×

On separation of variables for reflection algebras. (English) Zbl 1456.81244

Summary: We implement our new separation of variables (SoV) approach for open quantum integrable models associated to higher rank representations of the reflection algebras. We construct the (SoV) basis for the fundamental representations of the \(Y( gl_n)\) reflection algebra associated to general integrable boundary conditions. Moreover, we give the conditions on the boundary \(K\)-matrices allowing for the transfer matrix to be diagonalizable with simple spectrum. These SoV basis are then used to completely characterize the transfer matrix spectrum for the rank one and two reflection algebras. The rank one case is developed for both the rational and trigonometric fundamental representations of the 6-vertex reflection algebra. On the one hand, we extend the complete spectrum characterization to representations previously lying outside the SoV approach, e.g. those for which the standard algebraic Bethe Ansatz applies. On the other hand, we show that our new SoV construction can be reduced to the generalized Sklyanin’s one whenever it is applicable. The rank two case is developed explicitly for the fundamental representations of the \(Y( gl_3)\) reflection algebra associated to general integrable boundary conditions. For both rank one and two our SoV approach leads to a complete characterization of the transfer matrix spectrum in terms of a set of polynomial solutions to the corresponding quantum spectral curve equation. Those are finite difference functional equations of order equal to the rank plus one, i.e. here two and three respectively for the \(Y( gl_2)\) and \(Y( gl_3)\) reflection algebras.

MSC:

81R12 Groups and algebras in quantum theory and relations with integrable systems
81R05 Finite-dimensional groups and algebras motivated by physics and their representations
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
82B23 Exactly solvable models; Bethe ansatz
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Maillet J M and Niccoli G 2018 On quantum separation of variables J. Math. Phys.59 091417 · Zbl 1402.81174 · doi:10.1063/1.5050989
[2] Maillet J M and Niccoli G 2019 Complete spectrum of quantum integrable lattice models associated to Y(gln) by separation of variables SciPost Phys.6 071 · doi:10.21468/SciPostPhys.6.6.071
[3] Maillet J M and Niccoli G 2019 Complete spectrum of quantum integrable lattice models associated to Uq(gl^n) by separation of variables J. Phys. A: Math. Theor.52 315203 · Zbl 1509.81594 · doi:10.1088/1751-8121/ab2930
[4] Maillet J M and Niccoli G 2019 On quantum separation of variables: beyond fundamental representations (arXiv:1903.06618)
[5] Faddeev L D and Sklyanin E K 1978 Quantum-mechanical approach to completely integrable field theory models Sov. Phys.—Dokl.23 902-4
[6] Faddeev L D, Sklyanin E K and Takhtajan L A 1979 Quantum inverse problem method I Theor. Math. Phys.40 688-706 · Zbl 1138.37331 · doi:10.1007/BF01018718
[7] Faddeev L D, Sklyanin E K and Takhtajan L A 1979 Teor. Mat. Fiz.40 194-220 (translated from) · Zbl 1138.37331
[8] Takhtadzhan L A and Faddeev L D 1979 The quantum method of the inverse problem and the Heisenberg XYZ model Russ. Math. Surveys34 11 · Zbl 0449.35096 · doi:10.1070/RM1979v034n05ABEH003909
[9] Sklyanin E K 1979 Method of the inverse scattering problem and the non-linear quantum Schrödinger equation Sov. Phys.—Dokl.24 107-9
[10] Sklyanin E K 1979 On complete integrability of the Landau-Lifshitz equation preprint LOMI E-3-79 · Zbl 0449.35089
[11] Faddeev L D and Takhtajan L A 1981 Quantum inverse scattering method Sov. Sci. Rev. Math. C 1 107
