PHY6937: Difference between revisions
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The strategy is to take | The strategy is to take | ||
<math>S_{eff}[\Delta(r,\tau)]=S_{eff}[\Delta_{sp}+(\Delta(r,\tau)-\Delta_{sp})]</math> | <math>S_{eff}[\Delta(r,\tau)]=S_{eff}\left[\Delta_{sp}+(\Delta(r,\tau)-\Delta_{sp})\right]</math> | ||
where <math>\Delta_{sp}</math> solves (1) and (2) with LHS set to zero and expand in powers of <math>\Delta(r,\tau)-\Delta_{sp}</math>. So | where <math>\Delta_{sp}</math> solves (1) and (2) with LHS set to zero and expand in powers of <math>\Delta(r,\tau)-\Delta_{sp}</math>. So | ||
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<math>\begin{align} | <math>\begin{align} | ||
&\frac{\delta^2 S_{eff}}{\delta\Re e\Delta(r,\tau)\delta\Re e\Delta(r',\tau')}=\frac{2}{|g|}\delta(\tau-\tau')\delta(r-r')\\ | &\frac{\delta^2 S_{eff}}{\delta\Re e\Delta(r,\tau)\delta\Re e\Delta(r',\tau')}=\frac{2}{|g|}\delta(\tau-\tau')\delta(r-r')\\ | ||
&-\langle\left(\psi_{\downarrow}(r,\tau)\psi_{\uparrow}(r,\tau)+\psi^*_{\uparrow}(r,\tau)\psi^*_{\downarrow}(r,\tau)\right)\left(\psi_{\downarrow}(r',\tau')\psi_{\uparrow}(r',\tau')+\psi^*_{\uparrow}(r',\tau')\psi^*_{\downarrow}(r',\tau')\right)\rangle_{S_0+S_{int}}^{ | &-\langle\left(\psi_{\downarrow}(r,\tau)\psi_{\uparrow}(r,\tau)+\psi^*_{\uparrow}(r,\tau)\psi^*_{\downarrow}(r,\tau)\right)\left(\psi_{\downarrow}(r',\tau')\psi_{\uparrow}(r',\tau')+\psi^*_{\uparrow}(r',\tau')\psi^*_{\downarrow}(r',\tau')\right)\rangle_{S_0+S_{int}}^{\mbox{con}} | ||
\end{align}</math> | \end{align}</math> | ||
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<math>\begin{align} | <math>\begin{align} | ||
&\frac{\delta^2 S_{eff}}{\delta\Im m\Delta(r,\tau)\delta\Im m\Delta(r',\tau')}=\frac{2}{|g|}\delta(\tau-\tau')\delta(r-r')\\ | &\frac{\delta^2 S_{eff}}{\delta\Im m\Delta(r,\tau)\delta\Im m\Delta(r',\tau')}=\frac{2}{|g|}\delta(\tau-\tau')\delta(r-r')\\ | ||
&+\langle\left(\psi^*_{\uparrow}(r,\tau)\psi^*_{\downarrow}(r,\tau)-\psi_{\downarrow}(r,\tau)\psi_{\uparrow}(r,\tau)\right)\left(\psi^*_{\uparrow}(r',\tau')\psi^*_{\downarrow}(r',\tau')-\psi_{\downarrow}(r',\tau')\psi_{\uparrow}(r',\tau')\right)\rangle_{S_0+S_{int}}^{ | &+\langle\left(\psi^*_{\uparrow}(r,\tau)\psi^*_{\downarrow}(r,\tau)-\psi_{\downarrow}(r,\tau)\psi_{\uparrow}(r,\tau)\right)\left(\psi^*_{\uparrow}(r',\tau')\psi^*_{\downarrow}(r',\tau')-\psi_{\downarrow}(r',\tau')\psi_{\uparrow}(r',\tau')\right)\rangle_{S_0+S_{int}}^{\mbox{con}} | ||
\end{align}</math> | \end{align}</math> | ||
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<math> | <math> | ||
\frac{\delta^2 S_{eff}}{\delta\Im m\Delta(r,\tau)\delta\Re e\Delta(r',\tau')}=-i\langle\left(\psi^*_{\uparrow}(r,\tau)\psi^*_{\downarrow}(r,\tau)-\psi_{\downarrow}(r,\tau)\psi_{\uparrow}(r,\tau)\right)\left(\psi_{\downarrow}(r',\tau')\psi_{\uparrow}(r',\tau')+\psi^*_{\uparrow}(r',\tau')\psi^*_{\downarrow}(r',\tau')\right)\rangle_{S_0+S_{int}}^{ | \frac{\delta^2 S_{eff}}{\delta\Im m\Delta(r,\tau)\delta\Re e\Delta(r',\tau')}=-i\langle\left(\psi^*_{\uparrow}(r,\tau)\psi^*_{\downarrow}(r,\tau)-\psi_{\downarrow}(r,\tau)\psi_{\uparrow}(r,\tau)\right)\left(\psi_{\downarrow}(r',\tau')\psi_{\uparrow}(r',\tau')+\psi^*_{\uparrow}(r',\tau')\psi^*_{\downarrow}(r',\tau')\right)\rangle_{S_0+S_{int}}^{\mbox{con}} | ||
</math> | </math> | ||
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\end{align}</math> | \end{align}</math> | ||
<math>\begin{align} | |||
\frac{\delta^2S_{eff}}{\delta\Re e\Delta(r,\tau)\delta\Re e\Delta(r',\tau')}|_{\Delta_{sp}}&=\frac{2}{|g|}\delta(\tau-\tau')\delta(r-r')-\langle\Psi^*(r,\tau)\sigma_x\Psi(r,\tau)\Psi^*(r',\tau')\sigma_x\Psi(r',\tau')\rangle^{\mbox{con}}\\ | |||
&=\frac{2}{|g|}\delta(\tau-\tau')\delta(r-r')+\mbox{Tr}\left[\sigma_xG(r-r',\tau-\tau')\sigma_xG(r'-r,\tau'-\tau)\right] | |||
\end{align}</math> | |||
where the Green's functions are 2*2 matrices, | |||
<math>G^{-1}_k(i\omega_n)=\left(\begin{align}&-i\omega_n+\epsilon_k-\mu&\Delta_{sp}\\&\Delta_{sp}&-i\omega_n-\epsilon_k+\mu\end{align}\right)</math> | |||
Notice that this is a function of <math>\tau-\tau'</math> and <math>r-r'</math>. Let's call it <math>\Pi_{++}(r-r',\tau-\tau')</math> | |||
To determine the kinematics (of our collective modes) we need to expand <math>\Pi_{\mu\nu}(q, i\Omega_n)</math> in powers of <math>\vec{q}</math> and <math>\Omega_n</math>. Our small expansion parameters are | To determine the kinematics (of our collective modes) we need to expand <math>\Pi_{\mu\nu}(q, i\Omega_n)</math> in powers of <math>\vec{q}</math> and <math>\Omega_n</math>. Our small expansion parameters are | ||
Revision as of 22:51, 22 April 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
To see the origins of superconductivity, it is helpful to look at a toy system, which we already know will give us superconducting behavior. This is useful because the toy system is only a simple change to a non-interacting electron gas. By adding in some small attractive interaction, we will arrive at a superconducting system! This interaction need only occur between two electrons occupying the same position in space (and necessarily having opposite spin!). Additionally, we still find the interesting behaviour regardless of the size of the interaction; the only requirement is that it be non-zero!
We can write the Hamiltonian of the system as:
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 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 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 \ g<0} 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 \ |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:
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, 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^*\Delta + \Delta^*\psi_\uparrow \psi_\downarrow + \Delta\psi^\dagger_\downarrow \psi^\dagger_\uparrow+g\psi^\dagger_\downarrow \psi^\dagger_\uparrow \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 \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 (\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^\dagger_\downarrow (\tau, \vec{r})\psi^\dagger_\uparrow (\tau, \vec{r}) \rightarrow S_{int}\\ &-\frac{1}{g}\Delta^* (\tau, \vec{r})\Delta (\tau, \vec{r}) \} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \rightarrow S_{\Delta} \end{align}}
then, 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,\mathbf{r}),\psi_{\sigma}(\tau,\mathbf{r})]D[\Delta^{*}(\tau,\mathbf{r}),\Delta(\tau,\mathbf{r})]e^{-(S_{0}+S_{int.}+S_{\Delta})}} .
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 \left\langle e^{-S_{int.}}\right\rangle _{0}\cong exp[\frac{1}{2}\left\langle S_{int.}^{2}\right\rangle _{0}+\frac{1}{4!}(\left\langle S_{int.}^{4}\right\rangle _{0}-3\left\langle S_{int.}^{2}\right\rangle _{0}^{2})]} by cumulant expansion, which guarantees that until the 2nd order, it is accurate.
