[toc]

Bogoliubov transformation

Basics for bosons

• Bogoliubov transformations are linear transformation of creation/annihilation operators that preserve the algebraic relations among them. In the following, we deal with bosons that satisfies canonical communication.

  As a fact, generally, a Bogoliubov transformation $U$ is not unitary \(U^\dagger\neq U^{-1}\). Let us consider a general model with

  \[H= a^\dagger a +b^\dagger b +\gamma a^\dagger b^\dagger +\gamma a b\]

  where operators $a$ and $b$ are bosonic operators that satisfy canonical communication relation

  \[[a,a^\dagger]=1,[b,b^\dagger]=1\]

  Our aim is to find new quasiparticles \(\alpha,\beta\) under which $H$ is in a diagonal form. The first piece of requirement is to preserve the bosonic relation. Suppose

  \[\begin{align} \alpha& =M_{11}a+M_{12}b^\dagger \\ \beta^\dagger&=M_{21}a+M_{22}b^\dagger \end{align}\]

  Then

  \[\begin{align} [\alpha,\alpha^\dagger]& =[M_{11}a,M_{11}^*a^\dagger]+[M_{12}b^\dagger,M_{12}^*b]=|M_{11}|^2-|M_{12}|^2=1 \\ [\beta,\beta^\dagger]&=|M_{22}|^2-|M_{11}|^2=1 \end{align}\]

  when formulated into matrix form

  \[h= \begin{pmatrix} \epsilon_0 & \Delta \\ \Delta^* & \epsilon_0 \end{pmatrix}\]

The eigenenergy appears as a solution to the equation

\[\mathrm{Det}\left[\begin{pmatrix}\lambda & 0 \\ 0 & -\lambda \end{pmatrix} -h\sigma_z\right]=0\]

which yields the spectrum function for the canonical modes \(\alpha,\beta\)

\[\lambda_{1,2}= \sqrt{\epsilon^2-|\Delta|^2}\]

Further, one may write down the Bogoliubov transformation

\[\begin{align} \alpha & = \sqrt{\frac{\epsilon_0-\lambda}{2\epsilon_0}} a +e^{-\theta}\sqrt{\frac{\lambda+\epsilon_0}{2\epsilon_0}}b^\dagger \\ \beta & = e^{i\theta}\sqrt{\frac{\lambda+\epsilon_0}{2\epsilon_0}} a + \sqrt{\frac{\epsilon_0-\lambda}{2\epsilon_0}}b^\dagger \end{align}\]

where \(\Delta=\vert\Delta\vert e^{i\theta}\). As matter of fact, the case we often meet is that \(a_{k},a_{-k}^\dagger\) in the original Hamiltonian, in which, we have

\[\begin{align} \alpha_k & = \sqrt{\frac{\epsilon_0-\lambda}{2\epsilon_0}} a_k +e^{-\theta}\sqrt{\frac{\lambda+\epsilon_0}{2\epsilon_0}}a_{-k}^\dagger \\ \alpha^\dagger_{-k} & = e^{i\theta}\sqrt{\frac{\lambda+\epsilon_0}{2\epsilon_0}} a_{k} + \sqrt{\frac{\epsilon_0-\lambda}{2\epsilon_0}}a_{-k}^\dagger \end{align}\]

==Remark==

For a quantum field theory, the diagnolization procedures have some sublattices compared with the case above. For example, a scalar field with Hamiltonian

\[\mathcal L = \frac{1}{2}\partial_0 \phi \partial _0\phi-\frac{1}{2}\nabla\phi\nabla\phi\]

has a quadratic time derivative. However, in such a system, a unitary transformation cannot diagonalize it. Of course, one may introduce canonical creation and annihilation operators with the time derivative of order 1. We first derive the canonical quantization relation

\[\pi =\frac{\delta\mathcal L}{\delta \partial_0\phi}= \partial_0\phi\]

which the Hamiltonian

\[H=\pi \partial_0\phi-L=\frac{1}{2}(\pi^2+\nabla\phi\nabla\phi)\]

We introduce the raising and lowering operators

\[\begin{align} a^\dagger &=\frac{1}{\sqrt{2|\mathbf k|}}(\pi+i|\mathbf k|\phi)\\ a &=\frac{1}{\sqrt{2|\mathbf k|}}(\pi-i|\mathbf k|\phi) \end{align}\]

which gives the Hamiltonian

\[H=|\mathbf k|a^\dagger(\mathbf k)a(\mathbf k)+\frac{1}{2}|\mathbf k|\]

Another example is the QED for which one has to introduce

\[A_i(k)= \frac{\xi_i(k)}{\sqrt{\mathbf k^2}}(a_\mathbf k^\dagger+ia_\mathbf k)\]

with \(\xi_i(\mathbf k)\) being the polarized vector.

