Pavel Cejnar. of ESQPTs. Places to be visited: 1) Model examples 2) General features 3) Some implications 4) Experimental relevance

Transcription

1 01/3 The hitchhiker s guide to the landia Pavel Cejnar mff.cuni.cz Inst. of Particle & Nuclear Physics Faculty of Mathematics & Physics Charles University Prague, Czech Republic Places to be visited: 1) Model examples ) General features 3) Some implications 4) Experimental relevance of s Trento, September 015

2 0/3 definition Ground-State QPT is a non-analytic change of the ground-state energy & structure induced by variation of a non-thermal control parameter (e.g., external field strength, internal interaction parameter etc.) in the infinite size limit of the system. Ground-State QPTs happen at zero temperature => only quantum fluctuations can play a role E first-order E continuous (n th -order) λ λ Examples: Ferromagnet-paramagnet (& similar) transitions Insulator-superfluid transition Superradiant phase transition Structural phase transitions Topological (nonlocal) phase transitions Monographs: Sachdev 1999,011, Carr (ed.) 010

3 03/3 definition Excited-State QPT is a generalization of Ground-State QPT It shows up as a singularity in (nonanalytic only in the infinite size limit): a) density of energy levels as a function of energy E b) flow of energy levels with varying internal/external control parameter λ Energy, not temperature! E states with parity + and Double-to-single well transition in a parity-conserving Hamiltonian with the cusp catastrophe potential P.Cejnar, P.Stránský, Phys. Rev. E 78, (008) a) b) λ QPT ( nd order) states with parity +

4 04/3 definition is a generalization of ground-state QPT It shows up as a singularity in (nonanalytic only in the infinite size limit): a) density of energy levels as a function of energy E b) flow of energy levels with varying internal/external control parameter λ Note: s affect bunches of levels rather than individual ones. Their signatures are visible in averaged quantities characterizing ensembles of levels and not always in the behavior of each level Energy, not temperature! E a) b) Deformed-to-spherical transition in the nuclear geometric model P.Cejnar, P.Stránský, Phys. Rev. E 78, (008) λ QPT (1 st order)

5 05/3 examples 0 Dicke model: schematic description of the matter-field interaction that exhibits the superradiant phase transition N two-level atoms R.H. Dicke, Phys. Rev. 93,99 (1954) K. Hepp, E. Lieb, Ann. Phys. (NY) 76,360 (1973) ˆ ˆ ˆ Hˆ 0 Jˆ0 bˆ bˆ b b J Jˆ 4j j single-ω photons calculation: M.Kloc N infinite spectrum The Dicke QPT (earlier considered nonrealistic since the ~A term destroys criticality) was recently measured with BEC in optical cavity K. Baumann et al., Nature 464,1301 (010); Phys. Rev. Lett. 107,14040 (011) all atoms in lower state & no photons P. Pérez-Fernández et al., Phys. Rev. A 83, (011) T. Brandes, Phys. Rev. E 88,03133 (013) M.A. Bastarrachea-Magnani et al., Phys. Rev. A 89, 03101, 0310 (014) QPT ndorder atoms partly in higher state & abundant photons singular growth of the state density

6 06/3 examples Dicke model: schematic description of the matter-field interaction that exhibits the superradiant phase transition R.H. Dicke, Phys. Rev. 93,99 (1954) K. Hepp, E. Lieb, Ann. Phys. (NY) 76,360 (1973) Hˆ ˆ ˆ ˆ 0J0 b b b J b J b J 4 j ˆ ˆ ˆ ˆ ˆ ˆ bˆ Jˆ 0 N two-level atoms single-ω photons M. Kloc, P. Stránský, P. Cejnar, in preparation (015) E 0 / j Extension allowing one to cross between: δ=0: Tavis-Cummings integrable regime conserves δ=1: Dicke nonintegrable regime ˆ ˆ infinite size J ˆ0 b b QPT nd order 0.3 First finite size (a) (b) derivative of the state density (a) (b)

7 07/3 examples Interacting Boson SU(3) dynamical symmetry P. Cejnar, J.Phys. Conf.Ser. 3,0101 O(6)-U(5) transitional region P.Cejnar, M.Macek, S.Heinze, J.Jolie, J.Dobeš, J. Phys. A, 39, L515 (006) Extended Lipkin 1 st order QPT region P. Cejnar, P. Stránský, M. Kloc Ph.Scr. (015) Vibron models D. Larese, F. Pérez-Bernal, F. Iachello, J.Mol.Struct. 1051, 310 (013) SU(1,1) model P. Pérez-Fernández et al., Phys.Rev. A 83, (011)

