Now you are in the subtree of Special Topics in Many-Body Theory, Spring 2016 project. 

Experimental significance of Green's function

We explain how Green's functions can be related to several experimental quantities. The most direct relation for the one-particle Green's function, our major subject, is to tunneling and photoemission, which essentially measure the electron or hole density of states (the spectral functions $A$ and $B$). Experimental photoemission curves show how the spectral function sharpens up as the excitation energy nears the Fermi surface. Photoemission and tunneling experiments measure the addition and subtraction spectrum ($A$ and $B$) quite directly: the basic idea is that if we do Fermi's golden rule on a perturbation Hamiltonian of the form $R c$ or $R c^\dagger$, where $R$ is some remaining part describing the photon, tunneling leads, etc., then the transition rate will be related to a density of final states multipled by a squared matrix element of $c$ or $c^\dagger$. This is exactly the form of $A$ and $B$.

Many calculations have been done of perturbative results in the interaction strength, even though the relevance of these to experiments is unclear. We will say a bit more about such calculations in a moment, but first some examples of results. The interaction strength is meaured by the scattering length, which in the leading Born approximation is just
$$\begin{equation} a = {m U_0 \over 4 \pi \hbar^2}, \quad U_0 = \int U(r) d^3 r. \end{equation}$$
The small parameter in these perturbative expansions is $p_F a$. Note that $a$ as defined above only makes sense for a short-ranged interaction. (Recall that perturbation theory for the {\bf unscreened} Coulomb interaction is actually better at high density, since the Fermi energy blows up more rapidly in this limit than the Coulomb interaction energy. Here we are assuming some short-ranged interaction with a well-defined scattering length $a$, so that the perturbation theory is justified in the dilute limit.) The Fermi momentum is of order $\hbar / l$, where $l$ is the typical particle spacing. Hence the expansion is essentially in $a/l$.

The ground state energy is (Huang and Yang, 1957)
$$\begin{equation} E_0 = {3 {p_F}^2 N \over 10 m} \left[ 1 + {10 \over 9 \pi} {p_F a \over \hbar} + {4 (11 - 2 \log 2) \over 21 \pi^2} \left( {p_F a \over \hbar}\right)^2\right]. \end{equation}$$
The chemical potential can be evaluated from this as $\mu = (\partial E_0 / \partial N)$ at fixed volume.

The effective mass of quasiparticles is (Abrikosov and Khalatnikov, 1957)
$$\begin{equation} {m^* \over m} = 1 + {8 \over 15 \pi^2} (7 \log 2 - 1) \left({a p_F \over \hbar}\right)^2. \end{equation}$$
The velocity of sound is
$$\begin{equation} u^2 = {{p_F}^2 \over 3 m^2} \left[1 + {2 \over \pi} {a p_F \over \hbar} + {8 (11 - 2 \log 2) \over 15 \pi^2} \left( {a p_F \over \hbar}\right)^2\right]. \end{equation}$$
These can be used to determine the Fermi liquid parameters $F_0$ and $F_1$ for this dilute gas.

The main theoretical use of the Green's function is as a way to organize perturbation theory in the interaction potential $V$. In a moment we will quickly explain how this is done, although for the reasons given above, perturbation theory to finite order is not that useful. The real use of diagrammatic perturbation theory is when infinite classes of diagrams are summed: for instance, BCS theory can be obtained by summing an infinite series of so-called "cooperon" diagrams, even though the total set of all diagrams in the problem cannot be solved. Similar infinite sums of subclasses lead to various approximations like RPA (random phase approximation), NCA (non-crossing approximation), etc.

One could also imagine writing $V = V_{FL} + V_{res}$, where the potential is divided into both one part that gives Fermil liquid corrections, and then a possible small perturbation around the Fermi liquid. This is in fact how the diagrammatic calculations of superconductivity should be understood: the weak retarded attraction that gives rise to superconductivity acts on Fermi liquid quasiparticles, rather than noninteracting electrons.

