SYMMETRY AND PHASE TRANSITIONS IN NUCLEI. Francesco Iachello Yale University

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1 SYMMETRY AND PHASE TRANSITIONS IN NUCLEI Francesco Iachello Yale University Bochum, March 18, 2009

2 INTRODUCTION Phase diagram of nuclear matter in the r-t plane T(MeV) n(fm -3 ) 200 Critical point? Quark-gluon plasma? 100 Critical point? Nuclei Hadronic gas Color superconductor? Gas Solid? r(10 14 g/cm 3 ) Liquid Neutron stars

3 Finite nuclei: Liquid drops with radius R(θ,φ) Phase transitions between different shapes? Similar to phase transitions between different crystal modifications. Landau example: BaTiO 3 cubic O h tetragonal D 4h The phases have different symmetry SO(3) D 4h O h tetragonal( V ) cubic( VII ) Shape phase transitions in nuclei fl One of the best studied examples of phase transitions in physics Nuclei are finite systems fl finite size scaling

4 QUANTUM PHASE TRANSITIONS (QPTs) QPT s are phase transitions that occur as a function of coupling constants, x 1, x 2,, called control parameters, that appear in the quantum Hamiltonian, H, that describes the system H = ε ( H + ξ H + ξ H +...) Associated with phase transitions there are order parameters, the expectation values of suitable chosen operators that describe the state of the system OÚ. A simple example is the Ising model (a system of spins on a linear lattice) in a transverse magnetic field x z z H V = ξ ˆ σ ˆ ˆ i σiσ i i ij which has a second order QPT at x=x c =1/2. The order parameter is the magnetization M The two phases are (i) ferromagnet for x<x c and (ii) paramagnet for x>x c. [QPT s are also called ground state phase transitions and/or zero temperature phase transitions.] ˆ z σ R. Gilmore, J. Math. Phys. 20, 891 (1979).

5 N=n s +n d correlated pairs of nucleons (bosons) with J=0,2 (s,d) and with Hamiltonian with THEORY OF (SHAPE) QPTs IN NUCLEI: THE INTERACTING BOSON MODEL fl algebraic structure U(6). ξ ˆ χ ˆ ˆ χ H = ε0[(1 ξ) nd Q Q ] 4N ( ) ( ) χ ( ) nˆ = d d d Qˆ χ = d s+ s d + d d (2) (2) F. Iachello and A. Arima, The Interacting Boson Model, Cambridge University Press, The IBM has three phases, with dynamic symmetries U(5), SU(3), and SO(6), corresponding to the breaking of U(6) into its sub-algebras J=0 J=2 U(6) U(5) (I) SU(3) (II) ξ=0, χ=anything ξ=1, χ=- 7/2 SO(6) (III) ξ=1, χ=0 and two control parameters, ξ, χ. [The number of control parameters is the number of phases minus one.]

6 A convenient quantal order parameter is ν = 1 0 nˆ d 0 N [In general for U(n) models, the number of independent order parameters is n-1. If rotational invariance is imposed, the actual number is less than n-1, for this problem two. Order parameters are easily recognizable in the classical approach. Several suggestions have been made for the second quantal order parameter. One of them is ν 2 = 0 nˆ 0 0 nˆ d 2 1 d 1 N This order parameter is useful to study some properties of the system, but it requires knowledge of excited states.]

7 Phase transitions and their order Mean field: Number projected coherent states Bohr variables Euler angles N s d ; αμ = + αμ μ 0 μ N α ( βγθ,,, θ, θ) μ For s and d bosons the drop has ellipsoidal shape with radius θ i R = R0 1 + α μ Y2 μ ( θφ, ) μ U(5) Spherical U(6) SU(3) Deformed with axial symmetry SO(6) Deformed with tri-axial symmetry (γ-unstable)

8 (Ground state) energy functional EN ( ; βγ, ) = N; β, γ H N; β, γ N; β, γ N; β, γ also called Landau potential, V(β,γ). Minimization of V with respect to β,γ provides the equilibrium values β e,γ e (the classical order parameters) and the ground state energy V min Evaluation of V min and its derivatives with respect to the control parameters provides the order of the phase transition. Ehrenfest classification (discontinuities of the Landau potential as a function of the control parameters.) Vmin Vmin ξ 2 V ξ... min 2 0 th order 1 st order 2 nd order No discontinuity Crossover

9 Shape phase diagram of nuclei in the Interacting Boson Model 2 nd order SO(6) χ Deformed phase Spherical phase U(5) SU(3) 1 st order Coexistence region ξ D.H. Feng, R. Gilmore and S.R. Deans, Phys. Rev. C23, 1254 (1981).