[12] Sklyanin E K 1982 Quantum version of the inverse scattering problem method J. Sov. Math.19 1546-95 · Zbl 0497.35072 · doi:10.1007/BF01091462
[13] Faddeev L D 1984 Integrable models in (1+1)-dimensional quantum field theory Les Houches 1982, Recent Advances in Field Theory and Statistical Mechanics ed J B Zuber and R Stora (Amsterdam: Elsevier) pp 561-608
[14] Faddeev L D 1996 How algebraic Bethe ansatz works for integrable model Les Houches Lectures · Zbl 0934.35170
[15] Cherednik I V 1984 Factorizing particles on a half line and root systems Theor. Math. Phys.61 977 · Zbl 0575.22021 · doi:10.1007/BF01038545
[16] Sklyanin E K 1988 Boundary conditions for integrable quantum systems J. Phys. A: Math. Gen.21 2375 · Zbl 0685.58058 · doi:10.1088/0305-4470/21/10/015
[17] Gaudin M 1971 Boundary energy of a Bose gas in one dimension Phys. Rev. A 4 386 · doi:10.1103/PhysRevA.4.386
[18] Bariev R Z 1979 Correlation functions of the semi-infinite two-dimensional Ising model. I. Local magnetization Teort. Mat. Fiz.40 95 · doi:10.1007/BF01019245
[19] Bariev R Z 1980 Local magnetization of the semi-infinite XY-chain Physica A 103 363 · doi:10.1016/0378-4371(80)90224-1
[20] Bariev R Z 1980 Correlation functions of the semi-infinite two-dimensional Ising model. II. Two-point correlation functions Theor. Math. Phys.42 173 · doi:10.1007/BF01032121
[21] Schulz H 1985 Hubbard chain with reflecting ends J. Phys. C: Solid State Phys.18 581 · doi:10.1088/0022-3719/18/3/010
[22] Alcaraz F, Barber M, Batchelor M, Baxter R and Quispel G 1987 Surface exponents of the quantum XXZ Ashkin-Teller and Potts models J. Phys. A: Math. Gen.20 6397 · doi:10.1088/0305-4470/20/18/038
[23] Bariev R Z 1988 Correlation functions of the semi-infinite two-dimensional Ising model. III Influence of a ‘fixed’ boundary Teor. Mat. Fiz.77 127 · doi:10.1007/BF01028685
[24] Mezincescu L, Nepomechie R I and Rittenberg V 1990 Bethe ansatz solution of the Fateev-Zamolodchikov quantum spin chain with boundary terms Phys. Lett. A 147 70 · doi:10.1016/0375-9601(90)90016-H
[25] Pasquier V and Saleur H 1990 Common structures between finite systems and conformal field theories through quantum groups Nucl. Phys. B 330 523 · doi:10.1016/0550-3213(90)90122-T
[26] Batchelor M T, Mezincescu L, Nepomechie R I and Rittenberg V 1990 q-deformations of the O(3) symmetric spin-1 heisenberg chain J. Phys. A: Math. Gen.23 L141 · Zbl 0716.60130 · doi:10.1088/0305-4470/23/5/019
[27] Kulish P P and Sklyanin E K 1991 The general Uq(sl(2)) invariant XXZ integrable quantum spin chain J. Phys. A: Math. Gen.24 L435 · Zbl 0724.17019 · doi:10.1088/0305-4470/24/8/009
[28] Freidel L and Maillet J M 1991 Quadratic algebras and integrable systems Phys. Lett. B 262 278-84 · Zbl 1514.17016 · doi:10.1016/0370-2693(91)91566-E
[29] Mezincescu L and Nepomechie R I 1991 Integrability of open spin chains with quantum algebra symmetry Int. J. Mod. Phys. A 06 5231 · Zbl 0802.58075 · doi:10.1142/S0217751X91002458
[30] Mezincescu L and Nepomechie R 1992 Analytical Bethe Ansatz for quantum-algebra-invariant spin chains Nucl. Phys. B 372 597-621 (with addendum) · doi:10.1016/0550-3213(92)90367-K
[31] de Vega H J and Ruiz A G 1993 Boundary K-matrices for the six vertex and the n(2n−1) An−1 vertex models J. Phys. A: Math. Gen.26 L519 · Zbl 0786.58020 · doi:10.1088/0305-4470/26/3/015
[32] de Vega H J and González-Ruiz A 1993 The highest weight property for the SUq(n) invariant spin chains J. Phys. A: Math. Gen.26 L519 · Zbl 0786.58020 · doi:10.1088/0305-4470/26/12/007
[33] de Vega H J and Gonzalez-Ruiz A 1994 Boundary K-matrices for the XYZ XXZ and XXX spin chains J. Phys. A: Math. Gen.27 6129 · Zbl 0844.35121 · doi:10.1088/0305-4470/27/18/021
