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Antiperiodic XXZ Chains with Arbitrary Spins: Complete Eigenstate Construction by Functional Equations in Separation of Variables

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Abstract

Generic inhomogeneous integrable XXZ chains with arbitrary spins are studied by means of the quantum separation of variables (SOV) method. Within this framework, a complete description of the spectrum (eigenvalues and eigenstates) of the antiperiodic transfer matrix is derived in terms of discrete systems of equations involving the inhomogeneity parameters of the model. We show here that one can reformulate this discrete SOV characterization of the spectrum in terms of functional T − Q equations of Baxter’s type, hence proving the completeness of the solutions to the associated systems of Bethe-type equations. More precisely, we consider here two such reformulations. The first one is given in terms of Q-solutions, in the form of trigonometric polynomials of a given degree \({\mathsf{N}_s}\), of a one-parameter family of T − Q functional equations with an extra inhomogeneous term. The second one is given in terms of Q-solutions, again in the form of trigonometric polynomials of degree \({\mathsf{N}_s}\) but with double period, of Baxter’s usual (i.e., without extra term) T − Q functional equation. In both cases, we prove the precise equivalence of the discrete SOV characterization of the transfer matrix spectrum with the characterization following from the consideration of the particular class of Q-solutions of the functional T − Q equation: to each transfer matrix eigenvalue corresponds exactly one such Q-solution and vice versa, and this Q-solution can be used to construct the corresponding eigenstate.

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References

  1. Antonov, A., Feigin, B.: Quantum group representation and Baxter equation. Phys. Lett. B 392, 115–122 (1997). hep-th/9603105

  2. Batchelor M.T., Baxter R.J., O’Rourke M.J., Yung C.M.: Exact solution and interfacial tension of the six-vertex model with anti-periodic boundary conditions. J. Phys. A Math. Gen. 28, 2759–2770 (1995)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  3. Baxter R.J.: Partition function of the eight-vertex lattice model. Ann. Phys. 70, 193–228 (1972)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  4. Baxter R.J.: Exactly solved models in statistical mechanics. Academic Press, London (1982)

    MATH  Google Scholar 

  5. Bazhanov, V.V., Lukowski, T., Meneghelli, C., Staudacher, M.: A shortcut to the Q-operator. J. Stat. Mech. Theory Exp. P11002 (2010)

  6. Bazhanov, V.V., Lukyanov, S.L., Zamolodchikov, A.B.: Integrable structure of conformal field theory II. Q-operator and DDV equation. Commun. Math. Phys. 190, 247–278 (1997). hep-th/9604044

  7. Bazhanov, V.V., Lukyanov, S.L., Zamolodchikov, A.B.: Integrable structure of conformal field theory. III: The Yang-Baxter relation. Commun. Math. Phys. 200, 297–324 (1999). hep-th/9805008

  8. Belliard S., Crampé N.: Heisenberg XXX model with general boundaries: Eigenvectors from algebraic Bethe ansatz. SIGMA 9, 072 (2013)

    Google Scholar 

  9. Belliard, S., Pakuliak, S., Ragoucy, E., Slavnov, N. A.: Algebraic Bethe ansatz for scalar products in SU(3)-invariant integrable models. J. Stat. Mech. P10017 (2012). arXiv:1207.0956

  10. Cao J., Cui S., Yang W.-L., Shi K., Wang Y.: Spin-1/2 XYZ model revisit: general solutions via off-diagonal Bethe ansatz. Nucl. Phys. B 886, 185–201 (2014)

    Article  MathSciNet  ADS  Google Scholar 

  11. Cao J., Yang W.-L., Shi K., Wang Y.: Off-diagonal Bethe ansatz and exact solution of a topological spin ring. Phys. Rev. Lett. 111, 137201 (2013)

    Article  ADS  Google Scholar 

  12. Cao J., Yang W.-L., Shi K., Wang Y.: Off-diagonal Bethe ansatz solutions of the anisotropic spin-1/2 chains with arbitrary boundary fields. Nucl. Phys. B 877, 152–175 (2013)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  13. Cao J., Yang W.-L., Shi K., Wang Y.: Nested off-diagonal Bethe ansatz and exact solutions of the su(n) spin chain with generic integrable boundaries. JHEP 04, 143 (2014)

