Assignment 11 - Anderson model: mean-field solution - Class of 22/09 (Due: 29/09 23h55)
Condições de conclusão
\( \bar{G}^{R }_{d \sigma}(\omega^{+}) = \left(\omega^{+} - \varepsilon_0 - \Sigma^{(0)}(\omega^{+}) - U \langle \hat{n}_{d \bar{\sigma}} \rangle \right)^{-1} \)
Show that, at zero temperature and for \(\Sigma^{(0)}(\omega^{+}) \approx \Lambda - i \Delta \) with \(\Lambda\) and \(\Delta\) constants, the self-consistent equations for \(\langle \hat{n}_{d \sigma} \rangle\) takes the form:
\( \langle \hat{n}_{d \uparrow } \rangle = \frac{1}{2} - \frac{1}{\pi} \tan^{-1} \left( \frac{\bar{\varepsilon}_0 + U \langle \hat{n}_{d \downarrow } \rangle }{\Delta} \right) \)
\( \langle \hat{n}_{d \downarrow} \rangle = \frac{1}{2} - \frac{1}{\pi} \tan^{-1} \left( \frac{\bar{\varepsilon}_0 + U \langle \hat{n}_{d \uparrow} \rangle }{\Delta} \right) \)
where \(\bar{\varepsilon}_0 = \varepsilon_0 + \Lambda\).
Hint: If \(z(\omega)= \omega - z_0 \) and \(z = a + ib = |z|e^{i \theta}\), use the properties:
\( \frac{1}{z(\omega)}=\frac{d}{d \omega} \ln{(z(\omega))} \)
\( \ln{(z)} = \ln{(|z|)|} + i \theta \)
Write a code in your favorite script language (Python, Mathematica, etc.) to implement the above self-consistent equations (Upload the code here).
Choose values for \(U\) and \(\Delta\) and vary \(\bar{\varepsilon}_0\) between \(-1.5U\) and \(+U/2\). Show that there are solutions with spontaneous symmetry breaking (\(\langle \hat{n}_{d \uparrow} \rangle \neq \langle \hat{n}_{d \downarrow} \rangle\)) if \(U/\Delta\) is large and \(\bar{\varepsilon}_0 \approx -U/2 \).
Aberto: segunda-feira, 10 ago. 2020, 00:00
Vencimento: terça-feira, 29 set. 2020, 23:59
Problem 1 - Mean-field equations for the Anderson model
Consider the retarded Green's function for the Anderson model in the mean-field approximation derived in class:\( \bar{G}^{R }_{d \sigma}(\omega^{+}) = \left(\omega^{+} - \varepsilon_0 - \Sigma^{(0)}(\omega^{+}) - U \langle \hat{n}_{d \bar{\sigma}} \rangle \right)^{-1} \)
Show that, at zero temperature and for \(\Sigma^{(0)}(\omega^{+}) \approx \Lambda - i \Delta \) with \(\Lambda\) and \(\Delta\) constants, the self-consistent equations for \(\langle \hat{n}_{d \sigma} \rangle\) takes the form:
\( \langle \hat{n}_{d \uparrow } \rangle = \frac{1}{2} - \frac{1}{\pi} \tan^{-1} \left( \frac{\bar{\varepsilon}_0 + U \langle \hat{n}_{d \downarrow } \rangle }{\Delta} \right) \)
\( \langle \hat{n}_{d \downarrow} \rangle = \frac{1}{2} - \frac{1}{\pi} \tan^{-1} \left( \frac{\bar{\varepsilon}_0 + U \langle \hat{n}_{d \uparrow} \rangle }{\Delta} \right) \)
where \(\bar{\varepsilon}_0 = \varepsilon_0 + \Lambda\).
Hint: If \(z(\omega)= \omega - z_0 \) and \(z = a + ib = |z|e^{i \theta}\), use the properties:
\( \frac{1}{z(\omega)}=\frac{d}{d \omega} \ln{(z(\omega))} \)
\( \ln{(z)} = \ln{(|z|)|} + i \theta \)
Problem 2 - Solving the mean-field equations for the Anderson model
Write a code in your favorite script language (Python, Mathematica, etc.) to implement the above self-consistent equations (Upload the code here).
Choose values for \(U\) and \(\Delta\) and vary \(\bar{\varepsilon}_0\) between \(-1.5U\) and \(+U/2\). Show that there are solutions with spontaneous symmetry breaking (\(\langle \hat{n}_{d \uparrow} \rangle \neq \langle \hat{n}_{d \downarrow} \rangle\)) if \(U/\Delta\) is large and \(\bar{\varepsilon}_0 \approx -U/2 \).