Emission and collisional correlation in far-off equilibrium quantum systems

Abstract

We propose a scheme to describe dynamical correlations in finite fermion systems which are open in the sense that they can lose particles, electrons in the present case. It is built as an extension of recently developed schemes to describe dissipative dynamics in finite fermion systems, namely stochastic time-dependent adiabatic local-density approximation (STDLDA) and its averaged version, ASTDLDA, so far being applied only in closed systems. STDLDA and ASTDLDA are based on real-time real-space dynamics in terms of time-dependent density functional theory and add dynamical two-body collisions in a stochastic manner. The extension to systems that can emit electrons is achieved by complementing the complete (numerical) description of wave functions in “inner” space (inside the computation box) through a global description in “outer” space (outside the computation box) and a careful bookkeeping of flow between inner and outer regions. We test the method in a 1D model system, mimicking simple molecules or clusters. Two test cases are investigated: a metal-like system excited by an instantaneous dipole boost, and a covalent-like system excited by an instantaneous one-particle–one-hole transition. The dynamics is analyzed in terms of one-body observables (dipole moment, ionization, entropy). STDLDA and ASTDLDA exhibit clearly dissipative features at variance with mean-field dynamics. Unlike the pattern in closed systems, differences between STDLDA and ASTDLDA show up for ionization. They can be attributed to significantly larger fluctuations of the mean field in STDLDA.

Graphic Abstract

Appendices

The total energy in full space

The expression of the total energy, as written in Eq. (9), is justified by the fact that we can always write the total energy as:

$$\begin{aligned} {E} = {E}_\mathrm {rearr} + \sum _i n_i\varepsilon _i \end{aligned}$$

where ${E}_\mathrm {rearr}$ is the rearrangement energy. The “inner” part of the total energy, $E^\mathrm {in}$, is decomposed in a similar fashion:

$$\begin{aligned} {E}^\mathrm {in} = {E}_\mathrm {rearr}^\mathrm {in} + \sum _i n_i\varepsilon _i^\mathrm {in} \quad . \end{aligned}$$

Being defined inside the numerical box, we can compute each term entering this expression of $E^\mathrm{in}$.

We furthermore assume that there is no many-body interaction in the outer space. Therefore, we can write:

$$\begin{aligned} {E}_\mathrm {rearr} = {E}_\mathrm {rearr}^\mathrm {in} \quad . \end{aligned}$$

This finally yields the total energy in full space as

$$\begin{aligned} {E} = {E}^\mathrm {in} + \sum _i n_i(\varepsilon _i-\varepsilon _i^\mathrm {in}) = {E}^\mathrm {in} + \sum _i n_i\varepsilon _i^\mathrm {out} \quad . \end{aligned}$$

Sampling the one-body density operator in STDLDA

The computation of the one-body entropy in full space requires to know the one-body density operator in full space. This is given by construction in ASTDLDA. However, in STDLDA, the one-body density matrix is the average of an ensemble of $\mathcal N$ density matrices expressed in their own set of s.p. orbitals:

$$\begin{aligned} \widehat{\rho } = \frac{1}{\mathcal {N}} \sum _{\alpha =1}^\mathcal {N}\hat{\rho }^{(\alpha )} \quad \mathrm{with}\quad \hat{\rho }^{(\alpha )} = \sum _{i=1}^\varOmega {|{\varphi _i^{(\alpha )}}\rangle }n_i^{(\alpha )} {\langle {\varphi _i^{(\alpha )}}|} \ . \end{aligned}$$

The index $\alpha$ denotes each sample of the stochastic ensemble, $\{\varphi _i^{(\alpha )},i=1...\varOmega \}$ the associated basis set of s.p. orbitals and of size $\varOmega$, and $\hat{\rho }^{(\alpha )}$ the corresponding one-body density matrix. There is no reason that, from one sample to the other, the basis sets involve the same s.p. orbitals because the mean field is different from one sample to the other. Instead of working with $\mathcal N$ different basis sets, the idea is to recursively build a common set of natural orbitals while successively accumulating each sample of the ensemble in the computation of the one-body density operator. The size of this common set will thus grow accordingly with this accumulation.

