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:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z=\int D[\psi_\sigma ^{*} (\tau, \vec{r}), \psi_\sigma (\tau, \vec{r})]e^{-S_{BCS}}}

where, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (unknown function "\begin{align}"): {\displaystyle \begin{align}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 (\vec{r}) \\ &+\Delta^*(\tau, \vec{r})\psi_\uparrow (\tau, \vec{r})\psi_\downarrow (\tau, \vec{r}) \Delta (\tau, \vec{r})\psi^\dagger_\downarrow (\tau, \vec{r})\psi^\dagger_\uparrow (\tau, \vec{r})\\ &-\frac{1}{g}\Delta^* (\tau, \vec{r})\Delta (\tau, \vec{r}) \}} \end{align}</math>