C LASSIFICATION OF ESQPT S AND FINITE-SIZE EFFECTS Pavel Stránský ESQPT, Trento 22 September 2015 Institute of Particle and Nuclear Physics Charles University in Prague Czech Republic In collaboration with: Pavel Cejnar Michal Macek Yale University, New Haven, USA The Hebrew University, Jerusalem, Israel Amiram Leviatan

2.Models & results - CUSP potential (f=1) - Creagh-Whelan potential (f=2) - 3 coupled CUSPs (f=3) 3.Finite-size effects - separable system - oscillatory component of the level density, partial smoothing - effects of chaos 1.Stationary points - Effects on the smooth level density and flow rate - Morse theory - Nondegenerate and degenerate stationary points Content

1. Level density, flow rate Nonanalyticities induced by Hamiltonian stationary points

volume of the classical phase space oscillatory component Gutzwiller (Berry-Tabor) formula Level density spectrum: smooth component E E x (finite-size attribute of the system) smoothing function (Gaussian) Approximation

Flow rate Example: CUSP system Continuity equation flow rate – role of velocity (time)(coordinate) E control parameter discontinuity in the flow level dynamics: critical borderlines - averaged variations of energy eigenvalues with the system’s control parameter

Hamiltonian in the standard form Kinetic term quadratic in momenta No mixing of coordinates and momenta Analytic potential V Confined system (discrete spectrum) size parameter P. Stránský, M. Macek, P. Cejnar, Annals of Physics 345, 73 (2014) Smooth level density, flow rate and thermodynamical consequences in systems with f=2 studied extensively in:

Complicated kinetic terms - algebraic models have often very complicated semiclassical Hamiltonians that nontrivially mix coordinates and momenta Dicke model Jorge Hirsch... C. Emary, T. Brandes, Phys. Rev. E 67, (2003) P. Pérez-Fernández et al., Phys. Rev. A 83, (2011) Interacting Boson Model Michal Macek F. Iachello, A. Arima, The Interacting Boson Model (Cambridge University Press UK, 1987) A need for a more general approach

Morse lemma Let us have a smooth function (Hamiltonian) defined on a 2f-dimensional manifold (phase space). In the neighbourhood of a nondegenerate stationary point w one can find such a coordinate system that the function is locally quadratic in all directions: index of the stationary point E x2x2 x1x1  y1y1 y2y2 w Regular and irregular part of the smooth level density near energy smooth, given by the volume of the phase space far from w captures the nonanalytic properties due to the stationary point M. Kastner, Rev. Mod. Phys. 80, 167 (2008) Level density: Nondegenerate stationary point neighbourhood of w

Level density: Nondegenerate stationary point r evenr odd f integer f half-integer r evenr odd [f-1]-th derivative EEwEw jump EEwEw logarithmic divergence EEwEw inverse sqrt EEwEw Relevant for: lattices time-dependent Hamiltonian systems Michal Macek

Each singularity of the level density at a nondegenerate stationary point is uniquely classified by two numbers (f,r) Level density: Nondegenerate stationary point r evenr odd f integer f half-integer r evenr odd [f-1]-th derivative EEwEw jump EEwEw logarithmic divergence EEwEw inverse sqrt EEwEw Relevant for: lattices time-dependent Hamiltonian systems Michal Macek

(analytic only for m k even integer) Special class of separable Hamiltonians (flat minimum): - discontinuity of the -th derivative Example 1: We require discontinuity of the t-th derivative Example 2: Hamiltonian with the kinetic term of the standard form - satisfied when even in the thermodynamic limit the level density can be discontinuous - for infinitely flat potential -th derivative discontinuous Level density: Degenerate stationary point

(analytic only for m k even integer) Special class of separable Hamiltonians (flat minimum): - discontinuity of the -th derivative Example 1: We require discontinuity of the t-th derivative Example 2: Hamiltonian with the kinetic term of the standard form - satisfied when even in the thermodynamic limit the level density can be discontinuous - for infinitely flat potential -th derivative discontinuous Structural stability - an arbitrarily small perturbation converts any function into a Morse function: quadratic minimum M. Kastner, Rev. Mod. Phys. 80, 167 (2008) flat minimum Level density: Degenerate stationary point

