Skip to main content

Fermions Acting on Quasi-local Operators in the XXZ Model

  • Conference paper
Symmetries, Integrable Systems and Representations

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 40))

Abstract

This is a survey about the construction of fermions which act on the space of quasi-local operators in the XXZ model. We also include a proof of the anti-commutativity of fermionic creation operators.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. Boos, H., Jimbo, M., Miwa, T., Smirnov, F., Takeyama, Y.: Hidden Grassmann structure in the XXZ model II: creation operators. Commun. Math. Phys. 286, 875–932 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  2. Jimbo, M., Miwa, T., Smirnov, F.: Hidden Grassmann structure in the XXZ model III: introducing Matsubara direction. J. Phys. A 42, 304018 (2009)

    Article  MathSciNet  Google Scholar 

  3. Jimbo, M., Miwa, T., Miki, K., Nakayashiki, A.: Correlation functions of the XXZ model for Δ<−1. Phys. Lett. A 168, 256–263 (1992)

    Article  MathSciNet  Google Scholar 

  4. Jimbo, M., Miwa, T.: Quantum Knizhnik-Zamolodchikov equation at |q|=1 and correlation functions of the XXZ model in the gapless regime. J. Phys. A 29, 2923–2958 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  5. Kitanine, N., Maillet, J.-M., Terras, V.: Correlation functions of the XXZ Heisenberg spin-\(\frac{1}{2}\)-chain in a magnetic field. Nucl. Phys. B 567, 554–582 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  6. Göhmann, F., Klümper, A., Seel, A.: Integral representations for correlation functions of the XXZ chain at finite temperature. J. Phys. A 37, 7625–7651 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  7. Boos, H., Korepin, V.: Quantum spin chains and Riemann zeta functions with odd arguments. J. Phys. A 34, 5311–5316 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  8. Boos, H., Korepin, V., Smirnov, F.: Emptiness formation probability and quantum Knizhnik-Zamolodchikov equation. Nucl. Phys. B 658, 417–439 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  9. Kato, G., Shiroishi, M., Takahashi, M., Sakai, K.: Next nearest-neighbor correlation functions of the spin-1/2 XXZ chain at critical region. J. Phys. A 36, L337–L344 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  10. Sakai, K., Shiroishi, M., Nishiyama, Y., Takahashi, M.: Third-neighbor and other four-point correlation functions of spin-1/2 XXZ chain. J. Phys. A 37, 5097–5123 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  11. Boos, H., Jimbo, M., Miwa, T., Smirnov, F.: Completeness of a fermionic basis in the homogeneous XXZ model. J. Math. Phys. 50, 095206 (2009) (online)

    Article  MathSciNet  Google Scholar 

  12. Bazhanov, V., Lukyanov, S., Zamolodchikov, A.: Integrable structure of conformal field theory III: the Yang-Baxter relation. Commun. Math. Phys. 200, 297–324 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  13. Hernandez, D., Jimbo, M.: Asymptotic representations and Drinfeld rational fractions. Compos. Math. 148, 1593–1623 (2012)

    Article  MATH  Google Scholar 

  14. Slavnov, N.: Calculation of scalar products of wave functions and form-factors in the framework of the algebraic Bethe ansatz. Theor. Math. Phys. 79, 502–508 (1989)

    Article  MathSciNet  Google Scholar 

  15. Boos, H., Jimbo, M., Miwa, T., Smirnov, F.: Hidden Grassmann structure in the XXZ model IV: CFT limit. Commun. Math. Phys. 299, 825–866 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. Jimbo, M., Miwa, T., Smirnov, F.: On one-point functions of descendants in sine-Gordon model. In: New Trends in Quantum Integrable Systems: Proceedings of the Infinite Analysis 09, pp. 117–137. World Scientific, Singapore (2010)

    Google Scholar 

  17. Jimbo, M., Miwa, T., Smirnov, F.: Hidden Grassmann structure in the XXZ model V: sine-Gordon model. Lett. Math. Phys. 96, 325–365 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. Jimbo, M., Miwa, T., Smirnov, F.: Fermionic structure in the sine-Gordon model: form factors and null vectors. Nucl. Phys. B 852, 390–440 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  19. Boos, H., Göhmann, F.: On the physical part of the factorized correlation functions of the XXZ chain. J. Phys. A 42, 1–27 (2009)

    Article  Google Scholar 

  20. Boos, H., Göhmann, F.: Properties of linear integral equations related to the six-vertex model with disorder parameter II. Preprint arXiv:1201.2625 [hep-th]

Download references

Acknowledgements

Research of M.J. is supported by the Grant-in-Aid for Scientific Research B-23340039. Research of T.M. is supported by the Grant-in-Aid for Scientific Research B-22340031. Research of F.S. is supported by RFBR-CNRS grant 09-02-93106 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.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Michio Jimbo .

Editor information

Editors and Affiliations

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 〈Φ|Ψ〉≠0. Denote their eigenvalues and those for the Q-matrices as follows:

Introduce q-difference operators Δ ζ , \(\overline{D}_{\zeta}\) by

Hereafter we shall use the shorthand

Set

(29)

and define ω sym(ζ,ξ) by

As a function of ζ, ω(ζ,ξ) is characterised by the following two conditions.

  1. 1.

    ζ α T(ζ)(ω(ζ,ξ)−ω sym(ζ,ξ)) is a polynomial in ζ 2 of degree n,

  2. 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 ‘q-primitive’ \(\varDelta ^{-1}_{\zeta}\psi(\zeta/\xi)\).

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 Riemann-Hilbert problem has been formulated.

Appendix B: Anti-commutativity of Fermionic Creation Operators

In this appendix we prove the following anti-commutation relations between the creation operators.

Theorem 3

For all p,p′≥1, we have

(30)
(31)

Since the proofs are similar, we shall concentrate on the case (31).

The next Proposition says that the anti-commutation 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 anti-commutation relations between the creation and annihilation operators.

Similarly one calculates

Using the known anti-commutativity 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 ℂ2 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

(32)

and let (ζ 1,…,ζ l )∈(ℂ×)l be arbitrary. Then

$$\varOmega_{j,k}=\frac{a(\zeta_k)\prod_{i=1}^l(q^2\xi_i^2-\zeta_k^2)}{(\xi_j^2-\zeta_k^2)(q^2\xi_j^2-\zeta_k^2)} -q^{-2\kappa+n} \frac{d(\zeta_k)\prod_{i=1}^l(\xi_i^2-q^2\zeta_k^2)}{(\xi_j^2-\zeta_k^2)(\xi_j^2-q^2\zeta_k^2)}. $$

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 ,

(33)

is a solution of (32) for (q,τ)=(q 0,τ 0). If further κ is generic, then we have \(\xi^{2}_{j}\neq\pm1\), \(\xi^{2}_{j}\neq\pm \xi^{2}_{k}\) (jk).

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} withI=♯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 non-zero. 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

(34)

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≤ln. Under the assumption above, we have 〈Φ|Ψ〉≠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=LK+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. □

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer-Verlag London

About this paper

Cite this paper

Jimbo, M., Miwa, T., Smirnov, F. (2013). Fermions Acting on Quasi-local Operators in the XXZ Model. In: Iohara, K., Morier-Genoud, S., Rémy, B. (eds) Symmetries, Integrable Systems and Representations. Springer Proceedings in Mathematics & Statistics, vol 40. Springer, London. https://doi.org/10.1007/978-1-4471-4863-0_10

Download citation

Publish with us

Policies and ethics