Abstract
This is a survey about the construction of fermions which act on the space of quasilocal operators in the XXZ model. We also include a proof of the anticommutativity of fermionic creation operators.
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Acknowledgements
Research of M.J. is supported by the GrantinAid for Scientific Research B23340039. Research of T.M. is supported by the GrantinAid for Scientific Research B22340031. Research of F.S. is supported by RFBRCNRS grant 090293106 and DIADEMS program (ANR) contract number BLAN012004. The authors would like to thank the organisers of the workshops “Infinite Analysis 11—Frontier of Integrability—” at Tokyo and “Symmetries, Integrable Systems and Representations” at Lyon, for invitation and kind hospitality. With sincere gratitude for what they got from him, T.M. and F.S. would like to express their heartiest congratulations to Professor Michio Jimbo on his 60th birthday.
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Appendices
Appendix A: Formula for ω(ζ,ξ)
We quote an explicit formula for the function ω(ζ,ξ) from [2], Sect. 7. For that purpose we need to prepare some notation.
As in the text, we fix an eigencovector 〈Φ of T _{ M }(ζ,κ+α) and an eigenvector Ψ〉 of T _{ M }(ζ,κ) satisfying 〈ΦΨ〉≠0. Denote their eigenvalues and those for the Qmatrices as follows:
Introduce qdifference operators Δ _{ ζ }, \(\overline{D}_{\zeta}\) by
Hereafter we shall use the shorthand
Set
and define ω _{sym}(ζ,ξ) by
As a function of ζ, ω(ζ,ξ) is characterised by the following two conditions.

1.
ζ ^{−α} T(ζ)(ω(ζ,ξ)−ω _{sym}(ζ,ξ)) is a polynomial in ζ ^{2} of degree n,

2.
It satisfies the normalisation conditions for m=0,1,…,n:
Here Γ _{0} is a contour around ζ ^{2}=0, and for m=1,…,n, Γ _{ m } is a contour encircling \(\zeta^{2}=\tau_{m}^{2}, q^{2}\tau_{m}^{2}\).
As it is explained in [2], Sect. 5, the integral in (ii) does not depend on a particular choice of the ‘qprimitive’ \(\varDelta ^{1}_{\zeta}\psi(\zeta/\xi)\).
To be more explicit, consider the function
Then it has the form
where \(p^{+}_{m}(\zeta^{2})\) is a polynomial in ζ ^{2} of degree 2n. Using them we introduce (n+1)×(n+1) matrices \(\mathcal{A}\), \(\mathcal{B}\) by
The formula for ω(ζ,ξ) reads
where v ^{±}(ζ) denote column vectors with entries v ^{±}(ζ)_{ j }=ζ ^{±α+2j}.
For the purpose of studying various limits, it is more convenient to use an alternative expression in terms of solutions to integral equations [19]. The relevant formula can be found in [15], (3.11) (the function ω(ζ,ξ) in the present paper is denoted ω _{rat}(ζ,ξ) there, see [15], (2.11)). In this connection one should mention the recent paper [20] where a RiemannHilbert problem has been formulated.
Appendix B: Anticommutativity of Fermionic Creation Operators
In this appendix we prove the following anticommutation relations between the creation operators.
Theorem 3
For all p,p′≥1, we have
Since the proofs are similar, we shall concentrate on the case (31).
The next Proposition says that the anticommutation relation (31) holds in the sense of expectation values.
Proposition 1
Assume (27). Then for any \(X\in \mathcal{W}_{\alpha,0}\) we have
Proof
Abbreviating \(Z^{\kappa}_{\varPhi,\varPsi}\) to Z, we apply the Ward identities for the expectation values in Theorem 1,
In the second line we used the known anticommutation relations between the creation and annihilation operators.
Similarly one calculates
Using the known anticommutativity of b(ξ _{2}) and c(ξ _{1}), we arrive at
which is equivalent to the assertion of Proposition. □
Before proceeding, we recall a few facts from the algebraic Bethe ansatz. Normalising the \(\mathcal{L}\) operator as
we set
Let \(0\rangle=v_{+}^{\otimes n}\), \(\langle 0 =(v^{*}_{+})^{\otimes n}\) be the reference vector and covector respectively, where v _{+},v _{−} is the standard basis of ℂ^{2} and \(v_{+}^{*},v_{}^{*}\) is the dual basis. Let further l∈{0,1,…,n} and set for j=1,…,l
where a(ζ), d(ζ) are defined in (29).