To understand how this proceeds, we denote by $\widetilde{\rho }^{(M)}$ the one-body density operator built from the first M samples of the stochastic ensemble and we have:

$$\begin{aligned} \widetilde{\rho }^{(M)} = \frac{1}{M}\sum _{\alpha =1}^M\hat{\rho }^{(\alpha )} \; {\mathop {\xrightarrow {}}\limits ^{M\rightarrow \mathcal N}}\; \widetilde{\rho }^{(\mathcal {N})} = \widehat{\rho } \end{aligned}$$

where $\mathcal {N}$ is the size of the full ensemble.

We now detail the recursion algorithm. We start with the first sample of the stochastic ensemble, that is $\alpha =1$. The one-body density operator $\hat{\rho }^{(1)}=\widetilde{\rho }^{(1)}$ obtained at the end of the time evolution is that of a pure TDLDA state. It is diagonal in the propagation basis. This means that this basis is also the natural basis.

We now consider the recursive step $M\!-\!1\longrightarrow M$. We assume that we have a natural orbital representation of the one-body density operator at step $M-1$. We introduce the following notations:

$$\begin{aligned} \widetilde{\rho }^{(M-1)} = \sum _{i=1}^{\varOmega ^{(M-1)}}{|{\phi _i^{(M-1)}}\rangle }w_i^{(M-1)}{\langle {\phi _i^{(M-1)}}|} \end{aligned}$$

where the basis $\mathcal B(M-1,\varOmega ^{(M-1)})=\{\phi _i^{(M-1)},i=1...\varOmega ^{(M-1)}\}$ is the corresponding set of natural orbitals and $w_i^{(M-1)}$ the corresponding fractional occupation probabilities. The step to $M\!-\!1\longrightarrow M$ reads:

$$\begin{aligned} \widetilde{\rho }^{(M)}&= \frac{M-1}{M}\,\widetilde{\rho }^{(M-1)} + \frac{1}{M}\hat{\rho }^{({M})} \nonumber \\&= \frac{M-1}{M}\sum _{i=1}^{\varOmega ^{(M-1)}}{|{\phi _i^{(M-1)}}\rangle }w_i^{(M-1)}{\langle {\phi _i^{(M-1)}}|} \nonumber \\&\quad + \frac{1}{M}\sum _{j=1}^\varOmega {|{\varphi _j^{({M})}}\rangle }n_j^{(\alpha )}{\langle {\varphi _j^{({M})}}|} \end{aligned}$$

(26)

where $n_j^{(\alpha )}\in \{0,1\}$ are the occupation numbers of the pure state of the Mth sample $\hat{\rho }^{({M})}$. Mind that the notation $\phi$ stands for a natural orbital built from the accumulation of samples of the stochastic ensemble, while the notation $\varphi$ corresponds to the basis set associated to a pure state (and thus to a single sample).

We now concatenate the sets appearing in Eq. (26):

$$\begin{aligned} {\mathcal {C}}&= \{\psi _k^{(M)},k=1...\varOmega ^{(M-1)}+\varOmega \} \nonumber \\&= \{\phi _i^{(M-1)},i=1...\varOmega ^{(M-1)}\}\cup \{\varphi _j^{(\alpha {=M})},j=1...\varOmega \}. \end{aligned}$$

