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\begin{document}
\begin{center}
{\bf  Problems 2 Many Body Physics (Due on Thursday October  16 th)}
\end{center}

We are going to use the mathematica pogram which
we constructed last week to practice the evaluation
of several many body observables.

So take the one dimensional Hubbard model with 4
sites and periodic boundary conditions, at
half filling  (2 up particles
and 2 down particles).

Let us evaluate the following quantities for t=1,
and U=, 4 and 8
and plot them vs U.

a) kinetic energy per site
b) interaction energy per site
c) ${1 \over N_s} \sum_i  <S_i . S_{i+1}>$, average
spin exchange  energy.

Notice that observables are represented by operators,
and you are supposed to calculate their values by
taking expectation values in the GROUND STATE.

Can you explain qualitatively the trends you see  in
your results?

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1.  Consider the simple harmonic oscillator


\begin{displaymath}
\hat{H} =  {{p^2} \over {2m}} + {{ m \omega^2 x^2} \over {2}}
\end{displaymath}


Write it  in ``second quantized'' form , by expressing x and p in terms of creation
and annihilation operators.

\begin{displaymath}
\hat{H} = \hbar\omega(a^{\dagger} a + \frac{1}{2}).
\end{displaymath}

\bigskip

(i) Derive the equation of motion for $a$ and show that

\begin{displaymath}
a(t) = a(0)e^{- i \omega t}.
\end{displaymath}

Repeat this for $ a^{\dagger}$.

(ii) Using the relationship

\begin{displaymath}
a^{\dagger} = \frac{1}{\sqrt{2\hbar}} \left[ \frac{\hat{p}}{\sqrt{m \omega}} + i\hat{x} \sqrt{m \omega} \right].
\end{displaymath}

use the results of (i) to show that

%\begin{displaymath}
\begin{eqnarray*}\nonumber
\hat{p}(t) & =& \hat{p}(0) \cos \ \omega t - m\omega \hat{x}(o) sin \ \omega t \\
\hat{x}(t)&  =&  \hat{x}(0)\cos \ \omega t + \frac{\hat{p}(0)}{m \omega}sin \ \omega t.\\ \nonumber
\end{eqnarray*}
%\end{displaymath}

(iii) The above operator expressions appear identical to the classical equations of motion.

Why?

2.  Show that the Heisenberg spin operator S can be written in second quantized form as
\begin{eqnarray*}\nonumber
S_{x}& = &\phantom{+} \frac{1}{2}(c _{\uparrow}^{\dagger} c _{\downarrow} + c _{\downarrow}^{\dagger} c_{\uparrow}), \\
S_{y}& = &-\frac{i}{2}(c_{\uparrow}^{\dagger} c_{\downarrow} - c_{\downarrow}^{ \dagger} c_{\uparrow}), \\ \nonumber
S_{z}& = &\phantom{+} \frac{1}{2}(c_{\uparrow}^{\dagger} c_{\uparrow}   - c_{\downarrow}^{ \dagger} c_{\downarrow}). \\ \nonumber
\end{eqnarray*}

Check the commutation relations!

3.  Consider a system of fermions created by the field $\psi$ $^{\dag}$ $\vec{(r)}$ interacting under the Yukawa potential
\begin{displaymath}
V(r) = \frac{Ae^{-\lambda r}}{4 \pi r}.
\end{displaymath}

(i) Write the Hamiltonian in second quantized form, using the position basis.\\

(ii) Write the Hamiltonian
in  second quantized notation in the momentum basis, where
\begin{displaymath}
c_{\vec{k}} \ ^{\dagger} = \int d^{3}r \psi ^{\dagger}(\vec{r})e^{i \vec{k}. \vec{r}}.
\end{displaymath}


You will find it helpful to derive the Fourier representation
\begin{displaymath}
V(r) = \int \frac{d^{3}q} {(2\pi)^{3}}
e^{i\vec{q}.\vec{r} } \frac{A}{(q^{2} + \lambda^{2})}.
\end{displaymath}

4.  Let $\gamma$ be an oriented path from a to b in the complex plane.  If A(z) is an operator valued function of z, then define the line integral
\begin{displaymath}
\int_{\gamma} A(z)dz \equiv \sum_{i=1}^{N} A(z_{i})(z_{i+1}-z_{i})
\end{displaymath}


The time ordered product along the path $\gamma$ orders the operators along $\gamma$.  In the example
\begin{displaymath}
T_{\gamma}(A(z_{3})A(z_{2})A(z_{N})) = A(z_{N})A(z_{3})A(z_{1})
\end{displaymath}


Prove that

\begin{displaymath}
\left(
T_{\gamma} \exp \frac{1}{i} \int_{\gamma} H(\tau)d\tau
\right) ^{\dagger} =
\left(
T_{\gamma}\exp \frac{1}{i} \int_{\tilde\gamma} H(\tau)d\tau \right)
\end{displaymath}


where $\tilde \gamma$  is the path $\gamma$ traversed in the reverse direction (i.e. if $\gamma$ goes from $a$ to b, then $\tilde \gamma$ goes from b to $a$.)





