e xcited-state quantum phase transitions in systems with few degrees of freedom

EXCITED-STATE QUANTUM PHASE TRANSITIONS IN SYSTEMS WITH FEW

DEGREES OF FREEDOM

Pavel Stránský

Seminario Lunch Nuclear, Instituto de Física, UNAM 4th October 2013

Institute of Particle and Nuclear Phycics, Faculty of Mathematics and Physics, Charles University in Prague, Czech Republic

Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México

In collaboration with: Pavel Cejnar

www.pavelstransky.cz

Michal Macek, Amiram LeviatanRacah Institute of Physics, The Hebrew University, Jerusalem, Israel

1. Excited-state quantum phase transition

2. Models- CUSP potential (1 degree of freedom)- Creagh-Whelan potential (2 degrees of freedom)

3. Signatures of the ESQPT- level density and its derivatives- thermodynamical properties- flow rate

1. Excited-State Quantum Phase Transitions

What is a phase transition?A nonanalytic change of a system’s properties (order parameter, eg. volume, magnetization)as a result of some external conditions (control parameter, eg. pressure, temperature)

• First-order - latent heat (eg. melting ice)• Second-order (continuous) - divergent susceptibility, an infinite

correlation length (eg. ferromagnetic-paramagnetic transition)• Higher-order• Infinite-order (Kosterlitz-Thouless transition in 2D XY models)

Classification

<Magnetization>

Temperature

ferromagnetic phase

paramagnetic phase

finite-size effects

thermodynamical limit

Tc

What is a quantum phase transition?A nonanalytic change (in the infinite-size limit) of ground-state properties of a system by varying an external parameter at absolute zero temperature

Ei

1st orderground-state

QPT

2nd order (continuous)ground-state

QPT

Schematic example

spectrum

control parameter

potential surfacesorder parameter: ground-state energy E0

And what is an ESQPT?

Ei

nonanalyticity in the dependence of excitation energy

Ei (and the respective

wavefunction) on the control

parameter

Schematic example

nonanalyticity in level density as a function of

energy

nonanalyticity in level flow as a function the control

parameter

A natural extension of the ground state QPT to excited part of the spectra

ESQPT is related to the topological and structural changes of the phase space with varying the control parameter or energy.

The dimensionality of the system is crucial.

critical bordeline in the

x E plane

Finite models- size of the system

- number of independent components of the system

- number of degrees of freedom

Nonanalyticities in phase transitions occur only when the system’s size grows to infinity.

Finite model:

while f is maintained finite

Example: Interacting Boson Model

Generally, the number of degrees of freedom f grows with N. However, f is maintained in collective models described by some dynamical algebra A of rank r, for which

• f is related with the rank r (in s & x boson models f = r - 1)• is usually related with the considered irreducible representation of the

algebra

- b bosons (of the type s or d) – quasiparticles, generating a U(6) algebra (f = 5)- 3 degrees of freedom are always separated (conserving angular momentum)- (index of the representation of the U(6) algebra)

In quantized classical systems,

- this limit coincides with the classical limit

Examples of models with ESQPT

P. Cejnar, M. Macek, S. Heinze, J. Jolie, J. Dobeš, J. Phys. A: Math. Gen. 39, L515 (2006)

O(6)-U(5) transition in the Interacting Boson Model

M.A. Caprio, P. Cejnar, F. Iachello, Annals of Physics 323, 1106 (2008)

2-level fermionic pairing model

P. Cejnar, P. Stránský, Phys. Rev. E 78, 031130 (2008)

Geometric collective model of atomic nucleiLipkin model

P. Pérez-Fernández, A. Relaño, J.M. Arias, J. Dukelsky, J.E. García-

Ramos, Phys. Rev. A 80, 032111 (2009)P. Pérez-Fernández, P. Cejnar, J.M. Arias, J. Dukelsky,

J.E. García-Ramos, A. Relaño, Phys. Rev. A 83, 033802

(2009)M.A. Caprio, J.H. Skrabacz, F. Iachello, J. Phys. A 44, 075303

(2011)

2-level pairing models

Hamiltonian (the form under study in this work)

• Standard quadratic kinetic term

• No mixing of coordinates and momenta

• Potential V analytic and confining (discrete spectrum)

2. Models

CUSP 1D potential

R. Gilmore, Catastrophe Theory for Scientists and Engineers, Wiley, NY, 1981

Quantum level dynamics

E

Potential shapes

1st order ground-state QPT

2nd order ground-state QPT

Phase coexistenc

e

spinodal points

calculated with

E

(from the catastrophe theory)

