PHY6937: Difference between revisions
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<math>Z=\int D[\psi_\sigma ^{*} (\tau, \vec{r}), \psi_\sigma (\tau, \vec{r})]e^{-S_{BCS}}</math> | <math>Z=\int D[\psi_\sigma ^{*} (\tau, \vec{r}), \psi_\sigma (\tau, \vec{r})]e^{-S_{BCS}}</math> | ||
where, <math>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})]</math> | |||
It doesn't matter to multiply partition function by a constant: | |||
<math>Z\arrow Z=\int D[\psi_\sigma ^{*} (\tau, \vec{r}), \psi_\sigma (\tau, \vec{r})] D[\Delta^*(\tau, \vec{})] e^{-S_{BCS}}</math> | |||
== Microscopic derivation of the Giznburg-Landau functional == | == Microscopic derivation of the Giznburg-Landau functional == | ||
=== Little Parks experiment === | === Little Parks experiment === | ||
Revision as of 18:05, 7 February 2011
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
Outline of the course:
Pairing Hamiltonian and BCS instability
We can write the Hamiltonian of the system as:
![{\displaystyle 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}})]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8f15a6609668726cbac3dc59a49a5c5a175c3206)
in which, and
For this system, the partition function is:
![{\displaystyle Z=\int D[\psi _{\sigma }^{*}(\tau ,{\vec {r}}),\psi _{\sigma }(\tau ,{\vec {r}})]e^{-S_{BCS}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9f7ae5e19060e12decf735261a1a592318de0d4)
where,
![{\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}})]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a6f472d6c43fa87b81f3f5862bb1073dcc5641e)
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\arrow Z=\int D[\psi_\sigma ^{*} (\tau, \vec{r}), \psi_\sigma (\tau, \vec{r})] D[\Delta^*(\tau, \vec{})] e^{-S_{BCS}}}