8 08/3 genealogy Monodromy connected with an in the Tavis-Cummings model [Kloc et al. 015] Related concepts (special cases): Van Hove singularity [L.Van Hove 1953] singularities of single-particle (electronic/phononic) state density in crystals or other lattice systems Monodromy [J.J. Duistermaat 1980] topological defects in quantum spectra of pendulumlike D systems (e.g., H 0 and similar molecules) Separatrix [J. Cary et al. 1986] divergences of state densities related to borderlines between different types of 1D motions in the classical phase space Model-specific observations [D. Heiss et al. 005 ] Early work on the general concept: P. Cejnar, M. Macek, S. Heinze, J. Jolie, J. Dobeš, J. Phys. A, 39, L515 (006) Monodromy and excited-state quantum phase transitions in integrable systems M.A. Caprio, P. Cejnar, F. Iachello, Ann. Phys. (NY) 33, 1106 (008) Excited state quantum phase transitions in many-body systems P. Cejnar, P. Stránský, Phys. Rev. E 78, (008) Impact of quantum phase transitions on excited-level dynamics Closely related work in thermodynamics: M. Kastner, Rev. Mod. Phys. 80, 167 (008) Phase transitions and configuration space topology

9 09/3 FAQs Excited-State Quantum Phase Transition Why Quantum? s occur in quantum systems with a finite number of degrees of freedom. For such systems, the infinite-size limit coincides with the classical limit. s are rooted in classical description but their precursors show up in finite-size quantum spectra and are connected with genuinely quantum features Why Phase Transitions? In general, s in the ( parameter x energy ) map do not represent borderlines between some specific phases of the system. s typically represent a continuation of QPT critical points to the excited domain. The definition of phases can be developed, though usually not in terms of order parameters QPT critical point normal Quantum phase diagram of the extended Dicke model superradiant

10 10/3 Infinite-size finite models Finite number of degrees of freedom: Potentially infinite size parameter: f ~1,... e.g. N particles with f~1,.. collective DoFs Example: Array of N two-level atoms spin½ sites (Lipkin & Dicke models) N- N-1 N N Individual spin algebras Collective spin algebra.. Total phase space is N balls f N Collective phase space is a single ball f 1 Total Hilbert space N d 1 j 0 for for N odd N even d 0or general j with multiplicity N!( j 1) N N ( j 1)!( j)! d j1 j N 1 multiplicity N 1 d N1 collective Hilbert space j N d N1

11 11/3 Infinite-size finite models Finite number of degrees of freedom: Potentially infinite size parameter: f ~1,... e.g. N particles with f~1,.. collective DoFs Example: Array of N two-level atoms spin½ sites (Lipkin & Dicke models) General Hamiltonian conserving the total spin contains interactions across the whole lattice Hˆ a { x, y, z} N a ˆ i i1 ˆ a a N S a sˆ a, b { x, y, z} N a b ˆ i ˆ j i, j1 1 ˆ a ˆb S S N a b sˆ sˆ a, b, c { x, y, z} a b c ˆ i ˆ j ˆ k i, j, k 1 1 ˆa ˆb ˆc 3 S S S N a b c sˆ sˆ sˆ 1 ab abc can be written in terms of scaled generators with vanishing commutators in the limit N General definition of size parameter via phase-space volume Ω / ħ f in which the motion is confined for E< E 0 (selected energy scale) N 1 sˆ S ˆ N 0... infinite-size limit of such a system coincides with the classical limit: 1/ f

12 1/3 Stationary points & classification Classical Hamiltonian function of the system smooth part of quantum density of states i ( EE ) ( E) ( E) ~ ( E) i (, H q xp ) smooth & oscillatory components 1 f ( E) d x [ E H( x) ] ( ) f not relevant for sometimes very relevant for finite determines the 1 f1 f d ( x) f H( x) ( ) H( x) E Integral over (f 1)-dim energy hypersurface in the f-dim phase space. For analytic Hamiltonians it is nonanalytic only at stationary points. Nondegenerate (quadratic) stat. point [ gradient i H 0, Hessian det i j H 0 ] For such points the type of nonanalyticity in can be determined explicitly. It depends on the number of degrees of freedom f and on the index of stationary point r = number of negative eigenvalues of the Hessian matrix. 1 f E 1 f 1 ( ) ( ) r1 r ln E for r odd ( E) step function for r even pair of integers ( f, r ) gives unique classification of s connected with nondegenerate stationary points of a general H

13 13/3 Stationary points & classification Trivial examples P. Stránský, M. Macek, P. Cejnar, Ann. Phys. 345 (014) 73 Cusp potential V f 1 f Creagh-Whelan potential 4 1 x x x V ( x 1) 30 xy 0 y ( x 1) x 1D D A A (1,1) E (,1) (1,0) (1,0) (,0) (,0) QPT 1 st order λ λ

14 14/3 Stationary points & classification Trivial examples P. Stránský, M. Macek, P. Cejnar, Ann. Phys. 345 (014) 73 Cusp potential f 1 f Creagh-Whelan potential (1,1) (1,0) (,1) M (,0) (1,1) E (,1) (1,0) (1,0) (,0) (,0) QPT 1 50 st order 0 λ λ

