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E XCITED-STATE QUANTUM PHASE TRANSITIONS IN SYSTEMS WITH FEW DEGREES OF FREEDOM

E XCITED-STATE QUANTUM PHASE TRANSITIONS IN SYSTEMS WITH FEW DEGREES OF FREEDOM. Pavel Str ánský. www.pavelstransky.cz. Institut o de Ciencias Nucleares , Universidad Nacional Aut ó noma de M éxico. In collaboration with: Pavel Cejnar.

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E XCITED-STATE QUANTUM PHASE TRANSITIONS IN SYSTEMS WITH FEW DEGREES OF FREEDOM

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  1. EXCITED-STATE QUANTUM PHASE TRANSITIONS IN SYSTEMS WITH FEW DEGREES OF FREEDOM Pavel Stránský www.pavelstransky.cz Institutode Ciencias Nucleares, Universidad Nacional Autónoma de México In collaboration with: Pavel Cejnar Institute of Particle and Nuclear Phycics, Faculty of Mathematics and Physics, Charles University in Prague, Czech Republic Michal Macek, Amiram Leviatan Racah Institute of Physics, The Hebrew University, Jerusalem, Israel 4thOctober 2013 Seminario Lunch Nuclear, Instituto de Física, UNAM

  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. Excited-State Quantum Phase Transitions

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

  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 Schematic example Ei spectrum 2nd order (continuous) ground-state QPT 1st order ground-state QPT order parameter: ground-state energy E0 potential surfaces l control parameter

  6. And what is an ESQPT? A natural extension of the ground state QPT to excited part of the spectra nonanalyticity in level density ras a function of energy Schematic example Ei nonanalyticity in the dependence of excitation energy Ei(and the respective wavefunction) on the control parameter l nonanalyticity in level flow fas a function the control parameter critical bordeline in the l x E plane l The dimensionality of the system is crucial. ESQPT is related to the topological and structural changes of the phase space with varying the control parameter or energy.

  7. Finite models • size of the system • number of independent components of the system • number of degrees of freedom Nonanalyticities in phase transitions occur only when the system’s size grows to infinity. - this limit coincides with the classical limit Finite model: while f is maintained finite Generally, the number of degrees of freedom f grows with N. However, f is maintained in collective models described by some dynamical algebra A of rank r, for which • fis related with the rankr(in s & xboson models f = r - 1) • is usually related with the considered irreducible representation of the algebra In quantized classical systems, Example: Interacting Boson Model - 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) - (index of the representation of the U(6) algebra)

  8. Examples of models with ESQPT Geometric collective model of atomic nuclei O(6)-U(5) transition in the Interacting Boson Model 2-level fermionic pairing model 2-level pairing models Lipkin model M.A. Caprio, P. Cejnar, F. Iachello, Annals of Physics 323, 1106 (2008) M.A. Caprio, J.H. Skrabacz, F. Iachello, J. Phys. A 44, 075303 (2011) P. Cejnar, P. Stránský, Phys. Rev. E 78, 031130 (2008) 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) P. Cejnar, M. Macek, S. Heinze, J. Jolie, J. Dobeš, J. Phys. A: Math. Gen. 39, L515 (2006)

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

  12. Creagh-Whelan 2D potential Confinement conditions: • Extreemes lie on the line y= 0 • The potential profile on y = 0is the same as in the CUSP with B = -2 Other properties • integrable (separable) for B=C=0 • Bsqueezes one minimum and stretches the other • C squeezes both minima symmetrically • D squeezes the potential along x=0 axis Phase structure equals to the CUSP with B = -2 1st order ground-state QPT at A = 0 order parameter B=0, D=C=0.5 A = 2 A = -2 A = -1 A = 0 A = 1 y E potential for y = 0 E x S.C. Creagh, N.D. Whelan, Phys. Rev. Lett. 77, 4975 (1996); Phys. Rev. Lett. 82, 5237 (1999)

