1 / 35

Pavel Cejnar Institute of Particle & Nuclear Physics, Charles University, Prague

Monodromy & Excited-State Quantum Phase Transitions in Integrable Systems Collective Vibrations of Nuclei. Pavel Cejnar Institute of Particle & Nuclear Physics, Charles University, Prague cejnar@ipnp.troja.mff.cuni.cz. NIL DESPERANDUM !.

ike
Download Presentation

Pavel Cejnar Institute of Particle & Nuclear Physics, Charles University, Prague

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Monodromy & Excited-State Quantum Phase Transitions in Integrable Systems Collective Vibrations of Nuclei Pavel Cejnar Institute of Particle & Nuclear Physics, Charles University, Prague cejnar@ipnp.troja.mff.cuni.cz NIL DESPERANDUM !

  2. Monodromy(in classical & quantum mechanics): singularity in the phase space of a classical integrable system that prevents introduction of global analytic action-angle variables. Important consequences on the quantum level... Quantum phase transitions: abrupt changes of system’s ground-state properties with varying external parameters. The concept will be extended to excited states...

  3. Part 1/4: Monodromy

  4. Integrable systems Hamiltonian for f degrees of freedom: f integrals of motions “in involution” (compatible) Action-angle variables: The motions in phase space stick onto surfaces that are topologically equivalent to tori

  5. Monodromy in classical and quantum mechanics Etymology:Μονοδρoμια= “once around” Invented: JJ Duistermaat, Commun. PureAppl. Math. 33, 687 (1980). Promoted: RH Cushman, L Bates: Global Aspects of Classical Integrable Systems (Birkhäuser, Basel, 1997). Simplest example:spherical pendulum z Hamiltonian constraints y ρ x Conserved angular momentum: 2 compatible integrals of motions, 2 degrees of freedom (integrable system)

  6. Singular bundle of orbits: point of unstable equilibrium (trajectory needs infinite time to reach it) trajectories with E=1, Lz=0 “pinched torus” …corresponding lattice of quantum states:

  7. It is impossible to introduce a global system of 2 quantum numbers defining a smooth grid of states: q.num.#1: z-component of ang.momentum m q.num.#2: ??? candidates: “principal.q.num.” n, “ang.momentum”l, combination n+m low-E high-E m m m “crystal defect” of the quantum lattice K Efstathiou et al., Phys. Rev. A 69, 032504 (2004).

  8. Another example:Mexican hat (champagne bottle) potential MS Child, J. Phys. A 31, 657 (1998). V E=0 y Pinched torusof orbits: E=0, Lz=0 x radial q.num. n principal q.num. 2n+m Crystal defect of the quantum lattice

  9. Part 2/4: Quantum phase transitions & nuclear collective motions

  10. Ground-state quantum phase transition( T=0QPT ) The ground-state energyE0 may be a nonanalytic function of η(for ). For two typical QPT forms: 2nd order QPT 1st order QPT But the Ehrenfest classification is not always applicable...

  11. Geometric collective model quadrupole tensor of collective coordinates (2 shape param’s, 3 Euler angles ) …corresponding tensor of momenta neglect higher-order terms neglect … B oblate For zero angular momentum: spherical prolate motion in principal coordinate frame A y β γ 2D system x

  12. Interacting boson model(from now on) F Iachello, A Arima (1975) s-bosons (l=0) • “nucleon pairs with l = 0, 2” • “quanta of collective excitations” d-bosons (l=2) Dynamical algebra:U(6) …generators: …conserves: Subalgebras:U(5), O(6), O(5), O(3), SU(3),[O(6), SU(3)] Dynamical symmetries (extension of standard, invariant symmetries): U(5) O(6) SU(3) [O(6), SU(3)]

  13. inherent structure: triangle(s) D Warner, Nature 420, 614 (2002). The simplest, one-component version of the model, IBM-1

  14. IBM classical limit Method by: RL Hatch, S Levit, Phys. Rev. C 25, 614 (1982) Y Alhassid, N Whelan, Phys. Rev. C 43, 2637 (1991) ____________________________________________________________________________________ ● use of Glaubercoherent states ● classical Hamiltonian complex variables contain coordinates & momenta (12 real variables) ● boson number conservation (only in average) 10 real variables: (2 quadrupole deformation parameters, 3 Euler angles, 5 associated momenta) fixed ● classical limit: restricted phase-space domain ● angular momentum J=0 Euler angles irrelevant only 4D phase space 2 coordinates(x,y) or (β,γ) ● result: Similar to GCM but with position-dependent kinetic terms and higher-order potential terms

  15. GCM Phase diagramfor axially symmetric quadrupole deformation ground-state = minimum of the potential IBM critical exponent 1st order Triple point 2nd order Order parameter for axisym. quadrup. deformation: β=0 spherical, β>0 prolate,β<oblate. I II III 1st order

