Nonequilibrium Quasiparticles

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

2. Nonequilibrium Quasiparticles 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, 195125 (2007) Nonequilibrium Quasiparticles

3. Nonequilibrium Quasiparticles 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, 195125 (2007) Transients in Quantum Transport A. Kalvová: Time Partitioning … Chemnitz 2005 Nonequilibrium Quasiparticles

4. Nonequilibrium Quasiparticles • motivation • - definition Nonequilibrium Quasiparticles

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

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

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. a closed equation for (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 Quantum Transport Equation generalized collision term

9. a closed equation for (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 Quantum Transport Equation interaction term • existence, construction of • incorporation of the initial condition QUESTIONS

10. a closed equation for (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 not quasiparticle Quantum Transport Equation interaction term • existence, construction of • incorporation of the initial condition QUESTIONS

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

12. motivation 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 Nonequilibrium Quasiparticles

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

14. motivation (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 Nonequilibrium Quasiparticles

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

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

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

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

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

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

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

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

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

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

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

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

27. hierarchy of characteristic times non-equilibrium initial state ... arbitrary ... full description is necessary system evolves ... the processes… very fast, fast, slow very fastprocesses... smoothing, coherence loss, decay of the many-body correlation, chaotizationcharacteristic 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 satisfiesquantum transport equation Nonequilibrium Quasiparticles

28. motivation NeQP fromnon-equilibriumto „equilibrium“ KBA (Kadanoff Baym Ansatz) ... NGF ( )... in Wigner representation ...time dependent spectral density ... NGF quantum transport equationforQP distribution restriction ... processes… quasiclassical in time 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 quasi-classical in time

29. motivation NeQP fromnon-equilibriumto „equilibrium“ KBA (Kadanoff Baym Ansatz) ... NGF ( )... in Wigner representation ...time dependent spectral density ... NGF quantum transport equationforQP distribution restriction ... processes… quasiclassical in time GKBA (Generalized KBA)... quantum transport equationforreal particles(density matrix) causal structure rapid transients ...quasiparticles restriction ... renormalization (formation) …negligible during 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. motivation NeQP fromnon-equilibriumto „equilibrium“ KBA (Kadanoff Baym Ansatz) ... NGF ( )... in Wigner representation ...time dependent spectral density ... NGF quantum transport equationforQP distribution restriction ... processes… quasiclassical in time GKBA (Generalized KBA)... quantum transport equationforreal particles(density matrix) causal structure rapid transients ...quasiparticles restriction ... renormalization (formation) …negligible during QKBA (Quasiparticle KBA) ... quantum transport equationforreal particles dynamical(formation)processes …involved restriction ... if nonequilibrium QPsexist

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

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

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

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

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

38. definition NeQP NeQP propagator SGR...satisfied time local optical potential Nonequilibrium Quasiparticles

39. definition NeQP NeQP propagator SGR...satisfied time local optical potential Nonequilibrium Quasiparticles

40. definition NeQP NeQP propagator SGR...satisfied time local optical potential mean-field propagator unitaryevolutionindepend. „particles“ in external and internal fields Nonequilibrium Quasiparticles

41. definition NeQP NeQP propagator SGR...satisfied time local optical potential Nonequilibrium Quasiparticles

42. definition NeQP NeQP propagator SGR...satisfied time local optical potential NeQP propagator Nonequilibrium Quasiparticles

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

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

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

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

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

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

49. Discussion of the RSG rule • universal vertex, derived with almost no effort and no specific properties of the GF • off-diagonal vertex, linking in a smeared fashion propagation in the past and in the future • Similar to the Dyson Equation, but NO free GF • In fact, looks pretty much like a linear response • we would like to understand all these features FZÚ 14.2.2006 Semi-group property...

50. Discussion of the RSG rule • universal vertex, derived with almost no effort and no specific properties of the GF • off-diagonal vertex, linking in a smeared fashion propagation in the past and in the future • Similar to the Dyson Equation, but NO free GF • In fact, looks pretty much like a linear response • we would like to understand all these features FZÚ 14.2.2006 Semi-group property...