While each subset is orthonormal in itself:

$$\begin{aligned} {\langle {\phi _i^{(M-1)}}|}\phi _j^{(M-1)}\rangle&= \delta _{ij} \ \mathrm{with}\ i,j=1...\varOmega ^{(M-1)} \\ {\langle {\varphi _k^{({M})}}|}\varphi _l^{({M})}\rangle&= \delta _{kl} \ \mathrm{with}\ k,l=1...\varOmega \end{aligned}$$

the new set $\mathcal C$ is overcomplete and not orthonormal. We thus build a new orthonormal set of orbitals $\phi _i^{({M})}$ by diagonalizing the density matrix of $\widetilde{\rho }^{(M)}$ expressed in the overcomplete set $\mathcal C$. This gives us the natural set $\mathcal B(M,\varOmega ^{(M)})$ we are precisely looking for. The one-body density operator at step M now reads:

$$\begin{aligned} \widetilde{\rho }^{(M)}= & {} \sum _{i=1}^{\varOmega ^{(M)}} {|{\phi _i^{(M)}}\rangle }w_i^{(M)}{\langle {\phi _i^{(M)}}|}, \\ \phi _i^{(M)}= & {} \sum _{j=1}^{\varOmega ^{(M-1)}+\varOmega }\psi _j^{(M)}{R}_{ji}. \end{aligned}$$

The determination of the matrix R amounts to solve the following generalized eigenvalue problem:

$$\begin{aligned} \sum _m&{\langle {\psi _k^{(M)}}|}\widetilde{\rho }^{(M)}{|{\psi _m^{(M)}}\rangle } R_{ml} \nonumber \\&= w_l^{(M)} \sum _n {\langle {\psi _k^{(M)}}|}\psi _n^{(M)}\rangle R_{nl}. \end{aligned}$$

(27)

This ends the recursive step $M\!-\!1\longrightarrow M$.

This algorithm raises two problems. First, each recursive step appends $\varOmega$ new s.p. states to the basis which then quickly exceeds manageable sizes. Second, the overlaps ${\langle {\psi _k^{(M)}}|}\psi _{k'}^{(M)}\rangle$ needed to set up the entries of Eq. (27) are meant in full space, while we dispose explicitly of the s.p. wave functions in inner space only. The first problem finds a natural solution. Most of the states appended to build the natural set $\mathcal B(M,\varOmega ^{(M)})$ have a negligible occupancy $w_i^{(M)}$. We set a lower limit of typically $w_\mathrm {min}=10^{-3}$ and spread all states with $w_i^{(M)}<w_\mathrm {min}$ on the remaining states of index j, proportionally to $1-w_j^{(M)}$. This amounts to only a slight increase in the number of states in the natural basis, that is $\varOmega ^{(M-1)}\lesssim \varOmega ^{(M)}$. In addition, to keep the size manageable, we limit the total number of orbitals, by also skipping the least occupied orbitals when the total number of orbitals reaches typically 250.

To solve the second problem, we expect that the wave functions from different basis sets in outer space quickly develop independence. We thus assume that:

$$\begin{aligned} {\langle {\phi _i^{(M-1,\mathrm {out})}}|}\varphi _k^{({M},\mathrm {out})}\rangle = 0. \end{aligned}$$

allowing us to compute the overlap matrix in the $\mathcal C$ basis set. In other words, we assume that:

$$\begin{aligned} {\langle {\phi _i^{(M-1)}}|}\varphi _k^{(M)}\rangle&= {\langle {\phi _i^{(M-1,\mathrm {in})}}|}\varphi _k^{(M,\mathrm {in})}\rangle \ \mathrm{with}\\&\qquad i=1...\varOmega ^{(M-1)} \ , \ k=1...\varOmega \end{aligned}$$

With this recursion algorithm, at the end of the whole STDLDA procedure, the one-body density operator $\widehat{\rho }$ is expressed in a natural set, $\mathcal B(\mathcal N,\varOmega )$, and one can then use the corresponding occupation probabilities to compute the one-body entropy. It is worth noting that in closed systems, this method to compute the entropy gives results very close to those obtained by a more standard method, that is by computing occupation numbers by diagonalization of the space representation of the one-body density $\hat{\rho }(x,{t})$.