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 3: Hubbard model for a triangular molecule

Consider a triangular molecule made of identical atoms, each of which has an s-type valence orbital. The simplest model describing this system is the corresponding Hubbard model with hopping $t$ and on-site Coulomb repulsion $U$.

(a) Assuming that there are two spin-up electrons in the system, find their ground-state energy.

(b) In the absence of magnetic field, the eigenstates of 2 electrons can be classified either as being a singlet (total spin $S=0$) or triplet (total spin $S=1$). In (a) you found the triplet ground-state energy (for the component $S_z=+1$, but you can check that the $S_z=0,-1$ triplet solutions also have precisely the same groundstate energy). Find the singlet ground-state energy if $U\to \mathrm{\infty }$. Obviously infinite repulsion prevents the two electrons from ever being on the same site.

(c) Find the singlet ground-state energy for a finite $U$, i.e. when the two electrons can be at the same site.

Problem 4: BSC superconductor in Zeeman field

Consider a thin film superconductor whose thickness is small enough to allow the penetration of the magnetic field into the interior, while any coupling of magnetic field to orbital degrees of freedom is neglected. Its BCS Hamiltonian in the presence of a magnetic field $\mathbf{𝐁}=B{\mathbf{𝐞}}_z$, coupling only to the spin, can then be written as:

$\hat{H}=\sum _{\mathbf{𝐤},\sigma }({\epsilon }_{\mathbf{𝐤}}-\mu -\sigma {\mu }_{B}B){\hat{c}}_{\mathbf{𝐤}\sigma }^{†}{\hat{c}}_{\mathbf{𝐤}\sigma }+\sum _{\mathbf{𝐤},{\mathbf{𝐤}}^{\prime }}{V}_{\mathbf{𝐤},{\mathbf{𝐤}}^{\prime }}{\hat{c}}_{{\mathbf{𝐤}}^{\prime }\uparrow }^{†}{\hat{c}}_{-{\mathbf{𝐤}}^{\prime }\downarrow }^{†}{\hat{c}}_{-\mathbf{𝐤}\downarrow }{\hat{c}}_{\mathbf{𝐤}\uparrow }$

Here $V_{\mathbf{𝐤},{\mathbf{𝐤}}^{\prime }}=-V_0/\mathrm{\Omega }$ ($\mathrm{\Omega }$ is volume) for $\mathbf{𝐤},{\mathbf{𝐤}}^{\prime }$ within a shell of energy width $\hslash {\omega }_D$ on either side of the Fermi surfaace and zero otherwise; ${\mu }_B$ is the Bohr magneton; and $\sigma =+1$ for $\uparrow$ spin and $\sigma =-1$ for $\downarrow$ spin.

(a) Show that the expectation value of $\hat{H}$ in the BCS wavefunction

$|\mathrm{\Phi }⟩=\prod _{\mathbf{𝐤}}(u_{\mathbf{𝐤}}+v_{\mathbf{𝐤}}{\hat{c}}_{\mathbf{𝐤}\uparrow }^{†}{\hat{c}}_{-\mathbf{𝐤}\downarrow }^{†})|0⟩$ has the same energy as for $B=0$.

(b)

(c)