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Monodromy & Excited-State Quantum Phase Transitions in Integrable Systems Collective Vibrations of Nuclei Pavel Cejnar Institute of Particle & Nuclear Physics, Charles University, Prague [email protected] 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 H ( xi , pi ) Hamiltonian for f degrees of freedom: xi ( x1 , x2 ,..., x f ) pi ( p1 , p2 ,..., p f ) Ck ( xi , pi ) k 1... f f integrals of motions “in involution” (compatible) H , Ck 0, Ck , Cl 0 Action-angle variables: xi i pi I i i i d dt I i 0 d dt i(t ) i t i(0) I i (t ) const The motions in phase space stick onto surfaces that are topologically equivalent to tori Monodromy in classical and quantum mechanics Etymology: Μονοδρoμια = “once around” Invented: Promoted: JJ Duistermaat, Commun. Pure Appl. Math. 33, 687 (1980). RH Cushman, L Bates: Global Aspects of Classical Integrable Systems (Birkhäuser, Basel, 1997). spherical pendulum Simplest example: Hamiltonian constraints z H 12 px2 p y2 pz2 z x2 y 2 z 2 1 xpx yp y zp z 0 Conserved angular momentum: H , Lz Poisson 0 ρ Lz xpy ypx y x 2 compatible integrals of motions, 2 degrees of freedom (integrable system) p 2 Singular bundle of orbits: point of unstable equilibrium 1 (trajectory needs infinite time to reach it) trajectories with E=1, Lz=0 “pinched torus” -1 -0.5 0.5 -1 -2 …corresponding lattice of quantum states: 1 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 m “crystal defect” of the quantum lattice high-E m m K Efstathiou et al., Phys. Rev. A 69, 032504 (2004). Another example: Mexican hat (champagne bottle) potential MS Child, J. Phys. A 31, 657 (1998). V H 12 px2 p y2 a 2 b 4 E=0 y Pinched torus of orbits: E=0, Lz=0 0.6 0.4 x p 0.2 -1 -0.5 0.5 -0.2 -0.4 -0.6 Crystal defect of the quantum lattice 1 radial q.num. n principal q.num. 2n+m Part 2/4: Quantum phase transitions & nuclear collective motions Ground-state quantum phase transition ( T=0 QPT ) H H0 V (1 ) H (0) H (1) [ 0,1] H 0V H0 [ H0 , V ] 0 The ground-state energy E0 may be a nonanalytic function of η (for dim ). d E ( ) ( ) V ( ) V 0 0 d 0 d2 d 2 For V V 0 E0 ( ) 2 | ( ) V i 0 i 0 0 ( ) Ei ( ) E0 ( ) |2 0 0 two typical QPT forms: 2nd order QPT 0 0 V 1st order QPT 0 0 But the Ehrenfest classification is not always applicable... Geometric collective model quadrupole tensor of collective coordinates (2 shape param’s, 3 Euler angles ) …corresponding tensor of momenta H 5 35 [ ]( 0 ) ..... 5 A[ ]( 0 ) B [ ]( 2 ) 2K 2 (0) 5C [ ]( 2 ) For zero angular momentum: J i 10[ ] 0 * (1) B V A 2 B 3 cos 3 C 4 motion in principal coordinate frame x Re cos y y 2 Re 2 sin 2D system x ..... oblate 1 ( x2 y2 ) A( x 2 y 2 ) B( x 3 3 y 2 x) C ( x 2 y 2 ) 2 2K β γ 2 neglect … neglect higher-order terms H spherical prolate A Interacting boson model (from now on) H uijbi b j vijklbi b j bk bl i, j i i , j , k ,l s {d b F Iachello, A Arima (1975) s-bosons (l=0) • “nucleon pairs with l = 0, 2” • “quanta of collective excitations” 2,...,2 d-bosons (l=2) Dynamical algebra: U(6) …generators: Gij bib j …conserves: N b b i i i Subalgebras: U(5), O(6), O(5), O(3), SU(3), [O(6), SU(3)] H k0 k1C1[ U(5)] k2C2 [ U(5)] k3C2 [O(6)] k4C2 [O(5)] k5 C2 [O(3)] k6 C2 [SU(3)] Dynamical symmetries (extension of standard, invariant symmetries): U(6) U(5) O(5) O(3) O(6) O(5) SU(3) k3 k 6 0 U(5) k1 k2 k6 0 k1 k2 k3 k4 0 O(6) SU(3) [O(6), SU(3)] inherent structure: triangle(s) U(5) D Warner, Nature 420, 614 (2002). The simplest, one-component version of the model, IBM-1 O(6) SU(3) SU(3) O(6) 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 e 12 | |2 exp( s s d ) 0 ● classical Hamiltonian Hcl H complex variables contain coordinates & momenta (12 real variables) ● boson number conservation (only in average) N N | s | | | 2 ● classical limit: N 10 real variables: (2 quadrupole deformation parameters, 3 Euler angles, 5 associated momenta) fixed N 2 1 2 () q ip p q 2 restricted phase-space domain ● angular momentum J=0 Euler angles irrelevant only 4D phase space 2 coordinates (x,y) or (β,γ) ● result: T 1 2K 2 f ( , , 2 , ) V A 2 B 3 1 2 cos 3 C 4 2 Similar to GCM but with position-dependent kinetic terms and higher-order potential terms 0, 2 Phase diagram for axially symmetric quadrupole deformation GCM ground-state = minimum of the potential | 0 | 0 0 1 2 IBM critical exponent 1st order Triple point 2nd order E A 2 B 3 cos 3 C 4 ... 1 Order parameter for axisym. quadrup. deformation: β=0 spherical, I β>0 prolate, II β<oblate. III 1st order Part 3/4: Monodromy for integrable collective vibrations O(6)-U(5) transition ~ nd d d ~ Q ( 0) d s s d P 1 2 d d ss (…from now on) H ( , 0) 1 nd 2 Q(0) Q(0) N N N 1 ' ' 1 a nd 2 P P 2 C2 [ O(5)] N ( N 4) N N N ~ ~ a 4 3 C2 [O (5)] nd (nd 3) (d d )( d d ) ' v(v 3) “seniority” 4 3 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 : H H cl 2 (1 ) 2 2 5 4 2 (1 ) 4 N 2 2 kinetic energy Tcl potential energy Vcl 2 x2 y 2 2 x2 y2 2 2 [0, 2 ], [0, 2 ], [0,1] 1 N2 v N C2 [O(5)] 2 J 0 x y y x J=0 projected O(5) “angular momentum” O(6)-U(5) transition 1 N ~ ~ H N12 (d s s d ) (d s s d ) 1 N O(6) 0 1 N H H cl N 2 1 N nd H cl 12 2 12 2 H cl 2 2 42 2 4 deformed g.s. H spherical g.s. 4/5 nd 1N2 Q(0) Q(0) 2 (1 ) 2 2 524 2 (1 ) 4 U(5) 1 Μονοδρoμια Poincaré surfaces of sections: η=0.6 pinched torus 2 2 Available phase-space volume at given energy ( E ) E H ( p, q) dp f dq f connected to the smooth component of quantum level density ( E ) Tr E Hˆ smooth ( E ) osci llatory( E ) Ω(E) Volume of the “enveloping” torus: max( E) ( E ) 2 h( ; E ) d min (E ) 2 pmax( ; E ) max min d (E ) dE singular tangent E 0 E0 β (E ) T Classification of trajectories by the ratio R T oscillations in β and γ directions. For rational R of periods associated with the trajectory is periodic: M Macek, P Cejnar, J Jolie, S Heinze, Phys. Rev. C 73, 014307 (2006). R≈2 “bouncing-ball orbits” (like in spherical oscillator) Spectrum of orbits (obtained in a numerical simulation involving ≈ 50000 randomly selected trajectories) η=0.6 E At E=0 the motions change their character from O(6)- to U(5)-like type of trajectories E=0 R>3 “flower-like M Macek, P Cejnar, J Jolie, Sorbits” Heinze, R Phys. Rev. C 73,(Mexican-hat 014307 (2006). potential) O(6) transitional U(5) energy Lattice of J=0 states (N=40) Μονοδρoμια M Macek, P Cejnar, J Jolie, S Heinze, Phys. Rev. C 73, 014307 (2006). →seniority Part 4/4: Excited-state quantum phase transitions for integrable vibrations 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) J=0 level 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). 