The Yang-Baxter paradox. (English) Zbl 1519.81312
Summary: Consider the statement ‘Every Yang-Baxter integrable system is defined to be exactly-solvable’. To formalise this statement, definitions and axioms are introduced. Then, using a specific Yang-Baxter integrable bosonic system, it is shown that a paradox emerges. A generalisation for completely integrable bosonic systems is also developed.
MSC:
| 81R12 | Groups and algebras in quantum theory and relations with integrable systems |
| 82B23 | Exactly solvable models; Bethe ansatz |
| 82B20 | Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics |
| 16T25 | Yang-Baxter equations |
References:
| [1] | Au-Yang, H.; Perk, J. H H., Integrable Systems in Quantum Field Theory and Statistical Mechanics (1989), San Diego: Academic Press, San Diego |
| [2] | Avan, J.; Talon, M., Rational and trigonometric constant non-antisymmetric r-matrices, Phys. Lett. B, 241, 77 (1990) · doi:10.1016/0370-2693(90)91490-3 |
| [3] | Batchelor, M. T.; Foerster, A., Yang-Baxter integrable models in experiments: from condensed matter to ultracold atoms, J. Phys. A: Math. Theor., 49 (2016) · Zbl 1342.82017 · doi:10.1088/1751-8113/49/17/173001 |
| [4] | Batchelor, M. T.; Zhou, H-Q, Integrability versus exact solvability in the quantum Rabi and Dicke models, Phys. Rev. A, 91 (2015) · doi:10.1103/physreva.91.053808 |
| [5] | Baxter, R. J., Partition function of the eight-vertex lattice model, Ann. Phys., NY, 70, 193 (1972) · Zbl 0236.60070 · doi:10.1016/0003-4916(72)90335-1 |
| [6] | Baxter, R., Eight-vertex model in lattice statistics and one-dimensional anisotropic Heisenberg chain. I. Some fundamental eigenvectors, Ann. Phys., NY, 76, 1 (1973) · Zbl 1092.82511 · doi:10.1016/0003-4916(73)90439-9 |
| [7] | Baxter, R. J., Exactly Solved Models in Statistical Mechanics (1982), London: Academic, London · Zbl 0538.60093 |
| [8] | Bazhanov, V. V.; Sergeev, S. M., Zamolodchikov’s tetrahedron equation and hidden structure of quantum groups, J. Phys. A: Math. Gen., 39, 3295 (2006) · Zbl 1091.81047 · doi:10.1088/0305-4470/39/13/009 |
| [9] | Belavin, A. A.; Drinfel’d, V. G., Solutions of the classical Yang-Baxter equation for simple Lie algebras, Funct. Anal. Appl., 16, 159-180 (1982) · Zbl 0511.22011 · doi:10.1007/BF01081585 |
| [10] | Bernard, D.; Gaudin, M.; Haldane, F. D M.; Pasquier, V., Yang-Baxter equation in long-range interacting systems, J. Phys. A: Math. Gen., 26, 5219 (1993) · Zbl 0808.60086 · doi:10.1088/0305-4470/26/20/010 |
| [11] | Bethe, H., Zur theorie der metalle, Z. Phys., 71, 205 (1931) · JFM 57.1587.01 · doi:10.1007/bf01341708 |
| [12] | Braak, D., Integrability of the Rabi model, Phys. Rev. Lett., 107 (2011) · doi:10.1103/physrevlett.107.100401 |
| [13] | 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 (2013) · doi:10.1103/physrevlett.111.137201 |
| [14] | Caux, J-S; Mossel, J., Remarks on the notion of quantum integrability, J. Stat. Mech. (2011) · doi:10.1088/1742-5468/2011/02/p02023 |
| [15] | Choy, T. C.; Haldane, F. D M., Failure of Bethe-ansatz solutions of generalisations of the Hubbard chain to arbitrary permutation symmetry, Phys. Lett. A, 90, 83 (1982) · doi:10.1016/0375-9601(82)90057-3 |
