Exceptional Points and Quantum Phase Transitions · R Gilmore, Catastrophe Theory for Scientists...

Stellenbosch 2010

Pavel CejnarInstitute of Particle & Nuclear PhysicsFaculty of Mathematics & PhysicsCharles University, Prague, Czech Rep.

cejnar @ ipnp.troja.mff.cuni.cz

Exceptional Pointsand Quantum Phase

Transitions


Transitions

Stellenbosch 2010

Lecture IIExceptional Points – hidden machinery of quantum phase

transitionsb

Lecture IQuantum Phase Transitions – singularities in quantal spectra

affecting the ground state & excited states


Transitions

Stellenbosch 2010

1. Quantum phase transitions (QPTs)• QPTs in lattice-like systems (rough overview)• QPTs in collective many-body systems (finite algebraic models)2. Excited-state quantum phase transitions (ESQPTs)• ESQPTs & thermal phase transitions • ESQPTs & classical singularities• ESQPTs in 1D & 2D examples



“Now, Nina, do you think you could throw something into the sea?”“I think I could,” replied the child, “but I am sure that Pablo would throw it a great deal further than I can.”“Never mind, you shall try first.”Putting a fragment of ice into Nina’s hand, he addressed himself to Pablo: “Look out, Pablo; you shall see what a nice little fairy Nina is! Throw, Nina, throw, as hard as you can.”

Jules Verne: Off on a CometHector Sarvedac

Nina balanced the piece of ice two or three times in her hand, and threw it forward with all her strength.A sudden thrill seemed to vibrate across the motionless waters to the distant horizon, and the Gallian Sea had become a solid sheet of ice!

The old and the good 19th century...

1877

“Now, Nina, do you think you could throw something into the sea?”“I think I could,” replied the child, “but I am sure that Pablo would throw it a great deal further than I can.”“Never mind, you shall try first.”Putting a fragment of ice into Nina’s hand, he addressed himself to Pablo: “Look out, Pablo; you shall see what a nice little fairy Nina is! Throw, Nina, throw, as hard as you can.”

Jules Verne: Off on a CometHector Sarvedac

Nina balanced the piece of ice two or three times in her hand, and threw it forward with all her strength.A sudden thrill seemed to vibrate across the motionless waters to the distant horizon, and the Gallian Sea had become a solid sheet of ice!

The old and the good 19th century...

1877

first order (1st kind) (2nd kind) continuous

T

Φ

Tc

T

Φ

Tc

F F

thermal phase transitions

0ST

0ST

• liquid-gas p=pcrit

• ferromagnet• superfluid• etc.

• solid-liquid• liquid-gas

p<pcrit• superconductor 1st kind

• Bose-Einstein• etc.

order parameter

disordered disordered

ordered ordered

ousdiscontinu0

TF∂∂ continuous0

TF∂∂

The brave new world of the 20th century...

4He 3He

New types of phases & phase transitions

pE∂∂ 0

discontinuous at pc⇒ first ordercontinuous at pc⇒ continuous

(nth order if discont.) non-thermal control parameter

ground-state energyquantum fluctuations only!

⇒ Quantum Phase Transitions

(QPTs) n

n

pE

∂∂ 0

Quantum Phase Transitions („lattice systems“)

• infinite lattice-like systems with semi-local interactions• continuous phase transition of the type order – disorder• always accompanied by a universal “quantum-critical” domain at T>0• sometimes followed by a line of thermal phase transitions • relevant in numerous “new materials”

J Hertz, Phys. Rev. B 14, 1165 (1976)S Sachdev, Quantum Phase Transitions (Cambridge Univ.Press, 1999)M Vojta, Rep. Prog. Phys. 66, 2069 (2003)

Quantum Phase Transitions („lattice systems“)

• infinite lattice-like systems with local interactions• continuous phase transition of the type order – disorder• always accompanied by a universal “quantum-critical” domain at T>0• sometimes followed by a line of thermal phase transitions • relevant in numerous “new materials”

J Hertz, Phys. Rev. B 14, 1165 (1976)S Sachdev, Quantum Phase Transitions (Cambridge Univ.Press, 1999)M Vojta, Rep. Prog. Phys. 66, 2069 (2003)

Example: Ising model in transverse magnetic field

spin-spin interaction between neighboring sites

external field in x-direction

chh >>

0=h

∏ →≈Ψi

i0

∏ ↑=Ψi

i0 ∏ ↓=Ψi

i0or

( )−+ += iixi SSS 2

1 ...spin flips

LiHoF4

Quantum Phase Transitions („many-body systems“)

