Many Body Interactions–Green’s Function Method

Abstract

Let us assume that there is a complete orthogonal set of single particle states $\phi _i(\xi )$, where $\xi = \mathbf r,\sigma$. By this we mean that

$$\begin{aligned} \langle \phi _i\mid \phi _j\rangle =\delta _{ij}{\quad }\text {and}{\quad } \sum _i \mid \phi _i\rangle \langle \phi _i\mid = \underline{\mathbf 1}. \end{aligned}$$

Appendices

Problems

12.1

Show explicitly that

$$ \begin{array}{ll} &{}\int _{t_0}^t dt_1 \int _{t_0}^{t_1} dt_2 \int _{t_0}^{t_{3}} dt_3 H_\mathrm{I}(t_1)H_\mathrm{I}(t_2) H_\mathrm{I}(t_3) \\ &{}=\frac{1}{3!}\int _{t_0}^t dt_1 \int _{t_0}^{t} dt_2 \int _{t_0}^{t} dt_3 \mathrm{T} \{H_\mathrm{I}(t_1)H_\mathrm{I}(t_2) H_\mathrm{I}(t_3)\}. \end{array} $$

12.2

The complete first order contributions to $G(x, x^\prime )$ are shown in the figure.

• (a) Write each term $\delta G^{(1)}(x, y)$ out in terms of noninteraction two-particle Green’s function $G^{(0)}(x, y)$ and the interaction $V(x_1-x_2) \equiv U(\mathbf r_1 - \mathbf r_2)\delta (t_1-t_2)$. Here $x=(\mathbf r, t)$ and one may omit the spin to simplify the notation.

• (b) Let us now restore the spin labels $(\alpha ,\beta )$ and introduce the Fourier transform $G_{\alpha \beta }(k)$ of the $G_{\alpha \beta }(x, y)$ as follows:

  $$ G_{\alpha \beta }(x, x') = \frac{1}{(2\pi )^4}\int d^4k\,\mathrm e^{ik\cdot (x-x')} G_{\alpha \beta }(k) $$
  $$ G_{\alpha \beta }^{(0)}(x, x') = \frac{1}{(2\pi )^4}\int d^4k\,\mathrm e^{ik\cdot (x-x')} G_{\alpha \beta }^{(0)}(k), $$

  where $k=(\mathbf k,\omega )$, $d^4k \equiv d^3k\, d\omega$, and $k\cdot x\equiv \mathbf k\cdot \mathbf x -\omega t$. In addition, for the interaction given by $V(x_1-x_2) \equiv U(\mathbf r_1 - \mathbf r_2)\delta (t_1-t_2)$ we can write

  $$ \begin{array}{ll} V(x, x^\prime )_{\alpha \alpha ^\prime ,\beta \beta ^\prime } &{}= \frac{1}{(2\pi )^4}\int d^4k\,\mathrm e^{ik\cdot (x-x^\prime )} V(k)_{\alpha \alpha ^\prime ,\beta \beta ^\prime } \\ &{}= \frac{1}{(2\pi )^3}\int d^3 k\,\mathrm e^{i\mathbf \cdot (\mathbf x-\mathbf x^\prime )} U(\mathbf k)_{\alpha \alpha ^\prime ,\beta \beta ^\prime }\delta (t-t^\prime ), \end{array} $$

  where $V(k)_{\alpha \alpha ^\prime ,\beta \beta ^\prime }=U(\mathbf k)_{\alpha \alpha ^\prime ,\beta \beta ^\prime } =\frac{1}{(2\pi )^3}\int d^3 x\,\mathrm e^{-i\mathbf k\cdot \mathbf x} U(\mathbf k)_{\alpha \alpha ^\prime ,\beta \beta ^\prime }$ is the spatial Fourier transform of the interparticle potential. Express each term obtained in part (a) in terms of $G_{\alpha \beta }^{(0)}(k)$ and V(k) in the momentum space.

