MS Thesis Presentation: Keldysh Reading Group

Phonon Electron Equilibriation:
A Keldysh Field Theoretic Approach

A Presentation in Defense as a Partial Requirement of the in the BS-MS degree
June 25, 2021

Sagnik Ghosh

Dr Rajdeep Sensarma Lab

Dr Bijay K Agarwalla
Dr Sreejith GJ

INSPIRE-SHE Scholarship
& Contingency Grant

Consider any Material Lattice. It can be modelled as a collection of some electrons (fermions) acting in a cohesive way (Coulomb Interaction) and some vibrations (phonons).
Suppose you excite the electronic degrees of freedom (some or all) to very high energy states by some external means.
How does the whole system equilibrate?

Key points and Notes:

Physical Context:
Particle Detectors

Physical Context:
Pump-Probe Spectroscopy

Bibliography:

Dal Forno, S. & Lischner, J. Electron-phonon coupling and hot electron thermalization in titanium nitride. Phys. Rev. Materials 3, 115203 (11 Nov. 2019)
Elsayed-Ali, H. E., Norris, T. B., Pessot, M. A. & Mourou, G. A. Time-resolved observation of electron-phonon relaxation in copper. Phys. Rev. Lett. 58, 1212–1215. (12 Mar.1987)
Habib, A., Florio, F. & Sundararaman, R. Hot carrier dynamics in plasmonic transition metal nitrides. Journal of Optics 20,064001. (May 2018)

Two Temp Model
(Anisimov, 1974; Allen, 1987)

Two Temp Model

C_e (T_e) \frac{d T_e}{dt} = \nabla (\kappa_e \nabla T_e) - G(T_e, T_{ph})\;(T_e-T_{ph}) + S(t)

C_{ph} (T_{ph}) \frac{d T_{ph}}{dt} = \nabla (\kappa_{ph} \nabla T_{ph}) + G(T_e, T_{ph})\;(T_e-T_{ph})

\frac{d}{dt} (T_e-T_{ph})= -G(T_e, T_{ph}) \Big(\frac{1}{C_e (T_e)}+\frac{1}{C_{ph} (T_{ph})}\Big)(T_e-T_{ph})

Simplification: Assume Nano-material thin-films;

Two Temp Model

\frac{d}{dt} (T_e-T_{ph})= -G(T_e, T_{ph}) \Big(\frac{1}{C_e (T_e)}+\frac{1}{C_{ph} (T_{ph})}\Big)(T_e-T_{ph})

T_e(t)-T_{ph}(t)=[T^0_e-T^0_{ph}] e^{-\frac{t}{\tau}}

\frac{1}{\tau}=G \Big(\frac{1}{C_e }+\frac{1}{C_{ph} }\Big)

Solution (after simplification):

where,

Two Temp Model

\frac{1}{\tau_{e,ph}} := G(T_e, T_{ph}) \Big[\frac{1}{C_e (T_e)}+\frac{1}{C_{ph} (T_{ph})}\Big]

Defn (Timescale) :

T_e(t) \; \widetilde{=} \;\; T^0_e e^{-\frac{t}{\tau}} + T_{\infty}

Predicted Behaviour

Two Temp Model: Key Approximations

Electrons and Phonons Thermalize between themselves almost instantly (at all times).
Further phonon-mediated relaxation just updates the temperatures. Dynamics is essentially Quassi-static.

Inherent to the assumption of two temperatures,

But Do They?

Bibliography:

Anisimov, S., Kapeliovich, B., Perelman, T.,et al.Electron emission from metalsurfaces exposed to ultrashort laser pulses. Zh. Eksp. Teor. Fiz 66,375–377 (1974)
Phillip. B. Allen, Theory of thermal relaxation of electrons in metals, Phys. Rev. Lett, 59,1460 (28 September, 1987)

A Change is in Order

Motivation: Power Law tails in OQS

Bibliography:

Chakraborty, A. & Sensarma, R. Power-law tails and non-Markovian dynamics in open quantum systems: An exact solution from Keldysh field theory. Physical Review B97,104306 (2018).
Chakraborty, A., Gorantla, P. & Sensarma, R. Non-equilibrium field theory for dynamics starting from arbitrary athermal initial conditions. Physical Review B99,054306 (2019)

Keldysh Field Theory

A Tautology

\frac{1}{\mathbf{Tr}[\rho_t]}\mathbf{Tr}[U_{-\infty,\infty}U_{\infty,t}\rho_t U_{t-,\infty}]=\mathbb{I}

Time Evolution:

Z= \frac{1}{\mathbf{Tr}[\rho_0]} \int \prod_{j=1}^N d \phi_j \; e^{-i \sum_k\sum_{j=1}^{2N} \phi_j D^{-1}_{k, jj}\phi_j}\\ =\mathbb{I}

