CLASSIFICATION OF ESQPTSAND 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

www.pavelstransky.cz

Michal MacekYale 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 borderline

s

- 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 modelJorge Jorge HirschHirsch

......

C. Emary, T. Brandes, Phys. Rev. E 67, 066203 (2003)P. Pérez-Fernández et al., Phys. Rev. A 83, 033802 (2011)

Interacting Boson ModelMichal Michal MacekMacek

F. Iachello, A. Arima, The Interacting Boson Model (Cambridge University Press UK, 1987)

A need for a more general approach

Morse lemmaLet 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

x2

x1

y1

y2

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 even r oddf integer f half-integer

r even r odd

[f-1]-th derivative

EEw

jump

EEw

logarithmic divergence

EEw

inverse sqrt

EEw

Relevant for:• lattices

• time-dependent Hamiltonian systems

Michal Michal MacekMacek

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 even r oddf integer f half-integer

r even r odd

[f-1]-th derivative

EEw

jump

EEw

logarithmic divergence

EEw

inverse sqrt

EEw

Relevant for:• lattices

• time-dependent Hamiltonian systems

Michal Michal MacekMacek

(analytic only for mk 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 limitthe level density can be discontinuous

- for infinitely flat potential -th derivative discontinuous

Level density: Degenerate stationary point

(analytic only for mk 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 limitthe 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: SingularitiesContinuity 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)minimum12x(3,1)saddle 16x (3,2)saddle 21x (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

EElevel density 1st derivativeE

B = 30, C=D=20

(1,1)

(1,0)

(2,1)

(2,0)

(3,0)

(3,1)

(3,2)

(3,3)

Triple CUSP model

Flow rate in the CUSP system

positivepositive (levels (levels rise)rise)

negative negative (levels (levels fallfall))

approximatelapproximatelyy 0 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 systemflow 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)

3. Finite-size effects

P. Stránský, M. Macek, A. Leviatan, P. Cejnar, Annals of Physics 356, 57 (2015)

Separable systemLevel density given by a convolution

Creagh-Whelan with B=C=0:

Harmonic oscillator

xx((EExx))yy((EEyy))

Model

Ex

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 fraction of

regularityregularity

Classical dynamics

E

- Creagh-Whelan system

B=0C=30D=10

rigidity similar with the separable case

Nonintegrable system – Partial separability Poincaré

sectionsfraction of fraction of regularityregularity

Classical dynamics

E

x

px

x

px

E

Partially smoothed level density

corresponding patterns

Nonintegrable system – Partial separability Poincaré

sectionsfraction of fraction of regularityregularity

Classical dynamics

E

x

px

x

px

E

Partially smoothed level density

corresponding patterns

Level dynamics

Nonintegrable system – Chaossymmetric case

B=0, C=39, D=1

asymmetric case

B=39, 20, D=20

E

E

ffregreglevel 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)

http://www.pavelstransky.cz/cw.php

P. Stránský, M. Macek, A. Leviatan, P. Cejnar, Annals of Physics 356, 57 (2015)

References

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