Completeness of the Bethe ansatz for the periodic isotropic Heisenberg model. (English) Zbl 1434.82030
In this article the author studies the completeness of the Bethe ansatz for a periodic isotropic Heisenberg spin chain with arbitrary spins and inhomogeneities. The author provides a system of algebraic equations whose solutions are in bijection with the eigenvalues of the transfer matrix of the chain. The system describes pairs of polynomials with given discrete Wronskian. In this way, the author shows that the spectrum of the transfer matrix is necessarily simple modulo natural degeneration.
Reprint of [V. Tarasov, in: Ludwig Faddeev memorial volume. A life in mathematical physics. Hackensack, NJ: World Scientific. 549–566 (2018; Zbl 1397.82019)].
Reprint of [V. Tarasov, in: Ludwig Faddeev memorial volume. A life in mathematical physics. Hackensack, NJ: World Scientific. 549–566 (2018; Zbl 1397.82019)].
Reviewer: Nenad Manojlović (Faro)
Citations:
Zbl 1397.82019References:
| [1] | L. V. Avdeev and A. A. Vladimirov, Exceptional solutions of the Bethe ansatz equations, Teor. Matem. Fiz.69(2) (1986) 163-174 (in Russian); Theor. Math. Phys.69(2) (1987) 1071-1079. |
| [2] | Baxter, R., Exactly Solved Models in Statistical Mechanics (Academic Press, Inc., London, 1982). · Zbl 0538.60093 |
| [3] | Baxter, R., Completeness of the Bethe ansatz for the six- and eight-vertex models, J. Stat. Phys.108(1-2) (2002) 1-48. · Zbl 1067.82014 |
| [4] | Bethe, H., Zur Theorie der Metalle: I. Eigenwerte und Eigenfunktionen der linearen Atomkette, Z. Phys.71 (1931) 205-226. · JFM 57.1587.01 |
| [5] | Feigin, B., Frenkel, E. and Rybnikov, L., Opers with irregular singularity and spectra of the shift of argument subalgebra, Duke Math. J.155(2) (2010) 337-363. · Zbl 1226.22017 |
| [6] | L. D. Faddeev and L. A. Takhtajan, The quantum method for the inverse problem and the XYZ Heisenberg model, Uspekhi Matem. Nauk34(5) (1979) 13-63 (in Russian); Russian Math. Surveys34(5) (1979) 11-68. |
| [7] | L. D. Faddeev and L. A. Takhtajan, The spectrum and scattering of excitations in the one-dimensional isotropic Heisenberg model, Zap. Nauch. Semin. LOMI109 (1981) 134-178 (in Russian); J. Soviet Math.24 (1984) 241-267. · Zbl 0471.47009 |
| [8] | Giri, P. R. and Deguchi, T., Singular eigenstates in the even (odd) length Heisenberg spin chain, J. Phys. A48(17) (2015) 175207, 26 pp. · Zbl 1335.82008 |
| [9] | Deguchi, T. and Giri, P. R., Exact quantum numbers of collapsed and non-collapsed 2-string solutions in the Heisenberg spin chain, J. Phys. A49(17) (2016) 174001, 24 pp. · Zbl 1342.82024 |
| [10] | Gainutdinov, A. and Nepomechie, R., Algebraic Bethe ansatz for the quantum group invariant open XXZ chain at roots of unity, Nucl. Phys. B909 (2016) 796-839. · Zbl 1342.82030 |
| [11] | Hao, W., Nepomechie, R. and Sommes, A., Singular solutions, repeated roots and completeness for higher-spin chains, J. Stat. Mech. Theory Exp. (2014)(3) P03024, 20 pp. · Zbl 1456.82268 |
| [12] | A. N. Kirillov, Combinatorial identities and the completeness of states for Heisenberg magnet, Zap. Nauch. Semin. LOMI131 (1983) 88-105 (in Russian); J. Soviet Math.30 (1985) 2298-3310. |
| [13] | A. N. Kirillov, Completeness of the states for the generalized Heisenberg model, Zap. Nauch. Semin. LOMI134 (1985) 169-189 (in Russian); J. Soviet Math.36 (1987) 115-128. · Zbl 0538.58016 |
| [14] | Kirillov, A. N. and Sakamoto, R., Singular solutions to the Bethe ansatz equations and rigged configurations, J. Phys. A47(20) (2014) 205207, 20 pp. · Zbl 1302.82034 |
| [15] | Kirillov, A. N. and Sakamoto, R., Bethe’s quantum numbers and rigged configurations, Nucl. Phys. B905 (2016) 359-372. · Zbl 1332.82015 |
| [16] | Korepin, V. E., Calculation of norms of Bethe wave functions, Commun. Math. Phys.86(3) (1982) 391-418. · Zbl 0531.60096 |
