C3,3 0
C4,1
C4,2
C4,3
0
C4,4
M.P Pfannmfiller, H. Frahm/Nuclear Physics B 479 [FS] (1996) 575-593
593
where the non-vanishing entries are given by
( 8 1 ÷ 1~2)(~1 ÷ ~ 2 )
(/31 + & ) Finally the intertwiner Itl for the [bl + b2, 1 ] subrepresentation can be represented as It1 = id16 - It2 - It3.
(B.5)
References [ 1] [2] [31 [41 [51 [61 [71 [81 [91 [101 [ 111 1121 1131 1141
A.J. Bracken, M.D. Gould, J.R. Links and Y.-Z. Zhang, Phys. Rev. Lett. 74 (1995) 2768. Z. Maassarani, J. Phys. A 28 (1995) 1305. EH.L. Essler and V.E. Korepin, Phys. Rev. B 46 (1992) 9147. A. Foerster and M. Karowski, Nucl. Phys. B 396 (1993) 611. C.K. Lai, J. Math. Phys. 15 (1974) 1675. B. Sutherland, Phys. Rev. B 12 (1975) 3795. P. Schlottmann, Phys. Rev. B 36 (1987) 5177. I.N. Karnaukhov, Phys. Rev. Lett. 73 (1994) 1130. G. BediJrftig and H. Frahm, J. Phys. A 28 (1995) 4453. M. Scheunert, W. Nahm and V. Rittenberg, J. Math. Phys. 18 (1977) 155. M. Marcu, J. Math. Phys. 21 (1980) 1277. G.W. Delius, M.D. Gould, J.R. Links and Y.-Z. Zhang, Int. J. Mod. Phys. A 10 (1995) 3259. N.Yu. Reshetikhin, Lett. Math. Phys. 7 (1983) 205. P.B. Ramos and M.J. Martins, One parameter family of an integrable spl(2] 1) vertex model: Algebraic Bethe ansatz and ground state structure, preprint UFSCARF-TH-96-03, (1996).