Fundamental aspects of nuclear shape phase transitions

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1 Funamental aspects of nuclear shape phase transitions Pavel Cejnar Inst. of Particle an Nuclear Physics, Charles Univ., Prague, CZ ipnp.troja.mff.cuni.cz The stuy of shape phase transitions in nuclei an relate moels may enrich the general theory of quantum phase transitions.. The influence of aitional egrees of freeom We can stuy these phenomena in couple versions of Our Moels, such as protonneutron IBM, configuration-mixe IBM, Bose-Fermi IBFM. Mechanisms of first-orer an continuous QPTs Our Moels inclue both basic types of QPT within the same framework. We can perform comparative stuies of scaling properties, complex extensions 3. Quantum phase transitions for excite states Our Moels exhibit non-analytic evolutions of excite states relate to non-analytic changes of classical ynamics. We can analyze these effects to get eeper insight into the connection of QPTs with thermal phase transitions. Istanbul, September 9

2 Funamental aspects of nuclear shape phase transitions Pavel Cejnar Inst. of Particle an Nuclear Physics, Charles Univ., Prague, CZ ipnp.troja.mff.cuni.cz The stuy of shape phase transitions in nuclei an relate moels may enrich the general theory of quantum phase transitions.. The influence of aitional egrees of freeom We can stuy these phenomena in couple versions of Our Moels, such as protonneutron IBM, configuration-mixe IBM, Bose-Fermi IBFM. Mechanisms of first-orer an continuous QPTs Our Moels inclue both basic types of QPT within the same framework. We can perform comparative stuies of scaling properties, complex extensions 3. Quantum phase transitions for excite states Our Moels exhibit non-analytic evolutions of excite states relate to non-analytic changes of classical ynamics. We can analyze these effects to get eeper insight into the connection of QPTs with thermal phase transitions. Istanbul, September 9

3 . xtra egrees of freeom. xtra egrees of freeom combine systems combine systems proton proton-neutron IBM neutron IBM U5 SU3 O6 eforme axisymmetric spherical n orer st orer Caprio, Iachello 4,5 Arias, García-Ramos, Dukelsky 4 = = ζ ζ Q Q n = ] ~ [ ~ s s Q = ζ υ π ζ υ π υ π υ π υ π υ π υ π υ π ζ ζ Q Q Q Q Q Q n n = = υ π υ π = = V S coherent-state metho υ γ β π γ β υ υ υ π π π Γ Γ Ψ minimization of V=<Ψ Ψ> in β π,γ π,β ν,γ ν

. xtra egrees of freeom U5 = n Q Q ~ ~ Q = s s [ ] n orer ζ spherical ζ st orer IBFM withj 3/ fermion ζ B Q [ a a~

5,6,7, Liu 7, but no general phase iagram given.

quarupole parameter If : polarization of the core coherent-state metho to a triaxial shape an to Ψ Γ B c βγ ma jm

4 . xtra egrees of freeom U5 = n Q Q ~ ~ Q = s s [ ] n orer ζ spherical ζ st orer IBFM withj 3/ fermion ζ B Q [ a a~ = B F κ ] j j F O6 IBFM s QPTs stuie by Jolie et al. 4, Iachello 5, Alonso et al. 5,6,7, Liu 7, but no general phase iagram given.? eforme axisymmetric SU3 Bosonic core quarupole parameter sig B combine systems sig Boson-fermion interaction quarupole parameter If : polarization of the core coherent-state metho to a triaxial shape an to Ψ Γ B c βγ ma jm F the opposite axial shape m 3 j= minimization of V=<Ψ Ψ> in β,γ,c m = 7F Aβ Bβ cos3γ Cβ β β cos3γ 4 β V = κ ' κ' = κ 5 β β = = F κ'= prolate κ'= 4 triaxial oblate xample: SU3 bosonic core x fermion j=3/ κ'= 7