10 SHAPE PHASE TRANSITIONS IN NUCLEI Shape phase transitions have been observed unambiguously in two regions of the chart of nuclides: Region I : Nd-Sm-Gd-Dy Region II : Sr-Zr [The control parameter, ξ, is proportional to the number of valence nucleons, i.e. 2N, where N is the boson number.] Signatures of phase transitions that can be measured Ground state energy E 0 Separation energy Double separation energy Nuclear radius S N E N E N E ( ) 0 2n = ( 1) ( ) ξ 2 E0 S2n'( N) S2n( N 1) S2n( N) 2 = + ξ r = r + DN + F n ˆ c d 0 1 Isotope shift ( N ) N + 1 N ( N + 1) ( N ) δ r = r r = D+ F nd + n 0 d ˆ ˆ B(E2) BE ( 2;2 0 ) d + + ˆ 1 1 n 2

11 Shape quantum phase transitions observed in nuclei Region I: Nd-Sm-Gd-Dy Critical value Critical value E 0 ξ 2 nd order 1 st order Order parameter 2 nd order 1 st order Critical value Critical value Control parameter Control parameter

12 FINITE SIZE SCALING The behavior of the order parameter(s) can be investigated numerically Mean field ξ c = 0.5 Qualitatively: In finite systems the discontinuities are smoothed out 0 th orderø1 st order 1 st orderø2 nd order 2 nd orderøcrossover ξ c = 0.5

13 THEORY OF FINITE SIZE SCALING For second order transitions all physical quantities scale as a power law For example, the order parameter ν 1 scales as M.E. Fisher and M.N. Barber, Phys. Rev. Lett. 28, 1516 (1972). A 1 ( = ) = A N n ν ξ ξ AΦ Φ ( ξ = ξ ) N 1 c n0 c The U(5)-SO(6) transition has been studied theoretically by the method of Continuous Unitary Transformation (CUT) and compared with numerical studies Order parameter Scaling exponent, S nˆ 1 d ν = nˆ d N 1/3-2/3

14 EXCITED STATE QUANTUM PHASE TRANSITIONS (ESQPT) Phase transitions occur also as a function of excitation energy (also called temperature dependent phase transitions). The situation is best illustrated by means of a correlation diagram, where the energy levels are plotted as a function of ξ. Discontinuities are encountered when traversing the diagram both in the horizontal (QPT) and in the vertical (ESQPT) direction. T Separatrix QPT U(5) ξ c (0) ESQPT SO(6) M. Caprio, P. Cejnar and F. Iachello, Ann. Phys. (N.Y.) 323, 1106 (2008).

15 FINITE SIZE SCALING IN ESQPT Shape ESQPT are difficult to observe in nuclei because they rely on the measurement of highly excited states. [They have recently been observed in molecules.] A property of the correlation diagram that has been tested in nuclei is the collapse of all excitation energies to zero at the critical point. [An example of this collapse is shown here for the molecular U(3) model ] U(3) U(2) SO(3) F. Perez-Bernal and F. Iachello, Phys. Rev. A77, (2008). U(2) ξ c SO(3)

16 Infinite system (NØ ) In finite systems a gap develops E/ε 0 E/ε ξ c ξ Δ ξ ξ c Finite size scaling of the gap AΓ Δ ( ξ = ξ )~ N c Scaling exponent, S -1/3

17 Evidence for the gap in nuclei 600keV Δ = keV N 10 O. Scholten, F. Iachello and A. Arima, Ann. Phys. (N.Y.) 115, 325 (1978)

18 CRITICAL MATTER Matter at the critical value of a phase transition, ξ=ξ c. Expected to be rather complex, especially at the critical value of a first order transition, where two phases co-exist. Instead simple. Critical matter in the Interacting Boson Model SO(6) E(5) U(5) X(5) Critical matter SU(3)

19 Approximate analytic expressions for the energy spectra of nuclei at the critical point of the second order transition, called E(5), and along the line of first order transitions, called X(5): Zeros of Bessel functions. a. Critical point of a spherical to γ-unstable transition, U(5)-SO(6), called E(5) ( ) 2 s Es (, τν, Δ, LM, L) = E0 + Ax, τ Landau potential at ξ=ξ c flat with x=s-th zero of J λ λ = τ( τ + 3) b. Critical point of a spherical to axially deformed transition, U(5)-SU(3), called X(5) (,, γ,, ) 0 sl, ( ) 2 2 γ EsLn KM = E+ B x + An + CK with x s =s-th zero of J λ LL ( + 1) 9 λ = F. Iachello, Phys. Rev. Lett. 85, 3580 (2000). F. Iachello, Phys. Rev. Lett. 87, (2001).

