1. E XCITED-STATE QUANTUM PHASE TRANSITIONS IN SYSTEMS WITH FEW DEGREES OF FREEDOM Pavel Stránský Seminar at the Mathematical Modelling Lab, Northumbria University, Newcastle 6 th November 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 Leviatan Racah Institute of Physics, The Hebrew University, Jerusalem, Israel

2. 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

3. 1. E xcited- S tate Q uantum P hase T ransitions

4. What is a phase transition? A nonanalytic change of a system’s properties (order parameter, eg. density, magnetization) as a result of some external conditions (control parameter, eg. pressure, temperature) First-order - latent heat (eg. melting ice), order parameter discontinuous Second-order (continuous) - divergent susceptibility, infinite correlation length (eg. ferromagnetic-paramagnetic transition) Higher-order Infinite-order (Kosterlitz-Thouless transition in 2D XY models) Classification Temperature ferromagnetic phase paramagnetic phase finite-size effects thermodynamical limit TcTc

5. 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 EiEi 1 st order ground-state QPT 2 nd order (continuous) ground-state QPT Schematic example Quantum spectrum control parameter potential surfaces order parameter: ground-state energy E 0

6. And what is an ESQPT? EiEi nonanalyticity in the dependence of excitation energy E i (and the respective wavefunction) on the control parameter 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 Schematic example

7. Finite models -size 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 of atomic nuclei Generally, the number of degrees of freedom f grows with. 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) - (dimension of the irreducible representation of the U(6) algebra) In quantized classical systems, - this limit coincides with the classical limit

8. 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 s-boson condensate s+d-bosons condensate Examples of models with ESQPT 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 nuclei Lipkin 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 Graphene (ESQPT experimentally observed) B. Dietz, F. Iachello, M. Miski-Oglu, N. Pietralla, A. Richter, L. von Smekal, and J. Wambach, Phys. Rev. B 88, 104101 (2013)

9. 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)

10. 2. Models

11. CUSP 1D potential R. Gilmore, Catastrophe Theory for Scientists and Engineers, Wiley, NY, 1981 Quantum level dynamics E Potential shapes 1 st order ground-state QPT 2 nd order ground-state QPT Phase coexistence spinodal points calculated with E (from the catastrophe theory) P. Cejnar, P. Stránský, Phys. Rev. E 78, 031130 (2008)

12. 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 1 st 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 = -2A = -1A = 0A = 1 A = 2 E E B=0, D=C=0.5

13. 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 1 st 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 = -2A = -1A = 0A = 1 A = 2 E E B=0, D=C=1

14. 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 1 st 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 = -2A = -1A = 0A = 1 A = 2 E E B=0, D=C=4

15. 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 1 st 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 = -2A = -1A = 0A = 1 A = 2 E E B=0, D=C=20

16. 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 1 st 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 = -2A = -1A = 0A = 1 A = 2 E E B=0, C+D=4, D=1

17. 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 1 st 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 = -2A = -1A = 0A = 1 A = 2 E E B=0, C+D=4, D=2

18. 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 1 st 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 = -2A = -1A = 0A = 1 A = 2 E E B=0, C+D=4, D=4

19. 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 1 st 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 = -2A = -1A = 0A = 1 A = 2 E E B=0, D=C=20

20. 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 1 st 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 = -2A = -1A = 0A = 1 A = 2 E E B=30, D=C=20

21. 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 1 st 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 = -2A = -1A = 0A = 1 A = 2 E E B=39, D=C=20

22. 3. Signatures of ESQPT

23. 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.

24. Level density given by the volume of the classical phase space… … or by averaging the level density from the energy spectrum CUSP 1D systemCreagh-Whelan 2D system smooth part By approaching the infinite-size limit the averaged level density converges to the smooth semiclassical value

25. 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)

26. 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 = 3 f = 2

27. Isotropic minimum for arbitrary f the additional well behaves locally as  x k 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

28. k = 4 k = 2 (jump) Classification of singularities f = 1 local maximum local minimum E x 1. Local minimum 2. Local maximum (for quadratic minimum k = 2 ) 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

29. Classification of singularities f = 2 1. Local minimum (+) or maximum (-) quadraticseparableisotropic k = 4 k = 2 (break) isotropic minimum isotropic maximum k = 4 k = 2 2. Saddle point k = l = 2k = 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

30. Level density in the models CUSP potential ( f = 1 ) Creagh-Whelan potential ( f = 2 ) E A E A level densityderivative E A B = -2 B = 30, C=D=20

31. 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 rate infinite 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:

32. Flow rate in the CUSP system positive (levels rise) negative (levels fall) approximately 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

33. Flow rate in the Creagh-Whelan system The 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

34. Thermodynamical properties canonical ensemble - inverse temperature thermal distribution partition function smooth part Usually a single-peaked function. If the peak is narrow compared to the energy average, the microcanonical inverse temperature can be defined and calculated: Thermal anomalies can occur when ln  is not a monotonously increasing concave function of energy. Regular and irregular inverse temperature interested only in the irregular part caloric curve smooth part

35. Thermodynamics in the CUSP model secondary minimum saddle point

36. Thermodynamical properties for f = 2 local minimum – local maximumSaddle point 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.

37. 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

38. 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 - f = 2 Dicke model, Interacting boson model (with the rotations separated) 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).

39. Creagh-Whelan potential C = 0, D = 40C = 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

40. Conclusions The effect of more complicated kinetic terms Finite-size effects, multiple critical triangles Relation of the ESQPT with the chaotic dynamics Outlook 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 - f = 2 Dicke model, Interacting boson model (with the rotations separated) The nonanalytic features of ESQPT fade quickly with increasing f Singularities in the level density, thermal distribution function and flow rate are of the same type.

41. E Chaos and ESQPT C = D = 0.2C = 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. P. Pérez-Fernández, A. Relaño, J.M. Arias, P. Cejnar, J. Dukelsky, J.E. García-Ramos, Phys. Rev. E 83, 046208 (2011)

42. Conclusions The effect of more complicated kinetic terms Finite-size effects, multiple critical triangles Relation of the ESQPT with the chaotic dynamics T HANK YOU FOR YOUR ATTENTION More images on: http://www.pavelstransky.cz/cw.php Outlook 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 - f = 2 Dicke model, Interacting boson model (with the rotations separated) The nonanalytic features of ESQPT fade quickly with increasing f Singularities in the level density, thermal distribution function and flow rate are of the same type.

43. Conclusions The effect of more complicated kinetic terms Finite-size effects, multiple critical triangles Relation of the ESQPT with the chaotic dynamics T HANK YOU FOR YOUR ATTENTION More images on: http://www.pavelstransky.cz/cw.php Outlook 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 - f = 2 Dicke model, Interacting boson model (with the rotations separated) The nonanalytic features of ESQPT fade quickly with increasing f Singularities in the level density, thermal distribution function and flow rate are of the same type.