pines-demon
Gold Member
2024 Award
- 929
- 778
Everytime that I deal with Bogoliubov transformations I get some sign wrong or a missing assumption. Let's take the case for fermions
$$c_{\mathbf p \sigma}=u_{\mathbf p \sigma} b_{\mathbf p \sigma} +v_{\mathbf p \sigma} b^\dagger_{-\mathbf p \bar{\sigma}}$$
where ##\sigma## is the spin variable, ##\bar{\sigma}## the opposite one and ##b## are the Bogoliubov operators.
Per commutation relations
$$|u_{\mathbf p \sigma}|^2+|v_{\mathbf p \sigma}|^2=1.$$
Now my deal is how to derive more properties.
Take the example of the BCS s-Hamiltonian:
$$H=\sum_{\mathbf p \sigma}\epsilon c^\dagger_{\mathbf p \sigma}c_{\mathbf p \sigma} -\sum_{\mathbf p} \Delta c_{\mathbf p \uparrow}^\dagger c^\dagger_{-\mathbf p\downarrow}+\Delta^* c_{-\mathbf p -\downarrow}^\dagger c^\dagger_{\mathbf p\uparrow}$$
No spin dependence so I drop the ##\sigma## dependence of the ##u_{\mathbf p\sigma}=u_{\mathbf p}##, I guess that by particle-hole symmetry I can ##u_{-\mathbf p}=u_{\mathbf p}##. So that simplifies things, but the Hamiltonian has imaginary term ##\Delta## so I guess I can't choose ##u_{\mathbf p}## real.
Then I would naively try
$$\begin{pmatrix}c_{\mathbf p\uparrow}^\dagger& c_{-\mathbf p\downarrow} \end{pmatrix}\begin{pmatrix}\epsilon& -\Delta\\-\Delta^* & -\epsilon \end{pmatrix}\begin{pmatrix}c_{\mathbf p\uparrow}\\ c^\dagger_{-\mathbf p\downarrow} \end{pmatrix}=\begin{pmatrix}b_{\mathbf p\uparrow}^\dagger& b_{-\mathbf p\downarrow} \end{pmatrix}\bar{U}\begin{pmatrix}\epsilon& -\Delta\\-\Delta^* & -\epsilon \end{pmatrix}U\begin{pmatrix}b_{\mathbf p\uparrow} \\ b^\dagger_{-\mathbf p\downarrow} \end{pmatrix}$$
and then find diagonal and off-diagonal terms to fix the values of ##U##, where
$$U=\begin{pmatrix}u_{\mathbf p}&v_{\mathbf p}\\v^*_{\mathbf p} & u^*_{\mathbf p}\end{pmatrix}$$
now here is where more sources change signs. To me the matrix is ok as it is, if there is an extra sign in the elements, can be fixed by the complex conjugation.
Secondly, I understand that ##U## is not necessarily unitary, but then what is ##\bar{U}##? Is it ##U^{-1}##? Then I get:
$$U^{-1}=\frac{1}{|u_{\mathbf p}|^2-|v_{\mathbf p}|^2}\begin{pmatrix}u^*_{\mathbf p}&-v_{\mathbf p}\\-v^*_{\mathbf p} & u_{\mathbf p}\end{pmatrix}$$
What do I do with that determinant? I don't see it in most sources. Also how is this justified?
Clearly if
$$c_{\mathbf p \uparrow}=u_{\mathbf p} b_{\mathbf p \uparrow} +v_{\mathbf p} b^\dagger_{-\mathbf p \downarrow}$$
then taking complex conjugate
$$c_{\mathbf p \uparrow}^\dagger=u^*_{\mathbf p} b^\dagger_{\mathbf p \uparrow} +v^*_{\mathbf p} b_{-\mathbf p \downarrow}$$
but using ##\bar{U}=U^{-1}## as in the matrix equation above I get
$$c_{\mathbf p \uparrow}^\dagger=\frac{1}{|u_{\mathbf p}|^2-|v_{\mathbf p}|^2}(u^*_{\mathbf p} b^\dagger_{\mathbf p \uparrow} -v^*_{\mathbf p} b_{-\mathbf p \downarrow})$$
What is wrong here? How do I get Bogoliubov transformation right? What is a good source about general Bogoliubov transformations?
$$c_{\mathbf p \sigma}=u_{\mathbf p \sigma} b_{\mathbf p \sigma} +v_{\mathbf p \sigma} b^\dagger_{-\mathbf p \bar{\sigma}}$$
where ##\sigma## is the spin variable, ##\bar{\sigma}## the opposite one and ##b## are the Bogoliubov operators.
