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A TWO PARAMETER DEFORMATION OF THE SU(1/1) SUPERALGEBRA

arXiv:hep-th/9110074v1 30 Oct 1991CERN-TH.6299/91A TWO PARAMETER DEFORMATION OF THE SU(1/1) SUPERALGEBRAAND THE XY QUANTUM CHAIN IN A MAGNETIC FIELDHaye HinrichsenUniversit¨at Bonn, Physikalisches InstitutNussallee 12, W-5300 Bonn 1, FRGVladimir Rittenberg∗Theory Division, CERNCH-1211 Geneva 23, SwitzerlandAbstractWe show that the XY quantum chain in a magnetic field is invariant under a two parameterdeformation of the SU(1/1) superalgebra. One is led to an extension of the braid group and theHecke algebra which reduce to the known ones when the two parameter coincide.

The physicalsignificance of the two parameters is discussed.CERN-TH.6299/91October 1991∗Permanent address: Universit¨at Bonn, Physikalisches Institut, Nussallee 12, W-5300 Bonn 1, FRG

There were several attempts to extend the one-parameter quantum algebras to multiparam-eter ones [1]. As shown however by Reshetikhin [2] the link polynomials depend only on oneparameter.

One can state this result in a different way: if one has a one-dimensional quan-tum chain which is invariant under a multiparameter quantum algebra, one can do a similaritytransformation which eliminates all the parameters but one. As will be shown in this paper,the situation is different in the case of quantum superalgebras.

We will start with a physicalexample. Consider the quantum chainH = ΩqLXi=1σzi + Ωη2L−1Xi=1[(1 + u)σxi σxi+1 + (1 −u)σyi σyi+1] + B + S,(1)where σx, σy and σz are Pauli matrices inserted in the i-th position of the Kronecker productσki = 1 ⊗1 ⊗.

. .

⊗σk|{z}i⊗. .

. ⊗1 ⊗1(i = 1, 2, .

. .

L)(2)[σki , σlj] = 0. (i ̸= j)Ωq, Ωη and u are parameters, B and S are boundary and surface terms respectively.

This chainappears in the domain wall theory of two-dimensional commensurate-incommensurate phasetransitions [3] and in Glauber’s kinetic Ising model [4]. We will first make an important changeof notations, choose B = 0 (no periodic boundary conditions!) and fix S byΩq = q + q−12,Ωη = η + η−12,u = η −η−1η + η−1(3)S= 12 (−q−1σz1 −q σzL).With this change of notations we haveH = H(q, η) =L−1Xi=1Hi(q, η)(4)Hi(q, η) = 12 [η σxi σxi+1 + η−1σyi σyi+1 + q σzi + q−1σzi+1].A detailed discussion of the properties of the chain given by eq.

(4) will be given elsewhere [5],here we are going to mention only a few. First, there are the symmetry propertiesH(q, η) .= H(q−1, η) .= H(q, η−1).= H(η, q).

(5)The ”equality” among the Hamiltonians implies that the spectra are identical. The first twoequalities are obvious but not the last one which reminds of duality transformations of quantumchains [6].

In the continuum limit, one has the following phase structure [3, 5]:

Ωq ≤1,Ωη ≤1:massless-incommensurateΩq ≤1,Ωη > 1 or Ωq > 1,Ωη ≤1:massive incommensurateΩq > 1,Ωη > 1,Ωq ̸= Ωη:massiveΩq = Ωη,(Ωq > 1) :critical Ising typeΩq = Ωη = 1 :Pokrovsky-Talapov phase transitionIt is by now clear that the properties of the chain depend on both parameters q and η.We now perform a Jordan-Wigner transformation. First write σzj = −iσxj σyj and next defineτ x,yj= exp(iπ2j−1Xk=1(σzk + 1)) σx,yj(6){τ xi , τ xj } = {τ yi , τ yj } = {τ xi , τ yj } = 0.