Use Matsubara's Method
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_{\sigma}(\tau,\mathbf{r})=\frac{1}{\beta}\underset{\omega_{n}}{\sum}\underset{\mathbf{k}}{\sum}e^{i\mathbf{k}\cdot\mathbf{r}}e^{-i\omega_{n}\tau}\psi_{\sigma}(i\omega_{n},\mathbf{k}), \omega_{n}=(2n+1)\frac{\pi}{\beta};}
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(\tau,\mathbf{r})=\frac{1}{\beta}\underset{\Omega_{n}}{\sum}\underset{\mathbf{k}}{\sum}e^{i\mathbf{k}\cdot\mathbf{r}}e^{-i\Omega_{n}\tau}\Delta_{\mathbf{k}}(i\Omega_{n}), \omega_{n}=2n\frac{\pi}{\beta}.}
Then 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_{0}=\frac{L^{D}}{\beta}\underset{\omega_{n}}{\sum}\underset{\mathbf{k}}{\sum}[-i\omega_{n}+\varepsilon_{\mathbf{k}}-\mu]\psi_{\sigma}^{\dagger}(i\omega_{n},\mathbf{k})\psi_{\sigma}(i\omega_{n},\mathbf{k}).}
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_{int.}=\frac{L^{D}}{\beta^{2}}\underset{\omega_{n},\Omega_{n}}{\sum}\underset{\mathbf{k},\mathbf{q}}{\sum}[\Delta_{\mathbf{q}}^{*}(i\Omega_{n})\psi_{\uparrow}(i\Omega_{n}-i\omega_{n},\mathbf{\mathbf{q}-k})\psi_{\downarrow}(i\omega_{n},\mathbf{k})+\Delta_{\mathbf{q}}(i\Omega_{n})\psi_{\downarrow}^{\dagger}(i\omega_{n},\mathbf{k})\psi_{\uparrow}^{\dagger}(i\Omega_{n}-i\omega_{n},\mathbf{\mathbf{q}-k})].}
The Fourier transform of 1 body Green's 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 i=1,2} mean 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 {\mathbf{r}_{i},\tau_{i}}} ) 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 G(1-2)=\left\langle \psi(1)\psi^{*}(2)\right\rangle =\frac{1}{\beta}\underset{\omega_{n}}{\sum}\frac{1}{L^{D}}\underset{\mathbf{k}}{\sum}e^{-i\omega_{n}(\tau_{1}-\tau_{2})}e^{i\mathbf{k}\cdot(\mathbf{r}_{1}-\mathbf{r}_{2})}\frac{1}{-i\omega_{n}+\varepsilon_{\mathbf{k}}-\mu}} ,
so 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 G_{\sigma}^{0}(i\omega_{n},\mathbf{k})=\left\langle \psi_{\sigma}(i\omega_{n},\mathbf{k})\psi_{\sigma}^{\dagger}(i\omega_{n},\mathbf{k})\right\rangle _{0}=\frac{\beta}{L^{D}}\frac{1}{-i\omega_{n}+\varepsilon_{\mathbf{k}}-\mu}} .
Then 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 \left\langle S_{int.}^{2}\right\rangle _{0}=\frac{2L^{2D}}{\beta^{4}}\underset{\omega_{n},\Omega_{n}}{\sum}\underset{\mathbf{k},\mathbf{q}}{\sum}[G_{\uparrow}^{0}(i\omega_{n},\mathbf{k})G_{\downarrow}^{0}(i\Omega_{n}-i\omega_{n},\mathbf{q}-\mathbf{k})]\Delta_{\mathbf{q}}^{*}(i\Omega_{n})\Delta_{\mathbf{q}}(i\Omega_{n})=L^{D}\frac{2}{\beta}\underset{\Omega_{n},\mathbf{q}}{\sum}\chi_{p}(\mathbf{q},i\Omega_{n})\Delta_{\mathbf{q}}^{*}(i\Omega_{n})\Delta_{\mathbf{q}}(i\Omega_{n})} ,
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 \chi_{p}(\mathbf{q},i\Omega_{n})=\frac{L^{D}}{\beta^{3}}\underset{\omega_{n},\mathbf{k}}{\sum}G_{\uparrow}^{0}(i\omega_{n},\mathbf{k})G_{\downarrow}^{0}(i\Omega_{n}-i\omega_{n},\mathbf{q}-\mathbf{k})} is called pairing susceptibility.
Let's calculate it:
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 \chi_{p}(\mathbf{q},i\Omega_{n})=\frac{L^{D}}{\beta^{3}}\underset{\omega_{n},\mathbf{k}}{\sum}G_{\uparrow}^{0}(i\omega_{n},\mathbf{k})G_{\downarrow}^{0}(i\Omega_{n}-i\omega_{n},\mathbf{q}-\mathbf{k})=\frac{1}{L^{D}}\frac{1}{\beta}\underset{\omega_{n},\mathbf{k}}{\sum}\frac{-1}{i\omega_{n}-\varepsilon_{\mathbf{k}}+\mu}\times\frac{1}{i\omega_{n}-i\Omega_{n}+\varepsilon_{\mathbf{q}-\mathbf{k}}-\mu}} ,
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 \Rightarrow=\frac{1}{L^{D}}\frac{1}{\beta}\underset{\mathbf{k}}{\sum}\oint_{c}\frac{dz}{2\pi i}\frac{-1}{z-\varepsilon_{\mathbf{k}}+\mu}\times\frac{1}{z-i\Omega_{n}+\varepsilon_{\mathbf{q}-\mathbf{k}}-\mu}\frac{1}{e^{\beta z}+1}} .
Since 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 \frac{-1}{z-\varepsilon_{\mathbf{k}}+\mu}\times\frac{1}{z-i\Omega_{n}+\varepsilon_{\mathbf{q}-\mathbf{k}}-\mu}=\frac{1}{\varepsilon_{\mathbf{q}-\mathbf{k}}+\varepsilon_{\mathbf{k}}-2\mu-i\Omega_{n}}[\frac{1}{z-\varepsilon_{\mathbf{q}}+\mu}-\frac{1}{z-i\Omega_{n}+\varepsilon_{\mathbf{q}-\mathbf{k}}-\mu}]} ,
and change the integral path to
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 \Rightarrow=-\frac{1}{L^{D}}\frac{1}{\beta}\underset{\mathbf{k}}{\sum}\frac{1}{\varepsilon_{\mathbf{q}-\mathbf{k}}+\varepsilon_{\mathbf{k}}-2\mu-i\Omega_{n}}[\frac{1}{e^{\beta(\varepsilon_{\mathbf{q}}-\mu)}+1}-\frac{1}{e^{\beta(-\varepsilon_{\mathbf{q}-\mathbf{k}}+\mu)}+1}]=\int\frac{d^{D}k}{(2\pi)^{D}}\frac{1}{\varepsilon_{\mathbf{q}}+\varepsilon_{\mathbf{q}-\mathbf{k}}-2\mu-i\Omega_{n}}[1-f(\varepsilon_{\mathbf{k}})-f(\varepsilon_{\mathbf{q}-\mathbf{k}})].}
In the static (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 \ \Omega_{n}=0} ) and uniform (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 \mathbf{q}=0} ) limit,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 1-2f(\varepsilon_{\mathbf{k}})=Tanh[\frac{\beta}{2}(\varepsilon_{\mathbf{k}}-\mu)]} .
Then 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 \chi_{p}(0,0)=\int\frac{d^{D}k}{(2\pi)^{D}}\frac{Tanh[\frac{\beta}{2}(\varepsilon_{\mathbf{k}}-\mu)]}{2(\varepsilon_{\mathbf{k}}-\mu)}} .
In low energy, integrate the energy in the shell near Fermi energy:
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 \Rightarrow\chi_{p}(0,0)\cong N(0)\int_{\hbar\omega_{D}}^{-\hbar\omega_{D}}d\xi\frac{Tanh[\xi\beta/2]}{2\xi}\cong N(0)\int_{0}^{-\hbar\omega_{D}}d\xi\frac{Tanh[\xi\beta/2]}{\xi}=N(0)ln[\frac{2\hbar\omega_{D}e^{\gamma}}{\pi k_{B}T}].}
Then 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 \frac{1}{2}\left\langle S_{int.}^{2}\right\rangle _{0}=L^{D}\frac{1}{\beta}\chi_{p}(0,0)\underset{\Omega_{n},\mathbf{q}}{\sum}\Delta_{\mathbf{q}}^{*}(i\Omega_{n})\Delta_{\mathbf{q}}(i\Omega_{n})} .
