PHY6937

Welcome to Phy 6937 Superconductivity and superfluidity

PHY6937 is a one semester advanced graduate level course. Its aim is to introduce concepts and theoretical techniques for the description of superconductors and superfluids. This course is a natural continuation of the "many-body" course PHY5670 and will build on the logical framework introduced therein, i.e. broken symmetry and adiabatic continuity. The course will cover a range of topics, such as the connection between the phenomenological Ginzburg-Landau and the microscpic BCS theory, Migdal-Eliashberg treatment of phonon mediated superconductivity, unconventional superconductivity, superfluidity in He-4 and He-3, and Kosterlitz-Thouless theory of two dimensional superfluids.

The key component of the course is the collaborative student contribution to the course Wiki-textbook. Each team of students is responsible for BOTH writing the assigned chapter AND editing chapters of others.

Team assignments: Spring 2011 student teams

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Outline of the course:

Pairing Hamiltonian and BCS instability

We can write the Hamiltonian of the system as:

$H=\sum _{\vec {r}}[\psi _{\sigma }^{\dagger }({\vec {r}})(\epsilon _{\vec {p}}-\mu )\psi _{\sigma }^{\dagger }({\vec {r}})+g\psi _{\uparrow }^{\dagger }({\vec {r}})\psi _{\downarrow }^{\dagger }({\vec {r}})\psi _{\downarrow }({\vec {r}})\psi _{\uparrow }({\vec {r}})]$

in which, $g<0$ and $|g|<<\epsilon _{F}$

For this system, the partition function is:

$Z=\int D[\psi _{\sigma }^{*}(\tau ,{\vec {r}}),\psi _{\sigma }(\tau ,{\vec {r}})]e^{-S_{BCS}}$

where, $S_{BCS}=\int _{0}^{\beta }d\tau \sum _{\vec {r}}[\psi _{\sigma }^{\dagger }(\tau ,{\vec {r}})(\partial _{\tau }+\epsilon _{\vec {p}}-\mu )\psi _{\sigma }^{\dagger }({\vec {r}})+g\psi _{\uparrow }^{\dagger }({\vec {r}})\psi _{\downarrow }^{\dagger }({\vec {r}})\psi _{\downarrow }({\vec {r}})\psi _{\uparrow }({\vec {r}})]$

It doesn't matter to multiply partition function by a constant:

$Z\rightarrow Z=\int D[\psi _{\sigma }^{*}(\tau ,{\vec {r}}),\psi _{\sigma }(\tau ,{\vec {r}})]D[\Delta ^{*}(\tau ,{\vec {r}}),\Delta (\tau ,{\vec {r}})]e^{-S_{BCS}-S_{\Delta }}$

where, $S_{\Delta }=-\int _{0}^{\beta }d\tau \sum _{\vec {r}}{\frac {1}{g}}\Delta ^{*}(\tau ,{\vec {r}})\Delta (\tau ,{\vec {r}})$

Here, we need to pay attention: $\psi ^{\dagger }$ and $\psi$ are grassmann numbers. $\Delta ^{*}$ and $\Delta$ are constant. $\psi _{\uparrow }\psi _{\downarrow }$ and $\psi _{\downarrow }\psi _{\uparrow }$ behave like constant.

Let's make a shift of the constant:

$\Delta \rightarrow \Delta +g\psi _{\uparrow }\psi _{\downarrow }$

$\Delta ^{*}\rightarrow \Delta ^{*}+g\psi _{\downarrow }^{\dagger }\psi _{\uparrow }^{\dagger }$

Then, $S_{\Delta }=-\int _{0}^{\beta }d\tau \sum _{\vec {r}}{\{{\frac {1}{g}}\Delta ^{*}\Delta +\Delta ^{*}\psi _{\uparrow }\psi _{\downarrow }\Delta \psi _{\downarrow }^{\dagger }\psi _{\uparrow }^{\dagger }+g\psi _{\downarrow }^{\dagger }\psi _{\uparrow }^{\dagger }\psi _{\uparrow }\psi _{\downarrow }}\}$

Set ${\begin{aligned}S&=S_{BCS}+S_{\Delta }\\&=\int _{0}^{\beta }d\tau \sum _{\vec {r}}\{\psi _{\sigma }^{\dagger }(\tau ,{\vec {r}})(\partial _{\tau }+\epsilon _{\vec {p}}-\mu )\psi _{\sigma }^{\dagger }(\tau ,{\vec {r}})\rightarrow S_{0}\\&+\Delta ^{*}(\tau ,{\vec {r}})\psi _{\uparrow }(\tau ,{\vec {r}})\psi _{\downarrow }(\tau ,{\vec {r}})\Delta (\tau ,{\vec {r}})\psi _{\downarrow }^{\dagger }(\tau ,{\vec {r}})\psi _{\uparrow }^{\dagger }(\tau ,{\vec {r}})\rightarrow S_{int}\\&-{\frac {1}{g}}\Delta ^{*}(\tau ,{\vec {r}})\Delta (\tau ,{\vec {r}})\}\rightarrow S_{\Delta }\end{aligned}}$

then, $Z=\int D[\psi _{\sigma }^{*}(\tau ,{\vec {r}}),\psi _{\sigma }(\tau ,{\vec {r}})]D[\Sigma ^{*}(\tau ,{\vec {r}}),\Sigma (\tau ,{\vec {r}})]e^{-S}$