  13. Creagh-Whelan 2D potential Confinement conditions: • Extreemes lie on the line y= 0 • The potential profile on y = 0is the same as in the CUSP with B = -2 Other properties • integrable (separable) for B=C=0 • Bsqueezes one minimum and stretches the other • C squeezes both minima symmetrically • D squeezes the potential along x=0 axis Phase structure equals to the CUSP with B = -2 1st order ground-state QPT at A = 0 order parameter B=0, D=C=1 A = 2 A = -2 A = -1 A = 0 A = 1 y E potential for y = 0 E x S.C. Creagh, N.D. Whelan, Phys. Rev. Lett. 77, 4975 (1996); Phys. Rev. Lett. 82, 5237 (1999)

  14. Creagh-Whelan 2D potential Confinement conditions: • Extreemes lie on the line y= 0 • The potential profile on y = 0is the same as in the CUSP with B = -2 Other properties • integrable (separable) for B=C=0 • Bsqueezes one minimum and stretches the other • C squeezes both minima symmetrically • D squeezes the potential along x=0 axis Phase structure equals to the CUSP with B = -2 1st order ground-state QPT at A = 0 order parameter B=0, D=C=4 A = 2 A = -2 A = -1 A = 0 A = 1 y E potential for y = 0 E x S.C. Creagh, N.D. Whelan, Phys. Rev. Lett. 77, 4975 (1996); Phys. Rev. Lett. 82, 5237 (1999)

  15. Creagh-Whelan 2D potential Confinement conditions: • Extreemes lie on the line y= 0 • The potential profile on y = 0is the same as in the CUSP with B = -2 Other properties • integrable (separable) for B=C=0 • Bsqueezes one minimum and stretches the other • C squeezes both minima symmetrically • D squeezes the potential along x=0 axis Phase structure equals to the CUSP with B = -2 1st order ground-state QPT at A = 0 order parameter B=0, D=C=20 A = 2 A = -2 A = -1 A = 0 A = 1 y E potential for y = 0 E x S.C. Creagh, N.D. Whelan, Phys. Rev. Lett. 77, 4975 (1996); Phys. Rev. Lett. 82, 5237 (1999)

  16. Creagh-Whelan 2D potential Confinement conditions: • Extreemes lie on the line y= 0 • The potential profile on y = 0is the same as in the CUSP with B = -2 Other properties • integrable (separable) for B=C=0 • Bsqueezes one minimum and stretches the other • C squeezes both minima symmetrically • D squeezes the potential along x=0 axis Phase structure equals to the CUSP with B = -2 1st order ground-state QPT at A = 0 order parameter B=0, C+D=4, D=1 A = 2 A = -2 A = -1 A = 0 A = 1 y E potential for y = 0 E x S.C. Creagh, N.D. Whelan, Phys. Rev. Lett. 77, 4975 (1996); Phys. Rev. Lett. 82, 5237 (1999)

  17. Creagh-Whelan 2D potential Confinement conditions: • Extreemes lie on the line y= 0 • The potential profile on y = 0is the same as in the CUSP with B = -2 Other properties • integrable (separable) for B=C=0 • Bsqueezes one minimum and stretches the other • C squeezes both minima symmetrically • D squeezes the potential along x=0 axis Phase structure equals to the CUSP with B = -2 1st order ground-state QPT at A = 0 order parameter B=0, C+D=4, D=2 A = 2 A = -2 A = -1 A = 0 A = 1 y E potential for y = 0 E x S.C. Creagh, N.D. Whelan, Phys. Rev. Lett. 77, 4975 (1996); Phys. Rev. Lett. 82, 5237 (1999)

  18. Creagh-Whelan 2D potential Confinement conditions: • Extreemes lie on the line y= 0 • The potential profile on y = 0is the same as in the CUSP with B = -2 Other properties • integrable (separable) for B=C=0 • Bsqueezes one minimum and stretches the other • C squeezes both minima symmetrically • D squeezes the potential along x=0 axis Phase structure equals to the CUSP with B = -2 1st order ground-state QPT at A = 0 order parameter B=0, C+D=4, D=4 A = 2 A = -2 A = -1 A = 0 A = 1 y E potential for y = 0 E x S.C. Creagh, N.D. Whelan, Phys. Rev. Lett. 77, 4975 (1996); Phys. Rev. Lett. 82, 5237 (1999)

