diff --git a/doc/src/week47/LatexSlides/Greensfunctions.pdf b/doc/src/week47/LatexSlides/Greensfunctions.pdf index 32c75653..27e274c5 100644 Binary files a/doc/src/week47/LatexSlides/Greensfunctions.pdf and b/doc/src/week47/LatexSlides/Greensfunctions.pdf differ diff --git a/doc/src/week47/LatexSlides/Greensfunctions.tex b/doc/src/week47/LatexSlides/Greensfunctions.tex index a14a32c0..9795a9be 100644 --- a/doc/src/week47/LatexSlides/Greensfunctions.tex +++ b/doc/src/week47/LatexSlides/Greensfunctions.tex @@ -1177,13 +1177,20 @@ \frametitle{Single-particle Green's functions} \begin{small} {\scriptsize -Let us consider a particle in free space described by a single particle Hamiltonian $h_{1}$. Its eigenstates and eigenenergies are +We consider first a particle in free space described by a single particle Hamiltonian $\hat{h}$. Its eigenstates and eigenenergies are $$ -h_{1}\left|\phi_{n}\right\rangle=\varepsilon_{n}\left|\phi_{n}\right\rangle +\hat{h}\left|\phi_{n}\right\rangle=\varepsilon_{n}\left|\phi_{n}\right\rangle $$ -In general, if we put the particle in one of its $\left|\phi_{n}\right\rangle$ orbits, it will remain in the same state forever. Instead, we immagine to prepare the system in a generic state $\left|\psi_{t r}\right\rangle$ (tr stands for 'trial') and then follow its time evolution. If the trial state is created at time $t=0$, the wavefunction at a later time $t$ is given by [see Eq. (1.47)] +In general, if we put the particle in one of its +$\left|\phi_{n}\right\rangle$ orbits, it will remain in the same state +forever. + +We prepare now the system in a generic state +$\left|\psi_{\mathrm{T}}\right\rangle$ ($\mathrm{T}$ stands for trial) and then +follow its time evolution. If the trial state is created at time +$t=0$, the wavefunction at a later time $t$ is given by $$ \begin{aligned} @@ -1202,19 +1209,25 @@ \frametitle{Single-particle Green's functions} \begin{small} {\scriptsize -The second line shows that if one knows the eigentstates $\left|\phi_{n}\right\rangle$, it is relatively simple to compute the time evution: one expands $\left|\psi_{t r}\right\rangle$ in this basis and let every component propagate independently. Eventually, at time $t$, we want to know the probability amplitude that a measurement would find the particle at position $\mathbf{r}$, + +The above result shows that if one knows the eigentstates +$\left|\phi_{n}\right\rangle$, it is easy to compute the +time evution: one expands $\left|\psi_{\mathrm{T}}\right\rangle$ in this +basis and let every component propagate independently. Eventually, at +time $t$, we want to know the probability amplitude that a measurement +would find the particle at position $\mathbf{r}$, $$ \begin{aligned} -\langle\mathbf{r} \mid \psi(t)\rangle & =\left\langle\mathbf{r}\left|e^{-i h_{1} t / \hbar}\right| \psi_{t r}\right\rangle \\ -& =\int d \mathbf{r}^{\prime}\left\langle\mathbf{r}\left|e^{-i h_{1} t / \hbar}\right| \mathbf{r}^{\prime}\right\rangle\left\langle\mathbf{r}^{\prime} \mid \psi_{t r}\right\rangle +\langle\mathbf{r} \mid \psi(t)\rangle & =\left\langle\mathbf{r}\left|e^{-i \hat{h} t / \hbar}\right| \psi_{\mathrm{T}}\right\rangle \\ +& =\int d \mathbf{r}^{\prime}\left\langle\mathbf{r}\left|e^{-i \hat{h} t / \hbar}\right| \mathbf{r}^{\prime}\right\rangle\left\langle\mathbf{r}^{\prime} \mid \psi_{\mathrm{T}}\right\rangle \end{aligned} $$ $$ \begin{aligned} -& =\int d \mathbf{r}^{\prime} \sum_{n}\left\langle\mathbf{r} \mid \phi_{n}\right\rangle e^{-i \varepsilon_{n} t / \hbar}\left\langle\phi_{n} \mid \mathbf{r}^{\prime}\right\rangle\left\langle\mathbf{r}^{\prime} \mid \psi_{t r}\right\rangle \\ -& \equiv \int d \mathbf{r}^{\prime} G\left(\mathbf{r}, \mathbf{r}^{\prime} ; t\right) \psi_{t r}\left(\mathbf{r}^{\prime}\right), +& =\int d \mathbf{r}^{\prime} \sum_{n}\left\langle\mathbf{r} \mid \phi_{n}\right\rangle e^{-i \varepsilon_{n} t / \hbar}\left\langle\phi_{n} \mid \mathbf{r}^{\prime}\right\rangle\left\langle\mathbf{r}^{\prime} \mid \psi_{\mathrm{T}}\right\rangle \\ +& \equiv \int d \mathbf{r}^{\prime} G\left(\mathbf{r}, \mathbf{r}^{\prime} ; t\right) \psi_{\mathrm{T}}\left(\mathbf{r}^{\prime}\right), \end{aligned} $$ @@ -1229,24 +1242,26 @@ \frametitle{Single-particle Green's functions} \begin{small} {\scriptsize - Obviously, once $G\left(\mathbf{r}, \mathbf{r}^{\prime} ; t\right)$ + Once $G\left(\mathbf{r}, \mathbf{r}^{\prime} ; t\right)$ is known it can be used to calculate the evolution of any initial state. However,there is more information included in the - propagator. This is apparent from the expansion in the third line of - Eq (2.3): first, the braket $\left\langle\phi_{n} \mid + propagator. This is apparent from the expansion in the third line of the last equation. + First, the braket $\left\langle\phi_{n} \mid \mathbf{r}\right\rangle=\left\langle\phi_{n}\left|\psi^{\dagger}(\mathbf{r})\right| 0\right\rangle$ gives us the probability that putting a particle at position $\mathbf{r}$ and mesuring its energy rgiht away, would make the system to collapse into the eigenstate - $\left|\phi_{n}\right\rangle$. Second, the time evolution is a + $\left|\phi_{n}\right\rangle$. + + Second, the time evolution is a superposition of waves propagating with different energies and could - be inverted to find the eigenspectrum. Immagine an experiment in + be inverted to find the eigenspectrum. We could think of an experiment in which the particle is put at position $\mathbf{r}$ and picked up at - $\mathbf{r}^{\prime}$ after some time t. If one can do this for + $\mathbf{r}^{\prime}$ after some time $t$. If one can do this for different positions and elapsed timesand with good resolution-then a Fourier transform would simply give back the full eigenvalue - spectrum. Such an experiment is a lot of work to carry out! But would - give us complete information on our particle. + spectrum. Such an experiment + gives us complete information on our particle. } \end{small} @@ -1255,9 +1270,10 @@ \frame { -\frametitle{Single-particle Green's functions} +\frametitle{One-body Green's functions} \begin{small} {\scriptsize + We now want to apply the above ideas to see what we can learn by adding and removing a particle in an environment when many others are present. This can cause the particle to behave in an unxepected way, @@ -1274,16 +1290,17 @@ \frame { -\frametitle{Single-particle Green's functions} +\frametitle{Heisenberg