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Average-fluctuation separation in energy levels in many-particle quantum systems with k-body interactions using q-Hermite polynomials

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Abstract

Separation between average and fluctuation parts in state density in many-particle quantum systems with k-body interactions, modelled by the k-body embedded Gaussian orthogonal random matrices (EGOE(k)), is demonstrated using the method of normal mode decomposition of the spectra and verified using power spectrum analysis, for both fermions and bosons. The smoothed state density is represented by the q-normal distribution (\(f_{qN}\)) (with corrections) which is the weight function for q-Hermite polynomials. As the rank of interaction k increases, the fluctuations set in with smaller order of corrections in the smooth state density. They are found to be of GOE type, for all k values, for both fermion and boson systems.

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References

  1. T A Brody, J Flores, J B French, P A Mello, A Pandey and S S M Wong, Rev. Mod. Phys. 53, 385 (1981)

    Article  ADS  Google Scholar 

  2. J B French and V K B Kota, Annu. Rev. Nucl. Part. Sci. 32, 35 (1982)

    Article  ADS  Google Scholar 

  3. J B French, S Rab, J F Smith, R U Haq and V K B Kota, Can. J. Phys. 84, 677 (2006)

    Article  ADS  Google Scholar 

  4. J Karwowski, Int. J. Quantum Chem. 51, 425 (1994)

    Article  Google Scholar 

  5. V V Flambaum, A A Gribakina, G F Gribakin and I V Ponomarev, Physica D 131, 205 (1999)

    Article  ADS  Google Scholar 

  6. D Angom and V K B Kota, Phys. Rev. A 71, 042504 (2005)

    Article  ADS  Google Scholar 

  7. V K B Kota, Embedded random matrix ensembles in quantum physics (Springer-Verlag, Heidelberg, 2014)

    Book  MATH  Google Scholar 

  8. E P Wigner, Ann. Math. 62, 548 (1955)

    Article  MathSciNet  Google Scholar 

  9. O Bohigas, Random matrix theories and chaotic dynamics, Les Houches, Session LII, (1989), Chaos and Quantum Physics, Course 2, edited by M-J Giannoni et al (Elsevier Science Publishers B.V., North-Holland, 1991)

  10. H-J Stöckmann, Quantum chaos (Cambridge University Press, Cambridge, 1999)

  11. G Casati, F Valz-Gris and I Guarneri, Lett. Nuovo Cimento Soc. Ital. Fis. 28, 279 (1980)

    Article  Google Scholar 

  12. F Haake, Quantum signatures of chaos (Springer, New York, 2010)

    Book  MATH  Google Scholar 

  13. F J Dyson and M L Mehta, J. Math. Phys. 4, 701 (1963)

    Article  ADS  Google Scholar 

  14. A Polkovnikov, K Sengupta and A Silva, Rev. Mod. Phys. 83, 863 (2011)

    Article  ADS  Google Scholar 

  15. L DÁlessio, Y Kafri, A Polkovnikov and M Rigol, Adv. Phys. 65, 239 (2016)

  16. F Borgonovi, F M Izrailev, L F Santos and V G Zelevinsky, Phys. Rep. 626, 1 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  17. V K B Kota and N D Chavda, Int. J. Mod. Phys. E 27, 1830001 (2018)

    Article  ADS  Google Scholar 

  18. V K B Kota and N D Chavda, Entropy 20, 541 (2018)

    Article  ADS  Google Scholar 

  19. J S Cotler et al, J. High Energy Phys. 05, 118 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  20. L Benet, T Rupp and H A Weidenmüller, Ann. Phys. (N.Y.) 292, 67 (2001)

