Nonequilibrium Quasiparticles 1 Anděla Kalvová Institute of Physics, v.v.i. Academy of Sciences of the Czech Republic.

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Nonequilibrium Quasiparticles 1 Anděla Kalvová Institute of Physics, v.v.i. Academy of Sciences of the Czech Republic

Nonequilibrium Quasiparticles 2 Anděla Kalvová Institute of Physics, v.v.i. Academy of Sciences of the Czech Republic Quasi-Particle States of Electron Systems out of Equilibrium B. Velický, A. Kalvová, V. Špička; PRB 75, (2007)

Nonequilibrium Quasiparticles 3 Anděla Kalvová Institute of Physics, v.v.i. Academy of Sciences of the Czech Republic Quasi-Particle States of Electron Systems out of Equilibrium B. Velický, A. Kalvová, V. Špička; PRB 75, (2007) Transients in Quantum Transport A. Kalvová: Time Partitioning … Chemnitz 2005

Nonequilibrium Quasiparticles 4 - motivation - definition Nonequilibrium Quasiparticles

5 motivation First slide from: Transients in Quantum Transport A. Kalvová: Time Partitioning

6 (Non-linear) quantum transport non-equilibrium problem many-body Hamiltonian many-body density matrix additive operator Many-body system Initial state External disturbance

7 (Non-linear) quantum transport non-equilibrium problem Many-body system Initial state External disturbance Response many-body Hamiltonian many-body density matrix additive operator one-particle density matrix

8 (Non-linear) quantum transport non-equilibrium problem Quantum Transport Equation a closed equation for generalized collision term Many-body system Initial state External disturbance Response many-body Hamiltonian many-body density matrix additive operator one-particle density matrix

9 (Non-linear) quantum transport non-equilibrium problem Quantum Transport Equation a closed equation for Many-body system Initial state External disturbance Response many-body Hamiltonian many-body density matrix additive operator one-particle density matrix QUESTIONS  existence, construction of  incorporation of the initial condition interaction term

10 a closed equation for (Non-linear) quantum transport non-equilibrium problem Quantum Transport Equation Many-body system Initial state External disturbance Response many-body Hamiltonian many-body density matrix additive operator one-particle density matrix QUESTIONS  existence, construction of  incorporation of the initial condition interaction term not quasiparticle

Nonequilibrium Quasiparticles 11 quasiparticle way.... Landau theory for Fermi liquids motivation

Nonequilibrium Quasiparticles 12 Landau theory for Fermi liquids the original system of quantum particles with strong interactions transformed (purely phenomenologically in physical sense) to an effective system of quasiparticles - forming a weakly non-ideal gas - governed by a (slightly generalized) Boltzmann equation motivation

Nonequilibrium Quasiparticles 13 (Non-linear) quantum transport non-equilibrium problem Many-body system Initial state Weak extern. disturb. many-body Hamiltonian m.-b. equilibrium state additive operator motivation

Nonequilibrium Quasiparticles 14 (Non-linear) quantum transport non-equilibrium problem Many-body system Initial state Weak extern. disturb. Response many-body Hamiltonian mb equilibrium state additive operator distribution function for QP motivation

Nonequilibrium Quasiparticles 15 (Non-linear) quantum transport non-equilibrium problem Quantum Transport Eq. Quantum Boltzmann Eq. Many-body system Initial state Weak extern. disturb. Response many-body Hamiltonian m.-b. equilibrium state additive operator distribution function for QP motivation

16 (Non-linear) quantum transport non-equilibrium problem Quantum Transport Eq. Quantum Boltzmann Eq. Many-body system Initial state Weak extern. disturb. Response many-body Hamiltonian m.-b. equilibrium state additive operator distribution function for QP COLLISION INFLUENCE 1.evolution to equilibrium 2.time arrow = direction of the entropy growth 3.forgetting of the past – after ONE collision, (during average collision time)  from initial condition to “floating” initial condition motivation

Nonequilibrium Quasiparticles 17 linear transport... redistribution of the equilibrium QP (equilibrium GF) motivation

Nonequilibrium Quasiparticles 18 motivation non-linear transport... redistribution of non-equilibrium QP (non-equilibrium GF) ?? linear transport... redistribution of the equilibrium QP (equilibrium GF)

Nonequilibrium Quasiparticles 19 QP formation definition QP linear transport... redistribution of the equilibrium QP (equilibrium GF)

Nonequilibrium Quasiparticles 20 QP formation phenomenological constants linear transport... transfer equilibrium QP (equilibrium GF) definition QP

Nonequilibrium Quasiparticles 21 QP formation formation time linear transport... transfer equilibrium QP (equilibrium GF) phenomenological constants definition QP

Nonequilibrium Quasiparticles 22 QP formation renormal. constant linear transport... transfer equilibrium QP (equilibrium GF) phenomenological constants formation time definition QP

Nonequilibrium Quasiparticles 23 QP formation complex energy linear transport... transfer equilibrium QP (equilibrium GF) formation time renormal. constant definition QP phenomenological constants