15 15/3 Stationary points & classification Trivial examples P. Stránský, M. Macek, P. Cejnar, Ann. Phys. 345 (014) 73 Cusp potential E f 1 f Creagh-Whelan potential E E (1,1) (,1) (1,0) (,0)

16 7 stac.points 16/3 Stationary points & classification Trivial examples Triple cusp potential f 3 8x (3,0) minimum jump 1x (3,1) saddle1 log.divergence 6x (3,) saddle jump 1x (3,3) maximum log.divergence d de V 3D 4 ( x y 4 z 4 ) ( x y z ) x y 3z 4 of d de d de 000

17 E position 17/3 Flow of energy levels Level dynamics in the plane control parameter x energy can be treated as a flow of levels along the E-axis. s show up as singularities in the smoothed flow rate (velocity field) Continuity equation cf. ) normalization factor P. Stránský, M. Macek, P. Cejnar, Ann. Phys. 345 (014) 73 e.g.gaussian weighting function centered at energy E E λ time

18 17/3 Flow of energy levels Continuity equation cf. ) 1D cusp P. Stránský, M. Macek, P. Cejnar, Ann. Phys. 345 (014) 73 Level dynamics in the plane control parameter x energy can be treated as a flow of levels along the E-axis. s show up as singularities in the smoothed flow rate (velocity field) e.g.gaussian D Creagh-Whelan E singularities of state density induce related (generically the same type) singularities of flow rate (1,1) (,0) (,1) i ) ( E) dh i d ( E E ( E) i i (1,0)

19 18/3 Dynamical effects Relaxation dynamics after quantum quench Sudden change of parameter λ 1 λ => H 1 H at t =0. Response is monitored by the t >0 survival amplitude (probability) of the initial state 1 g.s. ( 1 ) SU(1,1) model Energy distribution of the initial state after the quanch: Survival probability: 1 H C a a b b ( ba b a ˆ 0 N critical quench critical quench critical quench fast decay & no revival t ) P. Pérez-Fernández, P. Cejnar, J.M. Arias, J. Dukelsky, J.E. García-Ramos, A. Relaño, Phys. Rev. A 83, (011) Survival amplitude depends on the distribution of the initial state in the eigenbasis of the final Hamiltonian ε QPT critical quench initial state 1 ( ) λ Related new results: L. Santos, F. Pérez-Bernal, Phys. Rev. A 9, (015)

20 19/3 Thermal properties Is there a general relation between s & Thermal Phase Transitions? We do not know yet Here we compare quantum and thermal phase diagrams of the extended Dicke model Hˆ Jˆ bˆ bˆ bˆ Jˆ bˆ Jˆ bˆ Jˆ bˆ Jˆ j Free energy F(α) as a function of photon-field coherent-state parameter α for decreasing temperatures normal superradiant transition appearance of saddles T c T 0 Im α normal N Re α superradiant D superradiant TC QPT critical point T c normal all-j model total Hilbert space T 0 single-j model only one subspace of the total Hilbert space superradiant

21 0/3 Thermal properties Microcanonical inverse temperature Thermal energy distribution P. Stránský, M. Macek, P. Cejnar, Ann. Phys. 345 (014) 73 Finite-f systems with show anomalous thermodynamic features 1 T canonical partition function D schematically: quadratic minimum (,0) quadratic maximum (,0) quadratic saddle point (,1) degenerate inflection point

22 1/3 Thermal properties P. Stránský, M. Macek, P. Cejnar, Ann. Phys. 345 (014) 73 Finite-f systems with show anomalous thermodynamic features 1 T D Creagh-Whelan w

23 /3 Realistic systems (experimental verification) Atomic nuclei The concept was originally formulated in nuclear collective models (Interacting Boson Model ). However, collective states showing signatures would too high in the spectrum. Some indirect dynamical signatures could perhaps be found in largeamplitude collective processes like fission Molecules -related anomalies (monodromy) in energy spectra have been observed in molecules like H O and others. However, there is no control parameter to probe the spectral flow properties See, e.g.: N.F. Zobov et al., Chem. Phys. Lett. 414,193 (005), D. Larese, F. Pérez-Bernal, F. Iachello, J. Mol. Struct. 1051,310 (013) Lattice systems like graphene Graphene exhibits van Hove singularities of the density of states, connected with the saddles of the single-electron dispersion relation. This has recently been interpreted as an and measured using the technique of Dirac microwave billiards ( photonic crystals ) B.Dietz, F.Iachello, A.Richter, Phys. Rev. B 88, (013) Synthetic quantum systems like cold atoms, BECs, quantum optical systems Such systems make it possible to tune control parameters and observe both key aspects of. They are experimentally available and potentially applicable in quantum information processing linear O H bent H O H H

24 3/3 Summary Excited State Quantum Phase Transitions are singularities ( asymptotically nonanalytic ) in quantum spectra of systems with a finite number of degrees of freedom reflect the presence of stationary points in classical dynamics affect the density and flow of quantum spectra influence thermal and dynamical properties Thank you very much for your benevolent attention