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

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) Dresden, Dec 2008 Non-equilibrium Ward Identity

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

...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

...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

...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

...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

...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

...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

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

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

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

Non-equilibrium Ward Identity (neq) Ward Identity: condensed matter derivation : Step3- 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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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 47

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

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, 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

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

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

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

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

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

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

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

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

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

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

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

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

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

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