Basic Fermion

The case for a system of fermions are much simple and we can just apply the unitary transformation which perserves the anticommunication relation. Here we present the transformations. Consider a BdG Hamiltonian (Please double check the sign before the order parameter!!!)

\[\begin{align} \mathcal{H}_{\mathrm{MF}} &=\sum_{k, \alpha} \xi_{k} c_{k, \alpha}^{\dagger} c_{k, \alpha}-\sum_{k}^{\prime}\left[\Delta e^{-i \varphi} c_{k, \uparrow} c_{-k, \downarrow}+\Delta e^{i \varphi} c_{-k, \downarrow}^{\dagger} c_{k, \uparrow}^{\dagger}\right]+\frac{V_{\mathrm{vol}} \Delta^{2}}{g} \\ &=\sum_{k}\left[c_{k, \uparrow}^{\dagger}, c_{-k, \downarrow}\right]\left[\begin{array}{cc} \xi_{k} & \Delta e^{i \varphi} \\ \Delta e^{-i \varphi} & -\xi_{k} \end{array}\right]\left[\begin{array}{c} c_{k, \uparrow} \\ c_{-k, \downarrow}^{\dagger} \end{array}\right]+\frac{V_{\mathrm{vol}} \Delta^{2}}{g} \end{align}\]

Then

\[E_{k}=\sqrt{\xi_{k}^{2}+\Delta^{2}}, \quad u_{k}=\sqrt{\frac{1}{2}\left(1+\frac{\xi_{k}}{E_{k}}\right)}, \quad v_{k}=\sqrt{\frac{1}{2}\left(1-\frac{\xi_{k}}{E_{k}}\right)}\]

with

\[\begin{align} &{\left[\begin{array}{c} \gamma_{\boldsymbol{k}, \uparrow} \\ \gamma_{-\boldsymbol{k}, \downarrow}^{\dagger} \end{array}\right]=\left[\begin{array}{cc} u_{\boldsymbol{k}} & -v_{\boldsymbol{k}} e^{i \varphi} \\ v_{\boldsymbol{k}} e^{-i \varphi} & u_{\boldsymbol{k}} \end{array}\right]\left[\begin{array}{c} c_{\boldsymbol{k}, \uparrow} \\ c_{-\boldsymbol{k}, \downarrow}^{\dagger} \end{array}\right],} \\ &{\left[\begin{array}{c} c_{\boldsymbol{k}, \uparrow} \\ c_{-\boldsymbol{k}, \downarrow}^{\dagger} \end{array}\right]=\left[\begin{array}{cc} u_{\boldsymbol{k}} & v_{\boldsymbol{k}} e^{i \varphi} \\ -v_{\boldsymbol{k}} e^{-i \varphi} & u_{\boldsymbol{k}} \end{array}\right]\left[\begin{array}{c} \gamma_{\boldsymbol{k}, \uparrow} \\ \gamma_{-\boldsymbol{k}, \downarrow}^{\dagger} \end{array}\right] .} \end{align}\]

Then we obtain the diagonalized Hamiltonian

\[\mathcal H_\mathrm{MF} = \sum_k (E_k\gamma_{k,\uparrow}^\dagger\gamma_{k,\uparrow }-E_k\gamma_{-k,\downarrow}\gamma_{-k,\downarrow}^\dagger)+\frac{V_\mathrm{vol}\Delta^2}{g}\]

and the according BCS can be constructed as

\[|\mathrm{BCS}\rangle = \prod_k\gamma_{k,\uparrow}\gamma_{k\downarrow}|\Omega\rangle\sim\prod_{k}(u_k+v_k c_{k,\uparrow}^\dagger c_{-k\downarrow}^\dagger)|\Omega\rangle\]

with \(\vert\Omega\rangle\) being the vacuum i.g. a Fermi sea with a Fermi surface. We can expect it for any \(k,\sigma\), with

\[\gamma_{k,\sigma}|\mathrm{BCS}\rangle =0\]

With the BCS ground state, one can evaluate the expectation value

\[\begin{align} & \langle c^\dagger_{k,\uparrow}c_{k,\uparrow}\rangle=\langle c^\dagger_{k,\downarrow}c_{k,\downarrow} \rangle = \upsilon_k^2 \\ & \langle c^\dagger_{k,\uparrow}c^\dagger_{-k,\downarrow}\rangle = u_k v_k \end{align}\]

Matsubara Summation

Pair Correlation function \(\chi_{n,\mathbf q}^c\)