    Article  ADS  Google Scholar 

  14. Castro-Alvaredo O.A., Maillet J.M.: Form factors of integrable Heisenberg (higher) spin chains. J. Phys. A Math. Theor. 40, 7451–7471 (2007)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  15. Caux, J.S., Hagemans, R., Maillet, J.M.: Computation of dynamical correlation functions of Heisenberg chains: the gapless anisotropic regime. J. Stat. Mech. Theory Exp. P09003 (2005)

  16. Caux J.S., Maillet J.M.: Computation of dynamical correlation functions of Heisenberg chains in a magnetic field. Phys. Rev. Lett. 95, 077201 (2005)

    Article  ADS  Google Scholar 

  17. Cherednik I.V.: Factorizing particles on a half line and root systems. Theor. Math. Phys. 61, 977–983 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  18. Deguchi, T.: Reduction formula of form factors for the integrable spin-s XXZ chains and application to correlation functions. J. Stat. Mech. Theory Exp. P04001 (2012)

  19. Deguchi, T., Matsui, C.: On the evaluation of form factors and correlation functions for the integrable spin-s XXZ chains via the fusion method. arXiv:1103.4206

  20. Derkachov, S.E.: Baxter’s Q-operator for the homogeneous XXX spin chain. J. Phys. A Math. Gen. 32, 5299–5316 (1999). solv-int/9902015

  21. Derkachov, S.E.: Factorization of R-matrix and Baxter’s Q-operator. J. Math. Sci. 151, 2848 (2008). math/0507252

  22. Dorlas T.C.: Orthogonality and completeness of the Bethe Ansatz eigenstates of the nonlinear Schrödinger model. Comm. Math. Phys. 154, 347–376 (1993)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  23. Drinfel’d, V.G.: Quantum groups. In: Proc. Internat. Congress of Math., Berkeley, USA, 1986, pp. 798–820, AMS (1987)

  24. Drinfel’d V.G.: Quasi-Hopf algebras. Leningrad Math. J. 1(6), 57–1419 (1990)

    MathSciNet  Google Scholar 

  25. Dugave, M., Göhmann, F., Kozlowski, K.K.: Thermal form factors of the XXZ chain and the large-distance asymptotics of its temperature dependent correlation functions. J. Stat. Mech. P07010 (2013)

  26. Faldella, S., Kitanine, N., Niccoli, G.: Complete spectrum and scalar products for the open spin-1/2 XXZ quantum chains with non-diagonal boundary terms. J. Stat. Mech. P01011 (2014)

  27. Galleas W.: Functional relations from the Yang-Baxter algebra: Eigenvalues of the XXZ model with non-diagonal twisted and open boundary conditions. Nucl. Phys. B 790, 524–542 (2008)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  28. Göhmann F., Klümper A., Seel A.: Integral representations for correlation functions of the XXZ chain at finite temperature. J. Phys. A 37, 7625–7652 (2004)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  29. Grosjean, N., Maillet, J.M., Niccoli, G.: On the form factors of local operators in the lattice sine-Gordon model. J. Stat. Mech. Theory Exp. P10006 (2012)

  30. Grosjean, N., Maillet, J.M., Niccoli, G.: On the form factors of local operators in the Bazhanov-Stroganov and chiral Potts models. Annales Henri Poincaré (2014). doi:10.1007/s00023-014-0358-9

  31. Grosjean, N., Niccoli, G.: The \({\tau_2}\)-model and the chiral Potts model revisited: completeness of Bethe equations from Sklyanin’s SOV method. J. Stat. Mech. P11005 (2012)

  32. Günther U., Rotter I., Samsonov B.: Projective Hilbert space structures at exceptional points. J. Phys. A Math. Theor. 40, 8815–8833 (2007)

    Article  MATH  ADS  Google Scholar 

  33. Izergin, A.G., Kitanine, N., Maillet, J.M., Terras, V.: Spontaneous magnetization of the XXZ Heisenberg spin-1/2 chain, Nucl. Phys. B 554 679–696 (1999). solv-int/9812021

  34. Izergin A.G., Korepin V.E.: Lattice versions of quantum field theory models in two dimensions. Nucl. Phys. B 205, 401–413 (1982)

    Article  MathSciNet  ADS  Google Scholar 

  35. Kirillov, A.N., Reshetikhin, N.Y.: Exact solution of the Heisenberg XXZ model of spin s, J. Sov. Math. 35, 2627–2643 (1986), translated from Zap. Nauch. Sem. LOMI 145, pp. 109–133, 1985