DJ Thouless, Nucl. Phys. 22, 78 (1961) ... ”collapse of RPA”HJ Lipkin, N Meshkov, AJ Glick, Nucl.Phys. 62, 188 (1965) … pseudospin systemR Gilmore, DH Feng, Nucl.Phys. A 301, 189 (1978) with a QPT R Gilmore, J. Math. Phys. 20, 891 (1979) ... nonspin systemsAEL Dieperink, O Scholten, F Iachello, Phys.Rev.Lett.44,1747(1980) with bothDH Feng, R Gilmore, SR Deans, Phys. Rev. C 23, 1254 (1981) 1st,2ndorder QPTsR Gilmore, Catastrophe Theory for Scientists and Engineers (Wiley, N.Y., 1981)

Mostly associated with the collective dynamics of many-body systems⇒ finite number of degrees of freedom => fundamental consequences (see below)

⇒ algebraic language based on dynamical symmetries & generalizationsphases → “quasi dynamical symmetries” [Rowe et al. 1988…]critical points → approx. “critical point solutions” [Iachello 2000…]

→ “partial dynamical symmetries” [Leviatan 1994…]


Atomic nuclei= finite objects

⇒ may show only precursors of real QPTs

Example: spherical-deformed transition



∑∑

∑∑Ω

=+

+−−

Ω

=−

+++

Ω

=+

++

Ω

=−

+−

==

+−=

11

121

121

iii

iii

iii

iiiz

aaJaaJ

aaaaJ

1) N=2j ≤ Ω fermions on 2 levels with capacity Ω

Lipkin model: SU(2)

3) N=2j interacting scalar/pseudoscalar bosons:

( )ssttJtsJstJ z+++

−+

+ −=== 21

−+

12 +zN J

λ

( )2221

−+ ++= JJJH Nz λ

B. Bartlett, nucl-th/03050522) N=2j spin1/2 particles:

∑∑∑ === −−

++

i

ziz

ii

ii SJSJSJ

cλ


critical for∞→j


∑∑

∑∑Ω

=+

+−−

Ω

=−

+++

Ω

=+

++

Ω

=−

+−

==

+−=

11

121

121

iii

iii

iii

iiiz

aaJaaJ

aaaaJ

1) N=2j ≤ Ω fermions on 2 levels with capacity Ω

Lipkin model: SU(2)

3) N=2j interacting scalar/pseudoscalar bosons:

( )ssttJtsJstJ z+++

−+

+ −=== 21

−+

( )2221

−+ ++= JJJH Nz λ

2) N=2j spin1/2 particles:

∑∑∑ === −−

++

i

ziz

ii

ii SJSJSJ

Sharpens with

phase Iphase II

T=0T>0

Bloch sphere


critical for∞→j


SU(3)

O(6)

U(5)

O(6)

deformed

spherical2ndorder

Interacting Boson Model: U(6)

SU(3)

1storder

∑=i

ii GCw ][

Dynamical Symmetries (special classes of Hamiltonians) => algebraic solutions, integrability...=> QPTs based on competing dynamical symmetries

+s+md

2,....,2 +−=m

∑∑ +++ +=lkji

lkjiijklji

jiij bbbbvbbuH,,,,

( ) 00Ns+∝Ψ


( ) 0000Nds ++ +∝Ψ β

critical for∞→N


SU(3)

O(6)

U(5)

O(6)

deformed

spherical2ndorder

Interacting Boson Model: U(6)

SU(3)

1storder

∑=i

ii GCw ][+s

+md

2,....,2 +−=m

∑∑ +++ +=lkji

lkjiijklji

jiij bbbbvbbuH,,,,

( ) 00Ns+∝Ψ

( ) 0000Nds ++ +∝Ψ β


critical for∞→N


IBM-like models: U(n)