  21. T Papenbrock and H A Weidenmüller, Rev. Mod. Phys. 79, 997 (2007)

    Article  ADS  Google Scholar 

  22. A M Garcia-Garcia and J J M Verbaarschot, Phys. Rev. D 94, 126010 (2016); Phys. Rev. D 96, 066012 (2017)

  23. Y Jia and J J M Verbaarschot, J. High Energy Phys. 07, 193 (2020)

    Article  ADS  Google Scholar 

  24. J B French and S S M Wong, Phys. Lett. B 33, 449 (1970)

    Article  ADS  Google Scholar 

  25. O Bohigas and J Flores, Phys. Lett. B 34, 261 (1971)

    Article  ADS  Google Scholar 

  26. V V Flambaum, G F Gribakin and F M Izrailev, Phys. Rev. E 53, 5729 (1996)

    Article  ADS  Google Scholar 

  27. V V Flambaum and F M Izrailev, Phys. Rev. E 56, 5144 (1997)

    Article  ADS  Google Scholar 

  28. K K Mon and J B French, Ann. Phys. (N.Y.) 95, 90 (1975)

  29. V K B Kota, Phys. Rep. 347, 223 (2001)

    Article  ADS  MathSciNet  Google Scholar 

  30. K Patel, M S Desai, V Potbhare and V K B Kota, Phys. Lett. A 275, 329 (2000)

    Article  ADS  Google Scholar 

  31. T Asaga, L Benet, T Rupp and H A Weidenmüller, Eur. Phys. Lett. 56, 340 (2001); Ann. Phys. (N.Y.) 298, 229 (2002)

  32. N D Chavda, V Potbhare and V K B Kota, Phys. Lett. A 311, 331 (2003); Phys. Lett. A 326, 47 (2004)

  33. N D Chavda, V K B Kota and V Potbhare, Phys. Lett. A 376, 2972 (2012)

    Article  ADS  Google Scholar 

  34. R J Leclair, R U Haq, V K B Kota and N D Chavda, Phys. Lett. A 372, 4373 (2008)

    Article  ADS  Google Scholar 

  35. G J H Laberge and R U Haq, Can. J. Phys. 68, 301 (1990)

    Article  ADS  Google Scholar 

  36. M Vyas and V K B Kota, J. Stat. Mech. Theor. Exp. 10, 103103 (2019)

    Article  Google Scholar 

  37. M Vyas and V K B Kota, J. Stat. Mech. Theor. Exp. 2020, 093101 (2020)

    Article  Google Scholar 

  38. P N Rao and N D Chavda, Phys. Lett. A 399, 127302 (2021)

    Article  Google Scholar 

  39. V K B Kota and M Vyas, J. Stat. Mech. Theor. Exp. 2021, 113103 (2021)

    Article  Google Scholar 

  40. K D Launey, T Dytrych and J P Draayer, Phys. Rev. C 85, 044003 (2012)

    Article  ADS  Google Scholar 

  41. D W E Blatt and B H J McKellar, Phys. Rev. C 11, 2040 (1975)

    Article  ADS  Google Scholar 

  42. H-W Hammer, A Nogga and A Schwenk, Rev. Mod. Phys. 85, 197 (2013)

    Article  ADS  Google Scholar 

  43. A M Garcia-Garcia, T Nosaka, D Rosa and J J M Verbaarschot, Phys. Rev. D 100, 026002 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  44. A M Garcia-Garcia, Y Jia and J J M Verbaarschot, Phys. Rev. D 97, 106003 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  45. A Ortega, M Vyas and L Benet, Ann. Phys. (Berlin) 527, 748 (2015)

    Article  ADS  Google Scholar 

  46. A Ortega and T Stegmann and L Benet, Phys. Rev. E 94, 042102 (2016); Phys. Rev. E 98, 012141 (2018)

  47. V K B Kota and R Sahu, Phys. Rev. E 64, 016219 (2001)

    Article  ADS  Google Scholar 

  48. J J M Verbaarschot and M R Zirnbauer, Ann. Phys. (N.Y.) 158, 78 (1984)

  49. T Papenbrock, Z Pluhar̀, J Tithof and H A Weidenmüller, Phys. Rev. E 83, 031130 (2011)

    Article  ADS  Google Scholar 

  50. L J Rogers, Proc. London Math. Soc. 24, 337 (1893); M E H Ismail, D Stanton and G Viennot, Eur. J. Combinatorics 8, 379 (1987)

    Article  Google Scholar 

  51. P J Szabowski, Electron. J. Probab. 15, 1296 (2010)

    MathSciNet  Google Scholar 

  52. G Szegö, Orthogonal polynomials, colloquium publications (Am. Math. Soc., Providence, 2003) Vol. 23

  53. H N Deota, N D Chavda and V Potbhare, Pramana – J. Phys. 81, 1045 (2013)

  54. P N Rao, H N Deota and N D Chavda, Pramana – J. Phys. 95, 34 (2021)

  55. A Relaño, J M G Gómez, R A Molina and J Retamosa, Phys. Rev. Lett. 89, 244102 (2002)

    Article  ADS  Google Scholar 

  56. E Faleiro, J M G Gómez, R A Molina, L Muñoz, A Relaño and J Retamosa, Phys. Rev. Lett. 93, 244101 (2004)

    Article  ADS  Google Scholar 

  57. J M G Gómez, A Relaño, J Retamosa, E Faleiro, L Salasnich, M Vranicar and M Robnik, Phys. Rev. Lett. 94, 084101 (2005)

    Article  ADS  Google Scholar 

  58. N R Lomb, Appl. Space Sci. 39, 447 (1976); J D Scargle, Astrophys. J. 263, 835 (1982)

  59. M L Mehta, Random matrices, 3rd Edn (Elsevier, Netherlands, 2004)

    MATH  Google Scholar 

  60. O Bohigas, R U Haq and A Pandey, in: Nuclear data for science and technology edited by K H Böckho (Reidel, Dordrecht, 1983) p. 809

    Book  Google Scholar 

Download references

Acknowledgements

Author thanks V K B Kota and V Potbhare for many useful discussions. This work is a part of University supported research project (Grant No: GCU/RCC/2021-22/20-32/508.

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  1. Department of Applied Physics, Faculty of Technology and Engineering, The Maharaja Sayajirao University of Baroda, Vadodara, 390 001, India

    N D Chavda

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Chavda, N.D. Average-fluctuation separation in energy levels in many-particle quantum systems with k-body interactions using q-Hermite polynomials. Pramana - J Phys 96, 223 (2022). https://doi.org/10.1007/s12043-022-02442-8

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  • DOI: https://doi.org/10.1007/s12043-022-02442-8

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