12.3

By definition the noninteracting fermion Green’s function is given by

$$ G_{\alpha \beta }^{(0)}(\mathbf x t,\mathbf x^\prime t^\prime )= -i<\varPhi |\mathrm{T}\{\psi _{\mathrm I\alpha }(\mathbf x t) \psi _{\mathrm I\beta }^{\dag }(\mathbf x^\prime t^\prime )\}\varPhi >, $$

the noninteracting ground state vector is taken to be normalized. Show that

$$ G_{\alpha \beta }^{(0)}(\mathbf k, \omega )=\delta _{\alpha \beta }\left[ \frac{\theta (k-k_\mathrm{F})}{\omega -\hbar ^{-1}\varepsilon _k+i\eta } + \frac{\theta (k_\mathrm{F}-k)}{\omega -\hbar ^{-1}\varepsilon _k-i\eta }\right] . $$

12.4

Let us define the phonon field operator $\varPhi (x)$ by

$$ \varPhi (x)=\sum _{\mathbf q} \gamma (\mathbf q) \mathrm e^{i\mathbf q\cdot \mathbf x}(b_{\mathbf q} +b_{-\mathbf q}^\dag ), $$

where $\gamma (q)=i\frac{4\pi Z e^2}{q}\left( \frac{\hbar N}{2M\omega _q}\right) ^{1/2}$. Then we can define the phonon propagator by $P_0(2,1)= -i\langle \mathrm{GS}\mid T\{\varPhi _I(x_2,t_2)\varPhi _I(x_1,t_1)\}\mid \mathrm{GS}\rangle$ where $\varPhi _I(x_2,t_2)=\mathrm e^{-i H_0t_2}\varPhi (x_2)\mathrm e^{iH_0t_2}$.

• (a) Take the Fourier transform of $P_0(2,1)$ to obtain $P_0(q,\omega )$, the wave vector–frequency space representation of $P_0(2,1)$.

• (b) Show that $P_0(q,\omega )$ can be written as

  $$ P_0(q,\omega )=\frac{2 \varOmega _\mathrm{p}|\gamma (q)|^2}{\omega ^2-\varOmega _\mathrm{p}^2+i\eta }, $$

  where $\varOmega _\mathrm{p}=\left( \frac{4\pi Z^2 e^2 N}{M}\right) ^{1/2}$ is the bare plasma frequency of the ions.

12.5

Let us consider a Dyson equation given by

$$ W(q,\omega )=V(q,\omega )-V(q,\omega )\chi _0(q,\omega )W(q,\omega ), $$

where the polarization function for the electron–hole pair is given by

$$ \chi _0(q,\omega )=2i\hbar ^{-1}\int \frac{d^3k_1 d\omega _1}{(2\pi )^4} G_0(k_1,\omega _1)G_0(k_1+q,\omega _1+\omega ). $$

• (a) Show that $\chi _0(q,\omega )$ can be written as

  $$\begin{array}{ll} \chi _0(q,\omega )=-\hbar ^{-1}\frac{2}{(2\pi )^3} \int d^3k &{} \theta (|\mathbf k+\mathbf q|-k_\mathrm{F})\theta (k_\mathrm{F}-|\mathbf k|) \\ &{} \times \left[ \frac{1}{\omega -(\omega _{\mathbf k+\mathbf q}-\omega _{\mathbf k})+i\eta }-\frac{1}{\omega -(\omega _{\mathbf k+\mathbf q}-\omega _{\mathbf k})-i\eta }\right] , \end{array} $$

  where $\hbar \omega _{\mathbf q}=\varepsilon (\mathbf q)=\frac{\hbar ^2 q^2}{2m}$. Note that the single particle propagator $G_0(q,\omega )$ is written as

  $$ G_0(q,\omega )=\frac{1}{\omega -\omega _{\mathbf q, k_\mathrm{F}}(1-i\eta )}~\text {with}~\omega _{\mathbf q, k_\mathrm{F}}=\frac{\hbar ^2(q^2-k_\mathrm{F}^2)}{2m}\gtrless 0. $$

• (b) Show that the solution of the Dyson equation given above is simply $W=\frac{V}{1+V\chi _0}$.

• (c) Show that $1+V\chi _0$ is the same as the Lindhard dielectric function $\epsilon (q,\omega )$.

12.6

Let us consider a model solid containing electrons as well as longitudinal optical phonons.

• (a) Show that effective propagator $D(\mathbf q,\omega )$ is given by

  $$ D(\mathbf q,\omega )=\frac{4\pi e^2}{q^2}\frac{1}{\left[ \epsilon (q,\omega )-\varOmega _\mathrm{p}^2/\omega ^2\right] }, $$

  where $\epsilon (q,\omega )$ and $\varOmega _\mathrm{p}$ are the Lindhard dielectric function and the bare plasma frequency of the ions.

• (b) Demonstrate that the effective electron–phonon coupling constant $\gamma ^\mathrm{eff}(q)$ is given by

  $$ \mid \gamma ^\mathrm{eff}(q)\mid =\frac{V(q)}{\varepsilon (\mathbf q,\omega _q)}\frac{\hbar \omega _q}{2}, $$

  where $\omega _q=\varOmega _\mathrm{p}\frac{q}{k_s}$, the renormalized plasmon frequency of the lattice in the long wavelength regime.

