Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 044, 29 pages      arXiv:1701.07279
Contribution to the Special Issue on Recent Advances in Quantum Integrable Systems

Integrable Structure of Multispecies Zero Range Process

Atsuo Kuniba a, Masato Okado b and Satoshi Watanabe a
a) Institute of Physics, Graduate School of Arts and Sciences, University of Tokyo, Komaba, Tokyo 153-8902, Japan
b) Department of Mathematics, Osaka City University, 3-3-138, Sugimoto, Sumiyoshi-ku, Osaka, 558-8585, Japan

Received January 26, 2017, in final form June 07, 2017; Published online June 17, 2017

We present a brief review on integrability of multispecies zero range process in one dimension introduced recently. The topics range over stochastic $R$ matrices of quantum affine algebra $U_q (A^{(1)}_n)$, matrix product construction of stationary states for periodic systems, $q$-boson representation of Zamolodchikov-Faddeev algebra, etc. We also introduce new commuting Markov transfer matrices having a mixed boundary condition and prove the factorization of a family of $R$ matrices associated with the tetrahedron equation and generalized quantum groups at a special point of the spectral parameter.

Key words: integrable zero range process; stochastic $R$ matrix; matrix product formula.

pdf (787 kb)   tex (201 kb)


  1. Alcaraz F.C., Lazo M.J., Exact solutions of exactly integrable quantum chains by a matrix product ansatz, J. Phys. A: Math. Gen. 37 (2004), 4149-4182, cond-mat/0312373.
  2. Arita C., Kuniba A., Sakai K., Sawabe T., Spectrum of a multi-species asymmetric simple exclusion process on a ring, J. Phys. A: Math. Theor. 42 (2009), 345002, 41 pages, arXiv:0904.1481.
  3. Baxter R.J., Exactly solved models in statistical mechanics, Dover Publications, Mineola, N.Y., 2007.
  4. Bazhanov V.V., Sergeev S.M., Zamolodchikov's tetrahedron equation and hidden structure of quantum groups, J. Phys. A: Math. Gen. 39 (2006), 3295-3310, hep-th/0509181.
  5. Belitsky V., Schütz G.M., Quantum algebra symmetry of the ASEP with second-class particles, J. Stat. Phys. 161 (2015), 821-842, arXiv:1504.06958.
  6. Blythe R.A., Evans M.R., Nonequilibrium steady states of matrix-product form: a solver's guide, J. Phys. A: Math. Theor. 40 (2007), R333-R441, arXiv:0706.1678.
  7. Borodin A., Corwin I., Gorin V., Stochastic six-vertex model, Duke Math. J. 165 (2016), 563-624, arXiv:1407.6729.
  8. Borodin A., Petrov L., Higher spin six vertex model and symmetric rational functions, arXiv:1601.05770.
  9. Bosnjak G., Mangazeev V.V., Construction of $R$-matrices for symmetric tensor representations related to $U_q(\widehat{sl_n})$, J. Phys. A: Math. Theor. 49 (2016), 495204, 19 pages, arXiv:1607.07968.
  10. Cantini L., de Gier J., Wheeler M., Matrix product formula for Macdonald polynomials, J. Phys. A: Math. Theor. 48 (2015), 384001, 25 pages, arXiv:1505.00287.
  11. Cantini L., Garbali A., de Gier J., Wheeler M., Koornwinder polynomials and the stationary multi-species asymmetric exclusion process with open boundaries, J. Phys. A: Math. Theor. 49 (2016), 444002, 23 pages, arXiv:1607.00039.
  12. Corwin I., The $q$-Hahn boson process and $q$-Hahn TASEP, Int. Math. Res. Not. 2015 (2015), 5577-5603.
