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

Compute the desity-density correlation function:

$D_{\sigma {\sigma }^{\prime }}(\mathbf{𝐫}-{\mathbf{𝐫}}^{\prime })=⟨\mathrm{G}\mathrm{S}|{\hat{\mathrm{\Psi }}}_{\sigma }^{†}(\mathbf{𝐫}){\hat{\mathrm{\Psi }}}_{\sigma }(\mathbf{𝐫}){\hat{\mathrm{\Psi }}}_{{\sigma }^{\prime }}^{†}({\mathbf{𝐫}}^{\prime }){\hat{\mathrm{\Psi }}}_{{\sigma }^{\prime }}({\mathbf{𝐫}}^{\prime })|\mathrm{G}\mathrm{S}⟩$

for a gas of noninteracting electrons in the Fermi sea ground state $|\mathrm{G}\mathrm{S}⟩$. Since Fock space is orthonormal, to get a non-zero result you should consider only those cases where particles are created and annihilated by the four operators in exactly the same states. Consider both cases with $|{\mathbf{𝐤}}_i|<{\mathbf{𝐤}}_F$ and $|{\mathbf{𝐤}}_i|>{\mathbf{𝐤}}_F$. Express your result in terms of pair-correlation function discussed in the class.

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 two 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 of 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 variational ansatz for ground state 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 $B=0$. Thus, minimization of energy with respect to the pair occupation amplitude gives the same result as at $B=0$.

(b) For small $B$, ignore the interaction term and calculate the lowering of the ground state energy of free electron gas caused by the Zeeman coupling, to second order in $B$.

(c) Combine (a) and (b) to obtain the reduction in the condensation energy of the superconductor caused by the Zeeman coupling. At what value of the magnetic field is superconductivity destroyed by the Zeeman coupling?

(d) Calculate the minimum excitation energy gap of the BCS superconductor as a function of the field at zero temperature $T=0$. When does the gap collapse? Compare this result with the one obtained in (c). This is the so-called "Pauli limiting field" also known as "Chandrasekhar-Clogston limit".