Summary

In this chapter we study Green’s function method – a formal theory of many body interactions. Green’s function is defined in terms of a matrix element of time-ordered Heisenberg operators in the exact interacting ground state. We then introduce the interaction representation of the state functions of many particle states and write the Green’s function in terms of time-ordered products of interaction operators. Wick’s theorem is introduced to write the exact Green’s function as a perturbation expansion involving only pairings of field operators in the interaction representation. Dyson equations for Green’s function and the screened interaction are illustrated and Fermi liquid picture of quasiparticle interactions is also discussed.

The Hamiltonian H of a many particle system can be divided into two parts $H_0$ and $H^\prime$, where $H^\prime$ represents the interparticle interactions given, in second quantized form, by

$$ H^\prime = \frac{1}{2}\int d^3r_1 d^3 r_2 \psi ^\dag (\mathbf r_1) \psi ^\dag (\mathbf r_2)U(\mathbf r_1 - \mathbf r_2)\psi (\mathbf r_2)\psi (\mathbf r_1). $$

Particle density at a position $\mathbf r_0$ and the total particle number N are written, respectively, as $n(\mathbf r_0)=\psi ^\dag (\mathbf r_0)\psi (\mathbf r_0); N=\int d^3 r \psi ^\dag (\mathbf r)\psi (\mathbf r).$

The Schrödinger equation of the many particle wave function $\varPsi (1, 2, \ldots , N)$ is $i\hbar \frac{\partial }{\partial t}\varPsi = H\varPsi ,$ where $\hbar =\equiv 1$ and $\varPsi (t) = \mathrm e^{-iHt}\varPsi _\mathrm{H}$. Here $\varPsi _\mathrm{H}$ is time independent. The state vector $\varPsi _\mathrm{I}(t)$ and operator $F_\mathrm{I}(t)$ in the interaction representation are written as

$$ \varPsi _\mathrm{I}(t) = \mathrm e^{iH_0t}\varPsi _\mathrm{S} (t); F_\mathrm{I}(t)=\mathrm e^{iH_0t}F_\mathrm{S}\mathrm e^{-iH_0t}. $$

The equation of motion for $F_\mathrm{I}(t)$ is $\frac{\partial F_\mathrm{I}}{\partial t} = i \left[ H_0, F_\mathrm{I}(t)\right]$ and the solution for $F_\mathrm{I}(t)$ can be expressed as $\varPsi _\mathrm{I}(t)=S(t, t_0)\varPsi _\mathrm{I}(t_0),$ where $S(t, t_0)$ is the S matrix given by

$$ S(t, t_0)=\mathrm{T}\left\{ \mathrm e^{-i\int _{t_0}^t H_\mathrm{I}(t^\prime ) dt^\prime }\right\} . $$

The eigenstates of the interacting system in the Heisenberg, Schrödinger, and interaction representation are related by

$$ \varPsi _\mathrm{H} (t)=\mathrm e^{iHt}\varPsi _\mathrm{S}(t) {\quad } \text {and} {\quad } \varPsi _\mathrm{I} (t)=\mathrm e^{iH_0t}\varPsi _\mathrm{S}(t). $$

At time t $=$ 0, $\varPsi _\mathrm{I}(t=0)=\varPsi _\mathrm{H}(t=0)=\varPsi _\mathrm{H}.$ $\varPsi _\mathrm{H}$ is the state vector of the fully interacting system in the Heisenberg representation: $\varPsi _\mathrm{H}=S(0,-\infty )\varPhi _\mathrm{H}$

The Green’s function $G_{\alpha \beta }(x, x^\prime )$ is defined, in terms of $\psi _{\alpha }^\mathrm{H}$ and $\psi _{\beta }^\mathrm{H\dag }$, by

$$ G_{\alpha \beta }(x, x^\prime )= -i\frac{\langle \varPsi _\mathrm{H}|\mathrm{T}\{\psi _{\alpha }^\mathrm{H}(x) \psi _{\beta }^\mathrm{H\dag }(x^\prime )\}|\varPsi _\mathrm{H}\rangle }{\langle \varPsi _\mathrm{H}|\varPsi _\mathrm{H}\rangle }, $$

where $x=\{\mathbf r, t\}$ and $\alpha$, $\beta$ are spin indices.

