zbMATH — the first resource for mathematics

Algebraic geometry and Bethe ansatz. I: The quotient ring for BAE. (English) Zbl 1388.81440
Summary: In this paper and upcoming ones, we initiate a systematic study of Bethe ansatz equations for integrable models by modern computational algebraic geometry. We show that algebraic geometry provides a natural mathematical language and powerful tools for understanding the structure of solution space of Bethe ansatz equations. In particular, we find novel efficient methods to count the number of solutions of Bethe ansatz equations based on Gröbner basis and quotient ring. We also develop analytical approach based on companion matrix to perform the sum of on-shell quantities over all physical solutions without solving Bethe ansatz equations explicitly. To demonstrate the power of our method, we revisit the completeness problem of Bethe ansatz of Heisenberg spin chain, and calculate the sum rules of OPE coefficients in planar \( \mathcal{N}=4 \) super-Yang-Mills theory.

81T25 Quantum field theory on lattices
81R12 Groups and algebras in quantum theory and relations with integrable systems
81T13 Yang-Mills and other gauge theories in quantum field theory
81T60 Supersymmetric field theories in quantum mechanics
FGb; GBLA; GitHub; SINGULAR; slimgb
Full Text: DOI arXiv
[1] H. Bethe, On the theory of metals. 1. Eigenvalues and eigenfunctions for the linear atomic chain (Zur theorie der metalle), Z. Phys.71 (1931) 205 [INSPIRE].
[2] V.E. Korepin, N.M. Bogoliubov and A.G. Izergin, Quantum inverse scattering method and correlation functions, Cambridge University Press, Cambridge U.K. (1993). · Zbl 0787.47006
[3] L.D. Faddeev, How algebraic Bethe ansatz works for integrable model, hep-th/9605187 [INSPIRE].
[4] Reshetikhin, NY, A method of functional equations in the theory of exactly solvable quantum systems, Lett. Math. Phys., 7, 205, (1983)
[5] Y. Wang, W.-L. Yang, J. Cao and K. Shi, Off-diagonal Bethe ansatz for exactly solvable models, Springer, Germany (2015). · Zbl 1341.82003
[6] Takahashi, M., One-dimensional Hubbard model at finite temperature, Prog. Theor. Phys., 47, 69, (1972)
[7] Sutherland, B., Low-lying eigenstates of the one-dimensional Heisenberg ferromagnet for any magnetization and momentum, Phys. Rev. Lett., 74, 816, (1995)
[8] Beisert, N.; Minahan, JA; Staudacher, M.; Zarembo, K., Stringing spins and spinning strings, JHEP, 09, 010, (2003)
[9] Kazakov, VA; Marshakov, A.; Minahan, JA; Zarembo, K., Classical/quantum integrability in AdS/CFT, JHEP, 05, 024, (2004)
[10] R.P. Langlands and Y. Saint-Aubin, Algebro-geometric aspects of the Bethe equations, in Strings and symmetries, G. Aktas et al. eds., Springer, Germany (1995).
[11] W. Hao, R.I. Nepomechie and A.J. Sommese, Completeness of solutions of Bethe’s equations, Phys. Rev.E 88 (2013) 052113 [arXiv:1308.4645] [INSPIRE].
[12] Vieira, P.; Wang, T., Tailoring non-compact spin chains, JHEP, 10, 035, (2014)
[13] B. Basso et al., Asymptotic four point functions, arXiv:1701.04462 [INSPIRE]. · Zbl 0591.65046
[14] Huang, R.; Rao, J.; Feng, B.; He, Y-H, An algebraic approach to the scattering equations, JHEP, 12, 056, (2015)
[15] M. Sogaard and Y. Zhang, Scattering equations and global duality of residues, Phys. Rev.D 93 (2016) 105009 [arXiv:1509.08897] [INSPIRE].
[16] J. Bosma, M. Sogaard and Y. Zhang, The polynomial form of the scattering equations is an H-basis, Phys. Rev.D 94 (2016) 041701 [arXiv:1605.08431] [INSPIRE]. · Zbl 0152.46301
[17] Cachazo, F.; He, S.; Yuan, EY, Scattering of massless particles in arbitrary dimensions, Phys. Rev. Lett., 113, 171601, (2014)
[18] F. Cachazo, S. He and E.Y. Yuan, Scattering equations and Kawai-Lewellen-Tye orthogonality, Phys. Rev.D 90 (2014) 065001 [arXiv:1306.6575] [INSPIRE].