+s+mx

…llm +−= ,......, ∑=

iii GCw ][

∑∑ +++ +=lkji

lkjiijklji

jiij bbbbvbbuH,,,,

l n f model0 (t ) 2 1 Lipkin model (……) / 1D vibron model (molecules)1/2 (τ ) 3 2 2D vibron model (molecules)1 (p ) 4 3 3D vibron model (molecules)3/2 (π ) 5 4 4D “vibron” model (???)2 (d ) 6 5 IBM (nuclei)5/2 (δ ) 7 63 ( f ) 8 7 Octupole collectivity (nuclei)7/2 (φ ) 9 84 ( g ) 10 9 Hexadecapol collectivity (nuclei)


critical for∞→N


Quantum optical models: HW (1) G

+b igG ≡

⊗( )[ ]−+

++−

++ +++++= bJJbabJJbJbbHj4

100 λωω

G f modelSU(2) 1 a=0 Jaynes-Cummings model

2 a=1 Dicke model: superradianceSU(1,1) 1 a=0 creation/disociac. 2-atom molecules

2 a=1

2ndorder QPT

ϑϑ 20 sin2

,2cos ==+

jbb

jJ exc.atoms

exc.field


critical for∞→kj,

Quantum Phase Transitions („many-body systems“)Some general features of finite algebraic models

kijkji gcgg =],[

02

→

ℵ ∞→ℵ

E

( ) igHH =

ℵ

=ℵ κ

igHH

TTT

TT

YXYXYX

XXX

⋅−⋅≡⋅

−≡ 222

dynamical algebra

Size parameter consistent with the following requirements:1) vanishing thermal fluctuation of scaled energy in the thermodynamic limit:

(canonical → microcanonical ensemble)

Hamiltonian (+ observables)

2) consistent scaling of Hamiltonian

0, →

ℵℵ ∞→ℵκκ

ji gg∞→ℵ

κ >0

scaled generators with vanishing commutators for

⇒ thermodynamic limit = classical limit

Consequence:

notation:

j, k, N… from previous examples

Excited-State Quantum Phase Transitions (ESQPTs)

Free energy

control parameter

phaseseparatrix

tem

pera

ture

canonical density matrix

thermal average of energy

partition function

Hamiltonian

QPT

entropy

'ˆˆˆ0 HHH λ+=


Free energy

control parameter

phaseseparatrix


entropythermal average of energy

partition function

Hamiltonian

QPT

[ ] [ ]εεεερε ε d)(lnd)(ln)( dd Ω∝∝S

level density

'ˆˆˆ0 HHH λ+=

~E


Free energy

control parameter

phaseseparatrix



partition function

Hamiltonian

QPT


level density phase space volume

= classical limit

'ˆˆˆ0 HHH λ+=

~E


Free energy

control parameter

phaseseparatrix



partition function

Hamiltonian'ˆˆˆ0 HHH λ+=

QPT

slope & curvature of individual levels


level density phase space volume

= classical limit

~E

)(

)(

)(

)()(

E

pdxdHEdEd

E

pdxdHEE ffff

Ω

−Θ=

Γ

−∝ ∫∫δρ

1E

2E

x

V

E1E 2E

1storder

“continuous”*no order

0E

0E

Γ∝ρ


The phase-space criterion

1D potential:

* Special type of ESQPT, in a sense stronger than 1st order. The same ESQPT results from a 1D inflection point with , see the inset**0=Vdx

d

**

)(

242

22

2ˆ

xV

bxaxxdxd

MH +++−= a

b2nd order

.

.1st order

„Cusp“ Hamiltonian – potential with the “cusp” type of catastrophe:basic 1D model for 1st & 2nd order QPTs

Textbook quantum mechanics!!!Below we present numerical diagonalization for 210=ℵ

M∝ℵ

http://upload.wikimedia.org/wikipedia/commons/5/5a/Cusp_shape.svg

a

±=π

+=π

b

1st order

2nd order

a=+1, b=0

V

V

x

x

bb 2

0ψ

24ψ

x

„Cusp“ Hamiltonian

210=ℵ

a

±=π

+=π

b2nd order

a=+1, b=0

V

V

x

x

„Cusp“ Hamiltonian

1st order 1st order1st order

“continuous”

“continuous”

spinodal region

QCP

QCP

Generic ESQPT structures accompanying 1st & 2nd order QPTs

210=ℵ

2D vibron model2ndorder QPT

“continuous” ESQPT

l=0,1,2,3,4,5

Examples

)()1( QQnH Nb ⋅−−= ξξ sbbsQ ++ += λλ~

ESQPT precursors scale with l/N

O(6) U(5)