20 Evidence for critical nuclei R.F. Casten and N.V. Zamfir, Phys. Rev. Lett. 87, (2001). All ratios parameter free! R.F. Casten and N.V. Zamfir, Phys. Rev. Lett. 85, 3584 (2000). Scale invariance at the critical value (Conformal invariance in QFT)

21 The situation can be summarized in the following diagram: left (algebraic dynamic symmetries and phase transitions), right (analytic descriptions and critical symmetries). Sm-isotopes Critical nucleus Other critical nuclei identified: 150 Nd and 154 Gd R. Krüken et al., Phys. Rev. Lett. 88, (2002).

22 MICROSCOPIC DESCRITIONS OF NUCLEAR SHAPE PHASE TRANSITIONS Several attempts at a microscopic description of nuclear shape phase transitions were made in the 1980 s. Yang Li-Ming et al. In view of the renewed interest in quantum phase transitions, new attempts are being made by making use of density functional theory, DFT: Rodriguez-Guzman et al. in Xe U(5)-SO(6) [Non relativistic mean field] Niksic, Vretenar, Lalazissis and Ring in Nd U(5)-SU(3) [Relativistic mean field] Nomura, Shimizu and Otsuka in Sm U(5)-SU(3) [Non relativistic mean field (SLy4)] Rodriguez and Egido in Nd [Non relativistic mean field] U(5)-SU(3) In all these approaches potential energy surfaces have been constructed. For a review see, F. Iachello and I. Talmi, Rev. Mod. Phys. 59, 339 (1987)

23 All DFT s appear to describe well the second order transition U(5)-SO(6) and produce potentials in agreement with those suggested by the IBM potential along the U(5)-SO(6) line and for the corresponding critical matter point E(5). R. Rodriguez- Guzman, private communication (2007); L. M. Robledo, R. Rodriguez-Guzman and P. Sarriguren, Phys. Rev. C 78, (2008). SO(6) Critical nucleus E(5) U(5)

24 Density functional theory appears however to describe the first order transition U(5)-SU(3) as a second order transitions (no coexistence), although a flat bottom potential appears at the critical value X(5). T. Niksic, D.Vretenar, G.A. Lalazissis and P. Ring, Phys. Rev. Lett. 99, (2007).

25 Several attempts have been made to construct spectra along the transition lines making use of different approximate methods: Generator coordinate method GCM Collective Model CM Interacting Boson Model IBM All calculations so far appear to produce spectra in accordance with the phase transitional behavior, but quantitatively in disagreement by a factor of ~1.5, especially for the gap Δ K. Nomura, N. Shimizu, and T. Otsuka, Phys. Rev. Lett (2008).

26 Surprisingly, they produce order parameters in agreement with experiment and with phenomenological IBM calculations Z.P. Li, T. Niksic, D. Vretenar, J. Meng, G.A. Lalazissis and P. Ring, private communication Square of the order parameter Towards an ab initio theory of quantum phase transitions?

27 QPT and ESQPT: Universal concept Universality a : All second order phase transitions of the type U(n) appear to have the same critical exponents, and the same scaling exponents. a P. Cejnar and F. Iachello, J. Phys. A: Math. Theor. 40, 581 (2007). U(n-1) SO(n) [Same method of analysis has been used to study QPT s in molecules within the framework of the Vibron Model, where also evidence for ESQPT have been found. For example, the molecule hydrogen peroxide, H 2 O 2, has a shape ESQPT from a-planar to trans-bent at ~250K~361cm -1 ]. F. Iachello and R.D. Levine, Algebraic Theory of Molecules, Oxford University Press, f H P 2 O O H P 1 B. Kuhn, T.R.Rizzo, D.Luckhaus, M. Quack, and M.A.Suhm, J. Chem. Phys.111,2565 (1999). f=0 0 cis-bent f=180 0 trans-bent f 0 0 or a-planar

28 CONCLUSIONS Shape Phase Transitions in nuclei: A paradigm for QPT s in physics Experimentally verified! Novel results: Theory of shape QPT and their scaling behavior Theory of ESQPT and their scaling behavior Experimental evidence for critical matter! (Scale invariance) Ab initio (DFT) theory of QPT Additional work not discussed here QPT in coupled systems (two fluids, especially proton-neutron) QPT in angle variables QPT in mixed Bose-Fermi systems QPT in configuration mixed models Symmetry plays a crucial role in QPT s and ESQPT s: It determines the phases of the system and the order of the phase transitions between them