Resolution of Identity:

Bosonic Partition Function:

\mathbb{I}= \int \; e^{-\phi^2}|\phi><\phi|\\

\begin{pmatrix} -1 & 1- i\omega_k & 0 & \cdots & 0 & 0 & \cdots & 0 & 0 & \rho_0(\omega_k) \\ 1- i\omega_k \delta t & -1 & 1- i\omega_k & \cdots & 0 & 0 & \cdots & 0 & 0 & 0 \\ 0 & 1- i\omega_k \delta t & -1 & \cdots & 1- i\omega_k & 0 & \cdots & 0 & 0 & 0 \\ \vdots & \vdots & \vdots & \cdots & \vdots & \vdots & \cdots & \vdots & \vdots & \vdots \\ 0 & 0 & 0 & \cdots & -1 & 1 & \cdots & 0 & 0 & 0 \\ \hline 0 & 0 & 0 & \cdots & 1 & -1 & \cdots & 0 & 0 & 0 \\ \vdots & \vdots & \vdots & \cdots & \vdots & \vdots & \cdots & \vdots & \vdots & \vdots \\ 0 & 0 & 0 & \cdots & 0 & 0 & \cdots & -1 & 1+ i\omega_k & 0 \\ 0 & 0 & 0 & \cdots & 0 & 0 & \cdots & 1+ i\omega_k \delta t & -1 & 1+ i\omega_k \\ 0 & 0 & 0 & \cdots & 0 & 0 & \cdots & 0 & 1+ i\omega_k \delta t & -1 \\ \end{pmatrix}

Structure of :

D^{-1}_{k,ij}

S [\phi] = \int_{-\infty}^{\infty} dt \sum_k \big[ \phi^{+}(t) (i \partial^2_t - \omega^2_k) \phi^{+}(t) -\phi^{-}(t) (i \partial^2_t - \omega^2_k) \phi(t)^{-}\big]

Continuum Limit (Keldysh Action) :

Z[\chi] = \int d \phi_j \; e^{-\sum_{i,j} \phi_iA_{ij}\phi_j + \phi_j\chi_j }\\ = \frac{1}{\mathbf{det}(A)} e^{-\sum_{i,j} \chi_iA^{-1}_{ij}\chi_j}

Gaussian Integral:

<\phi_a \phi_b> = \frac{1}{Z[0]}\frac{\partial^2 Z[ \chi]}{\partial \chi_b \partial \chi_a} = \;\;A^{-1}_{ab}\\ <\phi_a \phi_b \phi_c\phi_d>= \frac{1}{Z[0]}\frac{\partial^4 Z[ \chi]}{\partial \chi_d \partial \chi_c \partial\chi_b \partial\chi_a} = A^{-1}_{ac}A^{-1}_{bd}+ A^{-1}_{ad}A^{-1}_{bc}

Wick's Theorem (for Bosons):

\phi^{cl}_k(t)=(\phi^{+}_k(t) + \phi^{-}_k(t))/2 \; \; \; \; ;\; \; \; \phi^{q}_k(t)=(\phi^{+}_k(t) - \phi^{-}_k(t))/2\\ \psi^{1}_k(t)=(\psi^{+}_k(t) + \psi^{-}_k(t))/\sqrt{2} \; \; \; \; ;\; \; \; \psi^{2}_k(t)=(\psi^{+}_k(t) - \psi^{-}_k(t))/\sqrt{2}\\ \overline{\psi^{1}}_k(t)=(\overline{\psi^{+}}_k(t) - \overline{\psi^{-}}_k(t))/\sqrt{2} \; \; \; \; ;\; \; \; \overline{\psi^{2}}_k(t)=(\overline{\psi^{+}}_k(t) + \overline{\psi^{-}}_k(t))/\sqrt{2}

Coordinate Rotation:

G=\begin{pmatrix} G^R & G^K \\ 0 & G^A \end{pmatrix} \;\;\;;\;\;\; D=\begin{pmatrix} D^K & D^R \\ D^A & 0 \end{pmatrix}

Green's Function in Keldysh Field Theory:

\begin{pmatrix} D^K & D^R \\ D^A & 0 \end{pmatrix} = D^{\alpha\beta}_k(t,t') = - \int \prod_{j=1}^N d \phi_j \; \phi^{\alpha}_k(t)\phi^{\beta}_k(t')\;e^{i(S_0+S_{int})}

D_k(t,t') = D^0_k(t,t') + D^0_k(t,t')\circ\Sigma_k(t,t')\circ D^0_k(t,t')

+ D^0_k(t,t')\circ\Sigma_k(t,t')\circ D^0_k(t,t')\circ\Sigma_k(t,t')\circ D^0_k(t,t')