picture} \begin{small} {\scriptsize -In the following we consider the Heisenberg description of the field operators, + +In the following we consider the Heisenberg description of the field operators (see discussions earlier on different pictures)), $$ \psi_{s}^{\dagger}(\mathbf{r}, t)=e^{i H t / \hbar} \psi_{s}^{\dagger}(\mathbf{r}) e^{-i H t / \hbar} $$ -where the subscript $s$ serves to indicate possible internal degrees of freedom (spin, isospin, etc...). We omit the superscrips $\mathrm{H}$ (Hiesenberg) and S (Scrödinger) from the operators since the two pictures can be distinguished from the presence of the time variable, which appears only in the first case. Similarly, +where the subscript $s$ serves to indicate possible internal degrees of freedom (spin, etc...). We omit the superscrips $\mathrm{H}$ (Heisenberg) and $S$ (Schr\"odinger) from the operators since the two pictures distinguish themselves by the presence of the time variable, which appears only in the first case. Similarly, $$ \psi_{s}(\mathbf{r}, t)=e^{i H t / \hbar} \psi_{s}(\mathbf{r}) e^{-i H t / \hbar}, @@ -1296,19 +1313,23 @@ \frame { -\frametitle{Single-particle Green's functions} +\frametitle{Creation and annihilation operators} \begin{small} {\scriptsize -For the case of a general single-particle basis $\left\{u_{\alpha}(\mathbf{r})\right\}$ one uses the following creation and annihilation operators + +For the case of a general single-particle basis +$\left\{u_{\alpha}(\mathbf{r})\right\}$ one uses the following +creation and annihilation operators $$ \begin{aligned} -& c_{\alpha}^{\dagger}(t)=e^{i H t / \hbar} c_{\alpha}^{\dagger} e^{-i H t / \hbar}, \\ -& c_{\alpha}(t)=e^{i H t / \hbar} c_{\alpha} e^{-i H t / \hbar} +& a_{\alpha}^{\dagger}(t)=e^{i H t / \hbar} a_{\alpha}^{\dagger} e^{-i H t / \hbar}, \\ +& a_{\alpha}(t)=e^{i H t / \hbar} a_{\alpha} e^{-i H t / \hbar} \end{aligned} $$ -which are related to $\psi_{s}^{\dagger}(\mathbf{r}, t)$ and $\psi_{s}(\mathbf{r}, t)$ through Eqs. (1.14) and (1.15). +which are related to $\psi_{s}^{\dagger}(\mathbf{r}, t)$ and +$\psi_{s}(\mathbf{r}, t)$ (see whiteboard notes). } \end{small} @@ -1317,16 +1338,17 @@ \frame { -\frametitle{Single-particle Green's functions} +\frametitle{Eigenstates} \begin{small} {\scriptsize -In most applications the Hamiltonian is split in a unperturbed part $H_{0}$ and a residual interaction + + In most applications the Hamiltonian is split in an unperturbed part $\hat{H}_{0}$ and a residual interaction $\hat{H}_I$ $$ -H=H_{0}+V . +\hat{H}=\hat{H}_{0}+\hat{H}_I . $$ -The N-body eigenstates of the full Hamiltonian are indicated with $\left|\Psi_{n}^{N}\right\rangle$, while $\left|\Phi_{n}^{N}\right\rangle$ are the unperturbed ones +The $N$-body eigenstates of the full Hamiltonian are $\left|\Psi_{n}^{N}\right\rangle$, while $\left|\Phi_{n}^{N}\right\rangle$ are the corresponding unperturbed ones $$ \begin{aligned} @@ -1343,22 +1365,22 @@ \frame { -\frametitle{Single-particle Green's functions} +\frametitle{Definitions} \begin{small} {\scriptsize -The two-points Green's function describies the propagation of one particle or one hole on top of the ground state $\left|\Psi_{0}^{N}\right\rangle$. This is defined by +The two-points Green's function describes the propagation of one particle or one hole on top of the ground state $\left|\Psi_{0}^{N}\right\rangle$. This is defined by $$ g_{s s^{\prime}}\left(\mathbf{r}, t ; \mathbf{r}^{\prime}, t^{\prime}\right)=-\frac{i}{\hbar}\left\langle\Psi_{0}^{N}\left|T\left[\psi_{s}(\mathbf{r}, t) \psi_{s^{\prime}}^{\dagger}\left(\mathbf{r}^{\prime}, t^{\prime}\right)\right]\right| \Psi_{0}^{N}\right\rangle $$ -where $T[\cdots]$ is the time ordering operator that imposes a change of sing for each exchange of two fermion operators +where $T[\cdots]$ is the time ordering operator that imposes a change of sign for each exchange of two fermion operators $$ T\left[\psi_{s}(\mathbf{r}, t) \psi_{s^{\prime}}^{\dagger}\left(\mathbf{r}^{\prime}, t^{\prime}\right)\right]= \begin{cases}\psi_{s}(\mathbf{r}, t) \psi_{s^{\prime}}^{\dagger}\left(\mathbf{r}^{\prime}, t^{\prime}\right), & t>t^{\prime} \\ \pm \psi_{s^{\prime}}^{\dagger}\left(\mathbf{r}^{\prime}, t^{\prime}\right) \psi_{s}(\mathbf{r}, t), & t^{\prime}>t\end{cases} $$ -where the upper (lower) sign is for bosons (fermions). A similar definition can be given for the non interacting state $\left|\Phi_{0}^{N}\right\rangle$, in this case the Heisenberg operators (2.4) to (2.7) must evolve only according to $H_{0}$ and the notation $g^{(0)}$ is used. +where the upper (lower) sign is for bosons (fermions). A similar definition can be given for the non interacting state $\left|\Phi_{0}^{N}\right\rangle$, in this case the Heisenberg operators discussed above must evolve only according to $H_{0}$ and the notation $g^{(0)}$ is used. } \end{small} } @@ -1366,10 +1388,12 @@ \frame { -\frametitle{Single-particle Green's functions} +\frametitle{Fourier transform} \begin{small} {\scriptsize -If the Hamiltonian does not depend on time, the propagator (2.11) depends only on the difference $t-t^{\prime}$ + +If the Hamiltonian does not depend on time, the propagator defined above +depends only on the difference $t-t^{\prime}$ $$ \begin{aligned} @@ -1391,16 +1415,17 @@ \frame { -\frametitle{Single-particle Green's functions} +\frametitle{Rewrite of propagator} \begin{small} {\scriptsize -By using the relation + +Using the relation $$ \theta( \pm \tau)=\mp \lim _{\eta \rightarrow 0^{+}} \frac{1}{2 \pi i} \int_{-\infty}^{+\infty} d \omega \frac{e^{-i \omega \tau}}{\omega \pm i \eta} $$ -one obtains +we obtain $$ \begin{aligned} @@ -1416,23 +1441,38 @@ \frame { -\frametitle{Single-particle Green's functions} +\frametitle{Particle and hole propagators} \begin{small} {\scriptsize -In Eq. (2.16), $g^{p}$ propagates a particle from $\mathbf{r}^{\prime}$ to $\mathbf{r}$, while $g^{h}$ propagates a hole from $\mathbf{r}$ to $\mathbf{r}^{\prime}$. Note that the interpretation is that a particle is added at $\mathbf{r}^{\prime}$, and later on some (indistiguishable) particle is removed from $\mathbf{r}^{\prime}$ (and similarly for holes). In the mean time, it is the fully correlated $(N \pm 1)$ body system that propagates. We will discuss in the next chapter that in many cases-and especially in the vicinity of the Fermi surface-this motion mantains many