[34] Ghoshal S and Zamolodchikov A 1994 Boundary S-matrix and boundary state in two-dimensional integrable quantum field theory Int. J. Mod. Phys. A 9 3841 · Zbl 0985.81714 · doi:10.1142/S0217751X94001552
[35] Ghoshal S and Zamolodchikov A B 1994 Errata: Boundary S matrix and boundary state in two-dimensional integrable quantum field theory Int. J. Mod. Phys. A 9 4353 · Zbl 0985.81714 · doi:10.1142/S0217751X94002430
[36] Artz S, Mezincescu L and Nepomechie R I 1995 Analytical Bethe ansatz for a A2n−1(2), Bn(1), Cn(1), Dn(1) quantum-algebra-invariant open spin chains J. Phys. A: Math. Gen.28 5131 · Zbl 0865.17019 · doi:10.1088/0305-4470/28/18/006
[37] Jimbo M, Kedem R, Kojima T, Konno H and Miwa T 1995 XXZ chain with a boundary Nucl. Phys. B 441 437 · Zbl 0990.82515 · doi:10.1016/0550-3213(95)00062-W
[38] Jimbo M, Kedem R, Konno H, Miwa T and Weston R 1995 Difference equations in spin chains with a boundary Nucl. Phys. B 448 429 · Zbl 1009.82508 · doi:10.1016/0550-3213(95)00218-H
[39] Kulish P P 1996 Yang-Baxter equation and reflection equations in integrable models Low-Dimensional Models in Statistical Physics and Quantum Field Theory(Lecture Notes in Physics vol 469) ed H Grosse and L Pittner (Berlin: Springer) · Zbl 0879.58033 · doi:10.1007/BFb0102555
[40] Fan H, Hou B-Y, Shi K-J and Yang Z-X 1996 Algebraic Bethe ansatz for the eight-vertex model with general open boundary conditions Nucl. Phys. B 478 723 · Zbl 0925.82059 · doi:10.1016/0550-3213(96)00398-7
[41] Zhou H-Q 1996 Quantum integrability for the one-dimensional Hubbard open chain Phys. Rev. B 54 41 · doi:10.1103/PhysRevB.54.41
[42] Asakawa H and Suzuki M 1996 Finite-size corrections in the XXZ model and the Hubbard model with boundary fields J. Phys. A: Math. Gen.29 225 · Zbl 0943.82513 · doi:10.1088/0305-4470/29/2/004
[43] Zhou H-Q 1997 Graded reflection equations and the one-dimensional Hubbard open chain Phys. Lett. A 228 48 · Zbl 0962.82513 · doi:10.1016/S0375-9601(97)00035-2
[44] Guan X-W, Wang M-S and Yang S-D 1997 Lax pair and boundary K-matrices for the one-dimensional Hubbard model Nucl. Phys. B 485 685 · Zbl 0925.82046 · doi:10.1016/S0550-3213(96)00630-X
[45] Shiroishi M and Wadati M 1997 Bethe Ansatz equation for the Hubbard model with boundary fields J. Phys. Soc. Japan66 1 · Zbl 0942.82005 · doi:10.1143/JPSJ.66.1
[46] Tsuchiya O 1998 Determinant formula for the six-vertex model with reflecting end J. Math. Phys.39 5946 · Zbl 0938.82007 · doi:10.1063/1.532606
[47] Guan X-W 2000 Algebraic Bethe ansatz for the one-dimensional Hubbard model with open boundaries J. Phys. A: Math. Gen.33 5391 · Zbl 1046.82502 · doi:10.1088/0305-4470/33/30/309
[48] Mintchev M, Ragoucy E and Sorba P 2001 Spontaneous symmetry breaking in the gl(N)-NLS hierarchy on the half line J. Phys. A: Math. Gen.34 8345 · Zbl 1011.81024 · doi:10.1088/0305-4470/34/40/311
[49] Nepomechie R I 2002 Solving the open XXZ spin chain with nondiagonal boundary terms at roots of unity Nucl. Phys. B 622 615 · Zbl 1049.82010 · doi:10.1016/S0550-3213(01)00585-5
[50] Nepomechie R I and Ravanini F 2003 Completeness of the Bethe Ansatz solution of the open XXZ chain with nondiagonal boundary terms J. Phys. A: Math. Gen.36 11391 · Zbl 1039.82015 · doi:10.1088/0305-4470/36/45/003
[51] Cao J, Lin H-Q, Shi K-J and Wang Y 2003 Exact solution of XXZ spin chain with unparallel boundary fields Nucl. Phys. B 663 487 · Zbl 1023.82502 · doi:10.1016/S0550-3213(03)00372-9
[52] Doikou A 2003 Fused integrable lattice models with quantum impurities and open boundaries Nucl. Phys. B 668 447 · Zbl 1031.82011 · doi:10.1016/j.nuclphysb.2003.07.001
