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:

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

where $g<0$ 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:

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\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 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_\Delta=-\int_0^\beta d\tau\sum_{\vec{r}}\frac{1}{g}\Delta^*(\tau,\vec{r})\Delta(\tau,\vec{r})}

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 \psi^\dagger} and 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 \psi} are grassmann numbers. 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 \Delta^*} and 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 \Delta} are constant. 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 \psi_\uparrow\psi_\downarrow} and 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 \psi_\downarrow\psi_\uparrow} behave like constant.

Let's make a shift of the constant:

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 \Delta \rightarrow \Delta+g\psi_\uparrow\psi_\downarrow}

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 \Delta^*\rightarrow \Delta^*+g\psi^\dagger_\downarrow\psi^\dagger_\uparrow}

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

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

then, $Z=\int D[\psi _{\sigma }^{*}(\tau ,\mathbf {r} ),\psi _{\sigma }(\tau ,\mathbf {r} )]D[\Delta ^{*}(\tau ,\mathbf {r} ),\Delta (\tau ,\mathbf {r} )]e^{-(S_{0}+S_{int.}+S_{\Delta })}$