characteristics that are typical of a particle moving in free space, even if the motion itself could actually be a collective excitation of many constituents. But since it looks like a single particle state we may still refer to it as quasiparticle. + +In the last equation, $g^{p}$ propagates a particle from +$\mathbf{r}^{\prime}$ to $\mathbf{r}$, while $g^{h}$ propagates a hole +from $\mathbf{r}$ to $\mathbf{r}^{\prime}$. Note that the +interpretation is that a particle is added at $\mathbf{r}^{\prime}$, +and later on some (indistiguishable) particle is removed from +$\mathbf{r}^{\prime}$ (and similarly for holes). In the meantime, it +is the fully correlated $(N \pm 1)$ body system that propagates. + +In many cases, in particular in +the vicinity of the Fermi surface, this motion mantains many +characteristics that are typical of a particle moving in free space, +even if the motion itself could actually be a collective excitation of +many constituents. But since it looks like a single particle state we +may still refer to it as quasiparticle. } \end{small} } \frame { -\frametitle{Single-particle Green's functions} +\frametitle{Orthonormal basis set definitions} \begin{small} {\scriptsize + The same definitions can be made for any orthonormal basis $\{\alpha\}$, leading to the realtions $$ -g_{\alpha \beta}\left(t, t^{\prime}\right)=-\frac{i}{\hbar}\left\langle\Psi_{0}^{N}\left|T\left[c_{\alpha}(t) c_{\beta}^{\dagger}\left(t^{\prime}\right)\right]\right| \Psi_{0}^{N}\right\rangle +g_{\alpha \beta}\left(t, t^{\prime}\right)=-\frac{i}{\hbar}\left\langle\Psi_{0}^{N}\left|T\left[a_{\alpha}(t) a_{\beta}^{\dagger}\left(t^{\prime}\right)\right]\right| \Psi_{0}^{N}\right\rangle $$ where @@ -1445,8 +1485,8 @@ $$ \begin{aligned} -g_{\alpha \beta}(\omega)= & \left\langle\Psi_{0}^{N}\left|c_{\alpha} \frac{1}{\hbar \omega-\left(H-E_{0}^{N}\right)+i \eta} c_{\beta}^{\dagger}\right| \Psi_{0}^{N}\right\rangle \\ -& \mp\left\langle\Psi_{0}^{N}\left|c_{\beta}^{\dagger} \frac{1}{\hbar \omega+\left(H-E_{0}^{N}\right)-i \eta} c_{\alpha}\right| \Psi_{0}^{N}\right\rangle . +g_{\alpha \beta}(\omega)= & \left\langle\Psi_{0}^{N}\left|a_{\alpha} \frac{1}{\hbar \omega-\left(H-E_{0}^{N}\right)+i \eta} a_{\beta}^{\dagger}\right| \Psi_{0}^{N}\right\rangle \\ +& \mp\left\langle\Psi_{0}^{N}\left|a_{\beta}^{\dagger} \frac{1}{\hbar \omega+\left(H-E_{0}^{N}\right)-i \eta} a_{\alpha}\right| \Psi_{0}^{N}\right\rangle . \end{aligned} $$ @@ -1455,43 +1495,29 @@ } -\frame -{ -\frametitle{Single-particle Green's functions} -\begin{small} -{\scriptsize -Equations (2.17) and (2.19) are completely equivalent to the previous -ones. These may look a bit more abstract than the corresponding -Eqs. (2.11) and (2.16) but are more general since they show that the -formalism can be developed and applied in any orthonormal basis, -without restricting oneself to coordindate space. - -} -\end{small} -} - \frame { -\frametitle{Single-particle Green's functions} +\frametitle{Lehmann representation} \begin{small} {\scriptsize -As discussed in Sec. 2.1 for the one particle case, the information + +As discussed above for the one particle case, the information contained in the propagators becomes more clear if one Fourier transforms the time variable and inserts a completness for the intermediate states. This is so because it makes the spectrum and the transition amplitudes to apper explicitely. Using the completeness -relations for the $(N \pm 1)$-body systems in Eq. (2.19), one has +relations for the $(N \pm 1)$-body systems in the last equation, one has $$ \begin{aligned} -g_{\alpha \beta}(\omega)= & \sum_{n} \frac{\left\langle\Psi_{0}^{N}\left|c_{\alpha}\right| \Psi_{n}^{N+1}\right\rangle\left\langle\Psi_{n}^{N+1}\left|c_{\beta}^{\dagger}\right| \Psi_{0}^{N}\right\rangle}{\hbar \omega-\left(E_{n}^{N+1}-E_{0}^{N}\right)+i \eta} \\ -& \mp \sum_{k} \frac{\left\langle\Psi_{0}^{N}\left|c_{\beta}^{\dagger}\right| \Psi_{k}^{N-1}\right\rangle\left\langle\Psi_{k}^{N-1}\left|c_{\alpha}\right| \Psi_{0}^{N}\right\rangle}{\hbar \omega-\left(E_{0}^{N}-E_{k}^{N-1}\right)-i \eta} . +g_{\alpha \beta}(\omega)= & \sum_{n} \frac{\left\langle\Psi_{0}^{N}\left|a_{\alpha}\right| \Psi_{n}^{N+1}\right\rangle\left\langle\Psi_{n}^{N+1}\left|a_{\beta}^{\dagger}\right| \Psi_{0}^{N}\right\rangle}{\hbar \omega-\left(E_{n}^{N+1}-E_{0}^{N}\right)+i \eta} \\ +& \mp \sum_{k} \frac{\left\langle\Psi_{0}^{N}\left|a_{\beta}^{\dagger}\right| \Psi_{k}^{N-1}\right\rangle\left\langle\Psi_{k}^{N-1}\left|a_{\alpha}\right| \Psi_{0}^{N}\right\rangle}{\hbar \omega-\left(E_{0}^{N}-E_{k}^{N-1}\right)-i \eta} . \end{aligned} $$ which is known as the Lehmann repressentation of a many-body Green's -function ${ }^{1}$. Here, the first and second terms on the left hand +function. Here, the first and second terms on the left hand side describe the propagation of a (quasi)particle and a (quasi)hole excitations. @@ -1503,25 +1529,24 @@ \frametitle{Single-particle Green's functions} \begin{small} {\scriptsize -The poles in Eq. (2.20) are the energies relatives to the +The poles in last equation are the energies relatives to the $\left|\Psi_{0}^{N}\right\rangle$ ground state. Hence they give the energies actually relased in a capture reaction experiment to a bound state of $\left|\Psi_{n}^{N+1}\right\rangle$. The residues are transition amplitudes for the addition of a particle and take the name of spectroscopic amplitudes. They play the same role of the -$\left\langle\phi_{n} \mid \mathbf{r}\right\rangle$ wave function in -Eq. (2.3). In fact these energies and amplitudes are solutions of a -Schrödinger-like equation: the Dyson equation. The hole part of the +$\left\langle\phi_{n} \mid \mathbf{r}\right\rangle$ wave function. +In fact these energies and amplitudes are solutions of a +Schr\"odinger-like equation: the Dyson equation. The hole part of the propagator gives instead information on the process of particle emission, the poles being the exact energy absorbed in the -process. For example, in the single particle Green's function of a -molecule, the quasiparticle and quasihole poles are respectively the +process. For example, in the single particle Green's function of an atome, +the quasiparticle and quasihole poles are respectively the electron affinities an ionization energies. We will look at the physical significance of spectroscopic amplitudes below and derive the Dyson equation (which is the -fundamental equation in many-body Green's function theory) only later -on. +fundamental equation in many-body Green's function theory) below. } \end{small} @@ -1530,31 +1555,30 @@ \frame { -\frametitle{Single-particle Green's functions} +\frametitle{Spectral function} \begin{small} {\scriptsize -As a last definition, we rewrite the contents of Eq. (2.20) in a form +As a last definition, we rewrite the above Lehmann representation in a form that can compared more easily to experiments. By using the relation $$ \frac{1}{x \pm i \eta}=\mathcal{P} \frac{1}{x} \mp i \pi \delta(x) $$ -\footnotetext{${ }^{1}$ H. Lehmann, Nuovo Cimento 11, 324 (1954). } -it is immediate to extract the one-body spectral function +we can extract the one-body spectral function $$ S_{\alpha \beta}(\omega)=S_{\alpha \beta}^{p}(\omega)+S_{\alpha \beta}^{h}(\omega), $$ -where the partcle and hole components are +where the particle and hole components are $$ \begin{aligned} S_{\alpha \beta}^{p}(\omega) & =-\frac{1}{\pi} \operatorname{Im} g_{\alpha \beta}^{p}(\omega) \\ -& =\sum_{n}\left\langle\Psi_{0}^{N}\left|c_{\alpha}\right| \Psi_{n}^{N+1}\right\rangle\left\langle\Psi_{n}^{N+1}\left|c_{\beta}^{\dagger}\right| \Psi_{0}^{N}\right\rangle \delta\left(\hbar \omega-\left(E_{n}^{N+1}-E_{0}^{N}\right)\right) \\ +& =\sum_{n}\left\langle\Psi_{0}^{N}\left|a_{\alpha}\right| \Psi_{n}^{N+1}\right\rangle\left\langle\Psi_{n}^{N+1}\left|a_{\beta}^{\dagger}\right| \Psi_{0}^{N}\right\rangle \delta\left(\hbar \omega-\left(E_{n}^{N+1}-E_{0}^{N}\right)\right) \\ S_{\alpha \beta}^{h}(\omega) & =\frac{1}{\pi} \operatorname{Im} g_{\alpha \beta}^{h}(\omega) \\ -& =\mp \sum_{k}\left\langle\Psi_{0}^{N}\left|c_{\beta}^{\dagger}\right| \Psi_{k}^{N-1}\right\rangle\left\langle\Psi_{k}^{N-1}\left|c_{\alpha}\right| \Psi_{0}^{N}\right\rangle \delta\left(\hbar \omega-\left(E_{0}^{N}-E_{k}^{N-1}\right)\right) . +& =\mp \sum_{k}\left\langle\Psi_{0}^{N}\left|a_{\beta}^{\dagger}\right| \Psi_{k}^{N-1}\right\rangle\left\langle\Psi_{k}^{N-1}\left|a_{\alpha}\right| \Psi_{0}^{N}\right\rangle \delta\left(\hbar \omega-\left(E_{0}^{N}-E_{k}^{N-1}\right)\right) . \end{aligned} $$ @@ -1565,7 +1589,7 @@ \frame { -\frametitle{Single-particle Green's functions} +\frametitle{Dispersion relation} \begin{small} {\scriptsize The diagonal part of the spectral function is interpreted as the @@ -1574,30 +1598,41 @@ the state $\alpha$ leaving the residual system in a state of energy $\omega$. -By comparing Eqs. (2.23) and (2.24) to the Lehmann representation -(2.20), it is seen that the propagator is completely constrained by -its imaginary part. Indeed, +By comparing the last two equations to the Lehmann representation, we +see that the propagator is completely constrained by its imaginary +part. We have $$ g_{\alpha \beta}(\omega)=\int d \omega^{\prime} \frac{S_{\alpha \beta}^{p}\left(\omega^{\prime}\right)}{\omega-\omega^{\prime}+i \eta}+\int d \omega^{\prime} \frac{S_{\alpha \beta}^{h}\left(\omega^{\prime}\right)}{\omega-\omega^{\prime}-i \eta} . $$ -In general the single particle propagator of a finite system has isolated poles in correspondence to the bound eigenstates of the $(N+1)$-body system. For larger enegies, where $\left|\Psi_{n}^{N+1}\right\rangle$ are states in the continuum, it develops a branch cut. The particle propagator $g^{p}(\omega)$ is analytic in the upper half of the complex plane, and so is the full propagator (2.16) for $\omega \geq E_{0}^{N+1}-E_{0}^{N}$. Analogously, the hole propagator has poles for $\omega \leq E_{0}^{N}-E_{0}^{N-1}$ and is analytic in the lower complex plane. Note that high excitation energies in the (N-1)body system correspond to negative values of the poles $E_{0}^{N}-E_{k}^{N-1}$, so $g^{h}(\omega)$ develops a branch cut for large negative energies. +In general the single particle propagator of a finite system has +isolated poles in correspondence to the bound eigenstates of the +$(N+1)$-body system. For larger enegies, where +$\left|\Psi_{n}^{N+1}\right\rangle$ are states in the continuum, it +develops a branch cut. The particle propagator $g^{p}(\omega)$ is +analytic in the upper half of the complex plane, and so is the full +propagator for $\omega \geq +E_{0}^{N+1}-E_{0}^{N}$. Analogously, the hole propagator has poles for +$\omega \leq E_{0}^{N}-E_{0}^{N-1}$ and is analytic in the lower +complex plane. Note that high excitation energies in the (N-1)body +system correspond to negative values of the poles +$E_{0}^{N}-E_{k}^{N-1}$, so $g^{h}(\omega)$ develops a branch cut for +large negative energies. } \end{small} } \frame { -\frametitle{Single-particle Green's functions} +\frametitle{Expectation values} \begin{small} {\scriptsize -The one-body density matrix (1.40) can be obtained from the one-body -propagator. One simply chooses the appropriate time ordering in -Eq. (2.17) +The one-body density matrix can be obtained from the one-body +propagator. One simply chooses the appropriate time ordering $$ -\rho_{\alpha \beta}=\left\langle\Psi_{0}^{N}\left|c_{\beta}^{\dagger} c_{\alpha}\right| \Psi_{0}^{N}\right\rangle= \pm i \hbar \lim _{t^{\prime} \rightarrow t^{+}} g_{\alpha \beta}\left(t, t^{\prime}\right) +\rho_{\alpha \beta}=\left\langle\Psi_{0}^{N}\left|a_{\beta}^{\dagger} a_{\alpha}\right| \Psi_{0}^{N}\right\rangle= \pm i \hbar \lim _{t^{\prime} \rightarrow t^{+}} g_{\alpha \beta}\left(t, t^{\prime}\right) $$ (where the upper sign is for bosons and the lower one is for fermions). Alternatively, the hole spectral function can be used @@ -1616,9 +1651,8 @@ \frametitle{Single-particle Green's functions} \begin{small} {\scriptsize -Thus, the expectation value of a one-body operator, Eq. (1.41), on the -ground states $\left|\Psi_{0}^{N}\right\rangle$ is usually written in -one the following ways +Thus, the expectation value of a one-body operator for the +ground states $\left|\Psi_{0}^{N}\right\rangle$ is usually as $$ \begin{aligned} @@ -1627,7 +1661,7 @@ \end{aligned} $$ -which are equivalent. +and both terms are equivalent. } \end{small} @@ -1636,19 +1670,19 @@ \frame { -\frametitle{Single-particle Green's functions} +\frametitle{More on sum rules} \begin{small} {\scriptsize From the particle spectral function, one can extract the quantity $$ -d_{\alpha \beta}=\left\langle\Psi_{0}^{N}\left|c_{\alpha} c_{\beta}^{\dagger}\right| \Psi_{0}^{N}\right\rangle=\int d \omega S_{\beta \alpha}^{p}(\omega) +d_{\alpha \beta}=\left\langle\Psi_{0}^{N}\left|a_{\alpha} a_{\beta}^{\dagger}\right| \Psi_{0}^{N}\right\rangle=\int d \omega S_{\beta \alpha}^{p}(\omega) $$ which leads to the following sum rule $$ -\int d \omega S_{\alpha \beta}(\omega)=d_{\alpha \beta} \mp \rho_{\alpha \beta}=\left\langle\Psi_{0}^{N}\left|\left[c_{\alpha}, c_{\beta}^{\dagger}\right]_{\mp}\right| \Psi_{0}^{N}\right\rangle=\delta_{\alpha \beta} . +\int d \omega S_{\alpha \beta}(\omega)=d_{\alpha \beta} \mp \rho_{\alpha \beta}=\left\langle\Psi_{0}^{N}\left|\left[a_{\alpha}, a_{\beta}^{\dagger}\right]_{\mp}\right| \Psi_{0}^{N}\right\rangle=\delta_{\alpha \beta} . $$ } @@ -1656,28 +1690,31 @@ } \frame { -\frametitle{Single-particle Green's functions} +\frametitle{Sum rule for the energy} \begin{small} {\scriptsize -For the case of an Hamiltonian containing only two-body interactions, +For the case of an Hamiltonian containing up to two-body interactions only we have $$ \begin{aligned} H & =U+V \\ -& =\sum_{\alpha \beta} t_{\alpha \beta} c_{\alpha}^{\dagger} c_{\beta}+\frac{1}{4} \sum_{\alpha \beta \gamma \delta} v_{\alpha \beta, \gamma \delta} c_{\alpha}^{\dagger} c_{\beta}^{\dagger} c_{\delta} c_{\gamma}, +& =\sum_{\alpha \beta} t_{\alpha \beta} a_{\alpha}^{\dagger} a_{\beta}+\frac{1}{4} \sum_{\alpha \beta \gamma \delta} v_{\alpha \beta, \gamma \delta} a_{\alpha}^{\dagger} a_{\beta}^{\dagger} a_{\delta} a_{\gamma}, \end{aligned} $$ -there exist an important sum rule that relates the total energy of the state $\left|\Psi_{0}^{N}\right\rangle$ to its one-body Green's function. To derive this, one makes use of the equation of motion for Heisenberg operators (1.51), which gives +and we can derive an important sum rule that relates the total energy +of the state $\left|\Psi_{0}^{N}\right\rangle$ to its one-body Green's +function. To derive this, one makes use of the equation of motion for +Heisenberg operators, which gives $$ -i \hbar \frac{d}{d t} c_{\alpha}(t)=e^{i H t / \hbar}\left[c_{\alpha}, H\right] e^{-i H t / \hbar}, +i \hbar \frac{d}{d t} a_{\alpha}(t)=e^{i H t / \hbar}\left[a_{\alpha}, H\right] e^{-i H t / \hbar}, $$ with $^{2}$ $$ -\left[c_{\alpha}, H\right]=\sum_{\beta} t_{\alpha \beta} c_{\beta}+\frac{1}{2} \sum_{\beta \gamma \delta} v_{\alpha \beta \gamma \delta} c_{\beta}^{\dagger} c_{\delta} c_{\gamma} +\left[a_{\alpha}, H\right]=\sum_{\beta} t_{\alpha \beta} a_{\beta}+\frac{1}{2} \sum_{\beta \gamma \delta} v_{\alpha \beta \gamma \delta} a_{\beta}^{\dagger} a_{\delta} a_{\gamma} $$ which is valid for both fermions and bosons. @@ -1689,15 +1726,15 @@ \frame { -\frametitle{Single-particle Green's functions} +\frametitle{Green's function equation}} \begin{small} {\scriptsize -If one uses Eq. (2.33) and derives the propagator (2.17) with respect to time, +If one uses the last equation we can derive the propagator as function of time through $$ \begin{aligned} i \hbar \frac{\partial}{\partial t} g_{\alpha \beta}\left(t-t^{\prime}\right)= & \delta\left(t-t^{\prime}\right) \delta_{\alpha \beta}+\sum_{\gamma} t_{\alpha \gamma} g_{\gamma \beta}\left(t-t^{\prime}\right) \\ -& -\frac{i}{\hbar} \sum_{\eta \gamma \zeta} \frac{1}{2} v_{\alpha \eta, \gamma \zeta}\left\langle\Psi_{0}^{N}\left|T\left[c_{\eta}^{\dagger}(t) c_{\zeta}(t) c_{\gamma}(t) c_{\beta}^{\dagger}\left(t^{\prime}\right)\right]\right| \Psi_{0}^{N}\right\rangle +& -\frac{i}{\hbar} \sum_{\eta \gamma \zeta} \frac{1}{2} v_{\alpha \eta, \gamma \zeta}\left\langle\Psi_{0}^{N}\left|T\left[a_{\eta}^{\dagger}(t) a_{\zeta}(t) a_{\gamma}(t) a_{\beta}^{\dagger}\left(t^{\prime}\right)\right]\right| \Psi_{0}^{N}\right\rangle \end{aligned} $$ @@ -1708,18 +1745,33 @@ \frame { -\frametitle{Single-particle Green's functions} +\frametitle{More on energy sum rule} \begin{small} {\scriptsize -The braket in the last line contains the four points Green's function [see Eq. (2.38)], which can describe the simultaneous propagation of two particles. Thus, one sees that applying the equation of motion to a propagator leads to relations which contain Green's functions of higher order. This result is particularly important because it shows there exist a hierarchy between propagators, so that the exact equations that determine the one-body function will depend on the two-body one, the two-body function will contain contributions from three-body propagators, and so on. -For the moment we just want to select a particular order of the operators in Eq. (2.34) in order to extract the one- and two-body density matrices. To do this, we chose $t^{\prime}$ to be a later time than $t$ and take its limit to the latter from above. This yields +The braket in the last line contains the four-point Green's function to be discussed below. +The four-point Green's function can describe the simultaneous propagation of +two particles. Thus, one sees that applying the equation of motion to +a propagator leads to relations which contain Green's functions of +higher order. This result is particularly important because it shows +there exist a hierarchy between propagators, so that the exact +equations that determine the one-body function will depend on the +two-body one, the two-body function will contain contributions from +three-body propagators, and so on. + +For the moment we just want to select a particular order of the +operators in order to extract the one- and two-body +density matrices. To do this, we chose $t^{\prime}$ to be a later time +than $t$ and take its limit to the latter from above. This yields $$ \pm i \hbar \lim _{t^{\prime} \rightarrow t^{+}} \sum_{\alpha} \frac{\partial}{\partial t} g_{\alpha \alpha}\left(t-t^{\prime}\right)=\langle T\rangle+2\langle V\rangle $$ -(note that for $t \neq t^{\prime}$, the term $\delta\left(t-t^{\prime}\right)=0$ and it does not contribute to the limit). This result can also be expressed in energy representation by inverting the Fourier transformation (2.14), which gives +(note that for $t \neq t^{\prime}$, the term +$\delta\left(t-t^{\prime}\right)=0$ and it does not contribute to the +limit). This result can also be expressed in energy representation by +inverting the Fourier transformation. We have then $$ \lim _{\tau \rightarrow 0^{-}} \frac{\partial}{\partial \tau} g_{\alpha \beta}(\tau)=-\int d \omega \omega S_{\alpha \beta}^{h}(\omega) @@ -1730,10 +1782,10 @@ } \frame { -\frametitle{Single-particle Green's functions} +\frametitle{Expectation