5.  For the spin 1/2 fermions considered in question 3, prove the ``Hubbard Stratonovich'' identity
\begin{displaymath}
e^{-a (n_{\uparrow}-1/2)(n_{\downarrow} - 1/2)} = \frac{1}{2}e^{-a/4} \sum_{\sigma = \pm 1} e^{a\sigma (n_{\uparrow} -n_{\downarrow})}
\end{displaymath}

where $\mbox{cosh} a = e^{\frac{a}{4}}$ (Hint: consider the matrix representation of the operator
on the left hand side in the four dimensional Fock space of the up and down electrons.)
We shall later see how, considering $a \sim \delta t$ to be an infinitesimal time interval,
we can use this relation to treat interacting spins as spins moving in a fluctuating Ising
magnetic field.
\\

6.  Let $a^{\dagger}_{\alpha}$ , $a_{\alpha}$ be Fermion creation and annihilation operators, and assume
$$ H_{o} = \sum_{\alpha} \epsilon_{\alpha} a^{\dagger}_{\alpha}a_{\alpha}.$$

 (i) Compute $Tr[e^{\beta H_{o}}  a^{\dagger}_{\alpha} a_{\alpha}]$\\

(ii) Compute Tr [$e^{-\beta H_{o}} a_{\alpha}a^{\dagger}_{\alpha}$].\\

(iii) Compute $Tr [e^{-\beta H_{o}} a_{\alpha}a^{\dagger}_{\alpha}]$.  Using the identity $e^{-\beta H_{o}} a^{\dagger}_{\alpha} a_{\alpha} = -T \frac{\partial}{\partial \epsilon_{\alpha}} e^{-\beta H_{o}}$ check your answer to (i). \\

(iv) Confirm that the expectation
\begin{displaymath}
\langle n_{a} \rangle = Z^{-1} Tr[e^{-\beta H_{o}} a^{\dagger}_{\alpha} a_{\alpha} ] = f(\epsilon_{\alpha})
\end{displaymath}

where $f(x)$ is the Fermi function.\\

(v) Repeat the above procedure assuming that $a^{\dagger}_{a}$ and $a_{a}$ are boson operators.  What are the restrictions on the values of $\epsilon_{a}$ and what happens when one of these energies $\epsilon_{a}$ becomes zero?

(vi) Calculate


%\begin{equation}
$$
\left(
\begin{array}{l}
a^{\dagger}_{\alpha}(\tau) \\
a^{}_{\alpha}(\tau)
\end{array}
\right) =
e^{\tau H_o}
\left(
\begin{array}{l}
a^{\dagger}_{\alpha} \\
a^{}_{\alpha}
\end{array}
\right)
e^{-\tau H_o}
$$
%\end{equation}

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6.  Define an antitime ordering operator
$\tilde{T}$

\begin{equation}
\tilde T [A(t_1)B(t_2)]=
\left\{
\begin{array}{ll}
A(t_1) B(t_2) & (t_1 < t_2 )\\
B(t_2) A(t_1) & (t_1 > t_2 )
\end{array}
\right .
\end{equation}


Define

\begin{displaymath}
\tilde U (t,t_{o}) = \tilde T \exp \frac{1}{i} \int_{t_{o}}^{t} H(\tau)d\tau
\end{displaymath}

What is the differential equation satisfied by $\tilde{U}$(t,0)?\\
10.  Consider two different representations of a Fermion Fock space defined by the creation operators
\begin{displaymath}
\psi^{\dagger} a = c^{\dagger}_{\beta} \langle \beta|\alpha \rangle
\end{displaymath}

(i) Show that $\psi^{\dagger}_{\alpha},c^{\dagger}_{\beta} = \langle \beta|\alpha \rangle $.\\

(ii) Show that $\langle 0|\psi^{\dagger} _{\alpha} c^{\dagger}_{ \beta} = \langle \beta | \alpha \rangle$.\\

(III) Prove that if
\begin{displaymath}
| \phi \rangle = c^{\dagger}_{ \beta_{N}} c^{\dagger}_{ \beta_{N-1}}
 \ldots c^{\dagger}_{ \beta_{1}} |0 \rangle
\end{displaymath}
is a state of $N$ fermions that
$\phi(\alpha_{1}, \alpha_{2} \ldots \alpha_{N}) =  \langle 0|\phi_{\alpha_{1}} \phi_{\alpha_{2}} \ldots \phi_{\alpha_N} |\phi \rangle$ is a Slater determinant

\begin{displaymath}
\phi (\alpha_{1}, \alpha_{2} \ldots \alpha_{N}) = det \langle \alpha_{i}| \beta_{j} \rangle
\end{displaymath}

(iv) If $a \equiv \vec{r}$ and  $\beta \equiv \vec{k}$ so that $\Psi^{\dagger} (\vec{r})$ creates an electron in the position eigenstate and $c^{\dagger}_{ \vec{k}}$ creates an electron in a momentum eigenstate, what is the overlap matrix element
$\langle \vec{r}| \vec{k} \rangle$, for states normalized in a box of volume $V$? Explicitly write out the Slater determinant in the position basis for a three particle state $\phi = |\vec{k_{3}}, \vec{k_{2}}, \vec{k_{1}} \rangle$.
\\

11.  (i) If $M = M^{\dagger}$ is an Hermitian $N$ dimensional matrix, $c^{\dagger}_{i}$ and $c_{i}$ $(i = 1 \ldots N)$  are creation and annihilation operators for Fermions (i.e.$ {c_{i},c^{\dagger}_{j}} = \delta_{ij}$, show that

\begin{displaymath}
c_{i} ^{\dagger} (t) = \left[ e^{-\hat M t}c^{\dagger}e^{\hat Mt} \right] _{j} = c^{\dagger}_{j} \left[e^{-M} \right]_{ji}\;\;\;\;
(M = c^{\dagger}_{j} {M}_{jk}c_{k})
\end{displaymath}
(Hint consider the differential equation satisfied by $c^{\dagger}(t)$).
\end{document}
7.  let U$_{I}$(t,t$_{0}$) be the evolution operator in the interaction picture and
U$_{S}$(t,t$_{0}$) the corresponding operator in the Schr\"{d}inger picture.  Prove that

\begin{displaymath*}
U_{I}(t,0) = e^{iH_{o}t}U_{S}(t,0)
\end{displaymath*}
