Abstract
We consider the problem of analytically continuing energies computed with the Bethe ansatz, as posed by the study of non-compact integrable spin chains. By introducing an imaginary extensive twist in the Bethe equations, we show that one can expand the analytic continuation of energies in the scaling limit around another ‘pseudo-vacuum’ sitting at a negative number of Bethe roots, in the same way as around the usual pseudo-vacuum. We show that this method can be used to compute the energy levels of some states of the SL(2, ℂ) integrable spin chain in the infinite-volume limit, and as a proof of principle recover the ground-state value previously obtained in [1] (for the case of spins s = 0, \( \overline{s} \) = −1) by extrapolating results in small sizes. These results represent, as far as we know, the first (partial) description of the spectrum of SL(2, ℂ) non-compact spin chains in the thermodynamic limit.
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S.E. Derkachov, G.P. Korchemsky, J. Kotanski and A.N. Manashov, Noncompact Heisenberg spin magnets from high-energy QCD. 2. Quantization conditions and energy spectrum, Nucl. Phys. B 645 (2002) 237 [hep-th/0204124] [INSPIRE].
L.N. Lipatov, Asymptotic behavior of multicolor QCD at high energies in connection with exactly solvable spin models, JETP Lett. 59 (1994) 596 [Pisma Zh. Eksp. Teor. Fiz. 59 (1994) 571] [hep-th/9311037] [INSPIRE].
L.D. Faddeev and G.P. Korchemsky, High energy QCD as a completely integrable model, Phys. Lett. B 342 (1995) 311 [hep-th/9404173] [INSPIRE].
H. Bethe, On the theory of metals. 1. Eigenvalues and eigenfunctions for the linear atomic chain, Z. Phys. 71 (1931) 205 [INSPIRE].
L.D. Faddeev, E.K. Sklyanin and L.A. Takhtajan, The Quantum Inverse Problem Method. 1, Theor. Math. Phys. 40 (1980) 688 [Teor. Mat. Fiz. 40 (1979) 194] [INSPIRE].
L.A. Takhtajan and L.D. Faddeev, The Quantum method of the inverse problem and the Heisenberg XYZ model, Russ. Math. Surveys 34 (1979) 11 [Usp. Mat. Nauk 34 (1979) 13] [INSPIRE].
E.K. Sklyanin, Quantum version of the method of inverse scattering problem, J. Sov. Math. 19 (1982) 1546 [Zap. Nauchn. Semin. 95 (1980) 55] [INSPIRE].
R.J. Baxter, Exactly solved models in statistical mechanics, Academic Press (1982).
S.E. Derkachov, G.P. Korchemsky, J. Kotanski and A.N. Manashov, Noncompact Heisenberg spin magnets from high-energy QCD. 2. Quantization conditions and energy spectrum, Nucl. Phys. B 645 (2002) 237 [hep-th/0204124] [INSPIRE].
G.P. Korchemsky, Bethe ansatz for QCD Pomeron, Nucl. Phys. B 443 (1995) 255 [hep-ph/9501232] [INSPIRE].
S.E. Derkachov, G.P. Korchemsky and A.N. Manashov, Noncompact Heisenberg spin magnets from high-energy QCD. 3. Quasiclassical approach, Nucl. Phys. B 661 (2003) 533 [hep-th/0212169] [INSPIRE].
I. Affleck, On the Critical Behavior of Two-dimensional Systems With Continuous Symmetries, Phys. Rev. Lett. 55 (1985) 1355 [INSPIRE].
E. Granet, J.L. Jacobsen and H. Saleur, Analytical results on the Heisenberg spin chain in a magnetic field, J. Phys. A 52 (2019) 255302 [arXiv:1901.05878] [INSPIRE].
N. Beisert et al., Review of AdS/CFT Integrability: An Overview, Lett. Math. Phys. 99 (2012) 3 [arXiv:1012.3982] [INSPIRE].
A.V. Kotikov and L.N. Lipatov, DGLAP and BFKL equations in the N = 4 supersymmetric gauge theory, Nucl. Phys. B 661 (2003) 19 [Erratum ibid. 685 (2004) 405] [hep-ph/0208220] [INSPIRE].
A.V. Belitsky, V.M. Braun, A.S. Gorsky and G.P. Korchemsky, Integrability in QCD and beyond, Int. J. Mod. Phys. A 19 (2004) 4715 [hep-th/0407232] [INSPIRE].
M. Alfimov, N. Gromov and V. Kazakov, QCD Pomeron from AdS/CFT Quantum Spectral Curve, JHEP 07 (2015) 164 [arXiv:1408.2530] [INSPIRE].
N. Gromov, V. Kazakov, G.P. Korchemsky, S. Negro and G. Sizov, Integrability of Conformal Fishnet Theory, JHEP 01 (2018) 095 [arXiv:1706.04167] [INSPIRE].