P. Cejnar, P. Stránský, Phys. Rev. E 78, 031130 (2008)

Creagh-Whelan 2D potential

S.C. Creagh, N.D. Whelan, Phys. Rev. Lett. 77, 4975 (1996); Phys. Rev. Lett. 82, 5237 (1999)

• Extreemes lie on the line y = 0

• The potential profile on y = 0 is the same as in the CUSP with B = -2

Confinement conditions:

Phase structure equals to the CUSP with B = -2

1st order ground-state QPT at A = 0

order parameter

• 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

Other properties

x

y

potential for y = 0

A = -2 A = -1 A = 0 A = 1 A = 2

E

E

B=0, D=C=0.5








order parameter




axis

Other properties

x

y

potential for y = 0

A = -2 A = -1 A = 0 A = 1 A = 2

E

E

B=0, D=C=1








order parameter




axis

Other properties

x

y

potential for y = 0

A = -2 A = -1 A = 0 A = 1 A = 2

E

E

B=0, D=C=4








order parameter




axis

Other properties

x

y

potential for y = 0

A = -2 A = -1 A = 0 A = 1 A = 2

E

E

B=0, D=C=20








order parameter




axis

Other properties

x

y

potential for y = 0

A = -2 A = -1 A = 0 A = 1 A = 2

E

E

B=0, C+D=4, D=1








order parameter




axis

Other properties

x

y

potential for y = 0

A = -2 A = -1 A = 0 A = 1 A = 2

E

E

B=0, C+D=4, D=2








order parameter




axis

Other properties

x

y

potential for y = 0

A = -2 A = -1 A = 0 A = 1 A = 2

E

E

B=0, C+D=4, D=4








order parameter




axis

Other properties

x

y

potential for y = 0

A = -2 A = -1 A = 0 A = 1 A = 2

E

E

B=0, D=C=20








order parameter




axis

Other properties

x

y

potential for y = 0

A = -2 A = -1 A = 0 A = 1 A = 2

E

E

B=30, D=C=20








order parameter




axis

Other properties

x

y

potential for y = 0

A = -2 A = -1 A = 0 A = 1 A = 2

E

E

B=39, D=C=20

3. Signatures of ESQPT

smooth part

given by the volume of the classical phase space

oscillating part

Gutzwiller formula(given by the sum of all classical periodic orbits and their repetitions)

Level density

For the moment the focus will be only on the smooth part.

Level density

given by the volume of the classical phase space…

… or by averaging the level density from the energy spectrum

CUSP 1D system

Creagh-Whelan 2D system

smooth part

By approaching the infinite-size limit the averaged level density converges to the smooth semiclassical value

Smooth level density- integrated over momenta

surface of f-dimensional sphere with unit radius

Domain of the available configuration space at energy

E(union of all n disjunct parts)

f = 1:

f = 2:

Special cases

(sum over periods of all distinct trajectories at energy E)

(sum over the total coordinate area accessible at energy E)

(derivative of the total accessible phase-space area)

local maximum(saddle point)

local minimum

E

x

Stationary points of the potential

potential V is analytic

a nonanalyticity of is caused only by a singularity (discontinuity) in , which occurs at each stationary point of the potential

In the vicinity of the critical energy

the level density is decomposed as

Example:2 isotropic harmonic oscillators in f dimensions

With increasing number of freedom degrees, the level density is smoother and the singularities appear at higher derivatives

f = 1

f = 3f = 2

k = 4

k = 2 (jump)

Classification of singularities f = 1local maximum

local minimum

E

x

1. Local minimum

2. Local maximum

(for quadratic minimum k = 2)

for

for

for k = 2

k = 2 (logarithmic)

k = 4

In 1D systems, a singularity always appears in the level density (jump

or divergence)

infinite period of the motion on the top of the barrier - divergence

Classification of singularities f = 21. Local minimum (+) or maximum (-)quadratic separable isotropic

k = 4

k = 2 (break)isotropic minimu

m

isotropic maximum

k = 4

k = 2

2. Saddle point

k = l = 2 k = 2, l = 3

saddles

k = 2, l = 3

k = l = 2

In a 2D system, the level density is continuous; a singularity

(discontinuity) appears in the first energy derivative

Isotropic minimum for arbitrary f

the additional well behaves locally as xk

derivatives of the level density are continuous.

The more degrees of freedom, the more “analytic” the level density is.

Notes: • This conclusion holds qualitatively also for local maxima and saddle points of different types.