Nonequilibrium Quasiparticles 24 QP formation QP condition linear transport... transfer equilibrium QP (equilibrium GF) complex energy formation time renormal. constant definition QP phenomenological constants

Nonequilibrium Quasiparticles 25 QP formation linear transport... transfer equilibrium QP (equilibrium GF) definition QP phenomenological constants formation time renormal. constant complex energy

26... pole and residuum of the spectral representation of the equilibrium GF (one - el. propagator) QP formation... Wigner-Weisskopf (weak scatt.) linear transport... transfer equilibrium QP (equilibrium GF) definition QP phenomenological constants formation time renormal. constant complex energy

Nonequilibrium Quasiparticles 27 non-equilibrium initial state... arbitrary... full description is necessary system evolves... the processes… very fast, fast, slow very fast processes... smoothing, coherence loss, decay of the many-body correlation, chaotization characteristic time fast processes... collisions leading to the kinetic stage of transport characteristic time slow processes... compensating macroskopic inhomogeneities characteristic time BOGOLJUBOV POSTULATES initial stage, kinetic, hydrodynamic In kinetic stage …(quantum) distribution function provides complete description satisfies quantum transport equation hierarchy of characteristic times

28 from non-equilibrium to „equilibrium“ KBA (Kadanoff Baym Ansatz)... NGF ( )... in Wigner representation...time dependent spectral density... NGF quantum transport equation for QP distribution restriction... processes… quasiclassical in time motivation NeQP KB were inspired by equilibrium. Their ansatz leads to the distribution function of QP. Their dynamical behavior is described by equation of Bolzmann type. Processes in such MB system have to be quasiclassical in time

29 from non-equilibrium to „equilibrium“ KBA (Kadanoff Baym Ansatz)... NGF ( )... in Wigner representation...time dependent spectral density... NGF quantum transport equation for QP distribution restriction... processes… quasiclassical in time GKBA (Generalized KBA)... quantum transport equation for real particles (density matrix) causal structure rapid transients...quasiparticles restriction... renormalization (formation) …negligible during motivation NeQP In the opposite situation… in rapid transients, we construct the QTE for real particles. The ansatz GKBA is formulated fully in time domain. The using of GKBA is restricted too. It is possible to use it only if the formation processes during initial stage are negligible.

30 from non-equilibrium to „equilibrium“ KBA (Kadanoff Baym Ansatz)... NGF ( )... in Wigner representation...time dependent spectral density... NGF quantum transport equation for QP distribution restriction... processes… quasiclassical in time GKBA (Generalized KBA)... quantum transport equation for real particles (density matrix) causal structure rapid transients...quasiparticles restriction... renormalization (formation) …negligible during QKBA (Quasiparticle KBA)... quantum transport equation for real particles dynamical (formation) processes …involved restriction... if nonequilibrium QPs exist motivation NeQP

Nonequilibrium Quasiparticles 31 definition NeQP Any propagator … 1. should pass through a formation stage 2. then should represent a QP In nonstationary (transient, nonequilibrium) conditions … two questions 1.? formation time ? 2.? is it possible to construct … as QP counterpart to In transients, two processes compete permanently: formation and flight. Complex process results and we don’t know, if it is a flight of independent quasiparticles or if this concept fails. ??

Nonequilibrium Quasiparticles 32 QP formation functional relation nonequilibrium QP composition rule SGR QP comosition rule QCR equilibrium: definition NeQP

Nonequilibrium Quasiparticles 33 QP formation functional relation nonequilibrium QP composition rule SGR QP comosition rule QCR equilibrium: definition NeQP logarithmic derivative... time independent

Nonequilibrium Quasiparticles 34 nonequilibrium propagator:.... double-time function (matrix) selfenergy:.... definition NeQP

Nonequilibrium Quasiparticles 35 nonequilibrium propagator:.... double-time function (matrix) selfenergy:.... definition NeQP free-particle Ham. + external fields + mean-field

Nonequilibrium Quasiparticles 36 nonequilibrium propagator:.... double-time function (matrix) selfenergy:.... definition NeQP NeQP (formation time condition)..... It is natural to restrict our study to the case, when a NE extension of QP formation time exists. Therefore, we assume the existence formation time such, that the selfenergy is zero outside a double time region, which has the strip adjoing the time diagonal.

Nonequilibrium Quasiparticles 37 nonequilibrium propagator:.... double-time function (matrix) selfenergy:.... definition NeQP NeQP (formation time condition)..... NeQP propagator

Nonequilibrium Quasiparticles 38 NeQP propagator SGR... satisfiedtime local optical potential definition NeQP

Nonequilibrium Quasiparticles 39 NeQP propagator SGR... satisfiedtime local optical potential definition NeQP

Nonequilibrium Quasiparticles 40 NeQP propagator SGR... satisfiedtime local optical potential definition NeQP mean-field propagator unitary evolution independ. „particles“ in external and internal fields

Nonequilibrium Quasiparticles 41 NeQP propagator SGR... satisfiedtime local optical potential definition NeQP

Nonequilibrium Quasiparticles 42 NeQP propagator SGR... satisfiedtime local optical potential definition NeQP NeQP propagator