[53] Nepomechie R I 2004 Bethe ansatz solution of the open XXZ chain with nondiagonal boundary terms J. Phys. A: Math. Gen.37 433 · Zbl 1050.82011 · doi:10.1088/0305-4470/37/2/012
[54] de Gier J and Pyatov P 2004 Bethe ansatz for the Temperley-Lieb loop model with open boundaries J. Stat. Mech. P002 · Zbl 1145.82320 · doi:10.1088/1742-5468/2004/03/p002
[55] Arnaudon D, Avan J, Crampé N, Doikou A, Frappat L and Ragoucy E 2004 General boundary conditions for the sl(N) and sl(M|N) open spin chains J. Stat. Mech. P005 · Zbl 1072.82519 · doi:10.1088/1742-5468/2004/08/p08005
[56] Galleas W and Martins M J 2005 Solution of the SU(N) vertex model with non-diagonal open boundaries Phys. Lett. A 335 167 · Zbl 1123.82325 · doi:10.1016/j.physleta.2004.12.017
[57] Arnaudon D, Crampé N, Doikou A, Frappat L and Ragoucy E 2005 Analytical Bethe ansatz for closed and open gl(N)-spin chains in any representation J. Stat. Mech. P02007 · Zbl 1459.82026 · doi:10.1088/1742-5468/2005/02/p02007
[58] Arnaudon D, Crampé N, Doikou A, Frappat L and Ragoucy E 2006 Spectrum and Bethe ansatz equations for the Uq(gl(N)) closed and open spin chains in any representation Ann. Henri Poincaré7 1217 · Zbl 1115.82005 · doi:10.1007/s00023-006-0280-x
[59] Murgan R, Nepomechie R I and Shi C 2006 Exact solution of the open XXZ chain with general integrable boundary terms at roots of unity J. Stat. Mech. P08006 · doi:10.1088/1742-5468/2006/08/p08006
[60] Doikou A 2006 The open XXZ and associated models at q root of unity J. Stat. Mech. P09010 · Zbl 1456.82247 · doi:10.1088/1742-5468/2006/09/p09010
[61] Yang W-L, Nepomechie R I and Zhang Y-Z 2006 Q-operator and T-Q relation from the fusion hierarchy Phys. Lett. B 633 664 · Zbl 1247.82019 · doi:10.1016/j.physletb.2005.12.022
[62] Kitanine N, Kozlowski K K, Maillet J M, Niccoli G, Slavnov N A and Terras V 2007 Correlation functions of the open XXZ chain: I J. Stat. Mech. P10009 · Zbl 1456.82143 · doi:10.1088/1742-5468/2007/10/p10009
[63] Baseilhac P and Koizumi K 2007 Exact spectrum of the XXZ open spin chain from the q-Onsager algebra representation theory J. Stat. Mech. P09006 · Zbl 1456.82029 · doi:10.1088/1742-5468/2007/09/p09006
[64] Yang W-L and Zhang Y-Z 2007 On the second reference state and complete eigenstates of the open XXZ chain J. High Energy Phys.JHEP04(2007) 044 · doi:10.1088/1126-6708/2007/04/044
[65] Ragoucy E and Satta G 2007 Analytical bethe ansatz for closed and open gl(M|N) super-spin chains in arbitrary representations and for any Dynkin diagram J. High Energy Phys.JHEP09(2007) 001 · doi:10.1088/1126-6708/2007/09/001
[66] Frappat L, Nepomechie R I and Ragoucy E 2007 A complete Bethe ansatz solution for the open spin-s XXZ chain with general integrable boundary terms J. Stat. Mech. P09009 · Zbl 1456.82260 · doi:10.1088/1742-5468/2007/09/p09009
[67] Amico L, Frahm H, Osterloh A and Ribeiro G 2007 Integrable spin-boson models descending from rational six-vertex models Nucl. Phys. B 787 283 · Zbl 1225.82018 · doi:10.1016/j.nuclphysb.2007.07.022
[68] Kitanine N, Kozlowski K K, Maillet J M, Niccoli G, Slavnov N A and Terras V 2008 Correlation functions of the open XXZ chain: II J. Stat. Mech. P07010 · Zbl 1459.82071
[69] Galleas W 2008 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 · Zbl 1151.82005 · doi:10.1016/j.nuclphysb.2007.09.011
[70] Belliard S and Ragoucy E 2009 The nested Bethe Ansatz for ‘all’ open spin chains with diagonal boundary conditions J. Phys. A: Math. Theor.42 205203 · Zbl 1165.82008 · doi:10.1088/1751-8113/42/20/205203
[71] Nepomechie R I 2010 Nested algebraic Bethe ansatz for open GL(N) spin chains with projected K-matrices Nucl. Phys. B 831 429 · Zbl 1204.82011 · doi:10.1016/j.nuclphysb.2010.01.006
