Monodromy & Excited-State Quantum Phase Transitions in Integrable Systems Collective Vibrations of Nuclei Pavel Cejnar Institute of Particle & Nuclear Physics, Charles University, Prague NIL DESPERANDUM !

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...

Part 1/4:Monodromy

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

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

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

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 K Efstathiou et al., Phys. Rev. A 69, (2004). “crystal defect” of the quantum lattice mmm low-E high-E

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

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

Ground-state quantum phase transition ( T=0 QPT ) For two typical QPT forms: The ground-state energy E 0 may be a nonanalytic function of η (for ). 2 nd order QPT1 st order QPT But the Ehrenfest classification is not always applicable...

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

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

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

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 Glauber coherent states ● boson number conservation (only in average) complex variables contain coordinates & momenta ● classical limit: fixed 10 real variables: (2 quadrupole deformation parameters, 3 Euler angles, 5 associated momenta) ● angular momentum J=0 Euler angles irrelevant only 4D phase space (12 real variables) 2 coordinates (x,y) or (β,γ) restricted phase-space domain ● classical Hamiltonian ● result: Similar to GCM but with position-dependent kinetic terms and higher-order potential terms

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

Part 3/4: Monodromy for integrable collective vibrations

O(6)-U(5) transition (…from now on) “seniority” kinetic energy T cl potential energy V cl Classical limit for J=0 : J=0 projected O(5) “angular momentum” 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.

O(6)U(5) 01 4/5 O(6)-U(5) transition deformed g.s. spherical g.s.

η=0.6 pinched torus Poincaré surfaces of sections: Μονοδρoμια

Available phase-space volume at given energy connected to the smooth component of quantum level density β E 0 E0E0 singular tangent Volume of the “enveloping” torus:

Classification of trajectories by the ratioof 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, (2006).

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

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

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

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

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

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

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

x-dependence of velocity –1 ( classical limit of |ψ(x)| 2 ) At E=0 all v=0 states undergo a nonanalytic change. 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 i →0 (and i →0) as E→0.

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

i=10 i=20 i=30 i=11 maximum 10 maxima 20 maxima 30 maxima |Ψ(n d )| 2 v=0 ↓ E up =0 S Heinze, P Cejnar, J Jolie, M Macek, PRC 73, (2006).

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

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

N=80 v=0 v=18 2 nd ordercontinuous ground stateexcited states J=0 level dynamics for separate seniorities no phase transition (probably without Ehrenfest classif.)

Collaborators: Collaborators: Michal Macek (Prague), Jan Dobeš (Řež), Stefan Heinze, Jan Jolie (Cologne). Thanks to: Thanks to: David Rowe (Toronto), Pavel Stránský (Prague) … Conclusions: Quantum phase transitions in integrable systems: 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