Upload
others
View
3
Download
0
Embed Size (px)
Citation preview
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
Exceptional Pointsand Quantum Phase
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
Exceptional Pointsand Quantum Phase
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
Lecture IQuantum Phase Transitions – singularities in quantal spectra
affecting the ground state & excited states
“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…]
Quantum Phase Transitions („many-body systems“)
Atomic nuclei= finite objects
⇒ may show only precursors of real QPTs
Example: spherical-deformed transition
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)
Quantum Phase Transitions („many-body systems“)
∑∑
∑∑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
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)
critical for∞→j
Quantum Phase Transitions („many-body systems“)
∑∑
∑∑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
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)
critical for∞→j
Quantum Phase Transitions („many-body systems“)
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
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)
( ) 0000Nds ++ +∝Ψ β
critical for∞→N
Quantum Phase Transitions („many-body systems“)
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
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)
critical for∞→N
Quantum Phase Transitions („many-body systems“)
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)
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)
critical for∞→N
Quantum Phase Transitions („many-body systems“)
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
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)
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+=
Excited-State Quantum Phase Transitions (ESQPTs)
Free energy
control parameter
phaseseparatrix
canonical density matrix
entropythermal average of energy
partition function
Hamiltonian
QPT
[ ] [ ]εεεερε ε d)(lnd)(ln)( dd Ω∝∝S
level density
'ˆˆˆ0 HHH λ+=
~E
Excited-State Quantum Phase Transitions (ESQPTs)
Free energy
control parameter
phaseseparatrix
canonical density matrix
entropythermal average of energy
partition function
Hamiltonian
QPT
[ ] [ ]εεεερε ε d)(lnd)(ln)( dd Ω∝∝S
level density phase space volume
= classical limit
'ˆˆˆ0 HHH λ+=
~E
Excited-State Quantum Phase Transitions (ESQPTs)
Free energy
control parameter
phaseseparatrix
canonical density matrix
entropythermal average of energy
partition function
Hamiltonian'ˆˆˆ0 HHH λ+=
QPT
slope & curvature of individual levels
[ ] [ ]εεεερε ε d)(lnd)(ln)( dd Ω∝∝S
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
Excited-State Quantum Phase Transitions (ESQPTs)
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
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
Excited-State Quantum Phase Transitions (ESQPTs)
The phase-space criterion
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
Excited-State Quantum Phase Transitions (ESQPTs)
The phase-space criterion
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)
ESQPTscontinuous & 2ndorder
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)
ESQPTscontinuous & 2ndorder
sd-IBM
-2
deformed
spherical
A=C… saddle point & asymptotic value
0=η
60=Nρ
ε
A=C
QPT1storder
A=C… saddle point & asymptotic value
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
A=C… saddle point & asymptotic value
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
Lecture IQuantum Phase Transitions – singularities in quantal spectra
affecting the ground state & excited states
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