If we ignore the higher order in the cumulant expansion,
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_{eff}=-\underset{\mathbf{r}}{\sum}\int_{0}^{\beta}d\tau\frac{1}{g}\Delta_{\mathbf{q}}^{*}(i\Omega_{n})\Delta_{\mathbf{q}}(i\Omega_{n})-\frac{1}{2}\left\langle S_{int.}^{2}\right\rangle _{0}=\underset{\mathbf{r}}{\sum}\int_{0}^{\beta}d\tau[\frac{1}{\left|g\right|}-N(0)ln(\frac{2\hbar\omega_{D}e^{\gamma}}{\pi k_{B}T})]\Delta^{*}(\tau,\mathbf{r})\Delta(\tau,\mathbf{r})} .
Because the partition function 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\Delta^{*}D\Delta e^{-S_{eff}(\Delta)}} , if we only consider the 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} related factors.
The superconductivity phase transition temperature is the temperature makes 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 \frac{1}{\left|g\right|}-N(0)ln(\frac{2\hbar\omega_{D}e^{\gamma}}{\pi k_{B}T})=0} , which 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 T_{c}=\frac{\hbar\omega_{D}}{k_{B}}\frac{2}{\pi}e^{\gamma}e^{-\frac{1}{N(0)\left|g\right|}}=1.134\frac{\hbar\omega_{D}}{k_{B}}e^{-\frac{1}{N(0)\left|g\right|}}} .
Beyond the critical temperature, the 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} related factors in the partition function is just 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 1} , the same as no cooper pair, which is normal state; below the critical temperature, the 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} related factors in the partition function will diverge, which means superconductivity phase transition.
Finite 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 \vec{q}} (small) 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 \ (\Omega_n=0)}
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 \chi_p (q,0)-\chi_p (0,0)=\frac{1}{L^D} \sum_k \frac{1}{\beta} \sum_{i\omega_n}\frac{-1}{i\omega_n-\epsilon_k+\mu}(\frac{1}{i\omega_n+\epsilon_{q-k}-\mu}-\frac{1}{i\omega_n+\epsilon_{-k}-\mu}) }
for small 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 \vec{q}} , 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 \epsilon_{q-k}=\epsilon_{-k}+q\frac{\partial \epsilon_\rho}{\partial \rho}|_{\rho=-k}=\epsilon_\vec{k}+\vec{q}\cdot \vec{v}_{-k}=\epsilon_k-\vec{q}\cdot\vec{v}_k}
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 \frac{1}{i\omega_n+\epsilon_k-\mu-q\cdot v_k}=\frac{1}{i\omega_n+\epsilon_k-\mu}+\frac{q\cdot v_k}{(i\omega_n+\epsilon_k-\mu)^2}+\frac{(q\cdot v_k)^2}{(i\omega_n+\epsilon_k-\mu)^3}}
Thus,
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 \begin{align}\chi_p(q,0)-\chi_p(0,0) &=\frac{1}{L^D}\sum_k\frac{1}{\beta}\sum_{i\omega_n}\frac{-1}{i\omega_n-\epsilon_k+\mu}\left(\frac{q\cdot v_k}{(i\omega_n+\epsilon_k-\mu)^2}+\frac{(q\cdot v_k)^2}{(i\omega_n+\epsilon_k-\mu)^3}+...\right) \\ &=\frac{-1}{\beta}\sum_i\omega_n \int \frac{d^D k}{(2\pi)^D}\frac{(\vec{q}\cdot \vec{v}_k)^2}{(i\omega_n-\epsilon_k+\mu)(i\omega_n+\epsilon_k-\mu)^3} \end{align}}
Consider the states near the 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 \hbar \omega_D} shell near fermi surface, we have
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 \chi_p(q,0)-\chi_p(0,0) =\frac{1}{\beta}\sum_{i\omega_n}\int\frac{d\Omega_{F.S.}}{\Omega_D}(q\cdot v_F)^2\int_{-\infty}^{+\infty} d\xi N(\xi+\mu)\frac{1}{(\xi-i\omega_n)(\xi+i\omega_n)^3}}
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 \ \xi=\epsilon_k-\mu}
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 \begin{align} \int_{-\infty}^{+\infty} d\xi \frac{1}{(\xi-i\omega_n)(\xi+i\omega_n)^3} &=\frac{2\pi i}{(2i\omega_n)^3}\theta(\omega_n)-\frac{2\pi i}{(2i\omega_n)^3}\theta(-\omega_n)\\ &=\frac{2\pi i}{(2i|\omega|)^3} \end{align} }
So,
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 \begin{align} \chi_p(q,0)-\chi_p(0,0) &=\frac{1}{\beta}\sum_{i\omega_n}N(0)\int\frac{d\Omega}{\Omega_D}(q\cdot v_F)^2\frac{2\pi i}{(2i|\omega|)^3}\\ &=N(0)v_F^2|\vec{q}|^2\int\frac{d\Omega}{\Omega_D}(q\cdot v_F)^2 \frac{1}{\beta}\sum_{i\omega}\frac{2\pi i}{-i8|\frac{(2n+1)\pi}{\beta}|^3}\\ &=-\frac{1}{4}N(0)v_F^2q^2(<(\hat{q}\cdot \hat{v_F})>_{F.S.})\frac{\beta^2}{\pi^2}(\sum_{N=-\infty}^{+\infty}\frac{1}{|2n+1|^3}) \end{align} }
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 \sum_{N=-\infty}^{+\infty}\frac{1}{|2n+1|^3}=\sum_{n=0}^\infty\frac{2}{(2n+1)^3}=\frac{2}{\pi}\frac{7\zeta(3)}{8} }
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 \ \zeta(3)} is Riemann zeta function.
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 <(\hat{q}\cdot \hat{v}_F)^2>_{F.S.}=\frac{1}{D} }
For spherical F.S. in 3D,
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 \int\frac{d\Omega}{\Omega_D}(\hat{q}\cdot\hat{v}_F)^2=\frac{2\pi}{4\pi}\int_{-1}^{1}dcos\theta cos^2\theta = \frac{1}{3} }
For circular F.S. in 2D,
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 \int\frac{d\Omega}{\Omega_D}(\hat{q}\cdot\hat{v}_F)^2=\frac{1}{2\pi}\int_{0}^{2\pi}d\theta cos^2\theta = \frac{1}{2} }
Then
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 \begin{align} \chi_p(q,0)-\chi_p(0,0) &=-\frac{1}{4}N(0)v_{F}^{2}q^{2}\frac{1}{D}\frac{\beta^{2}}{\pi^{2}}\frac{2}{\pi}\frac{7\zeta(3)}{8} \\ &=-N(0)\frac{7\zeta(3)}{16D\pi^{2}}q^{2}\frac{1}{\pi \hbar^{2}}\left(\frac{\hbar v_{F}}{k_{B}T}\right)^{2} \\ &\equiv-N(0)q^{2}\xi^{2} \end{align} }
So
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 \begin{align} \frac{1}{2}\left\langle S_{int.}^{2}\right\rangle _{0}&=L^{D}\frac{1}{\beta}\underset{\Omega_{n},\mathbf{q}}{\sum}\chi_{p}(q,0)\Delta_{\mathbf{q}}^{*}(i\Omega_{n})\Delta_{\mathbf{q}}(i\Omega_{n}) \\ &=N(0)ln[\frac{2\hbar\omega_{D}e^{\gamma}}{\pi k_{B}T}]L^{D}\frac{1}{\beta}\underset{\Omega_{n},\mathbf{q}}{\sum}\Delta_{\mathbf{q}}^{*}(i\Omega_{n})\Delta_{\mathbf{q}}(i\Omega_{n})-L^{D}\frac{1}{\beta}\underset{\Omega_{n},\mathbf{q}}{\sum}N(0)q^{2}\xi^{2}\Delta_{\mathbf{q}}^{*}(i\Omega_{n})\Delta_{\mathbf{q}}(i\Omega_{n}) \end{align} } .
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_{eff}=\underset{\mathbf{r}}{\sum}\int_{0}^{\beta}d\tau\left[\left(\frac{1}{\left|g\right|}-N(0)ln(\frac{2\hbar\omega_{D}e^{\gamma}}{\pi k_{B}T})\right)\Delta^{*}(\tau,\mathbf{r})\Delta(\tau,\mathbf{r})-N(0)\xi^{2}(\nabla\cdot\Delta^{*}(\tau,\mathbf{r}))(\nabla\cdot\Delta(\tau,\mathbf{r}))\right]} .
Note that the last term in the expression tells us that 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_{eff} } would increase if gradient of 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 } is not zero.
Note that the above expression has a one-one correspondant to the Giznburg-Landau functional:
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 F=\int d^{D}r\left[ \alpha (T-T_{c}) |\Psi(\vec{r})|^{2}+\frac{\hbar^{2}}{2m^{*}}|\nabla \Psi(\vec{r})|^{2} \right] } ,
here 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(\vec{r}) } corresponds to 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 \ \frac{\Delta(\tau,\vec{r})}{|g|N(0)a_{0}} } in 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_{eff}} .