[72] Filali G and Kitanine N 2010 Partition function of the trigonometric SOS model with reflecting end J. Stat. Mech. L06001 · Zbl 1456.82256
[73] Filali G and Kitanine N 2011 Spin chains with non-diagonal boundaries and trigonometric SOS model with reflecting end SIGMA7 012 · Zbl 1217.82019 · doi:10.3842/SIGMA.2011.012
[74] Filali G 2011 Elliptic dynamical reflection algebra and partition function of SOS model with reflecting end J. Geom. Phys.61 1789 · Zbl 1226.82010 · doi:10.1016/j.geomphys.2011.01.002
[75] Crampe N and Ragoucy E 2012 Generalized coordinate Bethe ansatz for non-diagonal boundaries Nucl. Phys. B 858 502 · Zbl 1246.82014 · doi:10.1016/j.nuclphysb.2012.01.020
[76] Cao J, Yang W-L, Shi K and Wang Y 2013 Off-diagonal Bethe ansatz solutions of the anisotropic spin-1/2 chains with arbitrary boundary fields Nucl. Phys. B 877 152 · Zbl 1284.82017 · doi:10.1016/j.nuclphysb.2013.10.001
[77] Baseilhac P and Belliard S 2013 The half-infinite XXZ chain in Onsager’s approach Nucl. Phys. B 873 550 · Zbl 1282.82010 · doi:10.1016/j.nuclphysb.2013.05.003
[78] Belliard S and Crampé N 2013 Heisenberg XXX model with general boundaries: eigenvectors from algebraic Bethe Ansatz SIGMA9 072 · Zbl 1288.82020 · doi:10.3842/SIGMA.2013.072
[79] Belliard S, Crampé N and Ragoucy E 2013 Algebraic Bethe ansatz for open XXX model with triangular boundary matrices Lett. Math. Phys.103 493 · Zbl 1280.82003 · doi:10.1007/s11005-012-0601-6
[80] Cao J, Yang W-L, Shi K and Wang Y 2014 Nested off-diagonal Bethe ansatz and exact solutions of the su(n) spin chain with generic integrable boundaries J. High Energy Phys.JHEP04(2014) 143 · doi:10.1007/JHEP04(2014)143
[81] Belliard S 2015 Modified algebraic Bethe ansatz for XXZ chain on the segment—I—Triangular cases Nucl. Phys. B 892 1 · Zbl 1328.82017 · doi:10.1016/j.nuclphysb.2015.01.003
[82] Belliard S and Pimenta R A 2015 Modified algebraic Bethe ansatz for XXZ chain on the segment—II—general cases Nucl. Phys. B 894 527 · Zbl 1328.82009 · doi:10.1016/j.nuclphysb.2015.03.016
[83] Avan J, Belliard S, Grosjean N and Pimenta R 2015 Modified algebraic Bethe ansatz for XXZ chain on the segment—III—proof Nucl. Phys. B 899 229 · Zbl 1331.82011 · doi:10.1016/j.nuclphysb.2015.08.006
[84] Xu X, Hao K, Yang T, Cao J, Yang W-L and Shi K-J 2016 Bethe ansatz solutions of the τ2-model with arbitrary boundary fields J. High Energy Phys.JHEP11(2016) 80 · Zbl 1390.81233 · doi:10.1007/JHEP11(2016)080
[85] Baseilhac P and Belliard S 2017 Non-Abelian symmetries of the half-infinite XXZ spin chain Nucl. Phys. B 916 373 · Zbl 1356.82007 · doi:10.1016/j.nuclphysb.2017.01.012
[86] Derkachov S E, Korchemsky G P and Manashov A N 2003 Baxter Q-operator and separation of variables for the open SL(2,R) spin chain J. High Energy Phys.JHEP10(2003) 053 · doi:10.1088/1126-6708/2003/10/053
[87] Frahm H, Seel A and Wirth T 2008 Separation of variables in the open XXX chain Nucl. Phys. B 802 351 · Zbl 1190.82009 · doi:10.1016/j.nuclphysb.2008.04.008
[88] Amico L, Frahm H, Osterloh A and Wirth T 2010 Separation of variables for integrable spin-boson models Nucl. Phys. B 839 604 · Zbl 1206.82029 · doi:10.1016/j.nuclphysb.2010.07.005
[89] Frahm H, Grelik J H, Seel A and Wirth T 2011 Functional Bethe ansatz methods for the open XXX chain J. Phys A: Math. Theor.44 015001 · Zbl 1207.82010 · doi:10.1088/1751-8113/44/1/015001
[90] Niccoli G 2012 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 · Zbl 1456.81247