[19] Cachazo, F.; He, S.; Yuan, EY, Scattering of massless particles: scalars, gluons and gravitons, JHEP, 07, 033, (2014) · Zbl 1391.81198
[20] Cachazo, F.; He, S.; Yuan, EY, Einstein-Yang-Mills scattering amplitudes from scattering equations, JHEP, 01, 121, (2015) · Zbl 1388.81917
[21] Cachazo, F.; He, S.; Yuan, EY, Scattering equations and matrices: from Einstein to Yang-Mills, DBI and NLSM, JHEP, 07, 149, (2015) · Zbl 1388.83196
[22] R. Hartshorne, Algebraic geometry, Springer, Germany (1977). · Zbl 0367.14001
[23] P. Griffiths and J. Harris, Principles of algebraic geometry, Wiley Classics Library. John Wiley & Sons, Inc., New York U.S.A. (1994).
[24] D.A. Cox, J. Little and D. O’Shea, Ideals, varieties, and algorithms, Undergraduate Texts in Mathematics, Springer, Germany (2015).
[25] D.A. Cox, J.B. Little and D. O’Shea, Using algebraic geometry, Graduate texts in mathematics, Springer, Germany (1998).
[26] Y. Zhang, Lecture notes on multi-loop integral reduction and applied algebraic geometry, arXiv:1612.02249 [INSPIRE].
[27] Buchberger, B., A theoretical basis for the reduction of polynomials to canonical forms, SIGSAM Bull., 10, 19, (1976)
[28] Faugére, J-C, A new efficient algorithm for computing Gröbner bases (F4), J. Pure Appl. Alg., 139, 61, (1999) · Zbl 0930.68174
[29] J.C. Faugère, A new efficient algorithm for computing Gröbner bases without reduction to zero (F5), in the proceedings of the 2002 International Symposium on Symbolic and Algebraic Computation (ISSAC’02), July 7-10, Lille, France (2002).
[30] W. Decker, G.-M. Greuel, G. Pfister and H. Schönemann, Singular 4-1-0 — A computer algebra system for polynomial computations, http://www.singular.uni-kl.de (2016).
[31] W. Hao, R.I. Nepomechie and A.J. Sommese, Singular solutions, repeated roots and completeness for higher-spin chains, J. Stat. Mech.1403 (2014) P03024 [arXiv:1312.2982] [INSPIRE].
[32] Takahashi, M., One-dimensional Heisenberg model at finite temperature, Prog. Theor. Phys., 46, 401, (1971)
[33] Kirillov, A., Combinatorial identities, and completeness of eigenstates of the Heisenberg magnet, J. Math. Sci., 30, 2298, (1985)
[34] Avdeev, LV; Vladimirov, AA, Exceptional solutions to the Bethe ansatz equations, Theor. Math. Phys., 69, 1071, (1986)
[35] F.H.L. Essler, V.E. Korepin and K. Schoutens, Fine structure of the Bethe ansatz for the spin-1/2 Heisenberg XXX model, J. Phys.A 25 (1992) 4115.
[36] R. Siddharthan, Singularities in the Bethe solution of the XXX and XXZ Heisenberg spin chains, cond-mat/9804210. · Zbl 0930.68174
[37] J.D. Noh, D.S. Lee and D. Kim, Origin of the singular Bethe ansatz solutions for the Heisenberg XXZ spin chain, PhysicaA 287 (2000) 167 [cond-mat/0001175].
[38] K. Fabricius and B.M. McCoy, Bethe’s equation is incomplete for the XXZ model at roots of unity, J. Statist. Phys.103 (2001) 647 [cond-mat/0009279] [INSPIRE]. · Zbl 1019.82003
[39] R.J. Baxter, Completeness of the Bethe ansatz for the six and eight vertex models, J. Statist. Phys.108 (2002) 1 [cond-mat/0111188] [INSPIRE]. · Zbl 1067.82014
[40] E. Mukhin, V. Tarasov and A. Varchenko, Bethe algebra of homogeneous XXX Heisenberg model has simple spectrum, arXiv:0706.0688. · Zbl 1173.82006
[41] A.N. Kirillov and R. Sakamoto, Singular solutions to the Bethe ansatz equations and rigged configurations, J. Phys.A 47 (2014) 205207 [arXiv:1402.0651]. · Zbl 1302.82034
[42] A.N. Kirillov and R. Sakamoto, Some remarks on Nepomechie-Wang eigenstates for spin 1/2 XXX model, arXiv:1406.1958. · Zbl 1336.81040
[43] R.I. Nepomechie and C. Wang, Algebraic Bethe ansatz for singular solutions, J. Phys.A 46 (2013) 325002 [arXiv:1304.7978] [INSPIRE]. · Zbl 1276.82010
[44] Bernshtein, D., The number of roots of a system of equations, Funct. Anal. Appl., 9, 183, (1975) · Zbl 0328.32001
[45] Kushnirenko, A., Newton polytopes and the Bézout theorem, Funct. Anal. Appl., 10, 233, (1976) · Zbl 0341.32001
[46] Khovanskii, A., Newton polyhedra and the genus of complete intersection, Funct. Anal. Appl., 12, 38, (1978) · Zbl 0406.14035
[47] J.C. Faugère, FGb: a library for computing Gröbner bases, in Mathematical Software — ICMS 2010, K. Fukuda et al. eds., Lecture Notes in Computer Science volume 6327, Springer, Berlin, Germany (2010).
