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Non-equilibrium Ward Identity

Non-equilibrium Ward Identity. A. Kalvov á B. Velick ý, V. Špička Institute of Physics, Academy of Sciences of the Czech Republic. Non-equilibrium Ward Identity. A. Kalvov á B. Velick ý, V. Špička Institute of Physics, Academy of Sciences of the Czech Republic.

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Non-equilibrium Ward Identity

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  1. Non-equilibrium Ward Identity A. Kalvová B. Velický, V. Špička Institute of Physics, Academy of Sciences of the Czech Republic Non-equilibrium Ward Identity

  2. Non-equilibrium Ward Identity A. Kalvová B. Velický, V. Špička Institute of Physics, Academy of Sciences of the Czech Republic Phys. Rev. B77, 0411201 (2008) Non-equilibrium Ward Identity

  3. (eq) Ward Identity: quantum field theoryJ.S. Ward, Phys. Rev. 87, 182 (1950) Non-equilibrium Ward Identity

  4. F.J. Dyson, Phys. Rev. 75, 1736 (1949) (eq) Ward Identity: quantum field theoryJ.S. Ward, Phys. Rev. 87, 182 (1950) Non-equilibrium Ward Identity

  5. (eq) Ward Identity: quantum field theoryJ.S. Ward, Phys. Rev. 87, 182 (1950) Non-equilibrium Ward Identity

  6. (eq) Ward Identity: quantum field theoryJ.S. Ward, Phys. Rev. 87, 182 (1950) Non-equilibrium Ward Identity

  7. (eq) Ward Identity: quantum field theoryJ.S. Ward, Phys. Rev. 87, 182 (1950) Non-equilibrium Ward Identity

  8. (eq) Ward Identity: quantum field theoryJ.S. Ward, Phys. Rev. 87, 182 (1950) WI - relation between elmag. vertex for special values of argument and selfenergy Non-equilibrium Ward Identity

  9. (eq) Ward Identity: quantum field theoryJ.S. Ward, Phys. Rev. 87, 182 (1950) WI - relation between elmag. vertex for special values of argument and selfenergy gives a proof of two renormalization constants being equal Non-equilibrium Ward Identity

  10. (eq) Ward Identity: condensed matter Polaron electron in elmag. vacuum (QED) ... parallels ... electron in phonon vacuum (CM) Non-equilibrium Ward Identity

  11. (eq) Ward Identity: condensed matter Polaron electron in elmag. vacuum (QED) ... parallels ... electron in phonon vacuum (CM) POLARON • mass renormalisation only • no infinities (no cancelations) • - no polarisation of the el.-hole vacuum Non-equilibrium Ward Identity

  12. (eq) Ward Identity: condensed matter Polaron electron in elmag. vacuum (QED) ... parallels ... electron in phonon vacuum (CM) POLARON • mass renormalisation only • no infinities (no cancelations) • - no polarisation of the el.-hole vacuum = + + + + + ... Non-equilibrium Ward Identity

  13. (eq) Ward Identity: condensed matter Polaron electron in elmag. vacuum (QED) ... parallels ... electron in phonon vacuum (CM) POLARON • mass renormalisation only • no infinities (no cancelations) • - no polarisation of the el.-hole vacuum = + + + + + ... Dyson equation = = vertex Non-equilibrium Ward Identity

  14. (eq) Ward Identity: condensed matter Polaron electron in elmag. vacuum (QED) ... parallels ... electron in phonon vacuum (CM) POLARON • mass renormalisation only • no infinities (no cancelations) • - no polarisation of the el.-hole vacuum = + + + + + ... Dyson equation = in termsof = vertex WARD IDENTITY Non-equilibrium Ward Identity

  15. (eq) Ward Identity: condensed matter Polaron = = + + ... = + + + ... = + + ... lowest order contribution to the vertex ... lowest order contribution do the selfenergy = = Non-equilibrium Ward Identity

  16. (eq) Ward Identity: condensed matter Polaron = = Non-equilibrium Ward Identity