| [16] | Claeys, P. W.; De Baerdemacker, S.; Van Neck, D., Read-Green resonances in a topological superconductor coupled to a bath, Phys. Rev. B, 93 (2016) · doi:10.1103/physrevb.93.220503 |
| [17] | Crossley, J. N.; Ash, C. J.; Brickhill, C. J.; Stillwell, J. C.; Williams, N. H., What Is Mathematical Logic? (1972), Oxford: Oxford University Press, Oxford · Zbl 0251.02001 |
| [18] | Dahmen, S. R.; Links, J.; McKenzie, R. H.; Zhou, H-Q, Energy level statistics for models of coupled single-mode Bose-Einstein condensates, J. Stat. Mech. (2004) · Zbl 1073.82505 · doi:10.1088/1742-5468/2004/10/p10019 |
| [19] | Davies, B.; Foda, O.; Jimbo, M.; Miwa, T.; Nakayashiki, A., Diagonalization of the XXZ Hamiltonian by vertex operators, Commun. Math. Phys., 151, 89 (1993) · Zbl 0769.17020 · doi:10.1007/bf02096750 |
| [20] | Dunning, C.; Ibañez, M.; Links, J.; Sierra, G.; Zhao, S-Y, Exact solution of the p + ip pairing Hamiltonian and a hierarchy of integrable models, J. Stat. Mech. (2010) · doi:10.1088/1742-5468/2010/08/p08025 |
| [21] | Enol’skii, V. Z.; Kuznetsov, V. B.; Salerno, M., On the quantum inverse scattering method for the DST dimer, Physica D, 68, 138 (1993) · Zbl 0801.47005 · doi:10.1016/0167-2789(93)90039-4 |
| [22] | Enol’skii, V. Z.; Salerno, M.; Kostov, N. A.; Scott, A. C., Alternate quantizations of the discrete self-trapping dimer, Phys. Scr., 43, 229 (1991) · Zbl 1063.81605 · doi:10.1088/0031-8949/43/3/002 |
| [23] | Enol’skii, V. Z.; Salerno, M.; Scott, A. C.; Eilbeck, J. C., There’s more than one way to skin Schrödinger’s cat, Physica D, 59, 1-24 (1992) · Zbl 0760.58051 · doi:10.1016/0167-2789(92)90203-y |
| [24] | Erbe, B.; Schliemann, J., Different types of integrability and their relation to decoherence in central spin models, Phys. Rev. Lett., 105 (2010) · Zbl 1208.37048 · doi:10.1103/physrevlett.105.177602 |
| [25] | Essler, F. H L.; Fagotti, M., Quench dynamics and relaxation in isolated integrable quantum spin chains, J. Stat. Mech. (2016) · Zbl 1456.82585 · doi:10.1088/1742-5468/2016/06/064002 |
| [26] | Etingof, P.; Varchenko, A., Solutions of the quantum dynamical Yang-Baxter equation and dynamical quantum groups, Commun. Math. Phys., 196, 591 (1998) · Zbl 0957.17019 · doi:10.1007/s002200050437 |
| [27] | Frahm, H.; Grelik, J. H.; Seel, A.; Wirth, T., Functional Bethe ansatz methods for the open XXX chain, J. Phys. A: Math. Theor., 44 (2011) · Zbl 1207.82010 · doi:10.1088/1751-8113/44/1/015001 |
| [28] | Franzosi, R.; Penna, V., Chaotic behavior, collective modes, and self-trapping in the dynamics of three coupled Bose-Einstein condensates, Phys. Rev. E, 67 (2003) · doi:10.1103/physreve.67.046227 |
| [29] | Freidel, L.; Maillet, J. M., Quadratic algebras and integrable systems, Phys. Lett. B, 262, 278 (1991) · Zbl 1514.17016 · doi:10.1016/0370-2693(91)91566-e |
| [30] | Gaudin, M., Diagonalisation d’une classe d’Hamiltoniens de spin, J. Phys., 37, 1087-1098 (1976) · doi:10.1051/jphys:0197600370100108700 |
| [31] | Guan, K-L; Li, Z-M; Dunning, C.; Batchelor, M. T., The asymmetric quantum Rabi model and generalised Pöschl-Teller potentials, J. Phys. A: Math. Theor., 51 (2018) · Zbl 1397.81472 · doi:10.1088/1751-8121/aacb44 |