5 . xtra egrees of freeom U5 = n Q Q ~ ~ Q = s s [ ] n orer ζ spherical ζ st orer ζ O6 eforme axisymmetric SU3 combine systems = = Aitional types of bosons e.g., Devi, Kota 99 e.g., sg-ibm? ifferent configuration space other egrees of freeom than β,γ, higher eformations Aitional terms in the amiltonian igher-orer 3-boy boson interactions Van Isacker, Chen 98, Jolos 4, Thiamova, Cejnar 6 xternal rotation Cejnar,3 triaxial phase extene systems unchange V =

. xtra egrees of freeom Configuration mixing in IBM s-space for bosons projector P s-space for bosons projector P coherent-state metho Ψ = c P Γ βγ c P Γ βγ combine systems = P P P P = * * mix

6 . xtra egrees of freeom Configuration mixing in IBM s-space for bosons projector P s-space for bosons projector P coherent-state metho Ψ = c P Γ βγ c P Γ βγ combine systems = P P P P = * * mix minimization of V=<Ψ Ψ> in β,γ,c,c V = c Ψ Ψ Ψ Ψ c mix c Ψ Ψ Ψ Ψ mix De Coster, eye et al c This setup yiels phase structures beyon IBM, e.g., a possibility of prolate-oblate shape coexistence.* function of β,γ only; the lower eigenvalue forms the potential surface eigenpotential Frank, Van Isacker, Vargas 4 Frank, Van Isacker, Iachello 6 * Morales et al. 8 ellemans et al. 7,9

7 Funamental aspects of nuclear shape phase transitions Pavel Cejnar Inst. of Particle an Nuclear Physics, Charles Univ., Prague, CZ ipnp.troja.mff.cuni.cz The stuy of shape phase transitions in nuclei an relate moels may enrich the general theory of quantum phase transitions.. The influence of aitional egrees of freeom We can stuy these phenomena in couple versions of Our Moels, such as protonneutron IBM, configuration-mixe IBM, Bose-Fermi IBFM. Mechanisms of first-orer an continuous QPTs Our Moels inclue both basic types of QPT within the same framework. We can perform comparative stuies of scaling properties, complex extensions 3. Quantum phase transitions for excite states Our Moels exhibit non-analytic evolutions of excite states relate to non-analytic changes of classical ynamics. We can analyze these effects to get eeper insight into the connection of QPTs with thermal phase transitions. Istanbul, September 9

8 . QPT mechanisms xample: nergy gap τ = h finite-size scaling properties = st orer n orer * st excite groun etermines a characteristic time-scale of the system e a / 3 c * c * Lipkin amiltonian: η η = nt QQ n t t Q t s s t t t t =, = Vial et al., PRC 73, Nee for large- calculations => methos beyon the mean fiel Results for two-level boson moels: Dusuel, Vial, Arias, Dukelsky, García-Ramos 5-7

There is no global scaling of the whole spectrum at the critical point, but the h = M x n orer At the critical point the whole spectrum per

9 . QPT mechanisms finite xample: nergy gap c τ = h e a V st orer spectrum per boson consists of parity oublets with spacings > ħ = < exp a /ħ = exp a Quantum tunneling x 4 3 x x 8 x finite-size scaling properties = 4 st excite groun etermines a characteristic time-scale of the system There is no global scaling of the whole spectrum at the critical point, but the h = M x n orer At the critical point the whole spectrum per boson scales as 4/3. C x ~ 3 x = x 4 = 3 4 x V x c M ~ x 4/3 Pure quartic oscillator scales as ħ 4/3 C~ x Rowe et al ρ, ρ, low x D case low 4 3 x

10 . QPT mechanisms complex extension = λ & λ Λ= λ iµ Degeneracy of amiltonian eigenvalues: real case => rear solutions complex case => abunant solutions Complex egeneracies etermine the ynamics of real energy levels! Kato 966 Zirnbauer, Verbaaschot, Weienmüller 983 Shanley 988 eiss 988, eiss, Steeb 99, eiss, Sannino 99, eiss, Kotzé 99. Cejnar, einze, Dobeš 5, Cejnar, einze, Macek 7 λ