  16. Part 3/4: Monodromy for integrable collective vibrations

  17. O(6)-U(5) transition (…from now on) “seniority” The O(6)-U(5) transitional system is integrable: the O(5) Casimir invariant remains an integral of motion all the way and seniority v is a good quantum number. Classical limit for J=0 : kinetic energy Tcl potential energy Vcl J=0 projected O(5) “angular momentum”

  18. O(6)-U(5) transition spherical g.s. deformed g.s. O(6) U(5) 0 1 4/5

  19. Poincaré surfaces of sections: Μονοδρoμια η=0.6 pinched torus

  20. Available phase-space volume at given energy connected to the smooth component of quantum level density Volume of the “enveloping” torus: singular tangent E 0 E0 β

  21. Classification of trajectories by the ratio of periods associated with oscillations in β and γ directions. For rational the trajectory is periodic: M Macek, P Cejnar, J Jolie, S Heinze, Phys. Rev. C 73, 014307 (2006).

  22. R≈2 “bouncing-ball orbits” (like in spherical oscillator) Spectrum of orbits (obtained in a numerical simulation involving ≈ 50000 randomly selected trajectories) E η=0.6 At E=0 the motions change their character from O(6)- to U(5)-like type of trajectories E=0 R>3 “flower-like orbits” (Mexican-hat potential) M Macek, P Cejnar, J Jolie, S Heinze, Phys. Rev. C 73, 014307 (2006). R

  23. Lattice of J=0 states O(6) transitional U(5) (N=40) energy Μονοδρoμια M Macek, P Cejnar, J Jolie, S Heinze, Phys. Rev. C 73, 014307 (2006). →seniority

  24. Part 4/4: Excited-state quantum phase transitions for integrable vibrations

  25. N=80 all levels with J=0 1st order 2nd order E O(6)-U(5) What about phase transitions for excited states (if any) ??? This problem (independently of the model) solved at most for the lowest states. Difficulty: in the classical limit excited states loose their individuality... η ground-state phase transition(2nd order)

  26. J=0level dynamics across the O(6)-U(5) transition (all v’s) N=40 E=0 S Heinze, P Cejnar, J Jolie, M Macek, Phys. Rev. C 73, 014306 (2006).

  27. Wave functions in an oscillator approximation: DJ Rowe, Phys. Rev. Lett. 93, 122502 (2004), Nucl. Phys. A 745, 47 (2004). Method applicable along O(6)-U(5)transition for N→∞ and states with rel.seniority v/N=0: x may be treated as a continuous variable (N→∞) H oscillator with x-dependent mass: O(6) quasi-dynamical symmetrybreaks down once the edge of semiclassical wave function reaches thend=0ornd=Nlimits. O(6) limit O(6)-U(5) nd i=1 i=2 N=60, v=0

  28. we obtain: For v=0eigenstates of ground-state phase transition η=0.8 => approximation holds for energies below At E=0 all v=0 states undergo a nonanalytic change.

  29. x-dependence of velocity–1 ( classical limit of |ψ(x)|2 ) Effect of m(x)→∞for x → –¼ Similar effect appears in the β-dependence of velocity–1 in the Mexican hat at E=0 1/β-divergence In the N→∞limit the average <nd>i→0 (and <β >i →0) as E→0. At E=0 all v=0 states undergo a nonanalytic change.

  30. U(5) wave-function entropy i=1 i=10 N=80 |Ψ(nd )|2 i=20 i=30 v=0 ↓ Eup=0 S Heinze, P Cejnar, J Jolie, M Macek, PRC 73, 014306 (2006).

  31. i=1 1 maximum 10 maxima i=10 |Ψ(nd )|2 i=20 20 maxima v=0 ↓ Eup=0 i=30 30 maxima S Heinze, P Cejnar, J Jolie, M Macek, PRC 73, 014306 (2006).

  32. U(5) wave-function entropy i=1 i=10 quasi-O(6) quasi-U(5) N=80 |Ψ(nd )|2 i=20 i=30 v=0 ↓ Eup=0 S Heinze, P Cejnar, J Jolie, M Macek, PRC 73, 014306 (2006).

  33. Any phase transitions for nonzero seniorities? constant & centrifugal terms For δ≠0 fully analytic evolution of the minimum β0 and min.energy Veff(β0) =>no phase transition !!!

  34. J=0 level dynamics for separate seniorities N=80 excited states ground state v=0 continuous 2nd order (probably without Ehrenfest classif.) no phase transition v=18

  35. Conclusions: • Quantum phase transitionsin integrablesystems: connection with monodromy • Testing example: γ-soft nuclear vibrations [O(6)-U(5) IBM] - relation to other systems with Mexican-hat potential (Ginzburg-Landau model) • Concrete results on quantum phase transitions for individual excited states: • Open questions: • Connection with thermodynamic description of quantum phase transitions? • Extension to nonintegrable systems: is there an analog of monodromy? • E=0 phase separatrix for zero-seniority states • analytic evolutions for nonzero-seniority states Collaborators:Michal Macek(Prague), Jan Dobeš(Řež), Stefan Heinze, Jan Jolie(Cologne). Thanks to:David Rowe(Toronto), Pavel Stránský (Prague)…

More Related