values of the energy} \begin{small} {\scriptsize -By combining (2.35) with Eq. (2.28) one finally obtains +By combining the above results we arrive at $$ \begin{aligned} @@ -1741,36 +1793,45 @@ & =\mp \frac{1}{2} \sum_{\alpha \beta} \int d \omega\left\{\delta_{\alpha \beta} \omega+t_{\alpha \beta}\right\} S_{\beta \alpha}^{h}(\omega) \end{aligned} $$ -We use the relation $[A, B C]_{-}=[A, B] C-B[C, A]=\{A, B\} C-B\{C, A\}$ which is valid for both commutators and anticommutators +We used here the relation $[A, B C]_{-}=[A, B] C-B[C, A]=\{A, B\} +C-B\{C, A\}$ which is valid for both commutators and anticommutators + + +Surprisingly, for an Hamiltonian containing only two-body forces it is +possible to extract the ground state energy by knowing only the +one-body propagator. -Surprisingly, for an Hamiltonian containing only two-body forces it is possible to extract the ground state energy by knowing only the one-body propagator. This result was derived independently by Galitski and Migdal ${ }^{3}$ and by Kolutn ${ }^{4}$. When interactions among three or more particles are present, this relation has to be augmented to include additional terms. In these cases higher order Green's functions will appear explicitly. +When interactions among three or more particles are present, this +relation has to be augmented to include additional terms. In these +cases higher order Green's functions will appear explicitly. } \end{small} } \frame { -\frametitle{Single-particle Green's functions} +\frametitle{Higher order Green's functions} \begin{small} {\scriptsize -The definition (2.17) can be extended to Green's functions for the + +The definition of the one-body Green's function can be extended to Green's functions for the propagation of more than one particle. In general, for each additional particle it will be necessary to introduce one additional creation and one annihilation operator. Thus a $2 n$-points Green's function will propagate a maximum of $n$ quasiparticles. The explicit definition of -the 4-points propagator is +the four-point propagator is $$ -g_{\alpha \beta, \gamma \delta}^{4-p t}\left(t_{1}, t_{2} ; t_{1}^{\prime}, t_{2}^{\prime}\right)=-\frac{i}{\hbar}\left\langle\Psi_{0}^{N}\left|T\left[c_{\beta}\left(t_{2}\right) c_{\alpha}\left(t_{1}\right) c_{\gamma}^{\dagger}\left(t_{1}^{\prime}\right) c_{\delta}^{\dagger}\left(t_{2}^{\prime}\right)\right]\right| \Psi_{0}^{N}\right\rangle +g_{\alpha \beta, \gamma \delta}^{4-p t}\left(t_{1}, t_{2} ; t_{1}^{\prime}, t_{2}^{\prime}\right)=-\frac{i}{\hbar}\left\langle\Psi_{0}^{N}\left|T\left[a_{\beta}\left(t_{2}\right) a_{\alpha}\left(t_{1}\right) a_{\gamma}^{\dagger}\left(t_{1}^{\prime}\right) a_{\delta}^{\dagger}\left(t_{2}^{\prime}\right)\right]\right| \Psi_{0}^{N}\right\rangle $$ -while the 6 -point case is +while the six-point case is $$ \begin{aligned} & g_{\alpha \beta \gamma, \mu \nu \lambda}^{6-p t}\left(t_{1}, t_{2}, t_{3} ; t_{1}^{\prime}, t_{2}^{\prime}, t_{3}^{\prime}\right)= \\ -& \quad-\frac{i}{\hbar}\left\langle\Psi_{0}^{N}\left|T\left[c_{\gamma}\left(t_{3}\right) c_{\beta}\left(t_{2}\right) c_{\alpha}\left(t_{1}\right) c_{\mu}^{\dagger}\left(t_{1}^{\prime}\right) c_{\nu}^{\dagger}\left(t_{2}^{\prime}\right) c_{\lambda}^{\dagger}\left(t_{3}^{\prime}\right)\right]\right| \Psi_{0}^{N}\right\rangle, +& \quad-\frac{i}{\hbar}\left\langle\Psi_{0}^{N}\left|T\left[a_{\gamma}\left(t_{3}\right) a_{\beta}\left(t_{2}\right) a_{\alpha}\left(t_{1}\right) a_{\mu}^{\dagger}\left(t_{1}^{\prime}\right) a_{\nu}^{\dagger}\left(t_{2}^{\prime}\right) a_{\lambda}^{\dagger}\left(t_{3}^{\prime}\right)\right]\right| \Psi_{0}^{N}\right\rangle, \end{aligned} $$ @@ -1779,33 +1840,33 @@ } \frame { -\frametitle{Single-particle Green's functions} +\frametitle{Interpretations} \begin{small} {\scriptsize It should be noted that the actual number of particles that are propagated by these objects depends on the ordering of the time variables. Therefore the information on transitions between eigenstates of the systems with $N, N \pm 1$ and $N \pm 2$ bodies are -all encoded in Eq. (2.38), while additional states of $N \pm 3$-body -states are included in Eq. (2.38). Obviously, the presence of so many +all encoded in the four-point propagator, while additional states of $N \pm 3$-body +states are included in the six-point propagator. Obviously, the presence of so many time variables makes the use of these functions extremely difficult (and even impossible, in many cases). However, it is still useful to consider only certain time orderings which allow to extract the -information not included in the 2-point propagator. +information not included in the two-point propagator. } \end{small} } \frame { -\frametitle{Single-particle Green's functions} +\frametitle{Two-particle-two-hole propagator} \begin{small} {\scriptsize The two-particle-two-hole propagator is a two-times Green's function defined as $$ -g_{\alpha \beta, \gamma \delta}^{I I}\left(t, t^{\prime}\right)=-\frac{i}{\hbar}\left\langle\Psi_{0}^{N}\left|T\left[c_{\beta}(t) c_{\alpha}(t) c_{\gamma}^{\dagger}\left(t^{\prime}\right) c_{\delta}^{\dagger}\left(t^{\prime}\right)\right]\right| \Psi_{0}^{N}\right\rangle +g_{\alpha \beta, \gamma \delta}^{I I}\left(t, t^{\prime}\right)=-\frac{i}{\hbar}\left\langle\Psi_{0}^{N}\left|T\left[a_{\beta}(t) a_{\alpha}(t) a_{\gamma}^{\dagger}\left(t^{\prime}\right) a_{\delta}^{\dagger}\left(t^{\prime}\right)\right]\right| \Psi_{0}^{N}\right\rangle $$ which corresponds to the limit $t_{1}^{\prime}=t_{2}^{\prime+}$ and $t_{2}=t_{1}^{+}$of $g^{4-p t}$. @@ -1815,18 +1876,18 @@ } \frame { -\frametitle{Single-particle Green's functions} +\frametitle{More details} \begin{small} {\scriptsize As for the case of $g_{\alpha \beta}\left(t, t^{\prime}\right)$, if -the Hamiltonian is time-independent, Eq. (2.40) is a function of the +the Hamiltonian is time-independent, the last equation is a function of the time difference only. Therefore it has a Lehmann representation containing the exact spectrum of the $(N \pm 2)$-body systems $$ \begin{aligned} -g_{\alpha \beta, \gamma \delta}^{I I}(\omega) & =\sum_{n} \frac{\left\langle\Psi_{0}^{N}\left|c_{\beta} c_{\alpha}\right| \Psi_{n}^{N+2}\right\rangle\left\langle\Psi^{N+2}\left|c_{\gamma}^{\dagger} c_{\delta}^{\dagger}\right| \Psi_{0}^{N}\right\rangle}{\omega-\left(E_{n}^{N+2}-E_{0}^{N}\right)+i \eta} \\ -& -\sum_{k} \frac{\left\langle\Psi_{0}^{N}\left|c_{\gamma}^{\dagger} c_{\delta}^{\dagger}\right| \Psi_{k}^{N-2}\right\rangle\left\langle\Psi_{k}^{N-2}\left|c_{\beta} c_{\alpha}\right| \Psi_{0}^{N}\right\rangle}{\omega-\left(E_{0}^{N}-E_{k}^{N-2}\right)-i \eta} +g_{\alpha \beta, \gamma \delta}^{I I}(\omega) & =\sum_{n} \frac{\left\langle\Psi_{0}^{N}\left|a_{\beta} a_{\alpha}\right| \Psi_{n}^{N+2}\right\rangle\left\langle\Psi^{N+2}\left|a_{\gamma}^{\dagger} a_{\delta}^{\dagger}\right| \Psi_{0}^{N}\right\rangle}{\omega-\left(E_{n}^{N+2}-E_{0}^{N}\right)+i \eta} \\ +& -\sum_{k} \frac{\left\langle\Psi_{0}^{N}\left|a_{\gamma}^{\dagger} a_{\delta}^{\dagger}\right| \Psi_{k}^{N-2}\right\rangle\left\langle\Psi_{k}^{N-2}\left|a_{\beta} a_{\alpha}\right| \Psi_{0}^{N}\right\rangle}{\omega-\left(E_{0}^{N}-E_{k}^{N-2}\right)-i \eta} \end{aligned} $$ @@ -1835,7 +1896,7 @@ } \frame { -\frametitle{Single-particle Green's functions} +\frametitle{Spectral functions} \begin{small} {\scriptsize Similarly one defines the two-particle and two-hole spectral functions @@ -1849,9 +1910,9 @@ $$ \begin{aligned} S_{\alpha \beta, \gamma \delta}^{p p}(\omega) & =-\frac{1}{\pi} \operatorname{Im} g_{\alpha \beta, \gamma \delta}^{p p}(\omega) \\ -& =\sum_{n}\left\langle\Psi_{0}^{N}\left|c_{\beta} c_{\alpha}\right| \Psi_{n}^{N+2}\right\rangle\left\langle\Psi_{n}^{N+2}\left|c_{\gamma}^{\dagger} c_{\delta}^{\dagger}\right| \Psi_{0}^{N}\right\rangle \delta\left(\hbar \omega-\left(E_{n}^{N+2}-E_{0}^{N}\right)\right), \\ +& =\sum_{n}\left\langle\Psi_{0}^{N}\left|a_{\beta} a_{\alpha}\right| \Psi_{n}^{N+2}\right\rangle\left\langle\Psi_{n}^{N+2}\left|a_{\gamma}^{\dagger} a_{\delta}^{\dagger}\right| \Psi_{0}^{N}\right\rangle \delta\left(\hbar \omega-\left(E_{n}^{N+2}-E_{0}^{N}\right)\right), \\ S_{\alpha \beta, \gamma \delta}^{h h}(\omega) & =\frac{1}{\pi} \operatorname{Im} g_{\alpha \beta, \gamma \delta}^{h h}(\omega) \\ -& =-\sum_{k}\left\langle\Psi_{0}^{N}\left|c_{\gamma}^{\dagger} c_{\delta}^{\dagger}\right| \Psi_{k}^{N-2}\right\rangle\left\langle\Psi_{k}^{N-2}\left|c_{\beta} c_{\alpha}\right| \Psi_{0}^{N}\right\rangle \delta\left(\hbar \omega-\left(E_{0}^{N}-E_{k}^{N-2}\right)\right) . +& =-\sum_{k}\left\langle\Psi_{0}^{N}\left|a_{\gamma}^{\dagger} a_{\delta}^{\dagger}\right| \Psi_{k}^{N-2}\right\rangle\left\langle\Psi_{k}^{N-2}\left|a_{\beta} a_{\alpha}\right| \Psi_{0}^{N}\right\rangle \delta\left(\hbar \omega-\left(E_{0}^{N}-E_{k}^{N-2}\right)\right) . \end{aligned} $$ @@ -1860,13 +1921,13 @@ } \frame { -\frametitle{Single-particle Green's functions} +\frametitle{Two-body density functions} \begin{small} {\scriptsize -Following the demonstration of Sec. 2.3.1, it is immediate to obtain relations for the two-body density matrix (1.44) +Based on these equations it is easy to obtain relations for the two-body density matrix $$ -\Gamma_{\alpha \beta, \gamma \delta}=\left\langle\Psi^{N}\left|c_{\gamma}^{\dagger} c_{\delta}^{\dagger} c_{\beta} c_{\alpha}\right| \Psi^{N}\right\rangle=-\int d \omega S_{\alpha \beta, \gamma \delta}^{h h}(\omega) +\Gamma_{\alpha \beta, \gamma \delta}=\left\langle\Psi^{N}\left|a_{\gamma}^{\dagger} a_{\delta}^{\dagger} a_{\beta} a_{\alpha}\right| \Psi^{N}\right\rangle=-\int d \omega S_{\alpha \beta, \gamma \delta}^{h h}(\omega) $$ and, hence, for the expectation value of any two-body operator @@ -1883,7 +1944,7 @@ } \frame { -\frametitle{Single-particle Green's functions} +\frametitle{Polarization propagator} \begin{small} {\scriptsize The polarization propagator $\Pi_{\alpha \beta, \gamma \delta}$ @@ -1896,8 +1957,8 @@ $$ \begin{aligned} -\Pi_{\alpha \beta, \gamma \delta}\left(t, t^{\prime}\right)=- & \frac{i}{\hbar}\left\langle\Psi_{0}^{N}\left|T\left[c_{\beta}^{\dagger}(t) c_{\alpha}(t) c_{\gamma}^{\dagger}\left(t^{\prime}\right) c_{\delta}\left(t^{\prime}\right)\right]\right| \Psi_{0}^{N}\right\rangle \\ -& +\frac{i}{\hbar}\left\langle\Psi_{0}^{N}\left|c_{\beta}^{\dagger} c_{\alpha}\right| \Psi_{0}^{N}\right\rangle\left\langle\Psi_{0}^{N}\left|c_{\gamma}^{\dagger} c_{\delta}\right| \Psi_{0}^{N}\right\rangle +\Pi_{\alpha \beta, \gamma \delta}\left(t, t^{\prime}\right)=- & \frac{i}{\hbar}\left\langle\Psi_{0}^{N}\left|T\left[a_{\beta}^{\dagger}(t) a_{\alpha}(t) a_{\gamma}^{\dagger}\left(t^{\prime}\right) a_{\delta}\left(t^{\prime}\right)\right]\right| \Psi_{0}^{N}\right\rangle \\ +& +\frac{i}{\hbar}\left\langle\Psi_{0}^{N}\left|a_{\beta}^{\dagger} a_{\alpha}\right| \Psi_{0}^{N}\right\rangle\left\langle\Psi_{0}^{N}\left|a_{\gamma}^{\dagger} a_{\delta}\right| \Psi_{0}^{N}\right\rangle \end{aligned} $$ @@ -1906,19 +1967,19 @@ } \frame { -\frametitle{Single-particle Green's functions} +\frametitle{Including completeness} \begin{small} {\scriptsize After including a completeness of $\left|\Psi_{n}^{N}\right\rangle$ -states in (2.47), the contribution of to the ground states (at zero +states, the contribution of to the ground states (at zero energy) is cancelled by the last term in the equation. Thus one can Fourier transform to the Lehmann representation $$ \begin{aligned} -\Pi_{\alpha \beta, \gamma \delta}(\omega) & =\sum_{n \neq 0} \frac{\left\langle\Psi_{0}^{N}\left|c_{\beta}^{\dagger} c_{\alpha}\right| \Psi_{n}^{N}\right\rangle\left\langle\Psi_{n}^{N}\left|c_{\gamma}^{\dagger} c_{\delta}\right| \Psi_{0}^{N}\right\rangle}{\omega-\left(E_{n}^{N}-E_{0}^{N}\right)+i \eta} \\ -& -\sum_{n \neq 0} \frac{\left\langle\Psi_{0}^{N}\left|c_{\gamma}^{\dagger} c_{\delta}\right| \Psi_{n}^{N}\right\rangle\left\langle\Psi_{n}^{N}\left|c_{\beta}^{\dagger} c_{\alpha}\right| \Psi_{0}^{N}\right\rangle}{\omega+\left(E_{n}^{N}-E_{0}^{N}\right)-i \eta} +\Pi_{\alpha \beta, \gamma \delta}(\omega) & =\sum_{n \neq 0} \frac{\left\langle\Psi_{0}^{N}\left|a_{\beta}^{\dagger} a_{\alpha}\right| \Psi_{n}^{N}\right\rangle\left\langle\Psi_{n}^{N}\left|a_{\gamma}^{\dagger} a_{\delta}\right| \Psi_{0}^{N}\right\rangle}{\omega-\left(E_{n}^{N}-E_{0}^{N}\right)+i \eta} \\ +& -\sum_{n \neq 0} \frac{\left\langle\Psi_{0}^{N}\left|a_{\gamma}^{\dagger} a_{\delta}\right| \Psi_{n}^{N}\right\rangle\left\langle\Psi_{n}^{N}\left|a_{\beta}^{\dagger} a_{\alpha}\right| \Psi_{0}^{N}\right\rangle}{\omega+\left(E_{n}^{N}-E_{0}^{N}\right)-i \eta} \end{aligned} $$ @@ -1927,19 +1988,19 @@ } \frame { -\frametitle{Single-particle Green's functions} +\frametitle{Transition matrix elements} \begin{small} {\scriptsize Note that $\Pi_{\alpha \beta, \gamma \delta}(\omega)=\Pi_{\delta \gamma, \beta \alpha}(-\omega)$ due to time reversal symmetry. Also the forward and backward parts carry the same information. -Once again, the residues of the propagator (2.48) can be used to +Once again, the residues of the propagator can be used to calculate expectation values. In this case, given a one-body operator -(1.30) on obtains the transition matrix elements to any excited state +we obtain the transition matrix elements to any excited state $$ -\left\langle\Psi_{n}^{N}|O| \Psi_{0}^{N}\right\rangle=\sum_{\alpha \beta} o_{\beta \alpha}\left\langle\Psi_{n}^{N}\left|c_{\beta}^{\dagger} c_{\alpha}\right| \Psi_{0}^{N}\right\rangle +\left\langle\Psi_{n}^{N}|O| \Psi_{0}^{N}\right\rangle=\sum_{\alpha \beta} o_{\beta \alpha}\left\langle\Psi_{n}^{N}\left|a_{\beta}^{\dagger} a_{\alpha}\right| \Psi_{0}^{N}\right\rangle $$ } @@ -1947,13 +2008,20 @@ } \frame { -\frametitle{Single-particle Green's functions} +\frametitle{Coupling to experiment} \begin{small} {\scriptsize - Here we explore the connection between the information contained in various propagators and experimental data. The focus is on the experimental properties that are probed by the removal of particles. Also, from now on, we will only consider fermionic systems. -An important case is when the spectrum for the $N \pm 1$-particle system near the Fermi energy involves discrete bound states. This happens in finite system like nuclei or molecules. In these cases the main quantity of interest is the overlap wave function, which appears in the residues of Eq. (2.20) and in Eq. (2.24). This is +Here we explore the connection between the information contained in +various propagators and experimental data. The focus is on the +experimental properties that are probed by the removal of +particles. Also, from now on, we will only consider fermionic systems. + +An important case is when the spectrum for the $N \pm 1$-particle +system near the Fermi energy involves discrete bound states. This +happens in finite system like nuclei or molecules. In these cases the +main quantity of interest is the overlap wave function $$ \begin{aligned} @@ -1968,12 +2036,12 @@ } \frame { -\frametitle{Single-particle Green's functions} +\frametitle{Particle knock-out processes} \begin{small} {\scriptsize -The second line in Eq. (3.1) can be proved by using relations (1.19) -and (1.20). This integral comes out in the description of most -particle knock out processes because it represents the matrix element + +This integral comes out in the description of most +particle knock-out processes because it represents the matrix element between the initial and final states, in the case when the emitted particle is ejected with energy large enough the it interacts only weakly with the residual system. The quantity of interest here is the @@ -1994,12 +2062,12 @@ } \frame { -\frametitle{Single-particle Green's functions} +\frametitle{Spectroscopic strength from particle emission} \begin{small} {\scriptsize In order to make the connection with experimental data obtained from knockout reactions, it is useful to consider the response of a system -to a weak probe. The hole spectral function introduced in Eq. (2.24) +to a weak probe. The hole spectral function can be substantially "observed" these reactions. The general idea is to transfer a large amount of momentum and energy to a particle of a bound system in the ground state. This is then ejected from the @@ -2008,7 +2076,7 @@ particle it is then possible the reconstruct the spectral function of the system, provided that the interaction between the ejected particle and the remainder is sufficiently weak or treated in a controlled -fashion, $e . g$. by constraining this treatment with information from +fashion, for example by constraining this treatment with information from other experimental data. We assume that the $N$-particle system is initially in its ground state, @@ -2030,7 +2098,7 @@ } \frame { -\frametitle{Single-particle Green's functions} +\frametitle{Particle emission} \begin{small} {\scriptsize For simplicity we consider the transition matrix elements for a scalar external probe @@ -2050,7 +2118,7 @@ } \frame { -\frametitle{Single-particle Green's functions} +\frametitle{Transition matrix element} \begin{small} {\scriptsize The transition matrix element now becomes @@ -2068,13 +2136,25 @@ } \frame { -\frametitle{Single-particle Green's functions} +\frametitle{Impulse approximation} \begin{small} {\scriptsize -The last line is obtained in the so-called Impulse Approximation (or Sudden Approximation), where it is assumed that the ejected particle is the one that -has absorbed the momentum from the external field. This is a very good approximation whenever the momentum $\boldsymbol{p}$ of the ejectile is much larger than typical momenta for the particles in the bound states; the neglected term in Eq. (3.7) is then very small, as it involves the removal of a particle with momentum $\boldsymbol{p}$ from $\left|\Psi_{0}^{N}\right\rangle$. + +The last line is obtained in the so-called Impulse Approximation (or +Sudden Approximation), where it is assumed that the ejected particle +is the one that has absorbed the momentum from the external +field. This is a very good approximation whenever the momentum +$\boldsymbol{p}$ of the ejectile is much larger than typical momenta +for the particles in the bound states; the neglected term +is then very small, as it involves the removal of a particle with +momentum $\boldsymbol{p}$ from $\left|\Psi_{0}^{N}\right\rangle$. -There is one other assumption in the derivation: the fact that the final eigenstate of the $N$-particle system was written in the form of Eq. (3.4), i.e. a plane-wave state for the ejectile on top of an $(N-1)$-particle eigenstate. This is again a good approximation if the ejectile momentum is large enough, as can be understood by rewriting the Hamiltonian in the $N$-particle system as +There is one other assumption in the derivation: the fact that the +final eigenstate of the $N$-particle system was written in the form of +a plane-wave state for the ejectile on top of an $(N-1)$-particle +eigenstate. This is again a good approximation if the ejectile +momentum is large enough, as can be understood by rewriting the +Hamiltonian in the $N$-particle system as $$ H_{N}=\sum_{i=1}^{N} \frac{\boldsymbol{p}_{i}^{2}}{2 m}+\sum_{i