  36. Kirillov A.N., Reshetikhin N.Y.: Exact solution of the integrable XXZ Heisenberg model with arbitrary spin: I. The ground state and the excitation spectrum. J. Phys. A Math. Gen. 20, 1565–1585 (1987)

    Article  MathSciNet  ADS  Google Scholar 

  37. Kitanine N.: Correlation functions of the higher spin XXX chains. J. Phys. A Math. Gen. 34, 8151–8169 (2001)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  38. Kitanine, N., Kozlowski, K.K., Maillet, J.M., Niccoli, G., Slavnov, N.A., Terras, V.: Correlation functions of the open XXZ chain: I. J. Stat. Mech. Theory Exp. P10009 (2007)

  39. Kitanine, N., Kozlowski, K.K., Maillet, J.M., Niccoli, G., Slavnov, N.A., Terras, V.: Correlation functions of the open XXZ chain: II. J. Stat. Mech. Theory Exp. P07010 (2008)

  40. Kitanine, N., Kozlowski, K.K., Maillet, J.M., Slavnov, N.A., Terras, V.: Algebraic Bethe ansatz approach to the asymptotic behavior of correlation functions. J. Stat. Mech. Theory Exp. P04003 (2009)

  41. Kitanine N., Kozlowski K.K., Maillet J.M., Slavnov N.A., Terras V.: On the thermodynamic limit of form factors in the massless XXZ Heisenberg chain. J. Math. Phys. 50, 095209 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  42. Kitanine N., Kozlowski K.K., Maillet J.M., Slavnov N.A., Terras V.: Riemann-Hilbert approach to a generalized sine kernel and applications. Commun. Math. Phys. 291, 691–761 (2009)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  43. Kitanine, N., Kozlowski, K.K., Maillet, J.M., Slavnov, N.A., Terras, V.: A form factor approach to the asymptotic behavior of correlation functions in critical models. J. Stat. Mech. Theory Exp. P12010 (2011). arXiv:1110.0803

  44. Kitanine, N., Kozlowski, K.K., Maillet, J.M., Slavnov, N.A., Terras, V.: The thermodynamic limit of particle-hole form factors in the massless XXZ Heisenberg chain. J. Stat. Mech. Theory Exp. P05028 (2011)

  45. Kitanine, N., Kozlowski, K.K., Maillet, J.M., Slavnov, N.A., Terras, V.: Form factor approach to dynamical correlation functions in critical models. J. Stat. Mech. Theory Exp. P09001 (2012). arXiv:1206.2630

  46. Kitanine, N., Kozlowski, K.K., Maillet, J.M., Terras, V.: Large-distance asymptotic behaviour of multi-point correlation functions in massless quantum models. J. Stat. Mech. P05011 (2014)

  47. Kitanine, N., Maillet, J.M., Niccoli, G.: Open spin chains with generic integrable boundaries: Baxter equation and Bethe ansatz completeness from separation of variables. J. Stat. Mech. P05015 (2014)

  48. Kitanine N., Maillet J.M., Slavnov N.A., Terras V.: Spin-spin correlation functions of the XXZ-1/2 Heisenberg chain in a magnetic field. Nucl. Phys. B 641, 487–518 (2002)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  49. Kitanine, N., Maillet, J.M., Slavnov, N.A., Terras, V.: Dynamical correlation functions of the XXZ spin-1/2 chain. Nucl. Phys. B 729, 558–580 (2005). hep-th/0407108

  50. Kitanine, N., Maillet, J.M., Slavnov, N.A., Terras, V.: Master equation for spin-spin correlation functions of the XXZ chain. Nucl. Phys. B 712, 600–622 (2005). hep-th/0406190

  51. Kitanine N., Maillet J.M., Terras V.: Form factors of the XXZ Heisenberg spin-1/2 finite chain. Nucl. Phys. B 554, 647–678 (1999)