+\cdots

D_k(t,t') = D^0_k(t,t') + D^0_k(t,t')\circ\Sigma_k(t,t')\circ D_k(t,t')

Introducing Interaction

Dyson Equation

G=\begin{pmatrix} G^R & G^K \\ 0 & G^A \end{pmatrix}\;\;\;;\;\;\; \Sigma_{el}=\begin{pmatrix} \Sigma^R_{el} & \Sigma^K_{el} \\ 0 & \Sigma^A_{el} \end{pmatrix}

D=\begin{pmatrix} D^K & D^R \\ D^A & 0 \end{pmatrix} \;\;\;;\;\;\; \Sigma_{ph} =\begin{pmatrix} 0 & \Sigma_{ph}^A \\ \Sigma_{ph}^R & \Sigma_{ph}^K \end{pmatrix}

Causality Structure

D^R_0(t,t') = D^R_0(t,\tau)\overline{D}^R_0(\tau,t')+\overline{D}^R_0(t,\tau)D^R_0(\tau,t')\;\;\forall \;\; t>\tau>t'\\ D^K_k(t,t') = D^R_0(t,\tau)\overline{D}^K_0(\tau,t')+\overline{D}^R_0(t,\tau)D^K_0(\tau,t')\;\;\forall \;\; t>\tau>t'

G^R_0(t,t') = G^R_0(t,\tau)G^R_0(\tau,t')\;\;\forall \;\; t>\tau>t'\\ G^K_k(t,t') = G^R_0(t,\tau)G^K_0(\tau,t')\;\;\forall \;\; t>\tau>t'

Decomposition Theorem: (First Order)

Decomposition Theorem: (Second Order)

Bibliography:

Keldysh, L. V. et al. Diagram technique for non-equilibrium processes. Sov. Phys. JETP20,1018–1026 (1965).
Kamenev, A. Field theory of non-equilibrium systems
(Cambridge University Press, 2011).
Larkin, A. & Ovchinnikov, Y. Nonlinear conductivity of superconductors in the mixed state. Sov. Phys. JETP41,960–965 (1975).

The Grassmann Algebra

Consider a single fermionic level, which is spanned by the number states |0> and |1>

The annihilation and creation operators, $$c, c^{\dagger} $$ are defined by their acion as,

It follows that,

The number operator is defined as,

Consider a single fermionic level, which is spanned by the number states |0> and |1>

The annihilation and creation operators, $$c, c^{\dagger} $$ are defined by their acion as,

It follows that,

The number operator is defined as,

Operator Algebra:

The Idea

D^R(t,t'), D^K(t,t')

\Sigma_{ph}^R(t,t'), \Sigma_{ph}^K(t,t')

D^R(t+\epsilon,t'), D^K(t+\epsilon,t')\\ D^K(t+\epsilon,t+\epsilon)

D = D^0 + D^0\circ\Sigma\circ D

\Sigma_{el}^R(t,t'), \Sigma_{el}^K(t,t')\\ \Sigma_{el}^R(t+\epsilon,t'), \Sigma_{el}^K(t+\epsilon,t')

G^R(t,t'), G^K(t,t')

G^R(t+\epsilon,t'), G^K(t+\epsilon,t')\\ G^K(t+\epsilon,t+\epsilon)

G = G^0 + G^0\circ\Sigma\circ G

Form of Self Energies:

\Sigma_{el}(t,t') \sim -\lambda^2 G(t,t')D(t,t')

\Sigma_{ph}(t,t') \sim -\lambda^2 G(t,t')G(t,t')

Evolution Equations:

G^R(t+\epsilon , t') = G^R_0(t+\epsilon , t){G^R}(t , t')+ \frac{\epsilon}{2} G_0^R(t+\epsilon, t+\epsilon)\int^{t+\epsilon}_{t'} dt_2 \Sigma^R (t+\epsilon,t_2) G^R(t_2, t') \\ + \frac{\epsilon}{2} G_0^R(t+\epsilon, t)\int^{t}_{t'} dt_2 \Sigma^R (t,t_2) G^R(t_2, t')

D^R(t+\epsilon , t') = D^R_0(t+\epsilon , t){\overline{D^R}(t , t')}+\overline{D^R_0}(t+\epsilon , t)D^R(t , t') \\ + \frac{\epsilon}{2} D_0^R(t+\epsilon, t)\int^{t}_{t'} dt_2 \Sigma^R (t,t_2) D^R(t_2, t')

D^R(t,t'), D^K(t,t')

\Sigma_{ph}^R(t,t'), \Sigma_{ph}^K(t,t')

D^R(t+\epsilon,t'), D^K(t+\epsilon,t')\\ D^K(t+\epsilon,t+\epsilon)

D = D^0 + D^0\circ\Sigma\circ D

\Sigma_{el}^R(t,t'), \Sigma_{el}^K(t,t')\\ \Sigma_{el}^R(t+\epsilon,t'), \Sigma_{el}^K(t+\epsilon,t')

G_{thermal}^R(t,t'), G_{thermal}^K(t,t')

G_{thermal}^R(t+\epsilon,t'), G_{thermal}^K(t+\epsilon,t')\\ G_{thermal}^K(t+\epsilon,t+\epsilon)

G = G^0 + G^0\circ\Sigma\circ G

The Phonons
(Coupled to a Bath)

The Phonons (coupled to a Bath)

D^R_k(t,t')= -\frac{1}{2\omega_k} \sin[\omega_k(t-t')]\;\;\;\;\forall\;\;\;\; t>t'\\ = 0 \;\;\;\;\;\;\;\;\text{otherwise}

D^K_k(t,t')= -\frac{i}{2\omega_k} \cos[\omega_k(t-t')]\coth\big[\frac{\omega_k}{2\; T_{system}}\big]

Bare Green's Functions (time domain expressions):

The Phonons (coupled to a Bath)

J(\omega)=\eta\omega e^{-\frac{\omega^2}{\sigma^2}}

\Sigma^R(\omega)= -\lambda^2\frac{2}{\sqrt{\pi}}\omega DawsonF\big(\frac{\omega}{\sqrt{2}\sigma}\big)-i\lambda^2\omega \exp(-\frac{\omega^2}{\sigma^2})\\ \Sigma^K(\omega)=-i2\lambda^2\omega \exp(-\frac{\omega^2}{\sigma^2})\coth\big({\frac{\omega}{2T_{bath}}}\big)

DawsonF(x)=e^{-x^2}\int_0^x e^{y^2}dy

Bath Spectral Function:

Self-Energies (frequency domain):

Causality Structure (Recall):

D=\begin{pmatrix} D^K & D^R \\ D^A & 0 \end{pmatrix} \;\;\;;\;\;\; \Sigma_{ph} =\begin{pmatrix} 0 & \Sigma_{ph}^A \\ \Sigma_{ph}^R & \Sigma_{ph}^K \end{pmatrix}

\omega= \omega_0 \sqrt{\sin^2(\frac{ka}{2})}

System Dispersion Relation:

What to expect?

Energy spectrum will be re-normalised!
(Bigger the bath bandwidth, bigger should be the shift)
Higher the Bath Temperature is, we can expect population in a given level to go up.
(Bose-Einstein Distribution, No number conservation for phonons.)
With stronger coupling we should expect faster decay of transient phenomena.

Phononic Plots: Dynamical Behaviour

Section Take Away

Wider the Bath-Bandwidth is Further the system is driven away (initially) from the equilibrium value
Stronger the coupling, faster is the decay of transient oscillation
The transient oscillation is dictated by the level frequency
Higher Bath Temperature tend to shift the population to a higher value (Bosons!) and vice-versa!

The Electrons
(Coupled to a Bath)

The Electrons (coupled to a Bath)

G^R_k(t,t')= -i e^{-i\epsilon_k(t-t')}\;\;\;\;\forall\;\;\;\; t>t'\\ = 0 \;\;\;\;\;\;\;\;\text{otherwise}

G^K_k(t,t')= -i e^{-i\epsilon_k(t-t')}\tanh\big[\frac{\omega_k}{2\; T_{system}}\big]

Bare Green's Functions (time domain expressions):

The Electrons (coupled to a Bath)

J(\omega)=\theta(\omega^2-4\sigma^2)\frac{2}{\sigma} \sqrt{1-\frac{\omega^2}{4\sigma^2}}

\Sigma^R(\omega)= -i\lambda^2\theta(\omega^2-4\sigma^2)\frac{2}{\sigma} \sqrt{1-\frac{\omega^2}{4\sigma^2}}\\ \Sigma^K(\omega)=-i\lambda^2\theta(\omega^2-4\sigma^2)\frac{2}{\sigma} \sqrt{1-\frac{\omega^2}{4\sigma^2}}\tanh\big({\frac{\omega}{2T_{bath}}}\big)

Bath Spectral Function:

Self-Energies (frequency domain):

Computing Thermal Value

D_k^K(t,t)= \int_{-\infty}^{\infty}-\frac{2}{2\pi} \mathbf{Im}\Big[\frac{1}{\omega-\epsilon_k+i\eta-\Sigma^R(\omega)}\Big] d\omega

Causality Structure (Recall):

G=\begin{pmatrix} G^R & G^K \\ 0 & G^A \end{pmatrix}\;\;\;;\;\;\; \Sigma_{el}=\begin{pmatrix} \Sigma^R_{el} & \Sigma^K_{el} \\ 0 & \Sigma^A_{el} \end{pmatrix}