[48] Beisert, N.; etal., Review of AdS/CFT integrability: an overview, Lett. Math. Phys., 99, 3, (2012) · Zbl 1268.81126
[49] D. Chicherin, J. Drummond, P. Heslop and E. Sokatchev, All three-loop four-point correlators of half-BPS operators in planar\( \mathcal{N}=4 \)SYM, JHEP08 (2016) 053 [arXiv:1512.02926] [INSPIRE]. · Zbl 1390.81576
[50] C. Marboe and D. Volin, Fast analytic solver of rational Bethe equations, J. Phys.A 50 (2017) 204002 [arXiv:1608.06504] [INSPIRE]. · Zbl 1367.81081
[51] C. Marboe and D. Volin, The full spectrum of AdS5/CFT4 I: Representation theory and one-loop Q-system, arXiv:1701.03704 [INSPIRE]. · Zbl 1433.81123
[52] Y. Jiang and Y. Zhang, Algebraic geometry and bethe ansatz. Part II. The quotient ring of rational Q-systems, to appear.
[53] N. Beisert and M. Staudacher, Long-range PSU (2, 2|4) Bethe Ansatze for gauge theory and strings, Nucl. Phys.B 727 (2005) 1 [hep-th/0504190] [INSPIRE]. · Zbl 1126.81328
[54] Yang, C-N, Some exact results for the many body problems in one dimension with repulsive delta function interaction, Phys. Rev. Lett., 19, 1312, (1967) · Zbl 0152.46301
[55] P.P. Kulish and N. Yu. Reshetikhin, Diagonalization of GL(\(N\)) invariant transfer matrices and quantum N wave system (Lee model), J. Phys.A 16 (1983) L591 [INSPIRE].
[56] S. Belliard and É. Ragoucy, Nested Bethe ansatz for ‘all’ closed spin chains, J. Phys.A 41 (2008) 295202 [arXiv:0804.2822] [INSPIRE]. · Zbl 1142.82326
[57] E.K. Sklyanin, Boundary conditions for integrable quantum systems, J. Phys.A 21 (1988) 2375 [INSPIRE]. · Zbl 0685.58058
[58] R.I. Nepomechie and F. Ravanini, Completeness of the Bethe ansatz solution of the open XXZ chain with nondiagonal boundary terms, J. Phys.A 36 (2003) 11391 [hep-th/0307095] [INSPIRE]. · Zbl 1039.82015
[59] J. Cao, W.-L. Yang, K. Shi and Y. Wang, Off-diagonal Bethe ansatz solution of the XXX spin-chain with arbitrary boundary conditions, Nucl. Phys.B 875 (2013) 152 [arXiv:1306.1742] [INSPIRE]. · Zbl 1282.82011
[60] E. Brattain, N. Do and A. Saenz, The completeness of the Bethe ansatz for the periodic ASEP, arXiv:1511.03762.
[61] R.S. Vieira and A. Lima-Santos, Where are the roots of the Bethe Ansatz equations?, Phys. Lett.A 379 (2015) 2150 [arXiv:1502.05316]. · Zbl 1349.82019
[62] R.S. Vieira, On the number of roots of self-inversive polynomials on the complex unit circle, arXiv:1504.00615. · Zbl 1422.30013
[63] K. Fabricius and B.M. McCoy, Bethe’s equation is incomplete for the XXZ model at roots of unity, J. Statist. Phys.103 (2001) 647 [cond-mat/0009279] [INSPIRE]. · Zbl 1019.82003
[64] R. Suzuki, Refined counting of necklaces in one-loop\( \mathcal{N}=4 \)SYM, JHEP06 (2017) 055 [arXiv:1703.05798] [INSPIRE]. · Zbl 1380.81221
[65] Izergin, AG; Korepin, VE, Pauli principle for one-dimensional bosons and the algebraic Bethe ansatz, Lett. Math. Phys., 6, 283, (1982)
[66] N. Beisert, V. Dippel and M. Staudacher, A novel long range spin chain and planar N = 4 super Yang-Mills, JHEP07 (2004) 075 [hep-th/0405001] [INSPIRE].
[67] B. Boyer et al., GBLA — Gröbner Basis Linear Algebra packagearXiv:1602.06097.
[68] C. Eder, GB: Implementation of Faugère’s F4 algorithm using gbla for the linear algebra part, https://github.com/ederc/gb (2017).
[69] Brickenstein, M., Slimgb: Gröbner bases with slim polynomials, Rev. Mat. Complutense, 23, 453, (2010) · Zbl 1200.13044
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.