  13. Corwin I., Petrov L., Stochastic higher spin vertex models on the line, Comm. Math. Phys. 343 (2016), 651-700, arXiv:1502.07374.
  14. Crampe N., Ragoucy E., Vanicat M., Integrable approach to simple exclusion processes with boundaries. Review and progress, J. Stat. Mech. Theory Exp. (2014), P11032, 42 pages, arXiv:1408.5357.
  15. Derrida B., Evans M.R., Hakim V., Pasquier V., Exact solution of a $1$D asymmetric exclusion model using a matrix formulation, J. Phys. A: Math. Gen. 26 (1993), 1493-1517.
  16. Evans M.R., Hanney T., Nonequilibrium statistical mechanics of the zero-range process and related models, J. Phys. A: Math. Gen. 38 (2005), R195-R240, cond-mat/0501338.
  17. Evans M.R., Majumdar S.N., Zia R.K.P., Factorized steady states in mass transport models, J. Phys. A: Math. Gen. 37 (2004), L275-L280, cond-mat/0406524.
  18. Faddeyev L.D., Quantum completely integrable models in field theory, in Mathematical Physics Reviews, Vol. 1, Soviet Sci. Rev. Sect. C: Math. Phys. Rev., Vol. 1, Harwood Academic, Chur, 1980, 107-155.
  19. Frenkel I.B., Reshetikhin N.Yu., Quantum affine algebras and holonomic difference equations, Comm. Math. Phys. 146 (1992), 1-60.
  20. Garbali A., de Gier J., Wheeler M., A new generalisation of Macdonald polynomials, Comm. Math. Phys. 352 (2017), 773-804, arXiv:1605.07200.
  21. Großkinsky S., Schütz G.M., Spohn H., Condensation in the zero range process: stationary and dynamical properties, J. Stat. Phys. 113 (2003), 389-410, cond-mat/0302079.
  22. Gwa L.-H., Spohn H., Bethe solution for the dynamical-scaling exponent of the noisy Burgers equation, Phys. Rev. A 46 (1992), 844-854.
  23. Jimbo M., A $q$-analogue of $U({\mathfrak g}{\mathfrak l}(N+1))$, Hecke algebra, and the Yang-Baxter equation, Lett. Math. Phys. 11 (1986), 247-252.
  24. Jimbo M. (Editor), Yang-Baxter equation in integrable systems, Adv. Ser. Math. Phys., Vol. 10, World Sci. Publ., Teaneck, NJ, 1990.
  25. Kapranov M.M., Voevodsky V.A., $2$-categories and Zamolodchikov tetrahedra equations, in Algebraic Groups and their Generalizations: Quantum and Infinite-Dimensional Methods (University Park, PA, 1991), Proc. Sympos. Pure Math., Vol. 56, Amer. Math. Soc., Providence, RI, 1994, 177-259.
  26. Kipnis C., Landim C., Scaling limits of interacting particle systems, Grundlehren der Mathematischen Wissenschaften, Vol. 320, Springer-Verlag, Berlin, 1999.
  27. Kuan J., An algebraic construction of duality functions for the stochastic $U_q(A_n^{(1)})$ vertex model and its degenerations, arXiv:1701.04468.
  28. Kulish P.P., Reshetikhin N.Yu., Sklyanin E.K., Yang-Baxter equations and representation theory. I, Lett. Math. Phys. 5 (1981), 393-403.
  29. Kuniba A., Mangazeev V.V., Maruyama S., Okado M., Stochastic $R$ matrix for $U_q(A_n^{(1)})$, Nuclear Phys. B 913 (2016), 248-277, arXiv:1604.08304.
  30. Kuniba A., Maruyama S., Okado M., Multispecies TASEP and the tetrahedron equation, J. Phys. A: Math. Theor. 49 (2016), 114001, 22 pages, arXiv:1509.09018.
  31. Kuniba A., Maruyama S., Okado M., Multispecies totally asymmetric zero range process: I. Multiline process and combinatorial $R$, J. Integrable Syst. 1 (2016), xyw002, 30 pages, arXiv:1511.09168.