Little Parks experiment
Refer to the fig, a thin shell of superconductor with radius R is shown and a small uniform magnetic field is passing through the hollow center of the cylinder. The experiment intends to show the variation of the critical temperature with change of the magnetic field passing through the hollow superconductor cylinder.
Before showing it, we first have to rewrite the Giznburg-Landau functional to make it taken the presence of magnetic field into account. Hamiltonian for a free electron moving in a magnetic field can be written as:
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 \frac{1}{2m}(p+\frac{eA}{c})^{2}\psi + V\psi = E\psi }
The physical observable magnetic field B would remain the same if we choose a different vector potential 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 A\rightarrow A+ \nabla \chi } (ie perform gauge transformation). To maintain the same eigen-energy E which is observable, the wave function have to undergo a phase change: 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 \rightarrow e^{i\phi}\psi } 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 \phi=\frac{e}{c\hbar}\chi }
Now in our Hamiltonian, the wave function is arranged as 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^{*}(\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}) }
since 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 \rightarrow e^{i\phi}\psi } , so if we want the Hamiltonian to remind the same, 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 } has to transform as 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 e^{-2i\phi}\Delta }
Since 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 } corresponds to 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 } in the Giznburg-Landau functional, so the Giznburg-Landau functional is modified as
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 F=\int d^{D}r\left[ \alpha (T-T_{c}) |\Psi(\vec{r})|^{2}+\frac{1}{2m^{*}}| ( \frac{\hbar \nabla}{i} - \frac{2e}{c}A(\vec{r}) ) \Psi(\vec{r})|^{2} \right] }
choose symmetric gauge: 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 \vec{A}=\frac{1}{2}\vec{H}\times\vec{r}=\frac{1}{2}Hr\hat{\phi} }
In cylindrical coordinate: 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 \vec{\nabla}=\hat{r}\frac{\partial}{\partial r} + \frac{\hat{\phi}}{r}\frac{\partial}{\partial \phi} + \hat{z}\frac{\partial}{\partial z} }
define unit flux as 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 \Phi_{0}=\frac{hc}{2e} }
define fluxoid as 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 \Phi(R) = \pi HR^{2}\ } , so we have
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 \begin{align} F&=\int d^{D}r\left[ \alpha (T-T_{c})|\Psi(\vec{r})|^{2} +\frac{\hbar^{2}}{2m^{*}}| (\frac{1}{R}\frac{\partial}{\partial \phi} - \frac{ie}{\hbar c} HR )\Psi(\vec{r}) |^{2}+ \frac{\hbar^{2}}{2m^{*}}|\frac{\partial}{\partial z} \Psi(\vec{r}) |^{2} \right] \\ &=\int d^{D}r\left[ \alpha (T-T_{c})|\Psi(\vec{r})|^{2} +\frac{\hbar^{2}}{2m^{*}R^{2}}| (\frac{\partial}{\partial \phi} - \frac{i\Phi}{\Phi_{0}} )\Psi(\vec{r}) |^{2} \right] \\ \end{align} }
When 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 \Phi = N\Phi_{0}\ } , the critical temperature will remain the same and the phase of 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\ } is changed as 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 \rightarrow e^{iN\phi} \Psi } . When 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 \Phi \neq N\Phi_{0}\ } , the critical temperature is found to vary as 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 T_{c}^{new}=T_{c}- \frac{\hbar^{2}}{2m^{*}R^{2}\alpha}\left (N-\frac{\Phi}{\Phi_{0}}\right )^{2}} . See the fig.
Microscopic derivation of the Giznburg-Landau functional
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=Z_{0}< e^{-S_{int}} >}
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 Z_{0}=\int D\psi ^{*} D\psi D\Delta ^{*} D\Delta e^{-(S_{\Delta} +S_{0})}}
we can expand this average for smallFailed 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} nearFailed 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 T_{c}} , for this perpose we can assume asecond order phase transition so that it increases continiously from zero to finite number after 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 T_{c}}
we need to calculate the average of 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 e^{-s_{int}}} which can be calculated by Tylor expansion:
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 e^{-S_{int}}=<-S_{int}+\frac{1}{2}S_{int}^{2}-\frac{1}{3}S_{int}^{3}+\frac{1}{4!}S_{int}^{4}+...>}
=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 1-<S_{int}>+\frac{1}{2} < S_{int}^{2}> -\frac{1}{3!}< S_{int}^{2}> +\frac{1}{4!}< S_{int}^{4}> +...}
.......................the odd power terms are zero because 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}(r,\tau )\psi _{\downarrow}(r,\tau ) > =0 }
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 =e^{\frac{1}{2}< S_{int}^{2}>}e^{\frac{1}{4!}< S_{int}^{4}>-\lambda }}
if we expand these two terms in to the second order the following expression can be got:
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 (1+\frac{1}{2} < S_{int}^{2}>+\frac{1}{2}(<\frac{1}{2} S_{int}^{2}>)^{2} +...)(1+\frac{1}{4!}< S_{int}^{4}>+...)}
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 =1+\frac{1}{2} < S_{int}^{2}>+\frac{1}{8}(< S_{int}^{2}>)^{2} +...)+\frac{1}{4!}< S_{int}^{4}>-\lambda +...}
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 \lambda} can be choosed in such a way .......
so, 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 \lambda =\frac{1}{8}< S_{int}^{2}> ^{2} }
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 =e^{\frac{1}{2} < S_{int}^{2}>+\frac{1}{4!}(< S_{int}^{4}>-3<S_{int}^{2}>^{2})+...}}
according to the expression we got before:
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_{int}=\frac{L^{D}}{\beta ^{2}}\sum_{\omega _{n},\Omega _{n}}\sum _{k,q}[\Delta ^{*}_{q}(i\Omega _{n})\psi_{\downarrow}(i\Omega _{n}-i\omega _{n}),\vec{q}-\vec{k})\psi_{\uparrow}(i\omega _{n},k)+\Delta _{q}(i\Omega_{n})\psi_{\uparrow}^{\dagger }(i\omega _{n},k)\psi_{\downarrow}^{\dagger }(i\Omega _{n}-i\omega _{n}),\vec{q}-\vec{k})]}
let's write 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_{int}} in terems od 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 a} for simplification. 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 a=\int \Delta ^{*}(1)\psi_{\downarrow}(1) \psi_{\uparrow}(1)+\Delta (1)\psi_{\downarrow}^{*}(1) \psi_{\uparrow}^{*}(1)}
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 a_{1}} is a couple grassman number, so we do not need to be worry about the sign when these terms comute with other terms.