| [32] | Jurco, B., Classical Yang-Baxter equations and quantum integrable systems, J. Math. Phys., 30, 1289-1293 (1989) · Zbl 0692.58015 · doi:10.1063/1.528305 |
| [33] | Kitaev, A., Anyons in an exactly solved model and beyond, Ann. Phys., NY, 321, 2 (2006) · Zbl 1125.82009 · doi:10.1016/j.aop.2005.10.005 |
| [34] | Larson, J., Integrability versus quantum thermalization, J. Phys. B: At. Mol. Opt. Phys., 46 (2013) · doi:10.1088/0953-4075/46/22/224016 |
| [35] | Lieb, E. H.; Liniger, W., Exact analysis of an interacting Bose gas. I. The general solution and the ground state, Phys. Rev., 130, 1605 (1963) · Zbl 0138.23001 · doi:10.1103/physrev.130.1605 |
| [36] | Lieb, E.; Schultz, T.; Mattis, D., Two soluble models of an antiferromagnetic chain, Ann. Phys., 16, 407 (1961) · Zbl 0129.46401 · doi:10.1016/0003-4916(61)90115-4 |
| [37] | Lieb, E. H.; Wu, F. Y., Absence of Mott transition in an exact solution of the short-range, one-band model in one dimension, Phys. Rev. Lett., 20, 1445 (1968) · doi:10.1103/physrevlett.20.1445 |
| [38] | Links, J., Solution of the classical Yang-Baxter equation with an exotic symmetry, and integrability of a multi-species boson tunnelling model, Nucl. Phys. B, 916, 117 (2017) · Zbl 1356.81150 · doi:10.1016/j.nuclphysb.2017.01.005 |
| [39] | Links, J.; Foerster, A., On the construction of integrable closed chains with quantum supersymmetry, J. Phys. A: Math. Gen., 30, 2483 (1997) · Zbl 0980.37026 · doi:10.1088/0305-4470/30/7/026 |
| [40] | Links, J.; Hibberd, K. E., Bethe ansatz solutions of the Bose-Hubbard dimer, Symmetry, Integrability Geometry Methods Appl., 2, 095 (2006) · Zbl 1133.81031 · doi:10.3842/sigma.2006.095 |
| [41] | Links, J.; Zhou, H-Q; McKenzie, R. H.; Gould, M. D., Ladder operator for the one-dimensional Hubbard model, Phys. Rev. Lett., 86, 5096 (2001) · doi:10.1103/physrevlett.86.5096 |
| [42] | Lukyanenko, I.; Isaac, P. S.; Links, J., An integrable case of the p + ip pairing Hamiltonian interacting with its environment, J. Phys. A: Math. Theor., 49 (2016) · Zbl 1342.81732 · doi:10.1088/1751-8113/49/8/084001 |
| [43] | Martins, M. J.; Ramos, P. B., The quantum inverse scattering method for Hubbard-like models, Nucl. Phys. B, 522, 413 (1998) · Zbl 1047.82506 · doi:10.1016/s0550-3213(98)00199-0 |
| [44] | McGuire, J. B., Study of exactly soluble one-dimensional N-body problems, J. Math. Phys., 5, 622 (1964) · Zbl 0131.43804 · doi:10.1063/1.1704156 |
| [45] | Moroz, A., On the spectrum of a class of quantum models, Europhys. Lett., 100 (2012) · doi:10.1209/0295-5075/100/60010 |
| [46] | Nepomechie, R. I.; Guan, X-W, The spin-s homogeneous central spin model: exact spectrum and dynamics, J. Stat. Mech. (2018) · Zbl 1457.82037 · doi:10.1088/1742-5468/aae2d9 |
| [47] | Nepomechie, R. I.; Wang, C., Twisting singular solutions of Bethe’s equations, J. Phys. A: Math. Theor., 47 (2014) · Zbl 1369.81036 · doi:10.1088/1751-8113/47/50/505004 |
| [48] | Oelkers, N.; Links, J., Ground-state properties of the attractive one-dimensional Bose-Hubbard model, Phys. Rev. B, 75 (2007) · doi:10.1103/physrevb.75.115119 |
| [49] | Reshetikhin, N. Y., A method of functional equations in the theory of exactly solvable quantum systems, Lett. Math. Phys., 7, 205 (1983) · doi:10.1007/bf00400435 |