(i ̸= j)Using Eq. (4) and (6) we getHj = 12[ ητ xj τ xj+1 + η−1τ yj τ yj+1 −iqτ xj τ yj −iq−1τ xj+1τ yj+1].

(7)We now observe the following important identity[T X, H(q, η)] = [T Y , H(q, η)] = 0(8)withT X = ∆(τ x) =LXj=1α1−L2LXj=1αj−1τ xjT Y = ∆(τ y) =LXj=1β1−L2LXj=1βj−1τ yj(9){T X, T X} = 2[L]α,{T Y , T Y } = 2[L]β,{T X, T Y } = 0,(10)where L is the length of the chain andα = −qη,β = −qη,[L]λ = λL −λ−Lλ −λ−1 . (11)The equalities (8) come from the existence of a fermionic zero mode for a q and η. Theequations (10) togehter with the coproduct (9) give a representation of a Hopf algebra.

Beforewe proove this statement let us consider the case α = β = −q. We first notice that in this caseSz = 12PLi=1 σzi also commutes with H(q, η) We now remind the reader the Uα[SU(1/1)] algebra[7].

With A± = 12(T X ± iT Y ) we have

{A±, A±} = 0,{A+, A−} = [E]α,[Sz, A±] = ±A±(12)[E, Sz] = [E, A±] = 0with the coproduct∆(α, A±) = αE/2 ⊗A± + A± ⊗α−E/2∆(α, Sz)= Sz ⊗1 + 1 ⊗Sz(13)∆(α, E)= E ⊗1 + 1 ⊗E.The fermionic representations correspond to take E = 1,Sz =12σz,A± = a± and{a+, a−} = 1 in Eq (13).In this representation E in Eq. (12) is equal to L (the numberof sites).

Comparing now (12), (13) with Eqs. (9,10) we observe [8] that the quantum chain(4) with η = 1 is invariant under Uα[SU(1/1)] transformations.

It was also shown by Saleur [8]that the quantities Uj = ∆q −Hj(q, 1) are the generators of the Heck algebraU2j = 2 ∆q UjUjUj±1Uj −Uj = Uj±1UjUj±1 −Uj±1(14)UiUi±j = Ui±jUi. (j ̸= 1)Actually they correspond to a quotient of this algebra since the generators satisfy also therelations [9]UjUj+2Uj+1(2Ωq −Uj)(2Ωq −Uj+2) = 0.

(15)The generators ˇRj = q−q−12+ Hj(q, 1) satisfy the braiding relationsˇRj ˇRj±1 ˇRj = ˇRj±1 ˇRj ˇRj±1(16)withˇR2j = (q −q−1) ˇRj + 1. (17)Considering the matrices Rj = P ˇRj (P is the graded permutation operator) we have(see Eq.

(13))R∆(α)R−1 = ∆(α−1). (18)We now consider the case η ̸= 1.

As suggested by Eqs. (9,10) we define the two parameterdeformation of the SU(1/1) algebra as follows:{T X, T X} = 2 [E]α,{T Y , T Y } = 2 [E]β(19){T X, T Y } = 0[E, T X] = [E, T Y ] = 0

with the coproduct∆(α, β; T X) = αE/2 ⊗T X + T X ⊗α−E/2∆(α, β; T Y ) = βE/2 ⊗T Y + T Y ⊗β−E/2(20)∆(α, β; E) = E ⊗1 + 1 ⊗E.Notice that Sz does not appear in the algebra anymore. We denote this quantum algebraby Uα,β[SU(1/1)].

It is a Hopf algebra for the sam reasons as the Uα[SU(1/1)]. If we takethe fermionic representations E = 1, τ x = (a+ + a−),τ y = −i(a+ −a−), from Eq.

(20) wederive Eqs. (9,10).

The quantum chain H(q, η) is thus invariant under the quantum algebraUα,β[SU(1/1)]. We would like to see what replaces the relations (14-18) when we have twoparameters.