11 . QPT mechanisms complex extension = λ & λ Λ= λ iµ x example: e = v ± = [ ± ] v Λ e& e v& = v& eλe& = e& vλv& e vλv& Λe& = V 4V = a bλ cλ V Degeneracy = real case complex case conitions => rear solutions in λ conition => solution in Λ always exists local behavior close to complex egeneracy Λ =λ ±iµ real egeneracy Λ =λ ±i if any Λ Λ ± Λ Λ branch point iabolic point Passing a branch point along the real axis: λ λ 44 µ 443 R istance of the branch point from a given place on the real axis avoie crossing of levels actual crossing branch point µ =.5 branch point µ =. iabolic point µ =

12 . QPT mechanisms complex extension = λ & λ Λ= λ iµ x example: = v v Λ λ Sharpness of the crossing inicate by: C ln R= ln λ λ D electrostatic potential Coulomb force graient µ R Λ λ v => st orer QPT C v=. v=. v=.5 height= v fwhm v C v v Peak area: λ λ fwhm height v Cejnar, einze, Macek 7 c λ

13 . QPT mechanisms complex extension = λ & λ Λ= λ iµ Cusp amiltonian: st orer QPT h 4 = x ax bx x λ size parameter = ħ 5 levels inclue in the calculation, only the closest one gives an essential contribution st orer QPT is locally a -level process! height many-level system C = ln R = i ln λ λ i> fwhm x height µ i> Λ i R i i λ h =.7 C fwhm height fwhm Peak area: All branch points locate on the λ Riemann sheet corresponing to the groun state solution λ Cejnar, einze, Macek 7 fwhm height exp a / h c λ

14 . QPT mechanisms complex extension = λ & λ Λ= λ iµ Cusp amiltonian: n orer QPT h 4 = x ax bx x λ size parameter = ħ 5 levels inclue in the calculation, giving only slowly ecreasing contributions Cont. QPT is locally a many-level process! many-level system C = ln R = i ln λ λ i> height fwhm x height µ i> R i Λ i i λ h =. C fwhm height fwhm Peak area: fwhm height h All branch points locate on the λ Riemann sheet corresponing to the groun state solution λ Cejnar, einze, Macek 7 /3 / c λ

15 . QPT mechanisms complex extension Cusp amiltonian: n orer QPT h 4 = x ax bx x λ size parameter = ħ 5 levels inclue in the calculation, giving only slowly ecreasing contributions Cont. QPT is locally a many-level process! h =. C = λ & λ Λ= λ iµ fwhm height λ fwhm C = h ln R = i h ln λ λ i> height fwhm x height Peak area: scaling factor fwhm height h /3 The number of strongly contributing terms in the sum is proportional to ħ i> c i Thermoynamic analogy: Scale C is an analog of specific heat Branch points are like complex zeros of partition function [Yang, Lee 95] Cejnar, einze, Dobeš 5 zero latent heat in n orer phase transition for ħ The scaling factor oes not influence the exponential increase for the st orer QPT => infinite latent heat

16 . QPT mechanisms complex extension IBM O6-U5 amiltonian: n orer QPT scaling factor η η n n = n C = Q Q ln Ri = ln i n λ i= n λ i= size parameter ħ ~ ~ ~ n n = Q = s s [ ] C = λ & λ Λ= λ iµ states with seniority= fwhm x height Peak area: fwhm height / 3 n = no. of levels involve imension n = no. of branch points on the g.s. Riemann sheet Thermoynamic analogy: Scale C is an analog of specific heat Branch points are like complex zeros of partition function [Yang, Lee 95] Cejnar, einze, Dobeš 5 zero latent heat in n orer phase transition c for ħ Cejnar, einze, Macek 7