[91] Faldella S and Niccoli G 2014 SOV approach for integrable quantum models associated with general representations on spin-1/2 chains of the 8-vertex reflection algebra J. Phys. A: Math. Theor.47 115202 · Zbl 1288.82010 · doi:10.1088/1751-8113/47/11/115202
[92] Faldella S, Kitanine N and Niccoli G 2014 Complete spectrum and scalar products for the open spin-1/2 XXZ quantum chains with non-diagonal boundary terms J. Stat. Mech. P01011 · Zbl 1456.82044
[93] Kitanine N, Maillet J M and Niccoli G 2014 Open spin chains with generic integrable boundaries: Baxter equation and Bethe ansatz completeness from separation of variables J. Stat. Mech. P05015 · Zbl 1456.82057
[94] Maillet J M, Niccoli G and Pezelier B 2017 Transfer matrix spectrum for cyclic representations of the 6-vertex reflection algebra I SciPost Phys.2 009 · doi:10.21468/SciPostPhys.2.1.009
[95] Kitanine N, Maillet J M, Niccoli G and Terras V 2017 The open XXX spin chain in the SoV framework: scalar product of separate states J. Phys. A: Math. Theor.50 224001 · Zbl 1373.82020 · doi:10.1088/1751-8121/aa6cc9
[96] Maillet J M, Niccoli G and Pezelier B 2018 Transfer matrix spectrum for cyclic representations of the 6-vertex reflection algebra II SciPost Phys.5 026 · doi:10.21468/SciPostPhys.5.3.026
[97] Kitanine N, Maillet J M, Niccoli G and Terras V 2018 The open XXZ spin chain in the SoV framework: scalar product of separate states J. Phys. A: Math. Theor.51 485201 · Zbl 1411.82015 · doi:10.1088/1751-8121/aae76f
[98] de Gier J and Essler F H L 2005 Bethe Ansatz solution of the asymmetric exclusion process with open boundaries Phys. Rev. Lett.95 240601 · doi:10.1103/PhysRevLett.95.240601
[99] de Gier J and Essler F H L 2006 Exact spectral gaps of the asymmetric exclusion process with open boundaries J. Stat. Mech. P12011 · Zbl 07078059
[100] Kinoshita T, Wenger T and Weiss D S 2006 A quantum Newton’s cradle Nature440 900EP · doi:10.1038/nature04693
[101] Hofferberth S, Lesanovsky I, Fischer B, Schumm T and Schmiedmayer J 2007 Non-equilibrium coherence dynamics in one-dimensional Bose gases Nature449 324EP · doi:10.1038/nature06149
[102] Bloch I, Dalibard J and Zwerger W 2008 Many-body physics with ultracold gases Rev. Mod. Phys.80 885 · doi:10.1103/RevModPhys.80.885
[103] Crampé N, Ragoucy E and Simon D 2010 Eigenvectors of open XXZ and ASEP models for a class of non-diagonal boundary conditions J. Stat. Mech. P11038
[104] Prosen T 2011 Open XXZ spin chain: nonequilibrium steady state and a strict bound on ballistic transport Phys. Rev. Lett.106 217206 · doi:10.1103/PhysRevLett.106.217206
[105] Trotzky S, Chen Y-A, Flesch A, McCulloch I P, Schollwöck U, Eisert J and Bloch I 2012 Probing the relaxation towards equilibrium in an isolated strongly correlated one-dimensional bose gas Nat. Phys.8 325EP · doi:10.1038/nphys2232
[106] Schneider U et al 2012 Fermionic transport and out-of-equilibrium dynamics in a homogeneous Hubbard model with ultracold atoms Nat. Phys.8 213EP · doi:10.1038/nphys2205
[107] Ronzheimer J P, Schreiber M, Braun S, Hodgman S S, Langer S, McCulloch I P, Heidrich-Meisner F, Bloch I and Schneider U 2013 Expansion dynamics of interacting bosons in homogeneous lattices in one and two dimensions Phys. Rev. Lett.110 205301 · doi:10.1103/PhysRevLett.110.205301
[108] Eisert J, Friesdorf M and Gogolin C 2015 Quantum many-body systems out of equilibrium Nat. Phys.11 124EP · doi:10.1038/nphys3215
[109] Sklyanin E K 1985 The quantum Toda chain Non-Linear Equations in Classical and Quantum Field Theory ed N Sanchez (Berlin: Springer) pp 196-233 · Zbl 0601.58039 · doi:10.1007/3-540-15213-X_80
[110] Sklyanin E K 1985 Goryachev-Chaplygin top and the inverse scattering method J. Sov. Math.31 3417 · Zbl 0604.70005 · doi:10.1007/BF02107243
