1. Non-equilibrium Ward Identity
A. Kalvová
B. Velický, V. Špička
Institute of Physics,
Academy of Sciences of the Czech Republic
Dresden, Dec 2008
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, (2008)
Dresden, Dec 2008
Non-equilibrium Ward Identity

3. Non-equilibrium Ward Identity
(eq) Ward Identity: quantum field theory J.S. Ward, Phys. Rev. 87, 182 (1950)
Dresden, Dec 2008
Non-equilibrium Ward Identity

4. Non-equilibrium Ward Identity
(eq) Ward Identity: quantum field theory J.S. Ward, Phys. Rev. 87, 182 (1950)
F.J. Dyson, Phys. Rev. 75, 1736 (1949)
Dresden, Dec 2008
Non-equilibrium Ward Identity

5. Non-equilibrium Ward Identity
(eq) Ward Identity: quantum field theory J.S. Ward, Phys. Rev. 87, 182 (1950)
Dresden, Dec 2008
Non-equilibrium Ward Identity

6. Non-equilibrium Ward Identity
(eq) Ward Identity: quantum field theory J.S. Ward, Phys. Rev. 87, 182 (1950)
Dresden, Dec 2008
Non-equilibrium Ward Identity

7. Non-equilibrium Ward Identity
(eq) Ward Identity: quantum field theory J.S. Ward, Phys. Rev. 87, 182 (1950)
Dresden, Dec 2008
Non-equilibrium Ward Identity

8. Non-equilibrium Ward Identity
(eq) Ward Identity: quantum field theory J.S. Ward, Phys. Rev. 87, 182 (1950)
WI - relation between elmag. vertex for special values of argument and selfenergy
Dresden, Dec 2008
Non-equilibrium Ward Identity

9. Non-equilibrium Ward Identity
(eq) Ward Identity: quantum field theory J.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
Dresden, Dec 2008
Non-equilibrium Ward Identity

10. Non-equilibrium Ward Identity
(eq) Ward Identity: condensed matter
Polaron
electron in elmag. vacuum (QED) ... parallels ... electron in phonon vacuum (CM)
Dresden, Dec 2008
Non-equilibrium Ward Identity

11. Non-equilibrium Ward Identity
(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
Dresden, Dec 2008
Non-equilibrium Ward Identity

12. Non-equilibrium Ward Identity
(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 = + + + +
Dresden, Dec 2008
Non-equilibrium Ward Identity

13. Non-equilibrium Ward Identity
(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
Dresden, Dec 2008
Non-equilibrium Ward Identity

14. Non-equilibrium Ward Identity
(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 terms of =
vertex
WARD IDENTITY
Dresden, Dec 2008
Non-equilibrium Ward Identity

15. Non-equilibrium Ward Identity
(eq) Ward Identity: condensed matter
Polaron = = +
+ ... = + +
+ ... = +
+ ...
lowest order contribution to the vertex ... lowest order contribution do the selfenergy = =
Dresden, Dec 2008
Non-equilibrium Ward Identity

16. Non-equilibrium Ward Identity
(eq) Ward Identity: condensed matter
Polaron = =
Dresden, Dec 2008
Non-equilibrium Ward Identity

17. Non-equilibrium Ward Identity
(eq) Ward Identity: condensed matter
Polaron = =
WARD IDENTITY for
Dresden, Dec 2008
Non-equilibrium Ward Identity

18. Non-equilibrium Ward Identity
(eq) Ward Identity: condensed matter
Polaron = =
WARD IDENTITY for
Dresden, Dec 2008
Non-equilibrium Ward Identity

19. Non-equilibrium Ward Identity
(eq) Ward Identity: condensed matter
Polaron = =
WARD IDENTITY for
for
Dresden, Dec 2008
Non-equilibrium Ward Identity

20. Non-equilibrium Ward Identity
(neq) Ward Identity: condensed matter
general notes
(neq) Ward Identity
meaning ... connects the scalar transport vertex
with one particle selfenergy
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
consequences ... set of eqs. for one particle NGF
so called renormalized multiplicative laws,
extended reconstruction equations
Dresden, Dec 2008
Non-equilibrium Ward Identity

21. ...time ordering operator
(neq) Ward Identity: condensed matter
Lecture on NGF
Equilibrium:
...time ordering operator
Dresden, Dec 2008
Non-equilibrium Ward Identity

22. ...time ordering operator
(neq) Ward Identity: condensed matter
Lecture on NGF
equilibrium:
...time ordering operator
...acting on time contour
non-quilibrium:
Dresden, Dec 2008
Non-equilibrium Ward Identity