F.H.L. Essler, H. Frahm and H. Saleur, Continuum limit of the integrable sl(2/1) 3–\( \overline{3} \) superspin chain, Nucl. Phys. B 712 (2005) 513 [cond-mat/0501197] [INSPIRE].
H. Frahm, F.H.L. Essler and H. Saleur, The integrable sl(2/1) superspin chain and the spin quantum Hall effect, in Advances in Solid State Physics 45, Springer (2005), pp. 185–197.
Y. Ikhlef, J.L. Jacobsen and H. Saleur, A staggered six-vertex model with non-compact continuum limit, Nucl. Phys. B 789 (2008) 483 [INSPIRE].
Y. Ikhlef, J.L. Jacobsen and H. Saleur, An Integrable spin chain for the SL(2, ℝ)/U(1) black hole σ-model, Phys. Rev. Lett. 108 (2012) 081601 [arXiv:1109.1119] [INSPIRE].
V.V. Bazhanov, G.A. Kotousov, S.M. Koval and S.L. Lukyanov, On the scaling behaviour of the alternating spin chain, JHEP 08 (2019) 087 [arXiv:1903.05033] [INSPIRE].
E. Vernier, J.L. Jacobsen and H. Saleur, Non compact conformal field theory and the \( {a}_2^{(2)} \) (Izergin-Korepin) model in regime III, J. Phys. A 47 (2014) 285202 [arXiv:1404.4497] [INSPIRE].
E. Vernier, J.L. Jacobsen and H. Saleur, Non compact continuum limit of two coupled Potts models, J. Stat. Mech. 1410 (2014) P10003 [arXiv:1406.1353].
E. Vernier, J.L. Jacobsen and H. Saleur, A new look at the collapse of two-dimensional polymers, J. Stat. Mech. 1509 (2015) P09001.
N.F. Robertson and J.L. Jacobsen and H. Saleur, SL(2, ℂ), in preparation (2020).
W. Toda, Wave propagation in anharmonic lattices, J. Phys. Soc. Jpn. 23 (1967) 501.
M. Gaudin, La fonction d’onde de Bethe, Masson (1983).
J.T. Chalker and P.D. Coddington, Percolation, quantum tunnelling and the integer Hall effect, J. Phys. C 21 (1988) 2665.
R. Frassek, C. Giardinà and J. Kurchan, Non-compact quantum spin chains as integrable stochastic particle processes, J. Stat. Phys. 180 (2020) 135 [arXiv:1904.01048] [INSPIRE].
Y. Ikhlef, P. Fendley and J. Cardy, An Integrable modification of the critical Chalker-Coddington network model, Phys. Rev. B 84 (2011) 144201 [arXiv:1103.3368] [INSPIRE].
R. Couvreur, E. Vernier, J.L. Jacobsen and H. Saleur, On truncations of the Chalker-Coddington model, Nucl. Phys. B 941 (2019) 507 [arXiv:1809.07429] [INSPIRE].
N.A. Slavnov, Algebraic Bethe ansatz, arXiv:1804.07350 [INSPIRE].
P.P. Kulish, N.Y. Reshetikhin and E.K. Sklyanin, Yang-Baxter Equation and Representation Theory. 1, Lett. Math. Phys. 5 (1981) 393 [INSPIRE].
V.O. Tarasov, L.A. Takhtajan and L.D. Faddeev, Local Hamiltonians for integrable quantum models on a lattice, Theor. Math. Phys. 57 (1983) 1059 [Teor. Mat. Fiz. 57 (1983) 163] [INSPIRE].
W. Heisenberg, Zur Theorie des Ferromagnetismus, Z. Phys. 49 (1928) 619 [INSPIRE].
E. Granet and J.L. Jacobsen, On zero-remainder conditions in the Bethe ansatz, JHEP 03 (2020) 178 [arXiv:1910.07797] [INSPIRE].
I. Gelfand and M. Neumark, Unitary representations of the Lorentz group, Acad. Sci. USSR. J. Phys. 10 (1946) 93.
V. Bargmann, Irreducible unitary representations of the Lorentz group, Annals Math. 48 (1947) 568 [INSPIRE].
I.M. Gelfand, M.I. Graev and N.Y. Vilenkin, Generalized functions. Volume 5. Integral geometry and representation theory, NY Academic Press, New York U.S.A. (1966).
S.E. Derkachov, Baxter’s Q-operator for the homogeneous XXX spin chain, J. Phys. A 32 (1999) 5299 [solv-int/9902015] [INSPIRE].
B. Riemann, Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse, Ph.D. Thesis, University of Göttingen, Göttingen Germany (1851).