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_{int}^{4}> =\int_{1234} < (a_{1}^{*}+a_{1})(a_{2}^{*}+a_{2})(a_{3}^{*}+a_{3})(a_{4}^{*}+a_{4})>}
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 =(< a_{1}^{*}a_{2}^{*}a_{3}a_{4}>+ < a_{1}^{*}a_{2}a_{3}^{*}a_{4}>+< a_{1}^{*}a_{2}a_{3}a_{4}^{*}>+< a_{1}a_{2}^{*}a_{3}^{*}a_{4}> +< a_{1}a_{2}^{*}a_{3}a_{4}^{*}>)}
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 =6< a_{1}^{*}a_{2}^{*}a_{3}a_{4}>=6\int _{1234}\Delta ^{*}(1)\Delta ^{*}(2)\Delta(3)\Delta(4)< \psi_{\downarrow}(1)\psi_{\uparrow}(1)\psi_{\downarrow}(2)\psi_{\uparrow}(2)\psi_{\downarrow}^{*}(3)\psi_{\uparrow}^{*}(3)\psi_{\downarrow}^{*}(4)\psi_{\uparrow}^{*}(4)> }
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 3< S_{int}^{2}> ^{2}=3\int_{1,2}2< a_{1}^{*}a_{2}> \int_{3,4}2< a_{3}^{*}a_{4}>=12\int_{1,2,3,4}\Delta ^{*}(1)\Delta ^{*}(2)\Delta(3)\Delta(4)< \psi_{\downarrow}(1)\psi_{\uparrow}(1)\psi_{\uparrow}^{*}(3)\psi_{\downarrow}^{*}(3)>< \psi_{\downarrow}(2)\psi_{\uparrow}(2)\psi_{\uparrow}^{*}(4)\psi_{\downarrow}^{*}(4)>}
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 -2G(2-3) G(2-4)G(1-4)G(1-3)=-12\int_{1,2,3,4}\Delta ^{*}(1)\Delta _{*}(2)\Delta (3)\Delta (4)G(2-3)G(2-4)G(1-4)G(1-3)}
Recall
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 G(2-3)=< \psi (r_{2},\tau _{2})\psi ^{*}(r_{3},\tau _{3})>=\frac{1}{\beta } \sum_{\omega _{n}}\frac{1}{L^{D}}\sum_{k}e^{-i\omega _{n}(\tau _{2}-\tau _{3})}e^{ik.(r_{2}-r_{3})}\frac{1}{-i\omega _{n}+\epsilon _{k}-\mu }}
see the solution which are 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 \tau} independent
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 \frac{1}{\beta ^{4}}\sum_{\omega _{{n}_{1}}}...\sum_{\omega _{{n}_{4}}}\int_{0}^\beta {d\tau_{1}}\int_{0}^\beta {d\tau_{2}} \int_{0}^\beta {d\tau_{3}} \int_{0}^\beta {d\tau_{4}}e^{-i\omega _{{n}_{1}}(\tau _{1}-\tau _{3})} e^{-i\omega _{{n}_{2}}(\tau _{1}-\tau _{4})}e^{-i\omega _{{n}_{3}}(\tau _{2}-\tau _{3})}e^{-i\omega _{{n}_{4}}(\tau _{2}-\tau _{4})} G(i\omega _{{n}_{1}},r_{1}-r_{3})G(i\omega _{{n}_{2}},r_{1}-r_{4})G(i\omega _{{n}_{3}},r_{2}-r_{3})G(i\omega _{{n}_{4}},r_{2}-r_{4}) }
after getting integration over 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 \tau_{1}} we will get 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 \beta \delta (\omega _{n_{1}},-\omega _{n_{2}}) } and similarly by getting integration over 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 \tau_{2}} we have 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 \beta \delta (\omega _{n_{3}},-\omega _{n_{4}}) }
So, the final result can be written: 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 \sum_{i\omega _{n}}G(i\omega _{n},r_{1}-r_{3}G(-i\omega _{n},r_{2}-r_{3})G(-i\omega _{n},r_{1}-r_{4})G(i\omega _{n},r_{2}-r_{4}) }
Now, we wish to perform gradiant expansion:
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 ^{\ast }(1)=\Delta ^{\ast }(\frac{r_{1}+r_{4}}{2}+\frac{r_{1}-r_{4}}{2})=\Delta ^{\ast }(\frac{r_{1}+r_{4}}{2})+(\frac{r_{1}-r_{4}}{2})\nabla\Delta^{\ast }(\frac{r_{1}+r_{4}}{2})+... }
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 ^{\ast }(2)=\Delta ^{\ast }(\frac{r_{2}+r_{3}}{2}+\frac{r_{2}-r_{3}}{2})=\Delta ^{\ast }(\frac{r_{2}+r_{3}}{2})+(\frac{r_{2}-r_{3}}{2})\nabla\Delta^{\ast }(\frac{r_{2}+r_{3}}{2})+... } 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 -12\int d^{D}R_{1,4}d^{D}R_{2,3}d^{D}\mu _{1,4}d^{D}\mu_{2,3}\Delta ^{\ast }(R_{1,4})\Delta ^{\ast }(R_{2,3})\Delta (R_{2,3}) \Delta (R_{1,4})\sum_{\omega_{n}}G(i\omega_{n},R_{1,4}-R_{2,3}+\frac{1}{2}(\mu _{1,4}+\mu _{2,3}))}
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 R=\frac{1}{2}(R_{1,4}+R_{2,3})}
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 \mu =R_{1,4}-R_{2,3} }
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 r_{1}-r_{3}=\frac{1}{2}(\mu _{1,4}+\mu_{2,3})+\mu}
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 r_{2}-r_{4}=\frac{1}{2}(\mu _{1,4}+\mu_{2,3})-\mu}
Starting from the microscopic model, we found that 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\backsimeq Z_{0}\int D\Delta*D\Delta e^{-S_{eff}} }
,
where the 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 4^{th}}
order in 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 keeping only quadratic qradient terms, we have:
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_{eff}=\frac{1}{k_{B}T}\sum_{r}\left[\underset{A}{\underbrace{\left(\frac{1}{|g|}-N(0)In\left[\frac{2\hbar\omega_{D}e^{\gamma_{E}}}{\pi k_{B}T}\right]\right)}}|\Delta(r)|^{2}+N(0)\xi^{2}(\nabla\Delta*(r)).(\nabla\Delta(r))+\frac{1}{2}\underset{B}{\underbrace{\frac{7\zeta(3)N(0)}{8\pi^{2}k_{B}^{2}T^{2}}}}|\Delta(r)|^{4}\right]}
Effects of an applied magnetic field; Type I and Type II superconductivity
Derivation of the Ginzburg-Landau equations
Our starting point will be the Ginzburg-Landau (GL) free energy in the presence of an external magnetic field,
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 F=\int d^d \vec{r} \left [\alpha(T-T_c)|\Psi(\vec{r})|^2+\tfrac{1}{2}b|\Psi(\vec{r})|^4+\frac{\hbar^2}{2m}\left |\left (\nabla-i\frac{2e}{\hbar c}\vec{A}(\vec{r})\right )\Psi(\vec{r})\right |^2+\frac{1}{8\pi}(\nabla\times\vec{A}(\vec{r}))^2-\frac{1}{c}\vec{J}_{\text{ext}}(\vec{r})\cdot\vec{A}(\vec{r})\right ],}
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 \vec{A}} is the total vector potential 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 \vec{J}_{\text{ext}}} is an external current density, assumed to be controlled experimentally. This current satisfies
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 \nabla\times\vec{H}=\frac{4\pi}{c}\vec{J}_{\text{ext}},}
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 \vec{H}} is the external magnetic field. The expression is the sum of the energy due to the superconducting order parameter, with the magnetic field introduced via the gauge invariance argument given above, the energy of the magnetic field alone, and the work done by the superconductor to maintain the external current at a constant value.
Let us first derive the "saddle point" equations satisfied by the magnetic field in the normal state. In this case, we set 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} to zero everywhere and set
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 \left. \frac{\delta F}{\delta\vec{A}(\vec{r})}\right |_{\vec{A}=\vec{A}_\text{min}}=0.}
We will find this derivative by first finding the variation 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 F} in the free energy for this case, which 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 \delta F=\int d^d \vec{r}' \left [\frac{1}{4\pi}(\nabla\times\vec{A}(\vec{r}'))\cdot(\nabla\times\delta\vec{A}(\vec{r}'))-\frac{1}{c}\vec{J}_{\text{ext}}(\vec{r}')\cdot\delta\vec{A}(\vec{r}')\right ],}
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 \delta\vec{A}} is a small variation in the vector potential; we assume that it vanishes on the "surface" of our system. We now transform the first term using the identity,
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 (\nabla\times\vec{A})\cdot(\nabla\times\vec{B})=\nabla\cdot[\vec{A}\times(\nabla\times\vec{B})]+\vec{A}\cdot[\nabla\times(\nabla\times\vec{B})],}
obtaining
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 F=\int d^d \vec{r}' \left [\frac{1}{4\pi}\nabla\cdot[\delta\vec{A}(\vec{r}')\times(\nabla\times\vec{A}(\vec{r}'))]+\frac{1}{4\pi}\delta\vec{A}(\vec{r}')\cdot[\nabla\times(\nabla\times\vec{A}(\vec{r}'))]-\frac{1}{c}\vec{J}_{\text{ext}}(\vec{r}')\cdot\delta\vec{A}(\vec{r}')\right ].}
The first term is a "surface" term; since we assumed that 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\vec{A}} vanishes everywhere on the "surface", we are left with just
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 F=\int d^d \vec{r} \left [\frac{1}{4\pi}[\nabla\times(\nabla\times\vec{A}(\vec{r}'))]-\frac{1}{c}\vec{J}_{\text{ext}}(\vec{r}')\right ]\cdot\delta\vec{A}(\vec{r}').}
We conclude that the variational derivative that we are interested in 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 \frac{\delta F}{\delta\vec{A}(\vec{r})}=\frac{1}{4\pi}[\nabla\times(\nabla\times\vec{A}(\vec{r}'))]-\frac{1}{c}\vec{J}_{\text{ext}}(\vec{r}').}
At the "saddle point", this derivative is zero, so we obtain the equation,
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 \nabla\times(\nabla\times\vec{A})=\frac{4\pi}{c}\vec{J}_{\text{ext}}.}
We may introduce the total magnetic field 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 \vec{B}=\nabla\times\vec{A}} , thus obtaining
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 \nabla\times\vec{B}=\frac{4\pi}{c}\vec{J}_{\text{ext}}.}
Comparing this to the definition of 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 \vec{J}_{\text{ext}}} given above, we conclude that 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 \vec{B}=\vec{H}} in the normal state. In reality, this will only be approximately true due to para- or diamagnetic effects in the metal, but these effects will be small in comparison to those due to superconductivity, which we will now derive.