23. ...time ordering operator
(neq) Ward Identity: condensed matter
Lecture on NGF
equilibrium:
...time ordering operator
...acting on time contour
non-quilibrium:
Dresden, Dec 2008
Non-equilibrium Ward Identity

24. ...time ordering operator
(neq) Ward Identity: condensed matter
Lecture on NGF
equilibrium:
...time ordering operator
...acting on time contour
non-quilibrium:
Dresden, Dec 2008
Non-equilibrium Ward Identity

25. ...time ordering operator
(neq) Ward Identity: condensed matter
Lecture on NGF
equilibrium:
...time ordering operator
...acting on time contour
non-quilibrium:
Dresden, Dec 2008
Non-equilibrium Ward Identity

26. ...time ordering operator
(neq) Ward Identity: condensed matter
Lecture on NGF
equilibrium:
...time ordering operator
...acting on time contour
non-quilibrium:
Dresden, Dec 2008
Non-equilibrium Ward Identity

27. ...time ordering operator
(neq) Ward Identity: condensed matter
Lecture on NGF
equilibrium:
...time ordering operator
...acting on time contour
non-quilibrium:
real time NGF... choices
our choices:
Dresden, Dec 2008
Non-equilibrium Ward Identity

28. Non-equilibrium Ward Identity
(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
Dresden, Dec 2008
Non-equilibrium Ward Identity

29. Non-equilibrium Ward Identity
(neq) Ward Identity: condensed matter
derivation : Step1-
... arbitrary external time local disturbance
... Dyson eq. for field dependent NGF,
where
Dresden, Dec 2008
Non-equilibrium Ward Identity

30. Non-equilibrium Ward Identity
(neq) Ward Identity: condensed matter
derivation : Step2-
... arbitrary external time local disturbance
... Dyson eq. for field dependent NGF,
where
vertex...
in integral form as in the linear response of one electron GF to a small variation
Dresden, Dec 2008
Non-equilibrium Ward Identity

31. Non-equilibrium Ward Identity
(neq) Ward Identity: condensed matter
derivation : Step symmetry
... arbitrary external time local disturbance
... Dyson eq. for field dependent NGF,
where
vertex...
in integral form as in the linear response of one electron GF to a small variation
gauge invariance of the first kind (global symmetry)... time variable shift added to the one particle energy in free GF
...spatially homog. potential energy
...operator unity
Dresden, Dec 2008
Non-equilibrium Ward Identity

32. Non-equilibrium Ward Identity
(neq) Ward Identity: condensed matter
derivation : Step4-explicit time dep. DE
field dependence of the Dyson equation explicitly
Dresden, Dec 2008
Non-equilibrium Ward Identity

33. Non-equilibrium Ward Identity
(neq) Ward Identity: condensed matter
derivation : Step5-explicit time dep. DE
field dependence of the Dyson equation explicitly
time dependence of selfenergy , for a weak :
Dresden, Dec 2008
Non-equilibrium Ward Identity

34. Non-equilibrium Ward Identity
(neq) Ward Identity: condensed matter
derivation : Step5-explicit time dep. DE
field dependence of the Dyson equation explicitly
time dependence of selfenergy , for a weak :
explicitly:
Dresden, Dec 2008
Non-equilibrium Ward Identity

35. Non-equilibrium Ward Identity
(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
time dependence of selfenergy , for a weak :
explicitly:
Dresden, Dec 2008
Non-equilibrium Ward Identity

36. Non-equilibrium Ward Identity
(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
Dresden, Dec 2008
Non-equilibrium Ward Identity

37. Non-equilibrium Ward Identity
(neq) Ward Identity: condensed matter
equilibrium limit
nonequilibrium Ward Identity
equilibrium limit
(time homogeneity)
equilibrium Ward Identity
Dresden, Dec 2008
Non-equilibrium Ward Identity

38. Non-equilibrium Ward Identity
(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 Takahashi Identity
Dresden, Dec 2008
Non-equilibrium Ward Identity

39. Non-equilibrium Ward Identity
(neq) Ward Identity: condensed matter
consequence of neq WI
Dresden, Dec 2008
Non-equilibrium Ward Identity

40. Non-equilibrium Ward Identity
(neq) Ward Identity: condensed matter
consequence of neq WI
time dependence of one particle , for a weak :
Dresden, Dec 2008
Non-equilibrium Ward Identity