  32. Kuniba A., Maruyama S., Okado M., Multispecies totally asymmetric zero range process: II. Hat relation and tetrahedron equation, J. Integrable Syst. 1 (2016), xyw008, 20 pages, arXiv:1602.04574.
  33. Kuniba A., Okado M., Tetrahedron equation and quantum $R$ matrices for $q$-oscillator representations of $U_q(A^{(2)}_{2n})$, $U_q(C^{(1)}_n)$ and $U_q(D^{(2)}_{n+1})$, Comm. Math. Phys. 334 (2015), 1219-1244, arXiv:1311.4258.
  34. Kuniba A., Okado M., Matrix product formula for $U_q(A^{(1)}_2)$-zero range process, J. Phys. A: Math. Theor. 50 (2017), 044001, 20 pages, arXiv:1608.02779.
  35. Kuniba A., Okado M., A $q$-boson representation of Zamolodchikov-Faddeev algebra for stochastic $R$ matrix of $U_q\big(A^{(1)}_n\big)$, Lett. Math. Phys. 107 (2017), 1111-1130, arXiv:1610.00531.
  36. Kuniba A., Okado M., Sergeev S., Tetrahedron equation and generalized quantum groups, J. Phys. A: Math. Theor. 48 (2015), 304001, 38 pages, arXiv:1503.08536.
  37. Machida S., Quantized superalgebras and generalized quantum groups, Master Thesis, Osaka City University, 2017.
  38. Motegi K., Sakai K., Vertex models, TASEP and Grothendieck polynomials, J. Phys. A: Math. Theor. 46 (2013), 355201, 26 pages, arXiv:1305.3030.
  39. Povolotsky A.M., On the integrability of zero-range chipping models with factorized steady states, J. Phys. A: Math. Theor. 46 (2013), 465205, 25 pages, arXiv:1308.3250.
  40. Prolhac S., Evans M.R., Mallick K., The matrix product solution of the multispecies partially asymmetric exclusion process, J. Phys. A: Math. Theor. 42 (2009), 165004, 25 pages, arXiv:0812.3293.
  41. Sasamoto T., Wadati M., Stationary state of integrable systems in matrix product form, J. Phys. Soc. Japan 66 (1997), 2618-2627.
  42. Sasamoto T., Wadati M., Exact results for one-dimensional totally asymmetric diffusion models, J. Phys. A: Math. Gen. 31 (1998), 6057-6071.
  43. Sergeev S.M., Classical integrable field theories in discrete $(2+1)$-dimensional spacetime, J. Phys. A: Math. Theor. 42 (2009), 295206, 19 pages, arXiv:0902.4265.
  44. Spitzer F., Interaction of Markov processes, Adv. Math. 5 (1970), 246-290.
  45. Takeyama Y., A deformation of affine Hecke algebra and integrable stochastic particle system, J. Phys. A: Math. Theor. 47 (2014), 465203, 19 pages, arXiv:1407.1960.
  46. Takeyama Y., Algebraic construction of multi-species $q$-Boson system, arXiv:1507.02033.
  47. Tracy C.A., Widom H., On the asymmetric simple exclusion process with multiple species, J. Stat. Phys. 150 (2013), 457-470, arXiv:1105.4906.
  48. Yamane H., On defining relations of affine Lie superalgebras and affine quantized universal enveloping superalgebras, Publ. Res. Inst. Math. Sci. 35 (1999), 321-390, q-alg/9603015.
  49. Yang C.N., Some exact results for the many-body problem in one dimension with repulsive delta-function interaction, Phys. Rev. Lett. 19 (1967), 1312-1315.
  50. Zamolodchikov A.B., Tetrahedra equations and integrable systems in three-dimensional space, Soviet Phys. JETP 79 (1980), 641-664.
  51. Zamolodchikov A.B., Zamolodchikov A.B., Factorized $S$-matrices in two dimensions as the exact solutions of certain relativistic quantum field theory models, Ann. Physics 120 (1979), 253-291.

Previous article  Next article   Contents of Volume 13 (2017)