  17. WARD IDENTITY for (eq) Ward Identity: condensed matter Polaron = = Non-equilibrium Ward Identity

  18. WARD IDENTITY for (eq) Ward Identity: condensed matter Polaron = = Non-equilibrium Ward Identity

  19. WARD IDENTITY for (eq) Ward Identity: condensed matter Polaron = = for Non-equilibrium Ward Identity

  20. (neq) Ward Identity: condensed matter general notes derivation ... non-perturbative, doesnot depend on details of MB systems (external field, interactions, dimensionality,spin structure etc.) ... use the U(1) gauge symmetry of the neq Fermi Green’s Function following from the and Keldysh initial condition (neq) Ward Identity meaning ... connects the scalar transport vertex with one particle selfenergy consequences ... set of eqs. for one particle NGF so called renormalized multiplicative laws, extended reconstruction equations Non-equilibrium Ward Identity

  21. (neq) Ward Identity: condensed matter Lecture on NGF Equilibrium: ...time ordering operator Non-equilibrium Ward Identity

  22. (neq) Ward Identity: condensed matter Lecture on NGF equilibrium: ...time ordering operator ...acting on time contour non-quilibrium: Non-equilibrium Ward Identity

  23. (neq) Ward Identity: condensed matter Lecture on NGF equilibrium: ...time ordering operator ...acting on time contour non-quilibrium: Non-equilibrium Ward Identity

  24. (neq) Ward Identity: condensed matter Lecture on NGF equilibrium: ...time ordering operator ...acting on time contour non-quilibrium: Non-equilibrium Ward Identity

  25. (neq) Ward Identity: condensed matter Lecture on NGF equilibrium: ...time ordering operator ...acting on time contour non-quilibrium: Non-equilibrium Ward Identity

  26. (neq) Ward Identity: condensed matter Lecture on NGF equilibrium: ...time ordering operator ...acting on time contour non-quilibrium: Non-equilibrium Ward Identity

  27. (neq) Ward Identity: condensed matter Lecture on NGF equilibrium: ...time ordering operator ...acting on time contour non-quilibrium: real time NGF... choices our choices: Non-equilibrium Ward Identity

  28. (neq) Ward Identity: condensed matter NGF matrix formulation real time (Keldysh) matrix NGF and selfenergy specific physical approximation -- self-consistent form they satisfy Dyson equatin in matrix form explicitly Non-equilibrium Ward Identity

  29. (neq) Ward Identity: condensed matter derivation : Step1- ... arbitrary external time local disturbance ... Dyson eq. for field dependent NGF, where Non-equilibrium Ward Identity

  30. (neq) Ward Identity: condensed matter derivation : Step2- ... arbitrary external time local disturbance ... Dyson eq. for field dependent NGF, where vertex... in integral form asin the linear response of one electron GF to a small variation Non-equilibrium Ward Identity

  31. (neq) Ward Identity: condensed matter derivation : Step3- symmetry ... arbitrary external time local disturbance ... Dyson eq. for field dependent NGF, where vertex... in integral form asin the linear response of one electron GF to a small variation gauge invariance of the first kind (globalsymmetry)... time variable shift added to the one particle energy in free GF ...spatially homog. potential energy ...operator unity Non-equilibrium Ward Identity

  32. (neq) Ward Identity: condensed matter derivation : Step4-explicit time dep. DE field dependence of the Dyson equation explicitly Non-equilibrium Ward Identity

  33. time dependence of selfenergy , for a weak : (neq) Ward Identity: condensed matter derivation : Step5-explicit time dep. DE field dependence of the Dyson equation explicitly Non-equilibrium Ward Identity

  34. time dependence of selfenergy , for a weak : (neq) Ward Identity: condensed matter derivation : Step5-explicit time dep. DE field dependence of the Dyson equation explicitly explicitly: Non-equilibrium Ward Identity

  35. time dependence of selfenergy , for a weak : (neq) Ward Identity: condensed matter (neq) Ward Identity: condensed matter definition operator vertex correction derivation : Step5-explicit time dep. DE field dependence of the Dyson equation explicitly explicitly: Non-equilibrium Ward Identity

  36. (neq) Ward Identity: condensed matter definition operator vertex correction we define an operator vertex correction nonequilibrium Ward Identity three time terminals matrix equation; comprises three relations linking the transport vertex with one-particle quantities Non-equilibrium Ward Identity