O(5) ang.momentum v = 0O(3) ang.momentum l = 0

N=102

dssdQ ~0

++ +=

2ndorder QPT

“continuous” ESQPT

Examples

sd-IBM

002d1 QQN

nN

H ⋅−

−=ηη

ESQPT precursors scale with v/N

deformed

spherical

N=80Examples

sd-IBM

2-level fermion pairing model

Examples SU(1,1) model SU(2) Jaynes-Cummings model

Some references: Classical anomalies vs. singularities in quantal spectra:

1D: Cary, Rusu… 1993, 2D (monodromy): Cushman, Bates 1997, Child ……… Lipkin model: Heiss, Müller 2002, Leyvraz, Heiss… 2005, Pairing model: Reis, Terra Cunha, Oliviera, Nemes 2005 IBM: Cejnar, Heinze, Macek, Jolie, Dobeš 2006 2D vibron & pairing models: Caprio, Cejnar, Iachello 2008 3D vibron model: Pérez-Bernal, Iachello 2008 Cusp model: Cejnar, Stránský 2008 Quantum optical models: Pérez-Fernándes, Cejnar, Relaño, Dukelsky, Arias 2010 Other studies being performed in the context of quantum optics, BECs etc.

-1 -0.5 0.5 1

-0.2

0.2

0.4

0.6

24)1()|( xxxV ξξξ +−∝

x

based on the old-QM quantization scheme[Einstein-Brillouin-Keller, 1917-1926-1958][Bohr-Sommerfeld-Wilson, 1915]

-1 -0.5 0.5 1

-0.75

-0.5

-0.25

0.25

0.5

0.75

-1 -0.5 0.5 1

-0.6

-0.4

-0.2

0.2

0.4

0.6

-0.75 -0.5 -0.25 0.25 0.5 0.75

-0.4

-0.2

0.2

0.4

x

p

Action along periodic orbit

1.0+=E

1.0−=E

0=E

)|()(2

1 2 ξxVpxm

H +=

“Flow of levels”: a semiclassical approach

position-dependent mass term appears in all boson models

)(d)]()[(22c.domain

)(

ESxxVExmq

xp

=−∫∈

50 100 150 200

-1 -0.5 0.5 1

-0.75

-0.5

-0.25

0.25

0.5

0.75

-1 -0.5 0.5 1

-0.6

-0.4

-0.2

0.2

0.4

0.6

-0.75 -0.5 -0.25 0.25 0.5 0.75

-0.4

-0.2

0.2

0.4

x

p( )( )

h

n nxxpS π2d)( 41∫ +==

Action along periodic orbit

∞ 0const

1.0+=E

1.0−=E

0=E

based on the old-QM quantization scheme[Einstein-Brillouin-Keller, 1917-1926-1958][Bohr-Sommerfeld-Wilson, 1915]

“Flow of levels”: a semiclassical approach

)(d)]()[(22c.domain

)(

ESxxVExmq

xp

=−∫∈

Semiclassical energy curves⇒ contours of the dependence S(E,ξ)

)();( ξξ nn ESES ⇒=E

ξ

2D vibron modell = 0

divergentlevel density

divergentcurvature

of level energy

very roughly:

sharper than:)1,0(|| ∈∝ kE k

||logconst

EE

−≈

∂∂ξ

E∆=

1ρ

2

2

ξ∂∂ iE

E E

local minimum/maximumcontinuous (2ndorder) continuous (no order)*

saddle point

Γ∝ρ

)(

)(

)(

)()(

E

pdxdHEdEd

E

pdxdHEE ffff

Ω

−Θ=

Γ

−∝ ∫∫δρ



2D potential:

Γ∝ρ

* Precursors difficult to distinguish from the 1st order

E E

local minimum/maximumcontinuous (2ndorder) continuous (no order)*

saddle point

Γ∝ρ

)(

)(

)(

)()(

E

pdxdHEdEd

E

pdxdHEE ffff

Ω

−Θ=

Γ

−∝ ∫∫δρ



2D potential:

Γ∝ρ

* Precursors difficult to distinguish from the 1st order

Question of chaos: ESQPTs rely on structural changes induced by close approach of levels. However, generic multi-dimensional systems are chaotic and thus exhibit repulsion of levels! Therefore, further weakening of ESQPT signatures is expected.