  19. Creagh-Whelan 2D potential Confinement conditions: • Extreemes lie on the line y= 0 • The potential profile on y = 0is the same as in the CUSP with B = -2 Other properties • integrable (separable) for B=C=0 • Bsqueezes one minimum and stretches the other • C squeezes both minima symmetrically • D squeezes the potential along x=0 axis Phase structure equals to the CUSP with B = -2 1st order ground-state QPT at A = 0 order parameter B=0, D=C=20 A = 2 A = -2 A = -1 A = 0 A = 1 y E potential for y = 0 E x S.C. Creagh, N.D. Whelan, Phys. Rev. Lett. 77, 4975 (1996); Phys. Rev. Lett. 82, 5237 (1999)

  20. Creagh-Whelan 2D potential Confinement conditions: • Extreemes lie on the line y= 0 • The potential profile on y = 0is the same as in the CUSP with B = -2 Other properties • integrable (separable) for B=C=0 • Bsqueezes one minimum and stretches the other • C squeezes both minima symmetrically • D squeezes the potential along x=0 axis Phase structure equals to the CUSP with B = -2 1st order ground-state QPT at A = 0 order parameter B=30, D=C=20 A = 2 A = -2 A = -1 A = 0 A = 1 y E potential for y = 0 E x S.C. Creagh, N.D. Whelan, Phys. Rev. Lett. 77, 4975 (1996); Phys. Rev. Lett. 82, 5237 (1999)

  21. Creagh-Whelan 2D potential Confinement conditions: • Extreemes lie on the line y= 0 • The potential profile on y = 0is the same as in the CUSP with B = -2 Other properties • integrable (separable) for B=C=0 • Bsqueezes one minimum and stretches the other • C squeezes both minima symmetrically • D squeezes the potential along x=0 axis Phase structure equals to the CUSP with B = -2 1st order ground-state QPT at A = 0 order parameter B=39, D=C=20 A = 2 A = -2 A = -1 A = 0 A = 1 y E potential for y = 0 E x S.C. Creagh, N.D. Whelan, Phys. Rev. Lett. 77, 4975 (1996); Phys. Rev. Lett. 82, 5237 (1999)

  22. 3. Signatures of ESQPT

  23. Level density smooth part oscillating part given by the volume of the classical phase space Gutzwiller formula (given by the sum of all classical periodic orbits and their repetitions) For the moment the focus will be only on the smooth part.

  24. Level density smooth part … or by averaging the level density from the energy spectrum given by the volume of the classical phase space… CUSP 1D system Creagh-Whelan 2D system 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) Special cases (sum over periodsof all distinct trajectories at energy E) f = 1: (derivative of the total accessible phase-space area) (sum over the total coordinate area accessible at energy E) f = 2:

  26. Stationary points of the potential a nonanalyticity of is caused only by a singularity (discontinuity) in , which occurs at each stationary point of the potential potential V is analytic E In the vicinity of the critical energy local maximum (saddle point) the level density is decomposed as local minimum Example: 2 isotropic harmonic oscillators in f dimensions x f = 1 f = 3 f = 2 With increasing number of freedom degrees, the level density is smoother and the singularities appear at higher derivatives

  27. Classification of singularities f = 1 E 1. Local minimum local maximum for (for quadratic minimum k = 2) local minimum for x k = 4 In 1D systems, a singularity always appears in the level density (jump or divergence) k = 2 (jump) infinite period of the motion on the top of the barrier - divergence k = 4 2. Local maximum k = 2 (logarithmic) for k = 2

  28. Classification of singularities f = 2 1. Local minimum (+) or maximum (-) quadratic separable isotropic k = 2 (break) isotropic maximum isotropic minimum 2. Saddle point k = 4 k = 4 k = 2 k = l = 2 k = 2, l = 3 saddles In a 2D system, the level density is continuous; a singularity (discontinuity) appears in the first energy derivative k = 2, l = 3 k = l = 2

  29. Isotropic minimum for arbitrary f the additional well behaves locally as dxk 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

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

  31. Thermodynamical properties - inverse temperature canonical ensemble microcanonical ensemble smooth part partition function thermal distribution usually a single-peaked function whose maximum gives the microcanonical inverse temperature: Regular and irregular temperature caloric curve interested only in the irregular part Thermal anomalies can occur when ln r is not a monotonously increasing concave function of energy.