First, we will apply the "saddle point" condition for the superconducting order parameter, 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} , which 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 \left. \frac{\delta F}{\delta\Psi^{*}(\vec{r})}\right |_{\Psi=\Psi_{\text{min}}}=0.}
Again, we start by finding the variation in the free energy in terms of a small variation 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\Psi^{*}} in the order parameter:
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 F=\int d^d \vec{r}' \left [\alpha(T-T_c)\Psi(\vec{r}')\,\delta\Psi^{*}(\vec{r}')+b|\Psi(\vec{r}')|^2\Psi(\vec{r}')\,\delta\Psi^{*}(\vec{r}')-\frac{e}{mc}\vec{A}(\vec{r}')\cdot\left (\frac{\hbar}{i}\nabla-\frac{2e}{c}\vec{A}(\vec{r}')\right )\Psi(\vec{r}')\,\delta\Psi^{*}(\vec{r}')-\frac{1}{2m}\frac{\hbar}{i}\nabla\delta\Psi^{*}(\vec{r}')\cdot\left (\frac{\hbar}{i}\nabla-\frac{2e}{c}\vec{A}(\vec{r}')\right )\Psi(\vec{r}')\right ]}
The last term is equal to
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 -\frac{1}{2m}\left\{\frac{\hbar}{i}\nabla\cdot\left [\left (\frac{\hbar}{i}\nabla-\frac{2e}{c}\vec{A}(\vec{r}')\right )\Psi(\vec{r}')\,\delta\Psi^{*}(\vec{r}')\right ]-\left [\frac{\hbar}{i}\nabla\cdot\left (\frac{\hbar}{i}\nabla-\frac{2e}{c}\vec{A}(\vec{r}')\right )\Psi(\vec{r} ')\right ]\delta\Psi^{*}(\vec{r}')\right\}.}
The second term in this expression is a "surface" term. If we assume that 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\Psi^{*}} is zero on the "surface", then this term vanishes, leaving us with
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 F=\int d^d \vec{r}' \left\{\alpha(T-T_c)\Psi(\vec{r}')+b|\Psi(\vec{r}')|^2\Psi(\vec{r}')-\frac{e}{mc}\vec{A}(\vec{r}')\cdot\left (\frac{\hbar}{i}\nabla-\frac{2e}{c}\vec{A}(\vec{r}')\right )\Psi(\vec{r}')+\frac{1}{2m}\left [\frac{\hbar}{i}\nabla\cdot\left (\frac{\hbar}{i}\nabla-\frac{2e}{c}\vec{A}(\vec{r}')\right )\Psi(\vec{r}')\right ]\right\}\delta\Psi^{*}(\vec{r}').}
We can now immediately write down the variational derivative, which, upon being set to zero, gives us the first GL equation,
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 \frac{1}{2m}\left (\frac{\hbar}{i}\nabla-\frac{2e}{c}\vec{A}\right )^2\Psi+\alpha(T-T_c)\Psi+b|\Psi|^2\Psi=0.}
We also need to minimize the free energy with respect to the magnetic field. We have already done this for the normal case, and there is only one more term that we need to consider in the superconducting case; we will therefore only treat this term. We can quickly write down the variation in the superconducting part of the free energy 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 F_{SC}} , which 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 \delta F_{SC}=i\frac{e\hbar}{mc}\int d^d \vec{r}' \left [\Psi^{*}(\vec{r}')\left (\nabla-i\frac{2e}{\hbar c}\vec{A}(\vec{r}')\right )\Psi(\vec{r}')-\Psi(\vec{r}')\left (\nabla+i\frac{2e}{\hbar c}\vec{A}(\vec{r}')\right )\Psi^{*}(\vec{r}')\right ]\cdot\delta\vec{A}(\vec{r}').}
Combining this result with the previous result for the normal metal, we obtain the second GL equation,
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 \frac{1}{4\pi}\nabla\times(\nabla\times\vec{A})-\frac{1}{c}\vec{J}_{\text{ext}}-\frac{e}{mc}\left (\Psi^{*}\frac{\hbar}{i}\nabla\Psi-\Psi\frac{\hbar}{i}\nabla\Psi^{*}\right )+\frac{4e^2}{mc^2}|\Psi|^2\vec{A}=0,}
or, introducing 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 \vec{B}} 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 \vec{H}} ,
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 \frac{e}{m}\left (\Psi^{*}\frac{\hbar}{i}\nabla\Psi-\Psi\frac{\hbar}{i}\nabla\Psi^{*}\right )-\frac{4e^2}{mc}|\Psi|^2\vec{A}=\frac{c}{4\pi}\nabla\times(\vec{B}-\vec{H}).}
Given the definition of 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 \vec{H}} and the Maxwell equation (assuming static fields),
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 \nabla\times\vec{B}=\frac{4\pi}{c}\vec{J},}
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 \vec{J}} is the total current density, we conclude that the left-hand side of this equation is the current density induced inside the superconductor.
Let us now suppose that we do not assume that 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\Psi^{*}} vanishes on the surface. It may then be shown that the following boundary condition holds on the surface (see P. G. de Gennes, Superconductivity in Metals and Alloys):
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 \hat{n}\cdot\left (\frac{\hbar}{i}\nabla-\frac{2e}{c}\vec{A}\right )\Psi=\frac{i\hbar}{b_{dG}}\Psi.}
This relation holds for a superconductor-metal interface; for a superconductor-insulator interface, 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 b_{dG}\rightarrow\infty} . We may show that this condition implies that the normal component of the current density on the surface vanishes. If we multiply the above condition by 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^{*}} on both sides, we obtain
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 \hat{n}\cdot\Psi^{*}\left (\frac{\hbar}{i}\nabla-\frac{2e}{c}\vec{A}\right )\Psi=\frac{i\hbar}{b_{dG}}\Psi^{*}\Psi.}
Taking the complex conjugate of both sides gives us
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 \hat{n}\cdot\Psi\left (-\frac{\hbar}{i}\nabla-\frac{2e}{c}\vec{A}\right )\Psi^{*}=-\frac{i\hbar}{b_{dG}}\Psi^{*}\Psi.}
Adding these two equations together gives us
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 \hat{n}\cdot\left [\left (\Psi^{*}\frac{\hbar}{i}\nabla\Psi-\Psi\frac{\hbar}{i}\nabla\Psi^{*}\right )-\frac{4e}{c}|\Psi|^2\vec{A}\right ]=0.}
The left-hand side is proportional to the normal component of the current density inside the superconductor.
The GL Equations in Dimensionless Form
We will find it convenient to introduce dimensionless variables when working with the GL equations. We start by introducing a dimensionless order parameter, 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=\frac{\Psi}{\Psi_0}} , 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 \Psi_0^2=\frac{\alpha(T_C-T)}{b}.}
We may rewrite the first GL equation in terms of this parameter as
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 \frac{1}{2mb\Psi_0^2}\left (\frac{\hbar}{i}\nabla-\frac{2e}{c}\vec{A}\right )^2\psi-(\left |\psi\right |^2-1)\psi=0,}
and the second as
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 \frac{e\Psi_0^2}{mc}\left (\psi^{\ast}\frac{\hbar}{i}\nabla\psi-\psi\frac{\hbar}{i}\nabla\psi^{\ast}\right )-\frac{4e^2\Psi_0^2}{mc^2}\left |\psi\right |^2\vec{A}=\frac{1}{4\pi}\nabla\times[\nabla\times(\vec{A}-\vec{A}_0)],}
where we re-introduced 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 \vec{A}} into the right-hand side and also introduced 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 \vec{A}_0} , defined as
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 \vec{H}=\nabla\times\vec{A}_0.}
Next, we introduce a dimensionless position vector,
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 \tilde{r}=\frac{1}{\lambda}\vec{r},}
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 \lambda=\sqrt{\frac{mc^2}{16\pi e^2\Psi_0^2}}} is known as the penetration depth of the superconductor; we will see where this name comes from shortly. In terms of this vector, the first GL equation becomes
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 \left (\frac{1}{\Psi_0\lambda\sqrt{2mb}}\tilde{\nabla}-\frac{1}{\Psi_0\sqrt{2mb}}\frac{2e}{c}\vec{A}\right )^2\psi+(\left |\psi\right |^2-1)\psi=0}
and the second becomes
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 \frac{4\pi e\lambda\Psi_0^2}{mc}\left (\psi^{\ast}\frac{\hbar}{i}\tilde{\nabla}\psi-\psi\frac{\hbar}{i}\tilde{\nabla}\psi^{\ast}\right )-\left |\psi\right |^2\vec{A}=\tilde{\nabla}\times[\tilde{\nabla}\times(\vec{A}-\vec{A}_0)].}
Finally, we introduce a dimensionless vector potential,
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 \tilde{A}=\frac{1}{\Psi_0\sqrt{2mb}}\frac{2e}{c}\vec{A}}
and the dimensionless parameter,
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 \kappa=\frac{\Psi_0\lambda\sqrt{2mb}}{\hbar}.}
In terms of these, the first GL equation becomes
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 \left (-\frac{i}{\kappa}\tilde{\nabla}-\tilde{A}\right )^2\psi+(\left |\psi\right |^2-1)\psi=0}
and the second becomes
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 \frac{1}{2\kappa}\left (\psi^{\ast}\frac{\tilde{\nabla}}{i}\psi-\psi\frac{\tilde{\nabla}}{i}\psi^{\ast}\right )-\left |\psi\right |^2\vec{A}=\tilde{\nabla}\times[\tilde{\nabla}\times(\tilde{A}-\tilde{A}_0)].}
We see that our theory has a dimensionless parameter in it, namely 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 \kappa} , which is known as the Ginzburg-Landau parameter.