    Article  MATH  MathSciNet  ADS  Google Scholar 

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

    Article  MATH  MathSciNet  ADS  Google Scholar 

  53. Korff, C.: A Q-operator for the twisted XXX model. J. Phys. A Math. Gen. 39, 3203–3219 (2006). math-ph/0511022

  54. Korff C.: PT symmetry of the non-Hermitian XX spin-chain: non-local bulk interaction from complex boundary fields. J. Phys. A Math. Theor. 41, 295206 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  55. Korff C., Weston R.: PT symmetry on the lattice: the quantum group invariant XXZ spin chain. J. Phys. A Math. Theor. 40, 8845–8872 (2007)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  56. Kozlowski, K.K., Terras, V.: Long-time and large-distance asymptotic behavior of the current-current correlators in the non-linear Schrödinger model. J. Stat. Mech.: Theory Exp. P09013 (2011). arXiv:1101.0844

  57. Krichever I., Lipan O., Wiegmann P., Zabrodin A.: Quantum integrable models and discrete classical Hirota equations. Commun. Math. Phys. 188, 267–304 (1997)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  58. Kulish, P.P., Reshetikhin, N.Y.: Quantum linear problem for the sine-Gordon equation and higher representations, Zap. Nauch. Sem. LOMI 101, 101–110 (1981), translation in J. Sov. Math. 23, 2435–41 (1983)

  59. Kulish P.P., Sklyanin E.K.: Quantum spectral transform method. Recent developments. Lect. Notes Phys. 151, 61–119 (1982)

    MathSciNet  ADS  Google Scholar 

  60. Levy-Bencheton, D., Terras, V.: An algebraic Bethe ansatz approach to form factors and correlation functions of the cyclic eight-vertex solid-on-solid model. J. Stat. Mech. P04015 (2013). arXiv:1212.0246

  61. Li Y.-Y., Cao J., Yang W.-L., Shi K., Wang Y.: Exact solution of the one-dimensional Hubbard model with arbitrary boundary magnetic fields. Nucl. Phys. B 879, 98–109 (2014)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  62. Maillet, J.M., Sanchezde Santos, J.: Drinfel’d twists and algebraic Bethe Ansatz. In: L. D. Faddeev’s Seminar on Mathematical Physics, pp. 137–178, Amer. Math. Soc. Transl. Ser. 2, 201, Amer. Math. Soc., Providence (2000). q-alg/9612012

  63. Maillet, J.M., Terras, V.: On the quantum inverse scattering problem, Nucl. Phys. B 575, 627–644 (2000). hep-th/9911030

  64. Mangazeev V.V.: Q-operators in the six-vertex model. Nucl. Phys. B 886, 166–184 (2014)

    Article  MathSciNet  ADS  Google Scholar 

  65. Niccoli G.: Reconstruction of Baxter Q-operator from Sklyanin SOV for cyclic representations of integrable quantum models. Nucl. Phys. B 835, 263–283 (2010)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  66. Niccoli G.: Completeness of Bethe Ansatz by sklyanin SOV for cyclic representations of integrable quantum models. JHEP 03, 123 (2011)

    Article  MathSciNet  ADS  Google Scholar 

  67. Niccoli, G.: Non-diagonal open spin-1/2 XXZ quantum chains by separation of variables: Complete spectrum and matrix elements of some quasi-local operators. J. Stat. Mech. P10025 (2012)

  68. Niccoli, G.: An antiperiodic dynamical six-vertex model: I. complete spectrum by SOV, matrix elements of the identity on separate states and connections to the periodic eight-vertex model. J. Phys. A Math. Theor. 46, 075003 (2013). arXiv:1207.1928

  69. Niccoli, G.: Antiperiodic spin-1/2 XXZ quantum chains by separation of variables: Complete spectrum and form factors. Nucl. Phys. B 870, 397–420 (2013). arXiv:1205.4537

  70. Niccoli, G.: Form factors and complete spectrum of XXX antiperiodic higher spin chains by quantum separation of variables. J. Math. Phys. 053516 (2013)

  71. Niccoli, G., Teschner, J.: The Sine-Gordon model revisited I. J. Stat. Mech. P09014 (2010)

  72. Niekamp S., Wirth T., Frahm H.: The XXZ model with anti-periodic twisted boundary conditions. J. Phys. A Math. Theor. 42, 195008 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  73. Pasquier V., Gaudin M.: The periodic Toda chain and a matrix generalization of the Bessel function recursion relations. J. Phys. A Math. Gen. 25, 5243–5252 (1992)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  74. Pasquier V., Saleur H.: Common structures between finite systems and conformal field theories through quantum groups. Nucl. Phys. B 330, 523–556 (1990)

