Quantum chaos: An introduction via chains of interacting spins 1/2

American Journal of Physics

Volumn 80, Issue 3, 2012, Pages 246-251

  Gubin, Aviva ^a   Santos, Lea F ^a  

^a YESHIVA UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84857483890     PISSN: 00029505     EISSN: None     Source Type: Journal    
DOI: 10.1119/1.3671068     Document Type: Article

References (47)

1. Poincar H. Science and Method 1914, é, (Thomas Nelson and Sons, London)
2. Gleick J. Chaos: Making a New Science 1987, (Viking, New York)
3. Jensen R.V. Quantum chaos. Nature (London) 1992, 355:311-318. 10.1038/355311a0.
4. Gutzwiller M.C. Quantum chaos. Sci. Am. 1992, 206(1):78-84. 10.1038/scientificamerican0192-78.
5. Rudnick Z. Quantum chaos?. Noti. Am. Math. Soc. 2008, 55(1):32-34.
6. Gutzwiller M.C. Chaos in Classical and Quantum Mechanics 1990, (Springer-Verlag, New York)
7. Haake F. Quantum Signatures of Chaos 1991, (Springer-Verlag, Berlin)
8. Reichl L.E. The Transition to Chaos: Conservative Classical Systems and Quantum Manifestations 2004, (Springer, New York)
9. Stöckmann H.-J. Quantum Chaos: An Introduction 2007, (Cambridge U.P., Cambridge)
10. Berry M.V. Quantum scars of classical closed orbits in phase space. Proc. R. Soc. Lond. A 1989, 423:219-231. 10.1098/rspa.1989.0052.
11. Heller E.J. Bound-state eigenfunctions of classically chaotic Hamiltonian systems: Scars of periodic orbits. Phys. Rev. Lett. 1984, 53:1515-1518. 10.1103/PhysRevLett.53.1515.
12. McDonald S.W. Kaufman A.N. Spectrum and eigenfunctions for a Hamiltonian with stochastic trajectories. Phys. Rev. Lett. 1979, 42:1189-1191. 10.1103/PhysRevLett.42.1189.
13. Berry M.V. Quantizing a classically ergodic system: Sinai's billiard and the KKR method. Ann. Phys. (NY) 1981, 131:163-216. 10.1016/0003-4916(81)90189-5.
14. Bohigas O. Giannoni M.J. Schmit C. Characterization of chaotic quantum spectra and universality of level fluctuation laws. Phys. Rev. Lett. 1984, 52:1-4. 10.1103/PhysRevLett.52.1.
15. Guhr T. Mueller-Gröeling A. Weidenmüller H.A. Random matrix theories in quantum physics: Common concepts. Phys. Rep. 1998, 299:189-425. 10.1016/S0370-1573(97)00088-4.
16. Timberlake T. Random numbers and random matrices: Quantum chaos meets number theory. Am. J. Phys. 2006, 74:547-553. 10.1119/1.2198883.
17. Wigner E.P. On the statistical distribution of the widths and spacings of nuclear resonance levels. Proc. Cambridge Phil. Soc. 1951, 47:790-798. 10.1017/S0305004100027237.
18. Beenakker C.W. J. Random-matrix theory of quantum transport. Rev. Mod. Phys. 1997, 69:731-808. 10.1103/RevModPhys.69.731.
19. Alhassid Y. The statistical theory of quantum dots. Rev. Mod. Phys. 2000, 72:895-968. 10.1103/RevModPhys.72.895.
20. Berry M.V. Regular and irregular semiclassical wavefunctions. J. Phys. A 1977, 10:2083-2091. 10.1088/0305-4470/10/12/016.
21. Brody T.A. Flores J. French J.B. Mello P.A. Pandey A. Wong S.S. M. Random matrix physics: Spectrum and strength fluctuations. Rev. Mod. Phys 1981, 53:385-480. 10.1103/RevModPhys.53.385.
22. Wigner E.P. Characteristic vectors of bordered matrices with infinite dimension. Ann. Phys. 1955, 62:548-564.
23. French J.B. Wong S.S. M. Validity of random matrix theories for many-particle systems. Phys. Lett. B 1970, 33:449-452. 10.1016/0370-2693(70)90213-3.
24. Bohigas O. Flores J. Two-body random Hamiltonian and level density. Phys. Lett. B 1971, 34:261-263. 10.1016/0370-2693(71)90598-3.
25. Flores J. Horoi M. Müller M. Seligman T.H. Spectral statistics of the two-body random ensemble revisited. Phys. Rev. E 2001, 63:026204. 10.1103/PhysRevE.63.026204.
26. Kota V.K. B. Embedded random matrix ensembles for complexity and chaos in finite interacting particle systems. Phys. Rep. 2001, 347:223-288. 10.1016/S0370-1573(00)00113-7.