B

A

criticalspinodal

spherical0

(analog of 5D nuclear collective model for l=0)

saddle points

maximum

minima

oblate

prolate

2D collective model

ϕϕ

3cos112

1 3242

2

22 BrArrrr

rrr

H +++

∂∂

+∂∂

∂∂

ℵ−=

M∝ℵ

B

A

criticalspinodal

spherical0

saddle points

minimum

minima

oblate

prolate

(analog of 5D nuclear collective model for l=0)2D collective model

ϕϕ

3cos112

1 3242

2

22 BrArrrr

rrr

H +++

∂∂

+∂∂

∂∂

ℵ−=

M∝ℵ

B

A

criticalspinodal

spherical

oblate

prolate0

minimum

(analog of 5D nuclear collective model for l=0)2D collective model

ϕϕ

3cos112

1 3242

2

22 BrArrrr

rrr

H +++

∂∂

+∂∂

∂∂

ℵ−=

M∝ℵ

2nd order

2nd order

continuous

ground-state QPT

A

2D collective model 310,1 =ℵ=B

2nd order

2nd order

continuous

ground-state QPT

A

A

AA

Alevel density310,1 =ℵ=B

2−ℵ ]10,102[ 42⋅∈ℵ

A

A

AA

A

A

continuous

2nd order

2102,1 ⋅=ℵ=B2−ℵ level density ]10,102[ 42⋅∈ℵ

N=40l=0

ηphase-coexistence structures invisible for moderate N(because of zero-point motion)

1

0

-1

00

1d χχ

ηη QQN

nH ⋅−

−=

)2(]~[~ dddssdQ ×++= +++ χχ

27

0 −=χ

QPT1storder

U(5)

ESQPTscontinuous & 2ndorder

sd-IBM

-2

deformed

spherical

SU(3)

N=40l=0

η

phase-coexistence structures invisible for moderate N(because of zero-point motion)

1

0

-1

00

1d χχ

ηη QQN

nH ⋅−

−=

)2(]~[~ dddssdQ ×++= +++ χχ

27

0 −=χ

QPT1storder

U(5)


sd-IBM

-2

deformed

spherical

60=Nρ

εA B C

A… saddle pointB… local maximumC… asymptotic value

21=η

SU(3)

N=40l=0

η

1

0

-1

00

1d χχ

ηη QQN

nH ⋅−

−=

)2(]~[~ dddssdQ ×++= +++ χχ

27

0 −=χ

QPT1storder

SU(3) U(5)


sd-IBM

-2

deformed

spherical

A=C… saddle point & asymptotic value

0=η

60=Nρ

ε

A=C

QPT1storder


0=η

60=Nρ

ε

A=C

00

1d χχ

ηη QQN

nH ⋅−

−=

)2(]~[~ dddssdQ ×++= +++ χχ

27

0 −=χ

SU(3)

sd-IBM

[ ]NNNNNNNNµλµλµλε 3322

21 )()( ++++−=

NE

Nµ

)....4,182(),2,162(),0,122(

)....4,142(),2,102(),0,62()....4,82(),2,42(),0,2(),(

−−−−−−

−−=

NNNNNN

NNNµλ

Nλ

20

1

2−=ε1−=ε

5.0−=ε

Nµ

Nλ

N=20L=0 states

S

ερd

Sd∝

QPT1storder


60=Nρ

ε

A=C

00

1d χχ

ηη QQN

nH ⋅−

−=

)2(]~[~ dddssdQ ×++= +++ χχ

27

0 −=χ

SU(3)

sd-IBM

[ ]NNNNNNNNµλµλµλε 3322

21 )()( ++++−=

NE

20

1

2−=ε1−=ε

5.0−=ε

Nµ

NλS

ε−2 −1.5 −1 −0.5 0

singulargrowth

∞→N

ερd

Sd∝

Nµ

)....4,182(),2,162(),0,122(

)....4,142(),2,102(),0,62()....4,82(),2,42(),0,2(),(

−−−−−−

−−=

NNNNNN

NNNµλ

Nλ

N=20L=0 states

Dicke model

continuous ESQPT

saddle point



Some memos:• In “finite models”, infinite-size = classical !• Ground-State QPTs as changes of the potential minimum • Any singularity in the phase space above the minimum results in an

Excited-State QPT• ESQPTs are just a “microcanonical reformulation” of thermal phase

transitions !• Signatures of ESQPTs weaken with dimension (question of chaos)• ESQPTs have dramatic (or less dramatic) dynamical consequences !

Stellenbosch 2010

Date post:	25-Jul-2020
Category:	Documents
Upload:	others
View:	0 times
Download:	0 times

Exceptional Points and Quantum Phase Transitions · R Gilmore, Catastrophe Theory for Scientists...

Documents