  32. Thermodynamics in the CUSP model saddle point secondary minimum

  33. Thermodynamical properties for f = 2 local minimum – local maximum Saddle point for for for for (separable) these divergences affect the system for all b above a certain limiting value (isotropic) (regular component of the level density approximated in the figures by ) saddle – inflexion point quadratic saddle quadratic minimum – upward jump quadratic maximum – downward jump k = l = 2 k = 2, l = 3 The nonanalyticity in of the same type as in .

  34. Thermodynamics in the Creagh-Whelan model saddle point secondary minimum bimodal, but analytic (away from the critical region) Thermal distributions (populated energies in light shades) - higher temperature (lower b) brings the light upper in energy

  35. Flow rate of the spectrum - playing the role of velocity, it satisfies the Continuity equation weighted average of the expectation value of the perturbation: (connects the level density – with its singularities – and the flow rate) In our systems and this derivative equals x • The flow rate can be determined by • integrating the continuity equation: • using the Hellmann-Feynman formula from the wave functions: 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

  36. Flow rate in the CUSP system vanishes due to the potential symmetry approximately 0 positive (levels rise) negative (levels fall) 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 The wave function localized around the global minimum

  37. Flow rate in the Creagh-Whelan system flow rate energy derivative of the flow rate The 2D system is better studied by looking at the derivatives The singularities of the flow rate are of the same type as for the level density

  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, Tavis-Cummings model • - f = 2 Dicke model, Interacting boson model • Thenonanalyticfeaturesof ESQPT fade quicklywithincreasingf Singularities in the level density, thermal distribution function and flow rate are of the same type. Outlook • The effect of more complicated kinetic terms • Finite-size effects, multiple critical triangles • Relation of the ESQPT with the chaotic dynamics This work has been submitted to Annals of Physics (P. Stránský, P. Cejnar, M. Macek).

  39. Finite-size effects, multiple critical triangles, chaos C = 0, D = 40 C = 30, D = 10 C = 39, D = 1 E Creagh-Whelan potential A integrable (separable) Increasing chaos – decay of excited critical triangles The level dynamics is a superposition of shifted 1D CUSP-like critical triangles

  40. Conclusions • ESQPT originate in classical stationary pointsof 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, Tavis-Cummings model • - f = 2 Dicke model, Interacting boson model • Thenonanalyticfeaturesof ESQPT fade quicklywithincreasingf Singularities in the level density, thermal distribution function and flow rate are of the same type. Outlook • The effect of more complicated kinetic terms • Finite-size effects, multiple critical triangles • Relation of the ESQPT with the chaotic dynamics

  41. Chaos and ESQPT C = D = 0.2 C = D = 1 C = 39, D = 1 E Creagh-Whelan potential A Classical fraction of regularity black – regular, red - chaotic … but chaos is not so easy to besubdued. Itcanoccasionalybreakthrough. Many calculations indicate that the onset of chaos is related with the ESQPT (namely with the saddle point, serving as a shield against chaos)… P. Pérez-Fernández, A. Relaño, J.M. Arias, P. Cejnar, J. Dukelsky, J.E. García-Ramos, Phys. Rev. E 83, 046208 (2011)

  42. Conclusions • ESQPT originate in classical stationary pointsof 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, Tavis-Cummings model • - f = 2 Dicke model, Interacting boson model • Thenonanalyticfeaturesof ESQPT fade quicklywithincreasingf Singularities in the level density, thermal distribution function and flow rate are of the same type. Outlook • The effect of more complicated kinetic terms • Finite-size effects, multiple critical triangles • Relation of the ESQPT with the chaotic dynamics Thank you for your attention More images on: http://www.pavelstransky.cz/cw.php

  43. Conclusions • ESQPT originate in classical stationary pointsof 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, Tavis-Cummings model • - f = 2 Dicke model, Interacting boson model • Thenonanalyticfeaturesof ESQPT fade quicklywithincreasingf Singularities in the level density, thermal distribution function and flow rate are of the same type. Outlook • The effect of more complicated kinetic terms • Finite-size effects, multiple critical triangles • Relation of the ESQPT with the chaotic dynamics Thank you for your attention More images on: http://www.pavelstransky.cz/cw.php

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