27. Zelevinsky V. Brown B.A. Frazier N. Horoi M. The nuclear shell model as a testing ground for many-body quantum chaos. Phys. Rep. 1996, 276:85-174. 10.1016/S0370-1573(96)00007-5.
28. Izrailev F.M. Simple models of quantum chaos-Spectrum and eigenfunctions. Phys. Rep. 1990, 196:299-392. 10.1016/0370-1573(90)90067-C.
29. Casati G. Chirikov B.V. Guarneri I. Izrailev F.M. Band-random-matrix model for quantum localization in conservative systems. Phys. Rev. E 1993, 48:R1613-R1616. 10.1103/PhysRevE.48.R1613.
30. Flambaum V.V. Gribakina A.A. Gribakin G.F. Kozlov M.G. Structure of compound states in the chaotic spectrum of the Ce atom: Localization properties, matrix elements, and enhancement of weak perturbations. Phys. Rev. A 1994, 50:267-296. 10.1103/PhysRevA.50.267.
31. Santos L.F. Integrability of a disordered Heisenberg spin-1/2 chain. J. Phys. A 2004, 37:4723-4729. 10.1088/0305-4470/37/17/004.
32. Alcaraz F.C. Barber M.N. Batchelor M.T. Baxter R.J. Quispel G.R. W. Surface exponents of the quantum XXZ, Ashkin-Teller and Potts models. J. Phys. A 1987, 20:6397-6409. 10.1088/0305-4470/20/18/038.
33. Bethe H.A. On the theory of metal I. Eigenvalues and eigenfunctions of a linear chain of atoms. Z. Phys. 1931, 71:205-217. 10.1007/BF01341708.
34. Karbach M. Müller G. Introduction to the Bethe ansatz I. Comput. Phys. 1997, 11:36-44.
35. See the supplementary material at E-AJPIAS-80-003202 for codes written in Mathematica and Fortran to obtain the figures from the paper and also to solve additional exercises.
36. Hsu T.C. Angles d'Auriac J.C. Level repulsion in integrable and almost-integrable quantum spin models. Phys. Rev. B 1993, 47:14291-14296. 10.1103/PhysRevB.47.14291.
37. Kudo K. Deguchi T. Level statistics of XXZ spin chains with discrete symmetries: Analysis through finite-size effects. J. Phys. Soc. Jpn. 2005, 74:1992-2000. 10.1143/JPSJ.74.1992.
38. Santos L.F. Rigol M. Onset of quantum chaos in one-dimensional bosonic and fermionic systems and its relation to thermalization. Phys. Rev. E 2010, 81:036206-1. 10.1103/PhysRevE.81.036206.
39. Santos L.F. Transport and control in one-dimensional systems. J. Math. Phys. 2009, 50:095211. 10.1063/1.3181223.
40. Brown W.G. Santos L.F. Starling D.J. Viola L. Quantum chaos, localization, and entanglement in disordered Heisenberg models. Phys. Rev. E 2008, 77:021106. 10.1103/PhysRevE.77.021106.
41. Amico L. Fazio R. Osterloh A. Vedral V. Entanglement in many-body systems. Rev. Mod. Phys. 2008, 80:517-576. 10.1103/RevModPhys.80.517.
42. Vandersypen L.M. K. Chuang I.L. NMR techniques for quantum control and computation. Rev. Mod. Phys. 2005, 76:1037-1069. 10.1103/RevModPhys.76.1037.
43. Kane B.E. A silicon-based nuclear spin quantum computer. Nature (London) 1998, 393:133-137. 10.1038/30156.
44. Sologubenko A.V. Felder E. Giannò K. Ott H.R. Vietkine A. Revcolevschi A. Thermal conductivity and specific heat of the linear chain cuprate Sr2CuO3: Evidence for thermal transport via spinons. Phys. Rev. B 2001, 62:R6108. 10.1103/PhysRevB.62.R6108.
45. Trotzky S. Cheinet P. Fölling S. Feld M. Schnorrberger U. Rey A.M. Polkovnikov A. Demler E.A. Lukin M.D. Bloch I. Time-resolved observation and control of superexchange interactions with ultracold atoms in optical lattices. Science 2008, 319:295-299. 10.1126/science.1150841.
46. Simon J. Bakr W.S. Ma R. Tai M.E. Preiss P.M. Greiner M. Quantum simulation of antiferromagnetic spin chains in an optical lattice. Nature (London) 2011, 472:307-312. 10.1038/nature09994.
47. Chen Y.-A. Nascimbène S. Aidelsburger M. Atala M. Trotzky S. Bloch I. Controlling correlated tunneling and superexchange interactions with ac-driven optical lattices. Phys. Rev. Lett. 2011, 107:210405. 10.1103/PhysRevLett.107.210405.