In normal product of operators, all annihilation operators appear to the right of all creation operators: for example,

$$ \mathrm{N}\{\psi ^\dag (1)\psi (2)\}=\psi ^\dag (1)\psi (2){\quad }\text {while}{\quad } \mathrm{N}\{\psi (1)\psi ^\dag (2)\}=-\psi ^\dag (2)\psi (1). $$

Pairing or a contraction is the difference between a $\mathrm T$ product and an $\mathrm N$ product: $\mathrm{T}(AB)-\mathrm{N}(AB) = A^\mathrm{c} B^\mathrm{c}.$ The Wick’s theorem states that T product of operators $ABC\cdots$ can be expressed as the sum of all possible N products with all possible pairings.

Dyson equations for the interacting Green’s function G and the screened interaction W are written as $G=G^{(0)} + G^{(0)}\varSigma G; W=V+V\varPi W.$ Here $\varSigma$ and $\varPi$ denote the self energy and polarization part, and the simplest of which are given, respectively, by $\varSigma _0 = G^{(0)}W; \varPi _0 = G^{(0)}G^{(0)}.$ In the RPA, the G is replaced by $G^{(0)}$ and W is exactly equivalent to $\frac{V(q)}{\epsilon (q,\omega )}$, where $\epsilon (q,\omega )$ is the Lindhard dielectric function.

The Hamiltonian H of a system with the electron–phonon interaction is divided into three parts: $H=H_\mathrm{e} + H_\mathrm{N} +H_\mathrm{I},$ where

$$ H_\mathrm{e}= \sum _{k} \frac{\hbar ^2 k^2}{2m^*} c_k^\dag c_k, H_\mathrm{N} =\sum _{\alpha } \hbar \omega _\alpha \left( b_\alpha ^\dag b_\alpha + \frac{1}{2}\right) , $$

and $H_\mathrm{I}= \sum _{k,k', q} \frac{4\pi e^2}{\varOmega q^2} c_{k+q}^\dag c_{k'-q}^\dag c_{k'}c_k +\sum _{k,\alpha ,\mathbf G} \gamma (\alpha ,\mathbf G) (b_\alpha -b_{-\alpha }^\dag )c_{\mathbf k+\mathbf q+\mathbf G}^\dag c_{k}.$ Once we know $\varPsi (t_1)$ of the Schrödinger equation, $i\hbar \frac{\partial \varPsi }{\partial t}=H\varPsi ,$ we have

$$ \varPsi (x_2,t_2)=\int d^3 x_1 G_0(x_2,t_2;x_1,t_1)\varPsi (x_1,t_1). $$

For free electrons, $G_0(q,\omega )$ is the Fourier transform of $G_0(2,1)$:

$$ G_0(q,\omega )=\frac{1}{\omega -\varepsilon (q)(1-i\delta )}. $$

For a ‘model solid’ containing longitudinal phonons as well as electrons, two electrons can scatter via the virtual exchange of phonons and the total interaction, i.e. the sum of the Coulomb interaction and the interaction due to virtual exchange of phonons, is given, in terms of bare interaction $D_0$ and polarization $\chi _0$, by $D(\mathbf q,\omega )=\frac{D_0(\mathbf q,\omega )}{1+D_0(\mathbf q,\omega )\chi _0(\mathbf q,\omega )}.$

The Dyson equation for the Green’s function can be written

$$ G(k,\omega )=G^{(0)}(k,\omega )+G^{(0)}(k,\omega )\varSigma (k,\omega )G(k,\omega ). $$

The electron self energy is $\varSigma (k,\omega )=[G^{(0)}(k,\omega )]^{-1}-[G(k,\omega )]^{-1}$ and the energy of a quasiparticle is written as $E_p=\varepsilon _p +\varSigma (p,\omega )\mid _{\omega =E_p}.$ $\varSigma (\mathbf p, E_{\mathbf p})$ represents the interaction of a quasiparticle of momentum $\mathbf p$ with the ground state of the interacting electron gas. The energy of the state is written as

$$ E=E_0+\sum _{\mathbf p\sigma }\delta n_{\mathbf p\sigma } E_{\mathbf p\sigma }+\frac{1}{2}\sum _{\tiny \begin{array}{c} \mathbf p,\mathbf p^\prime \\ \sigma ,\sigma ^\prime \end{array}}f_{\sigma \sigma ^\prime } (\mathbf p,\mathbf p^\prime )\delta n_{n\sigma }\delta n_{\mathbf p^\prime \sigma ^\prime }. $$

The first term on the right is the ground state energy, the second is the quasiparticle energy $E_{\mathbf p\sigma }$ multiplied by the quasiparticle distribution function, and the third represents the interactions of the quasiparticles with one another.