Going below 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 T_c} with the Saddle Point Approximation
So, previously all of this work has shown us the behavious of a superconducting system near 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 T_c} only. If we want to go into lower temperatures, we will have to make a careful saddle-point approximation, following Bardeen, Cooper, and Schrieffer (BCS). Once again, we can start from our microscopic 'toy' Hamiltonian, and gain useful information.
Recall that the partition function can be written,
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 \mathbb{Z} = \int{d\Delta^* d\Delta} \left[ \int{D\psi^* D\psi \ e^{-S_{BCS} - S_{\Delta}}} \right] }
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} = \frac{-1}{g} \int_{0}^{\beta}{d\tau} \int{d^3r}\Delta^*(\vec{r},\tau) \Delta(\vec{r},\tau) }
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 S_{BCS} = S_0 + S_{int} \;\; \text{where} \;\; S_0 = \int_{0}^{\beta}{d\tau}\int{d^3r}\left[ \psi_{\sigma}^*(\vec{r},\tau) \left( \frac{\partial}{\partial \tau} + \epsilon - \mu \right) \psi_{\sigma}(\vec{r},\tau) \right] } 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 \text{and} \;\; S_{int} = \int_{0}^{\beta}{d\tau}\int{d^3r} \left[\Delta^*(\vec{r},\tau)\psi_{\downarrow}(\vec{r},\tau) \psi_{\uparrow}(\vec{r},\tau) + \Delta(\vec{r},\tau)\psi_{\downarrow}(\vec{r},\tau)\psi_{\uparrow}(\vec{r}.\tau) \right]}
Beyond saddle-point approximation, collective modes and response in the broken symmetry state
Recall we can write our partition function as
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\Delta^*D\Delta e^{-S_{eff}[\Delta]}}
or
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(\Re e\Delta)D(\Im m\Delta)e^{-S_{eff}[\Re e\Delta,\Im m\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_{eff}[\Delta]=\frac{1}{|g|}\int_0^\beta d\tau\int d^Dr\left[(\Re e\Delta(r,\tau))^2+(\Im m\Delta(r,\tau))^2\right]-\ln\left[\int D\psi^*D\psi e^{-S_0-S_{int}}\right]}
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 \begin{align} S_0&=\int_0^\beta d\tau\int d^Dr \psi^*(r,\tau)(\frac{\partial}{\partial\tau}+\epsilon_p-\mu)\psi(r,\tau)\\ S_{int}&=\int_0^\beta d\tau\int d^Dr(\Delta^*(r,\tau)\psi_\downarrow(r,\tau)\psi_{\uparrow}(r,\tau)+\Delta(r,\tau)\psi^*_\uparrow(r,\tau)\psi^*_{\downarrow}(r,\tau)) \end{align}}
We can rewrite the interaction term in the action as
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_{int}=\int_0^\beta d\tau\int d^Dr\left[\Re e(\Delta(r,\tau))(\psi_{\downarrow}\psi_{\uparrow}+\psi_{\uparrow}^*\psi_{\downarrow}^*)(r,\tau)+i*\Im m(\Delta(r,\tau))(\psi_{\uparrow}^*\psi_{\downarrow}^*-\psi_{\downarrow}\psi_{\uparrow})(r,\tau)\right]}
Consider now functional derivatives of 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_{eff}[\Delta]} :
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 \frac{\delta S_{eff}[\Delta]}{\delta\Re e\Delta(r,\tau)}=\frac{2}{|g|}\Re e\Delta(r,\tau)-\frac{1}{\int D\psi^*D\psi e^{-S_0-S_{int}}} \int D\psi^*D\psi e^{-S_0-S_{int}}\left(-\frac{\delta S_{int}}{\delta \Re e\Delta(r,\tau)}\right) }
The functional derivative of 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_{int}} with 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 \Re e\Delta(r,\tau)} 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 \frac{\delta S_{int}}{\delta \Re e\Delta(r,\tau)}=\psi_{\downarrow}(r,\tau)\psi_{\uparrow}(r,\tau)+\psi_{\uparrow}^*(r,\tau)\psi_{\downarrow}^*(r,\tau)}
Therefore, we have
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 \frac{\delta S_{eff}[\Delta]}{\delta \Re e\Delta(r,\tau)}=\frac{2}{|g|}\Re e\Delta(r,\tau)+\langle\psi_{\downarrow}(r,\tau)\psi_{\uparrow}(r,\tau)+\psi_{\uparrow}^*(r,\tau)\psi_{\downarrow}^*(r,\tau)\rangle\ \ (1)}
Similarly
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 \frac{\delta S_{eff}[\Delta]}{\delta \Im m\Delta(r,\tau)}=\frac{2}{|g|}\Im m\Delta(r,\tau)+i\langle\psi_{\uparrow}^*(r,\tau)\psi_{\downarrow}^*(r,\tau)-\psi_{\downarrow}(r,\tau)\psi_{\uparrow}(r,\tau)\rangle\ \ (2)}
If we were to set the LHS of the above two equations to zero, we would obtain our self-consistency conditions.
The strategy is to take
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_{eff}[\Delta(r,\tau)]=S_{eff}\left[\Delta_{sp}+(\Delta(r,\tau)-\Delta_{sp})\right]}
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 \Delta_{sp}} solves (1) and (2) with LHS set to zero and expand in powers of 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(r,\tau)-\Delta_{sp}} . So
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 \begin{align} &S_{eff}[\Delta_{sp}+(\Delta(r,\tau)-\Delta_{sp})]\\ &=S_{eff}[\Delta_{sp}]+\int_0^\beta d\tau\int d^3r\Re e(\Delta(r,\tau)-\Delta_{sp})\frac{\delta S_{eff}}{\delta\Re e\Delta(r,\tau)}|_{\Delta_{sp}}+\int_0^\beta d\tau\int d^3r\Im m(\Delta(r,\tau)-\Delta_{sp})\frac{\delta S_{eff}}{\delta\Im m\Delta(r,\tau)}|_{\Delta_{sp}}\\ &+\frac{1}{2}\int_0^\beta d\tau\int_0^\beta d\tau'\int d^3r\int d^3r'\Re e(\Delta(r,\tau)-\Delta_{sp})\Re e(\Delta(r',\tau')-\Delta_{sp})\frac{\delta^2 S_{eff}}{\delta\Re e\Delta(r,\tau)\delta\Re e\Delta(r',\tau')}|_{\Delta_{sp}}\\ &+\frac{1}{2}\int_0^\beta d\tau\int_0^\beta d\tau'\int d^3r\int d^3r'\Im m(\Delta(r,\tau)-\Delta_{sp})\Im m(\Delta(r',\tau')-\Delta_{sp})\frac{\delta^2 S_{eff}}{\delta\Im m\Delta(r,\tau)\delta\Im m\Delta(r',\tau')}|_{\Delta_{sp}}\\ &+\int_0^\beta d\tau\int_0^\beta d\tau'\int d^3r\int d^3r'\Re e(\Delta(r,\tau)-\Delta_{sp})\Im m(\Delta(r',\tau')-\Delta_{sp})\frac{\delta^2 S_{eff}}{\delta\Re e\Delta(r,\tau)\delta\Im m\Delta(r',\tau')}|_{\Delta_{sp}}+... \end{align}}
Since by definition of 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_{sp}} we have
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 \frac{\delta S_{eff}}{\delta\Re e\Delta(r,\tau)}|_{\Delta_{sp}}=\frac{\delta S_{eff}}{\delta\Im m\Delta(r,\tau)}|_{\Delta_{sp}}=0}
So only the 0th and 2nd order terms contribute. The 0th order term gave us the condensation energy, and the 2nd order term will give us information about collective modes (in the broken symmetry phase).