Nonequilibrium Quasiparticles 43 NeQP propagator SGR... satisfiedtime local optical potential definition NeQP NeQP propagator not a unitary evolution ( eff. single particle. nonhermit. H) flight of independent NeQP

Nonequilibrium Quasiparticles 44 starting point Dyson equations for to findclosed equation for... Phase Equation (subtraction) renormalized quasiparticle composition rule (RQCR) definition NeQP

Nonequilibrium Quasiparticles 45 starting point Dyson equations for to findclosed equation for... Phase Equation (subtraction) renormalized quasiparticle composition rule (RQCR) definition NeQP

Nonequilibrium Quasiparticles 46 starting point Dyson equations for to findclosed equation for... Phase Equation (subtraction) renormalized quasiparticle composition rule (RQCR) definition NeQP QP composition rule (QCR)

Nonequilibrium Quasiparticles 47 starting point Dyson equations for to findclosed equation for... Phase Equation (subtraction) renormalized quasiparticle composition rule (RQCR) definition NeQP renorm. vertex correction

Nonequilibrium Quasiparticles 48 starting point Dyson equations for to findclosed equation for... Phase Equation (subtraction) renormalized quasiparticle composition rule (RQCR) definition NeQP slide Transients in Quantum Transport B. Velický: Semigroup Property of Propagators...

49 Discussion of the RSG rule 1.universal vertex, derived with almost no effort and no specific properties of the GF 2.off-diagonal vertex, linking in a smeared fashion propagation in the past and in the future 3.Similar to the Dyson Equation, but NO free GF 4.In fact, looks pretty much like a linear response we would like to understand all these features Semi-group property... FZÚ

50 Discussion of the RSG rule 1.universal vertex, derived with almost no effort and no specific properties of the GF 2.off-diagonal vertex, linking in a smeared fashion propagation in the past and in the future 3.Similar to the Dyson Equation, but NO free GF 4.In fact, looks pretty much like a linear response we would like to understand all these features Semi-group property... FZÚ

Nonequilibrium Quasiparticles 51 starting point Dyson equations for to findclosed equation for... Phase Equation (subtraction) renormalized quasiparticle composition rule (RQCR) definition NeQP self-consistent condition for cancellation of the vertex explicit equation for

Nonequilibrium Quasiparticles 52 summary of notation of times :... minimum, maximum... evolution fromto() time- inhomogeneity... time between NEW... time, where QCR is valid... min independent of definition NeQP

Nonequilibrium Quasiparticles minimum, maximum... evolution fromto() time inhomogeneity... time between NEW... time, where QCR is valid... min definition NeQP implication: for fixed many exist... typically... identical NeQP independent phase equation independent of summary of notation of times :

Nonequilibrium Quasiparticles 54 cesta k fázové rovnici how to get the phase equation for an optical potential definition NeQP phase equation

Nonequilibrium Quasiparticles 55 how to get the phase equation for an optical potential definition NeQP phase equation

Nonequilibrium Quasiparticles 56 definice... how to get the phase equation for an optical potential definition NeQP phase equation

Nonequilibrium Quasiparticles 57 definice... how to get the phase equation for an optical potential definition NeQP phase equation

Nonequilibrium Quasiparticles 58 how to get the phase equation for an optical potential definition NeQP phase equation

Nonequilibrium Quasiparticles 59 ?... ? definition NeQP phase equation to findclosed equation for... Phase Equation

Nonequilibrium Quasiparticles 60 ?... ? phase equation definition NeQP phase equation far enough from

Nonequilibrium Quasiparticles 61 definition NeQP phase equation Initial condition ?... ?

Nonequilibrium Quasiparticles 62 definition NeQP phase equation It is natural to restrict our study to the case, when a NE extension of QP formation time exists. Therefore, we assume the existence formation time such, that the selfenergy is zero outside a double time region, which has the strip adjoing the time diagonal. Initial condition

Nonequilibrium Quasiparticles 63 ?... ? initial condition definition NeQP phase equation

Nonequilibrium Quasiparticles closed phase equation definition NeQP phase equation

Nonequilibrium Quasiparticles 65 why the term... phase equation ? why... and not ? condition for derivation of the phase equation... existence... „dominant strip“... analogue to the dominant strip for... existence … „forgotten times“,... decay of init. corr. (formation of the kinetic stage)... phase eq. doesn´t depend on (formation of QP) definition NeQP phase equation

Nonequilibrium Quasiparticles semilocal in time (...effective time locality) - anticausal time ordering (near future influence) - past influence...closed phase equation definition NeQP phase equation

Nonequilibrium Quasiparticles 67 Conclusion: consistent definition of the NeQP NeQP... two properties (equivalent), used for definition -corresponding propagator satisfies QP composition rule -corresponding propagator contains optical potencial existence of the NeQP... dependence on retard. selfenergy short formation time narrow „dominant strip“ in optical potential and NE retarded selfenergy connected with the phase eq. For its solution...necessity to know initial conditions... reflect time period of formation QP composition rule... „free term“ in (transf.) Dyson equation vanishing of the vertex correction validity of the phase equation definition NeQP phase equation