Publication: Thermalization in the two-body random ensemble
Full text at PDC
Advisors (or tutors)
Using the ergodicity principle for the expectation values of several types of observables, we investigate the thermalization process in isolated fermionic systems. These are described by the two-body random ensemble, which is a paradigmatic model to study quantum chaos and especially the dynamical transition from integrability to chaos. By means of exact diagonalizations we analyze the relevance of the eigenstate thermalization hypothesis as well as the influence of other factors, such as the energy and structure of the initial state, or the dimension of the Hilbert space. We also obtain analytical expressions linking the degree of thermalization for a given observable with the so-called number of principal components for transition strengths originating at a given energy, with the dimensions of the whole Hilbert space and microcanonical energy shell, and with the correlations generated by the observable. As the strength of the residual interaction is increased, an order-to-chaos transition takes place, and we show that the onset of Wigner spectral fluctuations, which is the standard signature of chaos, is not sufficient to guarantee thermalization in finite systems. When all the signatures of chaos are fulfilled, including the quasicomplete delocalization of eigenfunctions, the eigenstate thermalization hypothesis is the mechanism responsible for the thermalization of certain types of observables, such as (linear combinations of) occupancies and strength function operators. Our results also suggest that fully chaotic systems will thermalize relative to most observables in the thermodynamic limit.
© IOP Publishing. The authors A. R. and J. R. are thankful to L. Muñoz and J. M. G. Gómez for their collaboration and enlightening discussions. V. K. B. K. and M. V. thank Navinder Singh for useful discussions. This work is supported in part by Spanish Government grants for the research projects FIS2006-12783-C03-02, FIS2009-11621-C02-01, CSPD-2007-00042-Ingenio2010, and by the Universidad Complutense de Madrid grant UCM-910059. One of us, A. R., is supported by the spanish program JAE-Doc.
 L. E. Reichel, A Modern Course in Statistical Physics, University of Texas, Austin (1980).  Jördens R et al 2008 Nature 455 204.  Schneider U et al 2008 Science 322 1520.  Perfetti L et al 2006 Phys. Rev. Lett. 97 067402.  Konishita T, Wenger T and Weiss D S 2006 Nature 440 900.  Hofferberth S, Lesanovsky I, Fisher B, Schumm T and Schmiedmayer J 2007 Nature 449 324.  Zhang W M and Feng D H 1995 Phys. Rep. 252 1.  Relaño A, Dukelsky J, Gómez J M G and Retamosa J 2004, Phys. Rev. E 70 026208.  Stöckmann H-J (1999) Quantum Chaos (Cambridge: Cambridge University Press).  Haake F 2010 Quantum Signatures of Chaos (Heidelberg: Springer-Verlag).  Iglói F and Rieger H 2000 Phys. Rev. Lett. 85 3233.  Cazalilla M A 2006 Phys. Rev. Lett. 97 156403.  Calabrese P and Cardy J 2006 Phys. Rev. Lett. 96 136801.  Sengupta K, Powell S and Sachdev S 2004 Phys. Rev. A 69 053616.  Cherng R W and Levitov L S 2006 Phys. Rev. A 73 043614.  Tasaki H 1998 Phys. Rev. Lett. 80 1373.  Linden N, Popescu S, Short A J and Winter A 2009 Phys Rev. E 79 061103.  Reimann P 2008 Phys. Rev. Lett. 101 190403.  Goldstein S, Lebowitz J L, Mastrodonato C, Tumulka R and Zanghi N 2010 Phys. Rev. E 81 011109.  Kollath C, Läuchli A M and Altman E 2007 Phys. Rev. Lett. 98 180601.  Manmana S R, Wessel S, Noack R M and Muramatsu A 2007 Phys. Rev. Lett. 98 210405.  Rigol M, Dunjko V, Yurovsky V and Olshanii M 2008 Nature 452 854.  Rigol M 2009 Phys. Rev. Lett. 103 100403.  Rigol M 2009 Phys. Rev. A 80 053607.  Deutsch J M 1991 Phys. Rev. A 43 2046; arXiv:0911:0056.  Srednicki M 1994 Phys. Rev. E 50 888.  Berry M V 1977 J. Phys. A 10 2083.  Santos L F and Rigol M 2010 Phys. Rev. E 81 036206.  Relaño A 2010 J. Stat. Mech. P07016.  Olshanii M and Yurovsky V 2009 arXiv:0911.5587.  Neuenhahn C and Marquardt F 2010 arXiv:1007.5306.  J. B. French 1980 Moment Methods in Many Fermion Systems, (New York: Plenum).  Horoi M, Zelevinsky V and Brown B A 1995 Phys. Rev. Lett. 74 5194.  Kota V K B and Sahu R 2002 Phys. Rev. E 66 037103.  Brody T A, Flores J, French J B, Mello P A, Pandey A and Wong S S M 1981 Rev. Mod. Phys. 53 385.  Borgonovi F, Guarneri I, Izrailev F M and Casati G 1998 Phys. Lett. A 247 140.  Flambaum V V, Izrailev F M and Casati G 1996 Phys. Rev. E 54 2136.  Flambaum V V and Izrailev F M 1997 Phys. Rev. E 56 5144.  Benenti G, Casati G and Shepelyansky D L, 2001 Euro. Phys. J. D 17 265.  Kota V K B 2001 Phys. Rep. 347 223.  Gómez J M G, Kar K, Kota V K B, Molina R A, Relaño A and Retamosa J 2011 Phys. Rep. 499 103.  Berry M V and Tabor M 1977 Proc. R. Soc. London A 356 375.  Bohigas O, Giannoni M J and Schmit C 1984 Phys. Rev. Lett. 52 1.  Mehta M L 2004 Random Matrices (Amsterdam: Elsevier).  Heusler S, Müller S, Altland A, Braun P and Haake F 2007 Phys. Rev. Lett. 98 044103.  Vyas Manan, Kota V K B and Chavda N D 2010 Phys. Rev. E 81 036212.  Angom D, Ghosh S and Kota V K B 2004 Phys. Rev. E 70 016209.  Åberg S 1990 Phys. Rev. Lett. 64 3119.  Åberg S 1992 Prog. Part. Nucl. Phys. 28 11.  Jacquod Ph and Varga I 2002 Phys. Rev. Lett. 89 134101.  Jacquod Ph and Shepelyansky D L 1997 Phys. Rev. Lett. 79 1837.  It is also worth noting here that the actual value of the Brody parameter for GOE is not ω = 1, but ω = 0.965.  Sakhr J and Nieminen J M 2005 Phys. Rev. E 72 045204(R).  Berkovits R and Avishai Y 1998 Phys. Rev. Lett. 80 568.  Kota V K B and Sahu R 2001 Phys. Rev. E 64 016219.  Kota V K B 2003 Ann. Phys. (N.Y.) 306 58.  Kota V K B and Haq R U 2010 Spectral Distributions in Nuclei and Statistical Spectroscopy. (Singapore: World Scientific).  Santos L F and Rigol M 2010 Phys. Rev. E 82 031130.  Draayer J P, French J B and Wong S S M 1977 Ann. Phys. (N.Y.) 106 472.  Gómez J M G , Kar K, Kota V K B, Molina R A and Retamosa J 2004 Phys. Rev. C 69 057302.  Kota V K B and Sahu R 1998 Phys. Lett. B 429 1.