We first notice a remarkable identity satisfied by the Hj(q, η)[HjHj±1Hj −Hj±1HjHj±1 + (ν −1)(Hj −Hj±1)] (Hj −Hj±1) = µ(21)H2j = ν,whereν = α + α−12! β + β−12!= q + q−12!2+ η + η−12!2−1(22)µ = α + α−12−β + β−12!2= 4 q −q−12!2 η −η−12!2.We can now define a generalised Hecke algebra takingUi = √ν −Hi(p, q)(UiUi±1Ui −Ui±1UiUi±1 −Ui + Ui±1) (Ui −Ui±1) = µ(23)U2i= 2 √ν Ui.Notice that when η = 1, µ = 0 and we are back to the original Hecke algeb We did nothave the patience to find the equivalent of Eq.

(15) which gives the quotient of the generalisedHecke algebra (23) corresponding to the chain given by Eq. (4).

Another quotient is howeversuggested by the structure of Eq. (23):(UiUi±1Ui −Ui) (Ui −Ui±1) = µ2 .

(24)For µ = 0 one gets in this case the Temperley-Lieb algebra. We now turn our attention tothe generalised braid group algebra.

We take ˇRi = Hi(q, η) + √ν −1 and get( ˇRi ˇRi±1 ˇRi −ˇRi±1 ˇRi ˇRi±1) ( ˇRi −ˇRi±1) = µ(25)

withˇR2i = 1 +√ν −1 ˇRi . (26)In the basis where the σzi are diagonal (see Eq.

(4)) we haveˇRi =√ν −1 + q+q−1200η−η−120√ν −1 −q−q−12η+η−1200η+η−12√ν −1 + q+q−120η−η−1200√ν −1 −q+q−12. (27)We take the graded permutation matrix PP =10110−1(28)and define the matrix Ri = P ˇRi.

We now write the coproduct (20) in the original language ofPauli matrices∆(α, β; T X) = α−1/2(σy ⊗1) −α1/2(σz ⊗σy)∆(α, β; T Y ) = β−1/2(σx ⊗1) −β1/2(σz ⊗σx)(29)∆(α, β; E) = 1 ⊗1 + 1 ⊗1.It is trivial to check that similar to Eq. (18) we getR ∆(α, β) R−1 = ∆(α−1, β−1).

(30)Before concluding we would like to show that for Uα,β[SU(1/1)] one can introduce more thantwo parameters (as in the Lie algebra case when we had more than one). The most generalchain which has a zero mode for all its values of the parameters is [5]Hi=12 {Θ + Θ−12(ηζσxi σxi+1 + η−1ζ−1σyi σyi+1)+Θ −Θ−12(ηζ−1σxi σyi+1 + η−1ζσyi σxi+1)(31)+q σzi + q−1σzi+1) }.Hi depends on four parameters.

One can check however that Eq. (21) holds withν=q2 + q−24+ (η2 −η−2)(ζ2 −ζ−2)8+ (η2 + η−2)(ζ2 + ζ−2)(Θ2 + Θ−2)16

µ= (η2 −η−2)(ζ2 −ζ−2)4+ (η2 + η−2)(ζ2 + ζ−2)(Θ2 + Θ−2)8−1! (32)×(q −q−1)22+(Θ −Θ−1)2( ηζ −ζη)28which means that we are back to two parameters.To sum up we have shown that the two-parameter deformation of the SU(1/1) superalgebraleads to a new Hopf algebra.

One obtains then a generalisation of the Hecke algebra and of thebraid group (see Eqs. (23) and (25 keeping the structure of connection between the R matricesand the coproduct (see Eq.

(30)).Acknowledgements: We would like to thank Franz Gaehlen for writing the computer programwhich made us discover Eq. (21).We would also like to thank D. Altschuler, L. Alvarez-Gaum´e, D. Arnaudon, M. Chaichian, R. Coqueraux, C. Cinkovic and M. Scheunert for usefulldiscussions.References[1] E. E. Demidov, Yu.

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