A building block enters the calculation of the Cooper pair propagator

\[\begin{align} \chi_{n, \mathbf{q}}^{\mathrm{c}} \equiv-\frac{T}{L^{d}} \sum_{m, \mathbf{p}} G_{0}\left(\mathbf{p}, i \omega_{m}\right) G_{0}\left(-\mathbf{p}+\mathbf{q},-i \omega_{m}+i \omega_{n}\right) \\ =\frac{1}{L^{d}} \sum_{\mathbf{p}} \frac{1-n_{\mathrm{F}}\left(\xi_{\mathbf{p}}\right)-n_{\mathrm{F}}\left(\xi_{-\mathbf{p}+\mathbf{q}}\right)}{i \omega_{n}-\xi_{\mathbf{p}}-\xi_{-\mathbf{p}+\mathbf{q}}} \end{align}\]

where (around the transition point)

\[G_0(\mathbf p,i\omega_m)=\frac{1}{i\omega_m-\xi_\mathbf p}\]

and \(\omega_m = (2m+1)\pi T\) are fermionic Matsubara frequencies, while \(\omega_n =2\pi nT\) is a bosonic Matsubara frequency.

Density-Density response function \(\chi^d_{\mathbf q,\omega_n}\)

\[\begin{align} & \chi_{\mathbf{q}, \omega_{n}}^{\mathrm{d}} \equiv-\frac{T}{L^{d}} \sum_{\mathbf{p}, \omega_{m}} G_{0}\left(\mathbf{p}, i \omega_{m}\right) G_{0}\left(\mathbf{p}+\mathbf{q}, i \omega_{m}+i \omega_{n}\right) \\ = & -\frac{1}{L^{d}} \sum_{\mathbf{p}} \frac{n_{\mathrm{F}}\left(\xi_{\mathbf{p}}\right)-n_{\mathrm{F}}\left(\xi_{\mathbf{p}+\mathbf{q}}\right)}{i \omega_{n}+\xi_{\mathbf{p}}-\xi_{\mathbf{p}+\mathbf{q}}} \end{align}\]

In real calculation, we often take the static limit. That is

\[(\mathbf q,\omega_n)\rightarrow (\mathbf q,0)\]

It is convenient to change from an explicit matrix representation of the Gorkov Green function to an expression in terms of the Pauli matrices

\[\mathcal G_{0,p}=\frac{1}{i\sigma_0\omega_n-\sigma_3 \xi_p+\sigma_1\Delta_0}=\frac{-i\sigma_0\omega_n-\sigma_3\xi_p+\sigma_1\Delta_0}{\omega_n^2+\xi_p^2+\Delta_0^2}\]

where the Gorkov Green function takes the form as

\[\mathcal G^{-1}=\begin{pmatrix}-\partial_\tau-i\phi-\frac{1}{2m}(-i\nabla-\mathbf A)^2+\mu & \Delta_0e^{2i\theta} \\ \Delta_0e^{-2i\theta} & -\partial_\tau +i\phi +\frac{1}{2m}(-i\nabla-\mathbf A)-\mu \end{pmatrix}\]

On the other hand we have the expansion

\[\begin{align} \mathcal G^{-1} & =-\sigma_0\partial_\tau-\sigma_3 (i\phi+\frac{1}{2m}(-i\nabla-\sigma_3\mathbf A)^2-\mu)+\sigma_1\Delta_0 \\ & =-\sigma_0\partial_\tau -\sigma_3 (-\frac{\nabla^2}{2m}-\mu)+\sigma_1\Delta_0-i\sigma_3\phi+\frac{i}{2m}\sigma_0[\nabla,\mathbf A]_+-\sigma_3\frac{1}{2m}\mathbf A^2 \end{align}\]

• Gorkov Green function

  \[\begin{align} G_{0,p} & =\frac{u_k^2}{\omega - \epsilon_k}+\frac{v_k^2}{\omega+\epsilon_k}\\ F_{0,p} & =u_kv_k\frac{2\omega}{\omega^2+\epsilon_k^2} \end{align}\]

Conventions for Fourier transformation

• Fermionic operator (N is the lattice number)

  \[\begin{align} c(r)&=\frac{1}{\sqrt{N}}\sum_{k}c_{k}e^{ik\cdot r}\\ c(k)&=\frac{1}{\sqrt{N}}\sum_{r}c(r)e^{-ik\cdot r} \end{align}\]

• Order parameter formed by two Fermion operator

  For the superconducting order parameter \(\Delta(r)=c_\uparrow(r)c_\downarrow(r)\)

  \[\Delta (k) = \sum_r \Delta(r)e^{ik\cdot r}\]

• Function

  \[\begin{align} f(r) &=\sum_{k}f(k)e^{ik\cdot r} \\ f(k) &=\frac{1}{N}\sum_{r}f(r)e^{-ik\cdot r} \end{align}\]