One very important current use of diagrammatic methods is for Fermi liquids or superconductors in a disordered potential. We will say a little bit about localization by disorder in a few lectures, in the context of the quantum Hall effect, but will not need to use diagrammatic methods for the simple case we discuss. There is now a rich literature on diagrammatic methods for noninteracting or weakly interacting electrons in a random potential; the reader is referred to the review article of Lee and Ramakrishnan, and the textbook of Efetov.

Some of the most important experimental quantities, like the conductivity tensor, are related not to the one-particle Green's function but to the two-particle Green's function. This can be important to distinguish between a system where the electrons are localizde and one where the electrons are extended, for example: the one-particle Green's function is not sharply different, but the two-particle Green's function does differ. There is an important class of relations in nonequilibrium statistical mechanics called "fluctuation-dissipation relations" which relate the linear response of a system to a small perturbation to the fluctuations in its thermodynamic equilibrium. The classical example of such a relation is the Einstein relation between diffusion and velocity correlations of a Brownian particle (cf. Physics 212). The same linear-response idea can be used to relate the {\it linear response} conductivity tensor to fluctuations in the equilibrium state. Truly nonequilibrium calculations (beyond linear response) can be done using a Keldysh or "time-loop" formalism, which we will probably not have time to discuss in this course.

The Green-Kubo or Kubo-Greenwood formula relates the conductivity tensor as a function of $\omega$ and $q$ to the current-current correlation: (with $e=1$)
$$\begin{equation} \sigma_{\alpha \beta}(\omega,q) = {1 \over V k T} \int_0^\infty\,dt\,e^{i \omega t} \langle j_\alpha(t,q) j_\beta(0,-q) \rangle. \end{equation}$$
Here $\alpha$ and $\beta$ are spatial directions and the current density operator $j_\alpha(t,q)$ is the Heisenberg-representation generalization of the Schrodinger current operator
$$\begin{equation} j_\alpha(q) = \sum_{k,\sigma} k_\alpha c^\dagger_{k-q/2,\sigma} c_{k+q/2,\sigma} \end{equation}$$
which reduces to the ordinary current operator for single-particle quantum mechanics. Note that at $q=0$ we are just counting the occupancy of different plane-wave states, weighted by their momentum.

Now we start the process of expressing the desired Heisenberg representation Green's function in terms of the interaction representation operators: for $t_1 > t_2$,
$$\begin{eqnarray} G_{\alpha \beta}(t_1,r_1,t_2,r_2) &=& -i \langle \Psi_\alpha(t_1,r_1) \Psi_\beta^\dagger(t_2,r_2) \rangle \cr &&= -i \langle S^{-1}(t_1,-\infty) \Psi_{0 \alpha}(t_1,r_1) S(t_1,-\infty) S^{-1}(t_2,-\infty) \Psi_{0\beta}^\dagger(t_2,r_2) S(t_2,-\infty) \rangle \end{eqnarray}$$
Here the unitary $S$ operator is defined as
$$\begin{equation} S(t_1,t_2) = T \exp(-i \int_{t_2}^{t_1} V_0(t)\,dt), \end{equation}$$
where $V_0(t)$ is the interaction representation of the interaction part $V$ of the Hamiltonian. The purpose of $S$ is to connect the interaction and Heisenberg representations (we justify this in the next lecture): for a general operator $\Psi$ in Heisenberg representation,
$$\begin{equation} \Psi = S^{-1}(t,-\infty) \Psi_0 S(t,\infty). \end{equation}$$

The above expression for the Green's function is compactly written
$$\begin{equation} G_{\alpha \beta}(t_1,r_1,t_2,r_2) = -i \langle S^{-1} T\{ \Psi_{0 \alpha} (t_1,r_1) \Psi^\dagger_{0 \beta}(t_2,r_2) S\} \rangle. \end{equation}$$
Here $S$ with no indices means $S(\infty,-\infty)$. Further simplification is possible if we recall that there will be no transitions out of the ground state under adiabatic changes in the Hamiltonian, assuming a nondegenerate ground state.