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: 1 n 1 x d 2 N 2 nd i i ( x) x may be treated as a continuous variable (N→∞) H i Ei i nd H nd i ( x) nd H nd 2 i ( x N1 ) i ( x) N1 H N d dx i ( x N1 ) i ( x) 2 N1 2 d2 dx 2 nd H nd 2 i ( x N1 ) Ei i ( x) i ( x) oscillator with x-dependent mass: d2 2 H osc K ( x) 2 Lx x0 E0 dx O(6) limit O(6) quasi-dynamical symmetry breaks down once the edge of semiclassical wave function O(6)-U(5) i=1 nd N=60, v=0 i=2 reaches the nd=0 or nd=N limits. For v=0 eigenstates of H ( ) 1 nd 2 Q(0) Q(0) N N N 2 1 1 2 m ( x ) 1 1 we obtain: 1 1 V ( x) 2 2 2 5 4 2 1 16 x d H osc ( ) 16 [ x ] [ ] 4N 2 d x2 16( 1) 4 (1 ) 1 x0 ( ) 1 E0 ( ) V ( x ) 0 ground-state phase transition 0.8 0.2 0.6 0 η=0.8 -0.8 0.2 0.75 0.5 -1 E0 ( ) -0.2 x 12 nd N -0.2 x -0.1 0 0.1 0.2 0.2 x 0.25 0 0.8 -0.6 0.4 1 0.6 -0.2 -0.4 -0.25 -0.5 -0.75 -1 0.4 21 x 14 , 14 xmin , xmax => approximation holds for energies below Eup ( ) V ( xmin ) 0 At E=0 all v=0 states undergo a nonanalytic change. x-dependence of velocity–1 ( classical limit of |ψ(x)|2 ) E 0 E0 35 30 1 ( x xmin ) ∞ Effect of m(x)→ 25 1 ( x xend )1/ 2 20 for x → –¼ 15 Similar effect appears in the β-dependence of velocity–1 in the Mexican hat at E=0 x 14 nd 0 0 1/β-divergence 10 5 -0.25 1 4 -0.2 -0.15 -0.1 -0.05 x 0.05 0 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. SiU(5) Pi (nd ) ln Pi (nd ) nd i=1 i=10 U(5) wave-function entropy N=80 i=20 i=30 |Ψ(nd )|2 v=0 ↓ Eup=0 S Heinze, P Cejnar, J Jolie, M Macek, PRC 73, 014306 (2006). i=1 i=10 1 maximum 10 maxima |Ψ(nd )|2 i=20 20 maxima v=0 ↓ i=30 30 maxima Eup=0 S Heinze, P Cejnar, J Jolie, M Macek, PRC 73, 014306 (2006). SiU(5) Pi (nd ) ln Pi (nd ) nd i=1 i=10 U(5) wave-function entropy quasi-O(6) quasi-U(5) N=80 i=20 i=30 |Ψ(nd )|2 v=0 ↓ Eup=0 S Heinze, P Cejnar, J Jolie, M Macek, PRC 73, 014306 (2006). Any phase transitions for nonzero seniorities? 1 N 2 2 H H cl (1 ) J 0 2 C2 [O(5)] 1 N2 Nv 2 2 2 2 5 4 2 (1 ) 4 2 J 0 ( ) V 2 5 4 2 H cl (1 ) 2 2 (1 ) 2 (1 ) 4 2 2 2 2 Veff ( ) Veff constant & centrifugal terms 0 1 1 Veff 0.9 v0 0.6 0.5 v 0.6 N 4 0.5 0.4 0.4 0.3 0.3 0.3 0.2 0.2 0.2 0.1 0.1 0.1 v N 2 0.8 2 -0.1 2 -0.1 0.7 For δ≠0 fully analytic evolution of the minimum β0 and min.energy Veff(β0) 0.6 0.5 0 0 0.2 0.4 0.6 0.8 1 1 => no phase transition !!! 2 J=0 level dynamics for separate seniorities N=80 Eup 0 excited states continuous v=0 ground state 2nd order (probably without Ehrenfest classif.) no phase transition v=18 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: • • E=0 phase separatrix for zero-seniority states analytic evolutions for nonzero-seniority states Open questions: • • Connection with thermodynamic description of quantum phase transitions? Extension to nonintegrable systems: is there an analog of monodromy? Collaborators: Michal Macek (Prague), Jan Dobeš (Řež), Stefan Heinze, Jan Jolie (Cologne). Thanks to: David Rowe (Toronto), Pavel Stránský (Prague)…