17 Funamental aspects of nuclear shape phase transitions Pavel Cejnar Inst. of Particle an Nuclear Physics, Charles Univ., Prague, CZ ipnp.troja.mff.cuni.cz The stuy of shape phase transitions in nuclei an relate moels may enrich the general theory of quantum phase transitions.. The influence of aitional egrees of freeom We can stuy these phenomena in couple versions of Our Moels, such as protonneutron IBM, configuration-mixe IBM, Bose-Fermi IBFM. Mechanisms of first-orer an continuous QPTs Our Moels inclue both basic types of QPT within the same framework. We can perform comparative stuies of scaling properties, complex extensions 3. Quantum phase transitions for excite states Our Moels exhibit non-analytic evolutions of excite states relate to non-analytic changes of classical ynamics. We can analyze these effects to get eeper insight into the connection of QPTs with thermal phase transitions. Istanbul, September 9

18 3. xcite-state quantum phase transitions temperature ˆ thermal phase transition a singular evolution of the spectrum with the control parameter = ˆ ˆ λ ' phase separatrix Free energy control parameter S thermoynamic limit level ensity lnρ amiltonian thermal average of energy canonical ensity matrix entropy partition function Cejnar, Stránský 8

19 3. xcite-state quantum phase transitions energy ˆ thermal phase transition a singular evolution of the spectrum with the control parameter = ˆ ˆ λ ' Finite quantum systems systems with a phase separatrix control parameter finite number of quantum egrees of freeom: ħ Singular growth of the level ensity st orer: iscontinuity n orer: iscontinuous st erivative n th orer: iscontinuous n th erivative continuous no orer: singular erivative The λ x plane is more convenient in the finite case than the λ x T plane Singular evolution of the spectrum anomalous level ynamics sharp collisions avoie crossings of many levels iniviual levels may show coherent or chaotic ynamics Cejnar, Stránský 8

20 3. xcite-state quantum phase transitions η η = n n = Q= Q = ~ Q = s s [ einze, Macek, Cejnar, Jolie, Dobeš 6 Cejnar, einze, Macek 7 C k = n λ i k ln i k ~ ~ ] x O6-U5 leg of IBM SU3 O6 = n k [,] L= U5 fwhm height m x =. < /3 i η ρ D D L= v= v O6 SQPT continuous precursors scale with v/ = L= seniority v = QPT n orer η U5

21 3. xcite-state quantum phase transitions = Q Q 7 7 = = ~ ~ Q = s s [ ] [ ] λ µ λ µ 3 λ 3 µ = λ, µ =,, 4,, 8,4... 6,,,, 4,4... µ,, 6,, 8,4... M O = L= states SU3 limit of IBM SU3 O6 L= U5 µ =.5 = = λ level ensity ρ S Number of states with energy is prop. to area S S λ

22 3. xcite-state quantum phase transitions = Q Q 7 7 = = ~ ~ Q = s s [ ] [ ] λ µ λ µ 3 λ 3 µ = λ, µ =,, 4,, 8,4... 6,,,, 4,4... µ,, 6,, 8,4... M O = L= states Phase transition at =.5 continuous => softer than st orer singular => ρ harer than n orer => continuous phase transition with no hrenfest orer ρ SU3 limit of IBM SU3 O6 L= U5 µ = = λ critical contour =.5 S λ ρ level ensity ρ S singular growth.5.5

3. xcite-state quantum phase transitions = Q Q 7 7 = = ~ ~ Q = s s [ ] [ ] λ µ λ µ 3 λ 3 µ = 3 4 4β β cos3γ β classical potential coherent-state metho 3 V = β sale point & asymptotic value 3 Phase

23 3. xcite-state quantum phase transitions = Q Q 7 7 = = ~ ~ Q = s s [ ] [ ] λ µ λ µ 3 λ 3 µ = 3 4 4β β cos3γ β classical potential coherent-state metho 3 V = β sale point & asymptotic value 3 Phase transition at =.5 continuous => softer than st orer singular => ρ harer than n orer => continuous phase transition with no hrenfest orer ρ SU3 limit of IBM SU3 O6 ρ singular growth L= U5 ρ f f 4 δ x p= Γ level ensity phase-space volume f f Θ x p Ω.5.5