The following formula is well known [14].
Proposition 2
Assume that (ξ _{1},…,ξ _{ l })∈(ℂ^{×})^{l} is a solution of the Bethe equation
and let (ζ _{1},…,ζ _{ l })∈(ℂ^{×})^{l} be arbitrary. Then
We shall consider the specialisation of parameters q,τ=(τ _{1},…,τ _{ n }) to
Lemma 1
Define x _{ j }(κ) by
Then, for any subset I={i _{1},…,i _{ l }}⊂{1,…,n}, i _{1}<⋯<i _{ l },
is a solution of (32) for (q,τ)=(q _{0},τ _{0}). If further κ is generic, then we have \(\xi^{2}_{j}\neq\pm1\), \(\xi^{2}_{j}\neq\pm \xi^{2}_{k}\) (j≠k).
Proof is straightforward.
Hereafter we choose and fix a generic κ _{0}. Denote by \(\boldsymbol{\xi}_{0}^{(I)}\) the solution (33) at (κ,q,τ)=(κ _{0},q _{0},τ _{0}).
Lemma 2
We have
Proof
This follows from the calculation
where (32) is used. □
By Lemma 2 and the implicit function theorem, in a neighborhood of (κ,q,τ)=(κ _{0},q _{0},τ _{0}) there exists a unique branch \(\boldsymbol{\xi}^{(I)}(\kappa,q,\boldsymbol{\tau})=\{\xi^{2}_{1},\ldots,\xi^{2}_{l}\}\) of solutions to (32) such that \(\boldsymbol{\xi}^{(I)}(\kappa_{0},q_{0},\boldsymbol{\tau}_{0})=\boldsymbol{\xi}^{(I)}_{0}\). Denote by
the corresponding Bethe (co)vectors.
Lemma 3
In a neighborhood of (κ _{0},q _{0},τ _{0}), we have
for all I,J⊂{1,…,n} with ♯I=♯J=l.
Proof
We apply Proposition 2 at (q,τ)=(q _{0},τ _{0}). Setting ξ ^{(I)}(κ,q _{0},τ _{0})=(ξ _{1},…,ξ _{ l }) and ξ ^{(J)}(κ′,q _{0},τ _{0})=(ζ _{1},…,ζ _{ l }) we find
which is nonzero. Hence the scalar product does not vanish in some neighborhood of (κ _{0},q _{0},τ _{0}) and κ′≠κ. □
We finish the proof with the
Proposition 3
For any p,p′≥1 and \(X\in \mathcal{W}_{\alpha,0}\) we have
Proof
Denote the left hand side of (34) by Y. Take (κ,q,τ) in a neighborhood of (κ _{0},q _{0},τ _{0}) and α≠0 small enough. Choose 〈Φ=_{ I }〈κ+α,q,τ and Ψ〉=κ,q,τ〉_{ J }, where ♯I=♯J=l and 0≤l≤n. Under the assumption above, we have 〈ΦΨ〉≠0 by Lemma 3. Hence Proposition 1 is applicable, and we obtain that
Since the vectors {_{ I }〈κ+α,q,τ}, {κ,q,τ〉_{ I }} are bases of the spin n/2−l subspace, we find
If we choose n=L−K+1 and τ=τ _{0}, then \(\mathcal{T}_{[K,L],M}(1)\) becomes a permutation operator and the trace becomes simply \(q^{2\kappa S_{[K,L]}}Y\). We conclude that Y=0 provided (q,α) is close enough to (q _{0},0) and α≠0. But Y is rational in q,q ^{α}, so we must have that Y=0 identically. This completes the proof. □
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Jimbo, M., Miwa, T., Smirnov, F. (2013). Fermions Acting on Quasilocal Operators in the XXZ Model. In: Iohara, K., MorierGenoud, S., Rémy, B. (eds) Symmetries, Integrable Systems and Representations. Springer Proceedings in Mathematics & Statistics, vol 40. Springer, London. https://doi.org/10.1007/9781447148630_10
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DOI: https://doi.org/10.1007/9781447148630_10
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