| [17] | Korepin, V. E., Bogoliubov, N. M. and Izergin, A. G., Quantum Inverse Scattering Method and Correlation Functions (Cambridge University Press, 1993). · Zbl 0787.47006 |
| [18] | P. P. Kulish and N. Yu. Reshetikhin, Quantum linear problem for the sine-Gordon equation and higher representations, Zap. Nauch. Semin. LOMI101 (1981) 101-110 (in Russian); J. Soviet Math.23(4) (1983) 2435-2441. |
| [19] | Kulish, P. P., Reshetikhin, N. Yu. and Sklyanin, E. K., Yang-Baxter equations and representation theory I, Lett. Math. Phys.5(5) (1981) 393-403. · Zbl 0502.35074 |
| [20] | Langlands, R. P. and Saint-Aubin, Y., Combinational aspects of the Bethe equations, in Advances in Mathematical Sciences, , Vol. 11 (Amer. Math. Soc., 1997), pp. 231-301. · Zbl 1039.32504 |
| [21] | Li, J. R. and Tarasov, V., XXZ-type Bethe ansatz equations and quasi-polynomials, J. Phys.: Conf. Ser.411 (2013) 012020, 20 pp. |
| [22] | Mukhin, E., Tarasov, V. and Varchenko, A., Schubert calculus and representations of the general linear group, J. Amer. Math. Soc.22(4) (2009) 909-940. · Zbl 1205.17026 |
| [23] | Mukhin, E., Tarasov, V. and Varchenko, A., Generating operator of XXX or Gaudin transfer matrices has quasi-exponential kernel, SIGMA6 (2007) 060, 31 pp. · Zbl 1140.82015 |
| [24] | Mukhin, E., Tarasov, V. and Varchenko, A., Bethe algebra of homogeneous XXX Heisenberg model has simple spectrum, Comm. Math. Phys.288(1) (2009) 1-42. · Zbl 1173.82006 |
| [25] | Mukhin, E., Tarasov, V. and Varchenko, A., Bethe subalgebras of the group algebra of the symmetric group, Transform. Groups18(3) (2013) 767-801. · Zbl 1276.82014 |
| [26] | Mukhin, E., Tarasov, V. and Varchenko, A., Spaces of quasi-exponentials and representations of the Yangian \(Y(\mathfrak{g} \mathfrak{l}_N)\), Transform. Groups19(3) (2014) 861-885. · Zbl 1365.17006 |
| [27] | Mukhin, E., Tarasov, V. and Varchenko, A., On reality property of Wronski maps, Confluentes Math.1(2) (2009) 225-247. · Zbl 1181.81065 |
| [28] | Mukhin, E., Tarasov, V. and Varchenko, A., Reality property of discrete Wronski map with imaginary step, Lett. Math. Phys.100(2) (2012) 151-160. · Zbl 1299.81027 |
| [29] | Mukhin, E., Tarasov, V. and Varchenko, A., Bethe algebra of the \(\mathfrak{g} \mathfrak{l}_{N + 1}\) Gaudin model and algebra of functions on the critical set of the master function, in New Trends in Quantum Integrable Systems, eds. Feigin, B., Jimbo, M. and Okado, M. (World Scientific, 2011), pp. 307-324. · Zbl 1221.82038 |
| [30] | Mukhin, E. and Varchenko, A., Solutions to the XXX type Bethe ansatz equations and flag varieties, Cent. Eur. J. Math.1(2) (2003) 238-271. · Zbl 1029.82008 |
| [31] | Pronko, G. P. and Stroganov, Yu. G., Bethe equations “on the wrong side of equator”, J. Phys. A32(12) (1999) 2333-2340. · Zbl 0964.82013 |
| [32] | L. Rybnikov, A proof of the Gaudin Bethe ansatz conjecture, Preprint (2016) 15 pp; arXiv:1608.04625. · Zbl 1460.37057 |
| [33] | N. A. Slavnov, Calculation of scalar products of wave functions and form-factors in the framework of the algebraic Bethe ansatz, Teor. Matem. Fiz.79(2) (1989) 232-240 (in Russian); Theor. Math. Phys.79(2) (1989) 502-508. |
| [34] | Slavnov, N. A., Algebraic Bethe ansatz and quantum integrable systems, Russian Math. Surveys62(4) (2007) 727-766. · Zbl 1141.81012 |
| [35] | V. Tarasov, Irreducible monodromy matrices for the \(R\)-matrix of the XXZ-model and lattice local quantum Hamiltonians, Teor. Matem. Fiz.63(2) (1985) 175-196 (in Russian); Theor. Math. Phys.63(2) (1985) 440-454. |
| [36] | Tarasov, V. and Varchenko, A., Completeness of Bethe vectors and difference equations with regular singular points, Internat. Math. Res. Notices (1995)(13) 637-669. · Zbl 0860.17022 |
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