[111] Sklyanin E K 1992 Quantum inverse scattering method. Selected topics Quantum Group and Quantum Integrable Systems(Nankai Lectures in Mathematical Physics) ed M-L Ge (Singapore: World Scientific) pp 63-97
[112] Sklyanin E K 1995 Separation of variables. New trends Prog. Theor. Phys.118 35 · Zbl 0868.35002 · doi:10.1143/PTPS.118.35
[113] Baxter R J 1973 Eight-vertex model in lattice statistics and one-dimensional anisotropic Heisenberg chain. I, II, III Ann. Phys.76 1-24 25-47; 48-71 · Zbl 1092.82511 · doi:10.1016/0003-4916(73)90439-9
[114] Baxter R J 1982 Exactly Solved Models in Statistical Mechanics (London: Academic) · Zbl 0538.60093
[115] Babelon O, Bernard D and Smirnov F A 1996 Quantization of solitons and the restricted sine-Gordon model Commun. Math. Phys.182 319 · Zbl 0877.58029 · doi:10.1007/BF02517893
[116] Babelon O, Bernard D and Smirnov F A 1997 Null-vectors in integrable field theory Commun. Math. Phys.186 601 · Zbl 0878.35097 · doi:10.1007/s002200050122
[117] Smirnov F A 1998 Structure of matrix elements in the quantum Toda chain J. Phys. A: Math. Gen.31 8953 · Zbl 0954.81028 · doi:10.1088/0305-4470/31/44/019
[118] Derkachov S E, Korchemsky G P and Manashov A N 2003 Separation of variables for the quantum SL(2,R) spin chain J. High Energy Phys.JHEP07(2003) 047 · doi:10.1088/1126-6708/2003/07/047
[119] Bytsko A and Teschner J 2006 Quantization of models with non-compact quantum group symmetry: modular XXZ magnet and lattice sinh-Gordon model J. Phys. A: Mat. Gen.39 12927 · Zbl 1107.82008 · doi:10.1088/0305-4470/39/41/S11
[120] von Gehlen G, Iorgov N, Pakuliak S and Shadura V 2006 The Baxter-Bazhanov-Stroganov model: separation of variables and the Baxter equation J. Phys. A: Math. Gen.39 7257 · Zbl 1122.82012 · doi:10.1088/0305-4470/39/23/006
[121] von Gehlen G, Iorgov N, Pakuliak S and Shadura V 2009 Factorized finite-size Ising model spin matrix elements from separation of variables J. Phys. A: Math. Theor.42 304026 · Zbl 1179.82046 · doi:10.1088/1751-8113/42/30/304026
[122] von Gehlen G, Iorgov N, Pakuliak S, Shadura V and Tykhyy Y 2007 Form-factors in the Baxter-Bazhanov-Stroganov model I: norms and matrix elements J. Phys. A: Math. Theor.40 14117 · Zbl 1130.82010 · doi:10.1088/1751-8113/40/47/006
[123] von Gehlen G, Iorgov N, Pakuliak S, Shadura V and Tykhyy Y 2008 Form-factors in the Baxter-Bazhanov-Stroganov model II: Ising model on the finite lattice J. Phys. A: Math. Theor.41 095003 · Zbl 1139.82012 · doi:10.1088/1751-8113/41/9/095003
[124] Niccoli G and Teschner J 2010 The Sine-Gordon model revisited I J. Stat. Mech. P09014 · Zbl 1456.81248
[125] Niccoli G 2010 Reconstruction of Baxter Q-operator from Sklyanin SOV for cyclic representations of integrable quantum models Nucl. Phys. B 835 263 · Zbl 1204.81090 · doi:10.1016/j.nuclphysb.2010.03.009
[126] Grosjean N and Niccoli G 2012 The τ2 -model and the chiral Potts model revisited: completeness of Bethe equations from Sklyanin’s SOV method J. Stat. Mech. P11005
[127] Grosjean N, Maillet J M and Niccoli G 2012 On the form factors of local operators in the lattice sine-Gordon model J. Stat. Mech. P10006 · Zbl 1456.81235
[128] Niccoli G 2013 Antiperiodic spin-1/2 XXZ quantum chains by separation of variables: complete spectrum and form factors Nucl. Phys. B 870 397 · Zbl 1262.82011 · doi:10.1016/j.nuclphysb.2013.01.017
[129] Niccoli G 2013 Form factors and complete spectrum of XXX antiperiodic higher spin chains by quantum separation of variables J. Math. Phys.54 053516 · Zbl 1285.82021 · doi:10.1063/1.4807078
[130] Niccoli G 2013 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 · Zbl 1269.82012 · doi:10.1088/1751-8113/46/7/075003