41. Non-equilibrium Ward Identity
(neq) Ward Identity: condensed matter
consequence of neq WI
time dependence of one particle , for a weak :
Dresden, Dec 2008
Non-equilibrium Ward Identity

42. Non-equilibrium Ward Identity
(neq) Ward Identity: condensed matter
consequence of neq WI
renormalized multiplicative composition rule
Dresden, Dec 2008
Non-equilibrium Ward Identity

43. Non-equilibrium Ward Identity
(neq) Ward Identity: condensed matter
consequence of neq WI
neq renormalized semigroup rule (NE RSGR)
Dresden, Dec 2008
Non-equilibrium Ward Identity

44. Non-equilibrium Ward Identity
(neq) Ward Identity: condensed matter
consequence of neq WI
nonequilibrium multiplicative law
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
Dresden, Dec 2008
Non-equilibrium Ward Identity

45. neq renormalized semigroup rule (NE RSGR)
(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. neq renormalized semigroup rule (NE RSGR)
(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 renormalized semigroup rule (NE RSGR)
(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. Non-equilibrium Ward Identity
(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
Dresden, Dec 2008
Non-equilibrium Ward Identity

49. A. Kalvová, B. Velický, PRB 65, 155329 (2002)
(neq) Ward Identity: condensed matter
consequence of neq WI
Photoexcited transients in disordered semiconductors
A. Kalvová, B. Velický, PRB 65, (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. Photoexcited transients in disordered semiconductors
(neq) Ward Identity: condensed matter
consequence of neq WI
opt. pulse
Photoexcited transients in disordered semiconductors R
<
integral form
uncorrelated IC at

51. Steps to solve the equations in a direct fashion
(neq) Ward Identity: condensed matter
consequence of neq WI
opt. pulse
Photoexcited transients in disordered semiconductors
Steps to solve the equations in a direct fashion
Subtract the equilibrium part of the self-energy and solve only for the induced part. This provides finite time initial conditions, from which the integration starts (… the "dark polaron" idea of H. Haug)
integrate in the range [tm, ""] for each value of the right time t' > tm
The initial time must precede by at least the "collision duration time” characterizing the fast relaxation in the system
pulse envelope
equilibrium
observation period
transient process
Switch-on transient process with Keldysh initial condition
uncorrelated
initial state
correlated
(dressed) state

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

53. Photoexcited transients in disordered semiconductors
(neq) Ward Identity: condensed matter
consequence of neq WI
opt. pulse
Photoexcited transients in disordered semiconductors R
<
integral form
uncorrelated IC at
DARK ... equilibrium many-body ... known beforehand
INDUCED object of solution

54. time shape of optical pulse
(neq) Ward Identity: condensed matter
consequence of neq WI
opt. pulse
Photoexcited transients in disordered semiconductors
time shape of optical pulse

55. Photoexcited transients in disordered semiconductors
(neq) Ward Identity: condensed matter
consequence of neq WI
opt. pulse
Photoexcited transients in disordered semiconductors
time shape of optical pulse
Induced by optical pulse

56. Non-equilibrium Ward Identity
(neq) Ward Identity: condensed matter
consequence of neq WI
vertex corrections
Dresden, Dec 2008
Non-equilibrium Ward Identity

57. Non-equilibrium Ward Identity
(neq) Ward Identity: condensed matter
consequence of neq WI
vertex corrections
fixed
Dresden, Dec 2008
Non-equilibrium Ward Identity

58. Photoexcited transients in disordered semiconductors
(neq) Ward Identity: condensed matter
consequence of neq WI
opt. pulse
Photoexcited transients in disordered semiconductors
time shape of optical pulse
Induced by optical pulse

59. Photoexcited transients in disordered semiconductors
(neq) Ward Identity: condensed matter
consequence of neq WI
opt. pulse
Photoexcited transients in disordered semiconductors
time shape of optical pulse
Induced by optical pulse

60. Photoexcited transients in disordered semiconductors
(neq) Ward Identity: condensed matter
consequence of neq WI
opt. pulse
Photoexcited transients in disordered semiconductors
time shape of optical pulse
Induced by optical pulse

61. Photoexcited transients in disordered semiconductors
(neq) Ward Identity: condensed matter
consequence of neq WI
opt. pulse
Photoexcited transients in disordered semiconductors
time shape of optical pulse
Induced by optical pulse

62. Photoexcited transients in disordered semiconductors
(neq) Ward Identity: condensed matter
consequence of neq WI
opt. pulse
Photoexcited transients in disordered semiconductors
time shape of optical pulse
Induced by optical pulse

63. The end