  37. nonequilibrium Ward Identity equilibrium Ward Identity (neq) Ward Identity: condensed matter equilibrium limit equilibrium limit (time homogeneity) Non-equilibrium Ward Identity

  38. (neq) Ward Identity: condensed matter equilibrium limit...continuation equilibrium Ward Identity ?in matrix form? interpretation of the depends of the element of the Keldysh matrix in elements, correspond to the one particle renormalization in transport component connects both rims of the cut of selfenergy (fluctuation dissipation theorem gives original WI concerns the limit our neq WI in equilibrium limit gives finite energy transfer nonequilibrium Ward TakahashiIdentity Non-equilibrium Ward Identity

  39. (neq) Ward Identity: condensed matter consequence of neq WI Non-equilibrium Ward Identity

  40. time dependence of one particle , for a weak : (neq) Ward Identity: condensed matter consequence of neq WI Non-equilibrium Ward Identity

  41. time dependence of one particle , for a weak : (neq) Ward Identity: condensed matter consequence of neq WI Non-equilibrium Ward Identity

  42. (neq) Ward Identity: condensed matter consequence of neq WI renormalized multiplicative composition rule Non-equilibrium Ward Identity

  43. (neq) Ward Identity: condensed matter consequence of neq WI neq renormalized semigroup rule (NE RSGR) Non-equilibrium Ward Identity

  44. nonequilibrium multiplicative law (neq) Ward Identity: condensed matter consequence of neq WI neq renormalized semigroup rule (NE RSGR) single particle equation for and combined with self-consistent approximation implies the neq Ward Identity as a relation between the simpler single-particle and two particle transport vertex can be derived without explicit reference to the gauge symmetry. All what is required is the validity of the related Dyson Equation Non-equilibrium Ward Identity

  45. (neq) Ward Identity: condensed matter consequence of neq WI neq renormalized semigroup rule (NE RSGR) for propagators first term ... spliting in an intermediate “spliting time” propagation in the past and in the future ; can be called “now” second term ... blurring in time by the vertex correction; reflects coherence past-future; i.e. memory of the system Propagators  quantum coherence and memory in QTE blurring in time... is on the order of the quasiparticle formation time if it exists Non-equilibrium Ward Identity

  46. (neq) Ward Identity: condensed matter consequence of neq WI neq renormalized semigroup rule (NE RSGR) for propagators first term ... spliting in an intermediate “spliting time” propagation in the past and in the future ; can be called “now” second term ... blurring in time by the vertex correction; reflects coherence past-future; i.e. memory of the system Propagators  quantum coherence and memory in QTE blurring in time... is on the order of the quasiparticle formation time if it exists Dresden, Dec 2008 Non-equilibrium Ward Identity 46

  47. (neq) Ward Identity: condensed matter consequence of neq WI neq renormalized semigroup rule (NE RSGR) for propagators first term ... spliting in an intermediate “spliting time” propagation in the past and in the future ; can be called “now” second term ... blurring in time by the vertex correction; reflects coherence past-future; i.e. memory of the system Propagators  quantum coherence and memory in QTE blurring in time... is on the order of the quasiparticle formation time if it exists Dresden, Dec 2008 Non-equilibrium Ward Identity 47

  48. (neq) Ward Identity: condensed matter consequence of neq WI neq extended reconstruction equations for first term ... splitting at an intermediate “splitting time” propagation in the past and in the future ; can be called “now” second term ... blurring in time by the vertex correction; reflects coherence past-future; i.e. memory of the system Non-equilibrium Ward Identity

  49. (neq) Ward Identity: condensed matter consequence of neq WI Photoexcited transients in disordered semiconductors A. Kalvová, B. Velický, PRB 65, 155329 (2002) off-diagonal elements only c-band disordered v-band disordered opt. pulse in dark in light ideal bands sharp c-band empty sharp v-band full vertical transition realistic bands polaron effect non-vertical transition smeared&renorm. c-band empty smeared&renorm. v-band full

  50. R opt. pulse uncorrelated IC at (neq) Ward Identity: condensed matter consequence of neq WI Photoexcited transients in disordered semiconductors < integral form

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