Homework Set 3

Problem 1: Action on fermionic creation and annihilation operators on Fock states

Using the following convention

$|11111000\dots ⟩={\hat{c}}_5^{†}{\hat{c}}_4^{†}{\hat{c}}_3^{†}{\hat{c}}_2^{†}{\hat{c}}_1^{†}|\mathrm{v}\mathrm{a}\mathrm{c}\mathrm{u}\mathrm{u}\mathrm{m}⟩$

where $|\mathrm{v}\mathrm{a}\mathrm{c}\mathrm{u}\mathrm{u}\mathrm{m}>=|0000\dots ⟩$:

(a) Evaluate ${\hat{c}}_3^{†}{\hat{c}}_6{\hat{c}}_4{\hat{c}}_6^{†}{\hat{c}}_3|111110000\dots ⟩$.

(b) Write $|1101100100\dots ⟩$ in terms of excitations about the "filled Fermi sea" $|1111100000\dots ⟩$. Interpret your answer in terms of electron and hole excitations.

(c) Find $⟨\mathrm{\Phi }|\hat{N}|\mathrm{\Phi }⟩$, where $|\mathrm{\Phi }⟩=A|100⟩+B|111000⟩$ and $\hat{N}=\sum _i{\hat{c}}_i^{†}+{\hat{c}}_i$ is the operator of the total particle number.

Problem 2: Density-density correlation function for noninteracting electrons

The pair correlation function gives the relative probability of finding a particle at position ${\mathbf{𝐫}}^{\prime }$ if we know that there is one at position $\mathbf{𝐫}$. It is defined by:

$G(\mathbf{𝐫}-{\mathbf{𝐫}}^{\prime })=⟨\mathrm{\Phi }(\mathbf{𝐫})|{\hat{\mathrm{\Psi }}}^{†}({\mathbf{𝐫}}^{\prime })\hat{\mathrm{\Psi }}({\mathbf{𝐫}}^{\prime })|\mathrm{\Phi }(\mathbf{𝐫})⟩=⟨{\mathrm{\Phi }}_0|{\hat{\mathrm{\Psi }}}^{†}(\mathbf{𝐫}){\hat{\mathrm{\Psi }}}^{†}({\mathbf{𝐫}}^{\prime })\hat{\mathrm{\Psi }}({\mathbf{𝐫}}^{\prime })\hat{\mathrm{\Psi }}(\mathbf{𝐫})|{\mathrm{\Phi }}_0⟩$

Here $|\mathrm{\Phi }(\mathbf{𝐫})⟩=\hat{\mathrm{\Psi }}(\mathbf{𝐫})|{\mathrm{\Phi }}_0⟩$ is a state with $N-1$ particles, obtained after removal of one particle at position $\mathbf{𝐫}$, so that pair correlation function is computed as the expectation value of the density operator in this new quantum state. Compute the pair correlation function for a system of translationally invariant noninteracting spinless bosons in many-body state

$|{\mathrm{\Phi }}_0⟩=|n_{{\mathbf{𝐤}}_0},n_{{\mathbf{𝐤}}_1},\dots ⟩$

by transforming field operators to creation and annihilation operators in momentum representation while assuming that bosons are enclosed in a box of volume $V$ with periodic boundary conditions. Your final result should be a function of $\mathbf{𝐫},{\mathbf{𝐫}}^{\prime }$ and sums over $n_{\mathbf{𝐤}}$, where you can also use that

$⟨{\mathrm{\Phi }}_0|{\hat{\mathrm{\Psi }}}^{†}(\mathbf{𝐫})\hat{\mathrm{\Psi }}(\mathbf{𝐫})|{\mathrm{\Phi }}_0⟩=\frac{1}{V}\sum _{\mathbf{𝐤}}{n}_{\mathbf{𝐤}}=N$.

Problem 4: BSC superconductor in Zeeman field

Consider BCS Hamiltonian in the presence of a magnetic field $\mathbf{𝐁}=B{\mathbf{𝐞}}_z$ coupling only to the spin:

Failed to parse (syntax error): {\displaystyle \hat{H} = \sum_{\mathbf{k},\sigma} (\varepsilon_\mathbf{k} - \mu - \sigma \mu_B B) \hat{c}^\dagger_{\mathbf{k}\sigma} \hat{c}_{\mathbf{k}\sigma} + \sum_{\mathbf{k},\mathbf{k}'} V_{\mathbf{k},\mathbf{k}'} \hat{c}^\dagger_{\mathbf{k}' \uparrow} \hat{c}_{-\mathbf{k}' \downarrow}} \hat{c}_{-\mathbf{k} \downarrow} \hat{c}_{\mathbf{k} \uparrow}} }