    Article  MathSciNet  ADS  Google Scholar 

  75. Pereira R.G., Sirker J., Caux J.-S., Hagemans R., Maillet J.M., White S.R., Affleck I.: Dynamical spin structure factor for the anisotropic spin-1/2 Heisenberg chain. Phys. Rev. Lett. 96, 257202 (2006)

    Article  ADS  Google Scholar 

  76. Pereira, R.G., Sirker, J., Caux, J.-S., Hagemans, R., Maillet, J.M., White, S.R., Affleck, I.: Dynamical structure factor at small q for the XXZ spin-1/2 chain. J. Stat. Mech. P08022 (2007)

  77. Pronko, G.P.: On the Baxter’s Q operator for the XXX spin chain. Comm. Math. Phys. 212, 687–701 (2000). hep-th/9908179

  78. Reshetikhin N.Y.: The functional equation method in the theory of exactly soluble quantum systems. Sov. Phys. JETP 57, 691–696 (1983)

    MathSciNet  Google Scholar 

  79. Sklyanin, E.K.: The quantum Toda chain. Lect. Notes Phys. 226, 196–233 (1985)

  80. Sklyanin E.K.: Boundary conditions for integrable quantum systems. J. Phys. A Math. Gen. 21, 2375–2389 (1988)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  81. Sklyanin, E.K.: Functional Bethe Ansatz. In: Kupershmidt, B. Integrable and superintegrable systems, pp. 8–33. World Scientific, Singapore (1990)

  82. Sklyanin, E.K.: Quantum inverse scattering method. Selected topics. In: Ge, M.-L. (ed.) Quantum group and quantum integrable systems, pp. 63–97, Nankai Lectures in Mathematical Physics, World Scientific (1992). hep-th/9211111

  83. Sklyanin, E.K.: Separation of variables. New trends. Prog. Theor. Phys. 118, 35–60 (1995). solv-int/9504001

  84. Sklyanin E.K., Faddeev L.D.: Quantum mechanical approach to completely integrable field theory models. Sov. Phys. Dokl 23, 902–904 (1978)

    ADS  Google Scholar 

  85. Sklyanin, E.K., Takhtadzhyan, L.A., Faddeev, L.D.: Quantum inverse problem method I, Theor. Math. Phys. 40, 688–706 (1979), translated from Teor. Mat. Fiz. 40, 194–220 (1979)

  86. Slavnov N.A.: 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 

  87. Sokolov A.V., Andrianov A.A., Cannata F.: Non-Hermitian quantum mechanics of non-diagonalizable Hamiltonians: puzzles with self-orthogonal states. J. Phys. A Math. Gen. 39, 10207–10227 (2006)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  88. Takhtadzhan L.A., Faddeev L.D.: The quantum method of the inverse problem and the Heisenberg XYZ model. Russ. Math. Surv. 34(5), 11–68 (1979)

    Article  Google Scholar 

  89. Tarasov V., Varchenko A.: Completeness of Bethe vectors and difference equations with regular singular points. Int. Math. Res. Not. 1995(13), 637–669 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  90. Terras, V.: Drinfel’d twists and functional Bethe Ansatz. Lett. Math. Phys. 48, 263–276 (1999). math-ph/9902009

  91. Yung C.M., Batchelor M.T.: Exact solution for the spin-s XXZ quantum chain with non-diagonal twists. Nucl. Phys. B 446, 461–484 (1995)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  92. Zhang, X., Li, Y.-Y., Cao, J., Yang, W.-L., Shi, K., Wang, Y.: Retrieve the Bethe states of quantum integrable models solved via off-diagonal Bethe ansatz. arXiv:1407.5294

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  1. ENS Lyon, CNRS, Laboratoire de Physique, UMR 5672, Lyon, France

    Giuliano Niccoli

  2. Univ. Paris Sud, CNRS, LPTMS, UMR 8626, 91405, Orsay, France

    Véronique Terras

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  1. Giuliano Niccoli
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Niccoli, G., Terras, V. Antiperiodic XXZ Chains with Arbitrary Spins: Complete Eigenstate Construction by Functional Equations in Separation of Variables. Lett Math Phys 105, 989–1031 (2015). https://doi.org/10.1007/s11005-015-0759-9

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