| [50] | Sergeev, S., Integrability of q-oscillator lattice model, Phys. Lett. A, 357, 417 (2006) · Zbl 1236.82020 · doi:10.1016/j.physleta.2006.04.089 |
| [51] | Shastry, B. S., Exact integrability of the one-dimensional Hubbard model, Phys. Rev. Lett., 56, 2453 (1986) · Zbl 0940.82025 · doi:10.1103/physrevlett.56.2453 |
| [52] | Sklyanin, E. K., Separation of variables in the Gaudin model, J. Math. Sci., 47, 2473 (1989) · Zbl 0692.35107 · doi:10.1007/bf01840429 |
| [53] | Sklyanin, E. K., Boundary conditions for integrable quantum systems, J. Phys. A: Math. Gen., 21, 2375 (1988) · Zbl 0685.58058 · doi:10.1088/0305-4470/21/10/015 |
| [54] | Skrypnyk, T., Generalized Gaudin spin chains, nonskew symmetric r-matrices, and reflection equation algebras, J. Math. Phys., 48 (2007) · Zbl 1153.81435 · doi:10.1063/1.2816256 |
| [55] | Skrypnyk, T., Non-skew-symmetric classical r-matrices and integrable cases of the reduced BCS model, J. Phys. A: Math. Theor., 42 (2009) · Zbl 1181.81063 · doi:10.1088/1751-8113/42/47/472004 |
| [56] | Takahashi, M., One-dimensional Heisenberg model at finite temperature, Prog. Theor. Phys., 46, 401 (1971) · doi:10.1143/ptp.46.401 |
| [57] | Takhtadzhan, L. A.; Faddeev, L. D., The quantum method of the inverse problem and the Heisenberg XYZ model, Russ. Math. Surv., 34, 11 (1979) · Zbl 0449.35096 · doi:10.1070/rm1979v034n05abeh003909 |
| [58] | Torrielli, A., Classical integrability, J. Phys. A: Math. Theor., 49 (2016) · Zbl 1346.81051 · doi:10.1088/1751-8113/49/32/323001 |
| [59] | von Delft, J.; Poghossian, R., Algebraic Bethe ansatz for a discrete-state BCS pairing model, Phys. Rev. B, 66 (2002) · doi:10.1103/physrevb.66.134502 |
| [60] | Wiegmann, P. B.; Zabrodin, A. V., Bethe-ansatz for the Bloch electron in magnetic field, Phys. Rev. Lett., 72, 1890 (1994) · doi:10.1103/physrevlett.72.1890 |
| [61] | Yang, C. N., Some exact results for the many-body problem in one dimension with delta-function interaction, Phys. Rev. Lett., 19, 1312 (1967) · Zbl 0152.46301 · doi:10.1103/physrevlett.19.1312 |
| [62] | Ymai, L. H.; Tonel, A. P.; Foerster, A.; Links, J., Quantum integrable multi-well tunneling models, J. Phys. A: Math. Theor., 50 (2017) · Zbl 1368.82020 · doi:10.1088/1751-8121/aa7227 |
| [63] | Umeno, Y.; Shiroishi, M.; Wadati, M., Fermionic R-operator and integrability of the one-dimensional Hubbard model, J. Phys. Soc. Japan, 67, 2242 (1998) · Zbl 0943.82004 · doi:10.1143/jpsj.67.2242 |
| [64] | Yu, Y., Gauge symmetry in Kitaev-type spin models and index theorems on odd manifolds, Nucl. Phys. B, 799, 345 (2008) · Zbl 1292.82008 · doi:10.1016/j.nuclphysb.2008.01.029 |
| [65] | Zhang, Y-Z, On the solvability of the quantum Rabi model and its two-photon and two-mode generalizations, J. Math. Phys., 54 (2013) · Zbl 1284.81361 · doi:10.1063/1.4826356 |
| [66] | Zhou, H-Q; Links, J.; McKenzie, R. H.; Gould, M. D., Superconducting correlations in metallic nanoparticles: exact solution of the BCS model by the algebraic Bethe ansatz, Phys. Rev. B, 65 (2002) · doi:10.1103/physrevb.65.060502 |
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.