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 \begin{align} &\frac{\delta^2 S_{eff}}{\delta\Re e\Delta(r,\tau)\delta\Re e\Delta(r',\tau')}\\ &=\frac{2}{|g|}\delta(\tau-\tau')\delta(r-r')+\frac{\delta}{\delta\Re e\Delta(r,\tau)}\left[\frac{1}{\int D\psi^*D\psi e^{-S_0-S_{int}}}\int D\psi^*D\psi e^{-S_0-S_{int}}\left(\frac{\delta S_{int}}{\delta \Re e\Delta(r',\tau')}\right)\right]\\ &=\frac{2}{|g|}\delta(\tau-\tau')\delta(r-r')-\frac{\int D\psi^*D\psi e^{-S_0-S_{int}}\left(\frac{\delta S_{int}}{\delta \Re e\Delta(r,\tau)}\right)\left(\frac{\delta S_{int}}{\delta \Re e\Delta(r',\tau')}\right)}{\int D\psi^*D\psi e^{-S_0-S_{int}}}\\ &+\left(\frac{\int D\psi^*D\psi e^{-S_0-S_{int}}\left(\frac{\delta S_{int}}{\delta \Re e\Delta(r,\tau)}\right)}{\int D\psi^*D\psi e^{-S_0-S_{int}}}\right)\left(\frac{\int D\psi^*D\psi e^{-S_0-S_{int}}\left(\frac{\delta S_{int}}{\delta \Re e\Delta(r',\tau')}\right)}{\int D\psi^*D\psi e^{-S_0-S_{int}}}\right) \end{align}}
So
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 \begin{align} &\frac{\delta^2 S_{eff}}{\delta\Re e\Delta(r,\tau)\delta\Re e\Delta(r',\tau')}=\frac{2}{|g|}\delta(\tau-\tau')\delta(r-r')\\ &-\langle\left(\psi_{\downarrow}(r,\tau)\psi_{\uparrow}(r,\tau)+\psi^*_{\uparrow}(r,\tau)\psi^*_{\downarrow}(r,\tau)\right)\left(\psi_{\downarrow}(r',\tau')\psi_{\uparrow}(r',\tau')+\psi^*_{\uparrow}(r',\tau')\psi^*_{\downarrow}(r',\tau')\right)\rangle_{S_0+S_{int}}\\ &+\langle\psi_{\downarrow}(r,\tau)\psi_{\uparrow}(r,\tau)+\psi^*_{\uparrow}(r,\tau)\psi^*_{\downarrow}(r,\tau)\rangle\langle\psi_{\downarrow}(r',\tau')\psi_{\uparrow}(r',\tau')+\psi^*_{\uparrow}(r',\tau')\psi^*_{\downarrow}(r',\tau')\rangle_{S_0+S_{int}} \end{align}}
i.e.
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 \begin{align} &\frac{\delta^2 S_{eff}}{\delta\Re e\Delta(r,\tau)\delta\Re e\Delta(r',\tau')}=\frac{2}{|g|}\delta(\tau-\tau')\delta(r-r')\\ &-\langle\left(\psi_{\downarrow}(r,\tau)\psi_{\uparrow}(r,\tau)+\psi^*_{\uparrow}(r,\tau)\psi^*_{\downarrow}(r,\tau)\right)\left(\psi_{\downarrow}(r',\tau')\psi_{\uparrow}(r',\tau')+\psi^*_{\uparrow}(r',\tau')\psi^*_{\downarrow}(r',\tau')\right)\rangle_{S_0+S_{int}}^{\mbox{con}} \end{align}}
Similarly,
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 \begin{align} &\frac{\delta^2 S_{eff}}{\delta\Im m\Delta(r,\tau)\delta\Im m\Delta(r',\tau')}=\frac{2}{|g|}\delta(\tau-\tau')\delta(r-r')\\ &+\langle\left(\psi^*_{\uparrow}(r,\tau)\psi^*_{\downarrow}(r,\tau)-\psi_{\downarrow}(r,\tau)\psi_{\uparrow}(r,\tau)\right)\left(\psi^*_{\uparrow}(r',\tau')\psi^*_{\downarrow}(r',\tau')-\psi_{\downarrow}(r',\tau')\psi_{\uparrow}(r',\tau')\right)\rangle_{S_0+S_{int}}^{\mbox{con}} \end{align}}
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 \frac{\delta^2 S_{eff}}{\delta\Im m\Delta(r,\tau)\delta\Re e\Delta(r',\tau')}=-i\langle\left(\psi^*_{\uparrow}(r,\tau)\psi^*_{\downarrow}(r,\tau)-\psi_{\downarrow}(r,\tau)\psi_{\uparrow}(r,\tau)\right)\left(\psi_{\downarrow}(r',\tau')\psi_{\uparrow}(r',\tau')+\psi^*_{\uparrow}(r',\tau')\psi^*_{\downarrow}(r',\tau')\right)\rangle_{S_0+S_{int}}^{\mbox{con}} }
If we evaluate these functional derivatives at the saddle point 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_{sp}} we have
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 \begin{align} S_0+S_{int}&\rightarrow\int_0^\beta d\tau\int d^3r\sum_{\sigma=\uparrow,\downarrow}\psi^*_\sigma(r,\tau)\left(\frac{\partial}{\partial\tau}+\epsilon_p-\mu\right)\psi_{\sigma}(r,\tau)\\ &+\Delta_{sp}\int_0^\beta d\tau\int d^3r\left(\psi_{\downarrow}(r,\tau)\psi_{\uparrow}(r,\tau)+\psi_{\uparrow}^*(r,\tau)\psi_{\downarrow}^*(r,\tau)\right) \end{align}}
where we take the saddle point solution 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_{sp}} to be purely real. Arranging the Grassman fields into the Nambu spinor we have:
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)\left(\begin{align}&\frac{\partial}{\partial\tau}+\epsilon_p-\mu&\Delta_{sp}\\ &\Delta_{sp}&\frac{\partial}{\partial\tau}-\epsilon_p+\mu\end{align}\right) \left(\begin{align}&\psi_\uparrow\\&\psi^*_\downarrow\end{align}\right)}
where the Nambu spinor is defined as
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=\left(\begin{align}&\psi_\uparrow\\&\psi^*_\downarrow\end{align}\right)}
So
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 \begin{align} &\psi^*_\uparrow\psi^*_\downarrow+\psi_\downarrow\psi_\uparrow=\Psi^*\left(\begin{align}&0&1\\&1&0\end{align}\right)\Psi =\Psi^*\sigma_x\Psi\\ &\psi^*_\uparrow\psi^*_\downarrow-\psi_\downarrow\psi_\uparrow=\Psi^*\left(\begin{align}&0&1\\-&1&0\end{align}\right)\Psi =\Psi^*i\sigma_y\Psi \end{align}}
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 \begin{align} \frac{\delta^2S_{eff}}{\delta\Re e\Delta(r,\tau)\delta\Re e\Delta(r',\tau')}|_{\Delta_{sp}}&=\frac{2}{|g|}\delta(\tau-\tau')\delta(r-r')-\langle\Psi^*(r,\tau)\sigma_x\Psi(r,\tau)\Psi^*(r',\tau')\sigma_x\Psi(r',\tau')\rangle^{\mbox{con}}\\ &=\frac{2}{|g|}\delta(\tau-\tau')\delta(r-r')+\mbox{Tr}\left[\sigma_xG(r-r',\tau-\tau')\sigma_xG(r'-r,\tau'-\tau)\right] \end{align}}
where the Green's functions are 2*2 matrices,
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 G^{-1}_k(i\omega_n)=\left(\begin{align}&-i\omega_n+\epsilon_k-\mu&\Delta_{sp}\\&\Delta_{sp}&-i\omega_n-\epsilon_k+\mu\end{align}\right)}
Notice that this is a function of 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 \tau-\tau'} 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 r-r'} . Let's call it 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 \Pi_{++}(r-r',\tau-\tau')}
To determine the kinematics (of our collective modes) we need to expand 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 \Pi_{\mu\nu}(q, i\Omega_n)} in powers of 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 \vec{q}} 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 \Omega_n} . Our small expansion parameters are
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 \Omega_n \ll \Delta_{sp}} 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 q \ll \frac{\Delta_{sp}}{\hbar v_F}}
(Obviously 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 q \ll k_F} )
Start with 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 \Pi_{++}} : What we need 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 \Pi_{++}(q, i\Omega_n) - \Pi_{++}(0, 0)}
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 = \frac{2}{\beta}\sum_{\omega_n}{\int{\frac{d^3k}{(2\pi)^3}\left[\frac{1}{\omega_n^2 + E_k^2}\right]}}}