Per commutation relations
$$|u_{\mathbf p \sigma}|^2+|v_{\mathbf p \sigma}|^2=1.$$
Now my deal is how to derive more properties.
Take the example of the BCS s-Hamiltonian:
$$H=\sum_{\mathbf p \sigma}\epsilon c^\dagger_{\mathbf p \sigma}c_{\mathbf p \sigma} -\sum_{\mathbf p} \Delta c_{\mathbf p \uparrow}^\dagger c^\dagger_{-\mathbf p\downarrow}+\Delta^* c_{-\mathbf p -\downarrow}^\dagger c^\dagger_{\mathbf p\uparrow}$$
No spin dependence so I drop the ##\sigma## dependence of the ##u_{\mathbf p\sigma}=u_{\mathbf p}##, I guess that by particle-hole symmetry I can ##u_{-\mathbf p}=u_{\mathbf p}##. So that simplifies things, but the Hamiltonian has imaginary term ##\Delta## so I guess I can't choose ##u_{\mathbf p}## real.
Then I would naively try
$$\begin{pmatrix}c_{\mathbf p\uparrow}^\dagger& c_{-\mathbf p\downarrow} \end{pmatrix}\begin{pmatrix}\epsilon& -\Delta\\-\Delta^* & -\epsilon \end{pmatrix}\begin{pmatrix}c_{\mathbf p\uparrow}\\ c^\dagger_{-\mathbf p\downarrow} \end{pmatrix}=\begin{pmatrix}b_{\mathbf p\uparrow}^\dagger& b_{-\mathbf p\downarrow} \end{pmatrix}\bar{U}\begin{pmatrix}\epsilon& -\Delta\\-\Delta^* & -\epsilon \end{pmatrix}U\begin{pmatrix}b_{\mathbf p\uparrow} \\ b^\dagger_{-\mathbf p\downarrow} \end{pmatrix}$$
and then find diagonal and off-diagonal terms to fix the values of ##U##, where
$$U=\begin{pmatrix}u_{\mathbf p}&v_{\mathbf p}\\v^*_{\mathbf p} & u^*_{\mathbf p}\end{pmatrix}$$
now here is where more sources change signs. To me the matrix is ok as it is, if there is an extra sign in the elements, can be fixed by the complex conjugation.
Secondly, I understand that ##U## is not necessarily unitary, but then what is ##\bar{U}##? Is it ##U^{-1}##? Then I get:
$$U^{-1}=\frac{1}{|u_{\mathbf p}|^2-|v_{\mathbf p}|^2}\begin{pmatrix}u^*_{\mathbf p}&-v_{\mathbf p}\\-v^*_{\mathbf p} & u_{\mathbf p}\end{pmatrix}$$
What do I do with that determinant? I don't see it in most sources. Also how is this justified?
Clearly if
$$c_{\mathbf p \uparrow}=u_{\mathbf p} b_{\mathbf p \uparrow} +v_{\mathbf p} b^\dagger_{-\mathbf p \downarrow}$$
then taking complex conjugate
$$c_{\mathbf p \uparrow}^\dagger=u^*_{\mathbf p} b^\dagger_{\mathbf p \uparrow} +v^*_{\mathbf p} b_{-\mathbf p \downarrow}$$
but using ##\bar{U}=U^{-1}## as in the matrix equation above I get
$$c_{\mathbf p \uparrow}^\dagger=\frac{1}{|u_{\mathbf p}|^2-|v_{\mathbf p}|^2}(u^*_{\mathbf p} b^\dagger_{\mathbf p \uparrow} -v^*_{\mathbf p} b_{-\mathbf p \downarrow})$$
What is wrong here? How do I get Bogoliubov transformation right? What is a good source about general Bogoliubov transformations?
Last edited: