Introduction to Real Time propagation in Yambo

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Introduction

A new feature in yambo is the numerical integration of a time-dependent (TD) equation of motion (EOM), able to describe the evolution of the electronic system under the action of an external laser pulse. Similarly to the equilibrium case, the most diffuse ab initio approaches to real-time propagation are based on TD-DFT and there exist a number of GPL codes available to this end. On the contrary the implementation of real time propagation within MBPT is an almost unique feature of the yambo code. Two different schemes are available. In one case, the density matrix of the system, is propagated in time, as described in the first section. In the second case an effective time dependent Schrodinger equation for the occupied orbitals is propagated, as described in the second section.

Time Dependent Equation for the Reduced One--Body Density--Matrix

The first scheme is a Time Dependent equation for the one body reduced density matrix. The EOM for the density matrix projected in the space of the single particle wave-functions is derived from non-equilibrium (NEQ) many-body perturbation theory and reads

The time dependent equation of motion for the one body reduced density matrix as implmented in the Yambo code

Here we underline quantities which are vectors in the trans­ition space. The Hamiltonian contains the equilibrium eigenvalues plus the variation of the self-energy. It is a functional of the density matrix and can be written as follow:

Many-Body Hamiltonian

The variation of the self--energy can be efficiently constructed via the linearization of the self--energy (which is exact for most of the approximations available in yambo)

Expression for the many body self-energy doing a linear expansion in terms of the density matrix
.

K is the functional derivative of the self--energy with respect to the density matrix and corresponds to the kernel of the Bethe--Salpeter equation. In the language of the yambo code we will refer to it with the name "real time collisions" or simply "collisions". Choosing the HSEX approximation and using the GW energies for the quasi--particle corrections, the time propagation corresponds to a real-time version of the Bethe-Salpeter Equation[1] More details can be found in Ref. [2]

External field in length gauge.png represents the external potential written in the length gauge; shape, polarization, intensity (and eventually frequency) of the field E can be selected in input. r is the position operator. The coupling to the external field is exact up to first order.

Linear Response

From the knowledge of the density matrix, the first order polarization Polarization from rho.png is computed at each time step. The spectrum of the system can then be obtained by the Fourier transform of the polarization, which can be done as a post-processing step. Absorption is thus obtained via the dipole-dipole response function (equivalent to the length gauge in linear response). [math] \chi(\omega) = \frac{P(\omega)}{E(\omega)} [/math]

More details will be given during the tutorials:

Time Dependent Effective Schrödinger Equation

The second scheme is a Time Dependent Effective Schrödinger Equation(TD-ESE), where the coupling between electrons and the external field is described by means of the Modern Theory of Polarization:


Time-Dependent Schrödinger Equation


Where [math] | v_{i \mathbf{k}} \rangle [/math] are the time-dependent valence bands, [math]H_{rt}[/math] is the Hamiltonian, [math]\mathcal E(t)[/math] is the external field and the term [math] i e \partial k[/math] corresponds to the dipole operator in periodic systems. For more details on this last term see Ref. [3] and [4]. Differently from other codes, the equations of motion of Yambo are in the length gauge and not in the velocity one. If you want to know more about the advantages and disadvantages of the two gauges read section 2.7 of Ref. [5]. The Hamiltonian and the initial wave-functions are obtained from DFT. Then in order to obtain linear response, we probe the system with a delta function field, that excites all the frequencies at the same footing, and Fourier transforms.

The hamiltonian is defined as in the case of the equation of motion for the density matrix. Then the polarization is calculated by means of Berry's phase

Berry's polarization


Advantages and disadvantages of the two formalisms

The reason why in Yambo there are two distinct formalisms for studying the real-time response is due to the fact that these have particular advantages and disadvantages which are therefore suitable for some types of simulations and not others. The dynamics formulation in terms of density matrix or non-equilibrium Green's functions allows a systematic treatment of correlation effects and electron-phonon coupling[6], but the price to pay is that the coupling with the external field is correct only to the linear order. This formalism is important in all those phenomena where electronic relaxation plays a major role. On the other hand, in the TDSE formalism the coupling with the external field in an exact way even beyond the linear regime, and for this reason, it allows the description of phenomena coherent with the strong external fields, like the non-linear response.

Linear Response

In general the induced polarization is the sum of the responses of the system to all orders in the external field: [math]\bf{P}(t) = \sum_{n=-\infty}^{+\infty} \bf{P}_n (t) [/math] If the perturbation is weak enought the response is dominated by the first order term and from its Fourier transform we get the response function. Indeed

[math] \chi(\omega) = \frac{P(\omega)}{E(\omega)} [/math]

that is related to the dielectric constant through the relation [math]\epsilon(\omega) = 1 + 4 \pi \chi(\omega) [/math]. See also Ref. [4]

More details will be given during the tutorials:

Non Linear Response

Non-linear response calculations are similar to the linear response case, with the only difference that we excite the system with a sinusoidal external field. A perturbation with a specific frequency allows to expand the polarization in the form [math]\bf{P}(t) = \sum_{n=-\infty}^{+\infty} \bf{p}_n e^{-i\omega_n t} [/math] where the coefficient [math]\bf{p}_1,...,\bf{p}_n [/math] are related to [math]\chi^{(1)},...,\chi^{(n)} [/math]. In practice we use the scheme described in Ref. [4] and show below to extract the non-linear coefficients:

Schematic representation of real-time calculations

More details will be given during the tutorials:

References

  1. G. Strinati, La Rivista del Nuovo Cimento 11 (12), 1-86 (1988)
  2. C. Attaccalite, M. Grüning, and A. Marini PRB 84, 245110 (2011)
  3. I. Souza, J. Íñiguez, and D. Vanderbilt, PRB 69, 085106 (2004)
  4. 4.0 4.1 4.2 C. Attaccalite and M. Gruning Rev. B, 88, 235113 (2013)
  5. C. Attaccalite, arXiv 1609.09639 (2017)
  6. D. Sangalli, and A. Marini EPL 110, 47004 (2015)



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