Flow rate: Singularities Continuity equation: the same type of nonanalyticity independent of the crossing direction Derivatives We expect the same nonanalyticity in the flow rate and in the level density

2. Models and results

CUSP 1D model Creagh-Whelan 2D model Triple CUSP 3D model Separable combination of three CUSP systems parameter-free system 26 ESQPTs: 7x (3,0)minimum 12x(3,1)saddle 1 6x (3,2)saddle 2 1x (3,3)maximum E y x (2,0) (2,1) phase structure identical with CUSP Integrable (separable) for B=C=0 B squeezes one minimum and stretches the other C squeezes both minima symmetrically D squeezes the potential along x=0 axis Confinement conditions (1,0) (1,1) 2 ESQPTs (if ) E x

Level density in the models CUSP model Creagh-Whelan model E E level density1 st derivative E B = 30, C=D=20 (1,1) (1,0) (2,1)(2,1) (2,0)(2,0)

(3,0) (3,1)(3,1) (3,2)(3,2) (3,3)(3,3) Triple CUSP model

Flow rate in the CUSP system positive (levels rise) negative (levels fall) approximately 0 vanishes due to the potential symmetry The wave function localized around the global minimum Both minima accessible – the wave function is a mixture of states localized around and Singularly localized wave function at the top of the local maximum with Hellmann-Feynman formula (1,0) (1,1)

Flow rate in the Creagh-Whelan system flow rate energy derivative of the flow rate The singularities of the flow rate are of the same type as for the level density (2,0) (2,1)(2,1)

3. Finite-size effects P. Stránský, M. Macek, A. Leviatan, P. Cejnar, Annals of Physics 356, 57 (2015)

Separable system Level density given by a convolution Creagh-Whelan with B=C=0: Harmonic oscillator x(Ex)y(Ey)x(Ex)y(Ey)x(Ex)y(Ey)x(Ex)y(Ey) Model ExEx

Imbalanced system the time scale significantly differs in each degree of freedom The level dynamics is a superposition of shifted 1D CUSP-like critical triangles Rigidity (ratio of the mean level spacing in each direction) much bigger or smaller that one D = 40 E  can be chosen in such a way that it is big enough to smooth the level density in one degree of freedom and keep the oscillatory part in the other E Smoothing finite-size effect

Nonintegrable system – Partial separability fraction of regularity Classical dynamics E - Creagh-Whelan system B=0 C=30 D=10 rigidity similar with the separable case

Nonintegrable system – Partial separability Poincaré sections fraction of regularity Classical dynamics E x pxpx x pxpx E Partially smoothed level density corresponding patterns

Nonintegrable system – Partial separability Poincaré sections fraction of regularity Classical dynamics E x pxpx x pxpx E Partially smoothed level density corresponding patterns Level dynamics

Nonintegrable system – Chaos symmetric case B=0, C=39, D=1 asymmetric case B=39, 20, D=20 E E f reg level dynamics

Conclusions ESQPTs originate in stationary points of the classical Hamiltonian - Nondegenerate stationary points: singularities classified uniquely by (f – number of degrees of freedom, r – index of the stationary point); they occur in the ┌ f-1 ┐ -th derivative of the smooth level density or flow rate - Degenerate stationary points: higher flatness of the stationary point shifts the discontinuity towards lower derivatives Finite-size effects - Relevant if the motion in one degree of freedom of a separable system is much faster than in the other - Series of singularities belonging to the system of lower number f - Present even if the separability is only partial, wiped out only by complete chaos P. Stránský, M. Macek, P. Cejnar, Annals of Physics 345, 73 (2014) P. Stránský, M. Macek, A. Leviatan, P. Cejnar, Annals of Physics 356, 57 (2015) References T HANK YOU FOR YOUR ATTENTION