• The formula for works also for noninteger f

Level density in the modelsCUSP potential (f = 1)

Creagh-Whelan potential (f = 2)

E

A

E

A

level density derivativeE

A

B = -2 B = 30, C=D=20

Thermodynamical properties

canonical ensemble

microcanonical ensemble

- inverse temperature

thermal distribution

partition functionsmooth

part

usually a single-peaked function whose maximum gives the microcanonical inverse

temperature:

Thermal anomalies can occur when ln is not a monotonously

increasing concave function of energy.

Regular and irregular temperature

interested only in the irregular part

caloric curve

Thermodynamics in the CUSP model

secondary minimum

saddle point

Thermodynamical properties for f = 2local minimum – local maximum

Saddle point

for

for

for

for

(separable)

(isotropic)

quadratic minimum – upward jump

quadratic maximum – downward jump

quadratic saddle

k = l = 2

k = 2, l = 3

saddle – inflexion point

(regular component of the level density approximated in the figures by )

these divergences affect the system for all above a certain

limiting value

The nonanalyticity in of the same type as in .

Thermodynamics in the Creagh-Whelan model

(populated energies in light shades) - higher temperature (lower b) brings the light upper in energy

secondary minimum

saddle point

bimodal, but analytic(away from the critical

region)

Thermal distributions

Flow rate of the spectrum

- playing the role of velocity, it satisfies the

The flow rate can be determined by

•integrating the continuity equation:

•using the Hellmann-Feynman formula from the wave functions:

(connects the level density – with its singularities – and the flow rate)

Continuity equation

Nonanalyticities on the critical borderline1• f = 1: jump of level density opposite jump of the flow rate

divergence of the level density gives generally indeterminate result• f = 2: break of level density opposite break of the flow rateinfinite derivative of the level density

the opposite divergence of the absolute flow rate derivative

In our systems and this derivative equals x

weighted average of the expectation value of the perturbation:

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

Flow rate in the Creagh-Whelan systemThe 2D system

is better studied by looking at

the derivatives

flow rate energy derivative of the flow rate

The singularities of the flow rate are of the same type as

for the level density

Conclusions• ESQPT originate in classical stationary points of the potential

(local minima, maxima and saddle points)

• ESQPT are presented as - singularities in the smooth part of the level density- anomalies of the thermodynamical properties- nonanalytic spectral flow properties with changing control parameter

• ESQPT as presented are applicable to systems with finite number of degrees of freedom – models of collective dynamics of many-body systems, eg.:- f = 1 Lipkin-Meshkov-Glick model, Tavis-Cummings model- f = 2 Dicke model, Interacting boson model

• The nonanalytic features of ESQPT fade quickly with increasing f

• The effect of more complicated kinetic terms

• Finite-size effects, multiple critical triangles

• Relation of the ESQPT with the chaotic dynamics

Outlook

Singularities in the level density, thermal distribution function and flow rate are of the same type.

This work has been submitted to Annals of Physics (P. Stránský, P. Cejnar, M. Macek).

Creagh-Whelan potential

C = 0, D = 40 C = 30, D = 10 C = 39, D = 1

integrable(separable)

Increasing chaos – decay of excited critical triangles

The level dynamics is a superposition of shifted 1D CUSP-like critical triangles

A

E

Finite-size effects, multiple critical triangles, chaos

Conclusions



• Relation of the ESQPT with the chaotic dynamics

Outlook

• ESQPT originate in classical stationary points of the potential(local minima, maxima and saddle points)





E

Chaos and ESQPTC = D = 0.2 C = D = 1

A

Creagh-Whelan potential

Classical fraction of regularity

black – regular, red - chaotic

Many calculations indicate that the onset of chaos is related with the ESQPT (namely with the saddle point, serving as a shield against chaos)…

C = 39, D = 1

… but chaos is not so easy to be

subdued. It can occasionaly break

through.

… but chaos is not so easy to be

subdued. It can occasionaly break

through.

P. Pérez-Fernández, A. Relaño, J.M. Arias, P. Cejnar, J. Dukelsky, J.E. García-Ramos, Phys. Rev. E 83, 046208 (2011)

Conclusions



• Relation of the ESQPT with the chaotic dynamicsTHANK YOU FOR YOUR

ATTENTIONMore images on: http://www.pavelstransky.cz/cw.php

Outlook

• ESQPT originate in classical stationary points of the potential(local minima, maxima and saddle points)





e xcited-state quantum phase transitions in systems with few degrees of freedom

Documents

state properties

ground state qpt

groundstate energy e0and

external parameter

phase space

level flow f

creaghwhelan potential

infinitesize limit of