[131] Grosjean N, Maillet J M and Niccoli G 2015 On the form factors of local operators in the Bazhanov-Stroganov and chiral Potts models Ann. Henri Poincaré16 1103 · Zbl 1320.82012 · doi:10.1007/s00023-014-0358-9
[132] Niccoli G and Terras V 2015 Antiperiodic XXZ chains with arbitrary spins: Complete eigenstate construction by functional equations in separation of variables Lett. Math. Phys.105 989 · Zbl 1318.81032 · doi:10.1007/s11005-015-0759-9
[133] Niccoli G and Terras V 2016 The 8-vertex model with quasi-periodic boundary conditions J. Phys. A: Math. Theor.49 044001 · Zbl 1348.82023 · doi:10.1088/1751-8113/49/4/044001
[134] Levy-Bencheton D, Niccoli G and Terras V 2016 Antiperiodic dynamical 6-vertex model by separation of variables II: functional equations and form factors J. Stat. Mech. 033110 · Zbl 1457.82099 · doi:10.1088/1742-5468/2016/03/033110
[135] Kitanine N, Maillet J M, Niccoli G and Terras V 2016 On determinant representations of scalar products and form factors in the SoV approach: the XXX case J. Phys. A: Math. Theor.49 104002 · Zbl 1342.82036 · doi:10.1088/1751-8113/49/10/104002
[136] Smirnov F A 1998 Quasi-classical study of form factors in finite volume (arXiv:hep-th/9802132)
[137] Niekamp S, Wirth T and Frahm H 2009 The XXZ model with anti-periodic twisted boundary conditions J. Phys. A: Math. Theor.42 195008 · Zbl 1164.82003 · doi:10.1088/1751-8113/42/19/195008
[138] Jiang Y, Komatsu S, Kostov I and Serban D 2016 The hexagon in the mirror: the three-point function in the SoV representation J. Phys. A: Math. Theor.49 174007 · Zbl 1342.82033 · doi:10.1088/1751-8113/49/17/174007
[139] Ryan P and Volin D 2019 Separated variables and wave functions for rational gl(N) spin chains in the companion twist frame J. Math. Phys.60 032701 · Zbl 1426.82014 · doi:10.1063/1.5085387
[140] Gromov N, Levkovich-Maslyuk F and Sizov G 2017 New construction of eigenstates and separation of variables for SU(N) quantum spin chains J. High Energ. Phys.JHEP9(2017) 111 · Zbl 1382.81149 · doi:10.1007/JHEP09(2017)111
[141] Liashyk A and Slavnov N A 2019 On Bethe vectors in gl3-invariant integrable models J. High Energ. Phys.JHEP06(2018) 018 · Zbl 1395.81179 · doi:10.1007/JHEP06(2018)018
[142] Kulish P P, Reshetikhin N Yu and Sklyanin E K 1981 Yang-Baxter equation and representation theory: I Lett. Math. Phys.5 393-403 · Zbl 0502.35074 · doi:10.1007/BF02285311
[143] Kulish P P and Reshetikhin N Yu 1986 Gl3-invariant solutions of the Yang-Baxter equation and associated quantum systems J. Sov. Math.34 1948-71 · doi:10.1007/BF01095104
[144] Kulish P P and Reshetikhin N Yu 1982 Zap. Nauch. Sem. LOMI120 92-121 (translated from)
[145] Reshetikhin N Yu 1983 A method of functional equations in the theory of exactly solvable quantum systems Lett. Math. Phys.7 205 · doi:10.1007/BF00400435
[146] Kirillov A N and Reshetikhin N Yu 1986 Exact solution of the Heisenberg XXZ model of spin’s J. Sov. Math.35 2627-43 · Zbl 0612.35029 · doi:10.1007/BF01083768
[147] Kirillov A N and Reshetikhin N Yu 1985 Zap. Nauch. Sem. LOMI145 109-33 (translated from) · Zbl 0586.35032
[148] Izergin A G 1987 Partition function of the six-vertex model in a finite volume Sov. Phys.—Dokl.32 878 · Zbl 0875.82015
[149] Slavnov N A 1989 Calculation of scalar products of wave functions and form factors in the framework of the algebraic Bethe Ansatz Theor. Math. Phys.79 502-8 · doi:10.1007/BF01016531
[150] Foda O and Wheeler M 2012 Variations on Slavnov’s scalar product J. High Energy Phys.JHEP10(2012) 096 · Zbl 1397.81074 · doi:10.1007/JHEP10(2012)096
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.