S.E. Derkachov, G.P. Korchemsky and A.N. Manashov, Separation of variables for the quantum SL(2, ℝ) spin chain, JHEP 07 (2003) 047 [hep-th/0210216] [INSPIRE].
V.M. Braun, S.E. Derkachov and A.N. Manashov, Integrability of three particle evolution equations in QCD, Phys. Rev. Lett. 81 (1998) 2020 [hep-ph/9805225] [INSPIRE].
V.M. Braun, S.E. Derkachov, G.P. Korchemsky and A.N. Manashov, Baryon distribution amplitudes in QCD, Nucl. Phys. B 553 (1999) 355 [hep-ph/9902375] [INSPIRE].
A.S. Gorsky, I.I. Kogan and G.P. Korchemsky, High energy QCD: Stringy picture from hidden integrability, JHEP 05 (2002) 053 [hep-th/0204183] [INSPIRE].
A.V. Belitsky, Fine structure of spectrum of twist-three operators in QCD, Phys. Lett. B 453 (1999) 59 [hep-ph/9902361] [INSPIRE].
A.V. Belitsky, Integrability and WKB solution of twist-three evolution equations, Nucl. Phys. B 558 (1999) 259 [hep-ph/9903512] [INSPIRE].
A.V. Belitsky, Renormalization of twist-three operators and integrable lattice models, Nucl. Phys. B 574 (2000) 407 [hep-ph/9907420] [INSPIRE].
G.P. Korchemsky, Quasiclassical QCD Pomeron, Nucl. Phys. B 462 (1996) 333 [hep-th/9508025] [INSPIRE].
G.P. Korchemsky, Integrable structures and duality in high-energy QCD, Nucl. Phys. B 498 (1997) 68 [hep-th/9609123] [INSPIRE].
N. Beisert, S. Frolov, M. Staudacher and A.A. Tseytlin, Precision spectroscopy of AdS/CFT, JHEP 10 (2003) 037 [hep-th/0308117] [INSPIRE].
A.V. Belitsky, A.S. Gorsky and G.P. Korchemsky, Logarithmic scaling in gauge/string correspondence, Nucl. Phys. B 748 (2006) 24 [hep-th/0601112] [INSPIRE].
K. Hao, D. Kharzeev and V. Korepin, Bethe Ansatz for XXX chain with negative spin, Int. J. Mod. Phys. A 34 (2019) 1950197 [arXiv:1909.00800] [INSPIRE].
M. Kirch and A.N. Manashov, Noncompact SL(2, ℝ) spin chain, JHEP 06 (2004) 035 [hep-th/0405030] [INSPIRE].
P.A.M. Dirac, Unitary Representations of the Lorentz Group, Proc. Roy. Soc. Lond. A 183 (1945) 284 [INSPIRE].
H.J. De Vega and L.N. Lipatov, Interaction of reggeized gluons in the Baxter-Sklyanin representation, Phys. Rev. D 64 (2001) 114019 [hep-ph/0107225] [INSPIRE].
J. Wosiek and R.A. Janik, Solution of the odderon problem for arbitrary conformal weights, Phys. Rev. Lett. 79 (1997) 2935 [hep-th/9610208] [INSPIRE].
R.A. Janik and J. Wosiek, Solution of the odderon problem, Phys. Rev. Lett. 82 (1999) 1092 [hep-th/9802100] [INSPIRE].
V.E. Korepin, N.M. Bogoliubov and A.G. Izergin, Quantum inverse scattering method and correlation functions, Cambridge University Press (1993).
E. Granet, L. Budzynski, J. Dubail and J.L. Jacobsen, Inhomogeneous Gaussian Free Field inside the interacting arctic curve, J. Stat. Mech. 1901 (2019) 013102 [arXiv:1807.07927].
T. Fukui and N. Kawakami, Spectral flow of non-hermitian Heisenberg spin chain with complex twist, Nucl. Phys. B 519 (1998) 715 [cond-mat/9802128] [INSPIRE].
J.D. Noh and D. Kim, Finite-size scaling and the toroidal partition function of the critical asymmetric six-vertex model, Phys. Rev. E 53 (1996) 3225 [cond-mat/9511001].
C.-N. Yang and C.P. Yang, One-dimensional chain of anisotropic spin spin interactions. 1. Proof of Bethe’s hypothesis for ground state in a finite system, Phys. Rev. 150 (1966) 321 [INSPIRE].
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Granet, E., Jacobsen, J.L. & Saleur, H. Analytic continuation of Bethe energies and application to the thermodynamic limit of the SL(2, ℂ) non-compact spin chains. J. High Energ. Phys. 2020, 69 (2020). https://doi.org/10.1007/JHEP08(2020)069
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DOI: https://doi.org/10.1007/JHEP08(2020)069