24 3. xcite-state quantum phase transitions = Q Q 7 7 = = ~ ~ Q = s s [ ] [ ] λ µ λ µ 3 λ 3 µ = Finite- realization: SU3 limit of IBM Phase transition at =.5 continuous => softer than st orer singular => ρ harer than n orer => continuous phase transition with no hrenfest orer ρ SU3 O6 L= U5 ρ = 6 ρ.5.5

25 3. xcite-state quantum phase transitions = n Q = 7 Q = 7 SU3-U5 U5 leg of IBM SU3 L= 3 O6 U5 Finite- realization: =4 ρ = 6 A B C Types of phase transitions: A sale point continuous no orer B local maximum n orer C asymptotic value???

3. xcite-state quantum phase transitions = η n η Q 7 Q 7 SU3-U5 U5 leg of IBM SU3 = = L= O6 U5 highest energy C B A η =4 lowest energy = 6 st orer groun-state QPT More complex structures of the

26 3. xcite-state quantum phase transitions = η n η Q 7 Q 7 SU3-U5 U5 leg of IBM SU3 = = L= O6 U5 highest energy C B A η =4 lowest energy = 6 st orer groun-state QPT More complex structures of the potential energy surface aroun the st orer QPT. Their influence on the spectrum was not yet stuie in the IBM, but only in simpler moels. Types of phase transitions: A sale point continuous no orer B local maximum n orer C asymptotic value???

x st orer QPT b B local maximum continuous

27 3. xcite-state quantum phase transitions = a= h= h 4 x ax bx x V D cusp amiltonian aroun st orer groun-state QPT Cejnar, Stránský PR 78,33 8 b D B B B x D B ψ i D D x st orer QPT b B local maximum continuous no orerd n orer D D seconary minimum st orer D n orer D

28 3. xcite-state quantum phase transitions = b= h= h 4 x ax bx x D cusp amiltonian aroun n orer groun-state QPT Cejnar, Stránský PR 78,33 8 a = V B x - B B π =± π= a n orer QPT B local maximum continuous no orerd n orer D

29 3. xcite-state quantum phase transitions ξ = ξ n Q Q D angular momentum b ~ Q= s bλ bλ s B D vibron moel aroun n orer groun-state QPT V B r Pérez-Bernal, Iachello 8 Caprio, Cejnar, Iachello 8 Lipkin moel D eiss, Müller Leyvraz, eiss 5 Relaño, Arias, Dukelsky, García- Ramos, Pérez-Fernánez 8 vibron moel 3D D D IBM O6-U5 U5 5D D D Cejnar, einze, Jolie, Macek, Dobeš 6, 7 fermion pairing moel D Reis, Terra Cunha, Oliviera, Nemes 5 Caprio, Cejnar, Iachello 8 D vibron moel λ=½ ξ

30 Conclusions The stuy of shape phase transitions in nuclei an relate moels can enrich the general theory of quantum phase transitions.. The influence of aitional egrees of freeom Creation of new phases an phase transitions in combine systems. or cases. Mechanisms of first-orer & continuous QPTs Asymptotic behavior of complex branch points near the real axis of the moel control parameter etermines the type of the phase transition. st orer locally -level mechanism continuous many-level mechanism 3. Quantum phase transitions for excite states Non-analytic changes observe in spectra of excite states. These changes are roote in classical ynamics of the system. Frierich Justin Bertuch, 79-83, Bilerbuch für Kiner, source: Wikipeia singular growth of level ensity singular evolution of the spectrum with control parameter Istanbul, September 9

Pavel Cejnar Institute of Particle & Nuclear Physics, Charles University, Praha

and Nuclear Structure Pavel Cejnar Institute of Particle & Nuclear Physics, Charles University, Praha ha,, CZ cejnar @ ipnp.troja.mff.cuni.cz Program: > Shape phase transitions in nuclear structure data